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
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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)
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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
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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)
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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)
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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)
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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)
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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)
Page 16
Entanglement of 2 disjoint intervals in c = 1 theories16
Figure 6. Plot of c2(Lc) as function of Lc for different ? and L. Three points
on the self-dual line are reported: four-states Potts model, uncoupled Ising, and
SUSY. The dashed lines are fits to the function c2+ BL−KL
BL−2KL
c
for the 4-states Potts model) where KLis 1/2,1,4/3 respectively for the
four-states Potts model, Ising, and SUSY. In the inset we report cn for n = 3,4
for the SUSY point. The dashed lines are fit to A + BL2KL/n
fixed.
c
(c2+ AL−KL
c
+
c
, with KL= 4/3
algorithm. To check the implementation of the cluster observable, we report at ? = 10
the data obtained using the direct algorithm and Eq. (54). The perfect agreement
between the two results confirms the correctness of both implementations. Note that
Trρ2
Ais a monotonous function of ?, in contrast with the parity effects found for the
XXZ spin chain [34, 35] that also corresponds to a vertex model [45]. In the inset
we show the behavior of the statistical error of the observable (47) in the SUSY case
as function of the subsystem length ?. It agrees with the prediction in Eq. (52) and
its absolute value is extremely small, smaller than the size of the points in the main
plot in Fig. 5. Analogous results have been obtained for all the critical points on the
self-dual line using both algorithms.
The results for Trρ2
Ain a finite system are asymptotically described by the CFT
prediction (14) with n = 2 and c = 1. It is then natural to compute the ratio
Trρ2
A
(L
c2(Lc) =
πsin(π
L?))−1/4,(55)
that is expected to be asymptotically a function of the chord-length Lc= [L
This allows to extract the non-universal quantity c2 and to check the form of the
corrections to the scaling. In Fig. 6 we report the results for c2(Lc) for the SUSY
point, for the two uncoupled Ising models, and for the four states Potts model. It is
evident that for large Lc, c2(Lc) approaches a constant value around 0.5. This is a
first confirmation of the CFT predictions on the self-dual line.
πsin(π
L?)].
Page 17
Entanglement of 2 disjoint intervals in c = 1 theories17
Figure 7.
points are extrapolations obtained using the finite-size ansatz (57). The blue-
dashed line is the CFT prediction. Inset: Flat
of x used in the extrapolation (x = 0.134,0.25,0.5,0.587). The dashed lines are
fits to finite-size ansatz (57).
F2(x) versus the four point ratio x for the SUSY model. The red
2(x) vs 1/?−2/3for the four values
The previous results also provide a test for the theory of the corrections to the
scaling to S(n)
A.It has been shown [34, 35] that for gapless models described by
a Luttinger liquid theory, the corrections to the scaling have the form ?−2KL/n(or
L−2KL/n
c
for finite systems) where KLis the Luttinger parameter, related to the circle
compactification radius KL= 1/2η. On the basis of general CFT arguments [36], it
has been argued that this scenario is valid for any CFT and so also for the AT model
with KLreplaced by the dimension of a proper operator. It is then natural to expect
that for the AT model this dimension is KLin Eq. (31), also on the basis of the results
for the Ising model [34, 50]. The dashed lines in Fig. 6 are fits of c2(Lc) with the
function c2+ AL−KL
c
. The agreement is always very good, except for the four-state
Potts model, for which the exponent of the leading correction KLassumes the smallest
value and so subleading corrections enter (as elsewhere in similar circumstances, see
e.g. [35]). In fact, the fit with the function c2+ AL−KL
agreement with the data (but the presence of another fit parameter makes this result
not so robust). This analysis confirms that KLis the right exponent governing the
corrections to the scaling.
In the inset of Fig. 6 we also report the values of cnfor n = 3,4 as a function of
Lc. cnbecomes smaller as n increases as for the XXZ [34], XX [51], and Ising [50, 52]
c
+ BL−2KL
c
is in perfect
Page 18
Entanglement of 2 disjoint intervals in c = 1 theories18
Figure 8. Flat
parafermions, and four-states Potts model). The blue-dashed line is the CFT
prediction. The (colored) points close to the curve are extrapolations obtained
with the finite-size scaling ansatz (57). The black crosses are the Monte Carlo
data used for the fits. The block lengths used range from ? = 5 to ? = 80.
2(1/2) as function of η−1for different models (Ising, SUSY, Z4
spin-chains. The dashed lines are fits to the expected scaling behavior L−2KL/n
corrections, that reproduce perfectly the data.
c
of the
4.2. The entanglement entropy of two disjoint intervals.
In this section we investigate the entanglement entropy of two disjoint intervals and
check the correctness of our prediction (25) for the AT model on the self-dual line.
As for all other cases studied so far numerically (i.e. Heisenberg [17], Ising [24, 26],
and XY [26] chains), strong scaling corrections affect the determination of the scaling
function Fn(x). CFT predictions have been confirmed only using the general theory
of corrections to the scaling [34, 35, 36, 37].
In order to determine the function Fn(x), we consider the ratio
Trρn
A1∪A2
Trρn
A2
and, on the basis of the general CFT arguments [36], we expect that the the leading
correction to scaling can be effectively taken into account by the scaling ansatz
Flat
n
(x) + ?−2ω/nfn(x) + ... .
For the Ising model it has been found ω = 1/2 [24, 26]. Since for η = 2 the AT
Hamiltonian reduces to two uncoupled Ising models, one naively expects ω = KL/2
along the whole self-dual critical line of the AT model.
Hereafter we only consider Trρ2
A. We start our analysis from the SUSY point
that (assuming ω = KL/2) should have the smaller corrections to scaling. In Fig. 7
we show Monte Carlo data at ? = 10,20 (L = 120) for Flat
four point ratio x defined as in Eq. (15). We report with the blue dashed line the
Flat
n(x) =
A1Trρn
(1 − x)c(n−1/n)/6, (56)
n(x) = FCFT
(57)
2(x) plotted against the
Page 19
Entanglement of 2 disjoint intervals in c = 1 theories19
Figure 9. FCFT
finite-size scaling ansatz (57) fixing the value of FCFT
in log-log scale.
2
(1/2) − Flat
2(1/2) versus 1/?. The dashed lines are fits to the
2
(1/2). Left: the same plot
asymptotic CFT result (cf. Eq. (25)). As in all other cases considered in the literature
[24, 26], the curves for F2(x) at ? = 10,20 are not symmetric functions of x → 1 − x,
as instead the asymptotic CFT prediction must always be [17]. This is due to the
non-symmetrical finite-size corrections f2(x) in Eq. (57). We extrapolate the result
at ? → ∞ using the ansatz (57) and ω = 2/3. The extrapolations are reported as red
points in Fig. 7. There is a very good agreement between the extrapolations and the
theoretical curve. Since the correction exponent ω = 2/3 is rather large, and so the
corrections small, even small subsystems such as ? = 10,20 are enough to obtain a
good extrapolation. In the inset of Fig. 7 we report the Monte Carlo data for Flat
against ?−2/3. The linear behavior in this inset confirms the validity of the ansatz (57)
and the reported straight lines are the fits giving the extrapolations reported in the
main panel.
We also investigate other points on the self dual line, namely the 4-states Potts
model (η = 4), the parafermion Z4(η = 3), the uncoupled Isings (η = 2). In Fig. 8 we
report Flat
2
(x) at fixed x = 1/2 versus η−1for all the mentioned models.
we report x = 1/2 because it is the value of x providing the most stable estimate,
but also other values have been studied. Indeed, on one hand, the computational cost
of the simulations decreases going toward x = 1 (the reason being evident from the
definition of x for which smaller lattice sizes are needed). On the other hand, scaling
corrections become more severe in the region x ∼ 1, as clear from the results for the
SUSY model in Fig. 7. Thus x = 1/2 represents the best compromise between these
two drawbacks. The dashed curves in the left panel of Fig. 9 are fits of the data
with Eq. (57) obtained by fixing the value of FCFT
(25)). There is a very good agreement with the full theoretical picture, confirming
in particular the correctness of the exponent governing the leading correction to the
scaling. For the Z4parafermions and for the four-state Potts model, we needed very
large values of ? in order to show the correct asymptotic behavior (the range of ?
reported in the plot is in fact 5 ≤ ? ≤ 80). This is made clearer in the right panel
of Fig. 9 where the same data are shown in log-log scale. In Fig. 8 we reports the
2(x)
2(x) − FCFT
2
(x) to its predicted value (cf. Eq.
Page 20
Entanglement of 2 disjoint intervals in c = 1 theories20
Figure 10.
FCFT
2
models (SUSY, Z4 parafermions, Ising model and the model corresponding to
η−1= 0.74). The blue-dashed lines are asymptotic fits to Ax1/4.
Monte Carlo data for f2(x) obtained as f2(x) = (Flat
(x))?KL/2as function of x.We show data for ? = 10 and various
2(x) −
fits obtained by fixing only the exponent of the corrections ω = KL/2 and leaving
FCFT
2
(1/2) free. For all considered values of η, the extrapolation of Flat
? → ∞ is compatible (within error bars) with the expected result FCFT
We finally study the correction amplitude f2(x) in Eq. (57). This function is the
main reason of the asymmetry in x → 1−x for Flat
could greatly simplify future analyses. For the Ising model, it has been found that
f2(x) ∼ x1/4for small x, that is the same behavior of F2(x)−1. Since along the whole
self-dual line F2(x) − 1 ∼ x1/4, we would expect
f2(x) ∼ x1/4.
For the Ising model (i.e. η = 2), this scenario has been already verified with high
precision [24].
In Fig. 10 we report f2(x) obtained as f2(x) = (FCFT
of x (in logarithmic scale to highlight the small x behavior). All data correspond to
? = 10 and various values of L. For the two largest values of η (Z4 parafermionic
theory at η = 3 and for the Ising model at η = 2), we observe an excellent agreement
with our conjecture f2(x) ∼ x1/4. However decreasing the value of η, i.e. for the
SUSY model at η = 3/2 and for the model at η−1= 0.74, the behavior of f2(x) is not
as linear as before, especially for high value of x. Nonetheless for x < 0.4 the data
confirm the behavior x1/4. Furthermore, it seems that for any η ?= 2, subleading terms
in the expansion for small x appear and they are vanishing only for the Ising model.
2(1/2) to
(1/2).
2
2(x) and knowing its gross features
(58)
2
(x)−Flat
2)?KL/2as function
5. The Tree Tensor Network
This section is divided into two parts. First we explain in a self contained way how
to extract the spectrum of the reduced density matrix of some specific bipartitions
Page 21
Entanglement of 2 disjoint intervals in c = 1 theories 21
Figure 11. Examples of TTN for a N = 4 lattice and a N = 8 lattice.
of a pure state encoded in a Tree Tensor Network (TTN). We only recall the
basic definitions introduced in Ref. [53] and refer the reader to the literature for
complementary works on the subject [54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66].
Secondly we quickly recall how to use TTN to calculate the ground state of the
anisotropic Heisenberg spin-chain.
5.1. Tree Tensor network and reduced density matrices.
We consider a one dimensional lattice L made of N sites, where each site is described
by a local Hilbert space V of finite dimension d. In this work the state is the ground
state |ΨGS? of some local Hamiltonian H defined on L, but in general it could be an
arbitrary pure state |Ψ? ∈ V⊗Ndefined on the lattice L.
A generic state |Ψ? ∈ V⊗Ncan always be expanded as
d
?
where the dN
coefficients Ti1i2···iN
{|1s?,|2s?,···,|ds?} denote a local basis on the site s ∈ L. We refer to the index
isthat labels a local basis for site s (is= 1,···,d) as a physical index.
In the case we are interested in, the tensor of coefficients Ti1i2···iNin Eq. (59)
is the result of the contraction of a TTN. As shown in Fig. 11 for lattices of N = 4
and N = 8 sites, a TTN decomposition of Ti1i2···iNconsists of a collection of tensors
w that have both bond indices and physical indices. The tensors are interconnected
by the bond indices according to a tree pattern. The N physical indices correspond
to the leaves of the tree. Upon summing over all the bond indices, the TTN produces
the dNcomplex coefficients Ti1i2···iNof Eq. (59).
|Ψ? =
i1=1
d
?
i2=1
···
d
?
iN=1
Ti1i2···iN|i1?|i2?···|iN?,(59)
are complex numbers and the vectors
Page 22
Entanglement of 2 disjoint intervals in c = 1 theories22
Figure 12. (i) Diagrammatic representation of the two types of isometric tensors
in the TTN for a N = 4 lattice in Fig. 11. (ii) Graphical representation of the
constraints in Eqs. (61) and (62) fulfilled by the isometric tensors.
The tensors in the TTN will be constrained to be isometric, in the following sense.
As shown in Fig. 12 for the N = 4 lattice of Fig. 11, each tensor w in a TTN has
at most one upper leg/index α and two lower indices/legs β1,β2, so that its entries
read (w)α
β1,β2(everything can be generalized to tensors with more upper and lower
legs [53]). Then we impose that
?
For clarity, throughout this paper we use diagrams to represent tensors networks as
well as tensor manipulations. For instance, the constraints for the tensors w1and w2
of the TTN of Fig. 11 for a N = 4 lattice, namely
?
?
are represented as the diagrams in Fig. 12(ii). We refer to a tensor w that fulfills Eq.
(60) as an isometry.
An intuitive interpretation of the use of a TTN to represent a state |Ψ? can be
obtained in terms of a coarse-graining transformation for the lattice L. Notice that
the isometries w in Fig. 11 are organized in layers. The bond indices between two
layers can be interpreted as defining the sites of an effective lattice. In other words,
the TTN defines a sequence of increasingly coarser lattices {L0,L1,···,LT−1}, where
L0≡ L and each site of lattice Lτis defined in terms of several sites of Lτ−1by means
of an isometry wτ, see Fig. 13. In this picture, a site of the lattice Lτ effectively
corresponds to some number nτ of sites of the original lattice L0. For instance, each
of the two sites of L2in Fig. 13 corresponds to 8 sites of L0. Similarly, each site of
lattice L1corresponds to 4 sites of L0.
The use of isometric tensors, and the fact that each bond unambiguously defines
two parts (A : B) of the chain which are connected only through that bond as displayed
β1,β2
(w)α
β1,β2(w†)β1,β2
α?
= δαα?. (60)
β1β2
(w1)α
β1β2(w†
1)β1β2
α?
= δαα?,(61)
β1β2
(w2)β1β2(w†
2)β1β2= 1,(62)
Page 23
Entanglement of 2 disjoint intervals in c = 1 theories 23
Figure 13. The isometric TTN of Fig. 11 for a N = 8 lattice L0 with periodic
boundary conditions (the blue external circle) is associated with a coarse-graining
transformation that generates a sequence of increasingly coarse-grained lattices
L1, L2and L3(the inner circles). Notice that in this example we have added an
extra index to the top isometry w3, corresponding to the single site of an extra
top lattice L3, which we can use to encode in the TTN a whole subspace of V⊗N
instead of a single state |Ψ?.
in Fig. 14, implies that the rank of that bond in the TTN is given by the Schmidt
rank χ(A : B) of the partition (A : B) [59]. Thus the reduced density matrix ρAfor
a set A of sites of L is
ρA= trB|Ψ??Ψ| =
α
where pαare the eigenvalues of ρA. It follows then the R´ enyi entanglement entropies
S(n)
A
are
1
1 − nlogTrρn
and for n = 1
S(1)
A= −tr(ρAlogρA) = −
?
pα|ΨA
α??ΨA
α|,(63)
S(n)
A
=
A=
1
1 − nlog
?
α
pn
α,(64)
?
α
pαlogpα. (65)
In the following we denote the ranks of the tensor wτ, α,β1,β2as ,χτ,χτ−1,χτ−1.
In general, they fulfill
χτ< (χτ−1)2,(66)
meaning that wτ projects states in Vτ−1⊗ Vτ−1into the smaller Hilbert space Vτ.
Page 24
Entanglement of 2 disjoint intervals in c = 1 theories24
Figure 14. By erasing one of the indices in the TTN the spin chain is always
divided in two parts A and B [59]. Here we show that in the case of the N = 8
lattice of Fig. 11 there are three classes of indices, identified by their position in
the TTN. i) physical bonds connect a single spin with the rest of the lattice, ii)
bond indices of the first layer connect a block of two adjacent spins to the rest of
the lattice, iii) bond indices of the third layer of the lattice connect four adjacent
spins, to the other half. This implies that the rank of the index is the Schmidt
rank of the respective partition.
For a critical chain, the logarithmic scaling of the entanglement entropy (cf. Eq.
(2)) implies that the rank of the isometries should at least grow proportionally to the
length of the block represented by the effective spins
χτ∝ nτ, (67)
which means that while moving to higher layer of the tensor network the rank of
the isometries increases. This also implies that the leading cost of the computation is
concentrated in contracting the first few layers of the TTN. If N = 2Tand we describe
a pure state (so that the rank of the ατis one) the maximal rank of the tensors in the
TTN is
χτ= χT−1.χ = max
τ
(68)
In Ref. [53] it has been shown that i) a TTN description of the ground state of
chain of length N with periodic boundary conditions can be obtained numerically with
a cost of order O(logNχ4). ii) From the TTN it is also straightforward to compute
the spectrum {pα} of the reduced density matrix ρA (cf. Eq. (63)) when A is a
block of contiguous sites corresponding to an effective site of any of the coarse-grained
lattices L1,···,LT−1. Fig. 15 illustrates the tensor network corresponding to ρAfor
the case when A is one half of the chain. Many pairs of isometries are annihilated.
In addition, the isometries contained within region A can be removed since they do
Page 25
Entanglement of 2 disjoint intervals in c = 1 theories25
Figure 15. Computation of the spectrum {pα} of the reduced density matrix
ρAfor a block A that corresponds to one of the coarse-grained sites. (i) Tensor
network corresponding to ρAwhere A is half of the lattice. (ii) Tensor network
left after several isometries are annihilated with their Hermitian conjugate. (iii)
since the spectrum of ρAis not changed by the isometries acting on A, we can
eliminate them and we are left with a network consisting of only two tensors, which
can now be contracted together. The cost of this computation is proportional to
O(χ?3χ2) ≤ O(χ5).
not affect the spectrum of ρA. From the spectrum {pα}, we can now obtain the R´ enyi
entropies S(n)
A. The leading cost for computing the spectrum of the reduced density
matrix ρAfor this class of bipartitions is due to the contractions of the first layers of
the TTN. When the bipartition is such that A is a quarter of the chain, this implies
a cost proportional to O(χ?3χ2) ≤ χ?χ4, where χ?= χT−2.
It is also possible to compute the reduced density matrix ρAwhen A is composed
of two disjoint subintervals A1and A2, where now each of the two intervals is a block
of contiguous sites corresponding to an effective site of the coarse grained lattice. The
cost of this computation is again dominated by contracting the upper part of the tensor
network, and the most expensive case is obtained by considering A as the collection of
two N/4 spins blocks, separated by N/4 spins. The tensor network corresponding to
this ρAis shown in Fig. 16. Also in this case many pairs of isometries are annihilated.
The isometries contained within the composed region A can also be removed since
they do not affect the spectrum of ρA. The cost of contracting this tensor network is
proportional to max[O(χ2χ?4),O(χ3χ?2)] < O(χ6).
5.2. The TTN and the anisotropic Heisenberg spin-chain
In the previous subsection we have shown how to extract the spectrum of the reduced
density matrix for a single and a double spin block from a TTN state.
manuscript we are interested in reduced density matrices calculated on the ground-
state of the anisotropic Heisenberg spin chain (XXZ model) in zero magnetic field,
In this
Page 26
Entanglement of 2 disjoint intervals in c = 1 theories 26
Figure 16. Computation of the spectrum {pα} of the reduced density matrix
ρA when A corresponds to two coarse-grained sites separated by one coarse
grained site from both sides. (i) Tensor network corresponding to ρA where
A is a quarter of the lattice. (ii) Tensor network left after several isometries
are annihilated with their Hermitian conjugate.
ρA is not changed by the isometries acting on A, we can eliminate them and
we are left with a network consisting of only few tensors, which can now be
contracted together. The cost of contracting this tensor network is proportional
to max[O(χ2χ?4),O(χ3χ?2)] < O(χ6).
(iii) Since the spectrum of
defined by the Hamiltonian
H =
L
?
j=1
[σx
jσx
j+1+ σy
jσy
j+1+ ∆σz
jσz
j+1],(69)
where σα
assumed.
−1 < ∆ ≤ 1. This phase is described by a free-bosonic CFT compactified on a
circle with radius that depends on the parameter ∆
1
2KL
where KLis the Luttinger liquid parameter. § The sign convention in the Hamiltonian
(69) is such that the model is (anti)ferromagnetic for ∆ < 0 (∆ > 0). Hamiltonian
(69) is diagonalizable by means of Bethe ansatz. However, obtaining the spectrum of
the reduced density matrix from Bethe ansatz is still a major problem and only results
for small subsystems are known [44, 67]. For this reason we exploit variational TTN
techniques to obtain the ground state.
Here we follow the variational procedure described in detail in Ref. [53], where
the generic technique (consisting of assuming a tensor network description of the
ground state and minimize the energy variationally improving the tensors one by one
as described, i.e., in Ref. [4]) has been specialized and optimized for the case of a
jare the Pauli matrices at the site j.
We are interested in gapless conformal phases of the model, that is
Periodic boundary conditions are
η = 2r2
circle=
=arccos(−∆)
π
,(70)
§ Notice similarities and differences between Eq. (70) and its analogous for the AT model (31). The
relation between η and r2and the relation between KLand ∆ are the same for both XXZ spin-chain
and AT model, but the relation between η and ∆ (or KLand r) is different.
Page 27
Entanglement of 2 disjoint intervals in c = 1 theories27
Figure 17. TTN data for the non universal constant c2(Lc) as function of the
chord length Lc for different values of ∆.
function A + BL−KL
c
. The reported data have been obtained with L = 128 for
∆ = 0,0.1,0.6 and L = 64 for the other values.
The dashed curves are fits to the
TTN. We exploit translation invariance by using the same tensor at each layer of the
TTN. One could also improve the efficiency further by exploiting the U(1) symmetry
of the Hamiltonian (69), i.e. the rotations around the z axis. However we did not
make use of this symmetry here.
6. The Block Entanglement of the Anisotropic Heisenberg spin-chain
In this section we report the TTN results for the R` enyi entropies in the XXZ spin-
chain for a single and a double interval. As a main advantage compared to the classical
Monte Carlo simulations performed for the AT model, with a single TTN simulation
we obtain the spectrum of the reduced density matrix and hence any R` enyi entropy,
including von Neumann S(1)
A. Oppositely with the Monte Carlo methods only R` enyi
entropies S(n)
A
of integer order n ≥ 2 can be obtained and each of them requires an
independent simulation.
6.1. The single interval.
We first present the TTN results for the single interval. These have been already
obtained with many numerical variational techniques [34, 68, 69, 38] and are reported
here only to test the accuracy of the TTN and to fix units/scales etc. Using variational
TTN, we find the ground-state of the XXZ Hamiltonian (69) and from this we extract
the spectrum of the reduced density matrix of the single block, as explained in the
previous section. We then numerically obtain Trρn
A. The maximum size of the chain
Page 28
Entanglement of 2 disjoint intervals in c = 1 theories28
Figure 18. TTN data for Flat
L = 16,32,64,128, subsystem lengths ? = 4,8,16,32, and ∆ = −0.3,−0.1,01,0.6.
Different values of ∆ are distinguished by different colors, while different symbols
denote different values of ?. The arrows denote the (asymptotically) increasing
subsystem sizes ?.
2(x) as function of x for various sizes of the chain
that we consider is L = 128. The subsystem lengths considered are ? = 2,4,8,16,32.
Notice that with the TTN method, using a binary tree as we are doing, we can
effectively access only subsystems sizes of the form 2mwith m arbitrary integer, as it
should be clear from the previous section. In particular this limits the calculation to
even values of ? and we can not study the parity effects reported in Ref. [34, 35].
We considered different values of the anisotropy parameter ∆, namely ∆ =
−0.3,−0.1,0,0.1,0.2,0.4,0.6,0.8,1.
∆ ≤ −0.5. This can be easily traced back to the smallness of the finite-size gap that in
the minimization process causes the algorithm to be stuck in meta-stable states when
the system size is large enough. This drawback could be cured by using larger values
of χ (and so larger computational cost), but as we shall see, the considered values
of ∆ suffice to draw a very general picture of the entanglement. For the isotropic
Heisenberg antiferromagnet at ∆ = 1 we ignore the presence of logarithmic corrections
to the scaling [68, 36], that have a minimal effect for all our aims.
As for the AT model, we study the quantity c2(Lc) defined by the ratio in Eq.
(55). The results are shown in Fig. 17 for all considered values of ∆. The scaling
corrections are evident, especially for larger values of ∆, as expected [34]. These
corrections for Trρn
The dashed lines reported in Fig. 17 are fits to this form for n = 2, showing the
agreement between TTN data and the fits. We checked that all the TTN data agree
with the ones obtained in Ref. [34] using density matrix renormalization group. The
agreement is perfect and for this reason we refer to the above paper for a detailed
The TTN becomes less effective for values of
Aare indeed of the form L−2KL/n
c
[34] (KLis defined in Eq. (70)).
Page 29
Entanglement of 2 disjoint intervals in c = 1 theories29
Figure 19. TTN data for Flat
∆. The dashed lines are fits to the function with the generalized finite-? ansatz
(71).
2(1/2) − FCFT
2
(1/2) as function of 1/? for various
study of Trρn
Afor n > 2.
6.2. Double interval: the n = 2 case.
We now consider a subsystem made of two parts A1and A2of equal length ?. We
start by studying the quantity Trρ2
A1∪A2for finite chains and extract the universal
function FCFT
2
(x) by proper extrapolation. Since we only consider even ?, corrections
to the scaling are expected to be monotonic in ? also for F2(x), oppositely to the case
of arbitrary ? parity [17, 26]. The CFT prediction for the function F2(x) for the XXZ
chain is Eq. (5) with η given by Eq. (70).
In Fig. 18 we report TTN data for Flat
Eq. (56)) as function of the cross ratio x for ∆ = −0.3,−0.1,0.1,0.6 and subsystem
sizes ? = 4,8,16,32. The different values of ∆ are denoted with different colors, while
the different symbols stand for the various ?. On the same figure we also show the
asymptotic FCFT
2
(x) as dashed lines. It is evident that strong scaling corrections
affect the data, as expected. Colored arrows denote the direction of (asymptotically)
increasing subsystem sizes.Very surprisingly, while for ∆ = −0.3,−0.1,0.1 the
asymptotic CFT result is approached from below, for ∆ = 0.6 it is approached from
above.Moreover, for ∆ = 0.6 the behavior of the data is not monotonic.
contrasts the results obtained for the AT model in the previous sections and the ones
obtained for the XX and Ising spin-chains [26].
In order to shed some light on this unexpected phenomenon, it is worth to look
at Flat
2(x) as functions of ? for fixed values of x. In Fig. 19 we report one of these
plots for x = 1/2. Analogous figures are obtained for other values of x. Corrections
2(x) (obtained with the ratio defined in
This
Page 30
Entanglement of 2 disjoint intervals in c = 1 theories30
Figure 20. TTN data for Flat
∆ = −0.3,0.1,0.6,1, and subsystem lengths ? = 4,8,16,32.
different symbols the values of ? and with different colors the various ∆. The
dashed curves are the theoretical results given by Eq. (7). The arrows denote the
(asymptotically) increasing subsystem sizes ?.
3(x) as function of x for various sizes of the chain,
We denote with
to the scaling are non-monotonic in the range 0.2 ≤ ∆ ≤ 0.7. This phenomenon can
be understood if further corrections to the scaling are taken into account. There are
two corrections that can be responsible of this behavior. On the one hand, corrections
of the form ?−mKL(from ?−2mKL/nat n = 2) for any integer m are know to be
present [35], on the other hand usual analytic corrections such as ?−1are generically
expected to exist for any quantity from general scaling arguments. Thus the most
general finite-? ansatz has the form
Flat
2(x) = FCFT
2
(x) +f2(x)
?KL
+fA(x)
?
+fB(x)
?2KL... , (71)
where the first correction is the unusual one employed also for the Ashkin-Teller model,
and the other two are the ones just discussed. The effect of subleading corrections
is enhanced by the fact the the amplitude functions f2(x) and fA(x) or fB(x) have
opposite signs determining the non-monotonic behavior. Unfortunately, for values of ∆
for which the effect of subleading corrections is more pronounced (i.e. 0.1 ≤ ∆ ≤ 0.6),
we have KL < 1 < 2KL, making difficult to disentangle corrections with close
exponents. Thus, in order to present analyses of a good quality, we ignore the last
correction (i.e. we fix fB(x) = 0). To check the proposed scenario, we performed the
fit of the data in Fig. 19 with the ansatz (71) and fB(x) = 0. The results of the fits
are reported in the same figure, showing perfect agreement with the data for all the
values of ∆. We repeated the same analysis for other values of x, finding the same
quality of fits as for x = 1/2. However, we cannot exclude that corrections of the form
?−2KLhave an important role.
Page 31
Entanglement of 2 disjoint intervals in c = 1 theories31
Figure 21.
? = 4,8,16,32, The considered values of ∆ are ∆ = −0.3,−0.1,01,0.6,1. The
dashed curves are fits with the ansatz (72)
TTN data for Flat
3(x) at fixed x = 1/2 as function of ?−1for
6.3. Double interval: the n = 3 case.
Now we report the same analysis performed for Trρ2
i.e. Trρ3
A. Again we consider finite-size XXZ spin-chains and extract the universal
function FCFT
3
(x) by finite-size analysis. The expected CFT result is given for general
n by Eq. (7). In Fig. 20 we show TTN data for Flat
∆ = −0.3,0.1,0.6,1 and subsystem sizes up to ? = 32. We also show the theoretical
curves given by Eq. (7). As for the n = 2, the asymptotic universal curve is approached
from below for ∆ ≤ 0.6, and from above for ∆ ≥ 0.6. Furthermore, the behavior of
the numerical data for ∆ > 0.6 is non monotonic. This suggests that the ansatz in
Eq. (57) is not enough to describe accurately the TTN data and further corrections
to the scaling should be included as for Trρ2
For n = 3, the leading corrections to the scaling are described by the ansatz (57),
i.e. the leading exponent is 2KL/3. Thus, for the cases when subleading corrections are
more important (i.e. for ∆ ≥ 0.6) the ordering of the exponents is 2KL/3 < 4KL/3 < 1
and so it is reasonable to ignore the analytic correction. Thus we fit TTN data with
the function
(x) = f3(x)?−2KL/3+ fB(x)?−4KL/3.
Afor the third moment of ρA,
3(x) (obtained from Eq. (56)) at
A.
Flat
3(x) − FCFT
3
(72)
In Fig. 21 we report TTN data for Flat
of ∆. The dashed lines are fits with the finite-size ansatz (72), that perfectly reproduce
the data.
3(x)−FCFT
3
(x) for x = 1/2 and several values
6.4. Double interval: The von Neumann entropy.
TTN gives access to the full spectrum of the reduced density matrix of A1∪A2and so
to the entanglement entropy S(n)
1
as well. In Fig. 22 we report the function Flat
V N(x)
Page 32
Entanglement of 2 disjoint intervals in c = 1 theories32
Figure 22. TTN data for the von Neumann entropy for various values of ∆
in the interval [−0.3,1]. We show with different symbols the values of ∆ while
different colors stand for different ? and lattice sizes.
defined as
Flat
V N(x) = S(1)
A1∪A2− S(1)
A1− S(1)
A2−1
3log(1 − x),(73)
for ∆ in the interval [−0.3,1] for various L up to 128 and subsystem sizes ? =
2,4,8,16,32,64. We indicate with different symbols different values of ∆, while the
colors are for various sizes ?. As known from many other investigations on single
and double intervals (quantum Ising spin chain, XY model, XXZ) the von Neumann
entropy does not show oscillations with the parity of the subsystem and the corrections
are much smaller, actually negligible from any practical porpouse. Fig. 22 confirms
this observation for the two interval entanglement entropy for the XXZ spin-chain in
a wide range of ∆. Indeed, at fixed value of ∆ perfect data collapse is observed even
for very small values of ?.
Unfortunately, as already stated in the introduction, the CFT prediction for
FV N(x) is unknown for general x because the analytic continuation of Fn(x) to non-
integer n is not achievable. However, an expression for the leading term of the small
x expansion of FV N(x) has been recently extracted [30] from Eq. (13)
?x
where α = min[η,1/η] and Γ is the Euler function (not to be confused with the Γ
matrix in Eq. (8)). In order to check the correctness of this formula, in Fig. 23 we
report the same data for FV N(x) in a log-log scale to highlight the power-law behavior
for small x. We also report the small x expected from Eq. (74). For ∆ = −0.3 the
agreement is good, but it gets worse increasing ∆. The natural explanation is that
FV N(x) =
4
?α√π
Γ(α + 1)
2Γ(α + 3/2),(74)
Page 33
Entanglement of 2 disjoint intervals in c = 1 theories33
Figure 23. TTN data for the von Neumann entropy for ∆ = −0.3,0.1,0.6,0.8
(the data for different ∆ are denoted with different symbols) in log-log scale. We
used different colors to indicate the different block sizes ? and lattice sizes L. The
continuous lines are the small x behavior obtained from (74). The dashed lines
are the small x behavior where the O(x) term has been added as in Eq. (75).
the considered values of x are not small enough for the asymptotic Eq. (74) to be
valid. We should then include further terms in the small x expansion. As explained
in Ref. [30], further coefficients in the expansion for small x are difficult to obtain in
general. However, there is a term that is very easy to obtain and that (luckily enough)
is responsible of the previous disagreement. Indeed, as shown in Ref. [30] (cf. Eq. 70
and 71 there) the function Fn(x) has always (i.e. independently of η) a simple O(x)
contribution coming from the denominator in Eq. (7), i.e. |Θ(0|Γ)|2= 1+x(n−1/n)/6,
that can be easily analytically continued giving
?x
Notice that the added term becomes more important when α is close to 1, i.e. in the
XXZ spin-chain when ∆ approaches 1. In Fig. 23 we also report the prediction (75)
as dashed line, that is asymptotically in perfect agreement with the numerical data
for all values of ∆.
FV N(x) =
4
?α√π
Γ(α + 1)
2Γ(α + 3/2)−x
3+ O(x2α). (75)
7. Conclusions
In this manuscript we provided a number of results for the asymptotic scaling of the
R´ enyi entanglement entropies in strongly interacting lattice models described by CFTs
with c = 1. Schematically our results can be summarized as follows.
• We provided the analytic CFT result for the scaling function F2(x) for S(2)
the case of a free boson compactified on an orbifold describing, among the other
things, the scaling limit of the Ashkin-Teller model on the self-dual line. The
final result is given in Eq. (25).
A
in
Page 34
Entanglement of 2 disjoint intervals in c = 1 theories34
• We developed a cluster Monte Carlo algorithm for the two-dimensional Ashkin-
Teller model (generalizing the procedure of Caraglio and Gliozzi [18] for the
Ising model) that gives the scaling functions of the R´ enyi entanglement entropy
(for integer n) of the corresponding one-dimensional quantum model. With this
algorithm, we calculated numerically the scaling function F2(x) of the AT model
along the self-dual line and we confirm the validity of the CFT prediction. In
order to obtain a quantitative agreement, the corrections to scaling induced by
the finite length of the blocks are properly taken into account.
• We considered the XXZ spin chains by means of a tree tensor network (TTN)
algorithm. The low-energy excitations of model are described by a free boson
compactified on a circle for which CFT predictions are already available both for
n = 2 [17] and for general integer n [19]. Taking into account the corrections to
the scaling, we confirm these predictions (that resisted until now to quantitative
tests) for n = 2,3. Furthermore, we provide numerical determinations of the
scaling function of the von Neumann entropy (cf.
predictions do not exist yet for general x. For small x we confirm the recent
prediction of Ref. [30] (cf. Fig. 23).
Fig.22) for which CFT
The methods we employed (classical Monte Carlo with cluster observables and
TTN) are very general techniques that can be easily adapted to other models of
physical interest. On the CFT side, it must be mentioned that a closed form for the
functions Fn(x) at integer n for a free boson compactified on an orbifold is not yet
available, but work in this direction is in progress [70].
Acknowledgments
We thank John Cardy, Maurizio Fagotti, Erik Tonni, Ettore Vicari, and Guifre Vidal
for useful discussions. This work has been partly done when PC was guest of the
Galileo Galilei Institute in Florence whose hospitality is kindly acknowledged.
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