                                                                                                          0.8
  2.00           V    = 26                                                                                             V   = 26
                 V    = 28                                                                                0.7          V   = 28
  1.75                                                                                                                 V   = 210
                 V    = 210
                 V    = 212                                                                               0.6          V   = 212
  1.50
                 V    = 214                                                                                            V   = 214
                                                                                                          0.5          V   = 216
  1.25           V    = 216
                 V    = 218                                                                                            V   = 218




                                                                                                     hχχi
                                                                                                          0.4
hni




  1.00           V    = 220                                                                                            V   = 220

  0.75                                                                                                    0.3

  0.50                                                                                                    0.2

  0.25                                                                                                    0.1

  0.00                                                                                                    0.0

         0.1           0.2           0.3       0.4                0.5             0.6                            0.0        0.1      0.2      0.3          0.4          0.5          0.6      0.7
                                               µ                                                                                                     µ




                                                                                                                                                                                                      JHEP03(2025)027
         (a) ⟨n⟩ as a function of µ at m = 0.1.                                                                 (b) ⟨χχ⟩ as a function of µ at m = 0.1.

  2.00           V   = 26                                                                                              V   = 26
                                                                                                          0.6          V   = 28
                 V   = 28
  1.75           V   = 210                                                                                             V   = 210
                 V   = 212                                                                                             V   = 212
                                                                                                          0.5
                 V   = 214                                                                                             V   = 214
  1.50
                 V   = 216                                                                                             V   = 216                             0.6

                 V   = 218                                                                                0.4          V   = 218
  1.25
                 V   = 220                           2.0                                                               V   = 220
                                                                                                                                                             0.4
                                                                                                     hχχi
hni




  1.00
                                                     1.5                                                  0.3
                                                                                                                                                             0.2
  0.75
                                                     1.0
                                                                                                          0.2
  0.50                                                                                                                                                       0.0
                                                     0.5
                                                                                                                                                                   0.97 0.98 0.99 1.00 1.01 1.02
                                                                                                          0.1
  0.25
                                                     0.0

                                                           0.97   0.98   0.99   1.00   1.01
  0.00                                                                                                    0.0

          0.80       0.85     0.90     0.95   1.00     1.05         1.10        1.15          1.20              0.80       0.85    0.90    0.95     1.00         1.05         1.10     1.15    1.20
                                               µ                                                                                                     µ


           (c) ⟨n⟩ as a function of µ at m = 1.                                                                 (d) ⟨χχ⟩ as a function of µ at m = 1.

Figure 14. Volume dependence of physical quantities at β = 0.8. The bond dimension in the
calculations is D = 150. The sample size for the discretization of gauge group integrations is K = 14.
To evaluate the numerical differences in eqs. (4.3) and (4.7), we set ∆µ = 0.04 for m = 0.1, ∆µ = 0.02
for m = 1, and λ = ∆λ = 10−4 . At m = 1, the insets show the volume dependence of ⟨n⟩ and ⟨χχ⟩ in
the intermediate phase, where ⟨n⟩ is evaluated by ∆µ = 0.004.




     Our results encourage a future application of the TRG approach to the higher-dimensional
two-color QCD. It is possible to improve our construction of initial tensors such that more
SU(2) matrices are used to discretize the gauge group integration, i.e., a larger K is allowed,
without increasing the memory requirement significantly. In higher-dimensional cases where
spontaneous breaking of continuous symmetry can exist, it is instructive to apply the TRG
approach to evaluate the order parameters and perform extrapolations toward the chiral limit
(m = 0) and the vanishing λ limit. We also emphasize that our Grassmann tensor network
representation for the partition function can be extended to the three-color theory, which
suffers from the sign problem at finite density, without conceptual difficulties.

    Finally, we note that inhomogeneous phases in two-color QCD and related models [47, 66–
68] have attracted attention recently. As another future direction, we will explore the
applicability of the TRG approach in studying the spatial dependence of physical quantities.




                                                                                                 – 19 –
                                     number density, m = 0.1, V = 220 , K = 14
                   2.0         β   = 0, D = 84
                               β   = 0.4, D = 150
                               β   = 0.8, D = 150
                   1.5         β   = 1.2, D = 150
                               β   = 1.6, D = 150
                 hni




                   1.0




                                                                                                                JHEP03(2025)027
                   0.5



                   0.0

                         0.1        0.2         0.3      0.4       0.5           0.6
                                                         µ


Figure 15. Quark number density ⟨n⟩ as a function of chemical potential µ at m = 0.1, β =
0, 0.4, 0.8, 1.2, 1.6 in the thermodynamic limit. The bond dimension in the infinite coupling calculation
is D = 84, and the bond dimension in the finite β calculations is D = 150. At a finite β, the sample
size for the discretization of gauge group integrations is K = 14. To evaluate the numerical differences
in eq. (4.3), we set ∆µ = 0.04.


Acknowledgments
A part of the numerical calculation for the present work was carried out with ohtaka provided
by the Institute for Solid State Physics, the University of Tokyo. This work is supported by the
Endowed Project for Quantum Software Research and Education, the University of Tokyo [69],
and the Center of Innovations for Sustainable Quantum AI (JST Grant Number JPMJPF2221).
SA acknowledges the support from JSPS KAKENHI (JP23K13096, JP24H00214) and the Top
Runners in Strategy of Transborder Advanced Researches (TRiSTAR) program conducted as
the Strategic Professional Development Program for Young Researchers by the MEXT.

A     Coefficients of the Grassmann tensor F
Here, we discuss one method to derive the tensor elements of F in eq. (2.10). For simplicity, we
consider the two-color case where N = 2. We also label the spacetime direction ν by ν = x, t
not by ν = 1, 2. In this case, the integration over the original staggered fermions is written as
             Z
       F=        dχ1 dχ̄1 dχ2 dχ̄2 e−m(χ̄1 χ1 +χ̄2 χ2 ) (1 + χ̄1 A) (1 + χ1 B) (1 + χ̄2 C) (1 + χ2 D)

          = ABCD + mCD + mAB + m2 ,                                                                     (A.1)
where A, B, C, and D are sums of terms with one auxiliary Grassmann variable, and their
expressions are given by
    A = −ηx,1 − ηt,1 + (Ux† )11 ζ̄x,1 + (Ux† )12 ζ̄x,2 + (Ut† )11 ζ̄t,1 + (Ut† )12 ζ̄t,2 ,              (A.2)
        1h                                                                                       i
    B=     − (Ux )11 η̄x,1 − (Ux )21 η̄x,2 − a+ (Ut )11 η̄t,1 − a+ (Ut )21 η̄t,2 + ζx,1 + a− ζt,1 ,     (A.3)
        2



                                                      – 20 –
