Tensor network formulation for twodimensional lattice $$ \mathcal{} $$ = 1 WessZumino model
HJE
Tensor network formulation for twodimensional lattice
0 Institute for Theoretical Physics, Kanazawa University
1 Daisuke Kadoh
2 Center for Computational Sciences, University of Tsukuba
3 RIKEN Advanced Institute for Computational Science
4 Research and Educational Center for Natural Sciences, Keio University
Supersymmetric models with spontaneous supersymmetry breaking suffer from the notorious sign problem in stochastic approaches. By contrast, the tensor network approaches do not have such a problem since they are based on deterministic procedures. In this work, we present a tensor network formulation of the twodimensional lattice N = 1 WessZumino model while showing that numerical results agree with the exact solutions for the free case.
Field Theories in Lower Dimensions; Lattice Quantum Field Theory; Super

symmetry Breaking
3.3
Total tensor network
4
Numerical test in free theory
Some details
Free MajoranaWilson fermion
Free Wilson boson
Witten index of the free N = 1 WessZumino model
5
Summary and outlook A Coarsegraining step in Grassmann TRG 1 3
Tensor network representation of partition function
1 Introduction
2
3
2.1
2.2
3.1
3.2
4.1
4.2
4.3
4.4
1
Introduction
Supersymmetric field theories have attracted great attention because they provide a deep
insight about the nonperturbative physics [1–3] and have a close relation with the
gravitational theory [4]. The lattice simulations are promising approaches to obtain a further
understanding of them. However, it is generally difficult to use the standard Monte Carlo
techniques for the lattice supersymmetric theories on account of the sign problem, and the
theories with the supersymmetry breaking may be the most difficult cases as suggested
from the vanishing Witten index [5]. In this paper, we apply the tensor network approach,
which is free of the sign problem, to the twodimensional lattice N = 1 WessZumino model
in order to make a breakthrough on the issue.
The twodimensional N = 1 WessZumino model is a supersymmetric theory in which
a real scalar interacts with a Majorana fermion via the Yukawa term originate from the
– 1 –
superpotential [
6
]. The supersymmetry is spontaneously broken for the supersymmetric
φ4 theory in a finite volume [5], and the Witten index becomes zero because the fermion
Pfaffian has both the positive and negative signs. For the infinite volume case, the absence
of the nonrenormalization theorem suggests that the breaking may occur even at the
perturbative level [7, 8], and the theory has a rich phase structure which should be clarified
by numerical methods free from the sign problem.
Although the lattice regularization generally breaks the N = 1 supersymmetry for
the interacting theories in contrast to the case of N = 2 model [9–13],1 it is known that
the breaking term caused by the lattice cutoff disappears in the continuum limit for an
appropriate lattice action at least in the perturbation theory [18]. In the action, the Wilson
HJEP03(218)4
terms are included in both the fermion and the boson sectors, so that the supersymmetry is
exactly realized in the freetheory limit. Some numerical studies have been already done in
the lowdimensional WessZumino model [
19–27
]. In our study we use the tensor network
approach to investigate the N = 1 supersymmetric model much deeper.
The tensor renormalization group (TRG) is a coarsegraining algorithm for tensor
networks, which is based on the singular value decomposition (SVD). The TRG was originally
introduced in a twodimensional classical spin model [
28
]. Since the TRG was extended to
the Grassmann TRG for models including Grassmann variables [
29, 30
], some studies of
fermionic systems have been reported so far. In twodimensional quantum field theories, it
was already applied to the lattice φ4 theory [31] and to the lattice Schwinger model [32, 33]
and the lattice Nf = 1 GrossNeveu model [34], which are Dirac fermion systems. For the
lattice N = 1 WessZumino model, we have to clarify a method to construct a tensor
network representation for the Majorana fermions with the Yukawatype interaction and for
the case of nextnearestneighbor interacting bosons which originate from the Wilson term.
In this paper, we show that the partition function of the lattice N = 1 WessZumino
model can be expressed as a tensor network for any superpotential and any value of the
Wilson parameter r. Refining the known method for the Dirac fermions [34], we present
a way of making a tensor network representation for Majorana fermions. For the boson
action, we can change it to one with up to nearestneighbor interactions by introducing
two auxiliary fields. Then we also show a tensor network representation for bosons with a
new discretization scheme. In order to test our formulation, we compute the Witten index
by using the Grassmann TRG. Although we give a method of constructing tensors for any
interacting case, in numerical test we devote ourselves to the free WessZumino model,
which is the most suitable test bed for a tensor network representation. This is because
nontrivial structures of tensor arise from the hopping terms in the lattice action. This
point will be discussed along with the details of the tensor network representation in the
main part of this paper. The computation is done with r = 1/√2, so that one of the two
auxiliary fields is decoupled to reduce the computational cost.
This paper is organized as follows. We first recall the twodimensional N = 1
WessZumino model and its lattice version with the detailed notations in section 2. In section 3,
tensor network representation for the fermion part and the boson part are individually
1Nonlocal formulations of the WessZumino model have been studied in refs. [14–17].
– 2 –
constructed. By combining those two results, the tensor network representation for the
total partition function is also given. Section 4 shows the numerical results for the free
case, and we compare them with the exact ones. A summary and a future outlook are
given in section 5.
the corresponding action is given by
where γµ is the gamma matrix which satisfies
1
2
Scont. =
Z
d2x
(∂µ φ)2 +
1 ψ¯ γµ ∂µ + W ′′ (φ) ψ ,
{γµ , γν } = 2δµν ,
γµ = γµ† .
The Lorentz index µ takes two values 1 or 2, and the Einstein summation convention is
used throughout this paper. Showing the indices in the spinor space explicitly, γµ and ψ (x)
are written as (γµ )αβ and ψα (x) for α, β = 1, 2. The spinor index α and the spacetime
coordinate x are often suppressed without notice. W (φ) is an arbitrary real function of
φ, which is referred to as the superpotential in the superfield formalism, and gives the
Yukawa and φntype interactions with common coupling constants. W ′ (φ) is the first
differential of W (φ) with respect to φ, that is, W ′ (φ) ≡ (d/dφ) W (φ).
The Majorana fermion ψ satisfies
ψ¯ = −ψTC−1,
where C is the charge conjugation matrix which obeys
CT = −C,
C† = C−1,
C−1γµ C = −γµT.
For any W (φ), the action in eq. (2.1) is invariant under the supersymmetry transformation
δφ (x) = ǫ¯ψ (x) ,
δψ (x) = γµ ∂µ φ (x) − W ′ (φ (x)) ǫ,
where ǫ is a global Grassmann parameter with two components and ǫ¯ satisfies eq. (2.3).
2.2
Lattice theory
Let us consider a twodimensional square lattice with the lattice spacing a and the volume
V = aN1 × aN2, where N1, N2 ∈ N. In this paper, a is set to unity, and the lattice sites
are simply expressed by integers:
Γ = { (n1, n2)  nµ = 1, 2, . . . , Nµ for µ = 1, 2 } .
– 3 –
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
where µˆ is the unit vector along the µ direction, and the symmetric difference operator is
given by ∂µS = ∂µ + ∂µ
We define the lattice WessZumino model according to ref. [18]:
the forward one in the naive boson action
All of the fields live on the lattice sites n ∈ Γ and satisfy the periodic boundary conditions
in both directions. The forward and the backward difference operators, ∂µ and ∂µ∗, are
given by
∂µ φn = φn+µˆ − φn,
∂µ∗φn = φn − φn−µˆ,
δφn = ǫ¯ψn,
δψn = n
γµ ∂µSφn +
r
2
∂µ ∂µ∗φ n − W ′ (φn)o ǫ
even at a finite lattice spacing because SB has the similar structure with the Wilson–Dirac
operator D in eq. (2.11) in contrast to the naive one. For the interacting cases, however, the
– 4 –
invariance is explicitly broken owing to the lack of the Leibniz rule for the lattice difference
operators. The broken supersymmetry is shown to be restored in the continuum limit, at
least, at all orders of the perturbation [18].
The associated partition function is defined in the usual manner:
with the path integral measures
Z
Z
Z
Z =
Z
DφDψe−S
Dφ ≡
Dψ ≡
Y Z ∞ dφn
n∈Γ −∞
√2π
,
Y Z
Here dψn,α is a measure of the Grassmann integral defined in the following. The Grassmann
variable ξi and its measure dξi (i = 1, . . . , I) satisfy
{ξi, ξj } = {ξi, dξj } = {dξi, dξj } = 0
for all i, j.
Z
dξi1 = 0,
dξiξi = 1
for i = 1, 2, . . . , I,
The Grassmann integral is then defined by
which suggests that R dξi is equivalent to ∂/∂ξi.
respective partition functions are given by
In the free theory, the boson and the fermion are decoupled from each other, and the
ZB,exact =
ZF,exact =
Y
p1,p2
r
sign {m (m + 4r)}
ZB,exact
,
1
Pµ2=1 sin2 pµ +
m + 2r Pµ2=1 sin2 (pµ /2)
2
,
where pµ = 2πn/Nµ (n = 0, 1, 2, . . . , Nµ − 1) and the product in eq. (2.22) is taken for all
possible momenta [35]. Note that ZB = ∞ (ZF = 0) for m = 0, −2r, −4r when Nµ is an
even integer because the first term and the second term in the square root in eq. (2.22)
simultaneously vanish for certain combinations of p1 and p2. Thus we find that the Witten
index, which is defined as the partition function with periodic boundary conditions in a
finite volume,
reproduces the continuum one, sign {m}, for m ≪ 1.
After integrating the fermion field, the partition function can also be written as
Zexact = sign {m (m + 4r)}
Z =
Z
Dφe−SB Pf (C∗D) ,
– 5 –
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
where the Pfaffian of a 2I × 2I antisymmetric matrix A is defined by
this sign problem we employ the TRG method, whose first step is to represent eq. (2.17)
as a network of uniform tensors, which is explained in the next section.
Tensor network representation of partition function
We construct a tensor network representation for the fermion part of eq. (2.17)
basic idea follows from refs. [32, 34] which deal with the Dirac fermions. We describe the
procedure for the Majorana fermions with any value of the Wilson parameter r.
Now we use the following representations for γµ and C that satisfy eqs. (2.2) and (2.4):
where σi is the standard Pauli matrix. The method presented in this section is applicable
to any possible choice of γµ and C, and they just lead to different tensors. Then the
Majorana spinor takes the form
ψn =
ψn,1!
ψn,2
,
ψ¯n =
ψn,2, −ψn,1 ,
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
and we obtain
− 2
n∈Γ
1 X ψ¯nDψn = X
1 + r
2
n∈Γ
+
1 − r
2
where
which are local transformations of the field variable ψn. ψ˜n,α is introduced only to write
eq. (3.4) as simple as possible. Note that the second term in eq. (3.4) disappears for
ψn+1ˆ,2ψ˜n,1 + ψn+2ˆ,2ψn,1
˜
ψn+1ˆ,1ψ˜n,2 + ψn+2ˆ,1ψn,2 +
˜
W ′′ (φn) + 2r ψn,1ψn,2 ,
1
1
2
2
ψ˜n,1 = √ (ψn,2 + ψn,1) ,
ψ˜n,2 = √ (ψn,2 − ψn,1) ,
– 6 –
r = 1 because the hopping terms in eq. (2.11) are proportional to the projection operators,
(1 ± γµ ) /2.
Let us expand the four types of hopping factors in eq. (3.1):
e− 12 Pn∈Γ ψ¯nDψn
= Y
We will see that un, vn, pn, qn, which take 0 or 1 because of the nilpotency of ψn,α (and
ψ˜n,α), are regarded as the indices of tensors. The four types of hopping factors have the
same structure as Ψn+µˆΦn, where Ψn+µˆ and Φn are singlecomponent Grassmann numbers.
It is straightforward to show
Ψn+µˆΦn =
Z
Ψn+µˆdθ¯n+µˆ (Φndθn) θ¯n+µˆθn ,
where new independent Grassmann numbers θn, θ¯n+µˆ and the corresponding measures dθn,
dθ¯n+µˆ satisfy eqs. (2.20) and (2.21) with the periodic boundary conditions. By applying
this identity to each hopping factor in eq. (3.7) individually, one can make a tensor network
representation.
Then the fermion part of the partition function is represented as a product of tensors
DΞuvpq = Y dξnundχvnndηnpndζnqndξ¯nun−1ˆdχ¯vnn−1ˆdη¯npn−2ˆdζ¯nqn−2ˆ,
where ξn, ξ¯n, χn, χ¯n, ηn, η¯n, ζn, ζ¯n, and those with bars are singlecomponent Grassmann
numbers introduced in the manner of eq. (3.8), and P{u,v,··· } means the summation of
all possible configurations of the indices: Qn∈Γ
P1un=0 Pv1n=0 · · · . The new Grassmann
numbers and their corresponding measures satisfy the same anticommutation relations
and boundary conditions as those of the original ones. The tensor TF is defined as
TF (φ)uvpqabcd =
Z dΨdΦe(W ′′(φ)+2r)ΨΦ nΨdΦcΨ˜ bΦ˜ aΦqΨp Φ˜vΨ˜ uo
·
2
2
r 1 + r !u+p+a+c
r 1 − r !v+q+b+d
– 7 –
for all possible indices with singlecomponent Grassmann numbers Ψ, Φ, Ψ˜ = (Φ + Ψ) /√2,
Φ˜ = (Φ − Ψ) /√2. By integrating Ψ and Φ by hand, we can obtain the tensor elements. In
the case of r = 1, note that the indices vn and qn vanish and that the Grassmann fields χn
and ζn are decoupled because the second term in the r.h.s. of eq. (3.4) is absent. In that
case, the tensor network representation becomes much simpler:
{u,p} n∈Γ
ZFr=1 = X
Y TF (φn)unpnun−1ˆpn−2ˆ
Z
dξnun dηnpn dξ¯nun−1ˆ dη¯npn−2ˆ Y ξn+1ˆξn
¯
un η¯n+ˆ2ηn
where
TF (φ)ijkl =
Z
dΨdΦe(W ′′(φ)+2)ΨΦΦlΦ˜ kΨj Ψ˜ i.
(3.12)
(3.13)
HJEP03(218)4
It is rather straightforward to show that eq. (3.7) is reproduced from eq. (3.9) with
eqs. (3.10) and (3.11) and from the identity in eq. (3.8).
We now note that the eight
Grassmann measures in the r.h.s. of eq. (3.10) should be in this order and that the set
of measures at the site n commutes with ones at different lattice sites because they are
Grassmanneven as a set for nonzero elements of the tensor given in eq. (3.11).
The indices xn ≡ (un, vn) and the Grassmann fields ξn, χn carry the information of the
hopping factors with µ = 1 as indicated by the last factors in eq. (3.9) while tn ≡ (pn, qn)
and ηn, ζn are related to the hopping with µ = 2. In this sense, xn, tn, xn−1ˆ, tn−2ˆ, which
are the indices of the tensor in eq. (3.9), can be interpreted as being defined on the four
links which stem from the site n. Since each index is shared by two tensors which are
placed on the nearestneighbor lattice sites (see eq. (3.9)), we can find that the partition
function ZF is expressed as a network of the tensor TFxntnxn−1ˆtn−2ˆ on the twodimensional
square lattice Γ.
If one uses another representation of γµ and C, then the same partition function is
given by a different tensor. This means that the tensor network representation is not
uniquely determined.
3.2
Boson partition function
The tensor network representation is also constructed for the pure boson part of eq. (2.17)
tion because SB has the nextnearestneighbor interactions and φ is a noncompact field. A
popular way to avoid the former issue is to rewrite SB in a nearestneighbor form with the
aid of auxiliary fields. For the latter, we employ a new method using a discretization for
the integrals of φ.2 After these procedures, we find that a discretized version of eq. (3.14)
can be expressed as a tensor network for arbitrary discretization schemes.
2
A method for treating the noncompact field using a discretization is already proposed in the pioneering
work by Y. Shimizu [31]. We thank him for pointing out a new idea [36] presented in this paper.
– 8 –
Since the formulation is actually irrelevant to the details of the scalar theory, we will
derive a tensor network for a general theory:
the nearestneighbor, and that ϕn is a noncompact real field with N components. As seen
in section 3.2.5, it is very easy to extend it to the non PTsymmetric case.
We will show that eq. (3.14) can be expressed in the form of eq. (3.15) with N = 3 in
section 3.2.1. After decomposing the hopping terms of S˜B in section 3.2.2 and introducing
a formal discretization for the integrals of ϕ in section 3.2.3, we give the tensor network
representation for a discretized version of eq. (3.15) in section 3.2.4.
3.2.1
Introduction of auxiliary fields
The boson action SB in eq. (2.12) is transformed into a nearestneighbor form using two
real auxiliary fields G and H:
− rW ′ (φn) + αGn − βHn
φn+2ˆ + φn−2ˆ − 2φn
r ≤ 1/√
with SB,naive given in eq. (2.13), α = p(1 − 2r2)/2, and β = 1/√2. Note that α is real for
2 but becomes a pure imaginary for r > 1/√2. The integral measures for Gn
and Hn are defined in exactly the same way as φn in eq. (2.18). Although, in general, two
auxiliary fields are necessary for the nextnearestneighbor interactions in two directions,
values r = ±1/√2, and the required auxiliary field turns out to be only H.
it is somewhat surprising to find that G is decoupled from the other fields for particular
It is clear that S˜B has only the onsite and the nearestneighbor interactions which are
invariant under the PTtransformation
Defining a threecomponent field variable
φn, Hn, Gn
→
φ−n, H−n, G−n.
ϕn = (ϕn1, ϕn2, ϕn3) =
√2π
φn , √2π
Hn , √2π
Gn
we find that eq. (3.16) is just eq. (3.15) with N = 3.
– 9 –
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
In the previous section 3.2.1, we found that eq. (3.14) is a special case of eq. (3.15).
Hereafter we will try to derive a tensor network representation of general one (3.15). Before
that, let us see the hopping structure of the local Boltzmann weight, which is an important
building block of the tensor as shown in following sections 3.2.3 and 3.2.4.
It can be easily shown that S˜B is expressed as
S˜B = X L1 ϕn, ϕn+1ˆ + X L2 ϕn, ϕn+2ˆ ,
n∈Γ
n∈Γ
(3.20)
HJEP03(218)4
where Lµ is symmetric in the sense that Lµ (ϕ, ϕ′) = Lµ (ϕ′, ϕ) which is a consequence of
the PTinvariance of the action.3 All of the hopping terms with respect to the µ direction
are in Lµ (ϕn, ϕn+µˆ). This decomposition is actually not unique because the positions of
the onsite interactions and some constants are free to choose.
For our case in eq. (3.17), we find
Lµ (ϕn, ϕm) =
(φm − φn)2 +
W ′ (φn)2 + G2n + Hn2 + W ′ (φm)2 + G2m + Hm2
1
8
1
2
1
rW ′ (φn) + αGn + (−1)δµ 2 βHn
− rW ′ (φm) − αGm − (−1)δµ 2 βHm (φm − φn) .
Note that βHn and βHm have the different signs for µ = 2.
The Boltzmann factor e−S˜B can be written as
e−S˜B = Y
2
Y fµ (ϕn, ϕn+µˆ)
n∈Γ µ =1
f
µ ϕ, ϕ′ = e−Lµ (ϕ,ϕ′),
with
which is symmetric in the same sense as that of Lµ . This symmetric property plays an
important role in the subsequent discussion.
3.2.3
Discretization of noncompact field
The noncompactness of the variable ϕ is cumbersome in extracting the tensor structure
from fµ (ϕn, ϕn+µˆ) in practice. There are several possible ways to make the indices of the
tensor. In our method, we first carry out a discretization of the variable ϕ itself, which
automatically makes the partition function in eq. (3.15) into a discretized form.
Lµ (ϕ, ϕ′) = (Kµ (ϕ, ϕ′) + Kµ (ϕ′, ϕ)) /2.
2
3We can express the action as S˜B = Pn∈Γ Pµ =1 Kµ (ϕn, ϕn+µˆ) using a trial choice of Kµ .
Actually, Kµ (ϕn, ϕn+µˆ) transforms to Kµ (ϕ−n, ϕ−n−µˆ) by the PTtransformation, and the PTinvariance of
2
the action tells us that S˜B = Pn∈Γ Pµ =1 Kµ (ϕn+µˆ, ϕn). Thus the symmetric Lµ is always defined as
(3.21)
(3.22)
(3.23)
To make the discussion of the discretization clearly understood, let us begin with a
onedimensional integral
which converges for a given function f (x). We can formally approximate this integral with
a discretized form
Z ∞
−∞
I =
dxf (x) ,
I (K) =
X(disc.)
x∈SK
f (x) ,
for which I = limK→∞ I (K) is simply assumed. K is a parameter to control the
approximation of the integral by the sum. Now we suppose that SK is a set containing K
numbers, x1, x2, . . . , xK , which are given by a discretization scheme (disc.), and that P(xd∈isScK.)
is a summation of x ∈ SK with some factors, for instance, (disc.)dependent weights. A
multidimensional extension (SK → SKN ) is straightforward by defining SKN as a set of the
multidimensional discrete points.
The Gauss–Hermite quadrature gives a concrete example of this abstract definition.
The r.h.s. of eq. (3.25) is then defined as follows:
X(GH)
x∈SK
K
i=1
f (x) ≡
2
X wiexi f (xi) .
Here xi (i = 1, . . . , K) is the ith root of the Kth Hermite polynomial, and wi are the
weights given by the Hermite polynomial and xi. The r.h.s. of eq. (3.26) has an extra
exponential function because this quadrature is designed so that f (x) which has a damping
factor e−x2 is well approximated. In this case, we find that SK is a set of the roots and
the weight wiexi2 is the ingredient of P(GH). For a wellbehaved f (x), one can expect that
I = limK→∞ I (K).
With the prescriptions above, eq. (3.15) can be discretized as
√2π
N
X(disc.) Y2 fµ (ϕn, ϕn+µˆ)
ϕn∈SKN
µ =1
by replacing the measures for ϕn by P(ϕdnis∈cS.)N .4,5 Note that we use the same discretization
K
scheme for all components of ϕn. Here eq. (3.22) is also used, and K is the number of
discrete points. It is found that fµ (ϕn, ϕn+µˆ) is a matrix whose indices are ϕn and ϕn+µˆ
which take the KN discrete numbers in SKN . In this way, we now consider fµ as a matrix,
and this fact provides a benefit for a numerical treatment; that is, one can use linearalgebra
techniques instead of the functional analysis. The indices of the tensor will be naturally
derived from this matrix structure of fµ as will be seen in section 3.2.4.
4Here a onedimensional discretization is applied to each component of ϕn. We may also use a more
general scheme that cannot be written as the superposition of onedimensional discretization.
5In general one can set different discrete points for each direction: Ki 6= Kj for i 6= j, although in the
following we assume a common K just for the simplicity.
(3.24)
(3.25)
(3.26)
(3.27)
HJEP03(218)4
In order to derive the tensor network structure from eq. (3.27), one needs to separate ϕn
and ϕn+µˆ in fµ . If this separation works, the original field ϕn can be traced out at each n.
Since fµ is a symmetric matrix with complex entries in general, which is found in the
previous sections 3.2.2 and 3.2.3, we carry out the Takagi factorization: for ϕ, ϕ′ ∈ SKN ,
KN
w=1
KN
s=1
f1 ϕ, ϕ′ = X UϕwσwUwTϕ′ ,
f2 ϕ, ϕ′ = X VϕsρsVsTϕ′ ,
where U and V are unitary matrices, U T and V T are the transposes of U and V ,
respectively, and σw and ρs are nonnegative. Note that this factorization depends on the
discretization scheme, which determines the set SK . Instead of the Takagi factorization,
we can also use the SVD as seen in the next section.
We thus find that eq. (3.27) is written as
where
ZB (K) =
{w,s} n∈Γ
X
Y TB (K)wnsnwn−1ˆsn−2ˆ
,
√2π
N
√σiρj σkρl X (disc.)
UϕiVϕj UϕkVϕl
ϕ∈SKN
for all indices. One can verify eq. (3.30) from eq. (3.27) by applying the factorization in
eqs. (3.28) and (3.29) to fµ (ϕn, ϕn+µˆ) for µ = 1, 2 with the local indices wn, sn. Then the
index wn (sn) can be interpreted as a variable defined on the link which connects n and
n + 1ˆ (n + 2ˆ), so eq. (3.30) forms a tensor network on the twodimensional lattice Γ as with
the case of the fermion partition function in eq. (3.9). Here one finds the correspondence
between the tensor indices and the hopping structure of the lattice action as in the fermion
part. From this one can see that the tensor network structure is originated from the kinetic
terms for both fermions and bosons.
We expect that, in the large K limit, ZB (K) converges to ZB with an exact tensor
network representation
ZB =
X
Y TBwnsnwn−1ˆsn−2ˆ
{w,s} n∈Γ
(3.28)
(3.29)
(3.30)
(3.31)
(3.32)
if we can find a proper discretization scheme so that TB (K) converges to TB in K → ∞. In
practice one has to confirm that ZB (K) converges to ZB with increasing K in the choice
of a discretization scheme. We will see this point in section 4.3.
3.2.5
We give some miscellaneous remarks which may be important for future applications and
deeper understanding of the symmetry of the tensor network.
The tensor network representation in the form of eq. (3.32), which gives the boson
partition function in eq. (3.14), is not uniquely determined. Let F and G be regular K × K
matrices. We then find that eq. (3.32) also holds for another uniform tensor T˜B given by
˜
TBwsw′s′ = TBijklFiwGjsFk−w1′ Gl−s′1
(3.33)
HJEP03(218)4
for all indices.
Furthermore, by using Fn and Gn that are regular matrices
satisfying the periodic boundary conditions on the twodimensional lattice Γ and
transforming TBwnsnwn−1ˆsn−2ˆ by Fn, Gn, Fn−−11ˆ, and Gn−−12ˆ, ZB can also be written in terms of the
nonuniform tensors. This means that the tensor network representation of the partition
function is invariant under the gauge transformations for tensors.
The expression of eq. (3.30) is rather general in the sense that we can always find it for
twodimensional PTinvariant theories with the real scalars. It is very easy to generalize
this result to more complicated cases, non PTinvariant actions which, for example, have
only one of φ2n+µˆφn or φn−µˆφn terms or the theories with the complex scalars. For those
2
theories, although fµ is not symmetric in general, we can use the SVD instead of the Takagi
factorization. Then, U T and V T in eqs. (3.28) and (3.29) are replaced by other unitary
matrices, and we can express the partition function by a similar construction of the tensor
to eq. (3.31), where the second U and the second V are replaced with the other ones. An
extension to the higherdimensional theories is also straightforward.
We have much simpler expressions for the cases of SB,naive given in eq. (2.13) because
the auxiliary fields are not needed (N = 1) and Lµ is isotropic and given by a single L:
Equation (3.31) then becomes
L φ, φ′ =
1
2
φ − φ 2 +
′
1
8
TB,naive (K)ijkl = √2π
1
√σiσj σkσl X(disc.)
UφiUφj UφkUφl
because fµ ≡ f = e−L for µ = 1, 2 and because V = U and ρi = σi in eqs. (3.28) and (3.29).
In this case, instead of the Takagi factorization, we can use the SVD:
where O and P are real symmetric matrices. Then we have
K
w=1
f φ, φ′ = X OφwσwPwTφ′ ,
TB,naive (K)ijkl = √2π
1
√σiσj σkσl X(disc.)
OφiPφj OφkPφl.
φ∈SK
φ∈SK
(3.34)
(3.35)
(3.36)
(3.37)
We have seen that the fermion and the boson partition functions can be expressed as the
tensor networks in the previous two sections. By combining these results, we can also
express the total partition function as a tensor network.
Before presenting the total tensor, let us introduce combined indices Xn, Tn. Xn is
defined as Xn = (un, vn, wn), where (un, vn) and wn are indices of the fermion and the
boson tensors, respectively. Tn is also defined as Tn = (pn, qn, sn), and the dimension of
Xn and Tn is 2 × 2 × KN .
The total tensor is made by replacing e−SB(φ) in eq. (3.14) with e−SB(φ)ZF (φ) and
repeating the same procedure for making the tensor network representation of the boson
partition function. Additional contributions by ZF do not give any complexity. We find that
the total tensor network representation is given by the boson one in eq. (3.30) multiplied
by ZF (φ) from the right:
Z (K) =
X
Y T (K)XnTnXn−ˆ1Tn−2ˆ
with
T (K)XT X′T ′ =
1
√2π
ξn+1ˆξn
√σwρsσw′ ρs′ X (disc.)
ϕ∈SKN
UϕwVϕsUϕw′ Vϕs′ TF (φ)uvpqu′v′p′q′ , (3.39)
where U , V , σw, and ρs are given by eqs. (3.28) and (3.29). The measure DΞuvpq is given in
eq. (3.10), ξ, ξ¯, χ, χ¯, η, η¯, ζ, ζ¯ are onecomponent Grassmann numbers, and TF is the tensor
for the fermion part defined in eq. (3.11). Note that TF (φn) includes only φn which is a
component of ϕn. The total tensor T (K)XT X′T ′ is uniformly defined on the lattice.
Now the original partition function Z is expressed as a tensor network Z (K). We
have built it for a general superpotential by focusing on the hopping structure of the
lattice action. Introduction of local interaction terms does not change our formulation but
rather elements of tensor. Moreover, the same structure of the tensor network leads to the
same order of computational complexity.6 We will numerically verify that Z (K) indeed
converges to Z by using the TRG as a coarsegraining scheme for the tensor network in
the next section.
4
Numerical test in free theory
The partition function of the lattice N = 1 Wess–Zumino model has been expressed as
a tensor network in eq. (3.38). In this section, we test the expressions in the free theory
given by eq. (2.14) varying the mass for three lattice sizes V = 2 × 2, 8 × 8, 32 × 32 with
the periodic boundary conditions. Numerical tests in the free theory are effective to study
6If the superpotential contains hopping terms, the hopping structure of the lattice action changes and
one has to slightly modify the derivation of the tensor network representation.
whether the tensor is correctly given by our new formulation because the tensor network
structure is derived from the hopping terms in the action, i.e. the kinetic terms. The
computation is performed with the value of the Wilson parameter r = 1/√2 to reduce the
computational cost because the auxiliary field G is decoupled as seen in section 3.2.1.
In sections 4.2 and 4.3, we compute ZF and ZB individually using the (Grassmann) TRG
since they are independent with each other in the free theory. In section 4.3, the Witten
index given by the total partition function Z is computed by the Grassmann TRG. Since
the free theory is exactly solvable, we can compare an obtained result XTRG with the exact
solution Xexact by computing
In what follows, we briefly describe the TRG while introducing Dcut which defines the
truncated dimension of tensors. The SVD allows us to express a tensor Tijkl (i, j, k, l =
1, 2, · · · , N ) of which the tensor network representation of a partition function Z is made
as Tijkl = PIN=21 SijI σI (V †)Ikl, where S and V are unitary matrices and σI is the singular
σ2 ≥ σ3 ≥ · · · σN2 ≥ 0.7 In the TRG, Tijkl is approximately decomposed:
value of Tijkl. We assume that the singular values are sorted in descending order: σ1 ≥
(4.1)
(4.2)
(4.3)
Dcut
I=1
Dcut
I=1
Tijkl ≈
X SijI σI (V †)Ikl,
Tijkl ≈
X Sl′iI σI′ (V ′†)Ijk.
where Dcut, which is fixed throughout a computation, is used to truncate the dimension
of the tensor indices if it is smaller than N 2. If not so, the summation in eq. (4.2) is done
up to N 2 without the truncation. A similar decomposition can be done with a different
combination of the indices:
contracting the rankthree tensors √
σS, √σ′S′, √σV, √
The coarsegrained tensor TInJeKwL with I, J, K, L = 1, . . . , min{Dcut, N 2} is then given by
σ′V ′ and forms a network again as
with Tijkl. We can compute the partition function Z by repeating this procedure. Since
the number of tensors decreases through the coarsegraining, Z is finally given by a single
tensor for which the indices are contracted: Z = PID,Jcu=t 1 TInJeIwJ . More details are shown in
ref. [37], and appendix A is given for the Grassmann cases.
We employ the Gauss–Hermite quadrature (3.26) to discretize the integrals of φ and
H in (3.22):
X (GH) Y2 fµ (φn, Hn, φn+µˆ, Hn+µˆ) ,
(4.4)
7Strictly speaking, S and V are matrices with respect to the row specified by i, j and the column I, and
σI is the singular values of the matrix Tijkl with the row i, j and the column k, l. In addition, S and V are
taken to be real symmetric ones when Tijkl ∈ R for all i, j, k, l.
fµ (ϕn, ϕm) = exp − 2
1
φn/√2π, Hn/√2π is again used for the notational simplicity. SK is a set of the roots of
the Kth Hermite polynomial. We use the SVD to decompose fµ , which are K2
The twodimensional variable ϕn = (ϕn,1, ϕn,2) =
× K2 real
symmetric matrices, as
where
where
(4.6)
(4.7)
(4.8)
(4.9)
K2
w=1
K2
s=1
f1(ϕ, ϕ′) = X OϕwσwPϕ′w,
f2(ϕ, ϕ′) = X SϕsρsTϕ′s,
where σ1 ≥ σ2 ≥ . . . ≥ σK2 and ρ1 ≥ ρ2 ≥ . . . ≥ ρK2 . For reducing the memory usage
and the computational cost, we initially approximate the tensor network representation of
eq. (4.4) by Dinit ≤ K2:
Dinit Dinit
Y
X
n∈Γ wn=1 sn=1
ZB (K) ≈
X TB (K)wnsnwn−1ˆsn−2ˆ
,
Note that Dinit defines the bond dimension of the initial tensor. We will simply take
Dinit = Dcut for evaluating ZB in section 4.3 and Dinit = Dcut/2 for the Witten index in
section 4.4 because the bond dimension does not change after the coarsegraining steps
under these choices.
Here we mention the computational costs for the coarsegraining of tensor networks
and for the construction of tensors. Both of them are mainly consists of the SVD and the
contraction of tensor indices. Since the cost of the numerical SVD for square matrices is
proportional to the third power of the matrix dimension, the computational effort required
for the numerical decomposition described in eq. (4.2) and in eqs. (4.6) and (4.7) are in
proportion to N 6 and K6, respectively. A contraction of tensor indices are expressed as a
summation of them, so the cost of the contraction depends on the number of the tensor
indices. Then it is proportional to Dcut6 when contracting the rankthree tensors described
around eqs. (4.2) and (4.3), and is proportional to K2
× Dinit4 when building the tensor in
eq. (4.9). For the coarsegraining step, one can find that the volumedependence of the cost
is milder than Dcutdependence as follows. Since the TRG is a coarsegraining of
spacetime, one can reach a large spacetime volume by simply iterating the same local blocking
procedures.
More directly, the computational cost of the TRG is proportional to the
logarithm of the spacetime volume, i.e. the number of iterations. Summarizing the above,
ln V , and that for the construction of tensors is proportional to max K6, K2
the computational cost for the coarsegraining of tensor networks is proportional to Dcut6 ×
× Dcut4 ,
where N = Dinit = Dcut is simply assumed.
Free MajoranaWilson fermion
with varying m for V = 2 × 2 (top), 8 × 8 (center), 32 × 32 (bottom). The green, blue,
and yellow symbols denote the results for three different bond dimensions: Dcut = 8, 12,
16, and the solid and open ones indicate the positive and negative sign of the Pfaffian,
respectively. The purple curves represent the exact solutions given by eq. (2.23). Three
negative peaks at m = 0, −
√2, −2√2 correspond to the fermion zero modes, and the exact
Pfaffian has the negative sign for −2 2 < m < 0 as can be seen in eq. (2.23).
√
In the top plot of figure 1, the green symbols (Dcut = 8) around the peak at the center
are rather deviated from the exact solution, and they even have the opposite sign. The
deviation becomes smaller as Dcut increases, and the yellow symbols (Dcut = 16) have the
correct sign and agree well with the exact one even near the peak. The situation is further
improved by taking larger volumes even for the smallest Dcut, and the numerical results fit
well with the analytical curve in the center and the bottom figures.
These observation can also be clearly understood in figure 2, which shows the relative
errors δ (ln ZF) given by eq. (4.1). Note that the case for Dcut = 16 on V = 2 × 2 have
extremely small errors. This is because the maximal bond dimension of the coarsegrained
tensors on V = 2 × 2 lattice is less than or equal to Dcut. In other words, no truncation
occurs in the TRG steps. This striking feature is only found in the pure fermion case. In
contrast, the discretization error and the truncation error are inevitable in the boson case
since the approximation already enters in deriving the tensor network representation of the
boson partition function, and furthermore the tensor indices are truncated to carry out the
numerical evaluation as seen in previous section. For all volumes used in the computation,
the relative errors almost monotonically decreases as Dcut increases.
Thus we can conclude that the Pfaffian with the correct sign is reproduced from the
tensor network representation in eq. (3.9) with eq. (3.11) using the Grassmann TRG within
tiny errors O 10−3 for physically important parameters, m ≪ 1, and larger volumes.
4.3
Free Wilson boson
The boson partition function is given as a discretized form ZB (K) in eq. (4.4) by applying
the Gauss–Hermite quadrature to the integrals of φ and H. Then K is the number of
the discrete points. We prepare the initial tensor network approximately as eq. (4.8) and
compute it using the TRG for m > 0 because the adopted quadrature does not effectively
work for m < 0 (we will see this point later.). It is, however, sufficient to study the case
of m > 0 because the boson action does not depend on the sign of m, but on m2, in the
continuum theory.
HJEP03(218)4
V = 2 × 2 (top), 8 × 8 (center), 32 × 32 (bottom). The solid (open) symbols represent the positive
(negative) sign of ZF.
Figure 3 shows the logarithm of ZB (K) with fixed K = 64, and figure 4 shows the
corresponding relative errors defined by eq. (4.1). One can see that the TRG results are
consistent with the exact ones for large m in all of the lattice sizes and Dcut = 16, 24,
of ZF, respectively.
32. Figure 5 shows that the results are systematically improved by increasing Dcut as
one expects. The exponential improvement may be explained as follows. Usually the
singular values of the tensor are exponentially decaying; thus from a local point of view
– 20 –
figures show the results for V = 2 × 2, 8 × 8 and 32 × 32, respectively.
the truncation error gets exponentially smaller by increasing Dcut. Since the free energy
consists of the local tensors, it is likely that its error shows such a behavior as well.
The growth of the errors is observed near m = 0. Roughly speaking, this is because
the massless theory has no damping factors in fµ of eq. (4.5). We can show that fµ is
expressed as
f
µ ϕ, ϕ′ = exp
φ − φ′ 2 .
(4.10)
One can see that the damping factors are actually provided for m > 0 with the damping
rate m2 but is not for −4 2 < m < 0 on the line φ = −φ′, so the quadrature does not work
for m < 0. For m > 0, we have to take K larger as m decreases so that the quadrature
retains effective. That structure is encoded in the initial tensor in eq. (4.9) via the matrices
O, P, S, T and the singular values σw, ρs in eqs. (4.6) and (4.7). The singular values of the
initial tensor have unclear hierarchies for small masses as seen in figure 6. Thus we find
that, if m approaches zero from the right, we have to take K and Dcut as large as possible
to obtain the precise result.8
The Kdependence of the relative errors is investigated in figure 7. In order to purely
see the discretization effect due to finite K, we set the maximum bond dimension of the
tensor K2 and choose the lattice size V = 2×2 that allows us to carry out a full contraction
for the computation of the partition function. Although there are no other systematic errors
except for finite K, the value of K is practically restricted up to 10. Figure 7 shows that
the errors decrease by increasing K. From this we can say that a simple discretization
8Such a bad behavior could go away once the φ4 interaction term is introduced into the action because
it provides the fast damping factor in fµ .
scheme such as the Gauss–Hermite quadrature well approximates the original integrals if
K is sufficiently large, and that the tensor network representation reproduces the correct
values of the boson partition function.
4.4
Witten index of the free N = 1 WessZumino model
The Witten index computed by the Grassmann TRG is shown in figure 8. Figure 9 shows
the relative error of the Witten index. As discussed in section 2.2, the fermion and the
boson are decoupled from each other in the free case. In this section, however, we treat the
free Wess–Zumino model as a combined system of fermions and bosons; thus we perform the
Grassmann TRG for a single tensor network. One can see that the results tend to converge
to the exact values by increasing Dcut. The obtained indices with Dcut = 64 (yellow
symbols) take the values near one compared with those of Dcut = 32 (green symbols).
8 × 8 (center), 32 × 32 (bottom).
– 25 –
Thus we can conclude that eq. (3.38) gives a correct tensor network representation of
the twodimensional lattice N = 1 Wess–Zumino model. ZF and ZB become extremely
large and extremely small, respectively, for large spacetime volume. For instance, ZF are
of the order of O 10400 at m = 1 on V = 32 × 32 lattice as seen in figure 1. Surprisingly,
O (1) values are obtained as the Witten index as seen in figure 8. Namely, the boson effect
balancing huge ZF is correctly reproduced using the Grassmann TRG for the total tensor.
So we can say that the TRG is a very promising approach to study the supersymmetric
field theories.
5
Summary and outlook
We have shown that the twodimensional lattice N = 1 WessZumino model is expressed
as a tensor network. The known techniques of making a tensor were refined in the fermion
sector and generalized in the boson sector in the sense that it is possible to define a
tensor for any way of discretizing the integrals for scalar fields. We have also tested our
formulation in the free theory by estimating the Witten index and comparing it with the
exact solution. The resulting indices reproduce the exact one as Dcut, the dimension of the
truncated tensor indices in the TRG, increases.
Now we are tackling the issue on the supersymmetry breaking by estimating correlation
functions from the tensor network. Before investigating the physical breaking effects, we
have to show that the artificial ones by the lattice cutoff disappears in the continuum
limit beyond the arguments of the perturbation theory. We will estimate the expectation
value of the action, the supersymmetric WardTakahashi identity, and the mass spectra
of fermions and bosons to show it. We will then see the supersymmetry breaking in the
model with the doublewell potential by estimating several physical quantities and study
the phase structure in detail.
Although we have only dealt with the Wilson type discretization of derivatives, one may
use another way such as the domain wall discretization. In that case, partition functions
or Green’s functions will be represented as three dimensional tensor networks. For such
higher dimensional tensor networks, the higher order TRG was introduced in ref. [38], and
the Grassmann version was also proposed in ref. [39]. In this way one can in principle
go this direction; however, the computational cost could be severe. Therefore further
improvements of the algorithm might be needed for the actual computation in higher
dimensions.
We emphasize that the methodology of constructing the tensor is given for any
superpotential, that is, any interacting case, in this paper. Since the WessZumino model
consists of various building blocks: the scalar field, the Majorana fermion, and their
interactions such as the Yukawa and the φ4interactions, we expect that our method could be
very useful in TRG studies of other theories.
Coarsegraining step in Grassmann TRG
In this appendix, we describe the coarsegraining step in the Grassmann TRG for the
current bosonfermion system. We basically follow ref. [34], which deals with a pure fermionic
model (the Nf = 1 Gross–Neveu model) and show the method in our notation making the
difference that comes from the boson part clear.
We begin with the partition function that initially takes the following form:9
Z =
X
Y TXnTnXn−ˆ1Tn−2ˆ
Z
Y dΞunvpq
n∈Γ
· Y
n∈Γ
ξn+1ˆξn
where the local measure of Grassmann variables is defined as
dΞunvpq = dξnun dχvnn dηnpn dζnqn dξ¯nun−1ˆ dχ¯vnn−1ˆ dη¯npn−2ˆ dζ¯nqn−2ˆ ,
and the tensor elements are not zeros only when
un + vn + pn + qn + un−1ˆ + vn−1ˆ + pn−2ˆ + qn−2ˆ mod 2 = 0
holds.
The tensor T is made of the fermionic one TF (φn)unvnpnqnun−1ˆvn−1ˆpn−2ˆqn−2ˆ in
eq. (3.11) and the boson one TB (K)wnsnwn−1ˆsn−2ˆ in eq. (3.31) as in eq. (3.39). The indices
un, vn, pn, qn take two values 0 or 1 while wn, sn run from 1 to Dinit as seen in section 4.1.
The total indices Xn and Tn are given by Xn = (un, vn, wn) and Tn = (pn, qn, sn), and they
run from 1 to 2 × 2 × Dinit. As mentioned in section 4.1, we set Dinit = Dcut/2 for the
actual computations.
The coarsegraining of a tensor network mainly consists of three steps: the SVD of
tensors, a decomposition of Grassmann measures, and a contraction of the indices and
taking the integrals of Grassmann variables defined on Γ. The SVD and the decomposition
for Grassmann measures are performed in a different manner for even and odd sites. We
will see that the coarsegrained tensors take the same form as (A.1) with v = q = 0 and
are defined on the coarsegrained lattice
Γ⋆ =
n +
2
1 ˆ1 + 2ˆ
n = (n1, n2) ∈ Γ, where n1 + n2 is an even integer. .
(A.4)
between n and n⋆ is shown in figure 10.
This means that Γ⋆ is a set of the center of the plaquette (n, n + 1ˆ, n + 1ˆ + 2ˆ, n + 2ˆ) with
even sites n. The unit vectors of Γ⋆ are 1ˆ⋆ = 1ˆ + 2ˆ and 2ˆ⋆ = 1ˆ − 2ˆ. The correspondence
First, on even sites n ∈ Γ, we just take the truncated SVD of T like eq. (4.2):
TXnTnXn−ˆ1Tn−2ˆ ≈
Dcut
X
wn⋆−1ˆ⋆ =1
U(1XnTn)wn⋆−1ˆ⋆ σw13n⋆−ˆ1⋆ Vw3n†⋆−1ˆ⋆ (Xn−1ˆTn−2ˆ)
(A.5)
9Although Z and T depends on K as eqs. (3.38) and (3.39), K is simply abbreviated here.
(A.1)
(A.2)
(A.3)
HJEP03(218)4
where
where
S, Y are also shown. The blue symbols represent the tensors in the r.h.s. of eq. (A.1), and the red
ones represent the decomposed rankthree tensors appear in the following paragraphs.
n⋆ = n +
2
1 ˆ1 + 2ˆ ∈ Γ⋆.
The Grassmann measures are divided into two pieces as
dΞunvpq =
Z
Θn,unvnpnqn dξ¯nu⋆n⋆−ˆ1⋆
1
Θn,un−1ˆvn−1ˆpn−2ˆqn−2ˆ dξnu⋆n−⋆−1ˆˆ1⋆⋆
3
ξ¯n⋆ ξn⋆−ˆ1⋆
un⋆−1ˆ⋆ ,
and the new index un⋆−1ˆ⋆ is defined as
Θn,abcd = dξnadχbndηncdζnd,
1
Θn,abcd = dξ¯nadχ¯bndη¯ncdζ¯nd,
3
un⋆−1ˆ⋆ ≡ (un + vn + pn + qn) mod 2.
Note that each parenthesized factors on the r.h.s. of eq. (A.7) are Grassmanneven under
eqs. (A.3) and (A.10), and one can freely move them to make a new tensor. The tensor
in eq. (A.5) and the measures in eq. (A.7) have been decomposed into (XnTn)part and
(Xn−1ˆTn−2ˆ)part, and they are connected via the new indices (un⋆−ˆ1⋆ , wn⋆−ˆ1⋆ ).
For odd lattice sites n + 2ˆ next to even sites n, we take another decomposition:
TXn+ˆ2Tn+2ˆXn−1ˆ+ˆ2Tn ≈
Dcut
X
sn⋆−2ˆ⋆ =1
U(2TnXn+2ˆ)sn⋆−ˆ2⋆ σs2n4⋆−ˆ2⋆ Vs4n†⋆−ˆ2⋆ (Tn+2ˆXn−1ˆ+ˆ2)
(A.11)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
The Grassmann measure is also decomposed into (T X
n
)part and (T
uvpq
Z
ˆ
n+2,p q u
n n
v
ˆ ˆ
n+2 n+2
η¯ ⋆η
n
n −2
⋆ ˆ⋆
p
n −2
⋆ ˆ⋆
,
dη¯
p
n
Θ
4
ˆ
n+2,p
q
u
ˆ ˆ ˆ ˆ ˆ ˆ
n+2 n+2 n−1+2 n−1+2
v
dη
p
n −2
⋆ ˆ⋆
⋆
n −2
where
and p
n −2
⋆ ˆ⋆
is defined by
Θ
Θ
2
4
n,abcd
n,abcd
= ( 1)
a b c d
dη¯ dζ dξ dχ ,
n n n n
a b c d
= dη dζ dξ dχ¯ ,
n n n n
¯
p
n −2
⋆ ˆ⋆
p + q + u
n n
+ v
mod 2.
Note that the extra sign in eq. (A.13) arise from the rearrangement of the Grassmann
We thus find that the partition function can be expressed in terms of coarsegrained
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
HJEP03(218)4
Z
X
Y
{Y,S} n ∈Γ
⋆ ⋆
new
TY ⋆S ⋆Y
n n
S
n −1
⋆ ˆ⋆
n −2
⋆ ˆ⋆
Z
Y
n ∈Γ
⋆ ⋆
dΞ
up
n
n ∈Γ
⋆ ⋆
⋆ ˆ⋆ n
ξ ⋆
u ⋆
n
n +2
⋆ ˆ⋆ n
η ⋆
p ⋆
n
dΞ
up
n
= dξ
u ⋆
n
n
dη
p ⋆
n
n
dξ
u
¯ n −1
⋆
⋆ ˆ⋆
n
dη¯
p
n
,
defined by
ˆ
n+2
ˆ
n+2
ˆ
n+1
ˆ
n+1
) mod 2,u ⋆ (p
n
ˆ
n+1
ˆ
n+1
+u +v ) mod 2,p ⋆
n n n
· (u +v +p +q ) mod 2,u
n n n n
n −1
⋆ ˆ⋆
(p +q +u
n n
ˆ
n+2
) mod 2,p
n −2
⋆ ˆ⋆
(A.18)
The constraints described in eqs. (A.10) and (A.15) with eq. (A.3) are explicitly imposed
as Kronecker deltas.
new
TY ⋆S ⋆Y
n n
q
= σ
S
n −1
⋆ ˆ⋆
n −2
⋆ ˆ⋆
13
w ⋆ s ⋆
n n
σ
24
σ
13
w
X
X
X T X
n n
T
ˆ ˆ
n+2 n+1
σ
24
s
U
1
n −1
⋆ ˆ⋆
n −2
⋆ ˆ⋆
ξ
ξ
u
n
ξ
ˆ n
ˆ ˆ ˆ
n+1+2 n+2
ξ
u
ˆ
n+2
χ
ˆ n
Z
Θ
2
· (u
δ
Θ
1
v
n
ˆ ˆ ˆ
n+1+2 n+2
χ
δ
U
2
Θ
4
p
n
v
ˆ
n+2
δ
ˆ
n+2,p q u
n n
v
ˆ ˆ
n+2 n+2
n,u v p q
ˆ
n n n n n+1,p
q
ˆ
n+1 n+1
u v
ˆ n n
ˆ ˆ
n+1+2,u
p
v
q
ˆ ˆ ˆ ˆ
n+2 n+2 n+1 n+1
(X T )w
n n
n −1
⋆ ˆ⋆
(T X
n
w ⋆(X
n
T
ˆ ˆ
n+2 n+1
) s ⋆(T
n
n+1
ˆ n
X )
V
4†
q
n
η
V
3†
Θ
3
n+2
η
ˆ n
ζ
ζ
ˆ n
η¯
ˆ ˆ ˆ
n+1+2 n+1
p
n+1 ¯
ˆ
ζ
ˆ ˆ ˆ
n+1+2 n+1
ζ
q
ˆ
n+1
Owing to the similarity of the initial tensor and the resulting one, the procedure
described in this appendix can be simply iterated by setting eq. (A.18) as an initial tensor
for the next coarsegraining step.10 An important change is the absence of χ and ζ, so one
has to also set vn and qn to 0 for the following steps. Equation (A.18) and the initial tensor
have the different contents of indices, e.g. Xn = (un, vn, wn) reduces to Yn⋆ = (un⋆ , wn⋆ )
after the coarsegraining step. This means that the dimension of the tensor indices changes
from 2 × 2 × Dinit to 2 × Dcut. We take Dinit = Dcut/2 in section 4.4 to retain the size of
tensors for the sake of simplicity.
Note also that the definition of the unit vectors turns out to be proportional to original
ones after the next coarsegraining step, i.e. 1ˆ⋆⋆ = 1ˆ⋆ + 2ˆ⋆ = 2 · 1ˆ and 2ˆ⋆⋆ = 1ˆ⋆ − 2ˆ⋆ = 2 · 2ˆ.
Although the coarsegrained lattice Γ⋆ is not isotropic and the boundary conditions are not
the same as original ones, this strange situation will recover after the next coarsegraining
Acknowledgments
We thank Dr. Yuya Shimizu for his many helpful comments. This work is supported in
part by JSPS KAKENHI Grant Numbers JP16K05328, JP17K05411, GrantsinAid for
Scientific Research from the Ministry of Education, Culture, Sports, Science and
Technology (MEXT) (No. 15H03651), MEXT as “Exploratory Challenge on PostK computer
(Frontiers of Basic Science: Challenging the Limits)”, and the MEXTSupported Program
for the Strategic Research Foundation at Private Universities Topological Science (Grant
No. S1511006).
Open Access.
This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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