Calculation of critical exponents on fractal lattice Ising model by higher-order tensor renormalization group method Jozef Genzor

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Calculation of critical exponents on fractal lattice Ising model by higher-order tensor

renormalization group method

Jozef Genzor∗

Physics Division, National Center for Theoretical Sciences,

National Taiwan University, Taipei 10617, Taiwan

(Dated: March 22, 2023)

The critical behavior of the Ising model on a fractal lattice, which has the Hausdorﬀ dimension

log412 ≈1.792, is investigated using a modiﬁed higher-order tensor renormalization group algorithm

supplemented with automatic diﬀerentiation to compute relevant derivatives eﬃciently and accu-

rately. The complete set of critical exponents characteristic of a second-order phase transition was

obtained. Correlations near the critical temperature were analyzed through two impurity tensors

inserted into the system, which allowed us to obtain the correlation lengths and calculate the criti-

cal exponent ν. The critical exponent αwas found to be negative, consistent with the observation

that the speciﬁc heat does not diverge at the critical temperature. The extracted exponents sat-

isfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps

most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisﬁed very

well, assuming the Hausdorﬀ dimension takes the place of the spatial dimension. Moreover, using

automatic diﬀerentiation, we have extracted four critical exponents (α,β,γ, and δ) globally by

diﬀerentiating the free energy. Surprisingly, the global exponents diﬀer from those obtained locally

by the technique of the impurity tensors; however, the scaling relations remain satisﬁed even in the

case of the global exponents.

I. INTRODUCTION

The phase transition and critical phenomena are

prominent topics in condensed matter physics [1]. The

scaling behavior of physical quantities, such as the mag-

netic susceptibility and speciﬁc heat, is characterized by

the critical exponents when approaching the critical tem-

perature [2]. Various scaling assumptions give the rela-

tions between the critical exponents. One of such rela-

tions derived from the hyperscaling hypothesis, which is

expected to be valid for d≤4, involves the system di-

mension d. An intriguing question is the validity of the

hyperscaling hypothesis in the case of non-integer dimen-

sional systems such as fractals, where critical phenomena

remain understudied.

To some extent, the hyperscaling relation expressed in

terms of the ratios of the critical exponents β/ν and γ/ν

has already been considered in the literature

deﬀ = 2β

ν+γ

ν,(1)

where deﬀ is the eﬀective dimension that controls hyper-

scaling. Nevertheless, the question of whether the eﬀec-

tive dimension deﬀ is the same as the Hausdorﬀ dimen-

sion dHremains open for debate.

The validity of the hyperscaling relation was mostly

tested in the case of Sierpi´nski carpets. The Ising model

on Sierpinski carpets SC(3,1) of Hausdorﬀ dimension

dH= ln 8/ln 3 ≈1.8927 and SC(4,2) of Hausdorﬀ dimen-

sion dH= ln 12/ln 4 ≈1.7924 was studied using Monte

Carlo in conjunction with the ﬁnite-size scaling method

∗jozef.genzor@gmail.com

in Ref. [3]. The existence of an order-disorder transition

at ﬁnite temperature was clearly shown in both cases,

and the critical exponents, including their errors, were

estimated. In this case, the hyperscaling relation holds if

one assumes that the eﬀective dimension is the Hausdorﬀ

dimension. In case of SC(3,1), the exponent α, for which

the hyperscaling relation reads νdH= 2−α, was found to

be negative. Consistent with the previous conclusions is a

newer Monte Carlo study with ﬁnite-size scaling analysis

in Ref. [4] where the authors studied four diﬀerent Sier-

pinski carpets with the Hausdorﬀ dimension dHbetween

1.9746 and 1.7227, namely SC(5,1), SC(3,1), SC(4,2),

and SC(5,3). In the case of SC(3,1), the authors found

deﬀ to be only slightly smaller than dH. In Ref. [5], the

short-time dynamic evolution of an Ising model on Sier-

pinski carpet SC(3,1) was studied using the Monte Carlo

method. The authors concluded that the eﬀective dimen-

sion for the second order phase transition is noticeably

smaller than the Hausdorﬀ dimension deﬀ ∼1.77 < dH.

Another short-time critical dynamic scaling study in the

case of various inﬁnitely ramiﬁed fractals with Hausdorﬀ

dimension within the interval 1.67 ≤dH≤1.98 can be

found in Ref. [6]. Their results are consistent with the

convergence of the lower-critical dimension toward d= 1

for fractal substrates and suggest that the Hausdorﬀ di-

mension may diﬀer from the eﬀective dimension. The val-

ues for the diﬀerent sets of fractals depart from deﬀ =dH

for dH≤1.85. However, due to large error bars, the au-

thors cannot state a deﬁnitive conclusion on the actual

dependence of deﬀ on dH.

The Ising model on a fractal lattice with Hausdorﬀ

dimension dH= ln 12/ln 4 ≈1.792 (which is diﬀerent

from SC(4,2) of the same dH) depicted in Fig. 1 was al-

ready probed by two diﬀerent adaptations of the Higher-

Order Tensor Renormalization Group (HOTRG, intro-

arXiv:2210.15268v3 [cond-mat.stat-mech] 21 Mar 2023

FIG. 1: The layout of the fractal lattice after three extension

steps, n= 3. Tiny circles represent the two-state Ising spins.

The horizontal and vertical lines represent the spin-spin inter-

actions. The number of sites grows as 12nwith the number

of extension steps n, whereas the number of outgoing bonds

grows as 2n+2.

duced in Ref. [7]): (1) genuine fractal representation

(with no structure ﬁlling the gaps) [8, 9], and (2) J1-J2

(“tunable”) fractal constructed on a square-lattice frame

with two types of couplings, J1and J2[10]. Geomet-

rically, these two methods constitute the same fractal

when (J1, J2) = (1,0) albeit represented diﬀerently. Two

critical exponents, βand δ, were extracted in both cases

using the technique of local impurity tensors. There is

a slight discrepancy between the values of the exponents

between the two methods, which can be attributed to

the diﬀerence in the details of the calculation of the local

magnetization rather than the model representation. The

magnetization in (1) is calculated on a single site located

far from the system’s external boundary, whereas in (2),

a partial average over central sites is employed. This dis-

crepancy is interesting since it indicates that there is a

positional dependency, at least in the case of local magne-

tization. Another interesting ﬁnding is that the speciﬁc

heat does not exhibit singular behavior around its maxi-

mum; however, a sharp peak was observed in a numerical

derivative of the speciﬁc heat at the critical temperature.

The question about the value of the critical exponent α

associated with the speciﬁc heat has remained open until

now.

The position dependence of local thermodynamic func-

tions was studied in Ref. [11], where HOTRG was

adapted to the classical Ising model on SC(3,1). The

critical temperature Tcwas found to be positionally in-

dependent, whereas the (local) critical exponent βwas

found to vary by two orders of magnitude depending on

lattice location.

Let us mention that the Monte Carlo studies achieve

only a relatively modest maximal value of the segmenta-

tion steps k≤8. In contrast, in the case of the Higher

Order Tensor Renormalization Group (HOTRG) method

used in the study of SC(3,1) in Ref. [11], numerical con-

vergence of the physical observables is achieved at k∼35

iterative extensions (generations) of the system.

In Ref. [12], the quantum phase transition of the

transverse-ﬁeld Ising model on the Sierpi´nski fractal with

the Hausdorﬀ dimension log23≈1.585 was studied by

a modiﬁed HOTRG method. Ground-state energy and

order parameter were calculated and analyzed. The sys-

tem was found to exhibit a second-order phase transition.

From the order parameter, the critical exponents βand

δwere estimated.

Recently it was shown that the higher-order derivatives

for the tensor network algorithms could be calculated ac-

curately and eﬃciently using the technique of automatic

diﬀerentiation [13, 14], which emerged from deep learn-

ing. Automatic diﬀerentiation is based on the concept

of the computation graph, which is a directed acyclic

graph composed of elementary computation steps. This

technology propagates the gradients through the whole

computation process with machine precision. In the case

of the tensor network algorithms, an essential technical

ingredient is to implement numerically stable diﬀerentia-

tion through linear algebra operations such as the Singu-

lar Value Decomposition (SVD). Applying the automatic

diﬀerentiation on our tensor network fractal, we can now

calculate speciﬁc heat very accurately as the ﬁrst deriva-

tive of the bond energy with respect to temperature or as

a second derivative of the free energy with respect to tem-

perature without introducing numerical errors due to the

ﬁnite step as in the case of numerical derivatives. With

such an accurate method to obtain the speciﬁc heat, it

should be possible to extract the associated critical expo-

nent αﬁnally. Similarly, the magnetic susceptibility can

now be calculated as a ﬁrst derivative of the spontaneous

magnetization with respect to the external ﬁeld or as a

second derivative of the free energy with respect to the

external ﬁeld. Having calculated the magnetic suscepti-

bility, one can extract the critical exponent γ. Finally,

let us emphasize, that diﬀerentiation of the free energy

would yield global thermodynamic quantities, which were

not calculated before.

A question of high interest is to numerically estimate

the critical exponent ν, which appears in the hyperscal-

ing relation together with the spatial dimension d. The

critical exponent νcan be extracted from the correla-

tion length, which can be obtained from the correlation

function. The method for calculation of the correlation

function using the Tensor Renormalization Group (TRG)

method was introduced in Ref. [15]. This method was

implemented and tested in the case of the square lattice

Ising model in Ref. [16]. A similar approach is conceiv-

able in the case of the HOTRG method; therefore, it can

be used on the fractal lattice under study.

In this study, we have extracted the remaining four

critical exponents from our HOTRG calculations with

the local impurity tensors by calculating the correlation

function for obtaining the exponents νand ηand by aug-

menting our computations with the automatic diﬀeren-

tiation for αand γ. The critical exponent αwas found

to be negative (α≈ −0.87), which is consistent with the

observation that the speciﬁc heat does not diverge at the

critical temperature. The exponents we extracted satisfy

the known relations given by various scaling assumptions

with reasonable accuracy. Perhaps most interestingly,

the hyperscaling relation, which contains the spatial di-

mension, is satisﬁed very well, assuming the Hausdorﬀ

dimension takes the place of the spatial dimension. More-

over, using automatic diﬀerentiation, we have extracted

four critical exponents (α,β,γ, and δ) globally by dif-

ferentiating the free energy. Surprisingly, the global ex-

ponents are very diﬀerent from those obtained locally by

the technique of the impurity tensors (for example, the

global exponent βis more than ﬁve times larger than the

local β); however, the scaling relations remain satisﬁed

even in the case of the global exponents.

II. MODEL REPRESENTATION

We consider the nearest-neighbors fractal-lattice Ising

model with the Hamiltonian

H=−JX

hiji

σiσj−hX

σi,(2)

where J > 0 is the ferromagnetic coupling, and his the

uniform magnetic ﬁeld. At each site i, the Ising variable

σitakes only two values, +1 or −1. For brevity, we set

J= 1 and h= 0 in the following. The partition function

of the Ising model deﬁned on the fractal lattice can be

expressed in terms of tensor network states deﬁned by

four types of local tensors represented by T,X,Y, and

Txix0

iyiy0

x x

WξxiWξx0

iWξyiWξy0

i,(3)

Xxix0

iyi=

x x

WξxiWξx0

iWξyi,(4)

Yyiy0

ixi=

WξyiWξy0

iWξxi,(5)

Qxiyi=

WξxiWξyi,(6)

where Wis a 2×2 matrix determined by the bond weight

factorization. While the choice for Wis arbitrary to a

certain degree, here we choose an asymmetric factoriza-

tion

W=pcosh 1/T psinh 1/T

pcosh 1/T −psinh 1/T ,(7)

where Tis the temperature. Notice that when two local

tensors are contracted via non-physical (auxiliary) index

x, the bond weight WB(σi, σj) = exp (σiσj/T ) is cor-

rectly re-expressed

WB(σi, σj) =

x=0

WξixWξjx,(8)

where the ﬁrst matrix index ξi= (1 −σi)/2 takes values

of 0 and 1 when σi= 1 and σi=−1, respectively.

The coarse-graining renormalization procedure intro-

duced in Ref. [8, 9] is used to calculate the partition

function. We start counting the iteration steps from

zero; therefore, we denote the initial tensors in Eqs. (3) –

(6) as T(n=0) =T,X(n=0) =X,Y(n=0) =Y, and

Q(n=0) =Q. At each iterative step n, the new tensors

T(n+1),X(n+1),Y(n+1), and Q(n+1) are created from the

previous-iteration tensors T(n),X(n),Y(n), and Q(n), ac-

cording to the following extension relations

T(n+1)

(x1x2)(x0

1x0

2)(y1y2)(y0

1y0

2)=

x'1

y1y2

,(9)

X(n+1)

(x1x2)(x0

1x0

2)(y1y2)=

x'1

,(10)

Y(n+1)

(y1y2)(y0

1y0

2)(x1x2)=

y1y2

,(11)

Q(n+1)

(x1x2)(y1y2)=

.(12)

The partition function Zn(T) of the system after nex-

tensions is evaluated as

Zn(T) = X

T(n)

iijj ,(13)

where we impose the periodic boundary conditions.

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CalculationofcriticalexponentsonfractallatticeIsingmodelbyhigher-ordertensorrenormalizationgroupmethodJozefGenzorPhysicsDivision,NationalCenterforTheoreticalSciences,NationalTaiwanUniversity,Taipei10617,Taiwan(Dated:March22,2023)ThecriticalbehavioroftheIsingmodelonafractallattice,whichhastheHausdor...

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Calculation of critical exponents on fractal lattice Ising model by higher-order tensor renormalization group method Jozef Genzor

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