aainstitutetext: Institute for Physics of Intelligence, University of Tokyo,
Tokyo, 113-003, Japanbbinstitutetext: Center for Computational Sciences,
University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan

# Critical endpoint of (3+1)-dimensional finite density $\mathds{Z}_{3}$
gauge-Higgs model with tensor renormalization group

Shinichiro Akiyama<EMAIL_ADDRESS>b Yoshinobu Kuramashi
<EMAIL_ADDRESS>
###### Abstract

The critical endpoint of the (3+1)-dimensional $\mathds{Z}_{3}$ gauge-Higgs
model at finite density is determined by the tensor renormalization group
method. This work is an extension of the previous one on the $\mathds{Z}_{2}$
model. The vital difference between them is that the $\mathds{Z}_{3}$ model
suffers from the sign problem, while the $\mathds{Z}_{2}$ model does not. We
show that the tensor renormalization group method allows us to locate the
critical endpoint for the $\mathds{Z}_{3}$ gauge-Higgs model at finite
density, regardless of the sign problem.

††preprint: UTHEP-780, UTCCS-P-147

##  1 Introduction

The last decade was devoted to an initial stage to apply the tensor
renormalization group (TRG) method 111In this paper, the “TRG method” or the
“TRG approach” refers to not only the original numerical algorithm proposed by
Levin and Nave Levin:2006jai  but also its extensions Gu:2010yh ;
PhysRevB.86.045139 ; Shimizu:2014uva ; Sakai:2017jwp ; Adachi:2019paf ;
Kadoh:2019kqk ; Akiyama:2020soe ; PhysRevB.105.L060402 ; Kadoh:2021fri .,
which was originally proposed to study two-dimensional (2$d$) classical spin
systems in the field of condensed matter physics Levin:2006jai , to quantum
field theories consisting of scalar, fermion, and gauge fields Banuls:2019rao
; Meurice:2020pxc ; Okunishi:2021but . There were many attempts to confirm or
utilize the following expected advantages of the TRG method employing the
lower-dimensional models: (i) no sign problem Denbleyker:2013bea ;
Shimizu:2014uva ; Shimizu:2014fsa ; Takeda:2014vwa ; Kawauchi:2016xng ;
Shimizu:2017onf ; Kadoh:2018hqq ; Kadoh:2019ube ; Kuramashi:2019cgs ;
Butt:2019uul ; Takeda:2021mnc ; Nakayama:2021iyp , (ii) logarithmic
computational cost on the system size, (iii) direct manipulation of the
Grassmann variables Gu:2010yh ; Shimizu:2014uva ; Takeda:2014vwa ;
Sakai:2017jwp ; Yoshimura:2017jpk ; Kadoh:2018hqq ; Akiyama:2021xxr ;
Bloch:2022vqz , (iv) evaluation of the partition function or the path-integral
itself.

The first TRG calculation of the 4$d$ Ising model Akiyama:2019xzy  was the
trigger to explore various (3+1)$d$ quantum field theories with the TRG
method: complex $\phi^{4}$ theory at finite density Akiyama:2020ntf , real
$\phi^{4}$ theory Akiyama:2021zhf , Nambu$-$Jona-Lasinio (NJL) model at high
density and very low temperature Akiyama:2020soe , and $U(N)$ gauge theory
with the infinite-coupling limit Milde:2021vln . Recently, the phase structure
of the $\mathds{Z}_{2}$ gauge-Higgs model at finite density has been
investigated and its critical endpoint has been determined within the TRG
method Akiyama:2022eip . This is the first application of the TRG method to a
(3+1)$d$ lattice gauge theory beyond the (1+1)$d$ systems Shimizu:2014uva ;
Shimizu:2014fsa ; Shimizu:2017onf ; Unmuth-Yockey:2018ugm ; Kuramashi:2019cgs
; Bazavov:2019qih ; Fukuma:2021cni ; Hirasawa:2021qvh  and (2+1)$d$ ones
Dittrich:2014mxa ; Kuramashi:2018mmi ; Unmuth-Yockey:2018xak . In this paper,
we investigate the phase structure of the $\mathds{Z}_{3}$ gauge-Higgs model
at finite density. Figure 1 illustrates the expected phase diagram at
vanishing density, which is speculated from the numerical study of the phase
diagrams of the $\mathds{Z}_{N}$ gauge-Higgs models Creutz:1979he ;
Creutz:1983ev ; Baig:1987ka . We determine the critical endpoint following the
procedure employed in the $\mathds{Z}_{2}$ study Akiyama:2022eip . The
important difference between the $\mathds{Z}_{3}$ and $\mathds{Z}_{2}$ models
is that the former yields the sign problem at finite chemical potential
contrary to the latter. Therefore, this model has been investigated by dual
lattice simulations Gattringer:2012jt ; Langfeld:2018ykv . The purpose of this
work is to confirm the effectiveness of the TRG method for a (3+1)$d$ gauge
theory with the sign problem, which should serve as a better test bed for the
future study of the QCD at finite density.

Figure 1: Schematic phase diagram of (3+1)$d$ $\mathds{Z}_{3}$ gauge-Higgs
model at the vanishing chemical potential. $\beta$-axis denotes the inverse
gauge coupling and $\eta$ represents the spin-spin coupling. The pure
$\mathds{Z}_{3}$ gauge theory is characterized by $\eta=0$ whose transition
point is $\beta=2\ln(1+\sqrt{3})/3$ Balian:1974ts ; Balian:1974ir ;
Balian:1974xw ; KorthalsAltes:1978tp ; Yoneya:1978dt . The limit
$\beta\to\infty$ is equivalent to the three-state Potts model whose transition
point is obtained by the Monte Carlo renormalization group method
PhysRevLett.43.799 ; Caracciolo:1986ik , where the estimated transition point
of $4d$ Potts model is $2\eta=0.3875$.

This paper is organized as follows. In Sec. 2, we define the $\mathds{Z}_{3}$
gauge-Higgs model at finite density on a (3+1)$d$ lattice. In Sec. 3, we
provide a consistency check between the TRG approach and the dual lattice
simulations before we determine the critical endpoints at $\mu=0$, $1$, $2$ in
the (3+1)$d$ model and discuss to what extent they are shifted by the effect
of finite $\mu$. Section 4 is devoted to summary and outlook.

##  2 Formulation and numerical algorithm

We consider the path integral of the $\mathds{Z}_{3}$ gauge-Higgs model at
finite density on an isotropic hypercubic lattice
$\Lambda_{3+1}=\\{(n_{1},\dots,n_{4})\ |n_{\nu}=1,\dots,L\\}$ whose volume is
equal to $V=L^{4}$. The lattice spacing $a$ is set to $a=1$ without loss of
generality. The gauge fields $U_{\nu}(n)$ ($\nu=1,\dots,4$) reside on the
links and the matter fields $\sigma(n)$ are on the sites. Both variables
$U_{\nu}(n)$ and $\sigma(n)$ take their values on
$\mathds{Z}_{3}=\\{1,\exp\left({\rm i}\pi/3\right),\exp\left({\rm
i}2\pi/3\right)\\}$. The action $S$ is defined as

$\displaystyle S=$
$\displaystyle-{\beta}\sum_{n\in\Lambda_{3+1}}\sum_{\nu>\rho}\Re\left[U_{\nu}(n)U_{\rho}(n+\hat{\nu})U^{*}_{\nu}(n+\hat{\rho})U^{*}_{\rho}(n)\right]$
$\displaystyle-\eta\sum_{n}\sum_{\nu}\left[{\rm
e}^{\mu\delta_{\nu,4}}\sigma^{*}(n)U_{\nu}(n)\sigma(n+{\hat{\nu}})+{\rm
e}^{-\mu\delta_{\nu,4}}\sigma^{*}(n)U^{*}_{\nu}(n-{\hat{\nu}})\sigma(n-{\hat{\nu}})\right],$
(1)

where $\beta$ is the inverse gauge coupling, $\eta$ is the spin-spin coupling
and $\mu$ is the chemical potential. This parametrization follows Ref.
Gattringer:2012jt . We employ periodic boundary conditions for both the gauge
and matter fields in all directions. The path integral is then given by

$\displaystyle
Z=\left(\prod_{n,\nu}\sum_{U_{\nu}(n)\in\mathds{Z}_{3}}\right)\left(\prod_{n}\sum_{\sigma(n)\in\mathds{Z}_{3}}\right){\rm
e}^{-S},$ (2)

where the sum is taken over all possible field configurations. Since
$\sigma(n)\in\mathds{Z}_{3}$, one is allowed to choose the so-called unitary
gauge Creutz:1979he , which eliminates the matter field $\sigma(n)$ by
redefining the link variable $U_{\nu}(n)$ via

$\displaystyle\sigma^{*}(n)U_{\nu}(n)\sigma(n+{\hat{\nu}})\mapsto U_{\nu}(n).$
(3)

With the unitary gauge, Eq. (2) is reduced to be

$\displaystyle S=$
$\displaystyle-{\beta}\sum_{n\in\Lambda_{3+1}}\sum_{\nu>\rho}\Re\left[U_{\nu}(n)U_{\rho}(n+\hat{\nu})U^{*}_{\nu}(n+\hat{\rho})U^{*}_{\rho}(n)\right]$
$\displaystyle-2\eta\sum_{n}\sum_{\nu}\left\\{\cosh\left(\mu\delta_{\nu,4}\right)\Re\left[U_{\nu}(n)\right]+{\rm
i}\sinh\left(\mu\delta_{\nu,4}\right)\Im\left[U_{\nu}(n)\right]\right\\},$ (4)

whose path integral is given by

$\displaystyle
Z=\left(\prod_{n,\nu}\sum_{U_{\nu}(n)\in\mathds{Z}_{3}}\right){\rm e}^{-S},$
(5)

instead of Eq. (2). The construction of the tensor network representation for
the general $\mathds{Z}_{N}$ case is already described in Ref. Akiyama:2022eip
, which is based on the asymmetric construction established in Ref.
Liu:2013nsa . We also employ the anisotropic TRG (ATRG) Adachi:2019paf  with
the parallel computation as in Refs. Akiyama:2022eip ; Akiyama:2020ntf .

##  3 Numerical results

Figure 2:  $\beta$ dependence of the plaquette value (left) and its
susceptibility (right) at $\eta=0.5$ with $D=45$.

The path integral of Eq. (5) is evaluated using the parallelized ATRG
algorithm with the bond dimension $D$. Firstly, we make a consistency check
between our results and those obtained by the previous lattice calculation. In
Ref. Gattringer:2012jt , the dual lattice simulation gives $\beta$ dependence
of the plaquette value and its susceptibility choosing $\eta=0.5$, where no
phase transition is expected passing by the critical endpoint in Fig. 8 below.
We evaluate the plaquette value $\langle U\rangle$ defined by

$\displaystyle\langle U\rangle=\frac{1}{6V}\frac{\partial\ln
Z}{\partial\beta}$ (6)

using the impurity tensor method, setting $D=45$. To evaluate $\langle
U\rangle$ via the impurity tensor method, we construct the tensor network
representation differently from Ref. Akiyama:2022eip . The detail is explained
in Appendix A. In Fig. 2, we plot the plaquette value and its susceptibility
$\chi_{U}$ as a function of $\beta$ at $\eta=0.5$. We calculate $\chi_{U}$ via
the forward difference of $\langle U\rangle$. The results should be compared
with Fig. 5 in Ref. Gattringer:2012jt  obtained with the dual lattice
simulation. Both results show consistency with Ref. Gattringer:2012jt , where
no sign of the phase transition is observed. Therefore, we expect that $D=45$
is sufficiently large to calculate the thermodynamic quantities in the
finite-$\eta$ regime. Note that the peak height of $\chi_{U}$ is a little bit
smaller than that in Ref. Gattringer:2012jt . We think that this can be
attributed to the forward difference of $\langle U\rangle$.

Figure 3: $\eta$ dependence of $\langle L\rangle$ at $\mu=0$ for
$\beta\in[0.412,0.425]$. Figure 4: Convergence behavior of the transition
point of $\eta$ as a function of $D$ at $\beta=0.412$ with vanishing $\mu$.
Table 1: $\Delta\langle L\rangle$ and the first-order transition points of
$(\beta,\eta)$ at $\mu=0$, $1$, $2$. All the results are obtained with $D=50$.
$\mu=0$
---
$\beta$ | $\eta$ | $\Delta\langle L\rangle$
0.412 | 0.32511(4) | 0.1448570626
0.414 | 0.32353(3) | 0.1746984961
0.415 | 0.32266(4) | 0.1930650060
0.416 | 0.32172(4) | 0.2096426007
0.417 | 0.32084(4) | 0.2156841414
0.418 | 0.32003(4) | 0.2358818267
0.419 | 0.31922(4) | 0.2434405732
0.420 | 0.31834(4) | 0.2578042490
0.425 | 0.31422(2) | 0.3007002064
$\mu=1$
$\beta$ | $\eta$ | $\Delta\langle L\rangle$
0.415 | 0.28055(2) | 0.1443182714
0.416 | 0.27984(3) | 0.1815751895
0.417 | 0.27916(3) | 0.2059557920
0.418 | 0.27841(3) | 0.2232897426
0.419 | 0.27766(3) | 0.2468143002
0.420 | 0.27691(4) | 0.2614199552
0.421 | 0.27628(2) | 0.2735230025
$\mu=2$
$\beta$ | $\eta$ | $\Delta\langle L\rangle$
0.409 | 0.20972(3) | 0.0516254411
0.410 | 0.20894(3) | 0.1150573520
0.411 | 0.20816(4) | 0.1403103742
0.412 | 0.20741(3) | 0.1720190275
0.413 | 0.20661(3) | 0.1985852268
0.414 | 0.20584(3) | 0.2196185602
Figure 5: (Left) Fit of $\Delta\langle L\rangle$ at $\mu=0$ as a function of
$\eta$. (Right) Fit of $\Delta\langle L\rangle$ at $\mu=0$ as a function of
$\beta$. Table 2: Fit results for $\Delta\langle L\rangle$. All the results
are obtained with the TRG method whose bond dimension is set as $D=50$.
$\mu=0$
---
$A$ | $\beta_{\rm c}$ | $p$ | $B$ | $\eta_{\rm c}$ | $q$
2.1(3) | 0.409(7) | 0.47(4) | 2.3(4) | 0.3280(6) | 0.48(4)
$\mu=1$
$A$ | $\beta_{\rm c}$ | $p$ | $B$ | $\eta_{\rm c}$ | $q$
1.5(2) | 0.4139(2) | 0.35(7) | 1.7(2) | 0.2813(2) | 0.34(3)
$\mu=2$
$A$ | $\beta_{\rm c}$ | $p$ | $B$ | $\eta_{\rm c}$ | $q$
2.8(6) | 0.40873(7) | 0.48(4) | 3.2(8) | 0.20994(9) | 0.49(4)

Now, let us estimate the critical endpoint with the TRG approach. The first-
order phase transition line in the phase diagram terminates at the critical
endpoint $(\beta_{\rm c},\eta_{\rm c})$. We employ the average link defined by

$\displaystyle\langle L\rangle=\frac{1}{4V}\frac{\partial\ln
Z}{\partial(2\eta)}$ (7)

to detect the first-order phase transition. We regard the critical endpoint
$(\beta_{\rm c},\eta_{\rm c})$ as a point where the jump in $\langle
L\rangle$, as a function of $\eta$, vanishes. The factor $4V$ corresponds to
the number of links in $\Lambda_{3+1}$ with the periodic boundary condition.
We evaluate $\langle L\rangle$ with the impurity tensor method, whose
expression is given in the appendix of Ref. Akiyama:2022eip . In the
following, all the results are calculated by setting $D=50$ in the
thermodynamic limit, where the TRG computation converges for the system size.
We first determine the critical endpoint in the $\mu=0$ case. Figure 3 shows
the $\eta$ dependence of $\langle L\rangle$ at $\mu=0$ with the several
choices of $\beta$. We observe clear gaps in $\langle L\rangle$ at a certain
value of $\eta$ for $\beta\in[0.412,0.425]$. Values of these gaps in $\langle
L\rangle$, denoted by $\Delta\langle L\rangle$, are listed in Table 1,
together with the corresponding first-order transition point of
$(\beta,\eta)$. Although $\Delta\langle L\rangle$ is evaluated just by
$\langle L\rangle(\eta=\eta_{+})-\langle L\rangle(\eta=\eta_{-})$, where
$\eta_{+}$ and $\eta_{-}$ are chosen from different phases, we set
$\eta_{+}-\eta_{-}={\rm O}(10^{-5})$ for all $\beta$. The error for $\eta$ in
Table 1 is provided by the magnitude of $\eta_{+}-\eta_{-}$. Figure 4 shows
the typical $D$ dependence of the transition point $\eta$. The relative error
between the first-order transition points with $D=44$ and $D=50$ is 0.019%.
Hence, it may be expected that the finite-$D$ effect is well suppressed to
identify transition points. In order to determine the critical endpoint
$(\beta_{\rm c},\eta_{\rm c})$, we separately fit the data of $\Delta\langle
L\rangle$, assuming the functions $\Delta\langle L\rangle=A(\beta-\beta_{\rm
c})^{p}$ and $\Delta\langle L\rangle=B(\eta_{\rm c}-\eta)^{q}$, respectively,
where $A$, $B$, $\beta_{\rm c}$, $\eta_{\rm c}$, $p$, and $q$ are the fit
parameters. The fit results are drawn in Fig. 5 and their numerical values are
presented in Table 2. The fit provides us with $(\beta_{\rm c},\eta_{\rm
c})=(0.409(7),0.3280(6))$ as the critical endpoint at $\mu=0$.

Figure 6: $\eta$ dependence of $\langle L\rangle$ at $\mu=2$ for
$\beta\in[0.409,0.414]$. Figure 7: (Left) Fit of $\Delta\langle L\rangle$ at
$\mu=2$ as a function of $\eta$. (Right) Fit of $\Delta\langle L\rangle$ at
$\mu=2$ as a function of $\beta$.

Let us turn to the finite density cases with $\mu=1$ and 2, where the Monte
Carlo approach is ruled out by the sign problem. In Fig. 6 we plot the $\eta$
dependence of the link average with the several choices of $\beta$ at $\mu=2$.
This should be compared with Fig. 11 of Ref. Akiyama:2022eip  in the
$\mathds{Z}_{2}$ model. We find a similar quality of data for both cases,
which means that the TRG method works efficiently regardless of the sign
problem. Table 1 summarizes the finite values of $\Delta\langle L\rangle$ and
the transition points. $\Delta\langle L\rangle$ is fitted with the same
functions as in the case of $\mu=0$. The fit results at $\mu=2$ are shown in
Fig. 7. Their numerical values are presented in Table 2, together with the
results of $\mu=0$, 1. We obtain $(\beta_{\rm c},\eta_{\rm
c})=(0.4139(2),0.2813(2))$ and $(0.40873(7),0.20994(9))$ as the critical
endpoints at $\mu=1$ and $\mu=2$, respectively. Comparing the critical
endpoints at $\mu=0$, $1$, $2$, we find that $\beta_{\rm c}$ has little $\mu$
dependence, while $\eta_{\rm c}$ is sizably diminished as $\mu$ increases.
Notice that similar behavior has been observed in the $\mathds{Z}_{2}$ model
Akiyama:2022eip . We summarize these results in Fig. 8, where the critical
endpoints in the $\mathds{Z}_{3}$ model is plotted together with those in the
$\mathds{Z}_{2}$ model Akiyama:2022eip . According to Refs. Creutz:1979he ;
Baig:1987ka , the values of $\beta_{\rm c}$ and $\eta_{\rm c}$ are expected to
increase as $N$ in $\mathds{Z}_{N}$ increases. Therefore, the resulting
endpoints by the TRG method seem reasonable for the increase of $N$.

Figure 8:  Summary of the critical points in the (3+1)$d$ $\mathds{Z}_{3}$
gauge-Higgs model, together with the $\mathds{Z}_{2}$ model Akiyama:2022eip
for comparison. Black, blue, and green symbols represent the transition points
for $\mu=0$, $1$, and $2$, respectively. Open symbols denote the first-order
transition points. Closed symbols are for the critical endpoints estimated by
fitting. Diamonds and triangles denote the cases with $\mathds{Z}_{2}$ and
$\mathds{Z}_{3}$, respectively. Symbols on the $\beta$-axis are the pure gauge
transition points, which are $\ln(1+\sqrt{2})/2$ and $2\ln(1+\sqrt{3})/3$ for
$\mathds{Z}_{2}$ and $\mathds{Z}_{3}$, respectively Balian:1974ts ;
Balian:1974ir ; Balian:1974xw ; KorthalsAltes:1978tp ; Yoneya:1978dt . The
inset graph makes a closer look for transition points of the $\mathds{Z}_{3}$
model.

##  4 Summary and outlook

In this paper, we have determined the critical endpoints of (3+1)$d$
$\mathds{Z}_{3}$ gauge-Higgs model at finite density, where the sign problem
prohibits the conventional Monte Carlo approach. We have closely followed the
determination procedure employed in the previous work on the $\mathds{Z}_{2}$
model Akiyama:2022eip , which is free from the sign problem. The resulting
endpoints by the TRG method seem reasonable, compared with the previous work
of the $\mathds{Z}_{2}$ model Akiyama:2022eip . In addition, we have computed
the plaquette value as a function of $\beta$, which is comparable with the
previous study of the same model by the dual lattice simulation
Gattringer:2012jt . Our results show that the TRG method works efficiently for
both models regardless of the existence of the sign problem, so it is a
promising approach to the future investigation of the QCD at finite density.
The next step should be an extension of this study to the (3+1)$d$ lattice
gauge theories with continuous gauge groups, also including dynamical matter
fields.

###### Acknowledgements.

Numerical calculation for the present work was carried out with Oakbridge-CX
in the Information Technology Center at The University of Tokyo and the
computational resource was offered under the category of General Projects by
Research Institute for Information Technology in Kyushu University. We also
used the supercomputer Fugaku provided by RIKEN through the HPCI System
Research Project (Project ID: hp210204, hp220203) and computational resources
of Wisteria/BDEC-01 and Cygnus under the Multidisciplinary Cooperative
Research Program of Center for Computational Sciences, University of Tsukuba.
This work is supported in part by Grants-in-Aid for Scientific Research from
the Ministry of Education, Culture, Sports, Science and Technology (MEXT) (No.
20H00148). S. A. is supported by the Endowed Project for Quantum Software
Research and Education, the University of Tokyo (qsw ), and by JSPS KAKENHI
Grant Number JP23K13096.

##  Appendix A Impurity tensor method for $\langle U\rangle$

In section 3, $\langle U\rangle$ is calculated by the impurity tensor method,
where we slightly modify the asymmetric formulation in Ref. Liu:2013nsa . To
explain our modification, we first review the asymmetric formulation for
(3+1)$d$ $\mathds{Z}_{3}$ gauge-Higgs model. We regard the local Boltzmann
weight corresponding to the plaquette interaction in Eq. (2) as a four-rank
tensor,

$\displaystyle
W_{U_{\nu}(n)U_{\rho}(n+\hat{\nu})U_{\nu}(n+\hat{\rho})U_{\rho}(n)}$
$\displaystyle=\exp\left(\beta\Re\left[U_{\nu}(n)U_{\rho}(n+\hat{\nu})U^{*}_{\nu}(n+\hat{\rho})U^{*}_{\rho}(n)\right]\right)$
$\displaystyle\times\exp\left[\frac{\eta}{3}\left\\{\cosh\left(\mu\delta_{\nu,4}\right)\Re\left[U_{\nu}(n)\right]+{\rm
i}\sinh\left(\mu\delta_{\nu,4}\right)\Im\left[U_{\nu}(n)\right]\right\\},\right]$
$\displaystyle\times\exp\left[\frac{\eta}{3}\left\\{\cosh\left(\mu\delta_{\nu,4}\right)\Re\left[U_{\rho}(n+\hat{\nu})\right]+{\rm
i}\sinh\left(\mu\delta_{\nu,4}\right)\Im\left[U_{\rho}(n+\hat{\nu})\right]\right\\},\right]$
$\displaystyle\times\exp\left[\frac{\eta}{3}\left\\{\cosh\left(\mu\delta_{\nu,4}\right)\Re\left[U^{*}_{\nu}(n+\hat{\rho})\right]+{\rm
i}\sinh\left(\mu\delta_{\nu,4}\right)\Im\left[U^{*}_{\nu}(n+\hat{\rho})\right]\right\\},\right]$
$\displaystyle\times\exp\left[\frac{\eta}{3}\left\\{\cosh\left(\mu\delta_{\nu,4}\right)\Re\left[U^{*}_{\rho}(n)\right]+{\rm
i}\sinh\left(\mu\delta_{\nu,4}\right)\Im\left[U^{*}_{\rho}(n)\right]\right\\},\right].$
(8)

For later convenience, we have included spin-spin coupling terms associated
with the plaquette. Using the higher-order singular value decomposition
(HOSVD), we can decompose $W$ via

$\displaystyle
W_{U_{\nu}(n)U_{\rho}(n+\hat{\nu})U_{\nu}(n+\hat{\rho})U_{\rho}(n)}=\sum_{a,b,c,d}V_{U_{\nu}(n)a}V_{U_{\rho}(n+\hat{\nu})b}V_{U_{\nu}(n+\hat{\rho})c}V_{U_{\rho}(n),d}B_{abcd},$
(9)

where $V$’s are unitary matrices and $B$ is the core tensor. Integrating out
all link variables $U_{\nu}(n)$’s in Eq. (5) at each link independently, we
have a six-leg tensor $A$ at each link according to

$\displaystyle
A_{m_{1}m_{2}m_{3}m^{\prime}_{1}m^{\prime}_{2}m^{\prime}_{3}}=\sum_{U_{\nu}(n)}\left(\prod_{i=1}^{3}V_{U_{\nu}(n)m_{i}}V_{U_{\nu}(n)m^{\prime}_{i}}\right).$
(10)

Therefore, we have the tensor network representation for Eq. (5) as

$\displaystyle Z={\rm
tTr}\left[\left(\prod_{\ell}A_{\ell}\right)\left(\prod_{\square}B_{\square}\right)\right].$
(11)

Considering the tensor contraction among $A_{\ell}$’s with
$\ell=(n,1),(n,2),(n,3),(n,4)$ and $B_{\square}$’s associated with these
$A_{\ell}$’s, we can define a local tensor $\mathcal{T}_{n}$ at the site $n$.
This is the asymmetric construction, which gives us the uniform tensor network
representation such that

$\displaystyle Z={\rm
tTr}\left[\prod_{n\in\Lambda_{d+1}}\mathcal{T}_{n}\right].$ (12)

$\mathcal{T}_{n}$ can be regarded as an eight-leg tensor, say
$\mathcal{T}_{n;xyztx^{\prime}y^{\prime}z^{\prime}t^{\prime}}$, where each
index is constructed by three indices coming from the HOSVD.

Now, we re-express $\mathcal{T}_{n}$ in Eq. (12) using the original link
variables $U_{\nu}(n)$, not using the indices introduced by the HOSVD. This
can be easily done by re-expressing $A_{\ell}$’s in $\mathcal{T}_{n}$ as the
right-hand side of Eq. (10) and integrating out the indices coming from the
HOSVD. One can see that this derivation gives us a new local tensor in the
following form,

$\displaystyle\mathcal{T}_{n;(x_{1}x_{2}x_{3})(y_{1}y_{2}y_{3})(z_{1}z_{2}z_{3})(t_{1}t_{2}t_{3})(x^{\prime}_{1}x^{\prime}_{2}x^{\prime}_{3})(y^{\prime}_{1}y^{\prime}_{2}y^{\prime}_{3})(z^{\prime}_{1}z^{\prime}_{2}z^{\prime}_{3})(t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3})}=$
$\displaystyle\left(\prod_{\nu=1}^{4}\sum_{U_{\nu}(n)}\right)\delta_{U_{1}(n)y_{1}z_{1}t_{1}}\delta_{U_{2}(n)x_{2}z_{2}t_{2}}\delta_{U_{3}(n)x_{3}y_{3}t_{3}}\delta_{U_{4}(n)x_{4}y_{4}z_{4}}$
$\displaystyle\times
W_{U_{1}(n)x^{\prime}_{2}y^{\prime}_{1}U_{2}(n)}W_{U_{1}(n)x^{\prime}_{3}z^{\prime}_{1}U_{3}(n)}W_{U_{1}(n)x^{\prime}_{4}t^{\prime}_{1}U_{4}(n)}W_{U_{2}(n)y^{\prime}_{3}z^{\prime}_{2}U_{3}(n)}W_{U_{2}(n)y^{\prime}_{4}t^{\prime}_{2}U_{4}(n)}W_{U_{3}(n)z^{\prime}_{4}t^{\prime}_{3}U_{4}(n)}.$
(13)

Using this $\mathcal{T}_{n}$, the path integral is again represented as in Eq.
(12). Note that this construction has also been employed for 3$d$ $SU(2)$
gauge theory in Ref. Kuwahara:2022ubg . Thanks to Eq. (A), we can easily
introduce the impurity tensor to describe $\langle U\rangle$. For example, the
expectation value of the plaquette on the $12$-plane is expressed by the
following impurity tensor,

$\displaystyle\mathcal{S}^{[12]}_{n;(x_{1}x_{2}x_{3})(y_{1}y_{2}y_{3})(z_{1}z_{2}z_{3})(t_{1}t_{2}t_{3})(x^{\prime}_{1}x^{\prime}_{2}x^{\prime}_{3})(y^{\prime}_{1}y^{\prime}_{2}y^{\prime}_{3})(z^{\prime}_{1}z^{\prime}_{2}z^{\prime}_{3})(t^{\prime}_{1}t^{\prime}_{2}t^{\prime}_{3})}=$
$\displaystyle\left(\prod_{\nu=1}^{4}\sum_{U_{\nu}(n)}\right)\delta_{U_{1}(n)y_{1}z_{1}t_{1}}\delta_{U_{2}(n)x_{2}z_{2}t_{2}}\delta_{U_{3}(n)x_{3}y_{3}t_{3}}\delta_{U_{4}(n)x_{4}y_{4}z_{4}}\exp\left(\beta\Re\left[U_{1}(n)x^{\prime}_{2}(y^{\prime})^{*}_{1}U^{*}_{2}(n)\right]\right)$
$\displaystyle\times
W_{U_{1}(n)x^{\prime}_{2}y^{\prime}_{1}U_{2}(n)}W_{U_{1}(n)x^{\prime}_{3}z^{\prime}_{1}U_{3}(n)}W_{U_{1}(n)x^{\prime}_{4}t^{\prime}_{1}U_{4}(n)}W_{U_{2}(n)y^{\prime}_{3}z^{\prime}_{2}U_{3}(n)}W_{U_{2}(n)y^{\prime}_{4}t^{\prime}_{2}U_{4}(n)}W_{U_{3}(n)z^{\prime}_{4}t^{\prime}_{3}U_{4}(n)}.$
(14)

Therefore, we can finally express $\langle U\rangle$ as

$\displaystyle\langle U\rangle=\frac{1}{Z}{\rm
tTr}\left[\mathcal{S}_{c}\prod_{n\neq c}\mathcal{T}_{n}\right],$ (15)

by introducing the following impurity tensor,

$\displaystyle\mathcal{S}_{n}=\frac{1}{6}\left(\mathcal{S}^{[12]}_{n}+\mathcal{S}^{[13]}_{n}+\mathcal{S}^{[14]}_{n}+\mathcal{S}^{[23]}_{n}+\mathcal{S}^{[24]}_{n}+\mathcal{S}^{[34]}_{n}\right).$
(16)

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