# Static corrections versus dynamic correlation effects
in the valence band Compton profile spectra of Ni

L. Chioncela,b, D. Beneac,d, S. Mankovskyd, H. Ebertd, and J. Minárd,e a
Theoretical Physics III, Center for Electronic Correlations and Magnetism,
Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany  b
Augsburg Center for Innovative Technologies, University of Augsburg, D-86135
Augsburg, Germany  cFaculty of Physics, Babes-Bolyai University,
Kogălniceanustr 1, Ro-400084 Cluj-Napoca, Romania  dChemistry Department,
University Munich, Butenandstr. 5-13, D-81377 München, Germany  New
Technologies - Research Center, University of West Bohemia, Univerzitni 8, 306
14 Pilsen, Czech Republic

###### Abstract

We compute the Compton profile of Ni using the Local Density Approximation of
Density Functional Theory supplemented with electronic correlations treated at
different levels. The total/magnetic Compton profiles show not only
quantitative but also qualitative significant differences depending weather
Hubbard corrections are treated at a mean field +U or in a more sophisticated
dynamic way. Our aim is to discuss the range and capability of electronic
correlations to modify the kinetic energy along specific spatial directions.
The second and the fourth order moments of the difference in the Compton
profiles are discussed as a function of the strength of local Coulomb
interaction $U$.

##  I Introduction

The study of physical and chemical properties of transition metals is still an
extremely active experimental field and at the same time is the subject of
extensive theoretical studies. The fascinating aspect of $d$-electron systems
is the possible interplay of relativistic and electron correlation effects
that has long been questioned. Ab-initio methods provide the framework in
which relativity and correlations may be treated at equal footing. One notable
example is the magnetic anisotropy energy for Ni. Experimentally it is known
that the easy axis is along the [111] direction and the energy cost to rotate
the magnetic moment axis into [001] direction is about $3~{}\mu$eV per
atom[1]. LSDA+U calculations [2] accounting for spin-orbit and non-collinear
coupling have been employed and showed to reproduce these values for
relatively small value of the local Coulomb interaction $U=1.9$ eV and $J=1.2$
eV. Changes in the topology of the Fermi surface were discussed in the context
of magneto-crystalline anisotropy of Ni [3]. These changes were recently
addressed using the Gutzwiller variational theory with ab-initio parameters
which showed the importance of the spin-orbit coupling [4].

As a matter of fact, nickel is perhaps the most studied electronic system. In
the ordered ferromagnetic phase the vast majority of band structure
calculations within the Local Density Approximation to the Density Functional
Theory (DFT) converge to a value for the magnetic moment of $\approx
0.62~{}\mu_{B}$, which is a slightly overestimation of the experimental data.
The orbital contribution amounts up to $10\%$ and the spin moment is found to
be around $0.56~{}\mu_{B}$. The Generalized Gradient Approximation (GGA) add
gradient correction to the local density approximation, does not change upon
the value of the magnetic moment, however improve on the equilibrium lattice
parameter and bulk modulus. The exchange splitting in both LSDA/GGA is in the
range of $0.7$ to $0.75$ eV [5], while experimental data are situated between
$0.3-0.5$ eV [6, 7, 8, 9, 10]. The valence band photoemission spectra of Ni
shows a $3d$-band width that is about $30\%$ narrower than the value obtained
from the LSDA calculations. It is known that LSDA cannot reproduce the
dispersionless feature at about 6 eV binding energy, the so-called 6 eV
satellite [11]. An improved description of correlation effects for the $3d$
electrons via the combined Local Density Approximation and Dynamical Mean
Field Theory [12, 13, 14], LSDA+DMFT, gives the width of the occupied $3d$
bands of Ni properly, reproduce the exchange splitting and the 6 eV satellite
structure in the valence band [15, 16, 17, 18, 19, 20, 21, 22].

Momentum space quantities such as the spin-dependent electron momentum density
distribution have been calculated using various methods [23, 24] mostly
employing the LSDA. In addition to that, magnetic Compton scattering can
provide a sensitive method of investigating the spin-dependent properties. For
example, in Ni, it has already been shown that the negative polarization of
the $s$\- and $p$-like band electrons can be observed [23, 25]. Although the
total spin moment is well reproduced by theory, the degree of negative
polarization at low momentum, where these electrons contribute, is typically
underestimated. This discrepancy is often regarded as being due to the
insufficient treatment of correlation present in the LSDA exchange-correlation
functional at low momentum [23]. Early studies of electronic correlations in
band structure calculations for the Compton profiles in Li and Na (alkali
metals) have been performed by Eisenberger et al. [26] and by Lundqvist and
Lynden [27]. In the former study, the linear response theory to the atomic
potential in the random phase approximation is used [26], while in the later
the orthogonalized plane wave method for the homogeneous interacting electron
gas data [27] has been employed. Although both studies have been succesful to
describe the momentum densities and the Compton profiles they are not suitable
for transition-metal systems. Later on in the study of transition metals Bauer
et al. [28, 29, 30, 31] investigated extensively the role of local and non-
local DFT functionals for the problem of electron-electron correlation effects
pointing out several inconsistencies and improving the agreement of
theoretical difference profiles with the experimental data. We have studied
recently whitin the framework of LSDA+DMFT the directional Compton profile
$J({\bf p}_{z})$ for both Fe and Ni [32, 33]. The second moment of the Compton
profile difference allowed to quantify the momentum space anisotropy of the
electronic correlations of Fe and Ni. The changes in the shape and magnitude
of the anisotropy have been discussed as a function of the strength of the
Coulomb interaction $U$. According to our results Ni has a larger momentum
space anisotropy of the second moment of the total Compton profile in
comparison with Fe [33].

The aim of this paper is twofold. First, we perform a comparison of the
computed magnetic Compton profiles at a mean-field (LSDA+U) level and beyond
within the framework of dynamical mean field theory (LSDA+DMFT). Secondly, we
extent our previous work on computing moments of directional Compton profiles
[33] and analyze corrections to the kinetic energy. In particular, we compute
the magnitude of the fourth moment that is proportional to relativistic
kinetic energy corrections that arises from the variation of electron mass
with velocity. We discuss therefore the extend to which electronic
correlations can influence the relativistic correction to the kinetic energy.

In the following subsections we analyze the magnetic Compton (Sec. II.1), the
total Compton and the difference profiles (Sec. II.2). Subsection Sec. III
analyses the effects of electronic correlations upon the kinetic energy of
bonded electrons. We conclude the present paper in section Sec. IV.

##  II Magnetic and Total Compton profiles in the presence of electronic
correlations

We performed the electronic structure calculations using the spin-polarized
relativistic Korringa-Kohn-Rostoker (SPR-KKR) method in the atomic sphere
approximation (ASA) [34]. The exchange-correlation potentials parameterized by
Vosko, Wilk and Nusair [35] were used for the LSDA calculations. For
integration over the Brillouin zone the special points method has been used
[36]. Additional calculations have been performed with the many-body effects
described by means of dynamical mean field theory (DMFT) [12, 13, 14] using
the relativistic version of the so-called Spin-Polarized T-Matrix Fluctuation
Exchange approximation [37, 38] impurity solver. In this case a charge and
self-energy self-consistent LSDA+DMFT scheme for correlated systems based on
the KKR approach [17, 39, 40] has been used. The realistic multi-orbital
interaction has been parameterized by the average screened Coulomb interaction
$U$ and the Hund exchange interaction $J$. Despite the recent developments
allow to compute the dynamic electron-electron interaction matrix elements
exactly [41], we consider in the present work the values of $U$ and $J$ as
parameters for the sake of convenience of our discussions. It was shown that
the static limit of the screened-energy dependent Coulomb interaction leads to
a U parameter in the energy range of 1 and 3 eV for all $3d$ transition
metals. As the $J$ parameter is not affected by screening it can be calculated
directly within the LSDA and is approximately the same for all 3$d$ elements,
i.e $J\approx 0.9$ eV. In our calculations we used values for the Coulomb
parameter in the range of $U=2.0$ to 3.0 eV and the Hund exchange-interaction
$J=0.9$ eV.

The KKR Green function formalism allows to compute Compton profiles $J_{\bf
K}(p_{z})$ and magnetic Compton profiles $J_{mag,\bf K}(p_{z})$ (MCPs) in a
straightforward way [42, 43, 44]. In the case of a magnetic sample, the spin
resolved momentum densities are computed within the framework of LSDA and
LSDA+DMFT approaches using the Green’s functions in momentum space, as
follows:

$n_{m_{s}}(\vec{p})={-\frac{1}{\pi}\Im\int_{-\infty}^{E_{F}}G_{m_{s}}^{LSDA(+DMFT)}(\vec{p},\vec{p},E)dE}\,.$

where $m_{s}=\uparrow(\downarrow)$. The total electron
($n_{\uparrow}(\vec{p})+n_{\downarrow}(\vec{p})$) and spin
($n_{\uparrow}(\vec{p})-n_{\downarrow}(\vec{p})$) momentum densities projected
onto the direction ${\bf K}$ defined by the scattering vector, allows to
define the (Magnetic) Compton profile as a double integral in the momentum
plane perpendicular to the scattering momentum $\vec{p}_{z}||$ K:

$\displaystyle J_{\bf
K}^{LSDA(+DMFT)}(p_{z})=\int\int[n_{\uparrow}(\vec{p})+n_{\downarrow}(\vec{p})]dp_{x}dp_{y}$
$\displaystyle J_{mag,\bf
K}^{LSDA(+DMFT)}(p_{z})=\int\int[n_{\uparrow}(\vec{p})-n_{\downarrow}(\vec{p})]dp_{x}dp_{y}.$

A useful quantity in our analysis is the difference of Compton profiles taken
along the same momentum space direction with or without including electronic
correlations:

$\Delta J_{\bf K}(p_{z})=J_{\bf K}^{+U/DMFT}(p_{z})-J_{\bf K}^{LSDA}(p_{z}).$
(1)

In our further analysis the anisotropies of the Compton profile

$\Delta J_{\bf K,K^{\prime}}(p)=J_{\bf K}(p_{z})-J_{\bf K^{\prime}}(p_{z})$
(2)

are also studied using different local exchange-correlation potentials: the
“pure” LSDA, and the supplemented LSDA+U and LSDA+DMFT ones. The electron
momentum densities are usually calculated for the principal directions ${\bf
K}=[001],[110]$ and $[111]$ using an rectangular grid of 200 points in each
direction. The maximum value of the momentum in each direction is 8 a.u.. The
resultant Magnetic-Compton (Compton) profiles were normalized to their
respective calculated magnetic spin moments (number of valence electrons).

###  II.1 Magnetic Compton profiles

The computed magnetic Compton profiles are shown in Fig. 1 along the principal
directions [001], [110] and [111]. The profiles seen in the left/right columns
of Fig. 1 have been obtained using the values of $U=2/2.3$ eV for the Coulomb
and $J=0.9$ eV for the exchange parameters and the temperature of 400 K. The
calculated spin moment for LSDA is $0.61\mu_{B}$ while both LSDA(+U/DMFT)
results give $0.59\mu_{B}$. The theoretical MCPs have been convoluted with
experimental momentum resolution [24] and all the areas are normalized at the
coresponding spin moments. Our LSDA results [33] are in agreement with the
previous published results, obtained with LMTO [24] or the FLAPW-LSDA [23,
45].

The most obvious feature for all principal directions is a significant
discrepancy between experiment and theory for $p_{z}<2$ a.u. (see Fig. 1). We
notice the large dips in the [110] and [111] profiles near $p_{z}=0$ a.u.,
which were ascribed partially to the $s$\- and $p$-like electrons, but also to
a pronounced drop in the contribution from the fifth band [24].

Figure 1: (color on-line) Compton profiles of Ni along the principal
directions $[001]$, $[110]$ and $[111]$. Left column compares the LSDA+U/DMFT
for $U=2.0$ eV with the results for $U=2.3$ eV. For both calculations $J=0.9$
eV and $T=400$ K have been used. The computed profiles LSDA (black-dotted),
LSDA+U (blue-dashed) and LSDA+DMFT (red dashed-dot) are plotted in comparison
to the experimental spectra (black-dotted). The calculated MCP profiles were
convoluted with the experimental resolution 0.2 a.u.. Data were taken from
Dixon et. al. [24].

The results of computations for the average Coulomb parameter $U=2.0$ eV are
shown on the left column of Fig. 1. Along the [001] direction and around
$p_{z}=0$ a.u. all LSDA(+U/DMFT) results seem to get close to the experimental
data. Most significant differences are in the momentum range of $1$
a.u.$<p_{z}<2$ a.u., where also the maximum of the profile is located. Along
the $[110]$ direction, both LSDA+U and LSDA+DMFT give similar results,
overestimating the first maximum at around $0.5$ eV, show a minimum at around
$1$ eV, instead of a maximum seen in the experiment and underestimate the
experimental results in several regions above $2$ a.u. Along the $[111]$
direction, the maximum at $p_{z}=0$ a.u. is overestimated by all computations:
the slight improvement of DMFT is not really significant, LSDA+U get very
close to the maximum at around $2$ a.u. Overall dynamic correlations do really
not improve significantly the agreement with the experimental data, as already
at the level of LSDA reasonable good results has been obtained.

For a slightly larger value of $U=2.3$ eV the LSDA+U results start to depart
more from the experimental data, while on contrary the DMFT results improve
the agreement significantly. Along the $[001]$ in the entire low momentum
region $p_{z}<2$ a.u. LSDA+U overestimates the spectrum, however for larger
values of the momentum it captures the profile quite well. On the other hand,
DMFT improves the momentum dependence below $p_{z}<2$ a.u., however it
overestimates for values of the momentum in the range of 2 a.u. to 4 a.u. The
largest difference between the “+U” and “+DMFT” corrections are seen along the
[110] direction. This direction correspond to the shortest bond in the fcc
structure. Here DMFT captures the peaks at around $0.5$ a.u. and the main peak
at $2$ a.u., and continues very closely to the experimental data in the
complete range of the computed momenta. LSDA+U captures only the main maximum
and as a matter of fact produce worse results than LSDA. Along the $[111]$
direction DMFT get closer to the maximum at $p_{z}=0$ a.u. than LSDA+U. For
$p_{z}>0$ a.u., the dip at $0.5$ a.u. is captured better within DMFT, while
for higher momenta both the LSDA+U and LSDA+DMFT approaches follow essentially
the same behavior.

Although both static and dynamic corrections to the MCP spectra are rather
similar, we observe a clear tendency of LSDA+U to overestimate the
experimental data while LSDA+DMFT correct some discrepancies. The following
subsection presents the results for the difference in total Compton profiles
with respect to the LSDA results, where distinctions because of static and
dynamic corrections became more apparent.

###  II.2 Directional differences of Compton profiles

Fig. 2 shows the total Compton profiles differences computed according to Eq.
(1) along the $[001]$ direction. The upper/lower part represents the
DMFT/LSDA+U spectra after subtraction of LSDA results.

Figure 2: (color on-line) Computed difference Compton profiles of Ni along the
$[001]$ direction for different values of $U$. The difference spectra of
LSDA+DMFT with respect to LSDA, $U=2.0$ eV (black-dashed), $U=2.3$ eV (red-
solid).

One can easily recognize common features in comparing the LSDA+U with
LSDA+DMFT spectra. The Brillouin zone boundary along the [001] direction is
represented by the $X(1/2,0,1/2)$-high symmetry point. The zone boundary is
marked with the first dashed line and corresponds to the value of $k_{F}$ =
0.95 $a.u.$. The second dashed line is situated at $2k_{F}$ and is plotted to
facilitate the comparison between the spectra. As one can see the LSDA+U
spectra is sharper, since the DMFT self-energy contributes in smoothing out
the spectra, however the peaks remain in the same positions. No additional
broadening of the spectra has been applied. As a consequence of dynamic
correlations within the first Brillouin zone, the $\Delta J_{[001]}$ has
positive weight, on contrary to the LSDA+U results. We observe the Umklapp
features identified in the MCP spectra of Ni [001] in several studies (at
$\sim$ 1.2, 1.7, 2.7 and 3.7 a.u.) [24, 46, 47, 45] appearing in the $\Delta
J_{[001]}$ as sharp deeps. For the region with $1.9<p_{z}<3.8$ a.u. (third and
fourth Brillouin zone), electronic correlations produces negative difference
weights. Similar observation can be made for the different values of $U$.

Figure 3: (color on-line) Computed difference Compton profiles of Ni along the
[110] direction for different values of $U$. The difference spectra of
LSDA+DMFT with respect to LSDA, $U=2.0~{}eV$ (black-dashed), $U=2.3~{}eV$
(red-solid).

Along the $[110]$ direction the Brillouin zone is intercepted at the $K$-point
with the coordinate $3\pi/2a(1,1,0)$ in the Cartesian representation.
Similarly to the $[001]$ direction, the position of the main peaks of the
spectra are the same in the LSDA+U/DMFT calculations. The Umklapp features
identified in the MCP spectra by several studies at $\sim 2.0$, $3.3$ and
$4.6$ a.u. [24, 46, 47, 45] correspond to sharper peaks in the $\Delta
J_{[110]}$ spectra. Again in LSDA+DMFT the spectra have an overall positive
weight, due to correlation induced life time effects determined by the
imaginary part of the self-energy. Also, Umklapp features are more visible in
LSDA+U as no additional broadening is present.

Figure 4: (color on-line) Computed difference Compton profiles of Ni along the
$[111]$ direction for different values of $U$. The difference spectra of
LSDA+DMFT with respect to LSDA, $U=2.0$ eV (black-dashed), $U=2.3$ eV (red-
solid).

The same analysis can be performed upon the spectra in the $[111]$ direction.
For momenta $p_{z}<k_{F}$ within the first Brillouin zone, $\Delta J_{[111]}$
computed with DMFT/+U have a negative contribution, whilst for the second and
third zone the LSDA+DMFT difference spectra have weights with alternating
sign. The $\Delta J_{[111]}$ spectra are essentially negative only within the
first zone and from $p_{z}>1a.u.$ the LSDA+U spectra is positive.

The general tendency in both LSDA+U and LSDA+DMFT is that for larger $U$,
maxima and minima are slightly stretched out, while the modulations remain the
same. This is expected as the modulation of the Compton profile is connected
to the topology of the Fermi surface. As was previously demonstrated, cuts of
the momentum density remain unchanged with inclusion of correlation effects
[48]. The overall change in the shape of the Compton spectra comparing DMFT
versus LSDA/LSDA+U reflects the presence of the imaginary part of the dynamic
local self-energy.

The comparison between the Compton spectra taken for different U values, along
the same direction K allows to discuss the strength of local-correlation
effects. In the same time comparing spectra obtained for a fixed $U$ value
along different K, K′ directions may reveal possible non-local-correlations
effects. Although DMFT supplement the DFT-LDA part by a complex local and
dynamic self-energy $\Sigma(z)$, the charge self-consistency of LDA+DMFT
achieves indirect non-local effects. In the next subsection we analyze the
differences between pairs of directional profiles also known as Compton
profile anisotropies.

###  II.3 Compton profiles anisotropy and non-locality

Comparison between theoretical and experimental amplitudes of the Compton
profile anisotropies for Ni have already been performed [49, 50, 51]. The
computed anisotropy profiles $J_{[110]}-J_{[001]}$ and $J_{[111]}-J_{[001]}$
are shown in Fig. 5 that, in contrast with earlier work, reveal the role of
correlations effects. In the upper panel the LSDA results are presented, while
in the middle and lower panel the spectra obtained using LSDA+U and DMFT are
seen. A very similar behavior, independent of the various level of
sophistication to include the Coulomb interaction is visible.

Figure 5: (color on-line) Theoretical Compton profile anisotropies of Ni.
Upper panel (a) the anisotropy computed within LSDA. Lower pannels (b) and (c)
show the LSDA+U and LSDA+DMFT results for $U=2.3$ eV, $J=0.9$ eV and $T=400$
K. Comparison with the experimental anisotropy is presented in pannel (c).

In the lower panel (c) we compare the anisotropy spectra with the
corresponding experimental data of Anastassopoulos et. al. [51]. There is a
rather satisfying agreement between the theory and experiment, in particular
for the difference $J_{[110]}-J_{[001]}$ (Fig. 5 red lines) where the
theoretical calculation follow most of the maxima and minima seen in the
experiment. Here again no broadening has been used for the computed data. On
the other hand for $J_{[111]}-J_{[001]}$ (Fig. 5 green lines) differences may
be seen not only in the amplitude of the oscillation but also in the position
of the minima/maxima. In Fig. 6 we show the comparison on a reduced momentum
regime $0<p_{z}<1$ a.u. In the upper pannel of Fig. 6(a), the LSDA results are
seen to overestimate in the range $0<p_{z}<0.2$ a.u. the experimental spectra
plotted with dased lines. In the same momentum range the LSDA+U results
underestimate the experimental data as seen in Fig. 6(b). Figure 6(c) shows
the LSDA+DMFT results and one can see that the dynamic correlations capture at
best the behaviour of anisotropy of the Compton profile in the region around
the zero momentum $p_{z}<0.2$ a.u.

Figure 6: (color on-line) Theoretical Compton profile anisotropies of Ni in
comparison with the experimental measurements. LSDA/LSDA+U results, panel
(a)/(b) over/under-estimate the anisotropy data. DMFT results are in a good
agreement with the experimental spectra in the range $0<p_{z}<0.2$ a.u. LSDA+U
and LSDA+DMFT parameters are $U=2.3$ eV, $J=0.9$ eV and $T=400$ K.

Previous analysis attributed the discrepancies to the non-local correlation
effects [52, 51], although no quantitative evidence has been presented. One
possible alternative explanation for the results seen in Fig. 6 is that the
density functional exchange-correlation potentials misplace the position of
$d$-bands(orbitals). This agrees with the observation that LSDA overestimate
the exchange splitting. Including static corrections using LSDA+U the exchange
splitting is enhanced and therefore this does not correct upon the position of
the $d$-bands, and equally does not improve on the anisotropy spectra. On the
contrary, LSDA+DMFT is known to improve on the exchange splitting as a
consequence of a Fermi-liquid type of self-energy and we equally see in the
pannel of Fig. 6(c) although in a narrow momentum region $0<p_{z}<0.2$ a.u.,
an excellent agreement with the experimental anisotropies. As anisotropies in
the Compton profile measures differences $J_{\bf K}-J_{\bf K^{\prime}}$ they
indicate non-local effects, therefore Fig. 6(c) shows that local dynamic
correlations may capture non-locality in a very narrow region around zero
momentum as a consequence of full change self-consitency of LSDA+DMFT. In
addition our results show that a description of the non-local-correlation
effects is needed for a fruther improvement on the amplitudes of Compton
profiles anisotropies for larger momenta.

##  III Kinetic energy corrections from relativistic and electronic
correlations

The relativistic generalization of the Schrödinger theory of quantum mechanics
to describe particles with spin 1/2 was achieved using the Dirac equation (see
for example [53]). The construction of Dirac equation uses symmetry arguments
and energy considerations, and it starts from the general Hamiltonian:

$H=\frac{c}{i}{\bf\alpha}\cdot{\bf\nabla}+\frac{1}{2}\left(\beta-I\right)+V({\bf
r}),$ (3)

where ${\bf\alpha}$ and $\beta$ are standard Dirac matrices and $V({\bf r})$
represents a one particle effective potential. In the spirit of the non-
relativistic DFT the effective potential consists of the Hartree term, an
exchange correlation and a spin dependent part [54, 55]: $V({\bf
r})=V_{H}({\bf r})+V_{xc}({\bf r})+\beta\sigma_{z}B({\bf r})$. Rigorous four-
component relativistic many-electron calculations are hardly tractable in the
spirit of four-component Dirac relativistic quantum mechanics [56]. It is
important to note that many applications consider relativistic effects at one-
component level, which is frequently called the scalar relativistic approach
[57] which usually assume that corrections of higher order than $1/c^{2}$ can
be neglected for chemical accuracy. Furthermore it is assumed that the effect
of the spin-orbit coupling on the form of the orbitals may be neglected,
allowing a partition of the Hamiltonian into a spin-independent and a spin-
dependent part. The latter part is then only used in the final stage of the
calculation to couple to the correlated many-electron problem. The analysis of
components of the spin-orbit coupling was performed decoupling the
longitudinal and transversal contributions [54], allowing to identify the
source of the most important spin-orbit induced phenomena in solids.

There are a few methods available to quantitatively assess the interplay
between correlation and relativistic effects. Within the framework of DFT the
recently developed LSDA+DMFT scheme demonstrates a clear potential in this
direction. LSDA+DMFT has been systematically applied to $d$-, $f$-electron
systems with various DMFT solvers [58]. From a pragmatic point of view
perturbative solvers of DMFT written in adapted basis sets to include spin-
orbit effects [38] are efficient tools for realistic multi atom/orbital
calculations. This means that we in fact capture the interplay between
relativistic and correlation effects at a more economical level of the theory:
from the correlations point of view a perturbative solver is considered, while
for the relativistic part the four-component was replaced by a two component
formulation. For a single-particle in an effective potential $V_{eff}$, the
most common transformation of the Dirac Hamiltonian (Eq. (3)) into the two
component formulation: $H_{2comp}=UH_{4comp}U^{\dagger}$ is expressed as a
unitary transformation [57], followed by a Taylor expansion in the fine
structure constant $\propto 1/c^{2}$ and produce the following terms:

$\displaystyle H^{BP}$ $\displaystyle=$
$\displaystyle(mc^{2}+V_{eff}+\frac{p^{2}}{2m})-\frac{p^{4}}{8m^{3}c^{2}}$
$\displaystyle-$
$\displaystyle\frac{1}{8m^{2}c^{2}}(p^{2}V_{eff})+\frac{\hbar}{4m^{2}c^{2}}{\bf\sigma}(\nabla
V_{eff}\times\vec{p})+...$

The first terms in parenthesis in Eq. (III) represent the usual non-
relativistic Hamiltonian, then the second one is the so called mass-velocity
term, the third is called the Darwin term and the fourth operator describes
the spin-orbit coupling (interaction). It can be analytically proved that the
scalar mass-velocity and Darwin terms are unbounded from below. The resulting
Breit-Pauli Hamiltonian ($H^{BP}$) also known as the first order relativistic
Hamiltonian contain terms that are highly-singular and variationally instable.
Therefore this operator is suitable to be used in the low order perturbation
theory.

###  III.1 Moments of the differences of the Compton profiles

The measured Compton profile, or the momentum distribution enable in principle
to obtain averages $\langle p^{n}\rangle$ directly from the experiment. On the
computational side, the momentum space formulation allows to obtain the
Compton profiles within the LSDA [59] and LSDA+DMFT [32, 33] for many systems.
Further on additional information can be gained by taking moments of the
difference between the correlated and non-correlated Compton profiles along
different K-directions: $p_{z}||{\bf K}$

$\langle p^{n}\rangle_{\bf K}=\int_{0}^{\infty}p_{z}^{n}\left[J^{+U/DMFT}_{\bf
K}(p_{z})-J^{LSDA}_{\bf K}(p_{z})\right]dp_{z};$

Recently we have computed the second moments $\langle p^{2}\rangle$ [33] along
specific directions in Fe and Ni and discussed the effect of electronic
correlations upon the kinetic energy per bonds. Aside from the kinetic energy
$\langle p^{2}\rangle$, it is possible to obtain also relativistic energy
corrections in the form of $\langle p^{4}\rangle$. In the present work we are
interested to estimate the relativistic corrections to the kinetic energy that
arises from the variation of electron mass with velocity, and how it may vary
as a function of the local Coulomb interaction.

The second and the forth moments of the difference in the total Compton
profiles, along the bond directions would provide some specific terms from the
expansion (III). Namely we are going to evaluate the so called free particle
relativistic kinetic energy ($H_{0}$) in terms of the second moments and its
relativistic correction as the forth moment, along the bond directions of Ni.
In order to compare the magnitude of second and fourth order moments one has
to introduce a dimensionless quantity $p_{r}=p/mc$. In this reduced variable
$H_{0}=mc^{2}\left((1/2)p_{r}^{2}-(1/8)p_{r}^{4}+...\right)$ and the relevant
expectation values has the expression:

$\displaystyle\langle H_{0}\rangle_{\bf K}$ $\displaystyle=$
$\displaystyle\int H_{0}\Delta J^{+U/DMFT}_{\bf K}(p_{z})dp_{z};(p_{z}||{\bf
K})$ (5) $\displaystyle\approx$ $\displaystyle mc^{2}\left[\frac{1}{2}\langle
p_{r}^{2}\rangle_{\bf K}-\frac{1}{8}\langle p_{r}^{4}\rangle_{\bf K}\right]$

Table 1:  Expectation values $\langle p_{r}^{n}\rangle$ along different direction computed within LSDA+U |  | LSDA+U
---|---|---
$n$ | $U$ | [001] | [110] | [111]
| eV | 10-6 | 10-6 | 10-6
2 | 2.0 | 1.06 | 2.65 | 6.37
| 2.3 | -2.12 | 0.53 | 5.84
4 | 2.0 | -4.20 10-3 | -1.18 10-3 | 4.26 10-3
| 2.3 | -6.29 10-3 | -4.46 10-3 | 2.43 10-3

In the momentum space representation such integrals can be directly computed.
We performed calculations for the second and fourth moments for different
values of $U$ and different level of electronic correlations. The LSDA+U
results are given in the Table 1. One can clearly see that the second and
fourth order moments differ significantly along different directions. For
$U=2$ eV all second moments are positive for all directions, while for larger
$U=2.3$ eV along [001] the second moment is negative. In the reduced
representation $p/mc$ the magnitude of these moments is of order $10^{-6}$.
The forth order moments are 3 order of magnitude smaller than the second order
moments, and are negative along the [001] and [110] directions. Along the
[111] direction the forth order moment remain positive for all $U$ values.
Note that moments decrease in magnitude as the distance is increasing: the
largest moments are obtained for the nearest neighbors distances, which are
the shortest bonds. For Ni, this corresponds to the [110] direction.
Directional averaging over all second order moments provides the kinetic
energy, while a similar average over the 4-th order moments provides the
relativistic corrections to the kinetic energy.

The moments computed in LSDA+DMFT are given in Table 2. On contrary to the
LSDA+U, using DMFT produces a Compton profile having negative second moments
along all directions and for all studied values of $U=2$ eV and $2.3$ eV.

Table 2:  Expectation values $\langle p_{r}^{n}\rangle$ along different direction computed within LSDA+DMFT |  | LSDA+DMFT
---|---|---
$n$ | $U$ | [001] | [110] | [111]
| eV | 10-6 | 10-6 | 10-6
2 | 2.0 | -7.43 | -6.90 | -4.78
| 2.3 | -11.15 | -8.50 | -6.37
4 | 2.0 | 8.75 10-3 | -5.92 10-3 | -3.69 10-3
| 2.3 | -1.12 10-3 | -7.62 10-3 | -6.38 10-3

Note the qualitative difference between the LSDA+U and LSDA+DMFT second
moments: while the former mean field (LSDA+U) approach produce second moments
with different signs depending on the directions, within the later dynamic
(LSDA+DMFT) approach correction is always negative. Similarly to our previous
results [33] we see that the positive difference at low momentum region
$0<p_{z}<2$ a.u. is completely overruled by the negative weights at higher
momenta, which leads to the overall negative values for the correction
obtained in DMFT. In LSDA+U a negative second moment is obtained along [001]
and positive for the other two directions. The positive second moment is
obtained as the Compton profile computed in the mean field (LSDA+U) approach
always is larger than the corresponding LSDA profile, for any value of the
moment $p_{z}$. In order to discuss correction to the kinetic energies, we
computed the weighted sum of the nearest neighbors, i.e. six times the
contribution along [001], 12 times the contribution along [110] and 8 times
the contribution along [111] divided by the total number of neighbors (26).
Fig. 7 summarizes the computed results. The inset shows the directional
average of the second moment which is positive in LSDA+U and negative in DMFT,
whilst the corrections to the kinetic energy is negative in both +U/DMFT
calculations.

Figure 7: (color on-line) Averaged forth (second, in inset) order moments of
the difference Compton profiles computed within DMFT (red solid) and LSDA+U
(black dashed), along different directions and for the Coulomb parameters
$U=2$ eV$/2.3$ eV, $J=0.9$ eV and $T=400$ K.

##  IV Discussions and Conclusion

The influence of electronic correlations on the Compton profiles of Ni has
been discussed within the framework of DFT comparing the results of mean field
LSDA+U and beyond mean-field LSDA+DMFT. According to our results, the mean
field decoupling of the interaction (+U) overestimates slightly the MCP
spectra, while dynamic correlations improve the agreement with experiment. To
reveal differences between the LSDA+U and LSDA+DMFT approaches we studied the
directional differences, i.e. differences of Compton profiles with respect to
the LSDA spectra. Overall the difference spectra follow a similar momentum
dependence with visible deviations in the low momentum region. A qualitative
difference is evidenced in this region: within the mean field approach (+U)
negative differences are seen while in the dynamic case, the opposite result
is obtained. In other words, the mean field LSDA+U Compton spectra have a
smaller weight in the low energy region than the corresponding LSDA+DMFT
Compton spectra. According to our recent picture of momentum redistribution
because of interaction [33] we conclude that the weight from low momentum
distribution is shifted towards the higher momentum region in the LSDA+U
spectra. This is in agreement with the naive picture of the effects of LSDA+U
on the spectral weight distribution shifting weights towards higher energies.
In the Compton scattering language, photons would scatter accordingly on
moving electrons situated in higher energy bands, although this does not mean
that the electrons are moving faster, explaining the fact that there are no
dramatic changes in the Compton spectra (differences of order of $\pm 0.02$)
shown in Figs. 2, 3 and 4. On the contrary to the LSDA+U results, in the DMFT
calculations the Fermi liquid type of self-energy determines the spectral
weight transfer towards the low energy region, and accordingly the spectra of
photons scattering on the renormalized electronic structure would be
redistributed towards low momenta. Similar conclusions have been reached in
our previous studies [32, 33].

In the analysis of the Compton profile anisotropies we found that the
LSDA+DMFT results describe well the momentum region of $0<p_{z}<0.2$ a.u.
which is a consequence of the presence of a local and dynamic self-energy that
properly locates the position of Ni $d$-bands, as seen in various
calculations. The limited momentum range is due to the inherent DMFT
approximation that the self-energy neglect spatial fluctuations.

In order to assess the capability of electronic correlations to influence the
kinetic energy along specific directions we have computed the second and the
fourth order moments of the Compton spectra considering the reduced momentum
$p_{r}=p/mc$. Although the fourth order moments are significantly smaller,
$\langle p_{r}^{4}\rangle\propto 10^{-3}\langle p_{r}^{2}\rangle$, an overall
non-negligible contribution is obtained (see Eq.(5)). Within LSDA+U the second
moment has a positive sign, which is in agreement with the description of the
momentum redistribution towards higher momenta descried above, except for the
[001] direction. The overall energy correction is still positive in LSDA+U as
seen in the inset of Fig. 7. Negative second moments are obtained along all
directions in DMFT and produce a negative kinetic energy contribution. The
relativistic corrections to the kinetic energy are both negative and we see
that dynamic correlations (LSDA+DMFT) generate larger relativistic corrections
to the one-particle kinetic energy in comparison to their mean field (LSDA+U)
counter part.

As an overall conclusion in the range of the studied values of $U$ qualitative
and quantitative differences are seen in the Compton profiles depending
weather the LSDA is supplemented with static or dynamic many-body effects. An
important message is that relativistic effects and electronic correlations may
have a non-trivial interplay and dynamic correlations determine larger
relativistic corrections in the electronic structure of solids. Further
investigations are necessary for a quantitative assessments of such effects.

##  V acknowledgments

Financial support of the Deutsche Forschungsgemeinschaft through FOR 1346, the
DAAD and the CNCS - UEFISCDI (project number PN-II-ID-PCE-2012-4-0470) is
gratefully acknowledged. JM acknowledge also the CENTEM project, reg. no.
CZ.1.05/2.1.00/03.0088, co-funded by the ERDF as part of the Ministry of
Education, Youth and Sports OP RDI programme.

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# Extended lattice Gelfand-Dickey hierarchy

Kanehisa Takasaki
Department of Mathematics, Kindai University
3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan E-mail:
<EMAIL_ADDRESS>
###### Abstract

The lattice Gelfand-Dickey hierarchy is a lattice version of the Gelfand-
Dickey hierarchy. A special case is the lattice KdV hierarchy. Inspired by
recent work of Buryak and Rossi, we propose an extension of the lattice
Gelfand-Dickey hierarchy. The extended system has an infinite number of
logarithmic flows alongside the usual flows. We present the Lax, Sato and
Hirota equations and a factorization problem of difference operators that
captures the whole set of solutions. The construction of this system resembles
the extended 1D and bigraded Toda hierarchy, but exhibits several novel
features as well.

2010 Mathematics Subject Classification: 14N35, 37K10

Key words: cohomological field theory, Gelfand-Dickey hierarchy, KdV
hierarchy, KP hierarchy, Toda hierarchy, logarithmic flow

##  1 Introduction

Some years ago, Buryak and Rossi [1] pointed out that an extension of the
lattice KdV hierarchy emerges in a variant of cohomological field theory of
Witten’s $r$-spin class. The lattice KdV hierarchy (the “discrete” KdV
hierarchy in the terminology of Buryak and Rossi) is a system of Lax equations
for a difference operator of second order. The extended part comprises
evolution equations with an extra set of time variables and is constructed in
a rather abstract way. Finding a more explicit construction of these extra
flows, as well as a higher spin generalization on the basis of the Gelfand-
Dickey hierarchy, is an open problem raised therein.

In this paper, we present an extension of the lattice Gelfand-Dickey (GD for
short) hierarchy. Although evidence is fragmentary, we believe that this is
the integrable structure sought for by Buryak and Rossi. The lattice GD
hierarchy is a reduction of the lattice KP hierarchy (aka the discrete KP
hierarchy, the modified KP hierarchy, etc.) [2]. Our extended lattice GD
hierarchy is obtained by adding an infinite number of “logarithmic flows” to
the lattice GD hierarchy.

Logarithmic flows have been known for the 1D Toda hierarchy and its bigraded
generalizations. The name reflects the fact that the Lax equations contain the
logarithm of the Lax operator as a building block. These exotic flows were
first discovered in the large-$N$ limit of a matrix model of topological
string theory on $\mathbb{C}\mathbb{P}^{1}$ [3, 4]. The Lax equations were
accordingly formulated in the dispersionless limit of the 1D Toda hierarchy. A
dispersive version was constructed later [5] and further generalized to the
bigraded Toda hierarchy [6]. The meaning of the logarithm of the relevant Lax
operator, which is a difference operator, was also clarified therein.
Moreover, Hirota equations for the tau function in the presence of logarithmic
flows were also subsequently obtained [7, 8, 9]. These extended 1D and
bigraded Toda hierarchies are used to capture the all-genus Gromov-Witten
partition functions of $\mathbb{C}\mathbb{P}^{1}$ [10, 11] and its two-point
orbifolds [12, 13].

The construction of logarithmic flows in the lattice GD hierarchy is mostly
parallel to our previous work [9] on the extended 1D Toda hierarchy. The
somewhat unusual Lax formalism proposed therein turns out to be applicable to
the lattice GD hierarchy as well. Let us stress that the construction of
logarithmic flows by Carlet et al. [5, 6] is not very suited for this purpose.
The bigraded Toda hierarchy is labelled by two positive integers $N$ and
$\bar{N}$. The 1D Toda hierarchy is the case of $N=\bar{N}=1$. The lattice GD
hierarchy amounts to the case of $\bar{N}=0$, which is exceptional in the
perspectives of the bigraded Toda hierarchy.

This paper is organized as follows. Section 2 is a review of the notion of the
lattice KP hierarchy and its reduction to the lattice GD hierarchy. Section 3
presents the construction of the extended lattice GD hierarchy in the language
of the Lax and Sato equations. Section 4 deals with the Hirota equations for
the tau function. Section 5 is devoted to the factorization problem.

##  2 Lattice KP and GD hierarchies

The lattice KP hierarchy may be thought of as a subsystem of the 2D Toda
hierarchy with the second set
$\bar{\boldsymbol{t}}=(\bar{t}_{k})_{k=0}^{\infty}$ of the time variables
$(\boldsymbol{t},\bar{\boldsymbol{t}})$ being turned off to
$\bar{\boldsymbol{t}}=\boldsymbol{0}$ (see the recent review [14] for
generalities of the 2D Toda hierarchy). We now consider an $\hbar$-dependent
version [15] of these hierarchies. Let $s$ be the spatial coordinate therein,
which is now understood to be a continuous variable, and $\Lambda$ be the
shift operator

$\Lambda=e^{\hbar\partial_{s}},\quad\partial_{s}=\partial/\partial s,$

that acts on a function $f(s)$ of $s$ as $\Lambda f(s)=f(s+\hbar)$.

The lattice KP hierarchy comprises the Lax equations

$\hbar\frac{\partial L}{\partial t_{k}}=[B_{k},L]=[L,B_{k}^{-}],\quad
k=1,2,\ldots,$ (1)

for the difference (or pseudo-difference) Lax operator

$L=\Lambda+\sum_{n=1}^{\infty}u_{n}\Lambda^{1-n}.$

The coefficients $u_{n}$ depend on $\hbar$ as well as $s$ and the time
variables $\boldsymbol{t}=(t_{k})_{k=1}^{\infty}$. The generators $B_{k}$ and
$B_{k}^{-}$ of the flows are defined by the Lax operator as

$B_{k}=(L^{k})_{\geq 0},\quad B_{k}^{-}=(L^{k})_{<0},$

where $(\quad)_{\geq 0}$ and $(\quad)_{<0}$ denote the projection

$\left(\sum_{n=-\infty}^{\infty}a_{n}\Lambda^{n}\right)_{\geq 0}=\sum_{n\geq
0}a_{n}\Lambda^{n},\quad\left(\sum_{n=-\infty}^{\infty}a_{n}\Lambda^{n}\right)_{<0}=\sum_{n<0}a_{n}\Lambda^{n}.$

to the non-negative and negative powers of $\Lambda$.

The Lax equations can be converted to the Sato equations

$\hbar\frac{\partial W}{\partial t_{k}}=(W\Lambda^{k}W^{-1})_{\geq
0}W-W\Lambda^{k}=-(W\Lambda^{k}W^{-1})_{<0}W$ (2)

for the dressing operator

$W=1+\sum_{n=1}^{\infty}w_{n}\Lambda^{-n}.$

$L$ and $B_{k}$ are thereby expressed as

$L=W\Lambda W^{-1},\quad B_{k}=(W\Lambda^{k}W^{-1})_{\geq 0}.$

Thus the Sato equations (2) imply the Lax equations (1). Conversely, given a
Lax operator $L$ satisfying (1), one can find a solution $W$ of (2).

The wave function

$\Psi=\left(1+\sum_{n=1}^{\infty}w_{n}z^{-n}\right)z^{s/\hbar}e^{\xi(\boldsymbol{t},z)/\hbar},\quad\xi(\boldsymbol{t},z)=\sum_{k=1}^{\infty}t_{k}z^{k},$

satisfies the auxiliary linear equations

$\hbar\frac{\partial\Psi}{\partial t_{k}}=B_{k}\Psi,\quad L\Psi=z\Psi.$ (3)

The amplitude part of $\Psi$ can be expressed by the tau function
$\tau=\tau(\hbar,s,\boldsymbol{t})$ as

$1+\sum_{n=1}^{\infty}w_{n}z^{-n}=\frac{\tau(\hbar,s,\boldsymbol{t}-\hbar[z^{-1}])}{\tau(\hbar,s,\boldsymbol{t})},\quad[z]=\left(\frac{z^{k}}{k}\right)_{k=1}^{\infty}.$

Reduction to the lattice GD hierarchy [2] is induced by the reduction
condition

$(L^{N})_{<0}=0.$ (4)

The reduced Lax operator

$\mathfrak{L}=L^{N}=B_{N}=\Lambda^{N}+b_{1}\Lambda^{N-1}+\cdots+b_{N}$ (5)

satisfies the Lax equations

$\hbar\frac{\partial\mathfrak{L}}{\partial
t_{k}}=[B_{k},\mathfrak{L}]=[\mathfrak{L},B_{k}^{-}],\quad k=1,2,\ldots,$ (6)

of the lattice GD hierarchy. $B_{k}$ and $B_{k}^{-}$ can be rewritten in terms
of fractional powers of $\mathfrak{L}$ as

$B_{k}=(\mathfrak{L}^{k/N})_{\geq 0},\quad
B_{k}^{-}=(\mathfrak{L}^{k/N})_{<0}.$

The $N=2$ case is the lattice KdV hierarchy. The existence of two expressions
(6) of the Lax equations ensures that the reduced form (5) of $\mathfrak{L}$
is preserved by the flows. The flows with respect to $t_{kN}$’s become
trivial, i.e.,

$\frac{\partial\mathfrak{L}}{\partial t_{kN}}=0,\quad k=1,2,\ldots.$ (7)

The Sato equations persist to take the same form

$\hbar\frac{\partial W}{\partial t_{k}}=B_{k}W-W\Lambda^{k}=-B_{k}^{-}W$ (8)

in the reduction. The associated auxiliary linear equations read

$\hbar\frac{\partial\Psi}{\partial
t_{k}}=B_{k}\Psi,\quad\mathfrak{L}\Psi=z^{N}\Psi.$ (9)

######  Remark 1.

The usual KP and GD hierarchies are hidden in the lattice KP and GD
hierarchies. The first time variable $t_{1}$ plays the role of spatial
coordinate therein. Note that the lowest auxiliary linear equation

$\hbar\frac{\partial\Psi}{\partial t_{1}}=B_{1}\Psi,\quad B_{1}=\Lambda+b,$

implies the relation

$\Lambda\Psi=(\hbar\partial-b)\Psi,\quad\partial=\partial/\partial t_{1}.$
(10)

One can thereby rewrite the difference operators in the auxiliary linear
equations (3) and (9) into differential or pseudo-differential operators with
respect to $t_{1}$. In particular, the linear difference equation
$\mathfrak{L}\Psi=z^{k}\Psi$ can be thus converted to a linear differential
equation of the form

$Q\Psi=z^{N}\Psi,\quad
Q=(\hbar\partial)^{N}+a_{2}(\hbar\partial)^{N-2}+\cdots+a_{N},$ (11)

which may be thought of as the eigenvalue problem of the GD hierarchy. The Lax
equations (6) turn into Lax equations of the form

$\hbar\frac{\partial Q}{\partial t_{k}}=[A_{k},Q],\quad k=1,2,\ldots.$ (12)

$A_{k}$’s are differential operators obtained from $B_{k}$’s just as $Q$ is
converted from $\mathfrak{L}$. This is exactly the usual GD hierarchy, in
which the variable $s$ is left as a parameter. Shifting $s$ as $s\to s+\hbar$
amounts to Bäcklund-Darboux transformations of the GD hierarchy as the
relation (10) suggests.

##  3 Extended lattice GD hierarchy

We now extend the lattice GD hierarchy by an infinite number of logarithmic
flows. Let $\boldsymbol{x}=(x_{k})_{k=1}^{\infty}$ be the set of time
variables of these flows.

The Lax equations of the extended flows take seemingly the same form

$\hbar\frac{\partial\mathfrak{L}}{\partial
x_{k}}=\mathfrak{L}^{k}\hbar\frac{\partial\mathfrak{L}}{\partial
s}+[P_{k},\mathfrak{L}]=[\mathfrak{L}^{k}\hbar\partial_{s}+P_{k},\mathfrak{L}],\quad
k=1,2,\ldots,$ (13)

as those of the extended 1D/bigraded Toda hierarchy [9]. The structure of
$P_{k}$’s, however, is different from that case. In the present setup,
$P_{k}$’s are difference operators of the form

$P_{k}=-\left(\mathfrak{L}^{k}\hbar\frac{\partial W}{\partial
s}W^{-1}\right)_{\geq 0},$

thus having no negative powers of $\Lambda$. These extended Lax equations and
the foregoing Lax equations (6) constitute the extended lattice GD hierarchy.

These Lax equations can be converted to the Sato equations

$\hbar\frac{\partial W}{\partial x_{k}}=\mathfrak{L}^{k}\hbar\frac{\partial
W}{\partial
s}+P_{k}W=(\mathfrak{L}^{k}\hbar\partial_{s}+P_{k})W-W\Lambda^{kN}\hbar\partial_{s}$
(14)

for the dressing operator $W$. The associated wave function is redefined as

$\Psi=\left(1+\sum_{n=1}^{\infty}w_{n}z^{-n}\right)z^{s/\hbar}\exp\left(\hbar^{-1}\xi(\boldsymbol{t},z)+\hbar^{-1}\xi(\boldsymbol{x},z^{N})\log
z\right).$

The undressed wave function

$\Psi_{0}=z^{s/\hbar}\exp\left(\hbar^{-1}\xi(\boldsymbol{t},z)+\hbar^{-1}\xi(\boldsymbol{x},z^{N})\log
z\right)$

satisfies the linear equations

$\hbar\frac{\partial\Psi_{0}}{\partial
t_{k}}=\Lambda^{k}\Psi_{0},\quad\hbar\frac{\partial\Psi_{0}}{\partial
x_{k}}=\Lambda^{kN}\hbar\partial_{s}\Psi_{0}.$ (15)

These equations are transformed, via the dressing relation $\Psi=W\Psi_{0}$,
to the linear equations

$\hbar\frac{\partial\Psi}{\partial
t_{k}}=B_{k}\Psi,\quad\hbar\frac{\partial\Psi}{\partial
x_{k}}=(\mathfrak{L}^{k}\hbar\partial_{s}+P_{k})\Psi$ (16)

for the dressed wave function$\Psi$. These equations and the eigenvalue
problem $\mathfrak{L}\Psi=z^{N}\Psi$ give the auxiliary linear equations of
the extended lattice GD hierarchy.

The presence of the logarithmic factors $\log z$ and
$\log\Lambda=\hbar\partial_{s}$ in the wave functions and the auxiliary linear
equations explains the name “logarithmic flows”. Moreover the logarithm

$\log L=W\log\Lambda W^{-1}=W\hbar\partial_{s}W^{-1}$

of the Lax operator $L$, too, plays a role as follows.

One can rewrite the product $L^{kN}\log L$ of $L^{kN}$ and $\log L$as

$L^{kN}\log
L=W\Lambda^{kN}\hbar\partial_{s}W^{-1}=\mathfrak{L}^{k}\hbar\partial_{s}-\mathfrak{L}^{k}\hbar\frac{\partial
W}{\partial s}W^{-1},$ (17)

which splits into two pieces as

$L^{kN}\log L=(\mathfrak{L}^{k}\hbar\partial_{s}+P_{k})+P_{k}^{-},$ (18)

where

$P_{k}^{-}=-\left(\mathfrak{L}^{k}\hbar\frac{\partial W}{\partial
s}W^{-1}\right)_{<0}.$

The first piece $\mathfrak{L}^{k}\hbar\partial_{s}+P_{k}$ is the generator of
the $x_{k}$-flow in the Lax equation (13) and the Sato equation (14). Since
$L^{kN}\log L$ commute with $\mathfrak{L}$, one can rewrite the Lax equations
as

$\hbar\frac{\partial\mathfrak{L}}{\partial x_{k}}=[\mathfrak{L},P_{k}^{-}].$
(19)

In the same sense, the Sato equation turns out to have another expression as

$\hbar\frac{\partial W}{\partial x_{k}}=-P_{k}^{-}W.$ (20)

The operators $\mathfrak{L}^{k}\hbar\partial_{s}+P_{k}$ and $P_{k}^{-}$ may be
thought of as the $(\quad)_{\geq 0}$ and $(\quad)_{<0}$ parts of
$\mathfrak{L}^{k}\log L=L^{kN}\log L$ if these projectors are extended to
operators of the form $A\partial_{s}+B$, where $A$ and $B$ are genuine
difference operators, as

$(A\partial_{s}+B)_{\geq 0}=(A)_{\geq 0}\partial_{s}+(B)_{\geq
0},\quad(A\partial_{s}+B)_{<0}=(A)_{<0}\partial_{s}+(B)_{<0}.$

Although being logically problematical, such an interpretation seems to be
useful for the extended 1D/bigraded Toda hierarchies as well.

######  Remark 2.

The Lax operator of the bigraded Toda hierarchy of type $(N,\bar{N})$ is a
difference operator of the form

$\mathfrak{L}=\Lambda^{N}+b_{1}\Lambda^{N-1}+\cdots+b_{N+\bar{N}}\Lambda^{-\bar{N}},\quad
b_{N+\bar{N}}\not=0,$

where $N$ and $\bar{N}$ are positive integers. The 1D Toda hierarchy is the
case of $N=\bar{N}=1$. On the other hand, the Lax operator (5) of the lattice
GD hierarchy amounts to the case of $\bar{N}=0$. Carlet’s construction of
logarithmic flows [6] is based on the averaged (or subtracted) logarithm

$\mathrm{Log}\mathfrak{L}=\frac{1}{2N}\log_{+}\mathfrak{L}+\frac{1}{2\bar{N}}\log_{-}\mathfrak{L}=\frac{1}{2}W\log\Lambda
W^{-1}-\frac{1}{2}\bar{W}\log\Lambda\bar{W}^{-1}$

of $\mathfrak{L}$, where $\bar{W}$ is the second dressing operator of the 2D
Toda hierarchy, and does not work in the $\bar{N}=0$ case. To overcome this
difficulty, we have adopted our previous construction of logarithmic flows [9]
with suitable modifications. Let us note that our construction for the
$\bar{N}>0$ case is equivalent to Carlet’s construction.

######  Remark 3.

The building blocks of the Lax, Sato and auxiliary linear equations of the
$\boldsymbol{x}$-flows have the common factor $\hbar$. Although this factor
could be removed from these equations, we dare to keep it for symmetry with
the equations of the $\boldsymbol{t}$-flows.

######  Remark 4.

The dual expressions (13) and (19) of the Lax equations imply that the reduced
form (5) of the Lax operator is preserved by the $\boldsymbol{x}$-flows as
well as the $\boldsymbol{t}$-flows. Moreover one can show, in much the same
way as in the case of the extended bigraded Toda hierarchy [6], that the Lax
equations imply the zero-curvature equations

$\displaystyle\hbar\frac{\partial B_{k}}{\partial t_{l}}-\hbar\frac{\partial
B_{l}}{\partial t_{k}}+[B_{k},B_{l}]$ $\displaystyle=0,$ (21)
$\displaystyle\hbar\frac{\partial B_{k}}{\partial x_{l}}-\hbar\frac{\partial
C_{l}}{\partial t_{k}}+[B_{k},C_{l}]$ $\displaystyle=0,$ (22)
$\displaystyle\hbar\frac{\partial C_{k}}{\partial x_{l}}-\hbar\frac{\partial
C_{l}}{\partial x_{k}}+[C_{k},C_{l}]$ $\displaystyle=0,\quad k,l=1,2,\ldots,$
(23)

for $B_{k}$ and $C_{k}=\mathfrak{L}^{k}\hbar\partial_{s}+P_{k}$. These zero-
curvature equations ensure the consistency of the Lax equations.

######  Remark 5.

Equating the coefficients of powers of $\Lambda$ in both hand sides of the Lax
equations (6) and (13), one can derive a system of evolution equations for the
coefficients $b_{1},\cdots,b_{N}$ of $\mathfrak{L}$. In particular, the
$\Lambda^{0}$ terms give the evolution equations

$\frac{\partial b_{N}}{\partial t_{k}}=0,\quad\frac{\partial b_{N}}{\partial
x_{k}}={b_{N}}^{k}\frac{\partial b_{N}}{\partial s},\quad k=1,2,\ldots,$ (24)

for $b_{N}$. These equations in the case of $N=2$ agree, up to rescaling of
the time variables, with Buryak and Rossi’s result [1, Theorem4.2]. This is a
main piece of evidence suggesting that our extended lattice KdV hierarchy will
be the integrable structure sought for therein.

##  4 Hirota equations

As in the case of the extended 1D/bigraded Toda hierarchy [8, 9], Hirota
equations for the tau function can be derived from bilinear equations for the
wave functions. Let us first consider those bilinear equations for the wave
functions of the extended lattice GD hierarchy.

It is convenient to start from the bilinear equations

$\oint\frac{dz}{2\pi
i}\Psi(s,\boldsymbol{t},\boldsymbol{x},z)\Psi^{*}(s^{\prime},\boldsymbol{t}^{\prime},\boldsymbol{x},z)=0$
(25)

of the lattice KP hierarchy that holds under the condition

$(s-s^{\prime})/\hbar\in\mathbb{Z}_{\geq 0}=\\{0,1,2,\ldots\\}$ (26)

and for two independent copies $\boldsymbol{t}=(t_{k})_{k=1}^{\infty}$ and
$\boldsymbol{t}^{\prime}=(t^{\prime}_{k})_{k=1}^{\infty}$ of the
$\boldsymbol{t}$-variables. $\oint dz/2\pi i$ denotes the residue

$\oint\frac{dz}{2\pi i}\sum_{n=-\infty}^{\infty}f_{n}z^{n}=f_{-1}$

of formal Laurent series of $z$.
$\Psi^{*}=\Psi^{*}(s,\boldsymbol{t},\boldsymbol{x},z)$ is the dual wave
function

$\Psi^{*}=\left(1+\sum_{n=1}^{\infty}w^{*}_{n}z^{-n}\right)z^{-s/\hbar}\exp\left(-\hbar^{-1}\xi(\boldsymbol{t},z)-\hbar^{-1}\xi(\boldsymbol{x},z^{N})\log
z\right)$

defined in the same way as the dual wave function of the 2D Toda hierarchy.
Namely, the amplitude part is determined by the formal adjoint

$W^{*}=1+\sum_{n=1}^{\infty}\Lambda^{n}\cdot
w_{n}=1+\sum_{n=1}^{\infty}w_{n}|_{s\to s+n\hbar}\Lambda^{n}$

of $W$ as

$1+\sum_{n=1}^{\infty}w^{*}_{n}\Lambda^{n}=\Lambda^{-1}(W^{*})^{-1}\Lambda.$

(25) is actually the bilinear equations of the 2D Toda hierarchy restricted to
the range (26) of $s$ and $s^{\prime}$.

In the reduction to the lattice GD hierarchy, the auxiliary linear equations
for $t_{kN}$, $k=1,2,\ldots$, trivialize as

$\hbar\frac{\partial\Psi}{\partial t_{kN}}=\mathfrak{L}^{k}\Psi=z^{kN}\Psi.$

Accordingly, one can insert arbitrary positive powers of $z^{N}$ into the
bilinear equation (25) as

$\oint\frac{dz}{2\pi
i}z^{mN}\Psi(s,\boldsymbol{t},\boldsymbol{x},z)\Psi^{*}(s^{\prime},\boldsymbol{t}^{\prime},\boldsymbol{x},z)=0.$
(27)

These are bilinear equations that characterize the lattice GD hierarchy.

We can now use the technique developed for the extended 1D/bigraded Toda
hierarchy [9] to derive the following bilinear equation for the wave functions
of the extended lattice GD hierarchy.

######  Proposition 1.

The bilinear equation

$\oint\frac{dz}{2\pi
i}z^{mN}\Psi(s-\xi(\boldsymbol{\alpha},z^{N}),\boldsymbol{t},\boldsymbol{x}+\boldsymbol{\alpha},z)\Psi^{*}(s^{\prime}-\xi(\boldsymbol{\beta},z^{N}),\boldsymbol{t}^{\prime},\boldsymbol{x}+\boldsymbol{\beta},z)=0$
(28)

holds for $m,(s-s^{\prime})/\hbar\in\mathbb{Z}$ and two sets
$\boldsymbol{\alpha}=(\alpha_{k})_{k=1}^{\infty}$,
$\boldsymbol{\beta}=(\beta_{k})_{k=1}^{\infty}$ of arbitrary constants.

###### Proof.

One can rewrite the auxiliary linear equations with respect to $x_{k}$ in (16)
as

$\hbar\left(\frac{\partial}{\partial x_{k}}-z^{kN}\frac{\partial}{\partial
s}\right)\Psi=Q_{k}\Psi,$

where

$Q_{k}=P_{k}-\hbar\frac{\partial\mathfrak{L}^{k}}{\partial s}.$

These equations can be extended to higher orders as

$\prod_{k\geq 1}\left(\frac{\partial}{\partial
x_{k}}-z^{kN}\frac{\partial}{\partial
s}\right)^{m_{k}}\Psi=Q_{m_{1},m_{2},\ldots}\Psi,\quad
m_{1},m_{2},\ldots\in\mathbb{Z}_{\geq 0},$

where $Q_{m_{1},m_{2},\ldots}$’s are difference operators without negative
powers of $\Lambda$. Now let $Q_{m_{1},m_{2},\ldots}$ act on both sides of the
bilinear equation (27). Since this difference operator consists of only non-
negative powers of $\Lambda$, the outcome is a linear combination of (27) with
$m$ being shifted to the positive direction. Consequently, the bilinear
equation persists to hold under the action of $Q_{m_{1},m_{2},\ldots}$ as

$\oint\frac{dz}{2\pi
i}z^{mN}Q_{m_{1},m_{2},\ldots}\Psi(s,\boldsymbol{t},\boldsymbol{x},z)\cdot\Psi^{*}(s^{\prime},\boldsymbol{t}^{\prime},\boldsymbol{x},z)=0,$

equivalently,

$\oint\frac{dz}{2\pi i}z^{mN}\prod_{k\geq 1}\left(\frac{\partial}{\partial
x_{k}}-z^{kN}\frac{\partial}{\partial
s}\right)^{m_{k}}\Psi(s,\boldsymbol{t},\boldsymbol{x},z)\cdot\Psi^{*}(s^{\prime},\boldsymbol{t}^{\prime},\boldsymbol{x},z)=0.$

These equations can be packed into a generating functional form with the
auxiliary variables $\alpha_{1},\alpha_{2},\ldots$ as

$\oint\frac{dz}{2\pi
i}z^{mN}\exp\left(\sum_{k=1}^{\infty}\alpha_{k}\left(\frac{\partial}{\partial
x_{k}}-z^{kN}\frac{\partial}{\partial
s}\right)\right)\Psi(s,\boldsymbol{t},\boldsymbol{x},z)\cdot\Psi^{*}(s^{\prime},\boldsymbol{t}^{\prime},\boldsymbol{x},z)=0,$

i.e.,

$\oint\frac{dz}{2\pi
i}z^{mN}\Psi(s-\xi(\boldsymbol{\alpha},z^{N}),\boldsymbol{t},\boldsymbol{x}+\boldsymbol{\alpha},z)\Psi^{*}(s^{\prime},\boldsymbol{t}^{\prime},\boldsymbol{x},z)=0.$

Note that the last equation, too, holds for
$m,(s-s^{\prime})/\hbar\in\mathbb{Z}_{\geq 0}$. Therefore one can apply the
shift operator

$\exp\left(-\sum_{k=1}^{\infty}\beta_{k}z^{kN}(\partial_{s}+\partial_{s^{\prime}})^{k}\right)$

to the integrand to obtain the further deformed bilinear equation

$\oint\frac{dz}{2\pi
i}z^{mN}\Psi(s-\xi(\boldsymbol{\alpha}+\boldsymbol{\beta},z^{N}),\boldsymbol{t},\boldsymbol{x}+\boldsymbol{\alpha},z)\Psi^{*}(s^{\prime}-\xi(\boldsymbol{\beta},z^{N}),\boldsymbol{t}^{\prime},\boldsymbol{x},z)=0.$

This equation turns into (28) by substituting
$\boldsymbol{\alpha}\to\boldsymbol{\alpha}-\boldsymbol{\beta}$ and
$\boldsymbol{x}\to\boldsymbol{x}+\boldsymbol{\beta}$. ∎

Lastly, we rewrite the bilinear equation (28) in the language of the tau
function. As in the case of the extended 1D/bigraded Toda hierarchy [8, 9],
the wave functions are expressed by the tau function
$\tau=\tau(\hbar,s,\boldsymbol{t},\boldsymbol{x})$ as

$\displaystyle\Psi$
$\displaystyle=\frac{\tau(\hbar,s,\boldsymbol{t}-\hbar[z^{-1}],\boldsymbol{x})}{\tau(\hbar,s,\boldsymbol{t},\boldsymbol{x})}z^{s/\hbar}\exp\left(\hbar^{-1}\xi(\boldsymbol{t},z)+\hbar^{-1}\xi(\boldsymbol{x},z^{N})\log
z\right),$ $\displaystyle\Psi^{*}$
$\displaystyle=\frac{\tau(\hbar,s,\boldsymbol{t}+\hbar[z^{-1}],\boldsymbol{x})}{\tau(\hbar,s,\boldsymbol{t},\boldsymbol{x})}z^{-s/\hbar}\exp\left(-\hbar^{-1}\xi(\boldsymbol{t},z)-\hbar^{-1}\xi(\boldsymbol{x},z^{N})\log
z\right).$

Plugging these expressions into (28), we obtain the bilinear equation

$\displaystyle\oint\frac{dz}{2\pi
i}z^{mN+(s-s^{\prime})/\hbar}e^{\xi(\boldsymbol{t}-\boldsymbol{t}^{\prime},z)/\hbar}\tau(\hbar,s-\xi(\boldsymbol{\alpha},z^{N}),\boldsymbol{t}-\hbar[z^{-1}],\boldsymbol{x}+\boldsymbol{\alpha})$
$\displaystyle\qquad\mbox{}\times\tau(\hbar,s^{\prime}-\xi(\boldsymbol{\beta},z^{N}),\boldsymbol{t}^{\prime}+\hbar[z^{-1}],\boldsymbol{x}+\boldsymbol{\beta})=0$
(29)

that holds for $m,(s-s^{\prime})/\hbar\in\mathbb{Z}_{\geq 0}$. Note that the
factors
$\tau(\hbar,s-\xi(\boldsymbol{\alpha},z^{N}),\boldsymbol{t},\boldsymbol{x}+\boldsymbol{\alpha})$
and
$\tau(\hbar,s^{\prime}-\xi(\boldsymbol{\beta},z^{N}),\boldsymbol{t}^{\prime},\boldsymbol{x}+\beta)$
arising in the denominator has been removed by a simple trick [9, Section
2.4]. The bilinear equation (29) is a generating functional expression of an
infinite number of Hirota equations.

##  5 Factorization problem

We now turn to an analogue of the factorization problem proposed for solving
the extended 1D/bigraded Toda hierarchy [16]. Given an invertible difference
operator $U$ of infinite order that depends on the spatial coordinate and the
time variables in a particular manner, the problem is to factorize $U$ into
two difference operators of the form

$W=1+\sum_{n=1}^{\infty}w_{n}\Lambda^{-n},\quad\bar{W}=\sum_{n=0}^{\infty}\bar{w}_{n}\Lambda^{n}$

as

$U=W^{-1}\bar{W}.$ (30)

In the case of the extended lattice GD hierarchy, $U$ is assumed to satisfy
the differential equations

$\hbar\frac{\partial U}{\partial t_{k}}=\Lambda^{k}U,\quad\frac{\partial
U}{\partial x_{k}}=\Lambda^{kN}\frac{\partial U}{\partial s}$ (31)

and the algebraic relation

$\Lambda^{N}U=U\varphi(\Lambda),$ (32)

where $\varphi(\Lambda)$ is a difference operator of the form

$\varphi(\Lambda)=\sum_{n=0}^{\infty}\varphi_{n}\Lambda^{n}.$

######  Proposition 2.

If $W$ and $\bar{W}$ solve the factorization problem (30), $W$ is the dressing
operator of a solution of the extended lattice GD hierarchy. Namely,
$\mathfrak{L}=W\Lambda^{N}W^{-1}$ takes the reduced form (5), and $W$
satisfies the Sato equations (8) and (14). Moreover, $\bar{W}$ satisfies the
Sato equations

$\hbar\frac{\partial\bar{W}}{\partial
t_{k}}=B_{k}\bar{W},\quad\hbar\frac{\partial\bar{W}}{\partial
x_{k}}=\mathfrak{L}^{k}\hbar\frac{\partial\bar{W}}{\partial s}+P_{k}\bar{W}$
(33)

of a slightly different form.

###### Proof.

Substituting $U=W^{-1}\bar{W}$ in (32) yields the algebraic relation

$\mathfrak{L}=W\Lambda^{N}W^{-1}=\bar{W}\varphi(\Lambda)\bar{W}^{-1}.$ (34)

Since the right hand side does not contain negative powers of $\Lambda$,
$\mathfrak{L}$ turns out to take the reduced form (5). In the same way, (31)
yields the equations

$\hbar\frac{\partial W}{\partial
t_{k}}W^{-1}+W\Lambda^{k}W^{-1}=\hbar\frac{\partial\bar{W}}{\partial
t_{k}}\bar{W}^{-1}$

and

$\hbar\frac{\partial W}{\partial
x_{k}}W^{-1}-W\Lambda^{kN}W^{-1}\hbar\frac{\partial W}{\partial
s}W^{-1}=\hbar\frac{\partial\bar{W}}{\partial
x_{k}}\bar{W}^{-1}-W\Lambda^{kN}W^{-1}\hbar\frac{\partial\bar{W}}{\partial
s}\bar{W}^{-1},$

which one can rewrite as

$\hbar\frac{\partial W}{\partial
t_{k}}W^{-1}+\mathfrak{L}^{k/N}=\hbar\frac{\partial\bar{W}}{\partial
t_{k}}\bar{W}^{-1}$

and

$\hbar\frac{\partial W}{\partial
x_{k}}W^{-1}-\mathfrak{L}^{k}\hbar\frac{\partial W}{\partial
s}W^{-1}=\hbar\frac{\partial\bar{W}}{\partial
x_{k}}\bar{W}^{-1}-\mathfrak{L}^{k}\hbar\frac{\partial\bar{W}}{\partial
s}\bar{W}^{-1}.$

Let $B_{k}$ and $P_{k}$ be the difference operators defined by both hand sides
of these equations. Since the right hand side have no $(\cdots)_{<0}$ part,
$B_{k}$ and $P_{k}$ are equal to the $(\cdots)_{\geq 0}$ part of the left hand
side. One can thus identify these operators as

$B_{k}=\left(\hbar\frac{\partial W}{\partial
t_{k}}W^{-1}+\mathfrak{L}^{k/N}\right)_{\geq 0}=(\mathfrak{L}^{k/N})_{\geq 0}$

and

$P_{k}=\left(\hbar\frac{\partial W}{\partial
x_{k}}W^{-1}-\mathfrak{L}^{k}\hbar\frac{\partial W}{\partial
s}W^{-1}\right)_{\geq 0}=-\left(\mathfrak{L}^{k}\hbar\frac{\partial
W}{\partial s}W^{-1}\right)_{\geq 0}$

and obtain (8) and (14). ∎

Although looking arbitrarily given, the operator $\varphi(\Lambda)$ in the
algebraic relation (32) has to satisfy a set of differential equations. This
is a consequence of the differential equations (31) for $U$.

######  Proposition 3.

$\frac{\partial\varphi(\Lambda)}{\partial
t_{k}}=0,\quad\frac{\partial\varphi(\Lambda)}{\partial
x_{k}}=\varphi(\Lambda)^{k}\frac{\partial\varphi(\Lambda)}{\partial s},\quad
k=1,2,\ldots.$ (35)

###### Proof.

Differentiating both hand sides of (32) by $t_{k}$ gives

$\Lambda^{N}\hbar\frac{\partial U}{\partial t_{k}}=\hbar\frac{\partial
U}{\partial
t_{k}}\varphi(\Lambda)+U\hbar\frac{\partial\varphi(\Lambda)}{\partial t_{k}}.$

By (31), this equation turns into

$\Lambda^{N}U\varphi(\Lambda)=\Lambda^{N}U\varphi(\Lambda)+U\hbar\frac{\partial\varphi(\Lambda)}{\partial
t_{k}},$

which implies the first equation of (35). One can derive the second equation
of (35) by a similar reasoning as follows. Differentiating both hand sides of
(32) by $x_{k}$ and $s$ yields the two equations

$\Lambda^{N}\frac{\partial U}{\partial x_{k}}=\frac{\partial U}{\partial
x_{k}}\varphi(\Lambda)+U\frac{\partial\varphi(\Lambda)}{\partial
x_{k}},\quad\Lambda^{N}\frac{\partial U}{\partial s}=\frac{\partial
U}{\partial s}\varphi(\Lambda)+U\frac{\partial\varphi(\Lambda)}{\partial s}.$

By (31), the first equation can be rewritten as

$\Lambda^{N+kN}\frac{\partial U}{\partial s}=\Lambda^{kN}\frac{\partial
U}{\partial s}\varphi(\Lambda)+U\frac{\partial\varphi(\Lambda)}{\partial
x_{k}}.$

The second equation, multiplied with $\Lambda^{kN}$, becomes

$\Lambda^{N+kN}\frac{\partial U}{\partial s}=\Lambda^{kN}\frac{\partial
U}{\partial
s}\varphi(\Lambda)+\Lambda^{kN}U\frac{\partial\varphi(\Lambda)}{\partial s}.$

Subtracting this equation from the preceding equation and using (32), one
obtains the second equation of (35). ∎

Let us stress that the process from the factorization problem (30) to the
solution of the extended lattice GD hierarchy is reversible. Namely, given a
solution of the extended lattice GD hierarchy, one can reconstruct the
operators $U$ and $\varphi(\Lambda)$ as

$U=W^{-1}\bar{W},\quad\varphi(\Lambda)=\bar{W}^{-1}\mathfrak{L}\bar{W},$

which turn out to satisfy the differential equation (31) and the algebraic
condition (32). Thus the factorization problem can capture all solutions of
the extended lattice GD hierarchy.

######  Remark 6.

The $\Lambda^{0}$ part of (35) gives the equations

$\frac{\partial\varphi_{0}}{\partial
t_{k}}=0,\quad\frac{\partial\varphi_{0}}{\partial
x_{k}}={\varphi_{0}}^{k}\frac{\partial\varphi_{0}}{\partial s},\quad
k=1,2,\ldots.$ (36)

This is in accord with the equation (24) for $b_{N}$, because $b_{N}$ is equal
to $\varphi_{0}$ as (34) implies.

######  Remark 7.

If the coefficients $\varphi_{n}$ of $\varphi(\Lambda)$ are quasi-constants
(i.e., $\hbar$-periodic) with respect to $s$, $\varphi(\Lambda)$ commutes with
$\Lambda$, and the differential equations (31) can be reduced to the
differential equations

$\frac{\partial\varphi(z)}{\partial
t_{k}}=0,\quad\frac{\partial\varphi(z)}{\partial
x_{k}}=\varphi(z)^{k}\frac{\partial\varphi(z)}{\partial s},\quad k=1,2,\ldots$
(37)

for the generating function

$\varphi(z)=\sum_{n=0}^{\infty}\varphi_{n}z^{n}.$

Moreover, the algebraic relation (34) becomes the relation

$\mathfrak{L}=L^{N}=\varphi(\bar{L})=\sum_{n=0}^{\infty}\varphi_{n}\bar{L}^{n}$
(38)

connecting $L$ to the second Lax operator

$\bar{L}=\bar{W}\Lambda\bar{W}^{-1}.$

of the 2D Toda hierarchy hidden behind.

######  Remark 8.

Algebraic relations of the form (38) are commonly used as a reduction
condition in the 2D Toda hierarchy. The coefficients of such a reduction
condition, however, are constants. For instance, the bigraded Toda hierarchy
of type $(N,\bar{N})$ is induced by the reduction condition

$L^{N}=\bar{L}^{-\bar{N}}.$

In contrast, the coefficients $\varphi_{n}$ of (38), as well as those of (32)
and (34), are allowed to depend on the time variables of the logarithmic
flows. This is a remarkable feature of the logarithmic flows in the extended
lattice GD hierarchy.

##  6 Conclusion

We have constructed an extension of the lattice GD hierarchy with an infinite
number of logarithmic flows. This system is not a special case of the extended
bigraded Toda hierarchy [6] (even if the second set of the time variable
$(\boldsymbol{t},\bar{\boldsymbol{t}},\boldsymbol{x})$ therein is turned off
to $\bar{\boldsymbol{t}}=\boldsymbol{0}$). It is meaningless to let
$\bar{N}=0$ naively in the construction of the extended bigraded Toda
hierarchy of type $(N,\bar{N})$. Therefore we return to the slightly different
Lax formalism proposed in our previous work [9], and modify the generators
$P_{k}$ of logarithmic flows to fit into the $\bar{N}=0$ case. This leads to a
consistent system of Lax, Sato and Hirota equations as well as a factorization
problem that captures the whole set of solutions of this system.

The extended lattice GD hierarchy exhibits novel features. Firstly, the last
term $b_{N}$ of the Lax operator $\mathfrak{L}$ satisfies the simple evolution
equations (24) that are closed within themselves. These equations will
correspond to the extended sector of cohomological field theory studied by
Buryak and Rossi [1]. Secondly, the operator $U$ in the factorization problem
(30) is required to satisfy the algebraic condition (32) of an unusual form.
In particular, the coefficients $\varphi_{n}$ of the operator
$\varphi(\Lambda)$ on the right hand side are allowed to depend on the time
variables of the logarithmic flows. Last but not least, the structure of the
Lax and Hirota equations is different from the extended bigraded Toda
hierarchy.

### Acknowledgements

This work is partly supported by the JSPS Kakenhi Grant JP18K03350 and
JP21K03261.

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