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 theories 11
+ [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(ˆAi+1)+Cτ(ˆAi+1)−Cσ(ˆAi)−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)
Download full-text