Entanglement in fermionic systems at a quantum phase transition
We consider recent results on the use of the single-site entanglement measure for identifying and characterizing a quantum phase transition in systems of interacting fermions. We discuss the extension of these results to fermionic models where the single-site entanglement may fail to signal a quantu...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Entanglement in fermionic systems at a quantum phase transition / H. Johannesson, D. Larsson // Физика низких температур. — 2007. — Т. 33, № 11. — С. 1232-1242. — Бібліогр.: 73 назв. — англ. |
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Johannesson, H. Larsson, D. 2010-04-08T12:16:44Z 2010-04-08T12:16:44Z 2007 Entanglement in fermionic systems at a quantum phase transition / H. Johannesson, D. Larsson // Физика низких температур. — 2007. — Т. 33, № 11. — С. 1232-1242. — Бібліогр.: 73 назв. — англ. 0132-6414 PACS: 71.10.Fd; 03.65.Ud ; 03.67.Mn; 05.70.Jk https://nasplib.isofts.kiev.ua/handle/123456789/7715 We consider recent results on the use of the single-site entanglement measure for identifying and characterizing a quantum phase transition in systems of interacting fermions. We discuss the extension of these results to fermionic models where the single-site entanglement may fail to signal a quantum phase transition, with particular attention given to the one-dimensional extendedUV Hubbard model. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Entanglement in fermionic systems at a quantum phase transition Article published earlier |
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We consider recent results on the use of the single-site entanglement measure for identifying and characterizing a quantum phase transition in systems of interacting fermions. We discuss the extension of these results to fermionic models where the single-site entanglement may fail to signal a quantum phase transition, with particular attention given to the one-dimensional extendedUV Hubbard model.
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Entanglement in fermionic systems at a quantum phase transition / H. Johannesson, D. Larsson // Физика низких температур. — 2007. — Т. 33, № 11. — С. 1232-1242. — Бібліогр.: 73 назв. — англ. |
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Fizika Nizkikh Temperatur, 2007, v. 33, No. 11, p. 1232–1242
Entanglement in fermionic systems at a quantum phase
transition
Henrik Johannesson1 and Daniel Larsson2
1 Department of Physics, Göteborg University, Göteborg SE-412 96, Sweden
E-mail: johannesson@physics.gu.se
2 School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom
Received April 16, 2007
We consider recent results on the use of the single-site entanglement measure for identifying and charac-
terizing a quantum phase transition in systems of interacting fermions. We discuss the extension of these re-
sults to fermionic models where the single-site entanglement may fail to signal a quantum phase transition,
with particular attention given to the one-dimensional extended UV Hubbard model.
PACS: 71.10.Fd Lattice fermion models (Hubbard model, etc.);
03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell’s inequalities, GHZ
states, etc.);
03.67.Mn Entanglement production, characterization, and manipulation;
05.70.Jk Critical point phenomena.
Keywords: fermionic models, quantum phase transition, Hubbard model.
Introduction
In the last few years there has been a growing interest
in quantum many-particle systems from the point of view
of quantum information [1]. The interest is driven by the
need to go beyond the present understanding of how to
operate a few qubits and build scalable and fault-tolerant
devices that can be easily controlled and manipulated.
New ideas for carrying out quantum information tasks,
such as quantum state transfer, have also increased the
need to better understand the behavior of many-particle
systems. These goals have led to a vivid exchange of
ideas between the quantum information and condensed
matter communities. As a result, novel approaches in-
spired by quantum information theory are now actively
being pursued for attacking problems in condensed mat-
ter physics, in particular in the field of strongly correlated
electrons. Conversely, analytical and numerical methods
— as well as intuition and «know-how» — from con-
densed matter physics naturally find their way into quan-
tum information science when exploring various designs
of solid-state hardware for quantum information process-
ing.
One of the central concepts of quantum information
theory [2] is that of entanglement: A state of a composite
system is said to be entangled if it cannot be written as a
direct product of the individual states of its constituents.
As pointed out by Schrödinger in his famous «cat para-
dox» paper from 1935, entanglement (in German
«Verschränkung») lies at the very heart of quantum me-
chanics: «...[the fact that] the best possible knowledge of
a whole does not necessarily include the same for its
parts. [...] The whole is in a definite state, the parts taken
individually are not... [This is] not one, but the essential
trait of the new theory, the one which forces a complete
departure from all classical concepts» [3]. Today we un-
derstand that entanglement — and the non-local correla-
tions that go with it [4] — are not only intrinsic to the fab-
ric of reality, but can also be exploited as a resource for
processing quantum information. Much of current theo-
retical research aims at quantifying «how much» entan-
glement there is in a given quantum many-particle system
and how it is distributed over the system. This is a prereq-
uisite for identifying useful Hamiltonians to produce and
control entangled states, and also for exploring schemes
for quantum computing which rely on the entanglement
of a large number of degrees of freedom (such as topolog-
ical quantum computing [5] or «one-way» quantum com-
puting [6]). There is also a fundamental aspect to this en-
deavor: By studying entanglement properties of a
many-particle system one may extract information about
© Henrik Johannesson and Daniel Larsson, 2007
complex ground state wave functions without having to
calculate them explicitly!
One particular circle of problems that has attracted at-
tention in this context center around the use of entangle-
ment measures for identifying and characterizing quan-
tum phase transitions, following the pioneering work by
Osterloh et al. [7] and Osborne and Nielsen [8]. A quan-
tum phase transition (QPT) happens when the zero-tem-
perature quantum fluctuations in a quantum many-parti-
cle system cause a transition from one type of ground
state to another [9]. Such transitions are induced by the
change of a physical parameter (magnetic field, chemical
potential, pressure, ...) that enhances the quantum fluctu-
ations, or their effect on the system. The study of QPTs
has today become an important theme in condensed mat-
ter physics. The reason is that the very existence of a
quantum critical point, i.e. the point in the phase diagram
at which the QPT takes place, influences the physical
properties of the system also at experimentally accessible
temperatures, and opens up a quantum critical regime
with characteristics very different from what is expected
from conventional «text-book» theory. Theoretical sce-
narios invoking the existence of a (hypothetical) QPT to
explain certain anomalous, so called «non-Fermi liquid»
properties of a system are actively pursued in the field of
heavy-fermion physics [10]. Similar attempts are being
launched also at other problems in the physics of strongly
correlated quantum matter, from the study of complex
oxides [11] to ultra-cold gases trapped in optical lattices
[12].
The change of a ground state at a QPT is generically
associated with a non-analyticity in the ground state en-
ergy [9]. This is most often associated with an avoided
level crossing, where the non-analyticity develops as-
ymptotically in the thermodynamic limit. A special class
of QPTs are those where the transition is driven by a field
that couples to an operator that commutes with the rest of
the Hamiltonian. In such a case the non-analyticity simply
reflects the level crossing that goes with the transition.
Barring accidental cancellations, a nonanalyticity in the
energy automatically propagates into the elements of the
density matrix of the system. Since any measure of entan-
glement [13] is constructed from a (reduced) density ma-
trix one expects that the non-analyticity will somehow
show up also in the ground state entanglement. But how
exactly does it show up? And moreover, how does the
scale invariance at a continuous QPT (as happens when
there is an avoided level crossing, and the Hamiltonian
supports only local, or quasi-local, interactions) manifest
itself in the scaling of the entanglement as one approaches
the quantum critical point?
These are important questions, which are now just
beginning to be tackled. Some answers can already be
drawn from the large body of results for spin-1/2 models
in one dimension (interacting qubits on a 1D lattice). For
example, based on results for critical spin-1/2 chains, Wu
et al. [14] conjectured that a discontinuity [divergence] in
the [derivative of the] ground state concurrence is associ-
ated with a first [second] order QPT, barring the appear-
ance of accidental singularities [15]. The subsequent
proof that any entanglement measure can be expanded as
a unique functional of the first derivatives of the ground
state energy (with respect to the parameters that control
the QPT) puts this intuition on firm ground [16], as does
results from standard scaling theory [17]. Other related
results, employing the notion of localizable entanglement
[18], entanglement entropy [19], and generalized global
entanglement [20] have also been obtained. As to the
property of scale invariance at a continuous QPT, this is
strikingly seen in in the logarithmic divergence of the
block entanglement entropy with the length of the block
[21–23]. While some important issues remain to be clari-
fied — in particular about the connection between the
non-local correlations implied by entanglement and the
long-range («classical») correlations emerging at a QPT
[24] — the basic features of entanglement properties of
critical spin-1/2 systems are by now fairly well under-
stood. The frontier of this research area has advanced rap-
idly in the last year and is now making contact with poten-
tial applications in quantum information science, an
example being the use of spin chains for quantum state
transfer [25,26].
In contrast, less is known about the details of the en-
tanglement — QPT connection for systems of itinerant
particles. One difference from lattice qubit systems is that
the requirement of [anti-]symmetrization of the wave
function for indistinguishable [fermions] bosons implies
a physical Hilbert space that lacks a direct product struc-
ture. It is then no longer obvious how to define the very
notion of entanglement, i.e. the property that a many-par-
ticle wave function does not factorize into a product of
single-particle functions. There is an ongoing debate how
to unambiguously resolve this issue [27]. One possible
way to circumvent the problem and recover a direct prod-
uct structure of the Hilbert space, was suggested by
Zanardi [28]: Passing to an occupation number represen-
tation of local fermionic modes one takes as basis the 4 L
states | | |n n n L� � � � � �1 2 � , where, for spin-1/2 fer-
mions, | | , | , |n j j j j� � � �� ��0 , or |��� j is a local state at
site j, with L the number of sites on the lattice. Note that
the local Hilbert spaces are attached not to the individual
particles, but to the distinguishable sites of a lattice. Also
note that the labeling of occupied states by the spins of the
particles is not unique. Indeed, one could have opted in-
stead for a labeling in terms of the (crystal) momenta of
the particles. As this would imply infinite-dimensional
local Hilbert spaces, one usually makes the simpler
choice with spin quantum numbers as labels. Needless to
Entanglement in fermionic systems at a quantum phase transition
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1233
say, which local Hilbert space to pick is dictated by the
(Gedanken) experiment that is to be carried out: The
states that span the local Hilbert spaces are those which
diagonalize the operator that represents the observable
that is to be measured. Still, even with the simplest choice
of spin labeling, the problem is harder than for qubits
since now each lattice site is associated with four local
states instead of two. For pure states the entanglement
(von Neumnann) entropy remains a well-defined measure
of entanglement, but for mixed states one has to trade the
preferred concurrence measure (related to the entangle-
ment of formation) [29] for the less tractable measure of
negativity [30].
Entanglement behavior at a fermionic QPT Leaving
the difficult problem of mixed states aside, we shall here
focus on the use of the pure state entanglement entropy as
a marker of QPTs in systems of interacting spin-1/2 fer-
mions [31]. For that purpose we split our lattice fermion
system into two parts, A and B, and define as usual the en-
tanglement entropy � of the ground state |� 0� (with re-
spect to the chosen partition) by [32]
� � �Tr log( ) A A2 . (1)
The reduced density matrix A is obtained from the full
density matrix � �� �
| |0 0 by tracing over the local
states belonging to B: A B� Tr ( ). (Taking a trace over
the local states belonging to A gives the same result.) By
choosing A to be a single site, call it j, with B the rest of
the system, one thus arrives at the single-site entangle-
ment. Assuming translational invariance, and that the
ground state |� 0� is a superposition of basis states with
the same number of particles and with the same total spin
(as guaranteed by a translational invariant Hamiltonian
that conserves total spin and particle number), it is easy to
verify that the reduced ground state density matrix j for
the single site j is diagonal in the chosen basis. Intro-
ducing the expectation values that a single site is doubly
occupied ( )�2 , singly occupied by a fermion with spin-up
[spin-down], ( )[ ]�� � , or empty ( )�0 , we have that
� � �
� � � � �
�
2 0 0
0 0 2 2
2
�
�
�
� � � � �
�
� �
� �
�
| � � | ,
| � | ,
n n
n
n
m
j j
j
� � � � �
� � �
�� � � �
� �
0 0 2 2
0 2
2
1
| � | ,
.
n
n
m
n
j
(2)
Here � � �
†n c cj j j
� is the number operator that checks site
j for a fermion of spin
�� �, , n n nj j�
� �� �� �0 0| � � | is
the average ground state occupation number, and
m n nj j� �
� �� �( ) | � � |1 2 0 0� � is the ground state magneti-
zation per site. It follows that
� � � ��
�
j j j j j� �
� ���
��
� � �
�
0
2
, ,
| | | | . (3)
Combining Eqs. (1), (2), and (3) the single-site entangle-
ment can then be expressed as
� � � � �
�
�
�
�
�
� � �
�
�
�
�
�
� �
� � �
�
�
�
�
�
�
n
m
n
m
n
m
2 2
2
2 2 2
2
� �
�
log
log
log ( log ( .) )
2 2
2 2 2 2 2 2
2
1 1
n
m
n n
� �
�
�
�
�
�
� �
� � � � � �
�
� � � � (4)
Suppose that the single-site entanglement thus con-
structed is non-analytic, as signaled by a singularity in its
( )k �1 st derivative at a value, call it gc , of some control
parameter g (with all lower-order derivatives being con-
tinuous and finite). To single out g we write the
Hamiltonian �( )g of the system as � �( )g g� �0 � (with
� the conjugate operator to which g couples, and with all
other control parameters kept fixed and absorbed in �0).
To be specific, we shall assume that g is a magnetic field
strength ( )g h� , a chemical potential ( )g � � , or a local
on-site interaction ( )g U� . Note that g is precisely the
f ie ld conjugate to one of the order parameters
�g �
�� �0 0| |� that parameterize the reduced density
matrix: the magnetization per site m, the average occupa-
tion number n, or the expectation value �2 for double oc-
cupancy*.
Repeated differentiation of Eq. (4) yields
�
�
� �
�
�
� �
�
�
�
�
�
�
�
� � �
��
�
�
�
k
k
k
kg g
n
m
n
m
1
1
1
1 2 2 2
2 2
�
[ ] log� �
�
�
�
�
� �
�
�
�
� �
�
�
�
�
�
�
�
� � �
�
�
�
�
�
�
�
�
k
kg
n
m
n
m
1
1 2 2 2
2 2
[ ] log� � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
k
k
k
k
g
g
n
1
2
1 2 2
1
1 2 2
�
�
�
log ( )
[ ] log (1 2� � �n � ) � (5)
where «...» denote terms that contain lower-order deriva-
tives, all of which are continuous and finite (since other-
wise the ( )k �2 nd derivative of � would also be singular,
contrary to the assumption). A singularity in � � �� �k kg1 1
�
must hence reside in the terms in containing derivatives
of order k �1. Since the order parameter�g conjugate to g
is one of the parameters m n, or �2 (or possibly a linear
combination of m n, and�2), it follows that � � �� �k
g
kg1 1
�
also has a divergence or a discontinuity at g gc� . By the
1234 Fizika Nizkikh Temperatur, 2007, v. 33, No. 11
Henrik Johannesson and Daniel Larsson
* The condition on �g is much less restrictive than maybe first appears to be the case. In fact, a generic QPT in an interacting
fermion system is precisely driven by a change of magnetic field, chemical potential, or a local (screened) interaction (in turn
controlled by a change of the external pressure on the system, or by an applied voltage).
Hellman–Feynman theorem, � �0 0 0� � � � � � �� g e g,
we conclude that the ground state energy e0 has a singu-
larity in its kth derivative. But this is precisely what we
mean by a kth order QPT! Summarizing: a singularity in
the ( )k �1 st derivative of the single-site entanglement im-
plies a kth order QPT (with the proviso that the QPT is
«generic», that is, driven by a change in magnetic field,
chemical potential, or a local interaction) [31].
A few comments may here be in order. First, one could
think that the close link between the scaling of
� � �� �k kg1 1
� and that of � � �� �k
g
kg1 1
� would allow for
the critical exponent that controls �g to be directly read
off from � � �� �k kg1 1
� . This inference is invalid, though.
As a counter example, take a second order QPT ( )k � 2
w i t h �g ��2, w h e r e � � � � � ���
2
1u u u c~ | | a s
g g uc c� � . By inspection of Eq. (5) one notes that the
leading scaling of � � �� g will be governed by the same ex-
ponent only if m and n are independent of�2, or, they de-
pend on �2 as a power with exponent � 1. Whether this is
the case can only be determined on a case-to-case basis. A
second, important comment concerns the logarithmic fac-
tors in (5). These will cause logarithmic divergences if
one or several of the occupation parameters � � �0, ,� �
and �2 vanish at the transition (cf. the expression in (2)).
Such logarithmic corrections, multiplying the leading
scaling of � � �� �k kg1 1
� inherited from �g , thus signal a
change of the dimension of the accessible local Hilbert
space as the system undergoes the transition. This is a
useful and important property of the single-site entangle-
ment scaling not shared by the scaling of�g or its deriva-
tives. It is here important to note that a spurious signaling
of a kth order QPT by a divergence in � � �� �k kg1 1
� caused
by a vanishing occupation parameter is blocked by all
lower-order derivatives of � being continuous and finite.
Putting our result to use, are we sure to identify all
fermionic QPTs? In other words, is the non-analyticity in
the single-site entanglement not only a sufficient, but also
a necessary condition for the appearance of a QPT? The
answer comes with a negative signature. First, the diag-
nostics becomes fuzzy for a QPT of infinite order [33], a
Berezinskii–Kosterlitz–Thouless (BKT) type transition
being the paradigm case [34]. Although the essential sin-
gularity that here is present in the ground state is expected
to «infect» also the single-site entanglement via the re-
duced density matrix, its identification may be too diffi-
cult to serve as a useful tool. The situation becomes even
less transparent for other non-conventional QPTs that are
not associated with non-analyticities in the ground state
energy [35], the transition a between two quantum Hall
plateaus in the integer quantum Hall effect being an out-
standing example. While our result does not apply to
these cases, it is conceivable, maybe even expected, that
the change of ground state at the transition will still show
up as a non-analyticity in the entanglement. If and how
this happens is a question well worth further studies. A
class of QPTs where the single-site entanglement obvi-
ously fails as a marker for a QPT are those where the con-
trol parameter does not couple to a single-site. As we
shall discuss below, while such QPTs in principle can be
analyzed along the same lines as their simpler counter-
parts discussed above, the practical implementation of the
analysis may meet with certain difficulties. There is, how-
ever, another, more insidious way that the single-site en-
tanglement will fail to signal a QPT. This happens if all
local basis states | | , | , |n j j j j� � � �� ��0 , and |��� j be-
come equally populated as one approaches the transition.
As seen from (5), the ( )k �1 st derivative terms then vanish
identically, killing off the signal of the QPT. The simulta-
neous vanishing of � � �� g implies that � has a local
extremum at the transition (expected to be a maximum
since in this case all local basis states are equally repre-
sented in the make-up of the many-particle ground state).
However, one cannot a priori exclude that � is at an
extremum without the occurrence of a QPT. Hence, an
extremum of the single-site entanglement does not neces-
sarily signal a QPT. Whether a QPT is present or not in
this case requires information beyond that provided by
the entanglement measure. Unfortunately, this simple ob-
servation has been overlooked in some of the literature on
the subject, leading to unnecessary confusion and specu-
lations.
Having uncovered the general features of entangle-
ment scaling at a fermionic QPT, let us now look at a few
examples [31,36,37].
Case studies
As our first example we take the 1D Hubbard model
� � � �
�
� �
� �
�
�� �t c c U n nj
j
L
j j
j
L
j�
�
��
†
� �
1
1
1
. (6)
Here c j�
† c j� are the usual fermionic creation and annihi-
lation operators attached to site j of the lattice, with spin
� �� �, , and with �
†n c cj j j� � �� the corresponding number
operator. In the following we shall work with dimen-
sionless quantities u U t� � 4 and h H tB� �� , putting t �1
(of dimension energy), and assume periodic boundary
conditions. This model, which has long served as a para-
digm for strongly correlated electron systems [38], has
received renewed attention due to its possible realization
in 1D optical lattices with trapped ultra-cold gases of
fermionic atoms [39]. The sign and the strength of the
on-site interaction U and the tunneling rate t between
neighboring minima of the lattice potential can here be
chosen at will by tuning a Feshbach resonance, thus mak-
ing possible a fully «controllable» fermionic system gov-
erned by the Hubbard Hamiltonian in Eq. (6).
Entanglement in fermionic systems at a quantum phase transition
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1235
At half-filling of the lattice, n �1(i.e. with on average
one fermion/site), the model exhibits a QPT at u � 0, sepa-
rating a Mott insulating phase ( )u 0 from a metallic
phase ( )u ! 0 . The ground state energy density becomes
non-analytic at the transition, but allows for an asymp-
totic power series expansion with all derivatives being fi-
nite and continuous [40]. The QPT is thus of infinite or-
der, and can be shown to belong to the BKT universality
class [41]. As found by Gu et al. [42], the single-site en-
tanglement has a maximum at the transition, reflecting the
equipartition of empty-, singly- and doubly occupied lo-
cal states when u � 0 (non-interacting fermions). This
transition is thus special on two counts: it is of infinite or-
der and it supports an equipartition of local states. This
makes it an exceptional example of a fermionic QPT,
where no information can be deduced from the single-site
entanglement measure.
A metal-insulator transition can also be triggered when
u 0 by connecting the system to a particle reservoir and
t u n i n g t h e c h e m i c a l p o t e n t i a l g � � ( w i t h � a
dimensionless chemical potential multiplied by the hop-
ping amplitude t �1). The corresponding Hamiltonian is
given by that in Eq. (6), with the added term
�� �� � �
�
� ��( � � ).
j
L
j jn n
1
(7)
It is here important to point out that provided that there is
no interaction with the reservoir, a pure state entangle-
ment measure is still applicable at zero temperature. For
the case of repulsive on-site interaction, u 0, and with
n " 1, the system exhibits two quantum critical points
[43]: � c1 2� � and
� � � � �c J u d2 1
0
12 4 1 2� � �
�
�# ( )( [ exp ( )]) ,
with J 1( )� a first-order Bessel function. Both transi-
tions are second-order with diverging charge susceptibili-
ties $ � �Ci cic u i� � �� �( )| | , ,1 2 1 2 in the limits � �� �c1
(empty lattice transition) and � �� �c2 (Mott transition),
respectively (with c u( ) a positive u-dependent constant).
To obtain the single-site entanglement � we make the ob-
servation that �� conserves spin and particle number for
fixed �, and that hence the expression for � in (4) remains
valid. Recalling from the Lieb–Mattis theorem [44] that
the ground state has zero spin (for any n with nL an even
integer) we put m � 0 in (4). Using the Hellman–Feynman
theorem, the value of�2 can be extracted from the ground
state energy via the relation �2 0 4� � � � �( )E u L. Ex-
ploiting the Bethe Ansatz solution of the model [43], E0
can be expressed via a 1 � u expansion [45]:
E
L
n n
u
l
l
l
0
1
2 1
4
� � �
�
�
�
�
�
�
�
�
�%
% &sin ( ) ( ) . (8)
The values of & l n( ) are tabulated to fifth order in Ref. 45.
The ground state energy in (8) also determines the chemi-
cal potential as function of filling: �( )n E n� � � �0 . By in-
verting �( )n and inserting the resulting values for the
w-parameters from (2) into (4) we can plot � vs. � for any
value of u 1. Some representative plots are shown in
Fig. 1, together with the single-site entanglement for free
fermions (u � 0).
In order to analytically explore the quantum critical re-
gions � �� �c1 and � �� �c2 we first consider the u � �
l imit where �2 0� . In this limit (8) implies that
n( ) ( ) ( )� % �� � � �1 2arccos . Combining this expression
with (4) we obtain
�
�
� � � � �
�
�
$
� �( )
ln( )
(ln| | ), ,1
2 2
1 2i Ci
ci iconst (9)
for � �� �c1 and � �� �c2 , respectively. Turning to the
case of large but finite u, we focus on the Mott transition
� �� �c2 . A straightforward analysis, again using the
Bethe Ansatz result in (8), yields for the leading behavior
of the single-site entanglement:
�
�
� �
�
�
$C u C( ) ,2 (10)
with C u( ) a positive u-dependent constant.
The results in Eqs. (9) and (10) well illustrate our ge-
neral discussion in the previous section. For finite u the
logarithms in Eq. (5) add up to the u-dependent constant
C u( ), whereas in the limit u � � the entanglement
1236 Fizika Nizkikh Temperatur, 2007, v. 33, No. 11
Henrik Johannesson and Daniel Larsson
–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Chemical potential
S
in
g
le
-s
it
e
en
ta
n
g
le
m
en
t
u = 0
u = 2
u = 4
Fig. 1. Entanglement entropy � of a single site versus chemical
potential � for the repulsive Hubbard model. The plateaus cor-
respond to half-filling ( )n �1 , cut off at � � 2. The solid curve is
that for free fermions ( )u � 0 , plotted in the region 0 2" "n .
measure detects a change in the dimension of the local
Hilbert space, as signaled by the logarithmic correction
to the leading scaling. In the u � � limit the singly
occupied [empty] local states get suppressed when
� � � �� �� �c c1 2[ ], while for finite u both the metallic
( )� �! c2 and insulating (� � c2) ground states are super-
positions of all four types of local states | , | , |0� �� ��j j j ,
and |��� j .
Let us next study the case of a QPT driven by an ap-
plied magnetic field, again using a Bethe Ansatz approach
to the 1D Hubbard model as work horse. The Hamiltonian
is now written:
� � � � �
�
��
� �
�
�
�
� � �t c c U n n H Sj
j
L
j j
j
L
j B j
z
j
L
�
�
�� �†
1
1
1 1
. (11)
Here S n n
z
j j
j � � �� �( ) 2 is a spin-1/2 operator attached
to lattice site j. As before, we use dimensionless quanti-
ties: u U t� � 4 and h H tB� �� . Focusing on the limit
| |u 1 with u ! 0 (attractive interaction), and with n �1
(half-filling), we can again exploit the Hellman–Feyn-
man theorem. Together with the known Bethe Ansatz re-
sult for the ground state energy [46],
E L u m m u0 4 1 2 1 2 2 1� � � � � � � �( ) ( ) sin ( ) ( )% % � ,
we obtain
�2
0 21
4
1
2
1�
�
�
� � � �
L
E
u
m u�( ). (12)
Neglecting the �( )1 2� u corrections it follows immedi-
ately from Eq. (4) that
� � � � � � �2 2 1 2
1
2
02 2m m m m hlog ( ) ( ) log ( ), . (13)
The dependence of the magnetization on the applied field
can also be derived from the ground state energy, and one
finds
m h
h h
u
h
h h
c
c( )
, ,
,�
" !
� �
�
�
�
�
�
�
�
�
��
�
�
�� " "
0 0
1
2 4
1
1%
arccos h
h h
c
c
2
2
1
2
,
, ,!
'
(
)
))
*
)
)
)
(14)
with lower [upper] cri t ical f ield* h uc1 4 1� � +(| | )
+ � �[ (| | )]h uc2 4 1 . The single-site entanglement as a func-
tion of magnetic field, � �� ( )h , can now be read off from
(4) and (14). The result for the | |u 1 limit is plotted in
Fig. 2 for large values of h. Note that in this limit there are
two local states, |0� and |���, available to the system when
h hc! 1, implying that �( )h �1. In contrast, the fully mag-
netized state for h hc 2 is a direct product of local
spin-up states, and hence �( )h � 0. For comparison we
have plotted the single-site entanglement for free fer-
mions also in Fig. 2 (for both positive and negative values
of the magnetic field). This result is easily obtained from
Ref. 46 by noting that �2
21 4� � �m when u � 0, with
m h� � �( ) ( )1 4% arcsin in the interval � ! !4 4h .
The phase transitions at hc1 and hc2 are second
order, with diverging spin susceptibili t ies $ Si �
� � � �( | | )32 2 1 2% h hci , i �1 2, [15]. The plot in (2) indeed
suggests a corresponding divergence of � � �� h as
h hc� �1 and h hc� �2 as predicted by our general result
in the previous section. As an analytical check we write
u h h h ici
i� � � � � � � �4 4 1 1 2( ) ( ) , , and expand � � �� h
in h hc� 1 and h hc2 � , to obtain
�
�
� � � � �
�
h
h h ii Si
ci( )
ln
(ln | | ), ,1
2
1 2
$
const (15)
for h hc� �1 and h hc� �2 , respectively. This confirms
that � � �� h does diverge at the magnetic phase transitions,
in accordance with our general result.
Turning to the half-filled case with repulsive inter-
action, u 0, a QPT now occurs only at the value
of the field for which the magnetization saturates:
Entanglement in fermionic systems at a quantum phase transition
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1237
* The QPT at the upper critical field, the saturation point, is of a special type: Although being nontopological, there is no sym-
metry breaking at the transition. For a discussion, see [47].
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Magnetic field
S
in
g
le
-s
it
e
en
ta
n
g
le
m
en
t
4(|u|–1) 4(|u|) 4(|u|+1)
u = 0
|u| >> 1
Fig. 2. Entanglement entropy � of a single site versus magnetic
field h for the attractive Hubbard model with | |u 1 (dotted
curve). For comparison, the single-site entanglement for the
free case (u � 0) is shown by the solid curve (on a different
scale).
h u uc2
24 1� � �( ) [48]. As shown by Takahashi, the
ground state energy for any finite value of u 0 in the crit-
ical region h hc� �2 can be expanded in terms of the ex-
pectation value for single spin-down occupancy [49]:
E
L
u u n
u
n n
j
j j
0 2
0
2
2 0
3
0
4
4
1
24
1
1
� � � � �
�
�
�
�
� �
( )
( ) .
%
� (16)
With the same procedure as used for the attractive case
above, Eq. (16), together with (2) and (4), yield:
�
�
� � � � �
�
h
C
h h h hS c c
2 2
2 2
ln
const$ (ln | | ), . (17)
H e r e C u u� � � �2 1 2 , a n d 2 4 4 2 1 4%$ S u� � +�( )
+ � � �| |h hc2
1 2. The logarithmic correction in (17) now
signals the suppression of all but the spin-up states as one
approaches the saturation point hc2 from below.
The reason for the similarity of the scaling formulas in
Eqs. (15) and (17) can be made transparent by exploiting
a particle-hole transformation � for spin-up fermions:
� : ( ) ,
†c c
j
j
j� �, �1 (18)
(leaving the spin-down fermions untouched). This trans-
formation maps the zero-field repulsive Hubbard model
with a chemical potential onto the half-filled attractive
Hubbard model with an applied magnetic field. It follows
that the single-site entanglement at � �c c1 2( ) has the same
behavior as at h hc c2 1( ); cf. Figs. 3 and 4.
As a second example, let us briefly discuss how the
Mott–Hubbard transition in the 1D Hubbard model with
long-range hopping gets signaled by the single-site entan-
glement. The model is defined by [50]
�t m
m
L
m
l
L
t c c u n n� �
- �
�� �
�
�
�� ��
�
� � �
1 1
,
†
� � � � , (19)
with t i l mm
l m
� � � �� �( ) ( )( )1 1. The ground state energy
density at half-filling is given by
e un u n n uu
u u u u
c c
c c
0
3 2
1 4 1 24� � � � � � +
+ � � � �
[ ( ) ) ( ( )]
[( ) (( ) 4 3 2uu nc ) ]�
with u c � 2% the critical point [50]. This implies that
�2 0� � � �e u has a discontinuity in its second order deri-
vative with respect to u at u c and hence the transition
is third order. From Eq. (4) with n �1 it follows that
the s ing le s i te en tanglement can be wr i t ten as
� � � � � � �( ) log ( ) log ( )1 2 1 2 22 2 2 2 2 2� � � � w h e n n o
magnetic field is present (i.e. m � 0), and one immediately
verifies that � � �2 2
� u is also discontinuous at the transi-
tion point u c . Since the local basis states do not become
equally populated at u c — in contrast to the u � 0
metal–insulator transition of the ordinary Hubbard model
— the single-site entanglement here provides an accurate
diagnostics of the transition. A plot of the single-site en-
tanglement as function of u is shown in Fig. 5. This QPT
is unusual in exhibiting a discontinuity rather than a di-
vergence in a higher-order derivative of the ground state
energy. As one expects second- and higher-order phase
transitions to be continuous one may worry that some-
thing is askew. However, the reason for the «anomaly» is
simply that the Hamiltonian in [50] contains a long-range
hopping process, thus breaking scale invariance at large
distances (i.e. also in the «scaling limit»).
One can drive a Mott—Hubbard metal–insulator tran-
sition also by tuning the chemical potential when u u c ,
in exact analogy with the ordinary Hubbard model. Ex-
pressing n as a function of �, and applying the
Hellman–Feynman theorem to the ground state energy e0
above, one obtains a discontinuity in � � �n � at
1238 Fizika Nizkikh Temperatur, 2007, v. 33, No. 11
Henrik Johannesson and Daniel Larsson
–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Chemical potential
S
in
g
le
-s
it
e
en
ta
n
g
le
m
en
t
u = 2
Fig. 3. Entanglement entropy � of a single site vs. chemical
potential for u � 2.
4 5 6 7 8 9 10 11 12
0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Magnetic field
S
in
g
le
-s
it
e
en
ta
n
g
le
m
en
t
u = –2
Fig. 4. Entanglement entropy � of a single site vs. magnetic
field for u � �2 and at half-filling ( )n �1 , obtained from Fig. 3
via the particle-hole transformation � in Eq. (18).
� � %� �c
� � %� �c [51]. Eq. (5) immediately implies that � � �� � is
also discontinuous at� �� c , with the transition being sec-
ond order. In the limit u � � this discontinuity is multi-
plied by a logarithmic divergent factor when � �� �c , re-
flecting the suppression of empty states in this case.
Extensions of the theory: the one-dimensional
UV Hubbard model
The modification of the ordinary one-dimensional
Hubbard model obtained by allowing long-range hopping
is only one of several possible ones. A more realistic ex-
tension of the model that mimics the effect of a fi-
nite-range Coulomb interaction is the 1D extended Hub-
bard model [52,53], alias the 1D UV Hubbard model.
With the notation of Eq. (6), its Hamiltonian is written as
�UV j
j
L
j j
j
L
j j
j
L
t c c U n n V n� � � �
�
� �
� �
�
�
�
� � ��
�
��
†
� � � �
1
1
1 1
n j � 1 ,
(20)
where � � �n n nj j j� �� � . The term V is a nearest-neighbor
interaction that emulates a finite-range (screened) Cou-
lomb interaction. The inclusion of this term destroys the
integrability of the model, but its possible phases can be
extracted by exact diagonalization and variational tech-
niques, supplemented by exact results in various limits.
One finds a rich phase diagram in the UV -plane, with
phases exhibiting enhanced correlations for charge- and
spin density waves, singlet- and triplet superconducting
correlations, and a region of phase separation [53,54].
Some of the coexistence lines separating the various
phases were qualitatively reconstructed at half-filling by
numerically identifying the ridges of the single-site en-
tanglement as function of U and V [42]. Attempts to
improve upon this result — including an identification of
transitions to phases with enhanced pairing correlations
— were subsequently carried out via a study of the scaling
of the block entropy for the model [55].
Here we are more concerned with the detailed ana-
lyticity properties of the ground state entanglement as one
approaches a QPT. As long as the QPT is driven by an ex-
ternal field that couples locally to single sites on the lat-
tice, or by the on-site interaction ~ U , we can carry over
the approach above intact.
As an example, let us consider the model in the V � �
limit and at quarter filling ( )n � �1 2 . For this case only
every other site will be occupied since the energy cost to
put two particles on neighboring sites is infinite. There
are three distinct phases for this case: U 4, where every
second site will occupied by precisely one particle;
� ! !4 4U , with a mixture of doubly and singly occupied
sites; and U ! � 4, where all sites are doubly occupied. As
we tune the on-site interaction U , while staying in the
V � � limit, we expect that the single-site entanglement
will signal the QPTs that occur at U � 4 (metal–insulator
transition) and U � � 4 (transition to a spin-gapped phase)
respectively. To obtain an expression for the single-site
entanglement, we take off from the parameterization of
the reduced density matrix as given in Eq. (2). From
Ref. 54 we have that
�
%2 2 1
1
4
� � �
�
�
�
�
�
�
�
�( ) ,n
n U
arccos (21)
which at quarter filling (n � �1 2) takes the form
�
%2
1
2 4
� �
�
�
�
�
�
�arccos
U
. (22)
The Lieb–Mattis theorem [44] implies that the ground
state is a spin singlet (m = 0), and we thus conclude from
Eqs. (2) and (22) that
� �
%� �� � � �
�
�
�
�
�
�
1
4
1
2 4
arccos
U
, (23)
�
%0
1
2
1
2 4
� � �
�
�
�
�
�
�arccos
U
.
Using Eq. (4) we can immediately write down the sin-
gle-site entanglement:
�( ) logU
U U
� � � �
�
�
�
�
�
�
�
�
�
�
�
� � �
1
2
1
4
1
4
1
2
2% %
arccos arccos
4
1
2 4
1
2
2
�
�
�
�
�
�
�
�
�
�
�
� �
� �
�
�
�
�
�
�
�
�
�
�
�
�
% %
arccos arcc
U
log os
arccos
�
�
�
�
�
�
�
�
�
�
�
�
� �
� � �
�
�
�
�
�
�
�
�
�
�
�
�
U
U
4
1
2
1
2 4
2%
log
1
2
1
2 4
� �
�
�
�
�
�
�
�
�
�
�
�
�
%
arccos
U
. (25)
Entanglement in fermionic systems at a quantum phase transition
Fizika Nizkikh Temperatur, 2007, v. 33, No. 11 1239
2 4 6 8 10 12 14 16 18 20
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
On-site interaction
S
in
g
le
-s
it
e
en
ta
n
g
le
m
en
t
Fig. 5. Entanglement entropy � of a single site vs. on-site in-
teraction u in the Hubbard model with long-range hopping at
half-filling ( )n �1 . Note that the Mott–Hubbard transition at
uc � �% 2 is away from the maximum of the entanglement en-
tropy, in contrast to the case of the ordinary Hubbard model.
Differentiating with respect to U yields the following ex-
pression:
�
�
�
� �
� �
�
�
�
�
�
�
�
�
��
�
�
�
�
U U
U1
8
1
1 4
2
1
4
1
2 42
2% %( )
log arccos � �
'
(
*
� �
�
�
�
�
�
�
�
�
��
�
�
�� � �log log2 2
1
2 4
1
2
1
2% %
arccos arc
U
cos �
�
�
�
�
�
�
�
�
��
�
�
��
.
/
0
U
4
.
(26)
By inspection of Eq. (26) we pinpoint divergences at
U � � 4, signaling a second order QPT. Explicitly,
1 2�
�
3
�
�
�
U U U
U U
c
c
1
4 2
1
2
% | |
log (| | ) (27)
where we used the relations
1 4 2 16 22� � � � � � � � 3 � �( ) ( ) ( ) ( )u U U U U U Uc c c
when U c � 4,
and
1 4 2 16 22� � � � � � � � 3 � �( ) ( ) ( ) ( )U U U U U U Uc c c
when U c � � 4.
The divergences in (27) were expected, since it is already
known that U � � 4 define critical points for the two sec-
ond-order QPTs: from a metal to an insulator atU � 4, and
from an ordinary metal to a metallic phase with supercon-
ducting correlations at U � � 4 [54].
Suppose that we instead wish to explore a QPT driven
by a change in the nearest-neighbor interaction V . The
conjugate order-parameter is now that for nearest-neigh-
bor occupancies,
� �� �� �0 1 0 1| � � |n n n nj j j j , which does
not enter the reduced density matrix from which the sin-
gle-site entanglement measure is constructed (cf. Eq. (2)).
Hence, given our result from the previous section, the sin-
gle-site entanglement is not expected to signal a QPT
when driven by a change inV . The way out is obvious: We
have to construct a two-site entanglement measure, based
on the reduced density matrix for the two neighboring
sites j and j � 1. (In analogy with the analysis above,
which two sites that we choose is immaterial, provided
that the system is translationally invariant.) The extension
to a two-site measure is conceptually straightforward
[55–58], allowing us in principle to carry over our results
for non-analyticities in the entanglement unadorned.
However, the practical implementation of the theory
meets with certain obstacles. First, the reduced density
matrix now acts on a 16-dimensional space, with many
more entries to keep track on. Secondly, and more seri-
ously, there are no exact analytical results for the ground
state energy for arbitrary values of U and V that we can
draw upon.
Fortunately, in the limit U � � (still at quarter filling,
n � �1 2), the theory simplifies and we can make some
progress: When U � �, double occupancies of single
sites get suppressed, and as a consequence the dynamics
becomes insensitive to the spin of the fermions. The local
Hilbert spaces collapse to qubit spaces, spanned by the
two states |0� («empty») and |1� («singly occupied»).
Working with (dynamically) spinless fermions, one can
then exploit the time-honored Jordan–Wigner transfor-
mation [59], and map the model onto the Bethe Ansatz
solvable spin-1/2 XXZ chain [60]
� � � �
�
�
� � ��
i
x
i
i
x
i
y
i
y
i
z
i
z
1
1 1 14 . (28)
The Mott–Hubbard critical point at V � 2 [59] in this
way gets mapped onto the isotropic point 4 �1, separating
a spin liquid phase at � ! !1 14 from an Ising antiferro-
magnetic phase at 4 1. It follows that the desired infor-
mation about the critical entanglement properties can be
extracted by studying the two-site entanglement measure
for the XXZ model at 4 �1. The two-site reduced density
matrix for the XXZ model is well-known [61–63], and the
associated two-site entanglement is readily obtained [64],
revealing a local maximum at 4 �1, with all its higher-or-
der derivatives being finite and continuous. This atypical
behavior can be ascribed to the particular population of
local states at the Mott transition which conspires to kill
off the first-derivative of the two-site entanglement (with
respect to V ), in exact analogy to the vanishing of the
first-derivative of the single-site entanglement when the
local states are equally populated. This masks the ex-
pected non-analyticity of the entanglement, replacing it
by a local maximum. In fact, the QPT is here of BKT type,
just as for the Mott transition in the ordinary Hubbard
model [41], making the detection of the non-analyticity
(an essential singularity!) highly non-trivial.
As should be clear from this brief exposition, an ana-
lytical study of the complete phase diagram of the UV ex-
tended Hubbard model from the perspective of entangle-
ment scaling is not an easy task. Quite possibly it must
await further theoretical breakthroughs.
Outlook
Before concluding, one may ask: «What is the advan-
tage of probing the entanglement at a QPT rather than di-
rectly studying the behavior of the ground state energy?».
There are several answers to this question. First, as we
have already noted, certain QPTs can not be traced back
to a nonanalyticity in the ground state energy, a case in
point being the transitions between quantum Hall pla-
teaus [35]. There are reasons to expect that a singular be-
havior of some entanglement measure may still apply to
these transitions, and could serve as a convenient diag-
1240 Fizika Nizkikh Temperatur, 2007, v. 33, No. 11
Henrik Johannesson and Daniel Larsson
nostic tool. To find out, one needs to practice on the sim-
pler conventional QPTs, sorting out generic from acci-
dental non-analyticities. Secondly, to understand to what
extent non-local quantum correlations are implied in the
scaling and universal properties exhibited by systems un-
dergoing QPTs requires a detailed study of the associated
entanglement properties. Our results for the Hubbard
model [31,36,37], reviewed in the previous section, here
provide a rich backdrop, with the entanglement scaling
governed by the associated thermodynamic susceptibili-
ties (and with logarithmic corrections encoding a change
of dimension of the accessible local Hilbert spaces (see
also Ref. 65)). Moreover, recent theoretical developments
suggesting the use of many-particle entanglement for in-
formation processing may benefit from getting a firm
handle on entanglement properties at QPTs. For example,
certain schemes for quantum adiabatic computing [66] re-
lies on (the assumed) entanglement scaling properties
close to a QPT [67].
This is an area of research rich in opportunities. As it
comes to models of interacting fermions, there is cur-
rently an ongoing activity by several research groups to
explore QPTs in other extensions of the one-dimensional
Hubbard model. Progress has been made for the
bond-charge extended Hubbard model, where Anfossi
et al. [56,57] have obtained important results, clarifying
the role of multi-partite entanglement, as well as the rela-
tion between classical and quantum correlations at criti-
cality More work should also be done — for fermionic
and spin models — as it comes to unearthing effects from
disorder, impurities, local fields, and boundaries. A par-
ticularly promising class of models to practice on — hith-
erto unexplored in this context — are the integrable
multichain models discussed in the review by Zvyagin
[68]. Another fascinating direction of research would be
to study the quantum analogies [69,70] of the elusive and
highly nontrivial phase transitions «by breaking of
analyticity», known from commensurate-incommensu-
rate transitions [71,72]. A related class of phenomena in
the context of low-dimensional quantum spin models
have recently been discussed in Ref. 73. The study of en-
tanglement behavior may here open up new landscapes
for exploration.
We are grateful to K. Capelle and V.V. Franca for very
useful comments. H.J. acknowledges financial support
via a grant from the Swedish Research Council.
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