JHEP01(2021)121
𝑇
𝜇
1st
2nd Tricritical point
!
𝜓𝜓 ≠ 0 !
𝜓𝜓 = 0
Figure 1. Schematic view of expected phase diagram of the NJL model on the T-µ plane. Solid
and broken curves represent the first- and second-order phase transitions, respectively. Closed circle
denotes the tricritical point where the first-order phase transition line terminates.
where n = (n1, n2, n3, n4)(∈ Z4) specifies a position in the lattice Λ, with the lattice spacing
a. χ(n) and χ̄(n) are Grassmann-valued fields without the Dirac structure. Since they
describe the Kogut-Susskind fermions, χ(n) and χ̄(n) are single-component Grassmann
variables. ην(n) is the staggered sign function defined by ην(n) = (−1)n1+···+nν−1 with
η1(n) = 1. The partition function is defined in the ordinal manner:
Z =
Z


Y
n∈Λ
dχ(n)dχ̄(n)

 e−S
. (2.2)
For vanishing mass m, eq. (2.1) is invariant under the following continuous chiral transfor-
mation:
χ(n) → eiα(n)
χ(n), (2.3)
χ̄(n) → χ̄(n)eiα(n)
(2.4)
with α ∈ R and (n) = (−1)n1+n2+n3+n4 .
2.2 Tensor network representation
We introduce the tensor network representation for eq. (2.2) in a similar way with refs. [10,
11].3 Hereafter, we set a = 1 for simplicity. Firstly, we expand the local Boltzmann weights
3
See ref. [25] for a different TRG approach with the Kogut-Susskind fermion, where the TRG procedure
is applied to the Schwinger model after integrating out the fermion fields analytically.
– 3 –
JHEP01(2021)121
in the following manners to decompose the nearest-neighbor interactions:
exp
"
−
eµδν,4
2
ην(n)χ̄(n)χ(n + ν̂)
#
=
1
X
iν,1(n)=0
Z
e
µ
2
δν,4
√
2
ην(n)χ̄(n)dΦν(n)
·
e
µ
2
δν,4
√
2
χ(n + ν̂)dΦ̄ν(n + ν̂) · Φ̄ν(n + ν̂)Φν(n)
!iν,1(n)
, (2.5)
exp
"
e−µδν,4
2
ην(n)χ̄(n + ν̂)χ(n)
#
=
1
X
iν,2(n)=0
Z
e− µ
2
δν,4
√
2
ην(n)χ(n)dΨν(n)
·
e− µ
2
δν,4
√
2
χ̄(n + ν̂)dΨ̄ν(n + ν̂) · Ψ̄ν(n + ν̂)Ψν(n)
!iν,2(n)
, (2.6)
eg0χ̄(n)χ(n)χ̄(n+ν̂)χ(n+ν̂)
=
1
X
iν,3(n)=0
(
√
g0χ̄(n)χ(n) ·
√
g0χ̄(n + ν̂)χ(n + ν̂))iν,3(n)
. (2.7)
Secondly, integrating out χ and χ̄ at each lattice site n, we define
Tn;i4(n)i1(n)i2(n)i3(n)i4(n−4̂)i1(n−1̂)i2(n−2̂)i3(n−3̂)
=
Z
dχdχ̄ e−mχ̄χ
4
Y
ν=1
e
µ
2
δν,4
√
2
ην(n)χ̄dΦν(n)
!iν,1(n)
e
µ
2
δν,4
√
2
χdΦ̄ν(n)
!iν,1(n−ν̂)
×
e− µ
2
δν,4
√
2
ην(n)χdΨν(n)
!iν,2(n)
e− µ
2
δν,4
√
2
χ̄dΨ̄ν(n)
!iν,2(n−ν̂)
(
√
g0χ̄χ)iν,3(n)
× (
√
g0χ̄χ)iν,3(n−ν̂)

Φ̄ν(n + ν̂)Φν(n)
iν,1(n) 
Ψ̄ν(n + ν̂)Ψν(n)
iν,2(n)
. (2.8)
This serves as a change of variables from χ, χ̄ to the integer-valued fields iν = (iν,p)p=1,2,3
and alternative Grassmann variables Φν, Ψν. Renaming x = i1, y = i2, z = i3, t = i4,
eq. (2.2) is expressed in the form,
Z =
X
{t,x,y,z}
Z Y
n∈Λ
Tn;txyzt0x0y0z0 , (2.9)
which is the tensor network representation of this model.4 In current construction,
Tn;txyzt0x0y0z0 is factorized as
Tn;txyzt0x0y0z0 = In;txyzt0x0y0z0 Sn;txyzt0x0y0z0 Gn;txyzt0x0y0z0 . (2.10)
4
In eq. (2.9), we omit arguments in tensor indices and introduce shorthand notations such as x0
=
x(n − 1̂), y0
= y(n − 2̂), z0
= z(n − 3̂), t0
= t(n − 4̂)
– 4 –
