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 –
JHEP01(2021)121
In;txyzt0x0y0z0 denotes the contributions from the integration over χ(n) and χ̄(n). A straight-
forward calculation shows
In;txyzt0x0y0z0 = (−1)n1(y1+y2+z1+z2+t1+t2)+n2(z1+z2+t1+t2)+n3(t1+t2)
×

1
√
2
t1+t2+x1+x2+y1+y2+z1+z2+t0
1+t0
2+x0
1+x0
2+y0
1+y0
2+z0
1+z0
2
×
√
g0
t3+x3+y3+z3+t0
3+x0
3+y0
3+z0
3 e
µ
2
(t1−t2+t0
1−t0
2)
×
h
−m ¯
∆txyzt0x0y0z0,0∆txyzt0x0y0z0,0 + ¯
∆txyzt0x0y0z0,1∆txyzt0x0y0z0,1
i
, (2.11)
where
¯
∆txyzt0x0y0z0,q = δt1+t3+x1+x3+y1+y3+z1+z3+t0
2+t0
3+x0
2+x0
3+y0
2+y0
3+z0
2+z0
3,q, (2.12)
∆txyzt0x0y0z0,q = δt2+t3+x2+x3+y2+y3+z2+z3+t0
1+t0
3+x0
1+x0
3+y0
1+y0
3+z0
1+z0
3,q, (2.13)
with q = 0, 1. ¯
∆, ∆ are derived from χ̄-, χ-integration, respectively. The second line in
eq. (2.11) comes from the staggered sign factor ην(n). Consequently, In;txyzt0x0y0z0 does
depend on n ∈ Λ. Eq. (2.11) tells us that this tensor network is uniform in t-direction, but
has some periodic structure in x-,y-,z-directions. This periodicity corresponds to the parity
of the spatial lattice site n = (n1, n2, n3). A graphical representation of eq. (2.9) is shown
in figure 2(A). As a result of eqs. (2.5) and (2.6), some Grassmann variables are allowed
to exist in Tn;txyzt0x0y0z0 . These Grassmann variables have been denoted by Gn;txyzt0x0y0z0
in eq. (2.10). Some sign can arise reflecting on how we have arranged these Grassmann
variables in Gn;txyzt0x0y0z0 and we have set this sign Sn;txyzt0x0y0z0 in eq. (2.10). We now
assume that
Gn;txyzt0x0y0z0 =
dΦt1
4 dΨt2
4 dΦx1
1 dΨx2
1 dΦy1
2 dΨy2
2 dΦz1
3 dΨz2
3 dΨ̄
t0
2
4 dΦ̄
t0
1
4 dΨ̄
x0
2
1 dΦ̄
x0
1
1 dΨ̄
y0
2
2 dΦ̄
y0
1
2 dΨ̄
z0
2
3 dΦ̄
z0
1
3
×

Φ̄4(n + 4̂)Φ4(n)
t1

Ψ̄4(n + 4̂)Ψ4(n)
t2

Φ̄1(n + 1̂)Φ1(n)
x1

Ψ̄1(n + 1̂)Ψ1(n)
x2
×

Φ̄2(n + 2̂)Φ2(n)
y1

Ψ̄2(n + 2̂)Ψ2(n)
y2

Φ̄3(n + 3̂)Φ3(n)
z1

Ψ̄3(n + 3̂)Ψ3(n)
z2
,
(2.14)
where all the Grassmann measures depend on n and their arguments are omitted. Accord-
ing to this arrangement, Sn;txyzt0x0y0z0 is given by
Sn;txyzt0x0y0z0 = (−1)t1(t2+x2+y2+z2)+x1(x2+y2+z2)+y1(y2+z2)+z1z2
× (−1)t0
2(t0
1+x0
1+y0
1+z0
1)+x0
2(x0
1+y0
1+z0
1)+y0
2(y0
1+z0
1)+z0
2z0
1
× (−1)(t1+t2+x1+x2+y1+y2+z1+z2)(t0
1+x0
1+y0
1+z0
1)
. (2.15)
2.3 Grassmann ATRG
2.3.1 Procedure of the algorithm
We now formulate Grassmann ATRG (GATRG) algorithm to coarse grain the tensor net-
work defined by eq. (2.9). The basic idea is that we combine the ATRG procedure to
– 5 –
