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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
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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)
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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
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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)
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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 0.10.20.3 0.4
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)
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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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