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\` Edinburgh 99/1\
\` OUTP-99-12P\
[**Monopole clusters, Z(2) vortices and confinement in SU(2).\
**]{}\
A. Hart$^1$ and M. Teper$^2$.\
[**Abstract.**]{}
We extend our previous study of magnetic monopole currents in the maximally Abelian gauge [@hart98] to larger lattices at small lattice spacings ($20^4$ at $\beta = 2.5$ and $32^4$ at $\beta = 2.5115$). We confirm that at these weak couplings there continues to be one monopole cluster that is very much longer than the rest and that the string tension, $K$, is entirely due to it. The remaining clusters are compact objects whose population as a function of radius follows a power law that deviates from the scale invariant form, but much too weakly to suggest a link with the analytically calculable size distribution of small instantons. We also search for traces of Z(2) vortices in the Abelian projected fields; either as closed loops of ‘magnetic’ flux or through appropriate correlations amongst the monopoles. We find, by direct calculation, that there is no confining condensate of such flux loops. We also find, through the calculation of doubly charged Wilson loops within the monopole fields, that there is no suppression of the $q=2$ | 0 | non_member_996 |
effective string tension out to at distances of at least $r \simeq 1.6/ \sqrt{K}$, suggesting that if there are any vortices they are not encoded in the monopole fields.
PACS indices: 11.15.Ha, 12.38.Aw, 14.80.Hv.
Introduction {#sec_intro}
============
Many recent efforts to elucidate the mechanism of confinement in [qcd]{} and non–Abelian gauge theories have focused on isolating a reduced set of variables that are responsible for the confining behaviour. In the dual superconducting vacuum hypothesis [@mandelstam76; @thooft81] the crucial degrees of freedom are the magnetic monopoles revealed after Abelian projection. In the maximally Abelian gauge [@thooft81; @kronfeld87] one finds that the extracted U(1) fields possess a string tension that approximately equals the original SU(2) string tension (‘Abelian dominance’) [@suzuki90], and that this is almost entirely due to monopole currents in these Abelian fields (‘monopole dominance’) [@stack94; @bali96]. The magnetic currents observed in the maximally Abelian gauge are found to have non-trivial correlations with gauge-invariant quantities such as the action and topological charge densities (see for example [@feurstein97; @bakker98] and references therein) and this invites the hypothesis that the structures formed by the magnetic monopoles correspond to similar objects in the SU(2) vacuum, seen after gauge fixing and Abelian projection. If the | 0 | non_member_996 |
magnetic monopoles truly reflect the otherwise unknown infrared physics of the SU(2) vacuum, analysis of these structures may provide important information about the confinement mechanism.
The main purpose of this paper is to extend our previous study [@hart98] of monopole currents to lattices that are larger in physical units at the smallest lattice spacings. As reviewed in Sec. \[sec\_cl\_str\], we obtained in [@hart98] a strikingly simple monopole picture at $\beta = 2.3$, 2.4. When the magnetic monopole currents are organised into separate clusters, one finds in each field configuration one and only one cluster which is very much larger than the rest and which percolates throughout the entire lattice volume. Moreover this largest cluster is alone responsible for infrared physics such as the string tension. The remaining clusters are compact objects with radii varying with length roughly as $r \propto \sqrt{l}$ and with a population that follows a power law as a function of length. We found the exponent of this power law to be consistent with a universal value of 3. This simple pattern became more confused at $\beta = 2.5$. The scaling relations for cluster size that we established in [@hart98] suggested that our $L=16$ lattice at $\beta | 0 | non_member_996 |
= 2.5$ was simply too small. There was of course an alternative possibility: that the simple picture we found at lower $\beta$ failed as one approached the continuum limit. Clearly it is important to distinguish between these two possibilities, and this is what we propose to do in this paper. The cluster size scaling relations referred to above imply that an $L=32$ lattice at $\beta=2.5115$ should have a large enough volume. Such gauge fixed lattice fields were made available to us by G. Bali and we have used them, supplemented by calculations on an intermediate $L=20$ volume at $\beta=2.5$, to obtain evidence, as described in Secs. \[sec\_cl\_str\] and \[sec\_mon\_vor\_K\], that the monopole picture we found previously is in fact valid at these lattice spacings and that the deviations we found previously were due to too small a lattice size.
The fact that one has to go to space-time volumes that are ever larger, in physical units, as the lattice spacing decreases, hints at some kind of breakdown of ‘monopole dominance’ in the continuum limit. We finish Sec. \[sec\_cl\_str\], with a discussion of the form that this breakdown might take.
An attractive alternative to the dual superconducting vacuum as a mechanism | 0 | non_member_996 |
for confinement is vortex condensation [@thooft79; @mack80; @nielsen79; @kovacs98; @deldebbio98; @ambjorn98]. Here the confining degrees of freedom are the vortices created by the ’t Hooft dual disorder loops [@thooft79] and the confining disorder is located in the centre Z(N) of the SU(N) gauge group. When such a vortex intertwines a Wilson loop, the fields along the loop undergo a gauge transformation that varies from unity to a non-trivial element of the centre as one goes once around the Wilson loop. For SU(2) this means that the Wilson loop acquires a factor of $-1$. A condensate of such vortices will therefore completely disorder the Wilson loop and will lead to linear confinement. At the centre of the vortex, which will be a line in $D=2+1$ and a sheet in $D=3+1$, the fields are clearly singular (multivalued) if we demand that the vortex correspond to a gauge transformation almost everywhere. In a properly regularised and renormalised theory, this singularity will be smoothed out [@thooft79] into a core of finite size in which there is a non-trivial but finite action density, and whose size will be $O(1)$ in units of the physical length scale of the theory. One can either try to study | 0 | non_member_996 |
these vortices directly in the SU(2) gauge fields or one can go to the centre gauge [@deldebbio98; @ambjorn98], where one makes the gauge links as close to $+1$ or $-1$ as possible, and construct the corresponding fields where the link matrices take values in Z(2) (‘centre projection’) and where the only nontrivial fluctuations are singular Z(2) vortices. Just as a ’t Hooft–Polyakov monopole will appear as a singular Dirac monopole in the Abelian fields that one obtains after Abelian projection, one would expect the presence of a vortex in the SU(2) fields to appear as a singular Z(2) vortex after centre projection. This picture has received increasing attention recently and has, for example, proved successful in reproducing the static quark potential [@kovacs98; @deldebbio98] (‘centre dominance’). Our ability, in this paper, to address the question of how important are such vortices is constrained by the fact that we only work with Abelian projected SU(2) fields. So first we need to clarify how such vortices might be encoded in these Abelian fields and only then can we perform numerical tests to see whether there is any sign of their presence. This is the content of Sec. \[sec\_mon\_vor\_K\].
Finally there is a summary | 0 | non_member_996 |
of the results in Sec. \[sec\_summ\].
Monopole cluster structure {#sec_cl_str}
==========================
Background
----------
Fixing to the maximally Abelian gauge of SU(2) amounts to maximising with respect to gauge transformations the Morse functional R = - \_[n,]{} ( U\_(n) i\_3 U\^\_(n) i\_3 ). \[eqn\_R\] It is easy to see that this maximises the matrix elements $|[U_\mu(n)]_{11}|^2$ summed over all links. That is to say, it is the gauge in which the SU(2) link matrices are made to look as diagonal, and as Abelian, as possible — hence the name. Having fixed to this gauge, the link matrices are then written in a factored form and the U(1) link angle (just the phase of $[U_\mu(n)]_{11}$) is extracted. The U(1) field contains integer valued monopole currents [@degrand80], $\{ j_\mu(n) \}$, which satisfy a continuity relation, $\Delta_\mu j_\mu(n) = 0$, and may be unambiguously assigned to one of a set of mutually disconnected closed networks, or ‘clusters.’
In [@hart98] we found that the clusters may be divided into two classes on the basis of their length, where the length is obtained by simply summing the current in the cluster l = \_[n,]{} | j\_(n) |. The first class comprises the largest cluster, which is | 0 | non_member_996 |
physically the most interesting. It percolates the whole lattice volume and its length $l_{\max}$ is simply proportional to the volume $L^4$ (at least in the interval $2.3 \le \beta \le 2.5$) when these are re-expressed in physical units, [*i.e.*]{} $l_{\max}\sqrt{K} \propto (L\sqrt{K})^4$, where $K$ is the SU(2) lattice string tension in lattice units and we use $1/\sqrt{K}$ to set our physical length scale. We remark that over this range in $\beta$ there is a factor 2 change in the lattice spacing, and so one might have expected that the extra ultraviolet fluctuations on the finer lattice would lead to significant violations of the naïve scaling relation. That is to say, one might expect to need to coarse grain the currents at larger $\beta$ to obtain reasonable scaling. That this is not required is perhaps surprising.
The remaining clusters were found to be much shorter. Their population as a function of length (the ‘length spectrum’) is described by a power law N(l) = , \[eqn\_len\_spec\] where $\gamma \approx 3$ for all lattice spacings and sizes tested and the coefficient $c_l(\beta)$ is proportional to the lattice volume, $L^4$, and depends weakly on $\beta$. The radius of gyration of these clusters is small | 0 | non_member_996 |
and approximately proportional to the square root of the cluster length, just like a random walk. When folded with the length spectrum, this suggests [@hart98] that the ‘radius spectrum’ should also be described by a power law N(r) = , \[eqn\_rad\_spec\] with $\eta \approx 5$ and $c_r(\beta)$ weakly dependent on $\beta$. Such a spectrum is close to the scale invariant spectrum of 4–dimensional balls of radius $\rho$, $N(\rho) d\rho \sim d\rho/\rho
\times 1/\rho^4$, and so one might try and relate these clusters to the SU(2) instantons in the theory, which classically also possess a scale-invariant spectrum. It is well known, however, that the inclusion of quantum corrections renders the spectrum of the latter far from scale invariant, at least for the small instantons where perturbation theory can be trusted, and so such a connexion does not seem to be possible [@hart98]. On sufficiently large volumes the difference in length between the largest and second largest cluster is very marked, and where this gulf is clear one finds that the long range physics such as the monopole string tension arises solely from the largest cluster. This is the case at $\beta = 2.3$, $L \ge 10$ and at $\beta = 2.4$, | 0 | non_member_996 |
$L \ge 16$. On moving to a finer $L=16$ lattice at $\beta=2.5$ the gulf was found to disappear and the origin of the long range physics was no longer so clear cut. This could be a mere finite volume effect, or, much more seriously, it might signal the breakdown of this monopole picture in the weak coupling, continuum limit. Clearly this needs to be resolved and the only unambiguous way to do so is by performing the calculations on large enough lattices.
This calculation {#ssec_thiscalc}
----------------
The direct way to estimate the lattice size necessary at $\beta=2.5$ to restore (if that is possible) our picture is as follows. Suppose that the average size of the second largest cluster scales approximately as l\_ L\^()\^. We know that $l_{\max} \propto L^4 (\sqrt{K})^3$ to a good approximation for the largest cluster. So we will maintain the same ratio of lengths $l_{\second} / l_{\max}$, and a gulf between these, if = ( )\^[ -( )]{} \[eqn\_lK\_scale\] If we take our directly calculated values of $l_{\second}$, they seem to give roughly $\alpha \simeq 1$ and $\delta \simeq -2$. This suggests that we need to scale our lattice size with $\beta$ so as to keep $L(\sqrt{K})^{5/3}$ | 0 | non_member_996 |
constant. This estimate is not entirely reliable because, on smaller lattices, the distributions of the ‘largest’ and ‘second largest’ clusters overlap so that they exchange [*rôles*]{}. An alternative estimate can be obtained from the tail of the distribution in eqn. \[eqn\_len\_spec\] that integrates to unity. Doing so [@hart98] one obtains $\alpha \simeq 2$ and $0 < \delta < 0.25 $. This suggests that we scale our lattice size so as to keep $L(\sqrt{K})^{\{1.4 \to 1.5\}}$ constant. This estimate is also not very reliable, since it assumes that the distribution of secondary cluster sizes on different field configurations fluctuates no more than mildly about the average distribution given in eqn. \[eqn\_len\_spec\]. In fact the fluctuations are very large. \[As we can see immediately when we try to calculate $\langle l^2 \rangle$ in order to obtain a standard fluctuation — it diverges for a length spectrum with $N(l)\propto dl/l^3$.\] Nonetheless, the two very different estimates we have given above produce a very similar final criterion: to maintain the same gap between the largest and second largest clusters as $\beta$ is varied, one should choose $L$ so as to keep $L(\sqrt{K})^{\sim 1.5}$ constant.
So if we wish to match the clear picture on | 0 | non_member_996 |
an $L=10$ lattice at $\beta=2.3$ (where $K = 0.136$ (2)) we should work on a lattice that is roughly $L=28$ at $\beta=2.5$ (where $K = 0.0346$ (8)). In particular we note that an $L=32$ lattice at $\beta = 2.5115$ (where $K = 0.0324$ (10)) is more than large enough and an ensemble of 100 such configurations, already gauge fixed [@bali96], has been made available to us by the authors. The gauge fixing procedure used in obtaining these is somewhat different from the one we have used in our previous calculations (in its treatment of the Gribov copies — see below) and although this is not expected to affect the qualitative features that are our primary interest here, it will have some effect on detailed questions of scaling etc. We have therefore also performed a calculation on an ensemble of 100 gauge fixed $L=20$ field configurations at $\beta=2.5$. While the latter volume is not expected to be large enough to recreate a clear gulf between the largest and remaining clusters, we would expect to find smaller finite size corrections than with the $L=16$ lattice we used previously.
In gauge fixing a configuration we select a local maximum of the Morse functional, | 0 | non_member_996 |
$R$, of which on lattices large enough to support non–perturbative physics there are typically a very large number [@hart97a]. These correspond to the (lattice) Gribov copies. Gauge dependent quantities appear to vary by ${\cal O}(10\%)$ depending upon the Gribov copy chosen; this is true not only of local quantities such as the magnetic current density [@hioki91] but also of supposedly long range, physical numbers such as the Abelian and monopole string tensions [@bali96; @hart97a]. Some criterion must be employed for the selection of the maxima of $R$, and in the absence of a clear understanding of which maximum, if any, is the most ‘physical’, one maximum was selected at random in [@hart98]. An alternative strategy, used in gauge fixing the $L=32$ lattices at $\beta = 2.5115$, is to pursue the global maximum of $R$ [@bali96]. Each field configuration is fixed to the maximally Abelian gauge 10 times using a simulated annealing algorithm that already weights the distribution of maxima so selected towards those of higher $R$. The solution with the largest $R$ from these is selected. Details of this method are discussed in [@bali96]. The difference in procedures invites caution in comparing exact numbers between this ensemble and those studied | 0 | non_member_996 |
previously; for example a ${\cal O}(10\%)$ suppression in the string tension is observed. It is likely that cluster lengths will differ by a corresponding amount and this will prevent a quantitative scaling analysis using this ensemble. The power law indices do appear, however, to be robust [@hart97b] and it also seems likely that ratios of string tensions obtained on the same ensemble can be reliably compared with other ratios.
Cluster properties
------------------
The fact that the largest cluster does not belong to the same distribution as the smaller clusters is seen from the very different scaling properties of these clusters with volume [@hart98]. It is also apparent from the fact that the largest cluster is very much longer than the second largest cluster. Indeed for a large enough volume and for a reasonable size of the configuration ensemble, there will be a substantial gulf between the distribution of largest cluster lengths and that of the second largest clusters. By contrast the length distributions of the second and third largest clusters strongly overlap. This is the situation that prevailed for the larger lattices at $\beta=2.3$ and 2.4 but which broke down on the $L=16$ lattice at $\beta=2.5$. We can now compare | 0 | non_member_996 |
what we find on our $L=20$ and $L=32$ lattices with the latter. This is done in Table \[table\_length\]. There we show the longest and shortest cluster lengths for the largest, second largest and third largest cluster respectively over the ensemble. The ensemble sizes are not exactly the same, but it is nonetheless clear that there is a real gulf between the largest and second largest clusters on the $L=32$ lattice while there is significant overlap in the $L=16$ case. The $L=20$ lattice is a marginal case. We conclude from this that the apparent loss of a well separated largest cluster as seen in [@hart98] at $\beta=2.5$ was in fact a finite volume effect, and that our scaling analysis has proved reliable in predicting what volume one needs to use in order to regain the simple picture.
In Figure \[fig\_curr\_dens\] we show how the length of the largest cluster varies with the lattice volume when both are expressed in physical units (set by $\sqrt{K}$). To be specific, we have divided $l_{\max}\sqrt{K}$ by $(L\sqrt{K})^4$ and plotted the resulting numbers against $L\sqrt{K}$ for both our new and our old calculations. The fact that at fixed $\beta$ the values fall on a horizontal line | 0 | non_member_996 |
tells us that that the length of the largest cluster is proportional to the volume at fixed lattice spacing: $l_{\max}
\propto L^4$. The fact that the various horizontal lines almost coincide tells us that the current density in the largest cluster is consistent with scaling. That is to say, it has a finite non-zero value in the continuum limit. Thus the monopole whose world line traces out this largest cluster, percolates throughout the space–time volume and its world line is sufficiently smooth on short distance scales that its length does not show any sign of diverging as we take the continuum limit. We note that the $L=32$ lattice deviates by $\sim
10\%$ from the other values. This is consistent with what we might have expected from the different gauge fixing procedure used in that case.
Turning now to the secondary clusters, we display in Figure \[fig\_len\_spec\] the length spectrum that we obtain at $\beta=2.5115$. It is clearly well described by a power law as in eqn. \[eqn\_len\_spec\] and we fit the exponent to be $\gamma =
3.01$ (8). This is in accord with the universal value of 3 that was postulated in [@hart98] on the basis of calculations on coarser | 0 | non_member_996 |
lattices. The value one fits to the spectrum obtained on the $L=20$ lattice at $\beta=2.5$ is $\gamma = 2.98$ (7) and is equally consistent. We also examine the dependence on $\beta$ of the coefficient $c_l(\beta)$ in eqn. \[eqn\_len\_spec\] adding to the older work our calculations at $\beta=2.5$ on the $L=20$ lattice. \[We do not use the $L=32$ lattice for this purpose because of the different gauge fixing procedure used.\] If we assume a constant power (which is approximately the case), then $c_l(\beta)$ is just proportional to the total length of the secondary clusters. At fixed $\beta$ we find this length to be proportional to $L^4$ just as one might expect. \[Small clusters in very different parts of a large volume are presumably independent.\] The dependence on $\beta$, on the other hand, is much less clear. Between $\beta=2.3$ and $\beta=2.4$ it varies weakly, roughly as $K^{0.12 \pm 0.13}$. Between $\beta=2.4$ and $\beta=2.5$ it varies more strongly, roughly as $K^{0.48 \pm 0.09}$. We can try to summarise this by saying that c\_l() = L\^4 \^ \[eqn\_coeff\_spec\] where $\zeta = 0.5 \pm 0.5$, which is consistent with what was found previously [@hart98]. The smaller clusters are compact objects in $d=4$, and having determined | 0 | non_member_996 |
the cluster spectrum as a function of length we can then ask what is the spectrum when re-expressed as a function of the radius (of gyration) of the cluster. In [@hart98] we obtained this spectrum by determining the average radius as a function of length, and folding that in with the number density as a function of length. This is an approximate procedure (forced upon us by the fact that we did not foresee the interest of this spectrum during the processing of the clusters) and one can obtain the spectrum more accurately by calculating $r$ for each cluster and forming the spectrum directly. Doing so for the $L = 32$ lattice at $\beta =
2.5115$, also in Figure \[fig\_len\_spec\], we find a power law as in eqn. \[eqn\_rad\_spec\] with $\eta = 4.20$ (8). The spectrum on the $L=20$ lattice at $\beta=2.5$ yields $\eta = 4.27$ (6). We recall that in [@hart98] we claimed that the spectrum was consistent with the scale invariant result $dr/r \times 1/r^4$, [*i.e.*]{} $\eta = 5$. This followed from the fact that we found the the radius of the smaller clusters to vary with their length as $r(l) = s + t.l^{0.5}$ [*i.e.*]{} just what | 0 | non_member_996 |
one would expect from a random walk. Folded with a length spectrum $N(l)
\sim 1/l^3$, this gives $\eta = 5$. On the $L=32$ lattice we still find that the random walk [*ansatz*]{} provides an acceptable fit but we also find that $r(l) = s + t.l^{0.65}$ works equally well over similar ranges. The latter, when folded with $\gamma = 3$, gives $\eta
= 4.2$. It is clear that the direct calculation of $N(r)$ is much more accurate than the indirect approach.
Treating the power as a free parameter in the fit, $r(l) = s + t.l^u$, we find $u = 0.57$ (3) on $L=32$ at $\beta = 2.5115$, consistent with $u = 0.58$ (4) on $L=20$ at $\beta=2.5$. Thus both $u=0.5$ and $u=0.65$ lie within about two standard deviations from the fitted value. Note that what the fitted powers $\gamma$ and $u$ parameterise are the means of the distributions of lengths and radii respectively. That combining these does not give the directly calculated value of $\eta$ is not unexpected, and reflects the importance of fluctuations around the mean in the distributions.
If the secondary monopole clusters can be associated with localised excitations of the full SU(2) vacuum (‘4–balls’), it would | 0 | non_member_996 |
seem that such objects do not have an exactly scale invariant distribution in space–time, so that the number of larger radius objects is somewhat greater than would be expected were this the case. Now it is known that an isolated instanton (even with quantum fluctuations) is associated with a monopole cluster within its core (see [@hart96; @inst99] and references therein) and that the scale invariant semiclassical density of instantons acquires corrections due to quantum fluctuations. These corrections are, however, very large; in SU(2) the spectrum of small instantons (where perturbation theory is reliable) goes as $N(\rho) d\rho \propto
d\rho/\rho \times \rho^{10/3}$, where $\rho$ is the core size. The scale breaking we have observed for monopole clusters is negligible in comparison. Thus we cannot identify the ‘4–balls’ with instantons. Indeed, the fact that the monopole spectrum is so close to being scale invariant strongly suggests that these secondary clusters have no physical significance. In the next section we shall show explicitly that, in the large volume limit, they do not play any part in the long range confining physics.
Breakdown of ‘monopole dominance’?
----------------------------------
We finish this section by asking if there are hints from our cluster analysis that ‘monopole dominance’ | 0 | non_member_996 |
might be breaking down as we approach the continuum limit. This question is motivated by the observation that the monopoles are identified by a gauge fixing procedure which involves making the bare SU(2) fields as diagonal as possible. Since the theory is renormalisable, the long distance physics increasingly decouples from the fluctuations of the ultraviolet bare fields as we approach the continuum limit. For example, the ultraviolet contribution to the action density is $O(1/\beta)$ while the long distance contribution is $O(e^{-c\beta})$. Thus as $a\to 0$ the maximally Abelian gauge will be overwhelmingly driven by ultraviolet rather than by physical fluctuations. Moreover at the location of the monopoles the Abelian fields are far from unity and so one would expect the SU(2) fields also to be far from unity. Thus the number of monopoles would seem to be constrained by the probability of finding corresponding clumps of SU(2) fields with large plaquette values. This probability depends on the detailed form of the SU(2) lattice action far from the Gaussian minimum and one could easily choose an action where it is completely suppressed and yet which one would expect to be in the usual universality class. None of the above arguments are | 0 | non_member_996 |
completely compelling of course. In the Gaussian approximation, for example, the $O(1/\beta)$ ultraviolet fluctuations would not generate any monopoles at all, and in that case there would be no reason to expect any breakdown of monopole dominance. Nonetheless the arguments do suggest that it would be surprising if the long distance physics were to be usefully and simply encoded in the monopole structure (as defined on the smallest ultraviolet scales) all the way to the continuum limit.
There are different ways in which monopole dominance could be lost. The most extreme possibility is that as $a \to 0$ the fields simply cease to contain monopole clusters that are large enough to disorder large Wilson loops. That this is indeed so has been argued in [@grady98] where it has been claimed that the exponent $\gamma$ in our eqn. \[eqn\_len\_spec\] (but defined for loops rather than for clusters) increases rapidly with decreasing $a$. Of course this would not in itself preclude the existence of a large percolating cluster, as long as this cluster could be decomposed into a large number of small and correlated intersecting loops. Irrespective of this, we also note that the volumes used in [@grady98] are very small by | 0 | non_member_996 |
the criterion given in eqn. \[eqn\_lK\_scale\]. For example, from our scaling relations we would expect to need an $L
\simeq 46$ lattice at $\beta=2.6$ and an $L \simeq 70$ lattice at $\beta=2.7$ in order to resolve our simple monopole picture, if it still holds at these values of $\beta$. This contrasts with the $L=12$ and $L=20$ lattices actually employed in [@grady98]. So it appears to us that while the claims in [@grady98] are certainly interesting, further calculations on much larger lattices are required.
Our work suggests a somewhat different form of the breakdown to the one above. We see from eqn. \[eqn\_coeff\_spec\] that the ratio of the (total) monopole current residing in the physically irrelevant, smaller clusters to that residing in the large percolating cluster, increases rapidly as $a\to 0$ as $1/\sqrt{K}^{3-\zeta} \propto 1/a^{3-\zeta}$. This suggests that as $a\to 0$ a calculation of Wilson loops will become increasingly dominated by the fluctuating contribution of the unphysical monopoles that are ever denser on physical length scales, and that this will eventually prevent us from extracting a potential or string tension. That is to say: calculations in the maximally Abelian gauge will eventually acquire a similar problem to that which typically afflicts | 0 | non_member_996 |
Abelian projections using other gauges. In our case we can overcome this problem by going to a large enough volume that the physically relevant percolating cluster can be simply identified. \[The reason this cannot be done with other typical Abelian gauge fixings is that there the unphysical monopoles are dense on lattice scales making any meaningful separation into clusters impossible.\] We can then extract the string tension using, in our Wilson loop calculation, only this largest monopole cluster. The fact that the length of this cluster scales in physical units, with apparently no significant anomalous dimension, tells us that this calculation will not be drowned in ultraviolet ‘noise’ as we approach the continuum limit. Of course, the fact that we can only do this for volumes that diverge in physical units as $a\to 0$ is a symptom of the underlying breakdown of the Abelian projection.
The qualitative discussion in the previous paragraph over-estimates the effect of the secondary clusters; for example, the contribution that a cluster of fixed size in lattice units makes to a Wilson loop of a fixed physical size will clearly go to zero as $a\to 0$. So it is useful to ask how Wilson loops are | 0 | non_member_996 |
affected by the secondary clusters, and to do so using approximations that underestimate the effect of these smaller clusters. Consider an $R\times R$ Wilson loop. A monopole cluster that has an extent $r$ that is smaller then $R$ will affect it only weakly through higher multipole fields which cannot on their own give rise to an area law decay and a string tension. So we neglect such clusters and consider only those larger than $R$. Let us first neglect the observed breaking of scale invariance and simply assume that $r \propto \sqrt{l}$ and that $\gamma = 3$. We then find, by integrating eqn. \[eqn\_len\_spec\] and using eqn. \[eqn\_coeff\_spec\], that the number of secondary clusters with $r>R$ is proportional to $L^4 \sqrt{K}^{\zeta}/R^4$. We further assume that such clusters must be within a distance $\xi$ from the minimal surface of the Wilson loop, where $\xi$ is the screening length, if they are to disorder that loop significantly. The lattice volume this encompasses is the area of the planar loop, $R^2$, multiplied by a factor of $\xi$ for each of the two orthogonal directions in $d=4$. So the probability for this Wilson loop to be disordered thus decreases with $R$ as $(R^2\xi^2/L^4 \times | 0 | non_member_996 |
---
abstract: 'We propose a variant formulation of Hamiltonian systems by the use of variables including redundant degrees of freedom. We show that Hamiltonian systems can be described by extended dynamics whose master equation is the Nambu equation or its generalization. Partition functions associated with the extended dynamics in many degrees of freedom systems are given. Our formulation can also be applied to Hamiltonian systems with first class constraints.'
address: |
${}^\ast$\
${}^\dag$
author:
- and
title: |
Hidden Nambu mechanics:\
[A variant formulation of Hamiltonian systems ]{}
---
Introduction {#Introduction}
============
In general, we have a choice of variables describing a physical system. In most cases, we choose a set of variables whose number is same as the total number of degrees of freedom of the system so as to minimize the number of equations of motion. However, in some cases, it is quite useful to formulate the system by the use of variables including redundant ones. A system with gauge symmetry offers a typical example. To describe such a system, keeping the gauge symmetry manifest, we should employ a formulation that includes redundant variables. Although such a formulation is somewhat complicated, thanks to the symmetry, we can clearly | 0 | non_member_997 |
understand the important properties of the system such as conservation laws and form of interactions, and can also calculate physical quantities in a systematic way [@W1; @W2].
Therefore, it is interesting to explore the general features of formulations including redundant degrees of freedom. Here we base this on a principle (or brief) that [*physics should be independent of the choice of variables to describe it*]{}, and make an attempt to formulate Hamiltonian systems (systems of Hamiltonian dynamics) in terms of new sets of variables including redundant ones. What kind of dynamics describes the time evolution of the new variables?
Our strategy and conjecture are as follows. Consider a Hamiltonian system described by a canonical doublet $(q, p)$. Take $N(\ge 3)$ variables $(x_1, \cdots, x_{N})$ that are functions of the canonical doublet, and deal with them as fundamental variables to describe the system. If they contain redundant variables, constraints between some variables must be induced. To handle the constraints, Dirac formalism [@D1; @D2] provides a helpful perspective, where constraints with Lagrange multipliers are added to the original Hamiltonian. The induced constraints play a similar role to the Hamiltonian. As for the dynamics of $N$ variables, Nambu mechanics [@N] is quite suggestive. | 0 | non_member_997 |
In Nambu mechanics, fundamental variables form an $N$-plet, whose time evolution is generated by $N-1$ Hamiltonians according to the Nambu equations. Combining the advantages of the two theories, we conjecture that [*there is a formulation whose master equation has a form of the Nambu equation or its generalization, where the Hamiltonians consist of the original one and the induced constraints.*]{}
Nambu mechanics is a generalization of the Hamiltonian dynamics proposed by Nambu forty years ago [@N]. In his formulation, the dynamics of an $N$-plet is given by the Nambu equation, which is defined by $N-1$ Hamiltonians and the Nambu bracket, a generalization of the Poisson bracket. The structure of Nambu mechanics is so elegant that many authors have investigated its application. However, the applications have been limited to particular systems such as constrained systems, superintegrable systems, and hydrodynamic systems, because Nambu systems (systems of Nambu mechanics) should have multiple Hamiltonians or conserved quantities. For example, researchers have studied how Nambu mechanics can be embedded into constrained Hamiltonian systems [@BF; @CK; @R; @MS; @KT; @KT2] or how constrained systems can be described in terms of Nambu mechanics [@LJ].
In this article, we show that the structure of Nambu mechanics is, in | 0 | non_member_997 |
general, hidden in systems of Hamiltonian dynamics. That is, Hamiltonian systems can be described by Nambu mechanics or its generalization by means of a change of variables from canonical doublets to multiplets. Our formulation can be generalized to many degrees of freedom systems, and the associated partition functions are given. We also apply our formulation to systems with first class constraints. Our approach can be regarded as a complementary one to the previous works [@BF; @CK; @R; @MS; @KT; @KT2; @LJ].
The outline of this article is as follows. In the next section, we give a formulation of Hamiltonian systems using Nambu mechanics and its generalizations. As an application, Hamiltonian systems with first class constraints are also formulated as Nambu systems in Sect. 3. In the last section, we give conclusions and discussions on the direction of future work. In Appendix A, we derive the Nambu equation from the least action principle. In Appendix B, we show that a Nambu system of an $N$-plet can be described by Nambu mechanics with an $N+r$-plet ($r\ge1$).
Nambu systems hidden in Hamiltonian systems {#Nambu systems hidden in Hamiltonian systems}
===========================================
Review {#Review}
------
We begin with a brief review of Hamiltonian systems and | 0 | non_member_997 |
Nambu systems [@N]. A Hamiltonian system is a classical system described by a generalized coordinate $q=q(t)$ and its canonical conjugate momentum $p=p(t)$. These variables satisfy the Hamilton’s canonical equations of motion, $$\begin{aligned}
\frac{d q}{dt} = \frac{\partial H}{\partial p} ~,~~ \frac{d p}{dt}
= -\frac{\partial H}{\partial q}~,
\label{H-eq}\end{aligned}$$ where $H=H(q, p)$ is the Hamiltonian of this system. For any functions $A=A(q,p,t)$ and $B=B(q,p,t)$, the Poisson bracket is defined by means of the 2-dimensional Jacobian, $$\begin{aligned}
\{A, B\}_{\mbox{\tiny{PB}}} \equiv
\frac{\partial (A,B)}{\partial (q,p)}=
\frac{\partial A}{\partial q}\frac{\partial B}{\partial p}
- \frac{\partial A}{\partial p}\frac{\partial B}{\partial q}~.
\label{PB}\end{aligned}$$ In terms of the Poisson bracket, the Hamilton’s canonical equation of motion for any function $f=f(p,q)$ can be written as $$\begin{aligned}
\frac{d f}{dt} = \{f, H\}_{\mbox{\tiny{PB}}}~.
\label{H-eqf}\end{aligned}$$
On the other hand, a Nambu system is a classical system described by a multiplet. As the most simple example, let us consider a Nambu system described by a triplet $x=x(t)$, $y=y(t)$, and $z=z(t)$. These variables satisfy the Nambu equations $$\begin{aligned}
\frac{d x}{dt} = \frac{\partial ({H}_1, {H}_2)}{\partial (y, z)}~,~~
\frac{d y}{dt} = \frac{\partial ({H}_1, {H}_2)}{\partial (z, x)}~,~~
\frac{d z}{dt} = \frac{\partial ({H}_1, {H}_2)}{\partial (x, y)}~,
\label{N-eq}\end{aligned}$$ where ${H}_1(x, y, z)$ and ${H}_2(x, y, z)$ are $\lq\lq$Hamiltonians" of this system. For any functions | 0 | non_member_997 |
$A={A}(x, y, z, t)$, $B={B}(x, y, z, t)$, and $C={C}(x, y, z, t)$, the Nambu bracket is defined by means of the 3-dimensional Jacobian, $$\begin{aligned}
\{{A}, {B}, {C}\}_{\mbox{\tiny{NB}}}
\equiv \frac{\partial ({A}, {B}, {C})}{\partial (x, y, z)}~.
\label{NB}\end{aligned}$$ In terms of the Nambu bracket, the Nambu equation for any function $f=f(x,y,z)$ can be written as $$\begin{aligned}
\frac{d {f}}{dt} = \{{f}, {H}_1, {H}_2\}_{\mbox{\tiny{NB}}}~.
\label{N-eqf}\end{aligned}$$
It is straightforward to extend the above formalism to a system described by an $N$-plet $x_i$ $(i=1, 2, \cdots, N)$. These variables satisfy the Nambu equations $$\begin{aligned}
\frac{d x_i}{dt} = \sum_{i_1, \cdots, i_{N-1}=1}^{N}
\varepsilon_{i i_1 \cdots i_{N-1}}
\frac{\partial {H}_1}{\partial x_{i_1}} \cdots
\frac{\partial {H}_{N-1}}{\partial x_{i_{N-1}}}~,
\label{N-eq-N}\end{aligned}$$ where ${H}_a={H}_a(x_1, x_2, \cdots, x_N)$ $(a=1, \cdots, N-1)$ are $\lq\lq$Hamiltonians" of this system and $\varepsilon_{i i_1 \cdots i_{N-1}}$ is the $N$-dimensional Levi–Civita symbol, the antisymmetric tensor with $\varepsilon_{12 \cdots N} =1$. For any functions ${A}_{\alpha}={A}_{\alpha}(x_1, x_2, \cdots, x_N, t)$ $(\alpha=1, \cdots, N)$, the Nambu bracket is defined by means of the $N$-dimensional Jacobian, $$\begin{aligned}
\{{A}_1, {A}_2, \cdots, {A}_N\}_{\mbox{\tiny{NB}}}
&\equiv&
\frac{\partial ({A}_1, {A}_2, \cdots, {A}_N)}{\partial (x_1, x_2, \cdots, x_N)}
\nonumber \\
&=& \sum_{i_1, i_2, \cdots, i_N=1}^{N} \varepsilon_{i_1 i_2 \cdots i_N}
\frac{\partial {A}_1}{\partial x_{i_1}}\frac{\partial {A}_2}{\partial x_{i_2}}
\cdots \frac{\partial {A}_{N}}{\partial x_{i_N}}~.
\label{NB-N}\end{aligned}$$ In terms of the | 0 | non_member_997 |
Nambu bracket, the Nambu equation for any function ${f}={f}(x_1, x_2, \cdots, x_N)$ can be written as $$\begin{aligned}
\frac{d {f}}{dt} = \{{f}, {H}_1, {H}_2, \cdots, {H}_{N-1}\}_{\mbox{\tiny{NB}}}~.
\label{N-eqf-N}\end{aligned}$$
Hidden Nambu structure {#Hidden Nambu structure}
----------------------
Here let us describe a Hamiltonian system with a canonical doublet $(q, p)$ by means of $N$ variables $x_i=x_i(q, p)$ $(i=1, \cdots, N)$.
### Formulation {#Formulation}
First we study the case with $N=2$, for completeness. We assume that $x=x_1(q, p)$ and $y=x_2(q, p)$ satisfy $\{x, y\}_{\mbox{\tiny{PB}}} \ne 0$. In this case, the equation for a function $\tilde{f}(x, y) = f(q, p)$ is written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \frac{\partial(f, H)}{\partial(q, p)}
= \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x, y)}
\frac{\partial(x, y)}{\partial(q, p)}
= \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x, y)} \{x, y\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(N=2)}\end{aligned}$$ where $\tilde{H}(x, y) = H(q, p)$. If $\{x, y\}_{\mbox{\tiny{PB}}}=1$, the transformation $(q, p) \to (x, y)$ is the canonical transformation, and $(x, y)$ are canonical variables.
Next we study the case with $N=3$. We assume that variables $x=x_1(q, p)$, $y=x_2(q, p)$, and $z=x_3(q, p)$ satisfy at least two of the conditions $\{x, y\}_{\mbox{\tiny{PB}}} \ne 0$, $\{y, z\}_{\mbox{\tiny{PB}}} \ne 0$, and $\{z, x\}_{\mbox{\tiny{PB}}} \ne 0$. In this case, the equation for a function $\tilde{f}(x, y, z) = f(q, p)$ is written as $$\begin{aligned}
| 0 | non_member_997 |
\frac{d \tilde{f}}{dt} = \frac{\partial(f, H)}{\partial(q, p)}
= \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x, y)} \{x, y\}_{\mbox{\tiny{PB}}}
+ \frac{\partial(\tilde{f}, \tilde{H})}{\partial(y, z)} \{y, z\}_{\mbox{\tiny{PB}}}
+ \frac{\partial(\tilde{f}, \tilde{H})}{\partial(z, x)} \{z, x\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(N=3)}\end{aligned}$$ where $\tilde{H}(x, y, z) = H(q, p)$. Note that $q$, $p$, and $H$ are, in general, not uniquely determined as functions of $x$, $y$, and $z$.
Introducing a function $\tilde{G}=\tilde{G}(x, y, z)$ that satisfies the conditions $$\begin{aligned}
\frac{\partial \tilde{G}}{\partial x} = \frac{\partial(y, z)}{\partial(q, p)}~,~~
\frac{\partial \tilde{G}}{\partial y} = \frac{\partial(z, x)}{\partial(q, p)}~,~~
\frac{\partial \tilde{G}}{\partial z} = \frac{\partial(x, y)}{\partial(q, p)}~,
\label{xyzG}\end{aligned}$$ Eq. (\[H-eq(N=3)\]) is rewritten as the Nambu equation in the form of Eq. (\[N-eqf\]), $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \{\tilde{f}, \tilde{H}, \tilde{G}\}_{\mbox{\tiny{NB}}}~,
\label{H-eqf(N=3)}\end{aligned}$$ where we use the formula $$\begin{aligned}
\frac{\partial (\tilde{A}, \tilde{B}, \tilde{C})}{\partial (x, y, z)}
= \frac{\partial (\tilde{A}, \tilde{B})}{\partial (x, y)}
\frac{\partial \tilde{C}}{\partial z}
+ \frac{\partial (\tilde{A}, \tilde{B})}{\partial (y, z)}
\frac{\partial \tilde{C}}{\partial x}
+ \frac{\partial (\tilde{A}, \tilde{B})}{\partial (z, x)}
\frac{\partial \tilde{C}}{\partial y}~.
\label{J(N=3)}\end{aligned}$$ The conditions (\[xyzG\]) are compactly expressed as $$\begin{aligned}
\frac{\partial \tilde{G}}{\partial x_i}
= \frac{1}{2} \sum_{j, k=1}^3 \varepsilon_{ijk}\{x_j, x_k\}_{\mbox{\tiny{PB}}}~~~
\mbox{or}~~~
\sum_{k=1}^3 \varepsilon_{ijk} \frac{\partial \tilde{G}}{\partial x_k}
= \{x_i, x_j\}_{\mbox{\tiny{PB}}}~.
\label{xyzG2}\end{aligned}$$ In Appendix A, the Nambu equations in the form of Eq. (\[N-eq\]) are also derived from a Hamiltonian system with a canonical doublet $(q, p)$ using | 0 | non_member_997 |
the least action principle.
By the use of Eq. (\[xyzG2\]), it is shown that the Poisson bracket between $G(q,p)=\tilde{G}(x,y,z)$ and an arbitrary function $u(q,p)=\tilde{u}(x, y, z)$ vanishes such that $$\begin{aligned}
\{G, u\}_{\mbox{\tiny{PB}}}
&=& \frac{1}{2}
\sum_{i, j=1}^{3} \frac{\partial (\tilde{G}, \tilde{u})}
{\partial (x_i, x_j)} \{x_i, x_j\}_{\mbox{\tiny{PB}}}
= \frac{1}{2} \sum_{i, j, k=1}^{3} \varepsilon_{ijk}
\frac{\partial (\tilde{G}, \tilde{u})}{\partial (x_i, x_j)}
\frac{\partial \tilde{G}}{\partial x_k}\nonumber \\
&=& \frac{\partial (\tilde{G}, \tilde{u}, \tilde{G})}{\partial (x, y, z)} = 0~.
\label{Gu=0}\end{aligned}$$ This means that $G$ is a constant. We can eliminate the constant by redefining $G$, and the resulting $\tilde{G}(x,y,z)=0$ can be regarded as a $\it constraint$, which is induced by enlarging the phase space from $(q, p)$ to $(x, y, z)$.
Here we give two comments on the induced constraint $\tilde{G}(x,y,z)=0$. First, in the case in which $\partial \tilde{G}/\partial z \ne 0$, we can solve $\tilde{G}(x,y,z)=0$ for $z$ and obtain $z=z(x,y)$. Because the condition $\partial \tilde{G}/\partial z= \{x, y\}_{\mbox{\tiny{PB}}} \ne 0$ also enables us to express $q$ and $p$ as functions of $x$ and $y$, the expression $z=z(x,y)$ can also be obtained by inserting $q=q(x,y)$ and $p=p(x,y)$ into $z=z(q,p)$. Therefore the implicit form of the constraint $\tilde{G}(x,y,z)=0$ has an equivalent explicit form $z=z(x,y)$, which clearly shows that $z$ is a | 0 | non_member_997 |
redundant variable in this case. Second, $\tilde{H}(x,y,z)$ is not uniquely determined as a function of $x$, $y$, and $z$, i.e., we can add a term $\tilde{\lambda}(x,y,z)\tilde{G}(x,y,z)$ to $\tilde{H}(x,y,z)$, where $\tilde{\lambda}(x,y,z)$ is some function. If a Hamiltonian $\tilde{H}(x,y,z)$ satisfies $\tilde{H}(x,y,z)=H(q,p)$ and Eq. (\[H-eqf(N=3)\]), another Hamiltonian $\tilde{H}(x,y,z)+\tilde{\lambda}(x,y,z)\tilde{G}(x,y,z)$ also satisfies them. This is because the additional term $\tilde{\lambda}(x,y,z)\tilde{G}(x,y,z)$ always vanishes on the Nambu bracket.
It is straightforward to extend the above formulation to the case with general $N(\ge 3)$. We assume that at least $N-1$ of $\{x_i, x_j\}_{\mbox{\tiny{PB}}}$ $(i, j=1, \cdots, N)$ do not vanish. In this case, the equation for any function $\tilde{f}(x_1, \cdots, x_N) = f(q, p)$ is written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \frac{\partial(f, H)}{\partial(q, p)}
= \frac{1}{2} \sum_{i, j=1}^{N}
\frac{\partial (\tilde{f}, \tilde{H})}{\partial (x_{i}, x_{j})}
\{x_{i}, x_{j}\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(gN)}\end{aligned}$$ where $\tilde{H}(x_1, \cdots, x_N) = H(q, p)$.
Introducing functions $\tilde{G}_b=\tilde{G}_b(x_1, \cdots, x_N)$ $(b=1, \cdots, N-2)$ that satisfy the conditions $$\begin{aligned}
\frac{1}{(N-2)!} \sum_{i_3 \cdots i_{N}=1}^{N}
\varepsilon_{i_1 i_2 i_3 \cdots i_{N}}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{N-2})}
{\partial (x_{i_3}, \cdots, x_{i_{N}})}
= \{x_{i_1}, x_{i_2}\}_{\mbox{\tiny{PB}}}~,
\label{xiGb}\end{aligned}$$ Eq. (\[H-eq(gN)\]) is rewritten as the Nambu equation in the form of Eq. (\[N-eqf-N\]), $$\begin{aligned}
\frac{d \tilde{f}}{dt} =
\{\tilde{f}, \tilde{H}, \tilde{G}_1, \cdots, \tilde{G}_{N-2}\}_{\mbox{\tiny{NB}}}~,
\label{H-eqf(gN)}\end{aligned}$$ where we use the formula concerning Jacobians, | 0 | non_member_997 |
$$\begin{aligned}
\frac{\partial (\tilde{A}_1, \tilde{A}_2, \cdots, \tilde{A}_N)}
{\partial (x_1, x_2, \cdots, x_N)}
= \frac{1}{2(N-2)!} \sum_{i_1, i_2, i_3, \cdots i_{N}=1}^{N}
\varepsilon_{i_1 i_2 i_3 \cdots i_{N}}
\frac{\partial (\tilde{A}_1, \tilde{A}_2)}{\partial (x_{i_1}, x_{i_2})}
\frac{\partial (\tilde{A}_3, \cdots, \tilde{A}_{N})}
{\partial (x_{i_3}, \cdots, x_{i_{N}})}~.
\label{J(gN)}\end{aligned}$$
By the use of Eq. (\[xiGb\]), it is shown that the Poisson bracket between any of $N-2$ functions $G_b(q,p)=\tilde{G}_b(x_1, x_2, \cdots, x_{N})$ and an arbitrary function $u(q,p)=\tilde{u}(x_1, x_2, \cdots, x_{N})$ vanishes such that $$\begin{aligned}
\{G_b, u\}_{\mbox{\tiny{PB}}}
&=& \frac{1}{2} \sum_{i_1, i_2=1}^{N}
\frac{\partial (\tilde{G}_b, \tilde{u})}{\partial (x_{i_1}, x_{i_2})}
\{x_{i_1}, x_{i_2}\}_{\mbox{\tiny{PB}}}
\nonumber \\
&=& \frac{1}{2(N-2)!} \sum_{i_1, i_2, i_3, \cdots i_{N}=1}^{N}
\varepsilon_{i_1 i_2 i_3 \cdots i_{N}}
\frac{\partial (\tilde{G}_b, \tilde{u})}{\partial (x_{i_1}, x_{i_2})}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{N-2})}
{\partial (x_{i_3}, \cdots, x_{i_{N}})}
\nonumber \\
&=& \frac{\partial (\tilde{G}_b, \tilde{u}, \tilde{G}_1, \cdots, \tilde{G}_{N-2})}
{\partial (x_{1}, x_{2}, x_{3}, \cdots, x_{N})} = 0~.
\label{Gbu=0}\end{aligned}$$ Hence $G_b$ are constants. We can eliminate the constants by redefining $G_b$, and the resulting $\tilde{G}_b(x_1,x_2,\cdots, x_{N})=0$ can be regarded as [*induced constraints*]{}, which are associated with enlarging the phase space from $(q, p)$ to $(x_1,x_2,\cdots, x_{N})$.
In this way, [*Hamiltonian systems can be formulated as Nambu systems by the use of $N$ variables $x_i=x_i(q, p)$ $(i=1, 2, \cdots, N)$. The variables form an $N$-plet, and the $N-1$ Hamiltonians are given | 0 | non_member_997 |
by the original Hamiltonian $\tilde{H}(x_1,x_2,\cdots, x_{N})=H(q,p)$ and induced constraints $\tilde{G}_b(x_1,x_2,\cdots, x_{N})=0$ $(b=1, \cdots, N-2)$.*]{} Note that $\tilde{H}(x_1,x_2,\cdots, x_{N})$ is not uniquely determined, because of the freedom to add a term $\sum_b\tilde{\lambda}_b(x_1,x_2,\cdots, x_{N})\tilde{G}_b(x_1,x_2,\cdots, x_{N})$ to $\tilde{H}(x_1,x_2,\cdots, x_{N})$. Here $\tilde{\lambda}_b(x_1,x_2,\cdots, x_{N})$ are some functions.
### Examples {#Examples}
Here we present two simple examples to show how induced constraints are obtained for given multiplets.\
(a) $N=3$\
Consider composite variables, $$\begin{aligned}
x=\frac{1}{4}\left(q^2 - p^2\right)~,~~
y=\frac{1}{4}\left(q^2 + p^2\right)~,~~ z=\frac{1}{2}qp~,
\label{Ex1}\end{aligned}$$ which satisfy the following relations: $$\begin{aligned}
\{x, y\}_{\mbox{\tiny{PB}}} = z~,~~
\{y, z\}_{\mbox{\tiny{PB}}} = x~,~~ \{z, x\}_{\mbox{\tiny{PB}}} = -y~.
\label{Ex1-PB}\end{aligned}$$ Then the conditions (Eq. (\[xyzG\])) become $$\begin{aligned}
\frac{\partial \tilde{G}}{\partial x} = x~,~~
\frac{\partial \tilde{G}}{\partial y} = -y~,~~
\frac{\partial \tilde{G}}{\partial z} = z~,
\label{Ex1-dG}\end{aligned}$$ and $\tilde{G}$ is obtained by $$\begin{aligned}
\tilde{G}= \frac{1}{2}\left(x^2 - y^2 + z^2\right)+C~,
\label{Ex1-G}\end{aligned}$$ where $C$ is a constant. Redefining $\tilde{G}$ as $\tilde{G}-C$, we obtain the induced constraint $\tilde{G}(x,y,z)=G(q,p)=0$.
\
(b) $N=4$\
Consider variables including composite ones, $$\begin{aligned}
x_1=q~,~~ x_2=p~,~~ x_3=x_3(q, p)~,~~ x_4=x_4(q, p)~,
\label{Ex2}\end{aligned}$$ which satisfy the following relations: $$\begin{aligned}
&& \{x_1, x_2\}_{\mbox{\tiny{PB}}}
= 1~,~~ \{x_1, x_3\}_{\mbox{\tiny{PB}}} = \frac{\partial x_3}{\partial p}~,~~
\{x_1, x_4\}_{\mbox{\tiny{PB}}} = \frac{\partial x_4}{\partial p}~,~~
\nonumber \\
&& \{x_2, x_3\}_{\mbox{\tiny{PB}}} =
-\frac{\partial x_3}{\partial q}~,~~
\{x_2, x_4\}_{\mbox{\tiny{PB}}} = -\frac{\partial x_4}{\partial q}~,~~
\nonumber | 0 | non_member_997 |
\\
&&\{x_3, x_4\}_{\mbox{\tiny{PB}}} =
\frac{\partial x_3}{\partial q} \frac{\partial x_4}{\partial p}
- \frac{\partial x_3}{\partial p} \frac{\partial x_4}{\partial q}~.
\label{Ex2-PB}\end{aligned}$$ Then the conditions (Eq. (\[xiGb\])) become $$\begin{aligned}
\sum_{i_3, i_4=1}^{4} \varepsilon_{i_1 i_2 i_3 i_4}
\frac{\partial \tilde{G}_1}{\partial x_{i_3}}
\frac{\partial \tilde{G}_2}{\partial x_{i_4}}
= \{x_{i_1}, x_{i_2}\}_{\mbox{\tiny{PB}}}~,
\label{Ex2-dG}\end{aligned}$$ and $\tilde{G}_1$ and $\tilde{G}_2$ are given by $$\begin{aligned}
\tilde{G}_1= x_3-x_3(x_1, x_2)+C_1~,~~ \tilde{G}_2= x_4-x_4(x_1, x_2)+C_2~.
\label{Ex2-G}\end{aligned}$$ where $C_1$ and $C_2$ are constants. By redefining $G_1$ and $G_2$ to eliminate the constants, we obtain the induced constraints $\tilde{G}_1(x_1,x_2,x_3,x_4)=G_1(q,p)=0$ and $\tilde{G}_2(x_1,x_2,x_3,x_4)=G_2(q,p)=0$.
Many degrees of freedom systems {#Many degrees of freedom systems}
-------------------------------
Let us extend our formulation to Hamiltonian systems with many degrees of freedom. Consider a Hamiltonian system described by $n$ sets of canonical doublets $(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})$ (k$=1, 2, \cdots, n$). As is the case with $n=1$ given in Sect. 2.2, hidden Nambu structure can also be found in this system. Here we present the $N=3$ case, i.e., the case with $n$ sets of triplets $x_{i{\mbox{\tiny(k)}}} = x_{i{\mbox{\tiny(k)}}}
(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})$ $(i=1,2,3)$. Generalization to the $N(\ge 3)$ cases is straightforward.
### Dynamics {#Dynamics}
In this system, the Poisson bracket of $A$ and $B$ is defined as $$\begin{aligned}
\{A, B\}_{\mbox{\tiny{PB}}}
\equiv \sum_{{\rm k}=1}^{n}
\left(\frac{\partial A}{\partial q_{\mbox{\tiny(k)}}}
\frac{\partial B}{\partial p_{\mbox{\tiny(k)}}}
- \frac{\partial A}{\partial p_{\mbox{\tiny(k)}}}
| 0 | non_member_997 |
\frac{\partial B}{\partial q_{\mbox{\tiny(k)}}}\right)~,
\label{PB-n}\end{aligned}$$ and the Hamilton’s equation of motion for any function $f=f(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d f}{dt} = \{f, H\}_{\mbox{\tiny{PB}}}~,
\label{H-eqf-n}\end{aligned}$$ where $H=H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ is the Hamiltonian of the system. On the other hand, the Nambu bracket of $\tilde{A}$, $\tilde{B}$, and $\tilde{C}$ is defined as $$\begin{aligned}
\{\tilde{A}, \tilde{B}, \tilde{C}\}_{\mbox{\tiny{NB}}}
\equiv \sum_{{\rm k}=1}^{n}
\frac{\partial (\tilde{A}, \tilde{B}, \tilde{C})}
{\partial (x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})}~,
\label{NB-n}\end{aligned}$$ where $x_{\mbox{\tiny(k)}} = x_{1\mbox{\tiny(k)}}$, $y_{\mbox{\tiny(k)}} = x_{2\mbox{\tiny(k)}}$, and $z_{\mbox{\tiny(k)}} = x_{3\mbox{\tiny(k)}}$. Then the Nambu equation for any function $\tilde{f}=
\tilde{f}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, \cdots,
z_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \{\tilde{f}, \tilde{H}, \tilde{G}\}_{\mbox{\tiny{NB}}}~.
\label{N-eqf-n}\end{aligned}$$ Here $\tilde{H}=
\tilde{H}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}},
\cdots, z_{\mbox{\tiny({\it n})}})
=H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, \cdots,
q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ is the Hamiltonian and $\tilde{G}=\tilde{G}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}},
\cdots, z_{\mbox{\tiny({\it n})}})
= \sum_{\rm k} \tilde{G}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})$ is the sum of the induced constraints that satisfy the conditions $$\begin{aligned}
\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}{\partial x_{\mbox{\tiny(k)}}} =
\frac{\partial(y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})}
{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}~,~~
\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}{\partial y_{\mbox{\tiny(k)}}} =
\frac{\partial(z_{\mbox{\tiny(k)}}, x_{\mbox{\tiny(k)}})}
{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}~,~~
\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}{\partial z_{\mbox{\tiny(k)}}} =
\frac{\partial(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}})}
{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}~.
\label{xyzG-n}\end{aligned}$$ Note that the induced constraints are defined so as to be zero, $\tilde{G}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})
=G_{\mbox{\tiny(k)}}(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})=0$, and the Hamiltonian | 0 | non_member_997 |
is not uniquely determined because of the freedom to add a linear combination of $\tilde{G}_{\mbox{\tiny(k)}}$ to $\tilde{H}$.
The $3n$ variables $x_{i\mbox{\tiny(k)}}$ satisfy the relations $$\begin{aligned}
\{
x_{i_1{\mbox{\tiny($\rm k_1$)}}},
x_{i_2{\mbox{\tiny($\rm k_2$)}}},
x_{i_3{\mbox{\tiny($\rm k_3$)}}}
\}_{\mbox{\tiny{NB}}}
&=& \varepsilon_{i_1 i_2 i_3}~~~~\mbox{for ~$\rm k_1=k_2=k_3$}~,
\label{NB-n-rel1}\\
\{
x_{i_1{\mbox{\tiny($\rm k_1$)}}},
x_{i_2{\mbox{\tiny($\rm k_2$)}}},
x_{i_3{\mbox{\tiny($\rm k_3$)}}}
\}_{\mbox{\tiny{NB}}}
&=& 0~~~~~~~~~~\mbox{otherwise}.
\label{NB-n-rel2}\end{aligned}$$ The first type of relation (Eq. (\[NB-n-rel1\])) is invariant under the time evolution (Eq. (\[N-eqf-n\])) irrespective of the form of $\tilde{H}$. To be more specific, for infinitesimal transformations $x_{i\mbox{\tiny(k)}} \to x'_{i\mbox{\tiny(k)}}=
x_{i\mbox{\tiny(k)}}+(dx_{i\mbox{\tiny(k)}}/dt) dt$, $$\begin{aligned}
\{x'_{\mbox{\tiny(k)}}, y'_{\mbox{\tiny(k)}}, z'_{\mbox{\tiny(k)}}\}
_{\mbox{\tiny{NB}}}
= 1~\label{NB-n-rel'}\end{aligned}$$ hold. We can also show an important relation, $$\begin{aligned}
\frac{\partial(x'_{\mbox{\tiny(1)}}, y'_{\mbox{\tiny(1)}}, z'_{\mbox{\tiny(1)}},
\cdots,
x'_{\mbox{\tiny({\it n})}}, y'_{\mbox{\tiny({\it n})}}, z'_{\mbox{\tiny({\it n})}})}
{\partial(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, z_{\mbox{\tiny(1)}}, \cdots,
x_{\mbox{\tiny({\it n})}}, y_{\mbox{\tiny({\it n})}}, z_{\mbox{\tiny({\it n})}})}
= 1~,\label{LT}\end{aligned}$$ which guarantees the Liouville theorem, the conservation law of the phase space volume under time development. On the other hand, the second type of relation (Eq. (\[NB-n-rel2\])) does not always hold, unless there is no interaction between the $n$ subsystems, i.e., $\tilde{H}$ has a form such as $\tilde{H}(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}},
\cdots, z_{\mbox{\tiny({\it n})}})
= \sum_{\rm k} \tilde{H}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})$.
### Partition functions {#Partition functions}
It is well known that the partition function $Z_{\rm H}$ for a canonical ensemble | 0 | non_member_997 |
of the Hamiltonian system $(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, \cdots,
q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ is defined as $$\begin{aligned}
Z_{\rm H} \equiv
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n} dq_{\mbox{\tiny(k)}} dp_{\mbox{\tiny(k)}} e^{-\beta H}~,
\label{ZH}\end{aligned}$$ where $\beta = 1/(k_BT)$ is the inverse temperature made up of the Boltzmann constant $k_B$ and the temperature $T$. Here we study the partition function $Z_{\rm N}$ for an ensemble of the Nambu system $(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, \cdots, z_{\mbox{\tiny({\it n})}})$ hidden in the Hamiltonian system.
First let us conjecture the form of $Z_{\rm N}$ on physical grounds. Since $\tilde{H}=H$, $Z_{\rm N}$ must contain the $\lq\lq$Boltzmann weight" such as $e^{-\beta \tilde{H}}$. The other Hamiltonian $\tilde{G}$ is the sum of the constraints $\tilde{G}_{\mbox{\tiny(k)}}
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})
= G_{\mbox{\tiny(k)}} (q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})= 0$, and therefore there should be delta functions such as $\delta(\tilde{G}_{\mbox{\tiny(k)}})$ in $Z_{\rm N}$. Furthermore, $Z_{\rm N}$ must contain the volume element $\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}$ from the Liouville theorem.
On the basis of the above observations, it is expected that $Z_{\rm N}$ should have a form such that $$\begin{aligned}
Z_{\rm N} &\equiv&
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}
\delta(\tilde{G}_{\mbox{\tiny(k)}}) e^{-\beta \tilde{H}}
\label{ZN1}\\
&=& \iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}
\int_{-\infty}^{\infty} \frac{d\gamma_{\mbox{\tiny(k)}}}{2\pi}
e^{-\beta \tilde{H}
-i \sum_{\rm k} \gamma_{\mbox{\tiny(k)}} \tilde{G}_{\mbox{\tiny(k)}}}~.
\label{ZN2}\end{aligned}$$ We can derive $Z_{\rm H}$ (Eq. | 0 | non_member_997 |
(\[ZH\])) from this expression for $Z_{\rm N}$. For example, let us consider the case that $\partial \tilde{G}_{\mbox{\tiny(k)}}/\partial z_{\mbox{\tiny(k)}} \ne 0$. We assume that there are $N_{\mbox{\tiny k}}$ solutions of $\tilde{G}_{\mbox{\tiny(k)}}=0$, $z_{\mbox{\tiny(k)}}^{(a_{\mbox{\tiny k}})}$ $(a_{\mbox{\tiny k}} = 1, 2, \cdots, N_{\mbox{\tiny k}})$, and all of them satisfy the conditions (Eq. (\[xyzG-n\])). Then using the formula for the delta function and the change of variables, Eq. (\[ZN1\]) becomes $$\begin{aligned}
Z_{\rm N} &=&
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}
\sum_{a_{\mbox{\tiny k}}=1}^{N_{\mbox{\tiny k}}}
\delta(z_{\mbox{\tiny(k)}}-
z_{\mbox{\tiny(k)}}^{(a_{\mbox{\tiny k}})}(x_{\mbox{\tiny(k)}},y_{\mbox{\tiny(k)}}))
\left|\frac{\partial \tilde{G}_{\mbox{\tiny(k)}}}
{\partial z_{\mbox{\tiny(k)}}}\right|^{-1} e^{-\beta \tilde{H}}
\nonumber \\
&=& \iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}
\sum_{a_{\mbox{\tiny k}}=1}^{N_{\mbox{\tiny k}}}
\delta(z_{\mbox{\tiny(k)}}-
z_{\mbox{\tiny(k)}}^{(a_{\mbox{\tiny k}})}(x_{\mbox{\tiny(k)}},y_{\mbox{\tiny(k)}}))
\left|\frac{\partial (x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}})}
{\partial (q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}\right|^{-1}
e^{-\beta \tilde{H}}
\nonumber \\
&=& {\mathcal N}
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n} dq_{\mbox{\tiny(k)}} dp_{\mbox{\tiny(k)}} e^{-\beta H}
= {\mathcal N} Z_{\rm H}~,
\label{ZH=ZN}\end{aligned}$$ where ${\mathcal N}=\prod_{{\rm k}=1}^{n}N_{\mbox{\tiny k}}$ is a constant normalization factor. This factor is irrelevant to the evaluation of physical quantities.
It is natural to require that $Z_{\rm N}$ should agree with $Z_{\rm H}$ (up to some normalization factor), because we just describe the same physical system using different formulations. It should be noted here that both expressions for $Z_{\rm N}$ (Eq. (\[ZN1\]) or Eq. (\[ZN2\])) are different from that proposed in Ref. | 0 | non_member_997 |
[@N]. This comes from the fact that the Nambu mechanics considered here is an effective one induced by the redundancy of the variables.
Finally, we just give the result for the case of general $N(\ge 3)$. The possible form of the partition function is given by $$\begin{aligned}
Z_{\rm N} = \iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n}
dx_{1\mbox{\tiny{(k)}}} dx_{2\mbox{\tiny{(k)}}}
\cdots dx_{N\mbox{\tiny{(k)}}}
\delta(\tilde{G}_{1\mbox{\tiny{(k)}}}) \delta(\tilde{G}_{2\mbox{\tiny{(k)}}})
\cdots \delta(\tilde{G}_{N-2\mbox{\tiny{(k)}}}) e^{-\beta \tilde{H}}~,
\label{ZN-N}\end{aligned}$$ where $\tilde{G}_{b\mbox{\tiny{(k)}}}=0$ $(b = 1, 2, \cdots, N-2)$ are induced constraints. This expression should agree with $Z_{\rm H}$ (Eq. (\[ZH\])) up to some constant normalization factor.
Generalized Nambu equations {#Generalized Nambu equations}
---------------------------
We generalize our formulation to include a specific case that all multiplets share some variables. In such a case, a generalization of the Nambu equation would be required.
Let us describe a Hamiltonian system with $n$ sets of canonical doublets $(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})$ $({\rm k}=1, \cdots, n)$ using $2n+m$ variables $w_{\ell}$ $(\ell=1, \cdots, 2n+m)$. We classify the variables $w_{\ell}$ into two groups, $x_{a}$ $(a=1, \cdots, 2n)$ and $z_s$ $(s=1, \cdots, m)$, where $x_a$ are assumed to satisfy $\det \{x_a, x_b\}_{\mbox{\tiny{PB}}} \ne 0$. Note that the classification of variables is not unique.
First we study the case with $m=0$ for completeness. In this case, the | 0 | non_member_997 |
equation for any function $\tilde{f}(x_1, \cdots, x_{2n}) =
f(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \sum_{{\rm k}=1}^{n}
\frac{\partial(f, H)}{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}
= \frac{1}{2} \sum_{{\rm k}=1}^{n}
\sum_{{a, b}=1}^{2n} \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x_a, x_b)}
\frac{\partial(x_a, x_b)}{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}
= \sum_{{a, b}=1}^{2n} \tilde{g}_{ab}
\frac{\partial(\tilde{f}, \tilde{H})}{\partial(x_a, x_b)}~,
\label{H-eq(m=0)}\end{aligned}$$ where $\tilde{H}=\tilde{H}(x_1, \cdots, x_{2n}) =
H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ and $\tilde{g}_{ab}$ is defined as $$\begin{aligned}
\tilde{g}_{ab}(x_1, \cdots, x_{2n})
= g_{ab}(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
\equiv \frac{1}{2}\sum_{{\rm k}=1}^{n}
\frac{\partial(x_a, x_b)}{\partial(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})}
= \frac{1}{2}\{x_a, x_b\}_{\mbox{\tiny{PB}}}~.
\label{Gab}\end{aligned}$$ The $\tilde{g}_{ab}$ plays the role of a metric tensor, because it transforms under a change of variables $x_a \to x'_a$ as follows: $$\begin{aligned}
\tilde{g}'_{ab}(x'_1, \cdots, x'_{2n})
= \sum_{{c, d}=1}^{2n} \frac{\partial x'_a}{\partial x_c}
\frac{\partial x'_b}{\partial x_d} ~\tilde{g}_{cd}(x_1, \cdots, x_{2n})~.
\label{G'ab}\end{aligned}$$ In the case in which $\tilde{g}_{ab}$ depends on $x_a$, neither the transformation $(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
\to (x_1, \cdots, x_{2n})$ nor the time evolution of $x_a$ is a canonical transformation. The latter means that the Liouville theorem in general does not hold for the dynamics of $x_a$. This fact reminds us of the superiority of canonical variables.
Now let us proceed to the case with $m \ge 1$. The equation for | 0 | non_member_997 |
a function $\tilde{f}(w_1, \cdots, w_{2n+m}) =
f(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ can be written as $$\begin{aligned}
\frac{d \tilde{f}}{dt}
&=& \frac{1}{2} \sum_{{a, b}=1}^{2n} \frac{\partial(\tilde{f}, \tilde{H})}
{\partial(x_a, x_b)}\{x_a, x_b\}_{\mbox{\tiny{PB}}} \nonumber \\
&&+ \sum_{a=1}^{2n}\sum_{s=1}^{m} \frac{\partial(\tilde{f}, \tilde{H})}
{\partial(x_a, z_s)}\{x_a, z_s\}_{\mbox{\tiny{PB}}}
+ \frac{1}{2} \sum_{s, t=1}^{m} \frac{\partial(\tilde{f}, \tilde{H})}
{\partial(z_s, z_t)}\{z_s, z_t\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(m)}\end{aligned}$$ where $\tilde{H}= \tilde{H}(w_1, \cdots, w_{2n+m})
=H(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$. Introducing functions $\tilde{G}_s$ $(s=1, \cdots, m)$ and $\tilde{g}^{(m)}_{ab}$ that satisfy the following relations, $$\begin{aligned}
\frac{1}{2} \{x_a, x_b\}_{\mbox{\tiny{PB}}}
&=&
\tilde{g}^{(m)}_{ab} \frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (z_{1}, \cdots, z_{m})}~,~~
\label{phi-2n+m1}\\
\frac{1}{2}\{x_a, z_{s}\}_{\mbox{\tiny{PB}}}
&=&
-\sum_{b=1}^{2n} \tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{G}_1, \cdots, \tilde{G}_{s-1},
\tilde{G}_{s}, \tilde{G}_{s+1}, \cdots, \tilde{G}_{m})}
{\partial (z_{1}, \cdots, z_{s-1}, x_b, z_{s+1}, \cdots,z_{m})}~,~~
\label{phi-2n+m2}\\
\{z_s, z_t\}_{\mbox{\tiny{PB}}}
&=&\!\!\!\!
\sum_{a,b=1}^{2n}\tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{G}_1, \cdots,
\tilde{G}_{s-1}, \tilde{G}_{s}, \tilde{G}_{s+1}, \cdots,
\tilde{G}_{t-1}, \tilde{G}_{t}, \tilde{G}_{t+1}, \cdots, \tilde{G}_{m})}
{\partial (z_{1}, \cdots, z_{s-1}, x_{a}, z_{s+1},
\cdots, z_{t-1}, x_{b}, z_{t+1}, \cdots, z_{m})}~,
\label{phi-2n+m3}\end{aligned}$$ where $s<t$, Eq. (\[H-eq(m)\]) can be rewritten as $$\begin{aligned}
\frac{d \tilde{f}}{dt}
= \sum_{a, b=1}^{2n} \tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{f}, \tilde{H}, \tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (x_a, x_b, z_1, \cdots, z_m)}
\label{N-eqf(m)}\end{aligned}$$ using a formula concerning Jacobians.
By the use of Eqs. (\[phi-2n+m1\])–(\[phi-2n+m3\]), it is shown that the Poisson bracket between any of the $m$ functions $G_s(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
=\tilde{G}_s(w_1, \cdots, w_{2n+m})$ | 0 | non_member_997 |
and an arbitrary function $u(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
=\tilde{u}(w_1, \cdots, w_{2n+m})$ vanishes such that $$\begin{aligned}
\{G_s, u\}_{\mbox{\tiny{PB}}}
&=& \frac{1}{2} \sum_{a, b=1}^{2n}
\frac{\partial(\tilde{G}_s, \tilde{u})}
{\partial(x_a, x_b)}\{x_a, x_b\}_{\mbox{\tiny{PB}}}
\nonumber \\
&&
+ \sum_{a=1}^{2n} \sum_{s=1}^{m}
\frac{\partial(\tilde{G}_s, \tilde{u})}
{\partial(x_a, z_s)}\{x_a, z_s\}_{\mbox{\tiny{PB}}}
+ \frac{1}{2} \sum_{s, t=1}^{m}
\frac{\partial(\tilde{G}_s, \tilde{u})}
{\partial(z_s, z_t)}\{z_s, z_t\}_{\mbox{\tiny{PB}}}
\nonumber \\
&=& \sum_{a, b=1}^{2n}
\tilde{g}^{(m)}_{ab}
\frac{\partial (\tilde{G}_s, \tilde{u}, \tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (x_a, x_b, z_1, \cdots, z_m)} = 0~.
\label{phiu=0}\end{aligned}$$ Hence $G_s(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}},
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ are constants and, if necessary, we can define $G_s=\tilde{G}_s=0$ by shifting constants. We refer to Eq. (\[N-eqf(m)\]) as the generalized Nambu equation. Note that the Liouville theorem does not hold in general for the dynamics given by this equation. This unfavorable property is a result of two factors: Eq. (\[N-eqf(m)\]) has $x_{a}$-dependent $\tilde{g}^{(m)}_{ab}$ and multiplets in Eq. (\[N-eqf(m)\]) share common variables $z_{s}$. The latter means that it is difficult to define an appropriate phase space volume.
One of the non-vanishing components of $\tilde{g}^{(m)}_{ab}$ can be set to $\frac{1}{2}$ by redefinition of constraints $\tilde{G}_s$. For example, in the case in which $n=1$, we can set $\tilde{g}^{(m)}_{12}=\frac{1}{2}$ (and $\tilde{g}^{(m)}_{21}=-\frac{1}{2}$) by redefining $\tilde{G}_s$, and Eq. (\[N-eqf(m)\]) reduces to the Nambu equation (Eq. (\[H-eqf(gN)\])) with $N=2+m$.
| 0 | non_member_997 |
Finally, we consider the case in which the variables $x_a$ and $z_s$ are further classified into $M$ $\lq\lq$irreducible“ sets, $\{x_{a^1}^{\mbox{\tiny{({\rm 1})}}},
z_{s^1}^{\mbox{\tiny{({\rm 1})}}}\}
\bigoplus
\{x_{a^2}^{\mbox{\tiny{({\rm 2})}}},
z_{s^2}^{\mbox{\tiny{({\rm 2})}}}\}
\bigoplus \cdots \bigoplus
\{x_{a^M}^{\mbox{\tiny{({\it M})}}},
z_{s^M}^{\mbox{\tiny{({\it M})}}}\}$, where $a^i=1,\cdots,2n^i$ ($\sum_{i=1}^{M}n^i=n$) and $s^i=1,\cdots,m^i$ ($\sum_{i=1}^{M}m^i=m$). Here $\lq\lq$irreducible” means that the Poisson bracket between any two elements that belong to different sets vanishes, i.e., $\{x_{a^i}^{\mbox{\tiny{({\it i})}}},
x_{a^j}^{\mbox{\tiny{({\it j})}}}\}_{\mbox{\tiny{PB}}}=0$, $\{x_{a^i}^{\mbox{\tiny{({\it i})}}},
z_{s^j}^{\mbox{\tiny{({\it j})}}}\}_{\mbox{\tiny{PB}}}=0$, and $\{z_{s^i}^{\mbox{\tiny{({\it i})}}},
z_{s^j}^{\mbox{\tiny{({\it j})}}}\}_{\mbox{\tiny{PB}}}=0$ for $i \ne j$. Note that this classification is not unique, either. The equation of motion for any function $\tilde{f}(w_1, \cdots, w_{2n+m})$ can be expressed in the form of the generalized Nambu equation, $$\begin{aligned}
\frac{d \tilde{f}}{dt}
= \sum_{i=1}^{M}
\sum_{a^i, b^i=1}^{2n^i} \tilde{g}^{(m^i)}_{a^i b^i}
\frac{\partial (\tilde{f}, \tilde{H},
\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}},
\cdots, \tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (x_{a^i}^{\mbox{\tiny{({\it i})}}},
x_{b^i}^{\mbox{\tiny{({\it i})}}},
z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~.
\label{N-eqf(m)2}\end{aligned}$$ Here $\tilde{G}_{s^i}^{\mbox{\tiny{({\it i})}}}$ and $\tilde{g}^{(m^i)}_{a^ib^i}$ should satisfy the following conditions: $$\begin{aligned}
\frac{1}{2} \{x_{a^i}^{\mbox{\tiny{({\it i})}}},
x_{b^i}^{\mbox{\tiny{({\it i})}}}\}_{\mbox{\tiny{PB}}}
&=&
\tilde{g}^{(m^i)}_{a^ib^i}
\frac{\partial (\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}},
\cdots, \tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~,~~
\label{phi-2n+m4}\\
\frac{1}{2} \{x_{a^i}^{\mbox{\tiny{({\it i})}}},
z_{s^i}^{\mbox{\tiny{({\it i})}}}\}_{\mbox{\tiny{PB}}}
&=&
-\sum_{b^i=1}^{2n^i} \tilde{g}^{(m^i)}_{a^ib^i}
\frac{\partial (\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}},
\cdots, \tilde{G}_{s^i-1}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{s^i}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
\tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{s^i-1}^{\mbox{\tiny{({\it i})}}},
x_{b^i}^{\mbox{\tiny{({\it i})}}},
z_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~,~~
\label{phi-2n+m5}\\
\{z_{s^i}^{\mbox{\tiny{({\it | 0 | non_member_997 |
i})}}},
z_{t^i}^{\mbox{\tiny{({\it i})}}}\}_{\mbox{\tiny{PB}}}&=& \nonumber \\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\sum_{a^i,b^i=1}^{2n^i}\tilde{g}^{(m^i)}_{a^ib^i}
\frac{\partial (\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}},
\cdots, \tilde{G}_{s^i-1}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{s^i}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
\tilde{G}_{t^i-1}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{t^i}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{t^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
\tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{s^i-1}^{\mbox{\tiny{({\it i})}}},
x_{a^i}^{\mbox{\tiny{({\it i})}}},
z_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{t^i-1}^{\mbox{\tiny{({\it i})}}},
x_{b^i}^{\mbox{\tiny{({\it i})}}},
z_{t^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~,~~
\label{phi-2n+m6}\end{aligned}$$ where $s^i< t^i$. We refer to the systems where the master equations are given by Eq. (\[N-eqf(m)\]) or Eq. (\[N-eqf(m)2\]) as generalized Nambu systems.
Nambu systems in constrained Hamiltonian systems {#Nambu systems in constrained Hamiltonian systems}
================================================
Subject {#Subject}
-------
In the previous section, we found that a Hamiltonian system can be formulated as a Nambu system with multiplets including composite variables of $q$ and $p$. The main feature of our formulation is the existence of induced constraints that are required just for consistency between the variables. Together with the Hamiltonian of the original system, the induced constraints serve as Hamiltonians of the Nambu system. Therefore it is intriguing to study how constrained Hamiltonian systems, systems with [*physical*]{} constraints, are cast into Nambu systems in our formulation.
The relations between Nambu systems and constrained Hamiltonian systems have been investigated by many authors [@BF; @CK; @R; @MS; @KT; @KT2; @LJ]. To clarify the | 0 | non_member_997 |
difference between previous works and our approach, here we give a brief summary of the results obtained so far. In most works, Nambu systems are treated as the original systems, and studies have been carried out to find appropriate constrained Hamiltonian systems into which the Nambu systems can be embedded [@BF; @CK; @R; @MS; @KT; @KT2]. Specifically, it has been shown that Nambu equations (Eq. (\[N-eq\])) are compatible with the following equations: $$\begin{aligned}
&& p_i = H_1 \frac{\partial H_2}{\partial x_i}~,\label{N-eq-con1}\\
&& \sum_{i=1}^3 \frac{\partial(H_1, H_2)}{\partial(x_i, x_j)}\frac{dx_i}{dt} = 0~.
\label{N-eq-con2}\end{aligned}$$ Here $p_i$ ($i=1,2,3$) are the canonical conjugate momenta defined as $p_i \equiv \partial L/\partial \dot{x}_i$ with the Lagrangian $$\begin{aligned}
L = H_1 \sum_{i=1}^3 \frac{\partial H_2}{\partial x_i}\frac{dx_i}{dt} ~.
\label{N-eq-conL}\end{aligned}$$ Equation (\[N-eq-con2\]) can be derived as the Euler–Lagrange equation from this Lagrangian, and Eq. (\[N-eq-con1\]) leads to the relations $\phi_i \equiv p_i - H_1 {\partial H_2}/{\partial x_i} = 0$, which can be regarded as constraints. In this way, Nambu systems can be interpreted as Hamiltonian systems with specific constraints.
On the other hand, researchers have studied whether constrained systems can be described as Nambu systems or not. Specifically, it has been shown that constrained Hamiltonian systems can be formulated in terms of (a generalized | 0 | non_member_997 |
form of) Nambu mechanics by introducing an extra phase-space variable [@LJ]. For a system with canonical variables $(q_k, p_k)$ $(k=1, \cdots, n)$ and $m$ first class constraints $\phi_l(q_1, \cdots, p_n)=0$, the equations of motion are given by $$\begin{aligned}
\frac{d q_k}{d t} &=& \frac{\partial H}{\partial p_k}
+ \sum_{l=1}^m \left(\
\frac{\partial \lambda_l}{\partial p_k}\phi_l
+ \lambda_l \frac{\partial \phi_l}{\partial p_k}
\right)~,\label{eq-constraints1}\\~~
\frac{d p_k}{d t} &=& -\frac{\partial H}{\partial q_k}
- \sum_{l=1}^m \left(\
\frac{\partial \lambda_l}{\partial q_k}\phi_l
+ \lambda_l \frac{\partial \phi_l}{\partial q_k}
\right)~,
\label{eq-constraints2}\end{aligned}$$ where $\lambda_l$ are Lagrange multipliers. Equations (\[eq-constraints1\]) and (\[eq-constraints2\]) are derived from (a generalized form of) the Nambu equation $$\begin{aligned}
\frac{d f}{dt} = \sum_{k=1}^n \frac{\partial (f, H_1, H_2)}{\partial (q_k, p_k, r)}~,
\label{N-eq-constraints}\end{aligned}$$ where $f=f(q_1, \cdots, p_n)$, $r$ is an extra phase-space variable, and Hamiltonians are defined as $$\begin{aligned}
H_1 = H - r~,~~ H_2 = r+ \sum_{l=1}^m \lambda_l \phi_l~.
\label{H1H2-constraints}\end{aligned}$$ The equation for $r$ is given by $$\begin{aligned}
\frac{d r}{dt}= - \sum_{l=1}^m
\left(\lambda_l \{\phi_l, H\}_{\mbox{\tiny{PB}}}
+\phi_l \{\lambda_l, H\}_{\mbox{\tiny{PB}}}
\right)
= - \sum_{l=1}^m \lambda_l \frac{d \phi_l}{dt},
\label{r-eq-constraints}\end{aligned}$$ where the last equality holds after imposing constraints. Requiring the extra variable $r$ to decouple from the dynamics, i.e., $dr/dt = 0$, we obtain ${d \phi_l}/{dt} = 0$.
Our approach differs from these previous works. Our starting | 0 | non_member_997 |
---
abstract: 'In this work, we study the [A$_{x}$Fe$_{2-y}$Se$_2$]{} (A=K, Rb) superconductors using angle-resolved photoemission spectroscopy. In the low temperature state, we observe an orbital-dependent renormalization for the bands near the Fermi level in which the [$d_{xy}$]{} bands are heavily renormliazed compared to the [$d_{xz}$/$d_{yz}$]{} bands. Upon increasing temperature to above 150K, the system evolves into a state in which the [$d_{xy}$]{} bands have diminished spectral weight while the [$d_{xz}$/$d_{yz}$]{} bands remain metallic. Combined with theoretical calculations, our observations can be consistently understood as a temperature induced crossover from a metallic state at low temperature to an orbital-selective Mott phase (OSMP) at high temperatures. Furthermore, the fact that the superconducting state of [A$_{x}$Fe$_{2-y}$Se$_2$]{} is near the boundary of such an OSMP constraints the system to have sufficiently strong on-site Coulomb interactions and Hund’s coupling, and hence highlight the non-trivial role of electron correlation in this family of iron superconductors.'
author:
- 'M. Yi'
- 'D.H. Lu'
- 'R. Yu'
- 'S. C. Riggs'
- 'J.-H. Chu'
- 'B. Lv'
- 'Z. Liu'
- 'M. Lu'
- 'Y.-T. Cui'
- 'M. Hashimoto'
- 'S.-K. Mo'
- 'Z. Hussain'
- 'C. W. Chu'
- 'I.R. Fisher'
- 'Q. Si'
- 'Z.-X. Shen'
| 0 | non_member_998 |
title: 'Observation of Temperature-Induced Crossover to an Orbital-Selective Mott Phase in [A$_{x}$Fe$_{2-y}$Se$_2$]{} (A=K, Rb) Superconductors'
---
Electron correlation remains a central focus in the study of high temperature superconductors. The strongly correlated cuprate superconductors are understood as doped Mott insulators (MI) [@1LeePA06] while the iron-based superconductors (FeSCs) have been found to be at most moderately correlated [@2LuDH08; @3YangWL09; @4Qazibash09]. Even though the low energy electronic structures of different FeSCs families share the common Fe $3d$ bands, there are systematic variations in their physical properties, such as ordered magnetic moment and effective mass [@5YinZP11]. In general, electron correlation is the weakest in iron phosphides with relatively low mass renormalization [@2LuDH08], and moderate in the more extensively studied iron arsenides [@2LuDH08; @3YangWL09]. The Fe(Te,Se) chalcogenide family, in comparison, seems to harbor stronger correlation as inferred from larger ordered moment, yet metallic resistivity is still observed [@6Liu10]. The newest iron chalcogenide superconductors, [A$_{x}$Fe$_{2-y}$Se$_2$]{} (A=alkali metal) [@7Guo10; @8Krzton11; @9Li11; @10Fang11; @11Wang11] (AFS) hints at even stronger correlation with a large observed moment of 3.3[$\mu_{\tiny{\textrm{B}}}$]{} [@12Wei11] and insulating transport behavior in the phase diagram.
Another important factor in understanding the FeSCs lies in their multi-orbital nature. In such a system, orbital-dependent behavior as well as | 0 | non_member_998 |
competition between inter- and intra-orbital interactions could play a critical role in determining their physical properties. One example is the orbital anisotropy that onsets with the in-plane symmetry breaking structural transition as observed in the underdoped arsenides [@13Yi11]. Another example is the different pairing symmetry that could arise considering the relative strength of inter- and intra-orbital interactions extensively studied theoretically [@14chubukov12]. Theoretical models have considered correlation effects in the bad metal regime in terms of an incipient Mott picture [@5YinZP11; @15si08], and the proximity to the Mott transition may be orbital-dependent even for orbitally-independent Coulomb interactions [@16Yu11; @17Yu11b; @18Zhou11; @19craco11]. What arises from the model is an orbital selective Mott phase (OSMP), in which some orbitals are Mott-localized while others remain itinerant. First introduced in the context of the Ca$_{2-x}$Sr$_x$RuO$_4$ system, an OSMP may result from both the orbital-dependent kinetic energy and the combined effects of the Hund’s coupling and crystal level splittings [@20anisimov02; @21demedici]. An OSMP links naturally with models of coexisting itinerant and localized electrons that have been proposed to compensate for the shortcomings of both strong coupling and weak coupling approaches [@22You11; @23moon10]. However, to date, there has been no experimental evidence for OSMP in any FeSC.
| 0 | non_member_998 |
{width="90.00000%"}
In this paper, we present angle-resolved photoemission spectroscopy (ARPES) data from two superconducting AFSs, [K$_{x}$Fe$_{2-y}$Se$_2$]{} (KFS) and [Rb$_{x}$Fe$_{2-y}$Se$_2$]{} (RFS), with $T_C$ of 32K and 31K, respectively, as well as insulating and intermediate dopings (see SI). We observe the superconducting AFSs undergoing a temperature-induced crossover from a metallic state in which all three $t_{2g}$ orbitals-[$d_{xy}$]{}, [$d_{xz}$]{} and [$d_{yz}$]{} are present near the Fermi level ([$E_F$]{}) to a state in which the [$d_{xy}$]{} bands has diminished spectral weight while the [$d_{xz}$/$d_{yz}$]{} bands remain metallic. In addition, the intermediate doping shows stronger correlation than the superconducting doping, as seen in the further renormalization of the [$d_{xz}$/$d_{yz}$]{} bands and the much more suppressed [$d_{xy}$]{} intensity, while the insulating doping has no spectral weight near [$E_F$]{} in any orbitals. From comparison with our theoretical calculations, the ensemble of our observations are most consistent with the understanding that the presence of strong Coulomb interactions and Hund’s coupling places the superconducting AFSs near an OSMP at low temperatures, and crosses over into the OSMP via raised temperature, while the intermediate and insulating compounds are on the boundary of the OSMP and in the MI phase, respectively, suggesting the importance of electron correlation in this family of | 0 | non_member_998 |
FeSCs.
{width="95.00000%"}
The low temperature electronic structure of [K$_{x}$Fe$_{2-y}$Se$_2$]{}is shown in Fig. \[fig:fig1\]. The Fermi surface (FS) of KFS consists of large electron pockets at the Brillouin zone (BZ) corner-X point, and a small electron pocket at the BZ center-$\Gamma$ point (Fig. \[fig:fig1\](a)), consistent with previous ARPES reports [@24zhang11; @25qian11; @26mou11]. For the crystallographic 2-Fe unit cell, LDA calculations predict two electron pockets at the X point [@27nekrasov11] (Fig. \[fig:fig1\](i)). While FS map appears to show only one electron pocket at X, measurements under different polarizations (Fig. \[fig:fig1\](j)-(l)) reveal the expected two electron bands with nearly degenerate Fermi crossings ([$k_F$]{}) but different band bottom positions-a shallower one around -0.05eV and a deeper one that extends to the top of the [$d_{xz}$/$d_{yz}$]{} hole-like bands at -0.12eV. The Luttinger volume of the two electron pockets gives $\sim$0.16 electrons per Fe. Considering that the C$_4$ symmetry of the crystal dictates degeneracy of the [$d_{xz}$/$d_{yz}$]{} electron band bottom and hole band top at X, the shallower electron band that is not degenerate with the hole-like band at higher binding energy is most likely of [$d_{xy}$]{} character, and the deeper one [$d_{xz}$]{} along $\Gamma$-X and [$d_{yz}$]{} along the perpendicular direction. This observed orbital character assignment | 0 | non_member_998 |
seems to contradict the LDA prediction of the shallower electron band as [$d_{xz}$/$d_{yz}$]{} and the deeper one [$d_{xy}$]{} in FeSC [@28graser10] (Fig. \[fig:fig1\](i)), as observed in Co-doped [BaFe$_2$As$_2$]{} (Fig. \[fig:fig1\](f)-(g)). However, this assignment can be understood if we consider the KFS band structure as a whole. Three filled hole bands are observed near the $\Gamma$ point (Figs. \[fig:fig1\](d)-(e)), where the two lower ones can be identified as [$d_{xz}$/$d_{yz}$]{} and the higher one [$d_{xy}$]{}. Interestingly, the [$d_{xy}$]{} band is far more renormalized ($\sim$a factor of 10) compared to LDA than the [$d_{xz}$/$d_{yz}$]{} bands ($\sim$a factor of 3), indicating stronger correlation for the [$d_{xy}$]{} orbital. This is in contrast to the Co-doped [BaFe$_2$As$_2$]{} band structure, in which all orbitals are renormalized by roughly the same factor ($\sim$2) compared to LDA. Hence, our assignment of [$d_{xy}$]{} character to the shallower electron band is consistent with strong orbital-dependent renormalization, which brings the deeper [$d_{xy}$]{} electron band at X predicted by LDA to be shallower than the [$d_{xz}$/$d_{yz}$]{} band. This orbital-dependent renormalization behavior also emerges from our theoretical calculations as will be discussed later.
As the electronic structure at [$E_F$]{} is dominated by the large electron pockets at X, we focus on these features. Fig. | 0 | non_member_998 |
\[fig:fig2\] shows a temperature-dependent study of these bands taken in two polarization geometries. In Fig. \[fig:fig2\](a), the matrix element for [$d_{xy}$]{} is stronger than [$d_{xz}$]{}. At low temperatures, [$d_{xy}$]{} band is clearly resolved with weaker intensity for the [$d_{xz}$]{} band. With increasing temperature, the spectral weight of the [$d_{xy}$]{} band noticeably diminishes, eventually revealing the remaining [$d_{xz}$]{} band at high temperatures. In Fig. \[fig:fig2\](b), the deeper [$d_{yz}$]{} band has more intensity than the corresponding [$d_{xz}$]{} band in Fig. \[fig:fig2\](a) while the presence of the [$d_{xy}$]{} band is still very noticeable from the increased intensity where they significantly overlap, as well as from the energy distribution curves (EDCs) shown in Fig. \[fig:fig2\](c). With increasing temperature, again, the spectral weight of the [$d_{xy}$]{} band diminishes, leaving only the [$d_{yz}$]{} band at high temperatures, which clearly has a deeper band bottom than the shallow [$d_{xy}$]{} band (Fig. \[fig:fig2\](c)). In addition, we have artificially introduced a 210K thermal broadening to the 30K spectra as shown in the last column of Fig. \[fig:fig2\]. The clear contrast to the 210K data rules out a trivial thermal broadening as an origin for the observed diminishing of [$d_{xy}$]{} spectral weight.
{width="95.00000%"}
To capture this behavior quantitatively, we analyze | 0 | non_member_998 |
the temperature dependence of the EDCs at the X-point, stacked in Fig. \[fig:fig3\](c). At all temperatures, there is a large hump background corresponding to the large hole-like dispersion at high binding energy. At low temperatures, there is an additional peak around -0.05eV corresponding to the [$d_{xy}$]{} band bottom. We fit these EDCs with a Gaussian background for the large hump feature and a Lorentzian peak for the [$d_{xy}$]{} band. The integrated spectral weight for the fitted [$d_{xy}$]{} peak is plotted in Fig. \[fig:fig3\](d), which decreases toward zero with increasing temperature, seen as a non-trivial drop between 100K and 200K. As an independent check against trivial thermal effect, we choose small regions in the raw spectral image (marked in Fig. \[fig:fig3\](a)-(b)) dominated by [$d_{xy}$]{} (blue), [$d_{yz}$]{} (green), and mixed [$d_{xy}$]{} and [$d_{xz}$/$d_{yz}$]{} (magenta/cyan) characters and plot their integrated intensities as a function of temperature (Fig. \[fig:fig3\](e)). The spectral weight from [$d_{xy}$]{}-dominated region rapidly decreases, consistent with the fitted result in Fig. \[fig:fig3\](d), while that of [$d_{yz}$]{}-dominated region does not drop in a similar manner. The regions of mixed orbitals show a slower diminishing spectral weight compared to that for [$d_{xy}$]{}, reflecting the combined contributions from both [$d_{xz}$/$d_{yz}$]{} and [$d_{xy}$]{} orbitals. Although | 0 | non_member_998 |
this method has the uncertainty of small leakage of other orbitals into the chosen regions, which is the cause of the finite residual value for the [$d_{xy}$]{} curve, the contrasting behavior of the [$d_{xy}$]{} versus [$d_{xz}$/$d_{yz}$]{} orbitals is clearly demonstrated. A temperature cycle test was performed to exclude the possibility of sample aging (see SI). Measurements on the sister compound RFS reveal similar behavior (SI), suggesting the generality in this family of superconductors. This observation of a selected orbital that loses coherent spectral weight while the others remain metallic is reminiscent of a crossover into an OSMP in which selected orbitals become Mott localized while others remain metallic.
{width="95.00000%"}
To further understand this phenomenon, we perform theoretical calculations based on a five-orbital Hubbard model to study the metal-to-insulator transition (MIT) in the paramagnetic phase using a slave-spin mean-field method [@30yu; @31demedici05]. At commensurate electron filling n=6 per Fe (corresponding to Fe2$^+$ of the parent FeSC), we find the ground state of the system to be a metal, an OSMP or a MI depending on the intra-orbital repulsion U and the Hund’s coupling J. Furthermore, the MIT can be triggered by increasing temperature (Fig. \[fig:fig4\](a)) due to the larger entropy of | 0 | non_member_998 |
the insulating phase. At a fixed interaction strength (within a certain range, Fig. \[fig:fig4\](a)), the system goes from a metal to an OSMP and then to a MI with increasing temperature. The MI phase is suppressed by electron doping. By contrast, the OSMP can survive, as shown in Fig. \[fig:fig4\](b) for n=6.15, which roughly corresponds to the filling of the superconducting state from ARPES measurements. From the evolution of the orbitally resolved quasiparticle spectral weight, $Z_{\alpha}$, as a function of the temperature (Fig. \[fig:fig4\](c)), we show that the OSMP corresponds to the [$d_{xy}$]{} orbital being Mott localized ($Z=0$) and the rest of the $3d$ orbitals remaining delocalized ($Z>0$). This result originates primarily from a combined effect of the orbital dependence of the projected density of states and the interplay between the Hund’s coupling and crystal level splitting (see SI and Ref. ).
To compare with the ARPES measurements, we have calculated the quasiparticle spectral function A(k,E) in the 2-Fe BZ. At low temperature and in the non-interacting limit U=0 (Fig. \[fig:fig4\](d)), the electronic structure of the model agrees well with that from LDA, with the [$d_{xy}$]{} band deeper than the [$d_{xz}$/$d_{yz}$]{} band at X. This order switches with sufficiently strong | 0 | non_member_998 |
interaction. At U=3.75 eV (Fig. \[fig:fig4\](e)), the [$d_{xy}$]{}-dominated bands are pushed above their [$d_{xz}$/$d_{yz}$]{} counterparts by the strongly orbital-dependent mass renormalization, as reflected in the orbital dependent quasiparticle spectral weights (Fig. \[fig:fig4\](c)). The mass renormalization is the largest for the [$d_{xy}$]{} orbital ($\sim$10), and smaller for the [$d_{xz}$/$d_{yz}$]{} orbitals ($\sim$3), which is compatible with the low-temperature ARPES results (Fig. \[fig:fig1\](e)). The temperature-induced crossover to the OSMP is clearly seen from the suppression of the spectral weights in the [$d_{xy}$]{} orbital that accompanies the reduction of the weights in the other orbitals (Fig. \[fig:fig4\](c),(e)-(g)), in agreement with the ARPES results (Fig. \[fig:fig3\]).
The temperature-induced nature of the crossover constrains these AFS superconductors to be very close to the boundary of the OSMP in the zero temperature ground state, which is also the superconducting state. The best agreement between theory and experiments is achieved at U$\sim$3.75 eV. While the absolute value of this interaction is sensitive to the parameterization of the crystal levels and Hund’s coupling, it is instructive to make a qualitative comparison with the case of the iron pnictides. The enhanced correlation effects for AFS tracks the reduction of the width of the (U=0) [$d_{xy}$]{} band, which is about 0.7 | 0 | non_member_998 |
of its counterpart in 1111 iron arsenides.
One known concern for the AFS materials is the existence of mesoscopic phase separation into superconducting and insulating regions [@32li12], which would both contribute spectral intensity in ARPES data. From measurement on an insulating RFS sample (Fig. \[fig:figsi4\](d)), we see negligible spectral weight and no well-defined dispersions within 0.1eV from [$E_F$]{}, as expected for an insulator. Hence, the insulating regions in the superconducting compounds do not contribute spectral weight to the near-[$E_F$]{} energy range, in which the temperature-induced crossover is observed. We have also measured a KFS sample whose resistivity is intermediate between superconducting and insulating (Fig. \[fig:figsi4\](b)), and was previously proposed to be semiconducting containing both metallic and insulating phases [@33chen11]. Interestingly, its [$d_{xz}$/$d_{yz}$]{} bands, which must come from the metallic phase, appear further renormalized by a factor of 1.3 compared with those of the superconducting compounds. In addition, we resolve small but finite spectral weight for a very shallow [$d_{xy}$]{} electron band near the X-point (Fig. \[fig:figsi5\](c)). As expected, the peak position is even closer to [$E_F$]{}, consistent with additional renormalization for the shallow band near X, which is harder to discern with temperature dependent study. These observations are consistent with | 0 | non_member_998 |
the OSMP picture in that that the metallic phase in this KFS compound is likely even closer to the boundary of OSMP at low temperatures from the mass-diverging behavior of the [$d_{xy}$]{} bands. For the same interaction strength, calculation from our model also identifies the low temperature ground state of the superconducting, intermediate, and insulating phases to be located close to an OSMP, just at the boundary of an OSMP, and in a MI phase, respectively (see SI).
While we cannot completely rule out alternative explanations for the observations presented above, the consistency of the totality of the observations-including strongly orbital dependent band renormalization for [$d_{xy}$]{} versus [$d_{xz}$/$d_{yz}$]{} at both $\Gamma$ and X points in the low temperature metallic state, the non-trivial temperature-dependent spectral weight change for only the [$d_{xy}$]{} band, systematic doping dependence of the related effects in the intermediate and insulating compounds-and the theoretical calculations makes this understanding a most likely scenario, suggesting that the superconductivity in this AFS family exists in close proximity to Mott behavior.
We thank V. Brouet, W. Ku, B. Moritz and I. Mazin for helpful discussions. ARPES experiments were performed at the Stanford Synchrotron Radiation Lightsource and the Advanced Light Source, which are | 0 | non_member_998 |
both operated by the Office of Basic Energy Science, U.S. Department of Energy. The work at Stanford is supported by DOE Office of Basic Energy Science, Division of Materials Science and Engineering, under contract DE-AC02-76SF00515. The work at Rice has been supported by NSF Grant DMR-1006985 and the Robert A. Welch Foundation Grant No. C-1411. The work at Houston is supported in part by US Air Force Office of Scientific Research contract FA9550-09-1-0656, and the state of Texas through the Texas Center for Superconductivity at the University of Houston. MY thanks the NSF Graduate Research Fellowship Program for financial support.
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Supplementary Information
=========================
Materials and methods
---------------------
Single crystals of KFS were grown by the following method. Precursor FeSe was synthesized using Se powder (Alfa 99.999$\%$) and Fe powder (99.998$\%$) in a 1:1 molar ratio. The reagents were weighed and placed in a 2mL alumina crucible. The quartz tube was sealed after being flushed with argon and evacuated. The sealed quartz tube was placed in a furnace and heated from room temperature to 1050$^\circ$C in 12 hours. The furnace remained at 1050$^\circ$C for 12 hours, then was shut off and cooled to room temperature. Single crystals of KFS were obtained by a self-flux method with mixtures of K (Alfa, 99.95$\%$), and FeSe in molar ratios of 1:3 respectively. Potassium has a low boiling point so a small amount of additional K needs to be incorporated into the growth during synthesis. The reagents were weighed and placed in a 2mL alumina crucible, which | 0 | non_member_998 |
was then sealed in a 2mL quartz tube after being flushed with argon and evacuated. The 2mL quartz tube was then placed into a larger 5ml quartz tube and sealed after again being flushed with argon and evacuated. The double sealed quartz tube technique is employed because potassium attacks quartz. The sealed quartz tubes were placed in a furnace and heated from room temperature to 1040$^\circ$C over the course of 4 hours. After dwelling for 2 hours, the furnace was cooled to 850$^\circ$C. The quartz tube was then removed from the furnace, rotated 180$^\circ$C, and spun in a centrifuge for a few seconds to separate the self-flux from the single crystals. Crystals with dimensions up to approximately 3mm x 3mm x 50mm could readily be extracted from the crucible. The crystals have a platelike morphology, with the c-axis perpendicular to the plane. Two types of KFS were studied, one superconducting with TC onset of 32K and chemical composition as K$_{0.76}$Fe$_{1.72}$Se$_2$, and the other non-superconducting with composition as K$_{0.76}$Fe$_{1.78}$Se$_2$. Single crystals of RFS were grown as described elsewhere [@s1gooch]. Two types of RFS were studied: one superconducting with T$_C$ onset of 31K and chemical composition Rb$_{0.93}$Fe$_{1.70}$Se$_2$; and the other insulating with | 0 | non_member_998 |
composition Rb$_{0.90}$Fe$_{1.78}$Se$_2$.
As the crystals are sensitive to air, all sample preparations were preformed inside a nitrogen-flushed glove box. ARPES measurements were carried out at beamline 10.0.1 of the Advanced Light Source and beamline 5-4 of Stanford Synchrotron Radiation Lightsource using SCIENTA R4000 electron analyzers. The total energy resolution used was 25 meV or better and the angular resolution was 0.3$^\circ$C. Single crystals were cleaved in situ at low temperatures and measured in an ultra high vacuum chamber with a base pressure better than $3x10^{-11}$ Torr.
Discussion of orbital assignment
--------------------------------
The common band structure for FeSC in the Brillouin zone (BZ) corresponding to the two-Fe unit cell consists of three hole bands near the $\Gamma$ point and two electron bands at the X point [@s2graser]. Under C$_4$ rotational symmetry, the [$d_{xz}$]{} and [$d_{yz}$]{} bands are degenerate at the in-plane high symmetry points of the BZ-$\Gamma$ and X points. At the $\Gamma$ point, this means that the [$d_{xz}$]{} and [$d_{yz}$]{} hole bands must be degenerate. Hence we have assigned the two lower degenerate hole bands in Fig. \[fig:fig1\]e to [$d_{xz}$/$d_{yz}$]{} and the higher one to [$d_{xy}$]{}. At the X point, C$_4$ symmetry dictates the bottom of the [$d_{xz}$/$d_{yz}$]{} electron band | 0 | non_member_998 |
and the top of the [$d_{xz}$/$d_{yz}$]{} hole band be degenerate. Since the shallow electron band is separated from the hole band by more than 70meV, it is unlikely to be the [$d_{xz}$/$d_{yz}$]{} electron band, but is instead the [$d_{xy}$]{} band.
There is in principle the possibility of an alternative understanding. As described in Ref. [@s4Lin; @s4Brouet], the alternating arsenic atoms about the iron plane may induce parity-switching of certain orbitals when folding from the 1-Fe BZ to the 2-Fe BZ, which would swap the orbital characters of the [$d_{xz}$]{} electron band and the [$d_{xy}$]{} electron band at X along $\Gamma$-X direction, making the originally [$d_{xz}$]{} electron band to be more observable under an odd polarization (Fig. \[fig:fig1\]c). Under this understanding, the shallow band is still the [$d_{xz}$]{} band rather than the [$d_{xy}$]{} band. However, for this understanding to hold, the aforementioned degeneracy of the [$d_{xz}$/$d_{yz}$]{} bands must be lifted to account for the 70meV gap between the shallow electron band and the hole band at X.
There are two mechanisms that could lift this degeneracy. One is spin-orbit coupling. However, in this compound, the magnitude of the spin-orbit coupling is not strong enough to cause such a large gap. Moreover, | 0 | non_member_998 |
the effect of the spin-orbit coupling is stronger at the $\Gamma$ than at the X point [@s3mazin], and should cause a larger gap between the [$d_{xz}$/$d_{yz}$]{} hole bands at $\Gamma$, which is not observed. The second mechanism is in-plane symmetry breaking, which causes anisotropic shift of the [$d_{xz}$/$d_{yz}$]{} bands and hence lift the degeneracy of these two orbitals, as has been observed in the orthorhombic state of [BaFe$_2$As$_2$]{} [@s4Yi11]. Up to date, there has not yet been any report of in-plane symmetry breaking in the AFS superconductors. In addition, one should note that the orthorhombicity causes the existence of natural twinning in the crystals. Hence in an unstressed crystal, the ARPES signal is a combination of dispersions from both types of domains, and would see bands from $\Gamma$-X and $\Gamma$-Y directions simultaneously, as was observed in [BaFe$_2$As$_2$]{} [@s4Yi11]. For the microscopically phase separated AFS superconductors, such twinning effect should be even more noticeable than [BaFe$_2$As$_2$]{}, but is not observed in the dispersions of AFS superconductors.
Another factor that should be considered is the comparison between $\Gamma$ and X. The flat [$d_{xy}$]{} hole band observed at $\Gamma$ is renormalized by a factor of $\sim$10 compared to bare LDA while the [$d_{xz}$/$d_{yz}$]{} | 0 | non_member_998 |
hole bands are renormalized by a factor of $\sim$3. When such kind of orbital-dependent renormalization is considered for the bands at X, it would bring the [$d_{xy}$]{} electron band to be shallower than the [$d_{xz}$/$d_{yz}$]{} electron bands, which is more consistent with the assignment of the shallower electron band to [$d_{xy}$]{}. Further support for this assignment comes from the general consideration that a stronger mass renormalization would imply a stronger temperature dependence of the corresponding quasiparticle spectral weight, as has been shown in the main text.
To summarize, while the possibility in principle exists for other unknown mechanisms and alternative explanations, given the totality of the observations and considerations given above, the understanding of the shallow electron band as [$d_{xy}$]{} seems to be the most consistent overall.
Temperature cycle test
----------------------
To test that the disappearance of the [$d_{xy}$]{} orbital spectral weight with raised temperature is not due to sample aging, we have performed a temperature cycle test in which we cleaved the sample at 10K and took measurements as temperature is raised up to 300K and cooled back down again. Spectra images for selected temperatures as well as the EDC at X point in this temperature cycle are shown | 0 | non_member_998 |
in Fig. \[fig:figsi1\], where we see that the spectral weight for the [$d_{xy}$]{} orbital dominated shallow electron band diminishes at higher temperatures, and recovers when cooled back down. This shows that the observed temperature-induced crossover is not due to any surface effects and is a robust bulk phenomenon.
![\[fig:figsi1\]Temperature cycle test on [K$_{x}$Fe$_{2-y}$Se$_2$]{} (a) Spectral images across X point measured at selected temperatures in the temperature cycle from 10K to 300K to 40K. Polarization geometry and photon energy is the same as that in Fig. \[fig:fig2\](a). (b) EDC at X point taken in the temperature cycle, where the peak around -0.05eV is the [$d_{xy}$]{}band bottom, whose spectral weight diminishes up to 300K and recovers when cooled back down.](FigSI1){width="45.00000%"}
Results on [Rb$_{x}$Fe$_{2-y}$Se$_2$]{}
---------------------------------------
{width="90.00000%"}
We have done similar measurements on superconducting RFS as discussed in the main text for superconducting KFS. Fig. \[fig:figsi2\] summarizes the main results. RFS has similar electronic structure as KFS, including Fermi surface, band dispersions and their orbital characters, specifically there is a shallow electron band at X that is [$d_{xy}$]{} and a deep electron band that is [$d_{xz}$/$d_{yz}$]{}. Fig. S2d-f shows the temperature dependence of the electron bands, equivalent to Fig. \[fig:fig2\](a) for the KFS. Here | 0 | non_member_998 |
again we see that the shallow [$d_{xy}$]{} band is present at low temperatures, and its spectral weight diminishes with raised temperature. Similar quantitatively analysis is also done on the RFS. Fig. \[fig:figsi2\](g) shows the EDC at X point taken in the geometry of Fig. \[fig:figsi2\](d)-(f), taken from 10K to 200K. Similar to the case of KFS, the $\sim$-0.05eV hump shaded by blue is the [$d_{xy}$]{} band bottom, whose spectral weight diminishes with raised temperature. The EDCs are again fitted with a Gaussian background and a Lorentzian peak, whose integrated intensity as a function of temperature is plotted in Fig. S2h. The method of integrating intensity in boxed regions is also done on RFS. Here two regions were chosen (Fig. \[fig:figsi2\]d): blue box for a region dominated by [$d_{xy}$]{} band, and magenta box for a region of mixed [$d_{xy}$]{} and [$d_{xz}$]{}. The background for each region is taken as a box of same energy window away from dispersions, marked by dotted boxes in Fig. \[fig:figsi2\](d). The integrated intensity of these two regions as a function of temperature is plotted in Fig. \[fig:figsi2\](i), showing the diminishing trend of [$d_{xy}$]{}region and the slower trend of the mixed region. These behaviors in RFS are | 0 | non_member_998 |
very similar to those in KFS in the main text.
Orbital-selective Mott phase of the five-orbital Hubbard model for [K$_{x}$Fe$_{2-y}$Se$_2$]{}
----------------------------------------------------------------------------------------------
The five-orbital Hubbard model is given by $H=H_{\mathrm{kin}} + H_{\mathrm{int}}$, where $H_{\mathrm{kin}}$ and $H_{\mathrm{int}}$ respectively denote the kinetic and the on-site interaction parts of the Hamiltonian. $H_{\mathrm{int}}$ contains the intra- and inter-orbital Coulomb repulsion, as well as the Hund’s rule coupling and the pair hoppings.[@s8yu] The corresponding coupling strengths are respectively $U$, $U^\prime$, and $J$, which satisfy $U^\prime=U-2J$. [@s5castellani] For simplicity, we consider only the density-density interactions and neglect the spin-flip and pair-hopping terms. The results including these terms are qualitatively the same. The kinetic part $H_{\mathrm{kin}}$ is a tight-binding Hamiltonian, and is conveniently specified in the momentum space. In FeSCs, each unit cell contains two Fe ions. Hence, ideally the tight-binding Hamiltonian must be defined in the BZ corresponding to this two-Fe unit cell. However, the lattice symmetry of the FeSC system allows us to work in an unfolded BZ corresponding to one-Fe unit cell. Following Ref. [@s6wen] and notice that the ions are invariant under the transformation $P_zT_x$ and $P_zT_y$, where $T_{x(y)}$ is the translation along $x(y)$ direction by one Fe-lattice spacing, and $P_z$ refers to the | 0 | non_member_998 |
reflection about the Fe plane, we may define a pseudocrystal momentum $\bf{\tilde{k}}$ in the extended one-Fe BZ. This pseudocrystal momentum and the conventional momentum $\bf{k}$ are related by $\bf{\tilde{k}}=\bf{k}+\bf{Q}$ in [$d_{xz}$]{} and [$d_{yz}$]{} orbitals (where $\bf{Q}=(\pi,\pi)$), but $\bf{\tilde{k}}=\bf{k}$ in other orbitals. In the extended BZ, the tight-binding Hamiltonian reads
$$\label{eqn:eqn2}
H_{\mathrm{kin}}=\sum\limits_{\bf{\tilde k}\alpha\beta\sigma}[\xi_{\alpha\beta}(\bf{\tilde k})+(\Delta_\alpha-\mu)\delta_{\alpha\beta}]d^{\dagger}_{\bf{\tilde k}\alpha\sigma}d_{\bf{\tilde k}\beta\sigma}.$$
Here $\xi_{\alpha\beta}(\bf{\tilde{k}})$ is the hopping matrix in the momentum space associated with orbitals $\alpha$ and $\beta$, $\Delta_{\alpha}$ is the on-site energy reflecting the crystal field splitting, $\mu$ is the chemical potential, and $\delta_{\alpha\beta}$ is the Kronecker’s delta function. The expression of $\xi_{\alpha\beta}(\bf{\tilde{k}})$ is given in Ref. [@s7yu], and it has the same form as appeared in the appendix of Ref. [@s2graser]. We adopt the tight-binding parameters of Ref. [@s7yu], where they are obtained by fitting the LDA band structure of KFS. To better fit the LDA results, we have further tuned several hopping parameters by hand from their values in Ref. [@s7yu]. Using the same notation as in Ref. [@s2graser], the tight-binding parameters used in this paper for KFS are listed in Table \[Table:tab1\]. The chemical potential corresponding to the electron filling $n$=6.15 is $\mu$=-0.365 eV.
The five-orbital Hubbard model is studied using the recently | 0 | non_member_998 |
---
abstract: 'Intensity mapping is a promising technique for surveying the large scale structure of our Universe from $z=0$ to $z \sim 150$, using the brightness temperature field of spectral lines to directly observe previously unexplored portions of out cosmic timeline. Examples of targeted lines include the $21\,\textrm{cm}$ hyperfine transition of neutral hydrogen, rotational lines of carbon monoxide, and fine structure lines of singly ionized carbon. Recent efforts have focused on detections of the power spectrum of spatial fluctuations, but have been hindered by systematics such as foreground contamination. This has motivated the decomposition of data into Fourier modes perpendicular and parallel to the line-of-sight, which has been shown to be a particularly powerful way to diagnose systematics. However, such a method is well-defined only in the limit of a narrow-field, flat-sky approximation. This limits the sensitivity of intensity mapping experiments, as it means that wide surveys must be separately analyzed as a patchwork of smaller fields. In this paper, we develop a framework for analyzing intensity mapping data in a spherical Fourier-Bessel basis, which incorporates curved sky effects without difficulty. We use our framework to generalize a number of techniques in intensity mapping data analysis from the flat sky | 0 | non_member_999 |
to the curved sky. These include visibility-based estimators for the power spectrum, treatments of interloper lines, and the “foreground wedge" signature of spectrally smooth foregrounds.'
author:
- 'Adrian Liu$^{\dagger}$, Yunfan Zhang, Aaron R. Parsons'
bibliography:
- 'biblio.bib'
title: Spherical Harmonic Analyses of Intensity Mapping Power Spectra
---
Introduction {#sec:Intro}
============
[[^1]]{} In recent years, intensity mapping has been hailed as a promising method for conducting cosmological surveys of unprecedented volume. In an intensity mapping survey, the brightness temperature of an optically thin spectral line is mapped over a three-dimensional volume, with radial distance information provided by the observed frequency (and thus redshift) of the line. By observing brightness temperature fluctuations on cosmologically relevant scales (without resolving individual sources responsible for the emission or absorption), intensity mapping provides a relatively cheap way to survey our Universe. In addition, with an appropriate choice of spectral line and a suitably designed instrument, the volume accessible to an intensity mapping survey is enormous. This allows measurements to be made over a large number of independent cosmological modes, providing highly precise constraints on both astrophysical and cosmological models. For example, intensity mapping experiments tracing the $21\,\textrm{cm}$ hyperfine transition of hydrogen can easily access $\sim 10^9$ | 0 | non_member_999 |
independent modes, which is much greater than the $\sim 10^6$ accessible to the Cosmic Microwave Background, in principle unlocking a far greater portion of the available information in our observable Universe [@loeb_and_zaldarriaga2004; @mao_et_al2008; @tegmark_and_zaldarriaga2009; @ma_and_scott2016; @scott_et_al2016].
A large number of intensity mapping experiments are in operation, and more have been proposed. Post-reionization neutral hydrogen $21\,\textrm{cm}$ intensity mapping is being conducted by the Canadian Hydrogen Intensity Mapping Experiment [@bandura_et_al2014], the Green Bank Telescope [@masui_et_al2013], Tianlai telescope [@chen_et_al2012], Baryon Acoustic Oscillations from Integrated Neutral Gas Observations project [@battye_et_al2013], Hydrogen Intensity and Real-time Analysis eXperiment [@newburgh_et_al2016], and BAORadio [@ansari_et_al2012]. These experiments use neutral hydrogen as a tracer of the large scale density field, with a primary scientific goal of constraining dark energy via measurements of the baryon acoustic oscillation feature from $0 < z < 4$ [@wyithe_et_al2008; @chang_et_al2008; @pober_et_al2013a]. At $z \sim 2$ to $3.5$, data from the Sloan Digital Sky Survey have been used for Ly $\alpha$ intensity mapping [@croft_et_al2016]. Other experiments such as the CO Power Spectrum Survey [@keating_et_al2015; @keating_et_al2016] and the CO Mapping Array Pathfinder [@li_et_al2016] use CO as a tracer of molecular gas in the epoch of galaxy formation at roughly $z \sim 2$ to $3$. Using \[CII\] instead | 0 | non_member_999 |
is the Spectroscopic Terahertz Airborne Receiver for Far-InfraRed Exploration (operating at $0.5 < z < 1.5$; @uzgil_et_al2014), and the Tomographic Ionized carbon Mapping Experiment (operating at $5 < z < 9$; @crites_et_al2014). The highest redshift bins of the latter encroach upon the Epoch of Reionization (EoR), when the first galaxies systematically reionized the hydrogen content of the intergalactic medium. Extending into the EoR, intensity mapping efforts are mainly focused around the $21\,\textrm{cm}$ line. The Donald C. Backer Precision Array for Probing the Epoch of Reionzation array (PAPER; @parsons_et_al2010), the Low Frequency Array [@van_haarlem_et_al2013], the Murchison Widefield Array [@bowman_et_al2012; @tingay_et_al2013], the Giant Metrewave Radio Telescope [@paciga_et_al2013], the Long Wavelength Array (M. W. Eastwood et al., in prep.), 21 Centimeter Array [@huang_et_al2016; @zheng_et_al2016], and the Hydrogen Epoch of Reionization Array [@deboer_et_al2016] are radio interferometers that aim to use the $21\,\textrm{cm}$ line to probe the density, ionization state, and temperature of hydrogen in the range $6 < z < 13$ and beyond. The future Square Kilometre Array [@mellema_et_al2015] will provide yet more collecting area for $21\,\textrm{cm}$ intensity mapping to complement the aforementioned experiments. With such a large suite of instruments covering an expansive range in redshift, tremendous opportunities exist for understanding the formation | 0 | non_member_999 |
of the first stars and galaxies via direct measurements of the IGM during all the relevant epochs [@hogan_and_rees1979; @scott_and_rees1990; @madau_et_al1997; @tozzi_et_al2000], as well as fundamental cosmological parameters [@mcquinn_et_al2006; @mao_et_al2008; @visbal_et_al2009; @clesse_et_al2012; @liu_et_al2016] and exotic phenomena such as dark matter annihilations [@valdes_et_al2013; @evoli_et_al2014].
Despite its promise, intensity mapping is challenging, and to date the only positive detections have been tentative detections of Ly $\alpha$ at $z \sim 2$ to $3.5$ [@croft_et_al2016] and CO from $z\sim 2.3$ to $3.3$ [@keating_et_al2016], as well as detections of HI at $z\sim 0.8$ via cross-correlation with optical galaxies [@chang_et_al2010; @masui_et_al2013]. To realize the full potential of intensity mapping, it is necessary to overcome a large number of systematics. A prime example would be radiation from foreground astrophysical sources, which are particularly troublesome for HI intensity mapping. Especially at high redshifts, foregrounds add contaminant emission to the measurement that are orders of magnitude brighter than the cosmological signal [@dimatteo_et_al2002; @santos_et_al2005; @wang_et_al2006; @deOliveiraCosta_et_al2008; @sims_et_al2016]. Low frequency measurements (for instance, those targeting the $21\,\textrm{cm}$ EoR signal), are mainly contaminated by broadband foregrounds such as Galactic synchrotron emission or extragalactic point sources (whether they are bright and resolved or are part of a dim and unresolved continuum). These foregrounds are | 0 | non_member_999 |
typically less dominant at the higher frequencies and are thus easier (though still challening) to handle for CO or \[CII\] intensity mapping experiments. However, such experiments must also contend with the problem of interloper lines, where two spectral lines of different rest wavelengths may redshift into the same observation band, leading to confusion as to which spectral line has been observed.
In addition to astrophysical foregrounds, instrumental systematics must be well-controlled for a successful measurement of the cosmological signal. Among others, these systematics include beam-forming errors [@neben_et_al2016b], radio frequency interference [@offringa_et_al2013; @offringa_et_al2015; @huang_et_al2016], polarization leakage [@geil_et_al2011; @moore_et_al2013; @shaw_et_al2014b; @sutinjo_et_al2015; @asad_et_al2015; @moore_et_al2015; @kohn_et_al2016], calibration errors [@newburgh_et_al2014; @trott_and_wayth2016; @barry_et_al2016; @patil_et_al2016], and instrumental reflections [@neben_et_al2016a; @ewall-wice_et_al2016a; @thyagarajan_et_al2016].
In this paper, we focus specifically on measurements of the power spectrum $P(k)$ of spatial fluctuations in brightness temperature, where roughly speaking, the temperature field is Fourier transformed and then squared. In diagnosing the aforementioned systematics as they pertain to spatial fluctuation experiments, it is helpful to decompose the fluctuations into modes that separate purely angular fluctuations from purely radial fluctuations from those that are a mixture of both. In recent years, for example, simulations and measured upper limits of the $21\,\textrm{cm}$ power spectrum have often | 0 | non_member_999 |
been expressed as cylindrically binned power spectra. To form cylindrically binned power spectra, one begins with unbinned power spectra $P(\mathbf{k})$, where $\mathbf{k}$ is the three-dimensional wavevector of spatial Fourier modes. If the field of view is narrow, there exists a particular direction that can be identified as the line-of-sight (or radial) direction. One of the three components of $\mathbf{k}$ can then be chosen to lie along this direction and labeled $k_\parallel$ as a reminder that it is *parallel* to the line-of-sight. The remaining two components—which we arbitrarily designate $k_x$ and $k_y$ in this paper—describe transverse (i.e., angular fluctuations), and have a magnitude $k_\perp \equiv \sqrt{k_x^2 + k_y^2}$. Binning $P(\mathbf{k})$ along contours of constant $k_\perp$ gives $P(k_\perp, k_\parallel)$, the cylindrically binned power spectrum.
Expressing the power spectrum as a function of $k_\perp$ and $k_\parallel$ is a powerful diagnostic exercise because intensity mapping surveys probe line-of-sight fluctuations in a fundamentally different way than the way they probe angular fluctuations. Systematics are therefore usually anisotropic and have distinct signatures on the $k_\perp$-$k_\parallel$ plane [@morales_and_hewitt2004]. For example, cable reflections and bandpass calibration errors tend to appear as features parallel to the $k_\parallel$ axis [@dillon_et_al2015; @ewall-wice_et_al2016b; @jacobs_et_al2016]. Thus, the cylindrically binned power spectrum is a | 0 | non_member_999 |
useful intermediate quantity to compute before one performs a final binning along constant $k \equiv \sqrt{k_\perp^2 + k_\parallel^2}$ to give an isotropic power spectrum $P(k)$.
The diagnostic capability of $P(k_\perp, k_\parallel)$ is particularly apparent when considering foregrounds. Assuming that they are spectrally smooth, foregrounds preferentially contaminate low $k_\parallel$ modes, since $k_\parallel$ is the Fourier conjugate to line-of-sight distance, which is probed by the frequency spectrum in intensity mapping experiments. The situation is more complicated for the (large) subset of intensity mapping measurements that are performed on interferometers. Interferometers are inherently chromatic in nature, causing intrinsically smooth spectrum foregrounds to acquire spectral structure, which results in leakage to higher $k_\parallel$ modes. Even this leakage, however, has been shown in recent years to have a predictable “wedge" signature on the $k_\perp$-$k_\parallel$ plane, limiting the contaminated region to a triangular-shaped region at high $k_\perp$ and low $k_\parallel$ [@Datta2010; @Vedantham2012; @Morales2012; @Parsons_et_al2012b; @Trott2012; @Thyagarajan2013; @pober_et_al2013b; @dillon_et_al2014; @Hazelton2013; @Thyagarajan_et_al2015a; @Thyagarajan_et_al2015b; @liu_et_al2014a; @liu_et_al2014b; @chapman_et_al2016; @pober_et_al2016; @seo_and_hirata2016; @jensen_et_al2016; @kohn_et_al2016]. In fact, the foreground wedge is considered sufficiently robust that some instruments have been designed around it [@pober_et_al2014; @deboer_et_al2016; @dillon_et_al2016; @neben_et_al2016a; @ewall-wice_et_al2016a; @thyagarajan_et_al2016], implicitly pursuing a strategy of foreground avoidance where the power spectrum can be measured in | 0 | non_member_999 |
relatively uncontaminated Fourier modes outside the wedge. This mitigates the need for extremely detailed models of the foregrounds, providing a conservative path towards early detections of the power spectrum.
Despite its utility, the $k_\perp$-$k_\parallel$ power spectrum is limited in that it is ultimately a quantity that is only well-defined in the flat-sky, narrow field-of-view limit, where a single line-of-sight direction can be unambiguously defined. For surveys with wide fields of view, different portions of the survey have different lines of sight that point in different directions with respect to a cosmological reference frame. Note that this is a separate problem from that of wide-field imaging: even if one’s imaging software does not make any flat-sky approximations (so that the resulting images of emission within the survey volume are undistorted by any wide-field effects), the act of forming a power spectrum on a $k_\perp$-$k_\parallel$ invokes a narrow-field approximation. If one insists on forming $P(k_\perp, k_\parallel)$ as a diagnostic, the simplest way to do so is to split up the survey into multiple small patches that are individually small enough to warrant a narrow-field assumption. A separate power spectrum can then be formed from each patch by squaring the Fourier mode amplitudes, | 0 | non_member_999 |
and the resulting collection of power spectra can then be averaged together. While correct, such a “square-then-average" procedure results in lower signal-to-noise than a hypothetical “average-then-square" procedure whereby a single power spectrum is formed out of the entire survey. The latter allows the spatial modes of a survey to be averaged together coherently, which allows instrumental noise to be averaged down very quickly. Roughly speaking, if $N$ patches of sky are averaged in a coherent fashion to constrain a particular spatial mode, the noise on the measured mode amplitude averages down as $1/\sqrt{N}$. Squaring this amplitude to form a power spectrum then results in a quicker $1/N$ scaling of noise. In contrast, a “square-then-average" method combines $N$ independent pieces of information after squaring, and thus the power spectrum noise scales more slowly[^2] as $1/\sqrt{N}$. The result is a less sensitive statistic, whether for the diagnosis of systematics or for a cosmological measurement. To be fair, one could recover the lost sensitivity by also computing all cross-correlations between a series of small overlapping patches. However, the necessary geometric adjustments for such high precision mosaicking will likely be computationally wasteful, and it quickly becomes preferable to adopt an approach that incorporates the | 0 | non_member_999 |
curved sky from the beginning.
In this paper, we rectify the shortcomings of the $k_\perp$-$k_\parallel$ plane by introducing an alternative that is well-defined in the wide-field limit. Rather than expanding sky emission in a basis of rectilinear Fourier modes, we propose a spherical Fourier-Bessel basis. In this basis, the sky brightness temperature $T(\mathbf{r})$ of a survey (where $\mathbf{r}$ is the comoving position) is expressed in terms of $\overline{T}_{\ell m} (k)$, defined as[^3] $$\label{eq:TellmEverything}
\overline{T}_{\ell m} (k) \equiv \sqrt{\frac{2}{\pi}} \int \! d\Omega dr\, r^2 j_\ell (kr) Y_{\ell m}^* ({\hat{\mathbf{r}}}) T(\mathbf{r}),$$ where $k$ is the *total* wavenumber, $\ell$ and $m$ are the spherical harmonic indices, $Y_{\ell m}$ denotes the corresponding spherical harmonic, $r \equiv | \mathbf{r}|$ is the radial distance, $\mathbf{\hat{r}} \equiv \mathbf{r} / r$ is the angular direction unit vector[^4], and $j_\ell$ is the $\ell$th order spherical Bessel function of the first kind. The quantity $P(k_\perp, k_\parallel)$ is replaced by the analogous quantity $S_\ell (k)$, the spherical harmonic power spectrum, which roughly takes the form $$\label{eq:Sellkrough}
S_\ell (k) \propto \frac{1}{2 \ell + 1} \sum_{m = -\ell}^\ell |\overline{T}_{\ell m} (k)|^2,$$ where the sum over $m$ is analogous to the binning of $k_x$ and $k_y$ into $k_\perp$, and a more rigorous definition (with | 0 | non_member_999 |
constants of proportionality) will be defined in Section \[sec:SphericalPspecFormalism\]. Instead of the $k_\perp$-$k_\parallel$ plane, power spectrum measurements are now expressed on an $\ell$-$k$ plane. Now, we will show in Section \[sec:SphericalPspecFormalism\] that in the limit of a translationally invariant cosmological field, $S_\ell (k)$ reduces to $P(k)$. Therefore, just as $P(k_\perp, k_\parallel)$ can be averaged along contours of constant $k$ to form $P(k)$ once systematic effects are under control, the same can be done for $S_\ell (k)$ to form $P(k)$ by averaging over all values of $\ell$ for a particular $k$.
Spherical Fourier-Bessel methods have been explored in the past within the galaxy survey literature [@binney_quinn1991; @lahav_et_al1994; @fisher_et_al1994; @fisher_et_al1995; @heavens_taylor1995; @zaroubi_et_al1995; @castro_et_al2005; @leistedt_et_al2012; @rassat_refregier2012; @shapiro_et_al2012; @pratten_munshi2013; @yoo_desjacques2013]. In this paper, we build upon these methods and present a framework for implementing them in an analysis of intensity mapping data. We emphasize the way in which intensity mapping surveys have unique geometric properties, and how these properties affect spherical Fourier-Bessel methods. For instance, we pay special attention to the fact that particularly for the highest redshift observations, intensity mapping experiments probe survey volumes that are radially compressed but angularly expansive (as illustrated in Figure \[fig:surveyGeom\]). In harmonic space, this expectation is reversed, | 0 | non_member_999 |
and there is excellent spatial resolution along the line-of-sight (since high spectral resolution is relatively easy to achieve), but poor angular resolution. In addition to addressing these geometric peculiarities, we also show how interferometric data can be analyzed with spherical Fourier-Bessel methods. Importantly, we find that the foregrounds again appear as a wedge in interferometric measurements of $S_\ell (k)$, which suggests that the $\ell$-$k$ plane is at least as powerful a diagnostic tool as the $k_\perp$-$k_\parallel$ plane, particularly given the signal-to-noise advantages discussed above.[^5]
The rest of this paper is organized as follows. In Section \[sec:Notation\] we establish notational conventions for this paper. Section \[sec:SphericalPspecFormalism\] introduces spherical Fourier-Bessel methods for power spectrum estimation, with the complication of finite surveys (in both the angular and spectral directions) the subject of Section \[sec:FiniteVolume\]. In Section \[sec:Foregrounds\] we compute the signature of smooth spectrum foregrounds on the $\ell$-$k$ plane. Interloper lines are explored in Section \[sec:Interlopers\]. A framework for interferometric power spectrum estimation using spherical Fourier-Bessel methods (which includes a derivation of the foreground wedge) is presented in Section \[sec:Interferometry\]. To build intuition, we develop a parallel series of flat-sky, narrow field-of-view expressions in a series of Appendices. Our conclusions are summarized in | 0 | non_member_999 |
Section \[sec:Conclusions\]. Because of the large number of mathematical quantities defined in this paper, we provide a glossary of important symbols for the reader’s convenience in Table \[tab:Definitions\].
Quantity Meaning/Definition Context
------------------------------------------------- ------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------
$\mathbf{r}$ Comoving position Section \[sec:Intro\]
$\mathbf{\hat{r}}$ Angular direction unit vector Section \[sec:Intro\]
$\mathbf{r}_\perp$ Comoving transverse distance Eq.
$r(\nu)$ or $r_\nu$ Comoving radial distance Eq.
$s(r)$ Incorrect radial distance assumed for true radial distance $r$ due to interloper lines Eq.
$\nu(r)$ or $\nu_r$ Observed frequency of radio emission Section \[sec:Notation\]
$\alpha$ Linearized conversion factor between frequency and radial comoving distance Eq.
$ \boldsymbol \theta$ Sky image angle Eq.
$\mathbf{k}$ Wavevector of rectilinear spatial Fourier modes Section \[sec:Intro\]
$k_\perp$ Magnitude of wavevector components perpendicular to line of sight Section \[sec:Intro\]
$k_\parallel$ Magnitude of wavevector components parallel to line of sight Section \[sec:Intro\]
$k$ Total wavenumber/wavevector magnitude of rectilinear spatial Fourier modes Section \[sec:Intro\]
$\phi(\mathbf{r})$ Survey volume selection function Section \[sec:FiniteVolume\]
$\phi(r)$ Radial survey profile or survey volume selection function assuming full-sky covarage Section \[sec:FiniteVolume\]
$\Phi(r)$ Radial survey profile centered on radial midpoint of survey Section \[sec:MostlyRadialNoInterferometry\]
$T(\mathbf{r})$ or $T({\hat{\mathbf{r}}}, \nu)$ Sky temperature in configuration space Eq.
$ \overline{T}_{\ell m} (k)$ Sky temperature in spherical Fourier-Bessel space Eq.
$ | 0 | non_member_999 |
\overline{T}_{\ell m}^\textrm{meas} (k)$ Estimated sky temperature in spherical Fourier-Bessel space for finite-volume surveys Eq.
$\widetilde{T} (\mathbf{k})$ Sky temperature in rectilinear Fourier space Eq.
$\kappa (\nu)$ Frequency spectrum of foreground contaminants Eq.
$q_\ell (k)$ Frequency spectrum of foreground contaminants in radial spherical Bessel basis Eq.
$a_{\ell m} (\nu)$ Sky temperature in frequency/spherical harmonic space Eq.
$Y_{\ell m} $ Spherical harmonic function Section \[sec:SphericalPspecFormalism\]
$\psi_{\ell m} (k; {\hat{\mathbf{r}}}, \nu)$ Spherical Fourier-Bessel basis function in configuration space Eq.
$j_\ell (kr) $ $\ell$th order spherical Bessel function of the first kind Section \[sec:SphericalPspecFormalism\]
$C_\ell$ Angular power spectrum Section \[sec:RotationalInvarianceOnly\]
$P(\mathbf{k})$ Brightness temperature power spectrum Section \[sec:Intro\]
$P(k_\perp, k_\parallel)$ Brightness temperature power spectrum, assuming cylindrical symmetry Section \[sec:Intro\]
$P(k)$ Brightness temperature power spectrum, assuming isotropy Eq.
$S_\ell (k) $ Spherical harmonic power spectrum Eq.
$ \mathbf{b}$ Interferometer baseline vector Section \[sec:Interferometry\]
$\tau$ Interferometric time delay Eq.
$V(\mathbf{b}, \nu)$ Interferometric visibility Eq.
$\widetilde{V}(\mathbf{b}, \tau)$ Interferometric visibility in delay space Eq.
$A({\hat{\mathbf{r}}}, \nu)$ Primary beam of receiving elements of interferometer Eq.
$B({\hat{\mathbf{r}}}, \nu)$ Rescaled primary beam Eq.
$\overline{B^2}(\theta) $ Squared primary beam profile, averaged azimuthally about a baseline vector Eq.
$\gamma (\nu)$ Delay transform tapering function Eq.
$f_{\ell m} (\mathbf{b}, \nu)$ Response of baseline $\mathbf{b}$ at frequency | 0 | non_member_999 |
$\nu$ to unit perturbation of spherical harmonic mode $Y_{\ell m}$ Eq.
$g_{\ell m} (\mathbf{b}, \tau)$ Response of baseline $\mathbf{b}$ at delay $\tau$ to unit perturbation of spherical harmonic mode $Y_{\ell m}$ Eq.
$W_\ell (k; \mathbf{b}, \tau)$ Spherical harmonic power spectrum window function for a single baseline delay-based Eq.
power spectrum estimate
$\Theta(\nu)$ Re-centered frequency profile of the foregrounds as seen in the data, with finite bandwidth Section \[sec:CurvedSkyWedge\]
and tapering effects
$D(\mathbf{r})$ Survey volume selection function including primary beam, bandwidth, and data analysis Appendix \[sec:RectilinearInterferometerPspecNorm\]
tapering effects
Notational preliminaries {#sec:Notation}
========================
Suppose an intensity mapping survey has surveyed the brightness temperature field $T({\hat{\mathbf{r}}}, \nu)$ of a particular spectral line as a function of angle (specified here in terms of unit vector ${\hat{\mathbf{r}}}$) and frequency $\nu$. Such a quantity represents a three-dimensional survey of our Universe, since different frequencies of a spectral line map to different redshifts, and thus different radial distances from the observer. Explicitly, the comoving radial distance $r$ is given by $$\label{eq:ComovingDistDef}
r (\nu) = \frac{c}{H_0} \int_0^{z(\nu)} \frac{dz^\prime}{E(z^\prime)},$$ where $c$ is the speed of light, $H_0$ is the present day Hubble parameter, with $$1 + z \equiv \frac{\nu_\textrm{rest}}{\nu}\quad \textrm{and} \quad E(z) \equiv \sqrt{\Omega_\Lambda + \Omega_m (1+z)^3},$$ where $\nu_\textrm{rest}$ | 0 | non_member_999 |
is the rest frequency of the spectral line, $z$ is the redshift, $\Omega_\Lambda$ is the normalized dark energy density, and $\Omega_m$ is the normalized matter density. There is thus a one-to-one mapping between frequency and comoving radial distance, and as shorthand throughout this paper, we will adopt the notation $r_\nu \equiv r(\nu)$. Similarly, we will often use the symbol $\nu_r$ to denote frequency, with the subscript reminding us that the observed frequency is a function of the radial distance. Given a radial distance, transverse distances may also be computed given ${\hat{\mathbf{r}}}$ (or angle on the sky) using simple geometry.
If one’s survey occurs over a narrow radial range, the distance-frequency relation is often replaced by a linearized approximation where $$\label{eq:LinearDistanceApprox}
r - r_\textrm{ref} \approx - \alpha (\nu - \nu_\textrm{ref} ),$$ with $r_\textrm{ref}$ and $\nu_\textrm{ref}$ being a reference comoving radial distance and a reference frequency, respectively, with values constrained by Eq. , and $$\label{eq:AlphaConversion}
\alpha \equiv \frac{1}{\nu_\textrm{rest}} \frac{c}{H_0} \frac{(1+z_\textrm{ref})^2}{E(z_\textrm{ref})},$$ where $1 + z_\textrm{ref} = \nu_\textrm{rest} / \nu_\textrm{ref}$. In this paper, the symbols $\nu_r$ and $r_\nu$ will always refer to the exact nonlinear relations, and any invocations of the linearized approximations will be written out explicitly using Eq. . When using the | 0 | non_member_999 |
linearized approximation for the radial distance, we will often (though not always) also make the small angle approximation for converting between the angle $\boldsymbol \theta$ and the transverse comoving position $\mathbf{r}_\perp$ from some reference direction, where $$\label{eq:AngularConversion}
\mathbf{r}_\perp = r \boldsymbol \theta.$$
Given the well-defined prescriptions for converting between instrument-centric parameters (such as frequency $\nu$ and direction on the sky ${\hat{\mathbf{r}}}$) and cosmology-centric ones (such as $r$ and $\mathbf{r}_\perp$), we will often use both sets of parameters to describe the same quantities. For example, we will sometimes write the brightness temperature field as $T({\hat{\mathbf{r}}}, \nu)$, whereas other times we will write the same quantity as $T(\mathbf{r})$, where $\mathbf{r}$ is the comoving position. We will additionally find it useful to exhibit similar flexibility in our notation even for quantities that are not cosmological in nature, such as the primary beam of a radio telescope.
Spherical Fourier-Bessel Formalism {#sec:SphericalPspecFormalism}
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In this section we introduce the mathematical framework for describing the sky in terms of the spherical harmonic power spectrum. Our treatment here is essentially identical to that of @yoo_desjacques2013, albeit with different Fourier-Bessel transform conventions. No claims of originality are made in this section (except perhaps for Section \[sec:RotationalInvarianceOnly\]), and the | 0 | non_member_999 |
formalism is included only for completeness. We will, however, occasionally provide previews of how various parts of the framework are particularly helpful for intensity mapping and interferometry. In the spherical Fourier-Bessel basis, angular fluctuations are expressed by expanding the temperature field $T({\hat{\mathbf{r}}}, \nu)$ in spherical harmonics, such that $$\label{eq:SHTdef}
a_{\ell m} (\nu) \equiv \int d\Omega Y_{\ell m}^* ({\hat{\mathbf{r}}}) T({\hat{\mathbf{r}}}, \nu).$$ To capture modes along the line-of-sight, we perform a Fourier-Bessel transform along the frequency direction, yielding $$\label{eq:FBdef}
\overline{T}_{\ell m} (k) \equiv \sqrt{\frac{2}{\pi}} \int_0^\infty \! dr\, r^2 j_\ell (kr) a_{\ell m} (\nu_r),$$ with these last two expressions of course combining to give Eq. . The temperature field of the sky may therefore be thought of as being a linear combination of a set of basis functions $\psi_{\ell m } (k; {\hat{\mathbf{r}}}, \nu)$ that are indexed by $(k,\ell,m)$, so that $$\label{eq:InverseTrans}
T({\hat{\mathbf{r}}}, \nu) = \sum_{\ell m} \int dk\, \psi_{\ell m } (k; {\hat{\mathbf{r}}}, \nu) \overline{T}_{\ell m} (k),$$ where $$\label{eq:BasisFcts}
\psi_{\ell m } (k; {\hat{\mathbf{r}}}, \nu) \equiv k^2 \sqrt{\frac{2}{\pi}} j_\ell (kr_\nu) Y_{\ell m} ({\hat{\mathbf{r}}}).$$ Eqs. and are the forward transforms into the harmonic basis, while Eqs. and define the inverse transforms back into configuration space. This can be verified by substituting Eq. into | 0 | non_member_999 |
Eq. , and using orthonormality of spherical harmonics, as well as the analogous identity for spherical Bessel functions, given by $$\label{eq:BesselOrthog}
\int \! dr \,r^2 j_\ell (k r) j_\ell (k^\prime r) = \frac{\pi}{2 k k^\prime} \delta^D (k - k^\prime),$$ where $\delta^D$ is the Dirac delta function. Note that our convention for the radial transform differs from that of most works in the literature. From Eqs. and , one sees that our convention is symmetric in the following sense. Whether one is switching from $r$-space to $k$-space or vice versa, the prescription is always to multiply by $\sqrt{2 / \pi} j_\ell (kr)$ and the square of the coordinate (i.e., $r^2$ or $k^2$) of the original space before integrating over it. This makes our forward and backward transforms aesthetically and conveniently symmetric. Most previous works (e.g., @leistedt_et_al2012 [@rassat_refregier2012; @yoo_desjacques2013]), in contrast, opt for an asymmetric convention: an extra factor of $k$ is included in the forward transform from $r$ to $k$, and correspondingly there is one fewer factor of $k$ in the backwards transform.
Translationally invariant fields in the spherical Fourier-Bessel formalism {#eq:TransInvarFields}
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In some sense, the decision to expand fluctuation modes along the line of sight in terms of spherical | 0 | non_member_999 |
Bessel functions rather than some other set of basis functions is arbitrary. However, we will now show that spherical Bessel functions are a particularly good choice for describing temperature fields that are statistically translation invariant. Translation-invariant fields admit a representation in terms of their power spectrum $P(k)$, which we define implicitly via the equation[^6] $$\label{eq:RectilinearPspecDef}
\langle \widetilde{T} (\mathbf{k}) \widetilde{T}^* (\mathbf{k^\prime}) \rangle = (2 \pi)^3 \delta^D (\mathbf{k} - \mathbf{k}^\prime) P(k),$$ where the angled brackets $\langle \cdots \rangle$ signify an ensemble average over random realizations of the cosmological temperature field $T(\mathbf{r})$, whose Fourier transform $\widetilde{T} (\mathbf{k})$ we define by the convention $$\label{eq:forwardNormal}
\widetilde{T} (\mathbf{k}) = \int \! d^3 r \,e^{-i \mathbf{k} \cdot \mathbf{r}} T(\mathbf{r})$$ with the inverse transform given by $$\label{eq:inverseNormal}
T(\mathbf{r}) = \int \! \frac{d^3 k}{(2 \pi)^3} e^{i \mathbf{k} \cdot \mathbf{r}} \widetilde{T} (\mathbf{k}).$$ Unless otherwise stated, this Fourier convention for the temperature field will be the one used for all Fourier transforms in this paper. Ideally, our spherical Fourier-Bessel description should be directly relatable to $P(k)$, for it would be pointless if an estimation of the power spectrum required first returning to position space. We will now show that this requirement is met by our $\overline{T}_{\ell m} (k)$ modes.
To relate | 0 | non_member_999 |
$\overline{T}_{\ell m} (k)$ to $P(k)$, we combine Eqs. , , and to obtain $$\label{eq:YetAnotherTellm}
\overline{T}_{\ell m} (k) = \sqrt{\frac{2}{\pi}} \int \! \frac{d^3 k^\prime}{(2 \pi)^3} \widetilde{T} (\mathbf{k}^\prime) \int \! d^3 r\, j_\ell (kr) Y_{\ell m}^*({\hat{\mathbf{r}}}) e^{i \mathbf{k}^\prime \cdot \mathbf{r}}.$$ To simplify this, we expand $e^{i \mathbf{k}^\prime \cdot \mathbf{r}}$ in spherical harmonics using the identity $$\label{eq:PlaneWaveSphericalHarmonicExpansion}
e^{i \mathbf{k} \cdot \mathbf{r}} = 4\pi \sum_{\ell m} i^\ell j_\ell (kr) Y_{\ell m}^* ({\hat{\mathbf{k}}}) Y_{\ell m} ({\hat{\mathbf{r}}}),$$ which leads to $$\label{eq:TlmTkConversion}
\overline{T}_{\ell m} (k) = \frac{i^\ell}{(2\pi)^{\frac{3}{2}}} \int \frac{d^3 k^\prime}{k k^\prime} Y_{\ell m}^* ({\hat{\mathbf{k}}}^\prime) \delta^D (k - k^\prime) \widetilde{T} (\mathbf{k}^\prime).$$ This provides a link between the temperature field as expressed in our $(k,\ell, m)$ basis, and the same field in the rectilinear Fourier basis. Taking a cue from Eq. , where the power spectrum is closely related to the two-point correlation between different rectilinear Fourier modes, we may form a two-point correlator between different modes in our spherical Fourier-Bessel basis, giving $$\begin{aligned}
\label{eq:CurvedPspecDef}
\langle \overline{T}_{\ell m} (k) \overline{T}_{\ell^\prime m^\prime}^* (k^\prime) \rangle && = \frac{i^\ell (-i)^{\ell^\prime}}{(2\pi)^3} \!\! \int \frac{d^3 k_1}{k k_1} \frac{d^3 k_2}{k^\prime k_2} \nonumber \\
&& \qquad \times Y_{\ell m}^* ({\hat{\mathbf{k}}}_1) Y_{\ell^\prime m^\prime} ({\hat{\mathbf{k}}}_2) \langle \widetilde{T} (\mathbf{k}_1) \widetilde{T} (\mathbf{k}_2)^* \rangle\nonumber \\
&& \qquad \times \delta^D (k - | 0 | non_member_999 |
k_1) \delta^D (k^\prime - k_2) \nonumber \\
&& = \frac{\delta^D(k - k^\prime) }{k^2} \delta_{\ell \ell^\prime} \delta_{m m^\prime} P(k),\end{aligned}$$ where the last equality follows from Eq. and some algebraic simplifications. From this, we see that forming the power spectrum from $\overline{T}_{\ell m} (k)$ modes is remarkably similar to forming it from the rectilinear Fourier modes. Comparing Eqs. and , we see that if (roughly speaking) one can form $P(k)$ by squaring $\widetilde{T} (\mathbf{k})$ and normalizing appropriately, one can equally well form $P(k)$ by squaring $\overline{T}_{\ell m} (k)$ and normalizing (albeit with a different—and $k$ dependent—normalization that we will derive more explicitly in Section \[sec:FiniteVolume\]).
To understand why the squaring of $\overline{T}_{\ell m} (k)$ produces such a similar result to squaring $\widetilde{T} (k)$ (with both giving a result proportional to the power spectrum), notice that Eq. can be simplified to give $$\overline{T}_{\ell m} (k) = \frac{i^\ell}{(2\pi)^{\frac{3}{2}}} \int d\Omega_k Y^*_{\ell m} ({\hat{\mathbf{k}}}) \widetilde{T} (\mathbf{k})\bigg{|}_{|\mathbf{k}| = k},$$ where $ \widetilde{T} (\mathbf{k})$ is restricted to the shell where $|\mathbf{k}| = k$. In this form, one sees that an alternate way to understand our spherical harmonic Bessel modes is to view them as a spherical harmonic decomposition of $ \widetilde{T} (\mathbf{k})$ in Fourier space. In other | 0 | non_member_999 |
words, going from the rectilinear Fourier modes to spherical harmonic Bessel modes is simply a change of basis—to spherical harmonics—in angular Fourier coordinates. Now, suppose one were to form an estimate of $P(k)$ in by squaring $ \widetilde{T} (\mathbf{k})$ and then averaging over a shell of constant $|\mathbf{k}| = k$. Parseval’s theorem ensures that such a squaring and averaging operation is basis-independent. Thus, it does not matter whether the Fourier amplitudes on the shell of constant $|\mathbf{k}| = k$ are expressed in a spherical harmonic basis. Squaring and averaging $\overline{T}_{\ell m} (k)$ must therefore also yield the power spectrum, up to some $k$-dependent conversion factors to account for the radius of shells in Fourier space. Note that Eq. also cements the interpretation (suggested by our notation) that the quantity $k$ of our Fourier-Bessel basis is the total magnitude of the wavevector $\mathbf{k}$, rather than some wavenumber that only pertains to radial fluctuations.
Rotationally invariant fields in the spherical Fourier-Bessel formalism {#sec:RotationalInvarianceOnly}
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While the cosmological temperature field is expected to possess translationally invariant statistics, contaminants in an intensity mapping survey (such as foreground emission) will in general not possess such symmetry. This difference in symmetry will result in different signatures | 0 | non_member_999 |
on the $\ell$-$k$ plane that can in principle be used to separate contaminants from the cosmological signal.
To elucidate the contrast in these signatures, suppose we discard the assumption (from previous derivations) of translationally invariant statistics. In general, the two-point correlator will cease to exhibit the diagonal form given by Eq. . As a concrete example of this, consider a random temperature field that is statistically isotropic but not homogeneous. In the radial direction, suppose this field has some fixed (non-random and angular position-independent) radial dependence. Such a field would be an appropriate description for a (hypothetical) population of unresolved point sources. Under these assumptions, Eq. reduces to $$\overline{T}_{\ell m} (k) = a_{\ell m} q_\ell (k),$$ where $$\label{eq:qellk}
q_\ell (k) \equiv \sqrt{\frac{2}{\pi}} \int_0^\infty dr r^2 j_\ell (kr) \kappa (\nu_r),$$ with $\kappa (\nu_r)$ specifying the spectral (and therefore radial) dependence of our hypothetical sky as it appears in our data. The two-point correlator then becomes $$\langle \overline{T}_{\ell m} (k) \overline{T}_{\ell^\prime m^\prime}^* (k^\prime) \rangle = C_\ell q_\ell(k) q_\ell (k^\prime) \delta_{\ell \ell^\prime} \delta_{m m^\prime},$$ where statistical rotation invariance of the field allows us to invoke relation $\langle a_{\ell m} a_{\ell^\prime m^\prime}^* \rangle \equiv C_\ell \delta_{\ell \ell^\prime} \delta_{m m^\prime}$, with $C_\ell$ signifying the angular | 0 | non_member_999 |
Subsets and Splits