# Finite Size Corrections to Entanglement in Quantum Critical Systems

**ABSTRACT** We analyze the finite size corrections to entanglement in quantum critical systems. By using conformal symmetry and density functional theory, we discuss the structure of the finite size contributions to a general measure of ground state entanglement, which are ruled by the central charge of the underlying conformal field theory. More generally, we show that all conformal towers formed by an infinite number of excited states (as the size of the system $L \to \infty$) exhibit a unique pattern of entanglement, which differ only at leading order $(1/L)^2$. In this case, entanglement is also shown to obey a universal structure, given by the anomalous dimensions of the primary operators of the theory. As an illustration, we discuss the behavior of pairwise entanglement for the eigenspectrum of the spin-1/2 XXZ chain with an arbitrary length $L$ for both periodic and twisted boundary conditions.

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**ABSTRACT:**The properties of the entanglement entropy (EE) of two particle excited states in a one-dimensional ring are studied. For a clean system we show analytically that as long as the momenta of the two particles are not close, the EE is twice the value of the EE of any single particle state. For almost identical momenta the EE is lower than this value. The introduction of disorder is numerically shown to lead to a decrease in the median EE of a two particle excited state, while interactions (which have no effect for the clean case) mitigate the decrease. For a ring which is of the same size as the localization length, interaction increase the EE of a typical two particle excited state above the clean system EE value.Physical review. B, Condensed matter 02/2013; 87(7). · 3.77 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**The properties of the entanglement entropy (EE) of low-lying excitations in one-dimensional disordered interacting systems are studied. The ground state EE shows a clear signature of localization, while low-lying excitation shows a crossover from metallic behavior at short sample sizes to localized at longer length. The dependence of the crossover as function of interaction strength and sample length is studied using the density matrix renormalization group (DMRG). This behavior corresponds to the presence of the predicted many particle critical energy in the vicinity of the Fermi energy. Implications of these results to experiments are discussed.10/2013; - SourceAvailable from: de.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We study the Renyi entanglement entropy of an interval in a periodic spin chain, for a general eigenstate of a free, translational invariant Hamiltonian. In order to compute analytically the entropy we use two technical tools. The first one is used to reduce logarithmically the complexity of the problem and the second one to compute the R\'enyi entropy of the chosen subsystem. We introduce new strategies to perform the computations and derive new expressions for the entropy of these general states. Finally we show the perfect agreement of the analytical computations and the numerical results.Journal of Physics A Mathematical and Theoretical 01/2014; 47(24). · 1.77 Impact Factor

Page 1

arXiv:0808.0020v2 [quant-ph] 14 Oct 2008

Finite Size Corrections to Entanglement in Quantum Critical Systems

F. C. Alcaraz∗

Instituto de F´ ısica de S˜ ao Carlos, Universidade de S˜ ao Paulo,

Caixa Postal 369, 13560-590, S˜ ao Carlos, SP, Brazil.

M. S. Sarandy†

Instituto de F´ ısica, Universidade Federal Fluminense,

Av. Gal. Milton Tavares de Souza s/n, Gragoat´ a, 24210-346, Niter´ oi, RJ, Brazil.

(Dated: October 14, 2008)

We analyze the finite size corrections to entanglement in quantum critical systems. By using

conformal symmetry and density functional theory, we discuss the structure of the finite size con-

tributions to a general measure of ground state entanglement, which are ruled by the central charge

of the underlying conformal field theory. More generally, we show that all conformal towers formed

by an infinite number of excited states (as the size of the system L → ∞) exhibit a unique pattern

of entanglement, which differ only at leading order (1/L)2. In this case, entanglement is also shown

to obey a universal structure, given by the anomalous dimensions of the primary operators of the

theory. As an illustration, we discuss the behavior of pairwise entanglement for the eigenspectrum

of the spin-1/2 XXZ chain with an arbitrary length L for both periodic and twisted boundary

conditions.

PACS numbers: 03.65.Ud, 03.67.Mn, 75.10.Jm

I. INTRODUCTION

In recent years, the observation that entanglement

may play an important role at a quantum phase tran-

sition [1, 2, 3, 4] has motivated intensive research on

the characterization of critical phenomena via quantum

information concepts. In this direction, conformal invari-

ance has brought valuable information about the behav-

ior of block entanglement, as measured by the von Neu-

mann entropy, in critical models. Indeed, conformal field

theory (CFT) has been used as a powerful tool to deter-

mine universal properties of entanglement. Remarkably,

it was shown that the entanglement entropy obeys a uni-

versal logarithmic scaling law for one-dimensional criti-

cal models both at zero and finite temperatures [5, 6, 7],

which is governed by the central charge of the associ-

ated CFT. Moreover, corrections to the entanglement

entropy due to finite size effects have also been consid-

ered [6, 8] for periodic and open boundary conditions.

Together with approximative methods such as renormal-

ization group (see, e.g. Refs. [9, 10, 11, 12, 13]) and den-

sity functional theory (DFT) [14], CFT has been settled

as one of the most promising approaches for investigat-

ing the behavior of entanglement in many-body quantum

critical systems.

In this work, we will exploit in a new perspective the

impact of CFT methods for the evaluation of entangle-

ment at criticality. More specifically, our approach will

be based on the statement that finite size corrections to

the ground state expectation values of arbitrary observ-

∗Electronic address: alcaraz@ifsc.usp.br

†Electronic address: msarandy@if.uff.br

ables are ruled by conformal invariance. This conclusion

is indeed a consequence of two results: (1) Finite size cor-

rections to the energy spectrum of a critical theory are

determined by conformal invariance [15, 16, 17]; (2) DFT

techniques imply that, under certain conditions discussed

below, general observables can be evaluated as a function

of the first derivative of the ground state energy with re-

spect to a Hamiltonian coupling parameter [18, 19]. We

then simultaneously apply these two results to obtain

the finite size corrections to ground state entanglement

in critical models. As a by-product, conformal invari-

ance determines the structure of entanglement in the

presence of extra symmetries for certain higher energy

states, which are the lowest energy states in each sym-

metrically decoupled subspace of the Hilbert space. For

instance, if the Hamiltonian is translationally invariant

and has a U(1) symmetry due to its commutation with

the magnetization operator, we can split out the Hilbert

space in sectors of fixed momentum and magnetization.

More generally, we will also show that all conformal tow-

ers formed by an infinite number of excited states (as the

size of the system L → ∞) exhibit a unique pattern of

entanglement, which differ only at leading order (1/L)2.

This will be based on a generalization of the HK theo-

rem for individual states belonging to conformal towers

of critical systems. Finite size corrections to entangle-

ment in these excited states will obey a universal struc-

ture, given by the anomalous dimensions of the primary

operators of the theory.

Since our approach is applicable for any entanglement

measure, it allows in particular for the investigation of

the universality properties of pairwise entanglement mea-

sures, e.g., concurrence [20] and negativity [21]. For pair-

wise measures, criticality was first noticed through a di-

vergence in the derivative of entanglement, signaling a

Page 2

2

second-order phase transition [1]. For first-order phase

transitions, jumps in entanglement itself indicates quan-

tum critical points [22, 23]. A general explanation for this

distinct usual behavior of first-order and second-order

phase transitions has been provided in Refs. [19, 24] (for

an explicit discussion of examples which do not obey this

expected behavior, see Ref. [25]). From the point of view

of CFT, we will be able to explicitly work out the finite

size corrections to pairwise entanglement measures and

show how these corrections involve universal quantities,

such as the central charge or the anomalous dimension of

primary operators associated with the CFT. As an illus-

tration, we will consider the spin-1/2 XXZ chain, where

an analytical expression, valid up to o(L−2), will be pro-

vided for the negativity of nearest neighboring spins as a

function of the size L of the chain.

II.

EFFECTS IN CRITICAL QUANTUM SYSTEMS

ENERGY SPECTRUM AND FINITE SIZE

Let us consider a critical theory in a strip of finite

width L with periodic boundary conditions. The transfer

matrix of the theory is written as T = exp(aH), where

a denotes the lattice spacing and H is the Hamiltonian.

Then, for large L, the ground state energy density ε(L) =

E0(L)/L of H is provided by conformal invariance [15,

16], reading

ε(L) = ε∞−π cξ

6

L−2+ o(L−2), (1)

where ε∞is the energy density in the limit L → ∞ and

o(L−2) denotes terms of any order higher than L−2. In

Eq. (1), c is the central charge of the Virasoro algebra

(the conformal anomaly) and the parameter ξ must be

fixed in such a way that the equations of motion of the

theory are conformally invariant [26]. The structure of

the higher energy states is determined by the primary

operators of the theory [17]. For each operator Oαwith

anomalous dimension xα, there corresponds a tower of

states with energy densities εα

j,j′(L) given by

εα

j,j′(L) = ε(L) + 2π ξ(xα+ j + j′)L−2+ o(L−2), (2)

where j,j′= 0,1,... are indices labelling the tower of

states associated with the anomalous dimensions xα.

Higher-order corrections to Eqs. (1) and (2) as well as

convenient generalizations for more general boundary

conditions, e.g., twisted boundary conditions, may also

be obtained [27, 28].

III.

EXPECTATION VALUES OF OBSERVABLES

HOHENBERG-KOHN THEOREM AND

Let us turn now to the discussion on how DFT can

be allied with conformal invariance to extract informa-

tion about expectation values of observables from the

energy spectrum. DFT [29, 30] is originally based on the

Hohenberg-Kohn (HK) theorem [29] which, for a many-

electron system, establishes that the dependence of the

physical quantities on the external potential v(r) can be

replaced by a dependence on the particle density n(r).

The HK theorem can be extended for the context of a

generic quantum Hamiltonian H on a lattice (see, e.g.,

Refs. [18, 19]). In order to be specific, let us consider a

quantum spin chain of size L governed by the Hamilto-

nian

H = H0+ λ

L

?

i=1

Ai, (3)

where λ is a control parameter associated with the Hermi-

tian operators Aiwhich act on the site i, e.g., an observ-

able relevant to driving a quantum phase transition. Let

us take, for simplicity, a translationally invariant chain

(e.g., by assuming periodic boundary conditions). Then,

by taking the expectation value of Eq. (3), we obtain

?H? = ?H0? + λL?A?,(4)

where ?A? ≡ ?Ai? = ?Aj? (∀i,j) due to translation sym-

metry. Therefore

ε = ε0+ λ?A?, (5)

where ε = ?H?/L and ε0= ?H0?/L are the energy densi-

ties associated with H and H0, respectively. For a general

Hamiltonian such as given in Eq. (3), the HK theorem

can be generalized to the statement that there is a du-

ality (in the sense of a Legendre transform) between the

expectation value ?A? (corresponding to n(r)) and the

control parameter λ (corresponding to v(r)) [18, 19]. In

order to specify the conditions supporting this duality

let us separately consider the cases of nondegenerate and

degenerate Hamiltonians.

A. Nondegenerate case

Let λ and λ′be two fixed values of the coupling pa-

rameter in Eq. (3), which correspond to nondegenerate

ground states given by |ψ? and |ψ′?, respectively. We as-

sume that, for λ ?= λ′, we have that |ψ? ?= α|ψ′?, with

α a complex phase. This assumption means that differ-

ent values of the coupling parameter are associated with

distinct ground states. It reflects the requirement of the

uniqueness of the potential (see, e.g., Ref. [31]). A gen-

eral condition to ensure the uniqueness of the potential

for Hamiltonian (3) will be derived below. Then, by as-

suming a unique potential and taking two different cou-

plings λ and λ′, the Rayleigh-Ritz variational principle

allows us to write

?ψ|H|ψ? < ?ψ′|H|ψ′? = ?ψ′|H′|ψ′? + (λ − λ′)L?A?′, (6)

where ?A?′= ?ψ′|A|ψ′? and H and H′are the Hamilto-

nians associted with λ and λ′, respectively. Therefore

ε − ε′< (λ − λ′)?A?′. (7)

Page 3

3

Analogously, application of the variational principle for

the ground state |ψ′? results into

ε′− ε < (λ′− λ)?A?.

By adding Eqs. (7) and (8) we obtain

(8)

?A?′?= ?A?. (9)

Eq. (9) expresses the HK theorem for nondegenerate

ground states, stating that distinct densities are asso-

ciated with distinct potentials. In other words, we can

establish the map

λ ⇐⇒ |ψ? ⇐⇒ ?A? = ?ψ|A|ψ?. (10)

B. Degenerate case

In order to establish the HK theorem for degenerate

ground states, let us consider two fixed values of the cou-

pling constant, each of them associated with arbitrarily

degenerate ground states:

λ ←→ q − degenerate ground states:{|ψ1?,...,|ψq?} ,

λ′←→ q′− degenerate ground states:?|ψ′

Considering that any of the ground states are equally

likely, we can describe them by the uniformly distributed

density matrices

1?,...,|ψ′

q′??.

ρ =1

q

q

?

i=1

|ψi??ψi| , ρ′=1

q

q′

?

i=1

|ψ′

i??ψ′

i|. (11)

The requirement of uniqueness of the potential yields in

the degenerate case the condition that ρ and ρ′are dis-

tinct. Applying the variational principle, we obtain

Tr(ρH) < Tr(ρ′H) = Tr(ρ′H′) + (λ − λ′)L?A?′, (12)

where, here, ?A?′= Tr(ρ′A).

ε − ε′< (λ − λ′)?A?′. Therefore, as before, we use the

complementary equation ε′−ε < (λ′− λ)?A? and obtain

?A?′?= ?A?. The HK map in this case can be written as

λ ⇐⇒ ρ ⇐⇒ ?A? = Tr(ρA) .

Eq. (12) implies that

(13)

C. Uniqueness of the potential

As discussed above, the condition for the uniqueness

of the potential, which is fundamental for the derivation

of the HK theorem, is defined by the requirement that

different values of the coupling parameter λ are associ-

ated with distinct ground states of the Hamiltonian H.

Here we will show that a necessary and sufficient con-

dition for which different values of λ are associated with

distinct eigenstates of H is that the operators H0 and

?

λ and λ′yield the same eigenstate of H

?

?

iAi, as given in Eq. (3), do not have common eigen-

states. Sufficiency: Suppose that two distinct couplings

H0+ λ

?

i

Ai

?

?

|ψ? = E(λ)|ψ?

(14)

H0+ λ′?

i

Ai

|ψ? = E(λ′)|ψ?. (15)

Then, from Eqs. (14) and (15), we obtain

?

i

Ai|ψ? =E(λ) − E(λ′)

λ − λ′

|ψ?.(16)

Therefore, in this case, |ψ? is also an eigenstate of?

that H0 and?

Necessity: Let us suppose that H0 and?

H0|ψ? = E0|ψ?

?

Then we obtain that (H0+ λ?

ing the same eigenstate), which means that distinct cou-

plings will lead to the same eigenstate of H. Therefore,

the condition that H0and?

the potential. In conclusion, the sufficient and neces-

sary condition for the uniqueness of the potential can be

translated by the noncommutation relation

?

i

iAi

(as well as an eigenstate of H0). Hence, the condition

iAi do not have common eigenstates is

sufficient for ensuring the uniqueness of the potential.

iAi have a

common eigenstate

(17)

i

Ai|ψ? = a|ψ?

(18)

iAi)|ψ? = (E0+ λa)|ψ?.

Hence, by varying λ, we only change the eigenvalue (keep-

iAido not exhibit a com-

mon eigenstate is also necessary for the uniqueness of

H0,

?

Ai

?

?= 0 (19)

Naturally, we disregard in Eq. (19) the rather nonusual

situation where H0and?

state.

iAiare noncommuting observ-

iAi]|ψ? results in a vanishing quantum

ables, but [H0,?

D.HK theorem for conformal towers in quantum

critical models

Since the HK theorem is based on a variational princi-

ple, we cannot guarantee that the expectation values of

the observables in individual excited states are in general

a function of the derivative of the energy of the excited

state. Naturally, as previously mentioned in Sec. I, the

HK theorem can be applicable in the presence of symme-

tries to excited states that are the minimum energy states

in a given symmetric subspace of Hilbert space. In this

work, we show that, under certain conditions, the HK

theorem can also be extended for all the individual states

Page 4

4

of conformal towers in quantum critical models. We be-

gin by supposing a periodic chain governed by a Hamilto-

nian given by Eq. (3) which is conformally invariant in a

critical interval λc1≤ λ ≤ λc2. Moreover we will assume

the condition (19) for the uniqueness of the potential. Let

us denote by {|ψα

ated with the energy εα

j,j′(λ), with d = 1,...,D labelling

the D-fold degeneracy (see Eq. (2)). We take the system

in a uniformly distributed density matrix

j,j′;d(λ)?} the set of eigenstates associ-

ρα

j,j′(λ) =1

D

D

?

d=1

|ψα

j,j′;d(λ)??ψα

j,j′;d(λ)|. (20)

Our aim is to show that the potential λ uniquely specifies

the density

?A?α

j,j′ = Tr?ρα

j,j′(λ)A?=

j,j′/∂λ must be a monotonic

∂εα

∂λ

j,j′

. (21)

Therefore, the derivative ∂εα

function of λ. In order for this to occur, it is sufficient

that: (i) ∂εα

j,j′/∂λ is continuous in the interval λc1≤

λ ≤ λc2and (ii) ∂2εα

achieved for a smooth (well-behaved) energy. Concerning

condition (ii), let us take the derivative of Eq. (2), which

yields

∂2εα

j,j′

∂λ2

L2

The first term in the r.h.s. of Eq. (22) concerns the sec-

ond derivative of the ground state energy with respect

to λ. We can show that this term is strictly negative.

Indeed, from Eqs. (7) and (8), which hold for both de-

generate and non-degenerate ground states, we obtain

∂λ?A? =∂2ε

where ?A? denotes the expectation value of A taken in the

ground state. Concerning the second term in the r.h.s.

of Eq. (22), it is negligible for large L. Consequently, we

can write

∂2εα

j,j′

∂λ2

Hence, ∂2εα

tive ∂εα

j,j′/∂λ is monotonically related to λ. Therefore,

a D-fold degenerate (up to order L−2) eigenlevel given

by α, j, and j′defines a density matrix ρα

taken either as a function of λ or ?A?α

an extension of the HK theorem for arbitrary individual

eigenstates belonging to conformal towers in quantum

critical models.

j,j′/∂λ2?= 0. Condition (i) is usually

=∂2ε

∂λ2+2π

∂2

∂λ2[ξ(xα+ j + j′)]+o(L−2). (22)

∂

∂λ2< 0, (23)

≈∂2ε

∂λ2< 0 (large L). (24)

j,j′/∂λ2is non-vanishing and then the deriva-

j,j′ that can be

j,j′. This provides

IV.

ENTANGLEMENT IN CONFORMAL

INVARIANT MODELS

FINITE SIZE CORRECTIONS TO

The HK theorem implies a duality between the poten-

tial λ and the density ?A?. This behavior was revealed

specially useful for the investigation of entanglement in

the ground state of quantum systems undergoing quan-

tum phase transitions [19]. In particular, the dependence

of an arbitrary entanglement measure M on the param-

eter λ can be replaced by the dependence on the ground

state expectation value ?A? [19], which means that

M = M(λ) = M(?A?) = M(∂ε

∂λ),(25)

where the Hellmann-Feynman theorem [32, 33] has been

used in the last equality above. As discussed in the last

Section, in the case of critical models, the HK theorem

can also be applied to any state of conformal towers,

which allows us to write the entanglement of such states

as

Mα

j,j′ = Mα

j,j′(λ) = Mα

j,j′(?A?α

j,j′). (26)

Eq. (26) can be rewritten by observing that

?A?α

j,j′ =∂εα

j,j′

∂λ

= ?A? +2π

L2

∂

∂λ[ξ(xα+ j + j′)] + o(L−2)

(27)

By inserting Eq. (27) into Eq. (26) and performing a

series expansion, we obtain

Mα

j,j′ = M(?A?) +2π

L2

∂

∂λ[ξ(xα+ j + j′)]

?∂Mα

j,j′

∂λ

?

λ=?A?

+ o(L−2) (28)

This means that the entanglement corresponding to all

the eigenstates in the conformal towers are nearly the

same as that of the ground state, with corrections of or-

der L−2. Moreover, it follows that finite size effects for

an arbitrary measure of entanglement are ruled by con-

formal invariance.

In order to evaluate entanglement at a point λ = λc,

we should be able to perform a derivative of the energy

with respect to λ taken at λc. Therefore, note that we

will have that Eqs. (1) and (2) are the starting point to

determine finite size corrections to entanglement if and

only if the theory is critical in an interval around λc.

For the case of a single critical point (e.g., Ising spin-1/2

chain in a transverse field) instead of a critical region

(e.g, XXZ spin-1/2 chain in the anisotropy interval −1 <

∆ < 1), more general expressions for the energy should

be used, which take into account a mass spectrum (see,

e.g., Ref. [34]).

V.ENTANGLEMENT IN THE FINITE SIZE

SPIN-1/2 XXZ CHAIN

As an illustration of the previous results, let us con-

sider the spin-1/2 XXZ chain, whose Hamiltonian is given

by

HXXZ= −J

2

L

?

i=1

?σx

iσx

i+1+ σy

iσy

i+1+ ∆σz

iσz

i+1

?, (29)

Page 5

5

where periodic boundary conditions (PBC) are assumed.

We will set the energy scale such that J = 1. Entan-

glement for spin pairs can be quantified by the negativ-

ity [21], which is defined by

N(ρij) = 2 max(0,−min

α(µij

α)), (30)

where µij

pose ρij,TA of the density operator ρij, defined as

?αβ|ρTA|γδ? = ?γβ|ρ|αδ?. For the XXZ model, U(1)

invariance (?H,?

reads

where

?1 + 2Gz+ Gij

bij=

4

1

4

1

4

α

are the eigenvalues of the partial trans-

iσi

z

?

= 0) and translation invariance

ensure that the reduced density matrix for spins i and j

ρij=

aij

0

0

0

000

0

0

bij

zij∗bij

0

zij

0dij

,(31)

aij=

1

4

1

zz

?,

?1 − Gij

?1 − 2Gz+ Gij

??Gij

z? is the magnetization density (computed

for any site i) and Gij

the expectation value taken over an arbitrary quantum

state of the system. Moreover, invariance of HXXZunder

the discrete transformations σx → −σx, σy → σy, and

σz→ −σzimplies that Gij

the element zijin Eq. (32) is real, namely, zij= zij∗=

1/4?Gij

?

zz

?,

dij=

zz

?,

zij=

xx+ Gij

yy

?+ i?Gij

xy− Gij

yx

??, (32)

where Gz= ?σi

αβ= ?σi

ασj

β? (α,β = x,y,z), with

xy= 0 and Gij

yx= 0. Therefore,

xx+ Gij

yy

?. Then, evaluation of the negativity for

?

spins i and j from Eq. (31) yields

N(L) =1

2max

0,

4G2

z+

???Gij

xx+ Gij

yy

???

2

− Gij

zz− 1

?

(33)

.

From now on, we will be interested in computing the neg-

ativity for nearest neighbor spins. The generalized HK

theorem discussed in Section III implies that we can con-

sider ∆ as the external potential and ?σz

site i) as the relevant density. Thus, N(L) can be written

as a function of ∂ε/∂∆ for the ground state as well as for

any minimum energy state in a sector of magnetization

m (m = 0,±2,...,±L) and momentum P = (2π/L)p

(p = 0,1,...,L−1). In this direction, it is convenient to

write the correlation functions Gi,i+1

which results into

iσz

i+1? (for any

αβ

in terms of ∂ε/∂∆,

Gi,i+1

zz

= −2∂ε

∂∆,

?

Gi,,i+1

xx

+ Gi,i+1

yy

= −2ε − ∆∂ε

∂∆

?

. (34)

A. Ground state entanglement

For the ground state, we have that Gz= 0. Then, by

using Eq. (34) and Eq. (33), negativity reads

N(L) = −ε(L) + (∆ + 1)∂ε(L)

where we have used that |Gi,i+1

Gi,i+1

yy

≥ 0 (Marshall-Peierlsrule). Note that, in Eq. (35),

the energy density ε(L) can be seen as a function of

∂ε/∂∆ by the HK theorem, which is explicitly shown

in Fig. 1. Indeed, this implies that the negativity can be

taken as a function of ∂ε/∂∆, which illustrates the dual-

ity between potential and density established in Eq. (25)

for entanglement measures. The XXZ model is critical

∂∆

−1

2, (35)

zz

| ≤ 1 and Gi,i+1

xx

+

0 0.10.2 0.30.4

dε/d∆

-1

-0.9

-0.8

-0.7

-0.6

-0.5

ε

L = 4

L = 6

L = 8

L → ∞

-1-0.50

∆

0.51

-1

-0.9

-0.8

-0.7

-0.6

-0.5

ε

FIG. 1: (color online) Density energy ε as a function of ∂ε/∂∆

as given by the solution of the Bethe equations for finite size

chains as well as in the thermodynamic limit (values plotted

in the range −1 < ∆ < 1). For finite chains with lattice sizes

L > 8, the curves get nearly superposed with the curve for

the infinite chain. Inset: Density energy ε versus anisotropy

parameter ∆. Note that ε can be taken either as a function

of ∆ (the potential) or ∂ε/∂∆ (the density).

in the interval −1 ≤ ∆ < 1, with central charge c = 1.

Then, from Eq. (35), we can determine an approximate

analytical expression for the negativity in terms of en-

ergy as given by Eq. (1). The parameter ξ appearing

in Eq. (1) can be obtained analytically [35] for the XXZ

chain, reading

ξ = πsinγ

γ

, (36)

where γ is defined by

∆ = −cosγ ,γ ∈ [0,π)(37)

Then, substitution of Eq. (1) into Eq. (35) yields

NCFT(L) = N∞+

π2c

6γL2

?

sinγ +1 + ∆

γ

+∆√1 + ∆

√1 − ∆

?

+ o(L−2),(38)

Page 6

6

where N∞can be computed from Eq. (35), with ε∞and

∂ε∞/∂∆ directly given by the solution of the model at

the thermodynamic limit [36]. An exact value for the

negativity N(L) can be obtained from Eq. (35) by com-

puting ε(L) and ∂ε(L)/∂∆ via Bethe ansatz equations for

each length L. Naturally, this amounts to a much harder

computational effort for a general ∆, while Eq. (38) di-

rectly provides the negativity for a finite chain up to or-

der L−2with no need of solving the Bethe ansatz equa-

tions for each length L. A comparison between N(L)

and NCFT(L) for γ = π/2 and γ = π/3 is exhibited in

Tables I and II.

L

N(L)

NCFT(L)

4

8

16

32

64

128

256

512

1024 0.339263774123 0.339263774121

0.457106781187 0.446378653269

0.366669830087 0.366041268056

0.345995599194 0.345956921753

0.340938243195 0.340935835178

0.339680713890 0.339680563534

0.339366755018 0.339366745623

0.339288291732 0.339288291145

0.339268677562 0.339268677525

TABLE I: Comparison between N(L) and NCFT(L) for γ =

π/2 (the XX model).For an infinite chain, we have that

negativity is given by N(∞) = 0.339262139652.

L

N(L)

NCFT(L)

4

8

16

32

64

128

256

512

1024 0.375001580150 0.375001580143

0.489830037812 0.478556230132

0.401639244141 0.400889057533

0.381525197365 0.381472264383

0.376621871264 0.376618066096

0.375404791436 0.375404516524

0.375101148980 0.375101129131

0.375025283711 0.375025282283

0.375006320673 0.375006320571

TABLE II: Comparison between N(L) and NCFT(L) for γ =

π/3. For an infinite chain, we have that negativity is given

by N(∞) = 3/8 = 0.375.

B.Twisted boundary conditions

We can also use the results obtained for PBC to in-

vestigate the finite size corrections to the negativity with

more general boundary conditions. We will consider here

the so-called twisted boundary conditions (TBC), which

can be achieved as the effect of a magnetic flux through

a spin ring [37]. Remarkably, it has recently been shown

that TBC may improve multi-party quantum communi-

cation via spin chains [38]. In order to consider TBC,

it is convenient to rewrite the Hamiltonian in Eq. (29)

(with J = 1) in the following form

HXXZ= −1

2

L

?

j± iσy

i=1

?2?σ+

iσ−

i+1+ σ−

iσ+

i+1

?+ ∆σz

iσz

i+1

?,

(39)

where σ±

Φ < 2π), with Φ denoting a phase. The quantum chain

given by Eq. (39) is solvable by the Bethe ansatz [28]. In

presence of TBC, Eq. (1) still holds, but with an effective

central ˆ c(Φ) [28], which is given by

j

= (σx

j)/2 and σ±

L+1= e±iΦσ±

1 (0 ≤

ˆ c(Φ) = 1 −

3Φ2

2π(π − γ), (40)

with γ defined as in Eq. (37). Let us take the following

canonical transformations [39]

? σ±

j= e∓iΦj/Lσ±

j

, ? σz

j= σz

j (j = 1,...,L).(41)

In terms of this new set of operators, the original chain

with TBC is now given by the periodic chain

HXXZ = −1

2

L

?

j? σz

j±i? σy

L

?

?(? σx

j=1

?

e−iΦ/L? σ+

j+1

j? σ−

j+1+ eiΦ/L? σ−

j? σ+

j+1

+ ∆? σz

j= (? σx

?, (42)

where ? σ±

in the form

L+1= ? σ±

1. Defining the operators ? σx

?

L

jand ? σy

j

through ? σ±

HXXZ= −1

j)/2, the Hamiltonian can be put

2

j=1

cos

?Φ

?(? σx

j? σx

j? σx

j+1)

j+1+ ? σy

+ ∆? σz

j? σy

j+1)

2

− sin

?Φ

L

j? σy

j+1− ? σy

2

j? σz

j+1

?

. (43)

Note that the Hamiltonian in Eq. (43) is both U(1)

invariant ([H,?

Therefore, the two-spin reduced density matrix keeps the

form given in Eq. (31), with the correlation functions

Gij

nearest neighbor spins governed by Hamiltonian (43) can

be computed similarly as before. By using that Gz= 0

(ground state) and |Gi,i+1

N = 2max(0,|z| − a),

where a = (1+?Gzz)/4 and z = (?G?+i?G⊥)/4, with?Gzz=

(∀i).

derivatives of the energy density, it is convenient to define

HXXZ= HXXZ/cos(Φ/L). Then

??G?− η?G⊥+ ∆?Gzz

j? σz

j] = 0) and translationally invari-

ant (HXXZ exhibts PBC in terms of the set {? σ±

αβreplaced by?Gij

j,? σz

j}).

αβ= ?? σi

zz

α? σj

| ≤ 1 we obtain

β?. Then, the negativity for

(44)

?Gi,i+1

zz

,?G?=?Gi,i+1

xx

+?Gi,i+1

yy

, and?G⊥=?Gi,i+1

xy

−?Gi,i+1

yx

In order to write entanglement in terms of the

ε = −1

2

?

,(45)

Page 7

7

where ε = ?HXXZ?/L, η = tan(Φ/L), and ∆ =

∆/cos(Φ/L). From Eq. (45) we get

?Gzz = −2∂ε

∂∆,

?G⊥ = 2∂ε

?G? = 2

∂η,

?

−ε + η∂ε

∂η+ ∆∂ε

∂∆

?

. (46)

Therefore, the contribution (|z|−a) for expression for the

negativity in Eq. (44) reads

?

In order to obtain the results in terms of Φ and ∆, we

make use of the expressions

?Φ

∂ε

∂∆

|z| − a =

1

2

??

−ε + η∂ε

∂η+ ∆∂ε

∂∆

?2

+

?∂ε

∂η

?2

+

∂ε

∂∆−1

2

. (47)

∂ε

∂η

= cos

L

??

L∂ε

∂Φ+ tan

?Φ

L

?

ε

?

, (48)

=

∂ε

∂∆. (49)

Hence, finite size corrections to entanglement can be

found now by using Eq. (1) [replacing the central charge

c by the effective central charge ˆ c(Φ) as in Eq. (40)] into

Eq. (47). Examples comparing the negativity NCFT(L)

for nearest neighbors up to o(L−2) and the exact value of

the negativity N(L) (obtained through the numerical so-

lution of the Bethe ansatz equations derived in Ref. [28])

are exhibited in Tables III and IV below.

L

N(L)

NCFT(L)

4

8

16

32

64

128

256

512

1024 0.339263025109 0.339263774121

0.406774810601 0.446378653269

0.354315234931 0.366041268056

0.342922395530 0.345956921753

0.340170924101 0.340935835178

0.339488945731 0.339680563534

0.339318816833 0.339366745623

0.339276307427 0.339288291145

0.339265681501 0.339268677525

TABLE III: Comparison between the exact evaluation of

N(L) and the approximate expression NCFT(L) (up to or-

der L−2)) for γ = π/2 and Φ = π/2.

Note from Tables I and III that, for γ = π/2, NCFT(L)

gives the same result either for Φ = 0 (PBC) or Φ = π/2,

which is an indication that TBC should not affect the

negativity (up to o(L−2)) in the case of the XX model.

Indeed, this can be analytically proved. In this case, the

L

N(L)

NCFT(L)

4

8

16

32

64

128

256

512

1024 0.375000409414 0.375001178447

0.400000000000 0.452230707893

0.381121448251 0.394307676973

0.376577662094 0.379826919243

0.375405200439 0.376206729811

0.375102994373 0.375301682453

0.375025983614 0.375075420613

0.375006526789 0.375018855153

0.375001635654 0.375004713788

TABLE IV: Comparison between the exact evaluation of

N(L) and the approximate expression NCFT(L) (up to or-

der L−2)) for for γ = π/3 and Φ = 2π/3.

anisotropy is ∆ = 0, which implies that γ = π/2 and

ξ = 2. Then, from Eqs. (1) and (40), we have

ε(L) = ε∞−π ˆ c(Φ)

∂ε

∂Φπ

∂ε

∂∆∂∆

3

L−2+ o(L−2),

=

2

Φ

L2+ o(L−2),

∂ε∞

=

????

∆=0

+

2

3L2+ o(L−2).(50)

By inserting the above equations into Eqs. (48) and (49),

it can be shown that the negativity as given by Eq. (44)

gets

????

where |ε(Φ=0)| =

implies that the negativity for the XX model with TBC

is not affected by the phase Φ up to order 1/L2.

NCFT =??ε(Φ=0)

??+∂ε∞

?ε2

∂∆

∆=0

+

2

3L2−1

2+o(L−2), (51)

∞− 2ε∞πξ/(6L2). Hence, Eq. (51)

C.Excited states

Let us consider now the structure of the negativity

for the excited states in the XXZ model with PBC. The

U(1) and translation symmetries allow us the decompo-

sition of the associated eigenspace of HXXZ into dis-

joint sectors (fixed magnetization and momentum) la-

belled by the quantum numbers r = 0,1,2,...,L and

p = 1,2,...,L − 1, which give the number of spins up

in the σz basis and the eigenvalue of the momentum

P = (2π/L)p, respectively. An exact evaluation of the

negativity for nearest neighbor spins can be performed

from Eq. (33) by taking a non-vanishing magnetization

density Gz and by using Eq. (34), where the energy of

the excited state is obtained through the solution of the

Bethe ansatz equations.This is illustrated in Fig. 2,

where we plot the negativity between nearest neighbors

in a chain of length L = 256 sites for minimum energy

states with zero momentum in several magnetization sec-

tors. These states have anomalous dimensions xngiven

Page 8

8

-1-0.50

∆

0.51

0

0.1

0.2

0.3

0.4

N

n = 0

n = 4

n = 8

-0.4 -0.20 0.2 0.4

0.28

0.3

0.32

0.34

0.36

0.38

FIG. 2: (color online) Negativity for minimum energy states

as a function of the anisotropy ∆ for L = 256 sites. Note

that the curves are nearly the same, indicating a unique en-

tanglement pattern in the critical region. Inset: Negativity

as a function of ∆ in a larger zoom scale.

by [28]

xn= n2(π − γ)

2π

, (52)

where n = L/2 − r and j + j′= p = 0 in Eq. (2). Re-

markably, note that the negativities for the minimum

energy states plotted are nearly the same, indicating a

unique entanglement pattern in the critical region. In-

deed, this is a more general result, which holds also for

other excited states. For instance, let us take the so-

called marginal state [28], which is a state that will be

taken in the sector n = 0 with anomalous dimension

x = 2 (independently of γ) and j,j′= 0. Exact compu-

tation in Table V below shows that its negativity is also

close to the values found in Fig. 2. Indeed, we can show

∆ Marginal State Ground State

0.505023772863

0.205023772863

0.005023772863

-0.204976227137 0.357090720201 0.357706303640

-0.504976227137 0.374867541783 0.375473489099

0.265447369819 0.266151418398

0.315358910123 0.316005319520

0.338660739066 0.339288291732

TABLE V: Comparison between N(∆) for the marginal state

and and N(∆) for the ground state in a chain with L = 256

sites.

that entanglement in the critical region of the XXZ chain

will exhibit a unique pattern for all states accessible via

the CFT associated with the model. As discussed in Sec-

tion II, each primary operator of the theory corresponds

to a tower of states with energies given by Eq. (2). All

these states in the towers will have energies which differ

at order L−2[see Eqs. (1) and (2)]. According to Eq. (33),

such a difference is also reflected in the negativity of near-

est neighbor spins, which explains the behavior displayed

in both Fig. 2 and Table V. This can explicitly be shown

by inserting Eq. (2) into Eq. (33). As an illustration, we

take the minimum energy states with zero momentum in

a given magnetization sector labelled by n. For this case,

the negativity can be evaluated as

Nn(L) = N∞

n+|G∞|−1

6γL2

?wn

?

3γn2+ π sinγ G∞zn

??

+π (∆G∞+ |G∞|)

γ

− zncotγ+ o(L−2),

where

wn= π(c − 6n2), N∞

zn=?πc − 6n2(π − γ)?, G∞= −ε∞

with ε∞

ndenoting the energy density of the excited state

as L → ∞. Note that this unique pattern of entangle-

ment, which has been explicitly derived here, is in agree-

ment with the general discussion of Sec. IV. This is in-

deed exhibited in Eq. (28). Naturally, similar expressions

can be obtained for excited states higher than the mini-

mum energy states.

n = |G∞| + ∂ε∞

n/∂∆ − 1/2,

n+ ∆∂ε∞

n/∂∆.

VI.CONCLUSION

In conclusion, we have investigated the computation of

finite size corrections to entanglement in quantum criti-

cal systems. These corrections were shown to depend on

the central charge of the model as well as the anomalous

dimensions of the primary operators of the theory. Our

approach has naturally arisen as a general consequence

of the application of CFT and DFT methods in critical

theories. This framework has been illustrated in the XXZ

model, where we have shown that: (i) entanglement in

spin chains with arbitrary finite sizes can be analytically

computed up to order o(L−2) with no need of solving the

Bethe ansatz equations for each length L; (ii) Conformal

towers of excited states displays a unique pattern of en-

tanglement in the critical region. Indeed, we have been

able to provide a general argument according to which

this unique pattern of entanglement should appear in

all conformally invariant models. Further examples in

higher dimensional lattices and higher spin systems are

left for future investigation.

Acknowledgments

This work was supported by the Brazilian agencies

MCT/CNPq (F.C.A, M.S.S.), FAPESP (F.C.A.), and

FAPERJ (M.S.S.).

Page 9

9

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