# Structure and Phase Transitions of Metastable Hexagonal Uranium Thin Films

Rebecca Nicholls<EMAIL_ADDRESS>H.H. Wills Physics Laboratory
University of Bristol BS8 1TL, UK.  Chris Bell  H.H. Wills Physics Laboratory
University of Bristol BS8 1TL, UK.  Johann Bouchet  CEA, DES, IRESNE, DEC,
SESC, LM2C, F-13108 Saint-Paul-Lez-Durance, France  Ross Springell  H.H. Wills
Physics Laboratory
University of Bristol BS8 1TL, UK.  Gerard H. Lander  H.H. Wills Physics
Laboratory
University of Bristol BS8 1TL, UK.

###### Abstract

We report a simple technique for the synthesis of uniaxially textured,
metastable hexagonal close-packed-like uranium thin films with thicknesses
between $175-2800$ Å. The initial structure and texture of the layers have
been studied via X-ray diffraction and reflectivity and the time-dependent
transitions of the samples into various orientations of orthorhombic
$\alpha$-U have been mapped by similar techniques. The final crystallographic
orientations of each system and the timescales on which the transitions occur
are found to depend on the lattice parameters of the original layer. The
absence of the $\alpha$-U (001) orientation in the transition products
suggests that the transitions in these layers are mediated by mechanisms other
than the [110] transverse acoustic phonon mode previously suggested for the
cubic $\gamma$-U(110) to hcp-U(00.1) to $\alpha$-U(001) displacive phase
transition. Alternative transition pathways are discussed.

††preprint: APS/123-QED

##  I Introduction

Under ambient conditions, most elements crystallise in high-symmetry
structures where an isotropic local environment is provided by the
neighbouring atoms e.g. face-centred cubic (fcc), hexagonal close-packed (hcp)
and body-centred cubic (bcc) arrangements. In uranium metal, however, the
narrow 5$f$ electron bandwidth (1–3 eV) near the Fermi level drives the
formation of several uncommon low-symmetry structures in the bulk material.
The ground state, orthorhombic $\alpha$-U (Cmcm) structure is not adopted by
any other element under ambient pressure and hosts a series of charge density
wave states below 43 K, as well as superconductivity below 1 K [1]. Upon
heating, uranium transforms into the complex tetragonal $\beta$-U structure
(P$4_{2}$/$mnm$) at 935 K, crystallises in the body-centred cubic $\gamma$-U
structure (Im3m) at 1045 K and melts at 1405 K. Several computational studies
have calculated the energetics and relative stabilities of other theoretical
allotropes [2, 3, 4] but the most symmetric structures (e.g. hcp and fcc) have
never been observed experimentally in bulk, even under pressures of up to 100
GPa [5, 6, 7, 8].

In contrast, recent work using epitaxial engineering of thin films has shown
that strain effects can be exploited to stabilise energetically unfavourable
orientations and phases of metallic uranium. For example, the CDW transition
temperature in epitaxial $\alpha$-U has been increased to 120 K by straining
the $a$-axis [9], thin films of pure $\beta$-U have remained stable under
ambient conditions for over a year despite its instability in bulk at room
temperature [10] and single crystals of Mo-doped $\gamma$-U, which are yet to
be synthesised in bulk, can be epitaxially stabilised on Nb(110) [11]. There
is also evidence for a fourth ‘hidden’ allotrope of uranium metal that exists
only as a thin film.

The hexagonal phase of U metal was first alluded to when 70 Å of uranium
deposited onto single crystal W(110) substrates produced sharp, six-fold
symmetric LEED patterns [12, 13, 14]. Comparisons of the experimentally and
theoretically derived band structures suggested that the pattern is indeed
representative of a hexagonal close-packed (P63/$mmc$) structure, rather than
a twinned fcc state [14, 15, 16, 12]. However, no ex-situ structural
characterisation of these films was performed and the sub-surface structure
was not fully determined. Other studies later showed that the lattice
parameters of the phase can depend heavily on the both the layer thickness and
the underlying crystallographic character of the surface. The lattice
parameter ($c/a$) ratios in the few published reports of hcp-U vary from near-
ideal close-packed ratios ($1.63$) for $<100$ Å uranium layers grown onto
W(110) substrates, up to $c/a=1.90$ for a 500 Å layer deposited onto a
Gd(00.1) buffer layer [17]. Recent observations of hexagonal surface
reconstructions in epitaxial (001) and (110)-oriented $\alpha$-U thin films
complicates the situation further [18].

The hcp structure observed in the Gd/U system is of interest as the large
$c/a$ ratio makes the allotrope one of the most anisotropic elemental hcp
structures. Several calculations have predicted that an anisotropic bulk hcp
structure would order ferromagnetically and exhibit large spin-Hall angles
[17, 19, 20] whereas other studies refute claims of magnetic ordering in bulk
and suggest that strained thin films will not be magnetic [4]. These studies
did not take into consideration that the theoretical phonon dispersion for the
hcp phase, included in Section II of this paper, predicts that this high
symmetry allotrope is dynamically unstable at the $c/a$ value which minimises
its total energy.

Experiments which could elucidate the stabilisation mechanics of hcp-U, such
as inelastic X-ray scattering, require thicker single crystal layers than
those currently achieved in literature. However, the unpredictability of
uranium crystallisation and stability on both Gd(00.1) and W(110) is
problematic, as highlighted by the growth of unusual multi-domain $\alpha$-U
in Ref. [21] and by the preference for $\alpha$-U in thick (300-3000 Å) layers
deposited onto W(110) at growth temperatures of 300-800∘C [9, 18]. In light of
these limitations, it is important that other possible templates for hcp-U
growth are investigated to give insight into the wide range of $c/a$ ratios
and crystallographic character observed in the literature.

During the search for alternative buffer layers which promote reliable single
crystal growth, we developed a simple technique for the growth of metastable
hcp films of uranium metal onto textured Cu$\\{111\\}$ and Ir$\\{111\\}$
across a range of thicknesses. Intriguingly, we are able to measure the real-
time relaxation of the hcp-like structure into the orthorhombic $\alpha$-U
phase. We find that the hcp unit cell parameters, the rate of transformations,
and the crystallographic texture of the final $\alpha$-U films are all
dependent on the thickness of the uranium layer ($t_{U}$). These data shed
light on the relationship between the hexagonal and orthorhombic phases of
elemental uranium, suggesting that the soft phonon mode proposed by Axe is not
the driving mechanism for the transition in highly anisotropic hcp structures
[22].

The structure of this paper is as follows: Section II shows the theoretical
phonon dispersion and lattice parameters of relaxed hcp-U. Section III
presents details of the experimental sample growth and basic characterisation.
Section IV discusses more detailed X-ray diffraction (XRD) and X-ray
reflectivity (XRR) measurements which have been used to study the evolution of
the lattice parameters and microstructure with thickness. These techniques are
also used in Section V to track the transition of the samples from their
original state into $\alpha$-U. Finally, in Section VI we discuss alternative
mechanisms for the hcp-U to $\alpha$-U transitions based on the experimental
observations.

##  II Instability of hcp-U

One of the main motivations for the present effort to stabilise hcp-U was the
calculation of the phonon dispersion spectra in Fig. 1. Ab initio simulations
were performed using the ABINIT package [23, 24] in the framework of the
Projector Augmented Wave (PAW) method [25, 26] and by means of the generalized
gradient approximation (GGA) according to the parameterization of Perdew,
Burke and Ernzerhof (PBE) for the exchange-correlation energy and potential
[27]. Results are obtained using a plane-wave cutoff energy equal to 600 eV.
We considered 14 electrons in the valence and a cutoff radius of 2.85 a.u. for
U and a $10\times 10\times 5$ k-points mesh for the hcp structure. Dynamical
matrices were calculated for q-points on a similar grid. The relaxation of the
hcp structure gives a volume of 21.2 Å3 and a $c/a$ ratio of 1.84, in
agreement with calculations performed with the full-potential linear muffin-
tin-orbital (FPLMTO) method [17]. As expected, the energy of this high
symmetry structure exceeds the total energy of $\alpha$-U. A number of
imaginary modes in the optical branches (plotted with negative energy in Fig.
1) also indicate that hcp-U is dynamically unstable.

Figure 1: Calculated phonon dispersion for hcp-U ($c/a=1.84$ and $V=21.2$ Å3).
Imaginary phonon energies (plotted as negative) are found near the $\Gamma$
and M points. The volume of bulk $\alpha$-U at room temperature is 20.75 Å3
[28].

##  III Growth and Characterisation

Thin film samples were grown using a dedicated actinide D.C. magnetron
sputtering system (University of Bristol), which is a four-gun assembly housed
in an ultra-high vacuum chamber with a base pressure better than $1\times$10-9
mbar. All layers were deposited without intentional substrate heating using
$7.4-7.6\times$10-3 mbar argon gas as the sputtering medium.

Buffer layers of copper or iridium with nominal thicknesses of
$t_{\mathrm{Cu}}=175$ Å and $t_{\mathrm{Ir}}=145$ Å were deposited directly
onto epi-polished Corning glass substrates (MTI Corporation). Since Cu and Ir
are fcc metals, both tend to adopt the close-packed $\\{111\\}$ orientation
when deposited onto an amorphous surface at room temperature, providing a
rotationally disordered but locally ‘hexagonal’ template for the uranium
overlayer. If the hcp-U film perfectly and coherently strains to the buffer,
the Cu template generates $a_{hcp}=2.951$ Å and the Ir template generates
$a_{hcp}=3.135$ Å. These parameters ‘book-end’ the range of reliable lattice
parameters (2.96(2)-3.2(1) Å) found in literature, and the 500 Å thick hcp-U
single crystal from Ref. [17] gives the predicted epitaxial match and strains
illustrated in Fig. 2.

Figure 2: Schematics of the expected epitaxial matches between hcp-U(00.1) and
Cu(111)/Ir(111) using lattice parameters of $a=2.96$ Å and $c=5.625$ Å from
Ref. [17]. Values of predicted percentage strains for the U layer are shown
(compressive for Cu, tensile for Ir). The real space alignment
U[11.0]$\parallel$Cu[11-2] produces a match governed by the distance
$d_{\mathrm{U}}=2.563$ Å with $d_{\mathrm{Cu}}=2.565$ Å or
$d_{\mathrm{Ir}}=2.715$ Å. Figure 3: Evolution of the hcp-U (00.2) Bragg
reflections for various uranium thicknesses ($t_{\mathrm{U}}$) in Cu/U (top)
and Ir/U (bottom) bilayers, labelled with the $c$ parameter of the thickest
film (dashed vertical line). Insets show the fit to the 1400 Å reflection on a
linear scale, solid and dashed lines correspond to the Cu-K$\alpha_{1}$ and
Cu-K$\alpha_{2}$ wavelengths, respectively. Thicknesses labelled (*) were
directly derived from X-ray reflectivity profiles. Shoulder regions are
attributed to secondary X-rays from the Ni filter in the diffractometer.

Uranium metal was deposited onto the $\\{111\\}$-textured surfaces at room
temperature at a rate of 1.4 Å/s to produce layers with nominal thicknesses
($t_{\mathrm{U}}$) of 175, 350, 700, 1400 and 2800 Å, as calibrated by X-ray
reflectivity in relatively thin samples. Each bilayer was capped with a layer
of Nb (100 Å in the Cu systems, 200 Å in the Ir systems) to prevent oxidation
before being rapidly transferred to a Philips X’Pert diffractometer with a Cu-
Kα source. The out-of-plane texture of each bilayer was measured through
specular (2$\theta$-$\omega)$ and rocking curve ($\omega$) scans. Scans taken
within the first hour after growth showed only a series of strong asymmetric
reflections characteristic of the $(00.l)$ family ($l=2n$, where $n=1,2,3..$)
in an hcp-structure (see Supplemental Material). No other secondary phases or
orientations were observed Similar scans were repeated in the weeks following
growth to evaluate the stability of each system.

Figure 4: Evolution of the hcp-U lattice parameters and atomic volume and in
Cu/U and Ir/U bilayers as a function of $t_{\mathrm{U}}$. Left-hand inset is
an exaggerated illustration of the simultaneous structural relaxation and
columnar merging. Right-hand inset shows an hcp structure with the direction
of change in $a$ and $c$ labelled. The dashed line indicates the atomic volume
of room temperature, bulk $\alpha$-U (20.75 Å3) from Ref. [28].

Separate studies were conducted using single crystal Ir(111) and Cu(111)
buffer layers grown at relatively high temperatures onto single crystal
Al2O3(00.1) and MgO(111), respectively. Subsequent uranium deposition at
temperature intervals between RT$-500^{\circ}$C showed no evidence of single
crystal hcp-U formation on either buffer layer. At (nominal) room temperature,
an hcp-U layer formed with {00.1}-texture but did not adopt any in-plane
alignment on either buffer. Higher growth temperatures destroyed the hexagonal
state on both the Cu and Ir buffers and, in the case of Ir/U, caused
intermixing of the U and Ir layers.

The in-plane lattice parameters in the thickest Corning/Ir/U bilayers were
probed by applying offsets to the diffractometer to align an expected hcp off-
specular reflection (i.e. a Bragg peak with some in-plane component) with the
point detector. In a preferred orientation (00.1) system with perfectly random
in-plane ordering, the intensity of each off-specular reflection is invariant
to rotation about the [00.1] axis. Coupled $2\theta$-$\omega$ scans which
intersect this isotropic ring of intensity give an angular (2$\theta$)
position from which the in-plane lattice parameters of the layer can be
refined. Here we assume an ideal in-plane hexagonal structure, i.e. the two
sides of the primitive unit cell are the same length ($a=b$) and the angle
between them $\gamma=120^{\circ}$. Though a full structural refinement is
beyond the scope of this paper, we were able to find several off-specular
peaks in the thicker Ir/U samples that are strongly suggestive of a Mg-type
hcp structure. For the $c$-axis oriented 2800 Å Ir/U bilayer, least-squares
refinement of the $a$ parameter has been performed using the (10.4), (11.4),
(10.5) and (20.4) peak positions (see Supplemental Material for peak
positions). For the 1400 Å Ir/U system, the (10.4) and (11.4) positions have
been used. The existence of the hcp (11.4) reflection and its position in
reciprocal space distinguishes the structure from the textured $\beta$-U(001)
films in Ref. [10]. The off-specular reflections are relatively weak and were
not measured for the Cu/U series.

X-ray reflectivity was also used to determine the sample thickness, electronic
density and interfacial roughness of layers in samples with total thickness
$<1000$ Å. The reflectivity profiles (see Supplemental Material for profiles
and fitting parameters) were fitted using GenX, where error bars on each
fitting parameter are calculated from a 5% change in the optimal figure of
merit [29]. The $a$ lattice parameters of the thinnest uranium layers were
indirectly calculated by combining the XRR-determined electronic densities
with the $c$-parameters from the specular reflections, again assuming an ideal
hcp structure. The $a$ parameter of the 700 Å Ir-U sample was not determined
due to the low intensity off-specular reflections and an overall thickness too
close to the XRR limit.

##  IV Hexagonal Thin Films

As expected from the $\\{111\\}$-texture of the buffer, the as-deposited
uranium forms a textured hcp-like layer with the $c$-axis oriented out of
plane. Fig. 3 shows the U(00.2) reflection from each Cu/U and Ir/U bilayer,
fitted with asymmetric lineshapes. A plot showing the evolution of the full
width at half maximum (FWHM) of the specular peak ($\Delta 2\theta$) and the
associated rocking curve ($\Delta\omega$) with $t_{U}$ is included as
Supplemental Material. As expected from the Scherrer equation, the width of
the specular diffraction peak is inversely proportional to the layer
thickness, $t_{U}$. The sharp decrease and plateauing of the rocking curve
width with $t_{U}$ is suggestive of a microstructural evolution with
increasing thickness, as is observed in other metallic thin films systems
grown with low adatom mobility. In the context of the zone-dependent models of
References [30] and [31], these bilayers were grown in ‘Zone 1’, where
$T_{\mathrm{growth}}/T_{\mathrm{melting}}<0.3$ (in Kelvin). Films grown in
this regime are characterised by a low density ‘void’ network surrounding an
array of narrow, parallel columns which coalesce on a non-linear, thickness-
dependent scale. This increasing distance between grain boundaries would
generate the observed reduction in $\Delta\omega$. Evidence for coherent
column formation in the thin limit is discussed in Section V. We also note
that fibrous, columnar structures have been found in textured films of
$\beta$-U deposited onto Si(111) at room temperature [10].

Fig. 4 shows the evolution of the uranium lattice parameters with increasing
layer thickness, where the $1/t_{\mathrm{U}}$ functional forms in both $a$ and
$c$ are often seen in thin films undergoing plastic relaxation above a
critical thickness [32, 33]. The 1% overall reduction in the atomic volume is
driven by a simultaneous 2% contraction in $a$ and 2.2% expansion in $c$,
causing a contraction from $V=21.2$ Å3 to 20.8 Å3. This brings the hcp-U
atomic volume closer to bulk $\alpha$-U at 20.75 Å3 [34] as the $c/a$ ratio
increases from 1.85 to 1.93.

Fig. 5 then shows the influence of the $c/a$ ratio on the theoretical phonon
spectrum of hcp-U, where a stiffening of the phonon mode at the M point occurs
with increasing $c/a$. A larger $c/a$ ratio should improve the dynamical
stability of the structure, and this crossover to positive (real) phonon
energies between $c/a=1.84$ and $1.95$ correlates well with the experimental
plateau of $c/a$ at 1.93. The $\Gamma$ point instability is insensitive to
changes in $c/a$ (see Supplemental Material for calculated phonon dispersion
curves) but an additional expansion (+25%) of the unit cell volume can
eliminate all of the negative phonon energies. However, we note that atomic
volumes are fixed in the range $21-22$ Å3 for the bulk allotropes
($\alpha,\beta,\gamma$) and a large volume increase was not found
experimentally for the hcp-like layers.

Figure 5: Calculated phonon dispersion for hcp-U for $c/a$=1.84 (in blue) and
1.95 (in red) and a volume of $V=21.2$ Å3.

The combination of (i) the power law ($B\approx-1$) behaviour of the lattice
parameters across the entire $t_{\mathrm{U}}$ range and (ii) the strong right-
handed asymmetry on every specular (00.2) peak implies that there is a
continuous $c$-axis expansion normal to the sample surface. If the critical
thickness for relaxation is vanishingly small it is likely that, in the
thickest samples, a continuum of lattice parameters exists with unit cells
ranging from near-isotropic (e.g. $c/a=1.63$) at the interface to highly
anisotropic ($c/a=1.93$) at the surface.

Extrapolating the Ir/U power laws of Fig. 4 back to 70 Å (to represent the
thinnest films in published literature) gives $a=3.08$ Å and $c=5.35$ Å
($c/a=1.74$), which is consistent with a match of $a_{\textrm{Ir/U}}=3.135$ Å
at the interface (Section III). The significant difference between
$a_{\textrm{Cu/U}}=3.03(1)$ Å and the expected value of $a=2.95$ Å (from Fig.
2) in the thinnest Cu films may be driven by interfacial disorder (the Cu
roughness and uranium rocking curve FWHM are double those of Ir), an
alternative epitaxial match (such as a 30∘ relative rotation between the
uranium and Cu) or other microstructural differences in the buffer (different
crystallite shapes). We note that evaporation of uranium onto single crystal
Cu(111) in the sub-monolayer limit has shown that low energy uranium atoms can
embed themselves in the Cu surface by ejecting several substrate atoms and
this may cause significant intermixing at the interface [35]. The surface
stability of Ir(111) upon uranium deposition has not yet been studied in a
similar way.

Table 1 combines the results of this work with the lattice parameters of hcp-U
found in literature, highlighting the general agreement between all
theoretical calculations and the significant differences in experimentally
derived parameters. It is interesting to note that the pseudo-bulk parameters
of $c=5.64\pm 0.01$ Å (for U on the Ir buffer) and $c=5.65\pm 0.02$ Å (for U
on the Cu buffer) extracted from the asymptotes of the power laws are
comparable to the $c$-axis of $\beta$-U (5.656 Å) [36]. Although it is not
possible to determine from the electron diffraction profiles in References
[15] and [14] whether the hcp-U layers on W(110) are surface reconstructions
or true single crystals, a unit cell with $a=3.15$ Å or $c=5.4$ Å for
$t_{U}=70-80$ Å does not seem unreasonable in the context of this paper.

Surface |  $t_{\mathrm{U}}$ (Å) |  $a$ (Å) |  $c$ (Å) | $c/a$ | Ref.
---|---|---|---|---|---
W(110) | 70 | 3.15(1) | 5.1 | 1.62 | [15]
W(110) | 80 | 3.5(5) | 5.4(6) | $1.5(3)$ | [12]
W(110) | 80 | 3.2(1) | N/A | N/A | [14]
Gd(00.1) | 500 | 2.96(2) | 5.625(5) | 1.90(1) | [17]
Theory |  | 2.97 | 5.35 | 1.80 | [12]
Theory |  | 3.01 | 5.48 | 1.82 | [4]
Theory |  | 2.983 | 5.478 | 1.836 | [20]
Theory |  | 2.98 | 5.48 | 1.84 | [*]
Cu{111} | 201 | 3.03(1) | 5.47(1) | 1.804(7) |
Ir{111} | 175 | 2.98(2) | 5.507(3) | 1.85(2) |
Cu{111} | 2800 | N/A | 5.635(1) | N/A |
Ir{111} | 2800 | 2.919(1) | 5.626(2) | 1.928(1) |
Table 1: Summary of experimental and theoretical (relaxed) lattice parameters
of hcp-U from published literature and from this work (bottom three entries).
Error bars are given in brackets when available and refer to least significant
digit. [*] is the result from Section II of this paper.

##  V Transitions into $\alpha$-U

Figure 6: Specular scans of a 1400 Å hexagonal close-packed uranium film grown
onto a polycrystalline Cu{111} surface over the first 7 days following growth.
Arrows indicate the direction of growth/decay of the Bragg peaks. The dashed
line represents the position of bulk (002)α. These data have not been
corrected for absorption of the X-ray beam.

The samples from Section IV transitioned into $\alpha$-U on relatively slow,
thickness dependent timescales (from several hours up to several weeks). This
gradual conversion was probed via X-ray diffraction. In the following
discussion we use subscripts $(hkl)_{\alpha}$ and $(hk.l)_{hcp}$ to
distinguish the different crystallographic planes and directions for the two
phases. Fig. 6 shows the time-dependent XRD data for the $t_{\mathrm{U}}=1400$
Å sample grown on Cu, where the $(00.l)_{hcp}$ peak intensities decrease
monotonically with time and the $\alpha$-U peaks concomitantly increase in
intensity over a period of 7 days. The functional forms of the changes in
integrated intensity of the $(00.2)_{hcp}$, $(020)_{\alpha}$ and
$(110)_{\alpha}$ specular peaks are well-fitted by exponential functions with
similar time constants. If the transitions were propagating layer-by-layer
(e.g. if the transition nucleated at the buffer interface and travelled up
toward the surface) the hcp lineshape would be altered and/or shifted during
the transition. As neither the $(00.2)_{hcp}$ peak center nor the asymmetry
parameters change during the transition, it is likely that transitions are
occurring within individual, narrow, column-like grains.

Similar diffraction data were obtained for all other uranium thicknesses,
revealing that the final $\alpha$-U reflections present in the specular
direction change as a function of $t_{\mathrm{U}}$ and that no diffraction
peaks arise in the specular direction other than those related to $\alpha$-U.
No changes in the $\alpha$-U peak intensities were observed once the hcp-U
intensity had been fully depleted, suggesting that grains of $\alpha$-U are in
their final state and do not reorient post-transition. To graphically
represent the complex combinations and proportions of the final $\alpha$-U
orientations, the texture coefficient, $P_{hkl}$ has been used as an
approximate measure of the relative volume of grains belonging to each
orientation. $P_{hkl}$ is defined here as

$P_{hkl}=I_{hkl}/\Sigma_{hkl}I_{hkl}$ (1)

where $I_{hkl}$ is the integrated intensity of the peak in a coupled
$2\theta$-$\omega$ scan when aligned to the maximum of the respective rocking
curve, and the summation is over all diffraction peaks in the range
$2\theta=25-50^{\circ}$. This range encompasses the first allowed reflections
for the $\alpha$-U {001}α, {110}α, {021}α, {001}α and {111}α families and
excludes the {100}α orientation ($2\theta_{200}=65.34^{\circ}$) which was not
found in any sample.

Figure 7: Grain orientations in the uranium layers found in each bilayer
system post-transition, represented as an experimental texture coefficient to
be compared to a theoretical powder pattern. Each system has transitioned from
an initial $(00.1)_{hcp}$ oriented structure into various orientations of
orthorhombic $\alpha$-U shown in the legend.

The texture coefficients of the $\alpha$-U layers in both the Ir and Cu
systems are displayed in Fig. 7 as a function of $t_{\mathrm{U}}$. The
$(111)_{\alpha}$ reflection is consistently suppressed compared to powder and
the $(002)_{\alpha}$ reflection is absent in all but the thickest Ir/U
bilayers. The only ground state available to the thinnest films in both series
is the $(021)_{\alpha}$ orientation, while the mid-range layers can access the
$(110)_{\alpha}$ and $(010)_{\alpha}$ orientations. Notably, the 700 Å Cu/U
film becomes almost pure (010)α. The thickest films form more textured
$\alpha$-U with no single dominant orientation. It is important to note that
the texture coefficient is a semi-quantitative representation, as possible
extinction and absorption effects have not been taken into account. For Cu-Kα
radiation incident at 15∘ degrees, the $1/e$ attenuation length is $\sim 4000$
Å and the absorption effects should be small for the tested film thicknesses.
We also note that an orientation dependent mosaic texture is present and, if
the rocking curve intensity is combined multiplicatively with the specular
intensity, the texture coefficients of the (021)α and (110)α orientations in
the mid-range samples are altered (maximally by $\pm 0.2$). These caveats do
not impact the overall results of Fig. 7.

Match Direction | Comparable Spacing | Distance (Å)
---|---|---
(00.1)hcp | $d_{00.1}$ | 5.47(1)∗-5.635(1)
(111)α | 2$d_{111}$ | 4.558
(001)α | $d_{001}$ | 4.955
(021)α | 2$d_{021}$ | 5.049
(110)α | 2$d_{110}$ | 5.132
(010)α | $d_{010}$ | 5.869
Table 2: Comparison of the $c$-axis length in the hcp-U bilayers to similar
interplanar spacings in various orientations of $\alpha$-U. (*) denotes the
parameters from the 201 Å Cu/U sample, but thinner layers may have smaller $c$
axes.  Figure 8:  Specular scans of the Ir/U system with an XRR-derived
thickness of $t_{\mathrm{U}}=175$ Å in the month after growth. An
$(021)_{\alpha}$ reflection ($2d_{021}=5.037$ Å) with Laue fringes emerges at
the expense of the $(00.2)_{hcp}$ peak ($c=5.507$ Å). The cut-off intensity on
the right hand side of the figure is a broad (110) reflection from the Nb cap.

Further insight into the microstructure of these samples can be gained from
the time-resolved diffraction scans of thinner films. Fig. 8 shows several
specular scans of the $t_{\mathrm{U}}=175$ Å Ir/U bilayer taken over the month
following growth. The emergence of Laue fringes indicates that the original
$(00.1)_{hcp}$ layer has become a coherent (021)α layer with a sharp Ir-U
interface. These fringes were presumably suppressed in the initial hcp films
by the continuously changing lattice parameters, as discussed in Section IV.
As the periodicity of the fringes is related to the number of scattering
planes perpendicular to the surface normal, the thickness of the crystalline
ordered volume can be extracted from a fit to the oscillations. For the Ir/U
system with $t_{\mathrm{U}}=175$ Å, we find $t_{\mathrm{Laue}}=155\pm 2$ Å.
For the Cu/U system with $t_{\mathrm{U}}=201$ Å we find
$t_{\mathrm{Laue}}=189\pm 6$ Å. While both of these values are slightly
smaller than the XRR derived thicknesses, the close agreement supports the
previous picture of columns extending through the full thickness of the films.
Laue fringes in the $t_{U}=175-700$ Å high angle scans suggest that coherent
columns are maintained up to layer thicknesses of at least 700 Å for both the
Cu and Ir buffered systems.

We finally return to the observation that the hcp-U intensity in the thickest
films rapidly diminishes, while the intensity in the thinnest films persists
for several weeks. This suggests that there are kinetic barriers to these
transitions that are dependent on some microscopic parameter(s) that changes
with thickness, e.g. the $c/a$ ratio and the unit cell volume. The competition
between the $(010)_{\alpha}$, $(110)_{\alpha}$ and $(021)_{\alpha}$ ground
states and absence of the $(002)_{\alpha}$ reflection can be naively explained
by Table II, which shows that smaller $c_{hcp}$-axes (i.e. thinner hcp layers)
are more closely aligned with the $(021)_{\alpha}$ and $(110)_{\alpha}$
interplanar spacings and larger $c_{hcp}$-axes (i.e. thicker layers) lie
closer to the $b$-axis of $\alpha$-U. The in-plane match directions are more
complex and will be discussed in Section VI, along with potential lattice
distortions which drive the transitions into the three primary $\alpha$-U
orientations of Fig. 7.

##  VI Transition Pathways

The first attempt to geometrically relate the $\alpha$-U and hcp-U structures
can be found in Ref. [22], where an hcp structure arises as a by-product of a
displacive transition from the body-centred cubic $\gamma$-U phase to the
$\alpha$-U phase (ignoring the $\beta$-U structure). The Axe mechanism,
illustrated in Fig. 9, maps a single hcp-U unit cell onto a single $\alpha$-U
unit cell via the freeze-in of a transverse acoustic phonon. The displacement
of alternate $\gamma$-U (110) layers along the [1-10]γ direction generates an
intermediate, isotropic hexagonal close-packed structure ($c/a=1.64$) with its
$c$-axis parallel to the c-axis of $\alpha$-U.

The absence of the $(001)_{\alpha}$ orientation in the transition products
(Section V) and the large $c/a$ ratios (Section IV, $c/a=1.80$-$1.93$) suggest
that the Axe mode is not the mechanism governing the transitions in these
metastable Ir/U and Cu/U systems. For example, to convert the $t_{U}=2800$ Å
$(00.1)_{hcp}$ unit cell into $(001)_{\alpha}$, a $c$-axis contraction of 12%
would be required instead of the 0.9% expansion shown in Fig. 9. There are no
other displacive mechanisms which can convert a single, anisotropic
($c/a=1.93$) unit cell of hcp-U into $\alpha$-U with strains of under 10%. Any
instabilities pointing toward the Axe distortion have not yet been observed in
either bulk $\gamma$-U or $\alpha$-U. It is clear from Section V that several
other pathways linking the hcp structure to the orthorhombic ground state do
exist, and that there are well-defined relationships between the initial hcp
lattice parameters and the final orientations of $\alpha$-U.

Figure 9: Illustration of the Axe mode from Ref. [22], showing transformations
induced by [110] transverse acoustic displacements in a $bcc$-U lattice. Red
and black arrows represent in-plane displacements of hexagonal ‘sheets’ in
opposite directions. In the upper images, the light blue layer of atoms exist
in the plane below the dark blue atoms. These alternate planes (i.e. the (110)
planes in the $bcc$ structure) are shifted in opposite directions along the
[1-10] axis. The in-plane aspect ratio is stretched from $\sqrt{2}$ to
$\sqrt{3}$, giving rise to an intermediate hcp (00.1) structure (P63/$mmc$).
Stretching the ratio further and continuing the alternate layer shifts gives
the $\alpha$-U (001) structure. The estimated hcp-U lattice parameters are
taken as the midpoint between the two endpoint structures.

Fig. 10 shows an alternative, two-step phase transformation from (00.1)hcp
into (010)α. The pathway was determined using COMSUBS, a sub-program from the
ISOTROPY suite which finds pathways between parent and daughter phases via a
shared symmetry subgroup [37, 38]. Searches were initially limited to
combinations of strain modes with $\pm$10% of the parent lattice parameters,
displacive modes with a maximum atomic displacement of 1 Å and supercells
comprising up to six primitive cells of the parent structure. Fig. 10 shows
the only hcp-to-$\alpha$ pathway available within these limits for a parent
unit cell with $a=2.92$ Å and $c=5.63$ Å.

Figure 10: Schematic of the transformation from hcp-U (00.1) ($a=2.92$ Å and
$c=5.63$ Å) to $\alpha$-U (010) via the Pnmn structure. Red and black arrows
represent in-plane displacements of the ‘sheets’ (in opposite directions) and
orange arrows represent the out-of-plane displacements of alternate atoms
within these sheets. In the upper left and centre images, the lighter blue
layer of atoms exist in the plane below the darker blue atoms.

In the proposed (00.1)hcp to (010)α transition, a supercell comprised of two
primitive hcp cells is mapped onto an intermediate hcp-like orthorhombic
(Pnmn) structure via the motion of alternate ‘sheets’ in opposite directions
(red and black arrows) and the shearing of alternate rows of atoms within
these sheets in directions parallel to $c_{hcp}$ (orange arrows). Further
shearing, bonds breaking and reforming gives rise to the corrugated layers
characteristic of $\alpha$-U. Strain modes act on the supercell throughout,
with the overall transition resulting in a 4.2 % expansion of the original
$c_{hcp}$ axis, a 2.3 % contraction of the $a_{hcp}$ axis and a 3.4 %
contraction along the [11.0]hcp axis. These strains are significantly lower
than the 12 % distortion required by the Axe mode. If an entire sample volume
passed through this transition at a slow and linear rate, the initial specular
peak near $2\theta=32^{\circ}$ would shift gradually toward
$2\theta=30.44^{\circ}$ for bulk (020)α, while dropping rapidly in intensity.
No other peaks would appear in the specular direction during the transition,
consistent with experiment. If the Axe model proceeded in the same steady
fashion, the spectra would be dominated instead by a shift of the specular
peak toward $2\theta=36.23^{\circ}$ for bulk (001)α, coupled with a negligible
change in peak intensity.

It is interesting to note that the atomic motions indicated in Fig. 10 also
correspond to the displacements associated with the eigenvectors of the
imaginary modes in Fig. 1. Regardless of $q$-vector, the phonon modes are
related to optical displacements of the two atoms in the hcp cell in opposite
directions in the $x$ and $y$ directions, relative to the basal plane. It is
possible that temporary stability of an hcp-like structure is attained via
small distortions along these axes, whilst also paving the way for the gradual
transitions back into $\alpha$-U.

In order to explore alternative pathways, the COMSUBS boundary conditions were
relaxed to allow atomic displacements up to 1.4 Å and strains of $\pm$ 15% for
various starting structures (see Supplemental Material for a summary of all
pathways that do not require triclinic supercells). Each pathway has an
associated strain tensor, $T$ of the form [A% B% C%] which describes the
expansion or contraction of the supercell along the principal strain
directions. The magnitude of this strain tensor, $\mid
T\mid=\sqrt{A^{2}+B^{2}+C^{2}}$ has been plotted in Fig. 11 as a function of
thickness for each of the (001)α, (110)α, (021)α and (010)α transitions. The
maximum atomic shuffle in each transition is attached to the respective line.
As the transitions into (021)α (requiring a 4 unit cell supercell) and (110)α
(requiring a 2 unit cell supercell) are too complex to show in the main text,
please see Supplemental Material for 2D illustrations and animated figures.

Figure 11: Thickness dependence of the magnitude of the percentage strain
tensor, $T$ for various transition pathways. Attached to each line is the
maximum atomic displacement within the transition in Å. Data points correspond
to values of $a$ and $c$ extracted from the power law fit to the
experimentally derived Ir/U bilayer lattice parameters.

On the basis of the reduced atomic shuffle and strain, the (021)α transition
should be favoured in the thinnest films with the lowest $c/a$ ratios. For
unit cells with larger $c/a$ ratios, there is a rapid drop-off in the strain
required for the (010)α transition but the larger atomic shuffle may promote
competition with the (021)α and (110)α transitions. These results are
consistent with experimental observations, but a more detailed study using
density functional theory would allow the quantitative calculation of the
height of the energy barrier to each transition. The reaction kinetics could
also be explored by storing the samples at cryogenic temperatures and tracking
any increase in the hcp lifetime.

Finally we draw links to Ref. [21], where uranium metal deposited onto
Gd(00.1) at 600∘C formed domains of (021)α and (110)α with definite
crystalline order, but no discernible match to the hexagonal buffer layer. If
the system had initially grown as a two-domain hcp layer with a 30∘ relative
rotation between domains, as in Ref. [17], a conversion of Domain 1
(U[10.0]//Gd[10.0]) into (110)α and Domain 2 (U[21.0]//Gd[10.0]) into (021)α
via the mechanisms suggested here gives the orientations of $\alpha$-U found
in Ref. [21]. It is likely that a metastable, well-ordered hcp layer
transitioned into $\alpha$-U via the same pathways seen in these Cu/Ir
buffered systems, culminating in the unexplained epitaxial relationships.

##  VII Conclusions

We have shown that metastable hcp-like layers of uranium metal can be grown
onto {111}-textured Cu and Ir surfaces at room temperature, giving a highly
anisotropic ‘relaxed’ structure with experimental lattice parameters ($a=2.92$
and $c=5.63$ Å, $c/a=1.93$) that deviate significantly from the relaxed
theoretical hcp values ($a=2.98$ and $c=5.48$ Å, $c/a=1.84$). These films
gradually and predominantly relax into the (010), (110) and (021) orientations
of $\alpha$-U. We have suggested several two-step mechanisms for these
transitions and encourage more quantitative theoretical studies of the
pathways. The relatively long lifetimes of the thinner hcp-like layers open
the door to further studies of the electronic and magnetic properties of this
metastable structure, which could reveal magnetic or superconducting order at
lower temperatures. These results emphasize both the fascinating complexity of
this heavy elemental metal, and the power of thin film growth techniques to
explore novel crystal symmetries.

###### Acknowledgements.

This research was supported by the Engineering and Phys. Sciences Research
Council (EPSRC), UK, through the Centre for Doctoral Training in Condensed
Matter Physics (CDT-CMP) grant no. EP/L015544/1, and the National Nuclear User
Facility for Radioactive Materials Surfaces (NNUF-FaRMS), grant no.
EP/V035495/1. We acknowledge highly useful discussions with D. Chaney.

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# Antiferromagnetic fluctuations in the one-dimensional Hubbard model

Václav Janiš<EMAIL_ADDRESS>Antonín Klíč  Jiawei Yan  Institute of Physics, The
Czech Academy of Sciences, Na Slovance 2, CZ-18221 Praha 8, Czech Republic

###### Abstract

We study the low-temperature critical behavior of the one-dimensional Hubbard
model near half filling caused by enhanced antiferromagnetic fluctuations. We
use a mean-field-type approximation with a two-particle self-consistency
renormalizing the bare interaction. It allows us to control a transition from
high to low temperatures as well as from weak to strong-coupling. We show that
there is a crossover temperature $T_{0}=t\exp\\{-1/U\rho(0)\\}$ for arbitrary
interaction $U>0$ and the bare density of states at the Fermi energy
$\rho(0)>0$. The solution at lower temperatures goes over to strong coupling
and approaches a quantum critical point with the diverging staggered
susceptibility and a gap in the excitation spectrum at zero temperature.

††preprint: AIP/123-QED

##  I Introduction

The exactly solvable quantum many-body models set limiting situations for
testing non-perturbative approximate solutions in the strong-coupling regime.
There are three fundamental models with an exact low-temperature solution, the
Kondo model,Andrei et al. (1983) the single-impurity Anderson model
(SIAM),Tsvelick and Wiegmann (1983) and the one-dimensional Hubbard
model.Essler et al. (2005) The solutions of these models are baed on the
quantum-mechanical Bethe Ansatz.Bethe (1931) This algebraic method gives the
rigorous analytic expressions for the ground-state properties. The behavior of
the excited states at non-zero temperatures can then be determined from the
Bethe Ansatz only approximately.Takahashi (1972) In particular, the dynamics
of the elementary excitations can be obtained only with additional
approximations.

The most direct and versatile method for studying the dynamics of the many-
body quantum systems are the Green functions and Feynman diagrams.
Renormalizations must be introduced in the diagrammatic representations of
physical processes so that to allow for the transition from the weak-coupling
to the strong-coupling regime. The standard perturbation expansions use a
small parameter but their renormalizations pick up only selected classes of
contributions that can no longer be controlled by the expansion parameter.
They may be, however, justified in specific limits of the studied models. The
best way to prove reliability of specific self-consistent approximations is to
compare them with the exact solution, when available.

The SIAM in the strong-coupling Kondo regime is often used to check the
reliability of approximations on the local quantum dynamics which led to the
introduction of the dynamical mean-field theory (DMFT).Georges et al. (1996)
The quantum local models contain the full quantum dynamics but miss the
effects of spatial fluctuations. The latter are important for the formation of
the long-range order in low-dimensional models where one has to obey the
Mermin-Wagner theorem prohibiting continuous transitions to ordered states at
non-zero temperatures.Mermin and Wagner (1966) Since this theorem does not
restrict the long-range order at zero temperature, reliable approximations in
low dimensions must be able to distinguish between non-zero and zero
temperatures. Most of the approximate approaches to strong-electron
correlations fail to do so and either allow or prohibit transitions to ordered
phases uniformly for all temperatures.

The problem with the description of the low-energy excitations in the one-
dimensional Hubbard model is that one cannot start with the weak-coupling
Fermi liquid. The spectrum of the excitations in the half-filled band displays
a charge gap and the low-lying excitations do not have the fermionic
character. Instead, a Luttinger liquid, where spin and charge excitations are
separated, uses bosonic excitations, spinons and holons.Haldane (1981) The
critical behavior of the spin and charge correlation functions at zero
temperature were determined.Frahm and Korepin (1990); Schulz (1990) Using the
thermodynamic Bethe Ansatz the homogeneous thermodynamic properties of the
one-dimensional Hubbard model were also numerically determined.Kawakami and
Okiji (1989); Deguchi et al. (2000); Takahashi (1999); Takahashi and Shiroishi
(2002) Although antiferromagnetic short-range order near half filling was
discussed,Shiba (1972a, b) the low-temperature asymptotics of the staggered
susceptibility measuring the range of antiferromagnetic correlations has not
been determined yet. In particular, the way the Fermi liquid breaks down in
the low-temperature limit in the half-filled band without being
antiferromagnetically ordered has not been addressed within the many-body
perturbation theory.

We formulate the approximation with fermionic elementary excitations for non-
zero temperatures where the weak-coupling theory and/or high-temperature
expansion can be applied.Shiba (1972a); Takahashi and Shiroishi (2002) To
reach zero temperature one has, however, to use self-consistent non-
perturbative approximations where the solution at half filling goes over to an
insulator at arbitrary positive interaction. We developed a general mean-
field-like theory with two-particle self-consistency that suppresses spurious
transitions of the weak-coupling mean-field theory.Janiš and Augustinský
(2007); Janiš et al. (2017a); Janiš et al. (2019) It proved its reliability in
reproducing qualitatively correctly the zero-temperature Kondo limitJaniš et
al. (2017b); Janiš et al. (2019) as well as a crossover from high to low-
temperature regimes of the SIAM.Janiš and Klíč (2020) Most recently we
disclosed the existence of the Curie-Weiss law in the magnetic susceptibility
of the SIAM on an interval of intermediate temperatures for sufficiently
strong electron correlations.Janiš et al. (2020) The general approach is also
applicable to extended lattice systems.Janiš et al. (2019) The one-dimensional
Hubbard model offers now a possibility to check reliability of this
approximation for extended systems. The approximation primarily aims at two-
particle correlation functions and is generally justified in the critical
region of a response function. Here we use it to determine the low-temperature
limit of the staggered susceptibility in the charge-symmetric state of the
one-dimensional Hubbard model. The antiferromagnetic fluctuations or spin-flip
processes are decisive for the dynamics of the elementary excitations and
opening of the charge gap in the spectral function.

##  II Critical antiferromagnetic fluctuations

The Hamiltonian of the Hubbard model on a linear chain with $N$ lattice sites
with periodic boundary conditions is

$H_{I}=N\sum_{\sigma}\int_{-\pi/l_{0}}^{\pi/l_{0}}\frac{l_{0}dk}{2\pi}\epsilon(k)c^{\dagger}_{k\sigma}c^{\phantom{\dagger}}_{k\sigma}+U\sum_{\mathbf{i}}c^{\dagger}_{\uparrow\mathbf{i}}c_{\uparrow\mathbf{i}}c^{\dagger}_{\downarrow\mathbf{i}}c_{\downarrow\mathbf{i}}\,,$
(1)

with the dispersion relation

$\epsilon(k)=-2t\cos(l_{0}k)\,$ (2)

and $l_{0}$ being the interatomic distance.

The ground state of this model was solved exactly.Lieb and Wu (1968) The most
important result for the half-filled band was the absence of the Mott
transition at nonzero interactions U. The ground state is an insulator for any
$U>0$. It means that the weak-coupling expansion is not applicable and the
low-temperature state becomes increasingly strongly coupled with decreasing
temperature. The gap in the excitation spectrum is not, however, caused by a
long-range antiferromagnetic order. There is hence a mechanism driving the
system to the insulating state without becoming antiferromagnetically ordered.

It is evident that the antiferromagnetic fluctuations with dynamical spin-flip
processes will have the most important impact on the low-temperature behavior
for half filling with suppressed charge fluctuations. The general response of
the model on weak external magnetic excitations is the dynamical
susceptibility. If we assume that the bare interaction $U$ is renormalized,
screened to an effective local and static interaction $\Lambda$, the dynamical
susceptibility generally is

$\chi(\mathbf{q},z)=-\frac{2\phi(\mathbf{q},z)}{1+\Lambda\phi(\mathbf{q},z)}\,,$
(3)

where $z$ is a complex frequency and $\phi$ is the electron-hole bubble that
can be represented in the Matsubara formalism as a convolution of two
propagators

$\phi(\mathbf{q},i\nu_{m})=\frac{1}{N}\sum_{\mathbf{k}}\frac{1}{\beta}\sum_{\omega_{n}}G(\mathbf{q}+\mathbf{k},i\omega_{n+m})G(\mathbf{k},i\omega_{n})\,,$
(4)

where $\omega_{n}=(2n+1)\pi T$ are fermionic and $\nu_{m}=2m\pi T$ are bosonic
Matsubara frequencies at temperature $T$ and $\beta=1/T$. We set the Boltzmann
constant $k_{B}=1$.

The full one-electron propagator is

$\mathcal{G}(\mathbf{k},z)=\frac{1}{z+\bar{\mu}-\epsilon(\mathbf{k})-\Sigma^{Sp}(\mathbf{k},z)}\,$
(5)

with the effective chemical potential measuring the distance from half filling
$\bar{\mu}=\mu+Un/2$ and the spectral self-energy $\Sigma^{Sp}(\mathbf{k},z)$.
The approximation is then determined by the dependence of the effective
interaction $\Lambda$ and the self-energy $\Sigma^{Sp}(\mathbf{k},z)$ on the
bare interaction $U$. We generally split the approximation into the two-
particle part determining the homogeneous vertex $\Lambda$ and the one-
particle part determining the spectral self-energy
$\Sigma^{Sp}(\mathbf{k},z)$. We introduce a thermodynamic propagator
$G(\mathbf{k},i\omega_{n})$ with only the static, Hartree self-energy that we
use in the two-particle part determining the relation between the bare
interaction and vertex $\Lambda$. This restriction is chosen to derive
analytic expressions for the two-particle functions.

The static susceptibility at for $z=0$ has a maximum at a momentum
$\mathbf{q}=\mathbf{Q}$ determining the type of the low-temperature ordering.
It is the edge vector $Q=\pi/l_{0}$ for the one-dimensional half-filled
Hubbard model and $\mathbf{Q}=(\pi/l_{0},\pi/l_{0},\ldots)$ in higher
dimensions. We identified the momenta with the wave vectors and used
$\hbar=1$. The electron-hole bubble with the Hartree Green functions can be
analytically represented after summing over the Matsubara frequencies

$\phi(\mathbf{q},\omega_{+})=\frac{1}{N}\sum_{\mathbf{k}}\frac{f(\epsilon(\mathbf{k})-\bar{\mu})-f(\epsilon(\mathbf{k}+\mathbf{Q}+\mathbf{q})-\bar{\mu})}{\omega_{+}+\epsilon(\mathbf{k})-\epsilon(\mathbf{k}+\mathbf{Q}+\mathbf{q})}\,,$
(6)

where $f(x)=1/(1+e^{\beta x})$ is the Fermi function. We abbreviated the limit
of complex frequencies to the real axis $\omega_{\pm}=\omega\pm i0^{+}$.

Antiferromagnetic fluctuations are dominant at low temperatures around the
half-filled band $n=1$ with $\bar{\mu}=0$. They are controlled by small
transfer momenta of the two-particle bubble around the edge vector
$\mathbf{Q}$. It is then sufficient to expand the electron-hole bubble in
small momenta around $\mathbf{Q}$ to describe the critical antiferromagnetic
fluctuations,

$\phi(\mathbf{Q}+\mathbf{q},\omega_{+})\doteq\phi(\mathbf{Q},\omega_{+})+Dl_{0}^{2}q^{2}\,.$
(7)

We expand the dispersion relation and use only the first two terms

$\epsilon(\mathbf{k}+\mathbf{Q}+\mathbf{q})\doteq\epsilon(\mathbf{k}+\mathbf{Q})+l_{0}\mathbf{q}\cdot\nabla\epsilon(\mathbf{k}+\mathbf{Q})$
(8)

to determine the spatial expansion parameter $D>0$. We further use the
symmetries $\epsilon(\mathbf{k}+\mathbf{Q})=-\epsilon(\mathbf{k})$ and the
same one for the derivative
$\nabla\epsilon(\mathbf{k}+\mathbf{Q})=-\nabla\epsilon(\mathbf{k})$ for the
edge vector $\mathbf{Q}=(\pi/l_{0},\pi/l_{0},\ldots)$ to transform the
momentum integrals into energy integrals with the local density of states,
that in the one-dimensional model reads

$\rho(\epsilon)=\frac{1}{2\pi}\frac{\theta\left(4t^{2}-\epsilon^{2}\right)}{\sqrt{4t^{2}-\epsilon^{2}}}\,.$
(9)

Here $\theta(x)$ is the Heaviside step function. The homogeneous part of the
electron-hole bubble then is

$\phi(\mathbf{Q},\omega_{+})=P\int_{-\infty}^{\infty}d\epsilon\rho(\epsilon)\frac{f(\epsilon-\bar{\mu})-f(-\epsilon-\bar{\mu})}{2\epsilon+\omega_{+}}\\\
-\ \frac{i\pi}{2}\
\rho\left(-\frac{\omega}{2}\right))\left[f\left(-\frac{\omega}{2}-\bar{\mu}\right)-f\left(\frac{\omega}{2}-\bar{\mu}\right)\right]\,.$
(10)

and

$D=\int_{0}^{2t}d\epsilon\rho(\epsilon)\frac{4t^{2}-\epsilon^{2}}{4\epsilon}\left[\frac{f(\epsilon-\bar{\mu})-f(-\epsilon-\bar{\mu})}{2\epsilon^{2}}\right.\\\
\left.-\
\frac{f^{{}^{\prime}}(-\epsilon-\bar{\mu})}{\epsilon}-f^{{}^{\prime\prime}}(-\epsilon-\bar{\mu})\right]\,,$
(11)

with the prime denoting the derivative.

##  III Effective interaction

The gist of our approach is to determine a screening of the bare interaction
due to multiple scatterings of electron pairs. The renormalized interaction is
derived from the two-channel parquet equationsBickers (1991); Rohringer et al.
(2018) reduced so that to stay in the critical region of the response function
during the transition from weak to strong coupling.Janiš et al. (2019) The
critical behavior manifests itself as a singularity in the Bethe-Salpeter
equation with multiple electron-hole scatterings in the models with the
repulsive interaction.Janiš (1999) The bare interaction in this equation must
be replaced by a renormalized one to avoid spurious transition to a
magnetically ordered state in strong coupling. It is achieved by multiple
scatterings of electron pairs. The corresponding Bethe-Salpeter equation for
the effective interaction $\Lambda$ in this scheme isJaniš et al. (2019)

$\left[1-\frac{\Lambda^{2}}{N}\sum_{\mathbf{q}}\frac{1}{\beta}\sum_{\nu_{m}}\phi(-\mathbf{q},-i\nu_{m})\right.\\\
\left.\times\frac{G(\mathbf{k}+\mathbf{q},i\omega_{n+m})G(\mathbf{k}^{\prime}-\mathbf{q},i\omega_{n^{\prime}-m})}{1+\Lambda\phi(-\mathbf{q},-i\nu_{m})}\right]\Lambda=U\,.$
(12)

This equation cannot hold for the static vertex $\Lambda$ point-wise for all
fermionic momenta and frequencies, unless we turn the vertex dynamical. A
dynamical vertex would make the resulting approximation extremely complicated
and would lead to the loss of the analytic control of the critical behavior
that is the primary objective of our approximation. If we cannot satisfy Eq.
(12) fully we resort to an approximate solution in a mean sense. There are
several ways to do that. The critical behavior is not qualitatively changed by
fluctuations in the fermionic variables, since they are irrelevant in the
critical region of the pole in the response function. Different ways of
treating the fermionic fluctuations in Eq. (12) do not change the universal
critical behavior but affect the non-universal quantities. Here we multiply
both sides of Eq. (12) by the product of the one-particle thermodynamic
propagators
$G(\mathbf{k},i\omega_{n})G(\mathbf{k}^{\prime},i\omega_{n^{\prime}})$ and
sum/integrate over the fermionic variables $\mathbf{k},\omega_{n}$
$\mathbf{k}^{\prime},\omega_{n^{\prime}}$. Equation (12) with $n$ being the
charge density per site reduces after analytic continuation to real
frequencies to

$\Lambda=\frac{Un^{2}}{n^{2}+4\Lambda^{2}\mathcal{X}_{\mathbf{Q}}}\,,$ (13)

where

$\mathcal{X}_{\mathbf{Q}}=-S_{d}l_{0}^{d}P\int_{-\infty}^{\infty}\frac{d\omega}{\pi}b(\omega)\\\
\times\int_{0}^{L}\frac{q^{d-1}dq}{(2\pi)^{d}}\Im\left[\frac{\phi(\mathbf{Q},\omega_{+})^{3}}{1+\Lambda\left(\phi(\mathbf{Q},\omega_{+})+Dl_{0}^{2}q^{2}\right)}\right]>0\,$
(14)

is the term renormalizing the bare interaction due to multiple scattering of
singlet electron pairs. We denoted $b(x)=1/(1e^{\beta x}-1)$ the Bose
function, $S_{d}$ is the surface of the $d$-dimensional unit sphere and $L$ is
an appropriate cutoff for large momenta needed for dimensions $d>3$.
Positivity of integral $\mathcal{X}_{\mathbf{Q}}$ leads to a screening of the
bare interaction.

Equation (12) self-consistently determines the effective interaction $\Lambda$
that controls the critical behavior near the pole of the magnetic response
function, a singularity in the integrand in Eq. (14). The singularity emerges
for $a=1+\Lambda\phi(\mathbf{Q},0)=0$. This critical point can be reached only
if $\mathcal{X}_{\mathbf{Q}}<\infty$. It means that the singularity must be
integrable. The self-consistent equation for the effective interaction then
suppresses spurious poles of the random-phase approximation with the bare
interaction $U$ and correctly allows only for integrable singularities in the
response functions.

Equations (10)-(14) determine our general approximation of the mean-field
character with a two-particle self-consistency that can be applied in any
spatial dimension in the critical region of the diverging susceptibility. The
asymptotic integral over the transfer momentum $q$ in Eq. (14) can be
performed and its explicit value in the critical region of diverging
antiferromagnetic fluctuations in one dimension is

$\mathcal{X}=-\frac{1}{2\pi\sqrt{\Lambda D}}\int_{0}^{2t}d\omega\\\
\frac{\phi(0)^{3}\coth\left(\beta\omega/2\right)\sin\left(\alpha(\omega)/2\right)}{\left[\left(1+\Lambda\Re\phi(\omega_{+})\right)^{2}+\Lambda^{2}\Im\phi(\omega_{+})^{2}\right]^{1/4}}\,,$
(15)

where
$\alpha(\omega)=\arctan\left(-\Lambda\Im\phi(\omega_{+})/(1+\Lambda\Re\phi(\omega))\right)$
and we left out the dependence on the fixed edge vector $Q=\pi/l_{0}$ to
simplify the notation. Equation (13) for the effective interaction $\Lambda$
together with equation (15) for integral $\mathcal{X}$ can be solved
numerically in the whole phase space of the input parameters in the same way
as it was done in the SIAM.Janiš et al. (2019)

##  IV Low-temperature asymptotics of the one-dimensional Hubbard model

###  IV.1 General case

The most interesting situation in the Hubbard model is the low-temperature
limit of the half-filled band with enhanced antiferromagnetic fluctuations. It
is manifested by proximity of a singularity in the static staggered
susceptibility

$\displaystyle\chi^{AF}$ $\displaystyle=-\frac{2\phi(0)}{a}$ (16)

with the bubble $\phi(\omega)$ from Eq. (10) and a dimensionless Kondo scale
$a=1+\Lambda\phi(0)$ measuring the distance from the critical point. We can
use equation (13) as an equation for the Kondo scale $a\in(0,1)$ with
$\Lambda=(a-1)/\phi(0)$.

The dominant contribution to integral $\mathcal{X}$ in the critical region
with $a\ll 1$ comes from small frequencies and we can replace the bubble by
its low-frequency asymptotics $\phi(\omega_{+})\doteq\phi(0)-i\pi A\omega$. We
obtain from Eq. (10)

$\displaystyle A$
$\displaystyle=\frac{\beta}{2}f(\bar{\mu})f(-\bar{\mu})\rho(0)\,.$ (17)

Moreover, we can set $\omega=0$ in all regular functions of the integrand in
Eq. (15). We must, however, introduce a frequency cutoff. We choose the
hopping parameter t to suppress the irrelevant contribution from high
frequencies. We thereby do not affect the leading-order critical asymptotics.
We replace the integrand in Eq. (15) by its low-frequency asymptotics, that is

$\mathcal{X}=-\frac{\phi(0)^{3}}{2\pi\sqrt{\Lambda
D}}\int_{0}^{2t}d\omega\frac{\coth\left(\beta\omega/2\right)\sin\left(\alpha(\omega)/2\right)}{\left(a^{2}+\pi^{2}\Lambda^{2}A^{2}\omega^{2}\right)^{1/4}}\,,$
(18)

with $\alpha(\omega)=\arctan(\pi\Lambda A\omega/a)$.

We can analytically assess the integral in the leading order of the vanishing
scale $a\to 0$. After a few straightforward manipulations we end up with

$\mathcal{X}\doteq-\frac{\phi(0)^{3}}{2\pi\sqrt{\Lambda
D}}\left\\{\frac{2T}{\sqrt{a}}\int_{0}^{\infty}dx\frac{\sin\left(\arctan(x)/2\right)}{x\left[1+x^{2}\right]^{1/4}}\right.\\\
\left.+\ \frac{1}{3\pi\Lambda
A}\left[\left(a^{2}+4\pi^{2}\Lambda^{2}A^{2}t^{2}\right)^{3/4}\right.\right.\\\
\left.\left.-\
\left(a^{2}+4\pi^{2}\Lambda^{2}A^{2}T^{2}\right)^{3/4}\right]\right\\}\,.$
(19)

The numerical value of the integral in the first term is $\pi/2$. This
asymptotic form of the $\mathcal{X}$ integral holds only in the limit $a\to
0$. We compared in Fig. 1 the full, Eq. (18), and the asymptotic, Eq. (19),
expression of a dimensionless integral $\sqrt{a}\mathcal{X}/\phi(0)^{2}$ as a
function of the Kondo parameter $a\in(0,1)$ on the logarithmic scale at a
temperature $T=0.1t$ and with interaction $U=5t$. We can see quite a good
agreement in the limit $a\to 0$.

Integral $\mathcal{X}$ contains two contributions. The first one, proportional
to $T/\sqrt{a}$, is the dominant critical contribution at nonzero temperatures
and prevents the existence of the critical point with $a=0$ signaling the
transition to the antiferromagnetically ordered phase. The latter term in the
brackets is due to quantum fluctuations dominant at zero temperature where the
first term vanishes. It tells us that a transition to a Néel state, $a=0$, is
formally possible at zero temperature. But it is nonetheless unphysical, since
zero absolute temperature can be reached only asymptotically from finite non-
zero temperatures where the long-range order is suppressed by the first term
in Eq. (19).

Figure 1: Comparison of a dimensionless integral
$\sqrt{a}y=\sqrt{a}\mathcal{X}/\phi(0)^{2}$ calculated with the asymptotic
form, Eq. (19), black line, and the full expression, Eq. (18), red line, as a
function of the Kondo parameter $a$ on the logarithmic scale for $U=5t$ and
$T=0.01t$.

###  IV.2 Half-filled band

We choose the half-filled band with $\bar{\mu}=0$ to determine the explicit
temperature dependence of the Kondo scale $a$. It will directly lead via Eq.
(16) to the temperature dependence of the staggered susceptibility. We have in
this symmetric situation

$\displaystyle\Lambda$ $\displaystyle=-\frac{1-a}{\phi(0)}\,,$ (20a)
$\displaystyle A$ $\displaystyle=\frac{\rho(0)}{8T}\,,$ (20b)
$\displaystyle\phi(0)$
$\displaystyle=P\int_{-\infty}^{\infty}d\epsilon\rho(\epsilon)\frac{f(\epsilon)}{\epsilon}$
$\displaystyle\doteq-\rho(0)\ln\left(\frac{t}{T}\right)\,.$ (20c)

If we introduce dimensionless parameters $u=-U\phi(0)$ and
$y(a)=4\mathcal{X}(a)/\phi(0)^{2}$ the Kondo scale $a$ is then determined from
a cubic equation

$(1-a)^{3}y(a)+(1-a)=u\,.$ (21)

The solution of this equation in the critical region for $a\ll 1$ is
$y(a)=u-1$ with $y\propto T/\sqrt{a}$. The critical region can, however, be
reached only in the strong-coupling regime, that is if

$U|\phi(0)|\doteq U\rho(0)\ln\left(\frac{t}{T}\right)\gg 1\,.$ (22)

There is hence a crossover temperature for arbitrary interaction $U>0$ at
which the solution of the half-filled Hubbard model goes over to a strong-
coupling regime. This temperature is

$T_{0}=t\exp\left(-\frac{1}{U\rho(0)}\right)\,.$ (23)

The crossover temperature is reminiscent of the Kondo temperature from the
$sd$ Kondo model.Andrei et al. (1983)

Finally, the temperature dependence of the Kondo scale in the strong-coupling
regime for $T<T_{0}$ is

$a=\frac{4\rho(0)}{D}\frac{T^{2}\rho(0)^{2}\ln^{3}\left(t/T\right)}{\left(U\rho(0)\ln\left(t/T\right)-1\right)^{2}}\,.$
(24)

The low-temperature asymptotics of the staggered susceptibility for $T\ll
T_{0}$ then is

$\displaystyle\chi^{AF}$ $\displaystyle\doteq\frac{D}{2}\
\frac{U^{2}}{T^{2}}\,.$ (25)

The zero-temperature limit of the half-filled band is a quantum critical point
with the divergent staggered susceptibility. We plotted the temperature
dependence of its inverse for an intermediate coupling $U=5t$ in Fig. 2 It
shows that both its asymptotic limit from Eq. (25) and the full
susceptibility, Eq. (16), coincide in the critical region sufficiently below
the crossover temperature $T_{0}$ from Eq. (23) and approach zero.

Figure 2: Comparison of the temperature dependence of the staggered
susceptibility, Eq. (25), calculated for $U=5t$ with the full and asymptotic
expressions for the Kondo scale $a$, Eq. (24). We indicated the crossover
temperature $T_{0}=0.081t$ from Eq. (23).

##  V Spectral properties

The one-dimensional Hubbard model at half filling in the limit of zero
temperature is in the strong-coupling regime. It means that we cannot apply
the standard weak-coupling perturbation theory and the spectral properties at
zero temperature must be determined non-perturbatively. The fundamental
quantity determining the spectral properties of the correlated electron
systems is the spectral self-energy $\Sigma^{Sp}(\mathbf{k},\omega_{+})$
renormalizing the dispersion relation and the one-electron propagator from Eq.
(5). We used the Hartree propagator to determine the thermodynamic two-
particle properties of the model, which was necessary to comply with the Ward
identity and to keep the approximation with the static effective interaction
thermodynamically consistent.Janiš et al. (2019) The spectral self-energy, a
dynamical correction to the Hartree term, is generally determined from the
Schwinger-Dyson equation. The only restriction on the spectral self-energy is
thermodynamic consistency with the two-particle functions. It means that it
should share the critical behavior determined from the two-particle vertex to
maintain thermodynamic consistency. The Schwinger-Dyson equation must act in
the spin-symmetric subspace and use the spin-symmetric two-particle vertex
determined in the preceding sections to achieve the demanded thermodynamic
self-consistency.

The Schwinger-Dyson equation with the effective interaction $\Lambda$ as the
two-particle irreducible vertex reads

$\Sigma^{Sp}(k,\omega_{+})=-U\Lambda\int_{-\infty}^{\infty}\frac{l_{0}dq}{2\pi}\
P\int_{-\infty}^{\infty}\frac{dx}{\pi}\\\
\times\left\\{b(x)\mathcal{G}(k+q,\omega_{+}+x)\Im\left[\frac{\phi(q,x_{+})}{1+\Lambda\phi(q,x_{+})}\right]\right.\\\
\left.-\
f(x+\omega)\frac{\phi(q,x_{-})}{1+\Lambda\phi(q,x_{-})}\Im\mathcal{G}(q+k,x+\omega_{+})\right\\}\,.$
(26)

The bubble $\phi(q,\omega_{+})$ is determined from the thermodynamic
propagator $G(k,\omega_{+})$ not to affect the critical behavior of the model
with $a\to 0$. The one-electron Green function $\mathcal{G}(k,\omega_{+})$ in
this equation is, however the full Green function renormalized via the
resulting self-energy as defined in Eq. (5). It means that we use the full
one-particle renormalization for the one-particle Green function of the
Schwinger-Dyson equation determining the spin-symmetric self-energy.

Equation (26) is solved iteratively starting from the bare (Hartree) one-
electron propagator $G(k,\omega_{+})$. Similarly as in the case of the
staggered susceptibility we can perform the integral over momentum $q$ using
the expansion around the edge momentum $Q=\pi/l_{0}$ capturing the critical
antiferromagnetic fluctuations for $a\to 0$ in the low-temperature limit of
the half-filled band. The first iteration for the spectral self-energy then is

$\Im\Sigma^{Sp}_{0}(\epsilon,\omega_{+})=\frac{U\phi(0)}{2\pi\sqrt{D}}\left[b(\epsilon-\omega)+f(\epsilon)\right]\\\
\times\frac{\sin\left(\alpha(\epsilon-\omega)/2\right)}{\left[\left(\phi(0)-(1-a)\Re\phi(\epsilon-\omega)\right)^{2}+(1-a)^{2}\Im\phi(\epsilon-\omega)^{2}\right]^{1/4}}\,,$
(27)

where the bubble is taken from Eq. (10) with $\bar{\mu}=0$, the Kondo scale
$a$ from Eq. (24), and the angle

$\displaystyle\alpha(x)$
$\displaystyle=\arctan\left(-\frac{\pi\rho(0)^{2}x}{\phi(0)a}\right)\,.$ (28)

We also used the solution for the effective interaction
$\Lambda=(a-1)/\phi(0)$. The self-energy depends in this mean-field
approximation on momentum only via the dispersion relation $\epsilon(k)$ at
the critical point. Such a simplified dependence is generally expected for low
frequencies around the Fermi energy.

Already this non-self-consistent solution for the self-energy qualitatively
reveals the expected features of the spectral function at zero temperature.
Since the static bubble at half filling is logarithmically divergent at zero
temperature, $\phi(0)\propto\rho(0)\ln(t/T)$, the imaginary part of the self-
energy can either be zero or infinity. It appears that
$\Im\Sigma^{Sp}_{0}(\epsilon,\omega_{+})=\infty$ for
$|\epsilon|\leq|\omega|\leq 2t$. Consequently,
$\Im\mathcal{G}(\epsilon,\omega)=0$ and the spectral function has a gap on
this interval. The imaginary part of the self-energy at the edge point
$\epsilon=\omega$ has the low-temperature critical asymptotics

$\Im\Sigma^{Sp}_{0}(\omega,\omega_{+})\doteq-\
\frac{U^{4}D}{16T^{2}\sqrt{\ln(t/T)}}\,,$ (29)

while the asymptotics for $|\epsilon|<|\omega|$ is slower

$\Im\Sigma^{Sp}_{0}(\epsilon,\omega_{+})\doteq-\
\frac{U^{2}\sqrt{\ln(t/T)}}{2\sqrt{2}\pi T}\,.$ (30)

The actual critical asymptotics will be modified in the full self-consistent
solution of Eq. (26). The non-self-consistent solution is, nevertheless,
sufficient to illustrate the mechanism behind the opening of the gap in the
spectral function in this approximation.

##  VI Conclusions

The ground state and thermodynamics of the one-dimensional Hubbard model can
be determined within rigorous approaches. It is, however, very laborious to
get an overall picture of the behavior of this model from them. Moreover, the
Bethe Ansatz, on which the rigorous results are based, is not extensible to
higher dimensions. That is why approximate schemes applicable not only to
higher dimensional models but also to more realistic descriptions of
interacting electrons are being developed and tested.

We used a universal mean-field-like approximation with a renormalized static
effective interaction to study the critical antiferromagnetic fluctuations in
the half-filled band. The equilibrium state of our approximation remains
paramagnetic down to zero temperature as expected. The low-temperature limit
is, however, nontrivial with a crossover temperature below which the system
gets into the strong-coupling regime for arbitrary non-zero interaction $U>0$.
The solution at zero temperature then is at a quantum critical point with
infinite antiferromagnetic fluctuations and a diverging staggered
susceptibility the critical asymptotics of which was derived. We also showed
how a gap in the spectral function opens at zero temperature without being
antiferromagnetically ordered.

The exact spectrum of the Hubbard Hamiltonian in one dimension reveals that
the limit to the non-interacting case $U=0$ is not analytic. One hence cannot
rely on the picture of weakly interacting quasiparticles of the Fermi liquid
at low temperatures. Low temperatures can be described within many-body
approaches only via nonperturbative approximations renormalizing sufficiently
the bare interaction enabling a smooth extension to strong coupling. Our main
conclusion is that the zero-temperature state is strongly correlated and can
be approached within the standard many-body techniques with fermionic
excitations only asymptotically from high temperatures. Approximate solutions
derived within the many-body perturbation theory at strictly zero temperature
cannot be trusted in low-dimensional systems ($d<3$).

## Acknowledgment

The research was supported by Grant No. 19-13525S of the Czech Science
Foundation and INTER-COST LTC19045 of the Czech Ministry of Education, Youth
and Sports.

## Data Availability

We have not used any specific data needed to reproduce the conclusions of this
study beyond that available in the article.

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