Entanglement in a simple quantum phase transition
Tobias J. Osborne
1,2,
*
and Michael A. Nielsen
2,†
1
Department of Mathematics, University of Queensland 4072, Brisbane, Queensland, Australia
2
Centre for Quantum Computer Technology and Department of Physics, University of Queensland 4072, Brisbane,
Queensland, Australia
共Received 27 February 2002; published 23 September 2002兲
What entanglement is present in naturally occurring physical systems at thermal equilibrium? Most such
systems are intractable and it is desirable to study simple but realistic systems that can be solved. An example
of such a system is the one-dimensional infinite-lattice anisotropic XY model. This model is exactly solvable
using the Jordan-Wigner transform, and it is possible to calculate the two-site reduced density matrix for all
pairs of sites. Using the two-site density matrix, the entanglement of formation between any two sites is
calculated for all parameter values and temperatures. We also study the entanglement in the transverse Ising
model, a special case of the XY model, which exhibits a quantum phase transition. It is found that the
next-nearest-neighbor entanglement 共though not the nearest-neighbor entanglement兲 is a maximum at the
critical point. Furthermore, we show that the critical point in the transverse Ising model corresponds to a
transition in the behavior of the entanglement between a single site and the remainder of the lattice.
DOI: 10.1103/PhysRevA.66.032110 PACS number共s兲: 03.65.Ud, 73.43.Nq, 05.50.⫹q
I. INTRODUCTION
It seems to be a truism in quantum physics that strongly
entangled systems exhibit complicated behavior which is dif-
ficult to quantify. Two practical examples of this ‘‘principle’’
are the conventional superconductor 关1,2兴 and the fractional
quantum Hall effect 共FQHE兲关3兴. In both cases, for certain
parameter regimes, the system enters a very interesting en-
tangled state 共the BCS ground state for the superconductor
关4,5兴, and the Laughlin ground state for the FQHE 关6兴兲. For
many years these systems resisted attempts to understand
them using reasoning based on classical methods 关7兴.Itre-
quired a major breakthrough, the construction of an insight-
ful ground-state ansatz, to elucidate the physics of both the
FQHE and the superconductor. The key feature of both sys-
tems, which makes it hard to explain them classically, ap-
pears to be that their ground states are strongly entangled.
Entanglement is a uniquely quantum property of any non-
local superposition state of two or more quantum systems
关12–14兴. Such states are typified by the Bell state
兩
⌿
⫺
典
⫽ (1/
冑
2)(
兩
01
典
⫺
兩
10
典
). The many curious features of en-
tangled states have motivated considerable research. A re-
markable consequence of this work is the emerging under-
standing of entanglement as a resource 关12,15兴, like energy,
which can be used to accomplish interesting physical tasks.
The similarities between entanglement and energy appear
to be more than just superficial. It turns out to be possible to
quantify the entanglement present in a given quantum state.
This allows the development of quantitative high-level prin-
ciples governing the behavior of entangled states, indepen-
dent of their particular physical representation. These prin-
ciples can be seen as analogous to the laws of
thermodynamics governing the behavior of energy, indepen-
dent of the specific form in which it is given to us. We hope
that the quantitative theory of entanglement may provide a
powerful unifying framework for the understanding of com-
plex quantum systems. This is because, when viewed in
terms of their entanglement content, a large number of ap-
parently different states turn out to be equivalent.
This paper is one step in testing the hypothesis 关16–19兴
that the study of complex quantum systems may be simpli-
fied by first analyzing the static and dynamic entanglement
present in those systems. We will attempt to perform such an
analysis in a representative system chosen from condensed-
matter physics, specifically, the XY model 关20兴. The signa-
ture of complexity in this system is the occurrence of a quan-
tum phase transition.
Quantum phase transitions 共QPTs兲 are a qualitative
change in the ground state of a quantum many-body system
as some parameter is varied 关21,22兴. Unlike ordinary phase
transitions, which occur at a nonzero temperature, the fluc-
tuations in a QPT are fully quantum. Typically, at the critical
point in parameter space where a QPT takes place, long-
range correlations in the ground state also develop. The ex-
istence of a QPT in a quantum many-body system strongly
influences the behavior of the system near the critical point,
with the development of long-range correlations and a non-
zero expectation value for an order parameter 关21兴.
In Ref. 关16兴 it was argued that QPTs are genuinely quan-
tum mechanical in the sense that the property responsible for
the long-range correlations is entanglement. It was also ar-
gued that the system state is strongly entangled at the critical
point. It would be desirable, to begin with, to show that
systems near quantum critical points can be simply charac-
terized in terms of their entanglement content. Unfortunately,
such a proof seems very difficult. We need first to understand
the entanglement in such systems before proposing a classi-
fication scheme based on entanglement content. At the mo-
ment, the most promising technique to study entanglement in
critical quantum systems appears to be the renormalization
group, which is the standard way to obtain information about
systems at and near criticality.
*
Email address: osborne@physics.uq.edu.au
†
Email address: nielsen@physics.uq.edu.au
PHYSICAL REVIEW A, 66, 032110 共2002兲
1050-2947/2002/66共3兲/032110共14兲/$20.00 ©2002 The American Physical Society66 032110-1
The renormalization group 共RG兲 is based on the notion
that physics at small length scales 共and hence higher energy
scales兲 should not affect physics at much larger length scales.
The RG is, in fact, a family of methods which can be applied
to learn nonperturbative information about strongly interact-
ing systems. The development of the renormalization group
共see, for example, Refs. 关23,24兴 for a review兲 has shown that
phase transitions are universal in the sense that many prop-
erties of the system do not depend on the detailed dynamics
of the system under consideration. Instead, using RG tech-
niques, it has been shown that phase transitions depend only
on certain global properties, such as symmetry and dimen-
sion. We would like to apply the ideas of the RG to calculate
entanglement quantities in systems exhibiting a quantum
phase transition. To see if this is possible, it is desirable to
first carry out exact calculations in order to determine if
similar universality properties govern the entanglement
present in such systems. The purpose of this paper is there-
fore to do such calculations for the XY model.
Unfortunately the modern theory of entanglement 共see,
for example, Refs. 关25–28兴兲 is only partially developed, and
at the present time can only be applied in a limited number
of scenarios. In these limited scenarios, well-developed ana-
lytic tools exist to quantify the structure of entanglement
present in a system. Two important scenarios are 共a兲 the case
of a pure state of a bipartite system, that is, a system con-
sisting of only two components; and 共b兲 a mixed state of two
spin-
1
2
particles.
For this reason, we focus our investigation on two types
of calculation for the XY model. The first calculation is of
the entanglement between a single site in the lattice and the
rest of the system, for the ground state of the model. The
second calculation is of the entanglement between two sites
of the lattice at arbitrary temperatures and separations, allow-
ing us to determine whether there are truly quantum features
present in the two-body correlations in the system. Thus,
although we do not obtain an understanding of the three-
party and multiparty entanglement present in the system, we
do calculate significant partial information characterizing the
entanglement.
The entanglement present in condensed-matter systems
has been investigated previously by a number of authors
关17,19,29–37兴. It was considered by Nielsen 关17兴 who stud-
ied the Heisenberg model on two sites analytically. An ex-
pression for the ground-state entanglement in the infinite
one-dimensional 共1D兲 Heisenberg chain was obtained soon
after by Wootters 关30兴. Numerical calculations of entangle-
ment in the Heisenberg model on a small number of sites
were carried out by Arnesen et al. 关31兴. Arnesen et al. iden-
tified parameter regions where there is appreciable thermal
entanglement, which is entanglement present at nonzero tem-
peratures. Recent studies include the numerical calculation
of entanglement in the transverse Ising model on small num-
bers of sites 关33兴, and analytic computations of entanglement
in the XY model on two sites 关29兴 and three sites 关34兴. Ad-
ditional studies have been carried out on itinerant fermion
systems 关14兴 and other small condensed-matter systems re-
lated to the XY model 关29,35–37兴.
The structure of this paper is as follows. In Sec. II the
exact solution and calculation of the correlation functions for
the XY model is outlined using the Jordan-Wigner transform.
The thermal ground-state properties of this system are con-
sidered in Sec. III, focusing on the special case of the trans-
verse Ising model, and the role entanglement plays in the
quantum phase transition in this model. Thermal entangle-
ment in the transverse Ising model is then calculated in Sec.
IV. We conclude in Sec. V, and sketch some possible future
research directions.
II. EXACT SOLUTION OF THE XY MODEL
In this section we consider the exact solution of the XY
model on N sites, which is facilitated by use of the Jordan-
Wigner transform 关38兴. The observables that are important
for the calculation of the entanglement are evaluated in the
large-N or thermodynamic limit. The two fundamental ob-
jects constructed in this study are the one- and two-site den-
sity matrices. From knowledge of these matrices it is pos-
sible to calculate the one- and two-party entanglement
occurring in the XY model. The solution of the XY model is
well known, and the procedure outlined in this section to
solve it follows the standard method 关15,16,39,40兴. The main
result in this section is the explicit construction of the one-
and two-party density matrices for the XY model at thermal
equilibrium.
The Hamiltonian for the anisotropic XY model on a 1D
lattice with N sites in a transverse field is given by 关41兴
H⫽⫺
兺
j⫽ 0
N⫺ 1
冉
2
关共
1⫹
␥
兲
j
x
j⫹ 1
x
⫹
共
1⫺
␥
兲
j
y
j⫹ 1
y
兴
⫹
j
z
冊
,
共1兲
where
j
a
is the ath Pauli matrix (a⫽ x, y,orz) at site j,
␥
is the degree of anisotropy, and is the inverse strength of
the external field. We assume cyclic boundary conditions, so
that the Nth site is identified with the 0th site. The standard
procedure used to solve Eq. 共1兲 is to transform the spin op-
erators
j
a
into fermionic operators via the Jordan-Wigner
transform
c
i
⬅
兿
j⫽ 0
i⫺ 1
关
⫺
j
z
兴
i
⫺
, 共2兲
c
i
†
⫽
兿
j⫽ 0
i⫺ 1
关
⫺
j
z
兴
i
⫹
, 共3兲
where
i
⫹
⬅
1
2
共
i
x
⫹ i
i
y
兲
,
i
⫺
⬅
1
2
共
i
x
⫺ i
i
y
兲
. 共4兲
It is easy to verify that c
i
satisfy the fermionic anticommu-
tation relations
兵
c
i
,c
j
†
其
⫽
␦
ij
,
兵
c
i
,c
j
其
⫽ 0. 共5兲
In terms of the fermionic operators, Eqs. 共2兲 and 共3兲, the
Hamiltonian Eq. 共1兲 assumes the quadratic form
TOBIAS J. OSBORNE AND MICHAEL A. NIELSEN PHYSICAL REVIEW A 66, 032110 共2002兲
032110-2
H⫽
冉
兺
i,j⫽ 0
N⫺ 1
c
i
†
A
i,j
c
j
⫹
1
2
兺
i,j⫽ 0
N⫺ 1
共
c
i
†
B
i,j
c
j
†
⫹ H.c.
兲
冊
⫹ N, 共6兲
where A
i,i
⫽⫺1, A
i,i⫹ 1
⫽⫺
1
2
␥
⫽ A
i⫹ 1,i
, B
i,i⫹ 1
⫽⫺
1
2
␥
,
B
i⫹ 1,i
⫽
1
2
␥
, and all the other A
i,j
and B
i,j
are zero. The
quadratic Hamiltonian Eq. 共6兲 may be diagonalized by mak-
ing a linear transformation of the fermionic operators,
q
⫽
兺
i⫽ 0
N⫺ 1
共
g
qi
c
i
⫹ h
qi
c
i
†
兲
, 共7兲
q
†
⫽
兺
i⫽ 0
N⫺ 1
共
g
qi
c
i
†
⫹ h
qi
c
i
兲
, 共8兲
where q⫽⫺N/2,⫺ N/2⫹ 1,...,N/2⫺ 1 and the g
qi
and h
qi
can be chosen to be real. By requiring that the operators
q
obey fermionic anticommutation relations, and that the
Hamiltonian Eq. 共1兲 be manifestly diagonal when expressed
in terms of the fermionic modes
q
, the following two
coupled matrix equations must hold:
共
A⫺ B
兲
⌽
q
⫽
q
⌿
q
, 共9兲
共
A⫹ B
兲
⌿
q
⫽
q
⌽
q
, 共10兲
where the components of the two column vectors ⌽
q
and ⌿
q
are given by
关
⌽
q
兴
i
⫽ g
qi
⫹ h
qi
, 共11兲
关
⌿
q
兴
i
⫽ g
qi
⫺ h
qi
. 共12兲
The quadratic Hamiltonian Eq. 共6兲, when expressed in terms
of the operators
q
, takes the diagonal form
H⫽ 2
兺
q
q
q
†
q
⫺
兺
q
q
, 共13兲
where
q
⫽
冑
共
␥
sin
q
兲
2
⫹
共
1⫹ cos
q
兲
2
, 共14兲
and
q
⫽ 2
q/N.
Now that the XY Hamiltonian has been diagonalized we
can calculate the one- and two-site density matrices. Much of
the remainder of this paper is concerned with the case where
the system is at thermal equilibrium at temperature T. The
density matrix for the XY model at thermal equilibrium is
given by the canonical ensemble
⫽ e
⫺

H
/Z, where

⬅1/k
B
T, and Z⫽ tr(e
⫺

H
) is the partition function. The
thermal density matrix is diagonal when expressed in terms
of the Jordan-Wigner fermionic operators
q
. Our interest
lies in calculating the quantum correlations present in the
system as a function of the parameters

,
␥
, . In general,
this problem requires knowledge of all the possible spin-
correlation functions. These correlators are typically very
difficult to calculate from
as it is diagonal in terms of the
q
’s, which are complicated nonlocal functions of the origi-
nal spin operators. Fortunately, the only correlation functions
which we require are the one- and two-point correlation
functions. The evaluation of these functions has been carried
out previously 关40,42兴.
The one- and two-site density matrices may be con-
structed from the one- and two-point correlation functions,
using the operator expansion for the density matrix of a
system of N spin-
1
2
particles in terms of tensor products of
Pauli matrices. For the single-site density matrix
1
for the
first spin—equal, by translational symmetry, to the state
i
of
a single spin at an arbitrary site—the operator expansion
reads
1
⫽ tr
i
ˆ
共
兲
⫽
兺
␣
⫽ 0
3
q
␣
i
␣
2
, 共15兲
where tr
i
ˆ
is the partial trace over all degrees of freedom
except the spin at site i,
i
␣
are the Pauli matrices acting on
the site i with the convention
i
0
⫽ I
i
, and the coefficients q
␣
are real. The coefficients q
␣
are determined by the relation
q
␣
⫽ tr
共
␣
i
兲
⫽
具
␣
i
典
. 共16兲
To completely specify the single-site density matrix re-
quires knowledge of three expectation values (q
0
⫽ 1 be-
cause
1
must have trace unity兲. However, because the
Hamiltonian for the XY model Eq. 共1兲 possesses symmetries
it is possible to reduce this number to one. First of all, the
Hamiltonian is real, so that
1
*
⫽
1
. As the matrix
y
is
imaginary this means that q
2
must be zero. The second sym-
metry that the XY Hamiltonian possesses is the global phase-
flip symmetry
U
PF
⫽
兿
j⫽ 0
N⫺ 1
j
z
. 共17兲
This symmetry implies that
关
z
,
1
兴
⫽ 0, so forcing q
3
to be
zero. The single-site density matrix
1
is therefore deter-
mined solely by q
1
.
For the two-site density matrix, which is the joint state of
two spins at sites i and j, the operator expansion takes the
form
ij
⫽ tr
ij
ˆ
共
兲
⫽
兺
␣
,

⫽ 0
3
p
␣
i
␣
丢
j

4
. 共18兲
The coefficients are determined by the relation
p
␣
⫽ tr
共
i
␣
j

ij
兲
⫽
具
i
␣
j

典
, 共19兲
so that if the relevant correlation functions are known it is
possible to construct the two-site density matrix completely.
The operator expansion Eq. 共18兲 implies that we need
sixteen correlation functions to construct the two-site density
matrix. However, as in the case of the single-site density
matrix, this number can be reduced by appealing to the sym-
metries of the Hamiltonian. Translational invariance of the
ENTANGLEMENT IN A SIMPLE QUANTUM PHASE... PHYSICAL REVIEW A 66, 032110 共2002兲
032110-3
lattice means that the density matrix depends only on the
distance r⫽
兩
j⫺ i
兩
between the spins, that is,
ij
⫽
0r
. Re-
flection symmetry about any site also means that
ij
⫽
ji
.
Also, since the Hamiltonian is real,
ij
*
⫽
ij
. Finally, the
global phase-flip symmetry implies that
关
i
z
j
z
,
ij
兴
⫽ 0. The
symmetries of the XY model require that the only nonzero
coefficients in the operator expansion Eq. 共18兲 are p
00
, p
03
,
p
30
, p
11
, p
22
, and p
33
. Furthermore, p
00
⫽ 1 because the
density matrix must have trace unity, and p
03
⫽ p
30
.
In the thermodynamic limit, N→⬁, sums that appear in
the expectation values are replaced by integrals, and the cor-
relation functions for the XY model can be reduced to
quadratures 关20,40,42,43兴. The calculations are rather in-
volved, and we merely summarize the results here. In ther-
mal equilibrium, for arbitrary
␥
and , the transverse mag-
netization
具
z
典
is given by 关40兴
具
z
典
⫽⫺
1
冕
0
d
共
1⫹ cos
兲
tanh
冉
1
2

冊
, 共20兲
where we abuse notation and write
⬅
q
to indicate the
replacement of
q
with the continuous variable
which
results from the thermodynamic limit
q
→
.
The two-point correlation functions are given by 关42兴
具
0
x
r
x
典
⫽
冏
G
⫺ 1
G
⫺ 2
••• G
⫺ r
G
0
G
⫺ 1
••• G
⫺ r⫹ 1
⯗⯗ ⯗
G
r⫺ 2
G
r⫺ 3
••• G
⫺ 1
冏
, 共21兲
具
0
y
r
y
典
⫽
冏
G
1
G
0
••• G
⫺ r⫹ 2
G
2
G
1
••• G
⫺ r⫹ 3
⯗⯗ ⯗
G
r
G
r⫺ 1
••• G
1
冏
, 共22兲
具
0
z
r
z
典
⫽ 4
具
z
典
2
⫺ G
r
G
⫺ r
, 共23兲
where
G
r
⫽
1
冕
0
d
cos
共
r
兲
共
1⫹ cos
兲
tanh
冉
1
2

冊
⫺
␥
冕
0
d
sin
共
r
兲
sin
共
兲
tanh
冉
1
2

冊
. 共24兲
Summarizing, in the thermodynamic limit we may write
the single-site density matrix
1
entirely in terms of the
transverse magnetization, Eq. 共20兲,
1
⫽
I⫹
具
z
典
z
2
. 共25兲
Similarly, the two-site density matrix
0r
can be written en-
tirely in terms of the correlation functions Eq. 共21兲, Eq. 共22兲,
Eq. 共23兲, and the transverse magnetization,
0r
⫽
I
0r
⫹
具
z
典
共
0
z
⫹
r
z
兲
⫹
兺
k⫽ 1
3
具
0
k
r
k
典
0
k
r
k
4
. 共26兲
III. GROUND-STATE ENTANGLEMENT FOR THE
TRANSVERSE ISING AND XY MODELS
In this section we discuss the quantum correlations occur-
ring in the ground state of lattice systems undergoing a quan-
tum phase transition. We argue that the critical point corre-
sponds to the situation where the lattice is critically
entangled, where, somewhat loosely, we define critically en-
tangled to mean that entanglement is present on all length
scales. In Sec. III A we outline the properties of the ground
state of the transverse Ising model, which is a simple sub-
class of the anisotropic XY model. In Sec. III B the contri-
bution to the ground-state correlations from one- and two-
party entanglement in the XY model is calculated explicitly
in order to illustrate the sharp peak in the entanglement at the
critical point. Finally, in Sec. III C we discuss how the prop-
erties of shared entanglement may be related to critical quan-
tum lattice systems.
In Ref. 关16兴 it was argued that the physical origin of the
correlations which occur in systems exhibiting a quantum
phase transition is quantum entanglement. We reproduce the
argument of Ref. 关16兴 here in order that this study be self-
contained. For concreteness, we restrict our attention to a
lattice of spin-
1
2
particles.
Suppose the ground state of a quantum lattice system was
not entangled, that is, it is a product state. Then a simple
calculation shows that the spin-spin correlation function
具
i
␣
j

典
⫺
具
i
␣
典具
j

典
is identically zero. Thus, if the correla-
tion function is nonzero then the ground state must be en-
tangled. Furthermore, we conjecture that large values of the
correlation function imply a highly entangled ground state; it
is an interesting open problem to prove a precise form of this
conjecture.
For general quantum lattice systems the correlation func-
tion decays exponentially as a function of the separation
兩
i
⫺ j
兩
when the system is far from criticality 关21兴. When the
system is at a critical point, the correlations decay only as a
polynomial function of the separation. At this point a funda-
mental change in the ground state has occurred.
We believe that when a system approaches a critical point,
the structure of the entanglement in the ground state under-
goes a transition. Further, we conjecture that the nature of
this transition is governed by a change in the spatial extent of
the entanglement. The entanglement between a single spin
and the rest of the lattice away from the critical point must
be bounded in finite regions because the correlations are
damped exponentially. At the critical point correlations de-
velop on all length scales, and the physical property respon-
sible for these correlations, entanglement, should become
present at all length scales as well. We believe that a funda-
TOBIAS J. OSBORNE AND MICHAEL A. NIELSEN PHYSICAL REVIEW A 66, 032110 共2002兲
032110-4
mental transition in the nature of the entanglement in the
system occurs at this point; in some sense, at the critical
point the state is delocalized, compared to the local nature of
the entanglement away from the critical point. If this physi-
cal picture is correct, there should be evidence of entangle-
ment developing on all length scales in the one- and two-
party entanglement results.
As described in detail below, the ground state of the XY
model exhibits the features we have described in the previ-
ous paragraphs. That is, maximality of the entanglement at
criticality, and evidence that a transition in the entanglement
structure takes place at the critical point. Although much
work remains to be done to flesh out this physical picture, we
believe that further research will show that these are generic
properties of critical quantum systems.
A. Properties of the transverse Ising model ground state
The ground state of the XY model is very complicated
with many different regimes of behavior 关40,42兴. For the
sake of clarity, we focus most of our discussions on the trans-
verse Ising model, which arises as the zero-anisotropy limit
␥
→1 in Eq. 共1兲. The reason for this particular choice is
because the transverse Ising model is the simplest quantum
lattice system to exhibit a quantum phase transition 关21兴. The
central goal in this section is to illustrate the intimate rela-
tionship between the entanglement structure of the ground
state and the quantum phase transition. In particular, the cal-
culations for the transverse Ising model provide the clearest
evidence for the conjecture that the critical point corresponds
to the situation where the lattice is most entangled.
The Hamiltonian for the transverse Ising model may be
obtained from the XY model Hamiltonian, Eq. 共1兲, by setting
␥
⫽ 1,
H⫽⫺
兺
j⫽ 0
N⫺ 1
共
j
x
j⫹ 1
x
⫹
j
z
兲
. 共27兲
The structure of the transverse Ising model ground state
changes dramatically as the parameter is varied. The de-
pendence of the ground state on is quite complicated.
However, it is possible to investigate the ⫽ 0 and →⬁
limits exactly.
When approaches zero, the transverse Ising model
ground state becomes a product of spins pointing in the posi-
tive z direction,
兩
0
典
→0
⬇•••
兩
↑
典
j
兩
↑
典
j⫹ 1
•••. 共28兲
In the →⬁ limit the ground state again approaches a prod-
uct of spins pointing in the positive x direction,
兩
0
⫹
典
→⬁
⬇•••
兩
→
典
j
兩
→
典
j⫹ 1
•••. 共29兲
The →⬁ limit is fundamentally different from the ⫽ 0
case because the corresponding ground state is doubly de-
generate under the global phase flip, Eq. 共17兲, where
兩
0
⫺
典
→⬁
⬅U
PF
兩
0
⫹
典
→⬁
⬇•••
兩
←
典
j
兩
←
典
j⫹ 1
••• 共30兲
is a second ground state. The ⫽ 0 ground state is invariant
under the global phase flip. We note that in both limits the
ground state approaches a product state.
Using the solutions obtained for the limiting cases of
we can qualitatively describe the ground state as is varied.
When is small, the exchange term
j
x
j⫹ 1
x
may be re-
garded as a perturbation, and perturbation theory may be
used. In this case the ground state becomes a superposition
of the unperturbed ground state and low-lying excitations in
such a way that the small- ground state remains invariant
under the global phase flip.
When is much greater than one, 1/ is a small param-
eter and perturbation theory may again be used to show that
the now-degenerate ground states are a superposition of the
unperturbed ground states
兩
0
⫹ ,⫺
典
and low-lying excitations.
The degeneracy of the ground state under the global phase
flip remains for large. 共This degeneracy, along with the
invariance of the ground state
兩
0
典
under U
PF
may be estab-
lished nonperturbatively 关21兴.兲
When ⫽ 1 a fundamental transition in the form of the
ground state occurs. The symmetry under the global phase
flip breaks at this point and the system develops a nonzero
magnetization
具
x
典
⫽” 0 which grows as is increased. The
magnetization is the order parameter which identifies the
existence of a new phase.
Now that we have outlined the structure of the ground
state for the transverse Ising model as a function of ,we
have a basic physical picture with which to interpret the ex-
act results.
The calculation of the entanglement between a single site
and the rest of the lattice requires construction of the single-
site density matrix for the ground state. While the single-site
density matrix for the thermal state was constructed in Sec.
II, there is a distinction between the zero-temperature limit of
the thermal density matrix and the ground state, because of
the possible ground-state degeneracy. In the following, when
referring to the ground state of the system, we suppose the
system to be in one of the possible degenerate eigenstates
兩
0
⫹
典
or
兩
0
⫺
典
rather than any other linear combination. It
does not matter which of the two is chosen to be ‘‘the’’
ground state because all the entanglement quantities calcu-
lated in this paper do not depend on the choice, due to the
local symmetry connecting the two states. Therefore, without
loss of generality, when the system is in the ground state we
choose the system to be in the eigenstate
兩
0
⫹
典
for ⬎ 1 and
兩
0
典
for ⭐1. For simplicity, we will identify
兩
0
⫹
典
with
兩
0
典
when is greater than or equal to one.
The zero-temperature state,
0
, of the XY model may be
found by taking the limit

→⬁ of the canonical ensemble,
0
⫽ lim

→⬁
e
⫺

H
Z
. 共31兲
When the ground state is nondegenerate the zero-temperature
state is the same as the ground state of the system,
0
⫽
兩
0
典具
0
兩
. However, if the ground state is degenerate the
zero-temperature ensemble becomes an equal mixture of all
ENTANGLEMENT IN A SIMPLE QUANTUM PHASE... PHYSICAL REVIEW A 66, 032110 共2002兲
032110-5
