# Entanglement entropy of two disjoint intervals in c=1 theories

**ABSTRACT** We study the scaling of the Renyi entanglement entropy of two disjoint blocks

of critical lattice models described by conformal field theories with central

charge c=1. We provide the analytic conformal field theory result for the

second order Renyi entropy for a free boson compactified on an orbifold

describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual

line. We have checked this prediction in cluster Monte Carlo simulations of the

classical two dimensional AT model. We have also performed extensive numerical

simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor

network techniques that allowed to obtain the reduced density matrices of

disjoint blocks of the spin-chain and to check the correctness of the

predictions for Renyi and entanglement entropies from conformal field theory.

In order to match these predictions, we have extrapolated the numerical results

by properly taking into account the corrections induced by the finite length of

the blocks to the leading scaling behavior.

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**ABSTRACT:**We develop a systematic method to extract the negativity in the ground state of a 1+1 dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose ρ_{A}^{T_{2}} of the reduced density matrix of a subsystem A=A_{1}∪A_{2}, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity E=ln‖ρ_{A}^{T_{2}}‖. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result E∼(c/4)ln[ℓ_{1}ℓ_{2}/(ℓ_{1}+ℓ_{2})] for the case of two adjacent intervals of lengths ℓ_{1}, ℓ_{2} in an infinite system, where c is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain.Physical Review Letters 09/2012; 109(13):130502. · 7.73 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We study the R\'enyi entropies of N disjoint intervals in the conformal field theories given by the free compactified boson and the Ising model. They are computed as the 2N point function of twist fields, by employing the partition function of the model on a particular class of Riemann surfaces. The results are written in terms of Riemann theta functions. The prediction for the free boson in the decompactification regime is checked against exact results for the harmonic chain. For the Ising model, matrix product states computations agree with the conformal field theory result once the finite size corrections have been taken into account.Journal of Statistical Mechanics Theory and Experiment 09/2013; 2014(1). · 1.87 Impact Factor - SourceAvailable from: iopscience.iop.org[Show abstract] [Hide abstract]

**ABSTRACT:**We consider the mutual Renyi information I^n(A,B)=S^n_A+S^n_B-S^n_{AUB} of disjoint compact spatial regions A and B in the ground state of a d+1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes R_{A,B}. We show that in general I^n(A,B)\sim C^n_AC^n_B(R_AR_B/r^2)^a, where a the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants C^n_{A,B} depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where 2x=d-1, we show that C^2_AR_A^{d-1} is proportional to the capacitance of a thin conducting slab in the shape of A in d+1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere S^{d-1} or an ellipsoid. For spherical regions in d=2 and 3 we obtain explicit results for C^n for all n and hence for the leading term in the mutual information by taking n->1. We also compute a universal logarithmic correction to the area law for the Renyi entropies of a single spherical region for a scalar field theory with a small mass.Journal of Physics A Mathematical and Theoretical 04/2013; 46(28). · 1.77 Impact Factor

Page 1

Entanglement entropy of two disjoint intervals in

c = 1 theories

Vincenzo Alba1, Luca Tagliacozzo2, Pasquale Calabrese3

1Max Planck Institute for the Physics of Complex Systems, N¨ othnitzer Str. 38,

01187 Dresden, Germany,

2School of Mathematics and Physics, The University of Queensland, Australia,

ICFO, Insitut de Ciencias Fotonicas, 08860 Castelldefels (Barcelona) Spain

3Dipartimento di Fisica dell’Universit` a di Pisa and INFN, Pisa, Italy.

Abstract.

We study the scaling of the R´ enyi entanglement entropy of two disjoint blocks

of critical lattice models described by conformal field theories with central charge

c = 1. We provide the analytic conformal field theory result for the second order

R´ enyi entropy for a free boson compactified on an orbifold describing the scaling

limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this

prediction in cluster Monte Carlo simulations of the classical two dimensional AT

model. We have also performed extensive numerical simulations of the anisotropic

Heisenberg quantum spin-chain with tree-tensor network techniques that allowed

to obtain the reduced density matrices of disjoint blocks of the spin-chain and to

check the correctness of the predictions for R´ enyi and entanglement entropies from

conformal field theory. In order to match these predictions, we have extrapolated

the numerical results by properly taking into account the corrections induced by

the finite length of the blocks to the leading scaling behavior.

arXiv:1103.3166v1 [cond-mat.stat-mech] 16 Mar 2011

Page 2

Entanglement of 2 disjoint intervals in c = 1 theories2

1. Introduction

Let us imagine to divide the Hilbert space H of a given quantum system into two parts

HAand HBsuch that H = HA⊗ HB. When the system is in a pure state |Ψ?, the

bipartite entanglement between A and its complement B, can be measured in terms

of the R´ enyi entropies [1]

1

1 − nlogTrρn

where ρA= TrBρ is the reduced density matrix of the subsystem A, and ρ = |Ψ??Ψ|

is the density matrix of the whole system. The knowledge of S(n)

identifies univocally the full spectrum of non-zero eigenvalues of ρA[2], and provides

complementary information about the entanglement to the one obtained from the von

Neumann entanglement entropy S(1)

of A in the ground-state of a one-dimensional system is more suited than S(1)

understand if a faithful representation of the state in term of a matrix product state

can be or cannot be obtained with polynomial resources in the length of the chain

[3, 4].

For a one-dimensional critical system whose scaling limit is described by a

conformal field theory (CFT), in the case when A is an interval of length ? embedded

in an infinite system, the asymptotic large ? behavior of the quantities determining

the R´ enyi entropies is [5, 6, 7, 8]

??

where c is the central charge of the underlying CFT and a the inverse of an ultraviolet

cutoff (e.g. the lattice spacing). The prefactors cn(and so the additive constants c?

are non universal constants (that however satisfy universal relations [9]).

The central charge is an ubiquitous and fundamental feature of a conformal field

theory [10], but it does not always identify the universality class of the theory. A

relevant class of relativistic massless quantum field theories are the c = 1 models,

which describe many physical systems of experimental and theoretical interest. The

one-dimensional Bose gas with repulsive interaction, the (anisotropic) Heisenberg

spin chains, the Ashkin-Teller model and many others are all described (in their

gapless phases) by c = 1 theories. These are all free-bosonic field theories where

the boson field satisfies different periodicity constraints, i.e. it is compactified on

a specific target space.The two most notable examples are the compactification

on a circle (corresponding to the Luttinger liquid field theory) and on a Z2orbifold

(corresponding to the Ashkin-Teller model [11, 12, 13]). The critical exponents depend

in a continuous way on the compactification radius of the bosonic field. A survey

of the CFTs compactified on a circle or on a Z2 orbifold is given in Fig. 1, in a

standard representation [12, 13]. The horizontal axis is the compactification radius

on the circle rcircle, while the vertical axis represents the value of the Z2 orbifold

compactification radius rorb. The two axes cross in a single point, meaning that the

theories at rcircle=√2 and at rorb= 1/√2 are the same. (The graph is not a cartesian

plot, i.e. it has no meaning to have one rcircleand one rorbat the same time.) For

some values of rcircle and rorb, we report statistical mechanical models and/or field

theories to which they correspond. In the following we will consider the Ashkin-Teller

model that on the self-dual line is described by rorb∈ [?2/3,√2] and the XXZ spin

S(n)

A

=

A, (1)

A

as a function of n

A. Furthermore, the scaling of S(n)

A

with the size

A

to

Trρn

A? cn

a

?c(n−1/n)/6

,

⇒ S(n)

A

?c

6

?

1 +1

n

?

log?

a+ c?

n,(2)

n)

Page 3

Entanglement of 2 disjoint intervals in c = 1 theories3

Figure 1. Survey of c = 1 theories corresponding to a free boson compactified

on a circle (horizontal axis) and on an orbifold (vertical axis) as reported e.g.

in Refs. [12]. For some values of rcircleand rorb, the corresponding statistical

mechanical models are reported. The XXZ spin chain in zero magnetic field lies

on the horizontal axis in the interval rcircle∈ [0,1/√2]. The self-dual line of the

Ashkin-Teller model lies on the vertical axis in the interval rorb∈ [?

chain in zero magnetic field that is described by rcircle∈ [0,1/√2]. We mention that

different compactifications have been studied [14], but they correspond to more exotic

statistical mechanical models and will not be considered here.

According to Eq. (2), the central charge of the CFT can be extracted from the

scaling of both the R´ enyi and von Neumann entropies. In the last years, this idea has

overcome the previously available techniques of determining c, e.g. by measuring the

finite size corrections to the ground state energy of a spin chain [15]. However, the

dependence of the scaling of the entropies of a single block only on the central charge

prevents to extract from them other important parameter of the model such as the

compactification radius. It has been shown that instead the entanglement entropies of

disjoint intervals are sensitive to the full operator content of the CFT and in particular

they depend on the compactification radius and on the symmetries of the target space.

Thus they encode complementary information about the underlying conformal field

theory of a given critical quantum/statistical system to the knowledge of the central

charge present in the scaling of the single block entropies. (Oppositely in 2D systems

with conformal invariant wave-function, the entanglement entropy of a single region

depends on the compactification radius [16].)

This observation boosted an intense theoretical activity aimed at determining

R` enyi entropies of disjoint intervals both analytically and numerically [17, 18, 19, 20,

21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. A part of this paper is dedicated to consolidate

some of the results already provided in other works where they either have been studied

only on very small chains, with the impossibility of properly taking into account the

severe finite size corrections [17] or have been tested in the specific cases of spin chains

2/3,√2].

Page 4

Entanglement of 2 disjoint intervals in c = 1 theories4

equivalent to free fermionic models [24, 26]. An important point to recall when dealing

with more than one interval is that the R´ enyi entropies in Eq. (1) measure only the

entanglement of the disjoint intervals with the rest of the system. They do not measure

the entanglement of one interval with respect to the other, that instead requires the

definition of more complicated quantities because A1∪A2is in a mixed state (see e.g.

Refs. [31] for a discussion of this and examples). Furthermore, it must be mentioned

that some results about the entanglement of two disjoint intervals are at the basis of

a recent proposal to ”measure” the entanglement entropy [32].

1.1. Summary of some CFT results for the entanglement of two disjoint intervals

We consider the case of two disjoint intervals A = A1∪ A2= [u1,v1] ∪ [u2,v2]. By

global conformal invariance, in the thermodynamic limit, Trρn

?

where x is the four-point ratio (for real ujand vj, x is real)

Acan be written as

Trρn

A= c2

n

|u1− u2||v1− v2|

|u1− v1||u2− v2||u1− v2||u2− v1|

?c

6(n−1/n)

Fn(x), (3)

x =(u1− v1)(u2− v2)

(u1− u2)(v1− v2).(4)

The function Fn(x) is a universal function (after being normalized such that Fn(0) = 1)

that encodes all the information about the operator spectrum of the CFT and in

particular about the compactification radius. cnis the same non-universal constant

appearing in Eq. (2).

Furukawa, Pasquier, and Shiraishi [17] calculated F2(x) for a free boson

compactified on a circle of radius rcircle

F2(x) =θ3(ητ)θ3(τ/η)

[θ3(τ)]2

,(5)

where θνare Jacobi theta functions and the (pure-imaginary) τ is given by

?θ2(τ)

η is a universal critical exponent related to the compactification radius η = 2r2

This has been extended to general integers n ≥ 2 in Ref. [19]

Fn(x) =Θ?0|ηΓ?Θ?0|Γ/η?

where Γ is an (n − 1) × (n − 1) matrix with elements [19]

Γrs=2i

n

k =1

n

and

2F1(y,1 − y;1;1 − x)

2F1(y,1 − y;1;x)

‡ Because of the symmetry η → 1/η or rcircle→ 1/2rcirclefor any conformal property one could

also define η = 1/2r2

circleas sometimes done in the literature. However, corrections to scaling are

not symmetric in η → 1/η and this is often source of confusion. A lot of care should be used when

referring to one or another notation.

x =

θ3(τ)

?4

,τ(x) = i2F1(1/2,1/2;1;1 − x)

2F1(1/2,1/2;1;x)

.(6)

circle. ‡

[Θ?0|Γ?]2

?

,(7)

n−1

?

sinπk

?

βk/ncos

?

2πk

n(r − s)

?

,(8)

βy=.(9)

Page 5

Entanglement of 2 disjoint intervals in c = 1 theories5

η is the same as above, while Θ is the Riemann-Siegel theta function

?

The analytic continuation of Eq. (7) to real n for general values of η and x (to obtain

the von Neumann entanglement entropy) is still an open problem, but results for

x ? 1 and η ? 1 are analytically known [19, 30].

The function Fn(x) is known exactly for arbitrary integral n also for the critical

Ising field theory [30]. However, in the following we will need it only at n = 2 (i.e.

F2(x)) for which it assumes the simple form [24]

??(1 +√x)(1 +√1 − x)

In Ref. [30], it has been proved that in any CFT the function Fn(x) admits the

small x expansion

?x

where α is the lowest scaling dimension of the theory.

calculable from a modification of the short-distance expansion [30], and in particular

it has been found [30]

Θ(0|Γ) ≡

m∈Zα−1

exp?iπ mt· Γ · m?. (10)

FIs

2(x) =

1

√2

2

?1/2

+x1/4+((1−x)x)1/4+(1−x)1/4

?1/2

.(11)

Fn(x) = 1 +

4n2

?α

s2(n) +

?x

4n2

?2α

s4(n) + ... ,(12)

The functions sj(n) are

s2(n) = Nn

2

n−1

?

j=1

1

?sin?πj

n

??2α,(13)

where the integer N counts the number of inequivalent correlation functions giving

the same contribution. This expansion has been tested against the exact results for

the free compactified boson (Ising model) with α = min[η,1/η] (α = 1/4) and N = 2

(N = 1).

All the results we reported so far are valid for an infinite system. Numerical

simulations are instead performed for finite, but large, system sizes. According to

CFT [8], we obtain the correct result for a chain of finite length L by replacing all

distances uijwith the chord distance L/π sin(πuij/L) (but different finite size forms

exist for excited states [33]). In particular the single interval entanglement is [6]

?L

and for two intervals, in the case the two subsystems A1and A2have the same length

? and are placed at distance r, the four-point ratio x is

?

Trρn

A? cn

πasin

?π?

L

??−c(n−1/n)/6

,(14)

x =

sinπ?/L

sinπ(? + r)/L

?2

.(15)

1.2. Organization of the paper

In this paper we provide accurate numerical tests for the functions Fn(x) in truly

interacting lattice models described by a CFT with c = 1. In Sec. 2 we derive the CFT

prediction for the function F2(x) of a free boson compactified on an orbifold describing,

among the other things, the self-dual line of the AT model when rorb∈ [?2/3,√2].

in Sec. 3 based on the ideas introduced in Ref. [18]. This algorithm is used in Sec. 4

In order to check this result, we needed to develop a classical Monte Carlo algorithm

Page 6

Entanglement of 2 disjoint intervals in c = 1 theories6

to determine F2(x) for several points on the self-dual line. We also consider the XXZ

spin-chain in zero magnetic field to test the correctness of Eq. (7). In order to extend

the results of Ref. [17] to longer chains, we have used a tree tensor network algorithm

that has allowed us to study chains of length up to L = 128 with periodic boundary

conditions. In this way, we have been able to perform a detailed finite size analysis that

was difficult solely with the data from exact diagonalization reported in Ref. [17]. The

analysis also shows that only through the knowledge of the unusual corrections to the

leading scaling behavior [34, 35, 36, 37, 38, 26] we are able to perform a quantitative

test of Eq. (7). The tree tensor network algorithm is described in Sec. 5, while the

numerical results are presented in Sec. 6. The various sections are independent one

from each other, so that readers interested only in some results should have an easy

access to them without reading the whole paper.

2. n = 2 R` enyi entanglement entropy for two intervals in the

Ashkin-Teller model

In a quantum field theory Trρn

on an n-sheeted Riemann surface with branch cuts along the subsystem A, i.e.

Trρn

1where Zn(A) is the partition function of the field theory on a

conifold where n copies of the manifold R = system × R1are coupled along branch

cuts along each connected piece of A at a time-slice t = 0 [8, 39]. Specializing to CFT,

for a single interval on the infinite line, this equivalence leads to Eq. (2) [6], whose

analytic continuation to non-integer n is straightforward. When the subsystem A

consists of N disjoint intervals (always in an infinite system), the n-sheeted Riemann

surface Rn,N has genus (n − 1)(N − 1) and cannot be mapped to the complex plane

so that the CFT calculations become more complicated.

However, for two intervals (N = 2), when for a given theory the partition function

on a generic Riemann surface of genus g with arbitrary period matrix is known, Trρn

can be easily deduced exploiting the results of Refs. [19, 30]. In fact, a by-product

of the calculation for the free boson [19] is that the (n − 1) × (n − 1) period matrix

is always given by Eq. (8). Although derived for a free boson, the period matrix is

a pure geometrical object and it is only related to the structure of the world-sheet

Rn,2and so it is the same for any theory. This property has been used in Ref. [30] to

obtain Fn(x) for the Ising universality class for any n, in agreement with previously

known numerical results [26]. When also n = 2, the surface R2,2 is topologically

equivalent to a torus for which the partition function is known for most of the CFT.

The torus modular parameter τ is related to the four-point ratio by Eq. (6). Thus,

the function F2(x) is proportional to the torus partition function where τ is given by

Eq. (6) and with the proportionality constant fixed by requiring F2(0) = 1. This way

of calculating S(2)

A

has been used to obtain the first results both for the free compactified boson [17] and

for the Ising model [24].

For a conformal free bosonic theory with action

1

2π

the torus partition functions are known exactly both for circle and orbifold

compactification [41, 42, 12].

Afor integer n is proportional to the partition function

A= Zn(A)/Zn

A

is much easier than the general one for S(n)

A

[40, 19] and indeed it

S =

?

dzd¯ z ∂φ¯∂φ,(16)

Page 7

Entanglement of 2 disjoint intervals in c = 1 theories7

We now recall some well-known facts in order to fix the notations and derive

the function F2(x) for the Ashkin-Teller model. The bosonic field φ is said to be

compactified on a circle of radius rcirclewhen φ = φ + 2πrcircle. The torus partition

function (and the one on the n-sheeted Riemann surface) should be derived with this

constraint. It is a standard CFT exercise to calculate the resulting torus partition

function [41, 12]

Zcircle(η) =θ3(ητ)θ3(τ/η)

|ηD(τ)|2

,(17)

where ηD(τ) is the Dedekind eta function and η = 2r2

properties of the elliptic functions, Eq. (5) for F2(x) follows [17]. When specialized

at η = 1/2 (or η = 2), F2(x) has the simple form

?

that describes the XX spin-chain (that is equivalent to free fermions via the non-local

Jordan-Wigner transformation).

The concept of orbifold emerges naturally in the context of theories whose Hilbert

space admits some discrete symmetries. Let us assume that G is a discrete symmetry.

For the free bosonic theory, the simplest example is the one we are interested in, i.e.

the Z2symmetry. It acts on the point of the circle S1in the following way

circle. Using Eq. (6) and some

FXX

2

(x) =

(1 + x1/2)(1 + (1 − x)1/2)/2,(18)

g : φ → −φ.(19)

For the partition function of a theory on the torus, we introduce the notation [12]

±

±

(20)

where the ± denotes the boundary conditions on the two directions on the torus. The

full partition function, given a finite discrete group G, is

1

|G|

g,h∈G

h

where |G| denotes the number of elements in the group. The generalization to higher

genus Riemann surfaces is straightforward (but it is not so easy to obtain results, see

e.g. [13, 43]).

Now we specialize Eq. (21) to the case of the Z2symmetry. Since the action (16)

is invariant under g : φ → −φ, we have the torus partition function for the free boson

on the orbifold [41, 42, 12]

?

++

Standard CFT calculations lead to the result [12]

?

where all the τ arguments in θνand ηDare understood. At the special point η = 1/2

(or η = 2) we get

?|θ3|2+ |θ4|2+ |θ2|2

ZT /G=

?

g

(21)

Zorb=1

2

+

+

−

+

−

+

+

−

−

?

.(22)

Zorb(η) =1

2

Zcircle(η) +|θ3θ4|

ηD¯ ηD

+|θ2θ3|

ηD¯ ηD

+|θ2θ4|

ηD¯ ηD

?

,(23)

Zorb(η = 1/2) =1

22|ηD|2

+|θ3θ4|

ηD¯ ηD

+|θ2θ3|

ηD¯ ηD

+|θ2θ4|

ηD¯ ηD

?

= Z2

Ising.(24)

Page 8

Entanglement of 2 disjoint intervals in c = 1 theories8

Figure 2. F2(x) for the Ashkin-Teller model on the self-dual line for some values

of η. Inset: F2(x) − 1 in log-log scale to highlight the small x behavior. The

black-dashed line is ∼ x1/4.

Thus, from the orbifold partition function, using the last identity and normalizing

such that FAT

2

(0) = 1, we can write the funcion FAT

(x) =1

2

2

(x) as

FAT

2

?

F2(x) − FXX

2

(x)

?

+ (FIs

2(x))2, (25)

where F2(x) is given in Eq. (5), FXX

and FIs

2(x) is the result for Ising (cf. Eq. (11)). As a consequence of the η ↔ 1/η

symmetry of F2(x), also FAT

2

(x) displays the same invariance. For small x, recalling

that F2(x) − 1 ∼ xmin[η,η−1], FXX

?x1/4

xmin[η,η−1]

2

(x) is the same at η = 1/2 (cf. Eq. (18))

2

− 1 ∼ x1/2and FIs

2− 1 ∼ x1/4, we have

for η ≥ 1/4,

for η ≤ 1/4.

?2/3 < rorb <

FAT

2

(x) − 1 ∼

(26)

The critical Ashkin-Teller model lies in the interval

4/3 < η = 2r2

FAT

2

(x) for various values of η in the allowed range is reported in Fig. 2, where the

behavior for small x is highlighted in the inset to show the constant 1/4 exponent.

√2 and so

orb< 4. Thus we have FAT

2

(x) − 1 ∼ x1/4along the whole self-dual line.

3. The classical Ashkin-Teller model and the Monte Carlo simulation

The two dimensional Ashkin-Teller (AT) model on a square lattice is defined by the

Hamiltonian

?

H = J

?ij?

σiσj+ J??

?ij?

τiτj+ K

?

?ij?

σiσjτiτj,(27)

Page 9

Entanglement of 2 disjoint intervals in c = 1 theories9

where σiand τiare classical Ising variables (i.e. can assume only the values ±1). Also

the product στ can be considered as an Ising variable. The model has a rich phase

diagram whose features are reported in full details in Baxter’s book [45]. We review in

the following only the main features of this phase diagram. Under any permutation of

the variables σ,τ,στ the AT model is mapped onto itself. At the level of the coupling

constants, this implies that the model is invariant under any permutation of J,J?,K.

For K = 0, the AT model corresponds to two decoupled Ising models in σ and τ

variables. For K → ∞ it reduces to a single Ising model with coupling constant

J + J?. For J = J?= K it corresponds to the four-state Potts model. It is useful to

restrict to the symmetric Ashkin-Teller model where J = J?

?

The full phase diagram is reported in Fig. 3 (in units of the inverse temperature

β = 1). The model corresponds to two decoupled critical Ising models at K = 0 and

2J = log(1 +√2). For J = 0 it is equivalent to a critical Ising model in the variable

στ with critical points at 2K±= ±log(1 +√2). For K → ∞ there are two critical

Ising points at 2J = ±log(1 +√2). On the diagonal J = K the system corresponds

to a 4-state Potts model which is critical at K = (log3)/4. The different kinds of

orders appearing in the phase diagram are explained in the caption of Fig. 3. All

the continuous lines in Fig. 3 are critical lines. The blue lines C-Is are in the Ising

universality class. The line starting from AFIs belongs to the antiferromagnetic Ising

universality class. On the red line ABC the system is critical and the critical exponents

vary continuously [46, 45].

The AT model on a planar graph can be mapped to another AT model on the

dual graph. When specialized to the square lattice, the phase diagram is equivalent

to its dual on the self-dual line:

e−2K= sinh(2J).

H = J

?ij?

(σiσj+ τiτj) + K

?

?ij?

σiσjτiτj. (28)

(29)

On this line, the symmetric AT model maps onto an homogeneous six-vertex model

which is exactly solvable [45]. It follows that on the self-dual line the model is critical

for K ≤ (log3)/4 and its critical behavior is described by a CFT with c = 1. Along

the self-dual line the critical exponents vary countinuously and are exactly known.

For later convenience it is useful to parametrize the self dual line by a new parameter

∆

√2 − 2∆ + 1

√2 − 2∆ − 1,

with −1 < ∆ < 1/2. In terms of ∆, the orbifold compactification radius is [42]

η = 2r2

π

where KLis the equivalent of the Luttinger liquid parameter for the AT model.

e4J=

e4K= 1 − 2∆,(30)

orb=4arccos(−∆)

=

2

KL

, (31)

3.1. Cluster representation and Monte Carlo simulation

A Swendsen-Wang type cluster algorithm for the AT model has been proposed in Ref.

[47] and then re-derived in a simpler way by Salas and Sokal [48]. Here we partly follow

the derivation of Salas and Sokal and we restrict to the symmetric AT Hamiltonian

(28) and assume J ≥ |K|. Using the identities for Ising type variables

σiσj= 2δσiσj− 1,τiτj= 2δτiτj− 1,(32)

Page 10

Entanglement of 2 disjoint intervals in c = 1 theories10

Figure 3. Phase diagram of the 2D symmetric Ashkin-Teller model defined by

the Hamiltonian (28). The red ABC line is the self dual line. The point B at

K = 0 corresponds to two uncoupled Ising models. The point C is the critical

four-state Potts model at K = J = (log3)/4. At J = 0 there are two critical

Ising points at K = ±(log(1 +

(AFIs) antiferromagnetic. For K → ∞ there is another critical Ising point at

J = (log(1 +√2))/2. All continuous lines are critical. The blue lines C − Is

and the one starting at AFIs are in the Ising universality class. The red line

is critical with continuously varying critical exponents. The region denoted by I

corresponds to a ferromagnetic phase for all the variables. In the region II, σ, τ,

and στ are paramagnetic. In the region III only στ is ferromagnetic and in region

IV στ exhibits antiferromagnetic order while σ and τ are paramagnetic.

√2))/2, one (Is) ferromagnetic and the other

we can rewrite Eq. (28) as

?

For convenience we shift the interaction (28) by −4J. In order to write the Boltzmann

weight associated to a specific configuration we use exp(wδσiσj) = (exp(w)−1)δσiσj+1

and the analogous identity for the τ variables. The Boltzmann weight of a given link

?ij? is then

W?ij?(σi,σj,τi,τj) = e−4J+ [e−2(J+K)− e−4J][δσiσj+ δτiτj] +

+ [1 − 2e−2(J+K)+ e−4J]δσiσjδτiτj.

The key idea for the Swendsen-Wang algorithm is to introduce two new auxiliary

Ising-type variables mij and nij living on the link ?ij?. We redefine the Boltzmann

weight on the link ?ij? as [48]

W?ij?(σi,σj,τi,τj,mij,nij) = e−4Jδmij0δnij0+

− H = J

?ij?

(2δσiσj+ 2δτiτj− 2) + K

?

?ij?

(2δσiσj− 1)(2δτiτj− 1).(33)

(34)

Page 11

Entanglement of 2 disjoint intervals in c = 1 theories11

+ [e−2(J+K)− e−4J][δσiσjδmij1δnij0+ δτiτjδmij0δnij1] +

+ [1 − 2e−2(J+K)+ e−4J]δσiσjδτiτjδmij1δnij1.

Summing over mijand nijwe obtain the weight in Eq. (34). Eq. (35) has a graphical

interpretation in terms of clusters. In fact we can divide the links of the lattice in

“activated” (if mij = 1) or “inactive” (if mij = 0). The same considerations hold

for the nij variables. Therefore, each link of the lattice can be activated by setting

mij= 1 or nij= 1. The active links connect different lattice sites forming clusters.

There are clusters referring to the σ variables (called σ-clusters) and to the τ variables

(τ-clusters). Isolated lattice sites are clusters as well. Obviously, the lattice sites

belonging to the σ-clusters (τ-clusters) have the same value of σ (τ). The partition

function of the extended model defined by the weight (35) can be written as

?

We now proceed to the following definitions.

classes: we define l0 the total number of inactivated links; l1 the total number of

links connecting sites which belong only to one type of clusters either a σ-cluster or a

τ-cluster. We define l2the total number of links on which m and n are both equal to

1. Furthermore we introduce the quantities

B0≡ e−4J,

B1≡ [e−2(J+K)− e−4J],

B2≡ [1 − 2e−2(J+K)+ e−4J].

The following step is to perform the summation over σ,τ in Eq. (35). This is readily

done, obtaining the final expression for the partition function

?

where we denoted with Cσthe number of σ-clusters and with Cτthe total number of

τ-clusters. In the counting of τ-clusters (σ-clusters) we included all the lattice sites

connected by a link on which mij= 1 (nij= 1). Isolated sites (with respect to m or

n or both) count as single clusters. The links where mij = 1,nij = 1 contribute to

both types of clusters.

(35)

Z =

σ,τ=±1

?

m,n=±1

?

?ij?

W?ij?(σi,σj,τi,τj,mij,nij).(36)

We divide all the links into three

(37)

(38)

(39)

Z =

C{τ,σ}

Bl0

0Bl1

1Bl2

22Cσ+Cτ,(40)

3.2. Swendsen-Wang algorithm (the direct and embedded algorithms)

We are now in position to write the Swendsen-Wang algorithm for the symmetric AT

model. The Monte-Carlo procedure can be divided in two steps. In the first one,

given a configuration for (σ,τ) variables, we construct a configuration of the (m,n)

variables. In the second step we update the (σ,τ) variables at given (m,n). The

details of the step one are

• if σi= σjand τi= τj, we choose (mij,nij) with the following probabilities:

– (mij,nij) = (1,1) with p1= 1 − 2e−2(J+K)+ e−4J,

– (mij,nij) = (1,0) with p2= e−2(J+K)+ e−4J,

– (mij,nij) = (0,1) with p2= e−2(J+K)+ e−4J,

– (mij,nij) = (0,0) with p3= 1 − p1− 2p2,

• if σi= σjand τi= −τj, the probabilities are

– (mij,nij) = (1,0) with p1= 1 − e−2(J−K),

Page 12

Entanglement of 2 disjoint intervals in c = 1 theories12

Figure 4. A typical cluster configuration on a 12 × 12 lattice. Green lines are

σ-clusters and red dashed lines are τ-clusters. Links in blue are double links.

Periodic boundary conditions on both directions are used.

– (mij,nij) = (0,0) with p2= 1 − p1,

• if σi= −σjand τi= τj, the probabilities are

– (mij,nij) = (1,0) with p1= 1 − e−2(J−K),

– (mij,nij) = (0,0) with p2= 1 − p1,

• if σi= −σjand τi= −τjwe choose (mij,nij) = (0,0) with probability 1.

In the step two, given the configuration of (m,n) generated using the rules above we

build the connected σ-clusters and τ-clusters. The value of σ (τ) spins are required to

be equal within each σ-cluster (τ-cluster). We choose randomly the spin value in each

cluster and independently of the value assumed on the other clusters. This completes

the update scheme. (Note a typo in Ref. [48]: the minus sign in step 2 and 3 of the

update is missing.)

In Ref. [48] also the so called embedded version of the cluster algorithm is

introduced. Its implementation is slightly easier compared to the direct algorithm.

In the embedded algorithm instead of treating both σ and τ at the same time, one

deals with only one variable per time. Let us consider the Boltzmann weight of a link

?ij? at fixed configuration of τ

W?ij?(σi,σj,τi,τj) = e−2(J+Kτiτj)+ (1 − e−2(J+Kτiτj))δσiσj.

The model defined by this weight can be simulated with a standard Swendsen-Wang

algorithm for the Ising model using the effective coupling constant

Jeff

ij

= J + Kτiτj.

This is no longer translation invariant, but this does not affect the effectiveness of the

cluster algorithm for the Ising model as long as Jeff

to the case of fixed σ. Thus, the embedded algorithm is made of two steps

(41)

(42)

ij

≥ 0. The same reasoning applies

• For a given configuration of τ variables, we apply a standard Swendsen-Wang

algorithm to σ spins. The probability arising in the update step is pij =

1 − e−2(J+Kτiτj).

Page 13

Entanglement of 2 disjoint intervals in c = 1 theories13

• For a given configuration of σ variables, we update τ with the same algorithm

and probability pij= 1 − e−2(J+Kσjσi).

Direct and embedded algorithms are both extremely effective procedures to sample

the AT configurations. However, very important for the following, Eq. (40) for the

partition function does not hold anymore for a n-sheeted Riemann surface and we do

not know whether it is possible to write the embedded algorithm for this case.

3.3. R´ enyi entanglement entropies via Monte Carlo simulation of a classical system.

In this section we summarize the method introduced by Caraglio and Gliozzi [18] to

obtain the R´ enyi entropies via simulations of classical systems and we generalize it

to the AT model. The partition function Z = Tre−βHof a d-dimensional quantum

system at inverse temperature β can be written as an Euclidean path integral in d+1

dimensions [8]. Thus for the n-th power of the partition function one has

Zn=

?

n

?

k=1

D[φk]e

−

n ?

k=1

S(φk)

(43)

where φk≡ φk(? x,τ) is a field living on the k-th replica of the system and S(φk) is

the euclidean action (τ is the imaginary time.) The actual form of the action is not

important, but for the sake of simplicity we restrict to the case of nearest-neighbor

interactions

?

and the function F is arbitrary. We recall that Trρn

the euclidean partition function over a n-sheeted Riemann surface with branch cuts

along the subsystem A [8]. (This equivalence is also the basis of all quantum Monte

Carlo methods to simulate the block entanglement in any dimension [49].) Caraglio

and Gliozzi constructed this n-sheeted Riemann surface for the lattice model in the

following way. Let us consider a square lattice (for simplicity) and take the two points

of its dual lattice surrounding A (that in 1+1 dimension is just an interval with two

end-points). The straight line joining them defines the cut that we call λ. The length

of λ is equal to the length of A. Let us consider n independent copies of this lattice

with a cut. The n-sheeted Riemann lattice is defined by assuming that all the links of

the k-th replica intersecting the cut connect with the next replica k+1(modn). To get

the partition function over the n sheeted Riemann surface we define the corresponding

coupled action

?

This definition can be used in any dimension, even though we will use here only d = 2.

Finally, calling Zn(A) the partition function over the action (45), Trρn

S(φk) =

?ij?

F(φk(i),φk(j)),(44)

Acan be obtained by considering

Sn(φk) =

n

k=1

?

?ij?/ ∈λ

F(φk(i),φk(j)) +

?

?ij?∈λ

F(φk(i),φk+1(modn)(j)).(45)

Ais given by

Trρn

A=Zn(A)

Zn

.(46)

Following Ref. [18] we introduce the observable

O ≡ e−Sn(φ1,φ2,...,φn;λ)+?n

k=1S(φk;λ), (47)

Page 14

Entanglement of 2 disjoint intervals in c = 1 theories14

where Snand S are the euclidean actions of the model defined on the n-sheeted lattice

and on the n independent lattices respectively. The sum is restricted to links crossing

the cut, as the presence of λ in the arguments stresses. It then follows

?O?n≡Zn(A)

Zn

= Trρn

A,(48)

where ?·?nstands for the average taken onto the uncoupled action?n

The practical implementation of Eq. (47) to calculate Trρn

limitations: analyzing the Monte-Carlo evolution of the observable, one notices that

it shows a huge variance because it is defined by an exponential. Direct application

of Eq. (47) is possible then only for small lengths of the subsystem A. In order to

overcome this problem, let us consider the quantity Zn(A)/Znand imagine to divide

the subsystem in L parts to have A = A1∪A2...∪AL, with the lengths of the various

parts being arbitrary. Moreover we define a set of subsystemsˆAi≡ ∪i

holds

?

Eq. (49) is very useful because each term in the product can be simulated effectively

using a modified version of (47) if we choose the length of Aito be small enough. In

fact, by definition, we have

k=1S(φk).

We can now discuss our improvement to the procedure highlighted so far.

Ais plagued by severe

k=1Ai. Then it

Zn(A)

Zn

=

L

i=0

Zn(ˆAi+1)

Zn(ˆAi)

. (49)

?O(ˆAi)?Rn(ˆAi)≡Zn(ˆAi+1)

Zn(ˆAi)

, (50)

where O(ˆAi) is the modified observable

O(ˆAi) ≡ exp(−Sn(ˆAi+1) + Sn(ˆ Ai)).

We stress that in Eq. (50) the expectation value in the l.h.s must be taken on the

coupled action on the Riemann surface with cutˆAi. The disadvantage of Eq. (49) is

that, to simulate large subsystems, one has to perform L independent simulations and

then build the observable taking the product of the results. If the dimension of each

piece Aiis small this task requires a large computational effort. Another important

aspect is the estimation of the Monte Carlo error: if each term in (49) is obtained

independently, the error in the product is

?

i=0

O(ˆAi)

If the lengths of the intervals Aiare all equal, then the single terms of the summation

in Eq. (52) do not change much and the total error should scale as

Caraglio and Gliozzi [18] used another strategy to circumvent the problem with

the observable in Eq. (47). The trick was to consider the Fortuin-Kastelayn cluster

expansion of the partition function of the Ising model. The analogous for the AT

model was reported in the previous section

?

where Cσ,τare the σ/τ-cluster configurations. Going from n independent sheets to

the n-sheeted lattice, the type of links and their total number do not change, but the

(51)

σ(O)

O

=

?

?

?

L

?

σ2(O(ˆAi))

2

.(52)

√L.

Z =

C{σ,τ}

Bl0

0Bl1

1Bl2

22Cσ+Cτ,(53)

Page 15

Entanglement of 2 disjoint intervals in c = 1 theories15

Figure 5. Trρ2

L = 120.

embedded algorithm. The orange points correspond to the SUSY model and the

green ones to the Z4parafermions. The black crosses at ? = 10 are data obtained

using the direct algorithm. Inset: behavior of the statistical error of Trρ2

for the SUSY model. The blue-dashed line is the expected form A + B?1/2.

Afor a single interval of length ? in a finite system of length

Data have been obtained by Monte Carlo simulations using the

Avs ?

number of clusters does change, and so we get the cluster expression of observable

(47) for the AT model

O(ˆ Ai) = 2[Cσ(ˆ Ai+1)+Cτ(ˆAi+1)−Cσ(ˆ Ai)−Cτ(ˆAi)], (54)

where Cσ(ˆAi) (Cτ(ˆAi)) denote the total number of σ-clusters (τ-clusters) on the

Riemann surface with cutˆAi. Since the clusters are non local objects, they represent

“improved” observables and the variance for the Monte Carlo history of Eq. (54) is

much smaller than in the naive implementation.

4. The entanglement entropy in the Ashkin-Teller model

4.1. The single interval

We first present the results for the Ashkin-Teller model for a single interval. Although

these results do not provide any new information about the model, they are

fundamental checks for the effectiveness of the Monte Carlo algorithms. We performed

simulations using both algorithms described in the previous section: the direct cluster

algorithm and the embedded one. When using the direct algorithm, measures are

performed using the observable (54), while for the embedded algorithm we used the

observable in Eq. (47). In Fig. 5 we report the results of the simulations of Trρ2

the SUSY model (rorb=√3/2 in Fig. 1) and for the Z4parafermions (rorb=

both for L = 120. The orange and green points are obtained using the embedded

Afor

?3/2)

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- Available from Luca Tagliacozzo · May 16, 2014
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