and $b$. The wrong but crucial claim in the paper states that the probability
for two colors $x$ and $x^{\prime}$ to be hashed to the same color over the
randomness of $a$ and $b$, i.e., the probability of the event that
$h_{a,b}(x)=h_{a,b}(x^{\prime})$ holds, is at most $1/N$. But colors $x$ and
$x^{\prime}$ are mapped to the same color whenever $x=x^{\prime}\bmod p$,
regardless of the choice of $a$ and $b$. In contrast (besides other changes in
the design) in the core part of Lemma 10.5 we fix one distinct polynomial
$h_{x}(y)$ per color $x$ and evaluate it at a randomly chosen
$y\in\mathbb{F}_{y}$ to obtain a color in the smaller color space $[p]$. The
$O(\log\Delta+\log^{6}\log n)$ algorithm from [GGR21] uses the techniques of
[Gha19] in a black box manner, including the erroneous color space reduction.

It is interesting that the color space dependence also plays a role for
deterministic algorithms whose complexity is expressed as
$f(\Delta)+O(\log^{*}n)$; see the discussion in [MT20] that compares the color
space dependent results [MT20] with the related color space independent
results in [FHK16].

#### Acknowledgements.

This project was supported by the European Union’s Horizon 2020 Research and
Innovation Programme under grant agreement 755839 and by the Icelandic
Research Fund grants 174484 and 217965.

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##  Appendix A Concentration Bounds

We use the following variant of Chernoff bounds for dependent random
variables, that is obtained, e.g., as a corollary of Lemma 1.8.7 and Thms.
1.10.1 and 1.10.5 in [DN20].

######  Lemma A.1 (Generalized Chernoff).

Let $X_{1},\dots,X_{r}$ be binary random variables, and $X=\sum_{i}X_{i}$.

  1. 1.

If $Pr[X_{i}=1\mid X_{1}=x_{1},\dots,X_{i-1}=x_{i-1}]\leq q$, for all
$i\in[r]$ and $x_{1},\dots,x_{i-1}\in\\{0,1\\}$ with
$Pr[X_{1}=x_{1},\dots,X_{r}=x_{i-1}]>0$, then for any $\delta\in(0,1)$,

$Pr[X\leq(1-\delta)qr]\leq\exp(-\delta^{2}qr/2)\ .$ (6)

  2. 2.

If $Pr[X_{i}=1\mid X_{1}=x_{1},\dots,X_{i-1}=x_{i-1}]\geq q$, for all
$i\in[r]$ and $x_{1},\dots,x_{i-1}\in\\{0,1\\}$ with
$Pr[X_{1}=x_{1},\dots,X_{r}=x_{i-1}]>0$, then for any $\delta>0$,

$Pr[X\geq(1+\delta)qr]\leq\exp(-\min(\delta^{2},\delta)qr/3)\ .$ (7)

We will often use the following simple corollary of bounds (6-7).

######  Lemma A.2.

Let $S$ be a randomly sampled subset of vertices in $G$, each sampled
independently, with probability $q$. Let $\gamma\in(0,1)$ and $\alpha>0$ be
such that $\gamma^{2}\alpha q\Delta\geq c\log n$, for a sufficiently large
constant $c>1$. For every subset $T\subseteq V(G)$ of vertices of size
$|T|=\alpha\Delta$, we have $(1-\gamma)q\alpha\Delta\leq|T\cap
S|\leq(1+\gamma)q\alpha\Delta$, w.h.p.

We will also use the following variant of Talagrand’s inequality. A function
$f(x_{1},\dots,x_{n})$ is called _$c$ -Lipschitz_ iff changing any single
$x_{i}$ can affect the value of $f$ by at most $c$. Additionally, $f$ is
called _$r$ -certifiable_ iff whenever $f(x_{1},\dots,x_{n})\geq s$, there
exists at most $rs$ variables $x_{i_{1}},\dots,x_{i_{rs}}$ so that knowing the
values of these variables certifies $f\geq s$.

######  Lemma A.3.

[MR13][Talagrand’s Inequality II] Let $X_{1},\dots,X_{n}$ be $n$ independent
random variables and $f(X_{1},\dots,X_{n})$ be a $c$-Lipschitz $r$-certifiable
function. For any $b\geq 1$,

$P(|f-\mathbb{E}[f]|>b+60c\sqrt{r\mathbb{E}[f]})\leq
4\exp\left(-\frac{b^{2}}{8c^{2}r\mathbb{E}[f]}\right)\ .$


# Magnetic correlations of a doped and frustrated Hubbard model: benchmarking
the two-particle self-consistent theory against a quantum simulator

Guan-Hua Huang  Department of Physics, Southern University of Science and
Technology, Shenzhen, 518055, China  Zhigang Wu<EMAIL_ADDRESS>Shenzhen
Institute for Quantum Science and Engineering, Southern University of Science
and Technology, Shenzhen 518055, China.  Department of Physics, The University
of Hong Kong, Hong Kong, China.

###### Abstract

Recently a quantum simulator for the 2D Fermi-Hubbard model on an anisotropic
triangular lattice has been realized, where both geometrical frustration and
doping can be continuously tuned. Here we provide a comprehensive comparison
between the magnetic correlations calculated by the two-particle self-
consistent (TPSC) theory and those measured in this quantum simulator, at
temperatures comparable to the spin exchange energy and Hubbard interactions
comparable to the bandwidth. We find overall excellent agreements between the
TPSC calculations and the measurements from the quantum simulator at all
levels of frustration and doping. This is quite remarkable considering that
the Hubbard model is already in the intermediate to strong coupling regime,
for which very few methods yield reliable results. Our work showcases the
potential of TPSC as a theoretical approach capable of providing reasonably
accurate descriptions of the Hubbard model even at fairly strong interactions
and when the frustration is present.

Introduction.—Quantum simulation of the 2D Fermi-Hubbard model with ultracold
atomic gases has seen tremendous progress in the past decade [1, 2, 3]. These
systems have long been known to offer a pristine realization of the Hubbard
model through the use of optical lattices, where all the model parameters can
be precisely tuned [4]; but it is the advent of quantum gas microscope with
its unprecedented power of resolution [5, 6], combined with innovative ideas
of cooling [7, 8], that has made simulating strongly correlated regimes
possible. By now experiments have successfully reached temperatures as low as
half of the spin exchange energy and along the way simulated many equilibrium
and non-equilibrium properties of the Hubbard model. Among other things, these
include demonstration of metal-to-insulator transition [9], probe of short-
and long-range antiferromagnetic correlations [10, 11, 12, 13, 14],
measurement of equation of state [15, 16] and study of magnetic polarons [17,
18, 19]. In addition, various transport measurements have also been carried
out in the quantum simulator, such as charge transport in the bad metallic
regime [20], spin transport of the Mott insulator [21] as well as subdiffusion
and heat transport in the high temperature regime [22]. All these developments
are encouraging signs that the ultimate goal of simulating the 2D Hubbard
model at temperatures and dopings relevant to the physics of high Tc
superconductivity may finally be within reach.

However, the value of a quantum simulator lies not only in revealing the
properties of simulated models in regimes not accessible by classical
computation tools, but also in serving as an important platform for testing
various theoretical ideas used to unravel these models. For the 2D Fermi-
Hubbard model, an extraordinary amount of effort has been devoted to its
understanding [23] and various theoretical methods have been proposed for its
quantitative description [24, 25]. The so-called two-particle self-consistent
theory (TPSC), developed by Vilk and Tremblay [26, 27], is one of such
methods. TPSC is a non-perturbative approach which satisfies the Mermin-Wagner
theorem, the Pauli principle and various conservation and sum rules for the
charge and spin. It has been applied to a wide range of problems related to
the 2D Fermi-Hubbard model [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44] and was found to agree with Monte Carlo simulations on the
calculation of various physical quantities in the weak-to-intermediate
coupling regime [27], including the double occupancy, spin and charge
structure factors, and single-particle spectral functions. More importantly,
it predicts the opening of pseudogap [26] and the existence of d-wave
superconductivity in the model [30], two very consequential results. Although
calculations based on TPSC have been directly compared to measurements of
cuprate superconductors [31, 32, 33], the agreement or lack thereof should be
viewed along with the fact the single-band Hubbard model cannot possibly
capture the real materials perfectly. For this reason, it is important to
benchmark the methods proposed to solve the Hubbard model, such as the TPSC,
against a quantum simulator of the model.

In this paper, we perform such a benchmarking by comparing the magnetic
correlations calculated by TPSC to those measured in the quantum simulator of
the doped and frustrated Hubbard model on an anisotropic triangular lattice
[14]. To our surprise, we find that even for Hubbard interactions comparable
to or slightly larger than the bandwidth, TPSC still gives a fairly accurate
account of the magnetic correlations at all levels of doping and frustration.
Our results highlight the capacity of TPSC in understanding strongly-
correlated regimes of the Hubbard model that very few approaches can access.

Quantum simulator of a frustrated Hubbard model.—In the quantum simulator
[14], a two-component Fermi gas is loaded into a non-separable square optical
lattice potential, formed by the interference pattern of two laser beams. When
the intensities of the two beams are equal, this setup realizes the standard
Hubbard model on a square lattice with nearest-neighbor hopping $t$ and on-
site interaction $U$. Through varying the relative intensity of the two beams,
the horizontal diagonal hopping $t^{\prime}$ can be tuned from zero to $t$
continuously, thereby realizing a tunable triangular lattice potential (see
Fig. 1(a)). The Hamiltonian of the Hubbard model on such a lattice reads

$\hat{H}=-t\sum_{\langle
ij\rangle\sigma}\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma}-t^{\prime}\sum_{\langle\langle
ij\rangle\rangle,\sigma}\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma}+U\sum_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow},$
(1)

where $\langle\langle ij\rangle\rangle$ denotes the next-nearest neighbor
along the horizontal diagonal line with respect to the square lattice.

Figure 1: (a) Optical lattice potential used in Ref. [14] to realize the
quantum simulator of a tunable frustrated Hubbard model. (b) A sample of the
measured real-space magnetic correlation in the quantum simulator of Ref.
[14].

In these experiments, quantum gas microscope provides an in-situ, site-
resolved imaging of the atomic distribution after a spin-selective removal of
one component and in so doing measure the real-space magnetic correlation
function

$\displaystyle
C_{\bm{r}}=4\left(\langle\hat{S}_{j}^{z}\hat{S}_{i}^{z}\rangle-\langle\hat{S}_{j}^{z}\rangle\langle\hat{S}_{i}^{z}\rangle\right),$
(2)

where $\bm{r}$ denotes the position of site-$j$ relative to site-$i$ and
$\hat{S}_{i}^{z}=\frac{1}{2}(\hat{n}_{i\uparrow}-\hat{n}_{i\downarrow})$. A
sample data of thus measured magnetic correlation from this quantum simulator
is shown in Fig. 1(b).

TPSC and magnetic correlations.—For completeness, we review the basic ideas of
TPSC using the Luttinger-Ward functional formalism [27]. In this formalism,
the self-energy $\Sigma_{\sigma}[G_{\sigma}]$ is treated as a functional of
the single-particle Green’s function $G_{\sigma}$. Once a specific ansatz for
the self-energy is given by some conserving scheme, the single-particle
Green’s function is determined by Dyson’s equation. Preceding from here,
however, are two independent routes to obtaining the two-particle Green’s
functions. One is through Heisenberg equation of motion which relates the
single-particle Green’s function to the two-particle one. The other is through
Bethe-Salpeter equation, which expresses the susceptibilities, essentially the
two-particle Green’s function, in terms of the effective interactions (i.e.,
the irreducible vertices)
$\Gamma_{\sigma\sigma^{\prime}}=\delta\Sigma_{\sigma}/\delta
G_{\sigma^{\prime}}$. The central idea of TPSC is to enforce the consistency
in the calculations of the two-particle Green’s function on the choice of the
variational self-energy ansatz.

TPSC adopts a simple Hartree-like ansatz for the self-energy which assumes
momentum- and frequency-independent irreducible vertices
$\Gamma_{\uparrow\uparrow}$ and $\Gamma_{\uparrow\downarrow}$. In terms of the
spin vertex
$\Gamma_{sp}\equiv\Gamma_{\uparrow\downarrow}-\Gamma_{\uparrow\uparrow}$, the
single-particle Green’s function is given by

$\displaystyle
G^{(1)}_{\sigma}({\bm{k}},i\omega_{m})=\frac{1}{i\omega_{m}-(\epsilon_{\bm{k}}+n\Gamma_{sp}/2-\mu)}$
(3)

where $\omega_{m}=(2m+1)\pi T$ is the fermonic Matsubara frequency at
temperature $T$, $\epsilon_{{\bm{k}}}=-2t(\cos k_{x}+\cos
k_{y})-2t^{\prime}\cos(k_{x}+k_{y})$ is the single-particle dispersion, $n$ is
the density and $\mu$ is the chemical potential. The equation of motion for
the single-particle Green’s function gives the following relation between the
spin vertex $\Gamma_{sp}$ and the double occupancy
$\langle\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}\rangle$

$\Gamma_{sp}\langle\hat{n}_{i\uparrow}\rangle\langle\hat{n}_{i\downarrow}\rangle=U{\langle\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}\rangle}.$
(4)

The Bethe-Salpeter equation, which determines the spin susceptibility, leads
to another relation between them

$\displaystyle\frac{T}{N}\sum_{\nu}\sum_{{\bm{q}}}\frac{\chi^{(1)}(\bm{q},i\omega_{\nu})}{1-\frac{1}{2}\Gamma_{sp}\chi^{(1)}(\bm{q},i\omega_{\nu})}=n-2{\langle\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}\rangle},$
(5)

where $N$ is the number of lattice sites, $\omega_{\nu}=2\nu\pi T$ is the
bosonic Matsubara frequency and

$\displaystyle\chi^{(1)}\equiv\frac{T}{N}\sum_{m{\bm{k}}\sigma}G^{(1)}_{\sigma}({\bm{k}},i\omega_{m})G^{(1)}_{\sigma}({\bm{k}}+{\bm{q}},i\omega_{m}+i\omega_{\nu})$
(6)

is the so-called polarization bubble. For the single-particle Green’s function
given in Eq. (3), the latter takes the form of the Lindhard function
$\chi^{(1)}({\bm{q}},i\omega_{\nu})=-\frac{2}{N}\sum_{\bm{k}}\frac{f(\tilde{\epsilon}_{\bm{k}})-f(\tilde{\epsilon}_{\bm{k}+\bm{q}})}{i\omega_{\nu}+\tilde{\epsilon}_{\bm{k}}-\tilde{\epsilon}_{\bm{k}+\bm{q}}}$,
where $\tilde{\epsilon}_{\bm{k}}=\epsilon_{\bm{k}}+n\Gamma_{sp}/2$ and
$f(\tilde{\epsilon}_{\bm{k}})=1/(e^{(\tilde{\epsilon}_{\bm{k}}-\mu)/T}+1)$ is
the Fermi-Dirac distribution.

Figure 2: The real-space magnetic correlation $C_{\bm{r}}$ between the
nearest-, next-nearest and next-next-nearest neighbors, denoted by
$C_{(1,0)}$, $C_{(1,1)}$ and $C_{(2,1)}$ respectively. The dots with error
bars are experimental data and the bands are TPSC calculations. Here $U/t=9.5$
and $T/t=0.3\sim 0.35$.

To obtain the relation in Eq. (5), one first notes that the spin
susceptibility can be obtained from the Bethe-Salpeter equation as

$\displaystyle\chi_{sp}({\bm{q}},i\omega_{\nu})=\frac{\chi^{(1)}(\bm{q},i\omega_{\nu})}{1-\frac{1}{2}\Gamma_{sp}\chi^{(1)}(\bm{q},i\omega_{\nu})},$
(7)

which in turn determines the spin structure factor

$\displaystyle\tilde{C}_{\bm{q}}={T}\sum_{\nu}\chi_{sp}({\bm{q}},i\omega_{\nu}).$
(8)

Using the latter to calculate the real-space magnetic correlation function

$\displaystyle
C_{\bm{r}}=\frac{1}{N}\sum_{{\bm{q}}}\tilde{C}_{{\bm{q}}}e^{i{\bm{q}}\cdot\bm{r}}$
(9)

and observing that
$C_{\bm{r}=0}=n-2\langle\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}\rangle$, one
arrives at Eq. (5). Once $\Gamma_{sp}$ is determined self-consistently by
solving Eqs. (4)-(5), the spin structure factor in Eq. (8) and the real-space
magnetic correlation function in Eq. (9) can then be compared to experimental
measurements. Lastly, we point out that the complete theory of TPSC involves
additional steps to obtain a more accurate single-particle Green’s function,
but we omit this discussion as it is not relevant to our present purpose.

Benchmarking.—We first compare the magnetic correlations calculated by TPSC to
those measured in the quantum simulator in the case of half-filling. All
calculations are done on a $256\times 256$ lattice, where sparse sampling and
IR decomposition from the sparse-ir library are used to treat the temperature
Green’s functions [45, 46, 47]. As mentioned earlier, the experiments measure
directly the real-space correlation function $C_{\bm{r}}$ between any two
sites. Shown in Fig. 2 are comparisons between theory and experiment on the
nearest-, next-nearest and next-next-nearest neighbor correlations as a
function of $t^{\prime}/t$ at $U/t=9.5$. As the final temperature of the
atomic system varies slightly from one value of $t^{\prime}/t$ to another, we
show TPSC calculations (shaded bands in Fig. 2) for a range of temperatures
$T/t=0.3\sim 0.35$, consistent with that determined in the experiments. We
note that these temperatures are slightly lower than the spin exchange energy
$J=4t^{2}/U$. As we can see, the overall agreement between experiment and
theory is excellent. At half-filling, the suppression of antiferromagnetic
correlation by increasing frustration is clearly reflected by the reduction of
the spin vertex in the TPSC calculations as shown in Fig. 3.

Figure 3: The spin vertex $\Gamma_{sp}$ as a function of doping at various
degrees of frustration characterized by $t^{\prime}/t$. Here $U/t=9.5$ and
$T/t=0.35$. Figure 4: Spin structure factor of the Hubbard model at half-
filling and in increasing degree of frustration characterized by
$t^{\prime}/t$. The top panel contains results from the quantum simulator, the
middle panel calculations from the TPSC and the bottom panel comparisons
between them. The dots in the bottom panel are experimental data and lines
TPSC calculations. Here experimental parameters corresponding to different
$t^{\prime}/t$ are $T/t=0.26$ and $U/t=9.7$ for $t^{\prime}/t=0.0265$;
$T/t=0.34$ and $U/t=8.2$ for $t^{\prime}/t=0.57$; $T/t=0.32$ and $U/t=8.2$ for
$t^{\prime}/t=0.75$; and $T/t=0.39$ and $U/t=9.2$ for $t^{\prime}/t=0.97$. All
the TPSC calculations in the paper are done using experimental parameters
except for the case of $t^{\prime}/t=0.0265$, where we find an almost perfect
agreement with the experiment using a theoretical $T/t=0.39$. Figure 5: The
nearest- and next-nearest-neighbor correlations, $C_{(1,0)}$ and $C_{(1,1)}$,
as a function of doping and at various degrees of frustration characterized by
$t^{\prime}/t$. Here the experimental parameters are $U/t=9$, and $T/t=0.35$
for (a)-(c) and $T/t=0.4$ for (d)-(f).

In Ref. [14], the measured real-space magnetic correlations between all sites
are then used to construct the spin structure factor $\tilde{C}_{\bm{q}}$ by
inverting Eq. (9). There, to illustrate clearly the effects of frustration,
the physical lattice in Fig. 1 is mapped to a triangular lattice such that the
square-shaped first Brillouin zone of the physical lattice now stretches into
the rhombus-shaped one as illustrated in Fig. 4(a). Plotted as a function of
the crystal momentum corresponding to the triangular lattice, the experimental
$\tilde{C}_{\bm{q}}$ and the TPSC calculations are shown in Fig. 4. Here the
TPSC reproduces the full spin structure factor of the Hubbard model simulator
reasonably well; in particular, the splitting of the $\tilde{C}_{\bm{q}}$ peak
at the $M$ point into two parts at the $K$ points as the frustration increases
is well captured by the TPSC. We note that TPSC does produce a smaller peak
value of the spin structure factor in general compared to the experimental
measurements, which could be due to the limitation of the theory. Another
possible explanation is that because of the system’s non-uniform density the
experimental determination of the spin structure factor involves a cutoff of
real-space correlations beyond a certain distance, which may result in a less
accurate peak value 111Muqing Xu, private communications.

We next consider the case of finite doping where the particle density is given
by $n=1+\delta$. The comparisons between the TPSC calculations and the
experimental results on the nearest and next-nearest neighbor correlations as
a function of doping are shown in Fig. 5; again the agreements are on the
whole very good. Here an important characteristic of frustrated lattices is
the lack of particle-hole symmetry in contrast to the square lattice. This can
be seen from the fact that the system with particle doping $\delta>0$ can be
mapped to one with hole doping $-\delta$ and described by the same Hamiltonian
with $t^{\prime}$ replaced by $-t^{\prime}$ [49]. In fact, the magnetic
correlations with particle doping are calculated using this mapping. The
observed particle-hole asymmetry in frustrated lattices is well captured by
the TPSC calculations and, from the theoretical perspective, is due to the
reduction of the spin vertex by particle doping in the presence of frustration
(see Fig. 3). We notice that the experimental results in Fig. 5(c) has a
somewhat more pronounced particle-hole asymmetry than those of the TPSC
calculations. In particular, the agreement between theory and experiment is
almost perfect for the hole doping while some discrepancy exists for the
particle doping. However, this discrepancy is perhaps not as large as it
appears if we take into account the fact that the experimental results for the
square lattice already exhibits a slight but noticeable particle-hole
asymmetry (see Fig. 5(a)), due presumably to system errors.

Conclusions.—We have systematically benchmarked the TPSC calculations of the
magnetic correlations of a doped and frustrated Hubbard model against the
experimental results of a quantum simulator for $U/t=8\sim 10$. Although TPSC
is believed to be valid only for weak-to-intermediate coupling, i.e., for
$U/t\lesssim 8$, the overall excellent agreement with the copious data from
the quantum simulator indicates a wider range of applicability for this
theory. In the future, it would be highly desirable to test other predictions
of the TPSC, such as the opening of the pseudogap, in the next-generation of
quantum simulators of the Hubbard model where the temperature can be further
lowered. This can in principle be carried out by probing the single-particle
spectral weight of the atomic system using the photoemission spectroscopy [50,
51, 52], a tool analogous to the ARPES in solid-state systems [53].

Acknowledgement. We are grateful to Muqing Xu and Prof. Markus Greiner for
providing us the experimental data in Ref. [14]. We also want to thank Muqing
Xu and Lev Kendrick for stimulating discussions. This work is supported by
National Key R$\&$D Program of China (Grant No. 2022YFA1404103), NSFC (Grant
No. 11974161), Shenzhen Science and Technology Program (Grant No.
KQTD20200820113010023) and Key-Area Research and Development Program of
Guangdong Province (Grant No. 2019B030330001).

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Here the prefactors are chosen such that the conformal contribution is
$\sigma_{0}$.

The electron spectral weight is given in (76),

$\rho_{e}(\omega)=\rho_{e}^{c}(\omega)\left(1-\sum_{h}V_{h}^{\mathrm{A}}\mathcal{R}_{h}^{\textrm{A}}\big{(}\omega-\mathscr{E}_{e}\big{)}-V_{h}^{\mathrm{S}}\mathcal{R}_{h}^{\textrm{S}}\big{(}\omega-\mathscr{E}_{e}\big{)}\right),$
(152)

where

$V_{h}^{\mathrm{A/S}}=\frac{\alpha_{h}}{2(\beta J)^{h-1}}(v_{hf+}\pm
v_{hf-}\pm v_{hb+}+v_{hb-})\,.$ (153)

The conformal contribution is

$\text{Re}\sigma_{0}(\omega)=\sigma_{0}I(0^{+},\omega,\mathscr{E}_{e})\,,$
(154)

where the integral involved is of the type

$I(\epsilon,\omega,\mathscr{E})=\int{\rm
d}\Omega\frac{e^{\epsilon\Omega}}{\omega}\left(\frac{1}{e^{2\pi\Omega}+1}-\frac{1}{e^{2\pi(\Omega+\omega)}+1}\right)\frac{\cosh(\pi\Omega)\cosh(\pi(\Omega+\omega))}{\cosh(\pi(\Omega-\mathscr{E}))\cosh(\pi(\Omega+\omega-\mathscr{E}))}.$
(155)

We have add a convergence factor $e^{\epsilon\Omega}$ with $\epsilon\to 0^{+}$
eventually, and then split the two terms in the parenthesis. The individual
integrands are periodic under $\Omega\to\Omega+i$, so we can use an
rectangular contour between $\text{Im}\Omega=0$ and $\text{Im}\Omega=1$. The
encircled poles are at $\Omega=\mathscr{E}+\frac{i}{2}$ and
$\Omega=\omega+\mathscr{E}+\frac{i}{2}$. We obtain

$I(\epsilon,\omega,\mathscr{E})=\frac{1-e^{-\epsilon\omega}}{2\omega\sin\frac{\epsilon}{2}}e^{\epsilon\mathscr{E}},$
(156)
$\text{Re}\sigma_{0}(\omega)=\sigma_{0}I(0^{+},\omega,\mathscr{E})=\sigma_{0},$
(157)

which is independent of frequency as expected from scaling analysis.

Next we look at contribution due to corrections.

$\begin{split}\text{Re}\delta_{h}\sigma(\omega)&=-\sigma_{0}\int{\rm
d}\Omega\frac{e^{\epsilon\Omega}}{\omega}\left(\frac{1}{e^{2\pi\Omega}+1}-\frac{1}{e^{2\pi(\Omega+\omega)}+1}\right)\frac{\cosh(\pi\Omega)\cosh(\pi(\Omega+\omega))}{\cosh(\pi(\Omega-\mathscr{E}_{e}))\cosh(\pi(\Omega+\omega-\mathscr{E}_{e}))}\\\
&\times\left(V_{h}^{\mathrm{A}}\mathcal{R}_{h}^{\textrm{A}}\big{(}\Omega-\mathscr{E}_{e}\big{)}+V_{h}^{\mathrm{S}}\mathcal{R}_{h}^{\textrm{S}}\big{(}\Omega-\mathscr{E}_{e}\big{)}+V_{h}^{\mathrm{A}}\mathcal{R}_{h}^{\textrm{A}}\big{(}\Omega+\omega-\mathscr{E}_{e}\big{)}+V_{h}^{\mathrm{S}}\mathcal{R}_{h}^{\textrm{S}}\big{(}\Omega+\omega-\mathscr{E}_{e}\big{)}\right)\,.\end{split}$
(158)

Using explicit expression (78) for $\mathcal{R}_{h}^{\textrm{A/S}}$, we need
the following auxilary integral ($\epsilon\to 0^{+}$)

$\begin{split}J_{1}(\omega)&=\int{\rm
d}\Omega\frac{e^{\epsilon\Omega}}{\omega}\left(\frac{1}{e^{2\pi\Omega}+1}-\frac{1}{e^{2\pi(\Omega+\omega)}+1}\right)\frac{\cosh(\pi\Omega)\cosh(\pi(\Omega+\omega))}{\cosh(\pi(\Omega-\mathscr{E}_{e}))\cosh(\pi(\Omega+\omega-\mathscr{E}_{e}))}\\\
&\quad\times{}_{3}\textbf{F}_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{h\mskip
8.0mu1-h\mskip 8.0mu\frac{1}{2}+i(\Omega-\mathscr{E}_{e})}{1\mskip
8.0mu1};1\biggr{)}\\\
&={}_{3}\textbf{F}_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{h\mskip 8.0mu1-h\mskip
8.0mu1-i\omega}{1\mskip 8.0mu2};1\biggr{)}\,,\end{split}$ (159)

which is evaluated by expanding the hypergeometric function series, doing the
integral term by term with contour method as above, and then resumming.

Using $J_{1}$, we can express the correction as

$\text{Re}\delta_{h}\sigma(\omega)=-\sigma_{0}\frac{2\left(\frac{\pi}{2}\right)^{h}\Gamma(h)}{\sqrt{\pi}\sin\left(\frac{\pi
h}{2}\right)\Gamma(h-1/2)}\left(V_{h}^{\mathrm{A}}\text{Re}(J_{1}(\omega)+J_{1}(-\omega))+V_{h}^{\mathrm{S}}\text{Im}(J_{1}(\omega)+J_{1}(-\omega))\right)\,.$
(160)

Here the $J_{1}(-\omega)$ terms come from integrating
$\mathcal{R}_{h}^{\textrm{A/S}}(\Omega+\omega-\mathscr{E}_{e})$. Since
$J_{1}(\omega)=J_{1}(-\omega)^{*}$, only the $V_{h}^{\mathrm{A}}$ terms are
nonzero, and we obtain (90).

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*[CVRPs]: Capacitated Vehicle Routing Problem
  *[LK]: Lin-Kernighan
  *[LKH]: Lin-Kernighan-Helsgaun
  *[SA]: Simulated Annealing
  *[LNS]: Large Neighborhood Search
  *[GA]: Genetic Algorithm
  *[KGLS]: Knowledge Guided Local Search
  *[SISR]: Slack Induction by String Removal
  *[LM]: Language Model
  *[Prefix LM]: Prefix Language Model
  *[MHA]: Multi-Head Attention
  *[GPT]: Generative Pre-trained Transformer
  *[PNs]: Pointer Network
  *[RNN]: Recurrent Neural Network
  *[Seq2Seq]: Sequence-to-Sequence
  *[RL]: Reinforcement Learning
  *[DRL]: Deep Reinforcement Learning
  *[BFS]: Breadth-First Search
  *[LLaMA-2]: Large Language Model Meta AI 2
  *[NS]: nucleus sampling
  *[MCTS]: Monte Carlo Tree Search
  *[SIFT]: Scale Invariant Feature Transform
  *[HOG]: Histogram of Oriented Gradients
  *[CL]: Curriculum Learning
  *[SGD]: Stochastic Gradient Descent
  *[GS]: greedy sampling
  *[BKS]: best-known solution
  *[MITSC]: MIT SuperCloud
  *[AM]: Attention Model
  *[MLOps]: Machine Learning Operations
  *[GNN]: Graph Neural Network
  *[GNNs]: Graph Neural Network