Thermal entanglement of qubit pairs on the Shastry-Sutherland lattice
S. El Shawish
J. Stefan Institute, Ljubljana, Slovenia
A. Ramšak and J. Bonča
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia and
J. Stefan Institute, Ljubljana, Slovenia
(Dated: 2 April 2007)
We show that temperature and magnetic field properties of the entanglement between spins on the two-
dimensional Shastry-Sutherland lattice can be qualitatively described by analytical results for a qubit tetramer.
Exact diagonalization of clusters with up to 20 sites reveals that the regime of fully entangled neighboring pairs
coincides with the regime of finite spin gap in the spectrum. Additionally, the results for the regime of vanishing
spin gap are discussed and related to the Heisenberg limit of the model.
PACS numbers: 75.10.Jm, 03.65.Yz, 03.67.Mn
I. INTRODUCTION
In any physical system with subsystems in interaction, indi-
vidual parts of the system are to some extent entangled, even
if they are far apart, as realized already at the beginning of
modern quantum mechanics sixty years ago. Today it has be-
come appreciated that the ability to establish entanglement be-
tween quantum particles in a controlled manner is a crucial in-
gredient of any quantum information processing system1. On
the other hand, it turned out that the analysis of appropriately
quantified entanglement between parts of the system can also
be a very useful tool in the study of many body phenomena,
as is, e.g., the behavior of correlated systems in the vicinity of
crossovers between various regimes or even points of quantum
phase transition2.
Quantum entanglement of two distinguishable particles in a
pure state can be quantified through von Neuman entropy3,4,5.
Entanglement between two spin- 1
particles – qubit pair – can
be considered a physical resource, an essential ingredient of
algorithms suitable for quantum computation. For a pair of
subsystems A and B, each occupied by a single electron, an
appropriate entanglement measure is the entanglement of for-
mation, which can be quantified from the Wootters formula6.
In general, electron-qubits have the potential for even richer
variety of entanglement measure choices due to both their
charge and spin degrees of freedom. When entanglement is
quantified in systems of indistinguishable particles, the mea-
sure must account for the effect of exchange and it must ade-
quately deal with multiple occupancy states7,8,9,10,11,12. A typ-
ical example is the analysis of entanglement in lattice fermion
models (the Hubbard model, e.g.) where double occupancy
plays an essential role11.
In realistic hardware designed for quantum information
processing, several criteria for qubits must be fulfilled13: the
existence of multiple identifiable qubits, the ability to initial-
ize and manipulate qubits, small decoherence, and the ability
to measure qubits, i.e., to determine the outcome of compu-
tation. It seems that among several proposals for experimen-
tal realizations of such quantum information processing sys-
tems the criteria for scalable qubits can be met in solid state
structures consisting of coupled quantum dots14,15. Due to the
ability to precisely control the number of electrons in such
structures16, the entanglement has become experimentally ac-
cessible quantity. In particular, recent experiments on semi-
conductor double quantum dot devices have shown the evi-
dence of spin entangled states in GaAs based heterostuctures17
and it was shown that vertical-lateral double quantum dots
may be useful for achieving two-electron spin entanglement18.
It was also demonstrated recently that in double quantum dot
systems coherent qubit manipulation and projective readout is
possible19.
Qubit pairs to be used for quantum information processing
must be to a high degree isolated from their environment, oth-
erwise small decoherence requirement from the DiVincenzo’s
checklist can not be fulfilled. The entanglement, e.g., between
two antiferromagnetically coupled spins in contact with ther-
mal bath, is decreased at elevated temperatures and external
magnetic field20,21,22, and will inevitably vanish at some fi-
nite temperature23. Entanglement of a pair of electrons that
are confined in a double quantum dot is collapsed due to the
Kondo effect at low temperatures and for a very weak tunnel-
ing to the leads. At temperatures below the Kondo temper-
ature a spin-singlet state is formed between a confined elec-
tron and conduction electrons in the leads24. For other open
systems there are many possible sources of decoherence or
phase-breaking, for example coupling to phonon degrees of
freedom25.
The main purpose of the present paper is to analyze the ro-
bustness of the entanglement of spin qubit pairs in a planar
lattice of spins (qubits) with respect to frustration in magnetic
couplings, elevated temperatures as well as due to increasing
external magnetic field. The paper is organized as follows.
Sec. II introduces the model for two coupled qubit pairs –
qubit tetramer – and presents exact results for temperature and
magnetic field dependence of the entanglement between near-
est and next-nearest-neighboring spins in a tetrahedron topol-
ogy. In Sec. III the model is extended to infinite lattice of
qubit pairs described by the Shastry-Sutherland model26. This
model is convenient firstly, because of the existence of sta-
ble spin-singlet pairs in the ground state in the limit of weak
coupling between the qubit pairs, and secondly, due to a rel-
atively good understanding of the physics of the model in the
http://arxiv.org/abs/0704.0172v1
thermodynamic limit. Entanglement properties of the Shastry-
Sutherland model were so far not considered quantitatively.
Neverteless, several results concerning the role of entangle-
ment at a phase transition in other low-dimensional spin lattice
systems2,27,28,29,30,31,32, as well as in fermionic systems33,34,35
have been reported recently. Near a quantum phase transi-
tion in some cases entanglement even proves to be more ef-
ficient precursor of the transition compared to standard spin-
spin correlations35,36. In Sec. IV we discuss entanglement be-
tween nearest neighbors in the Heisenberg model, represent-
ing a limiting case of the Shastry-Sutherland model. Results
are summarized in Sec. V and some technical details are given
in Appendix A.
II. THERMAL ENTANGLEMENT OF A QUBIT
TETRAMER IN MAGNETIC FIELD
Consider first a double quantum dot composed of two adja-
cent quantum dots weakly coupled via a controllable electron-
hopping integral. By adjusting a global back-gate voltage,
precisely two electrons can be confined to the dots. The inter-
dot tunneling matrix element t determines the effective anti-
ferromagnetic (AFM) superexchange interaction J ∼ 4t2/U ,
where U is the scale of Coulomb interaction between two
electrons confined on the same dot. There are several possi-
ble configurations of coupling between such double quantum
dots. One of the simplest specific designs is shown schemat-
ically in Fig. 1(a): four qubits at vertices of a tetrahedron. In
addition to the coupling A-B, by appropriate arrangements of
gate electrodes the tunneling between A-C and A-D can as
well be switched on.
We consider here the case where J/U ≪ 1, thus double oc-
cupancy of individual dot is negligible and appropriate Hilbert
space is spanned by two dimers (qubit pairs): spins at sites
A-B and C-D are coupled by effective AFM Heisenberg mag-
netic exchange J and at sites A-C, B-C, A-D, B-D by J ′. The
corresponding hamiltonian of such a pair of dimers is given as
H4 = J(SA · SB + SC · SD) + (1)
+ 2J ′(SA · SC + SB · SC + SA · SD + SB · SD)−
− B(SzA + SzB + SzC + SzD),
where Si =
σi is spin operator corresponding to the site i
and B is external homogeneous magnetic field in the direction
of the z-axis. Factor 2 in Eq. (1) is introduced for convenience
– such a parameterization represents the simplest case of finite
Shastry-Sutherland lattice with periodic boundary conditions
studied in Sec. III.
A. Concurrence
We focus here on the entanglement properties of two cou-
pled qubit dimers. The entanglement of a pair of spin
qubits A and B may be defined through concurrence3, C =
2|α↑↑α↓↓ − α↑↓α↓↑|, if the system is in a pure state |ΨAB〉 =
ss′ αss′ |s〉A|s′〉B, where |s〉i corresponds to the basis | ↑ 〉i,
Figure 1: (Color online) (a) Two coupled qubit pairs (dimers) in tetra-
hedral topology. (b) Shastry-Sutherland lattice as realized, e.g., in the
SrCu2(BO3)2 compound.
| ↓ 〉i. Concurrence varies from C = 0 for an unentan-
gled state (for example | ↑ 〉A| ↑ 〉B) to C = 1 for com-
pletely entangled Bell states3 1√
(| ↑ 〉A| ↑ 〉B ± | ↓ 〉A| ↓ 〉B)
or 1√
(|↑ 〉A|↓ 〉B ± |↓ 〉A|↑ 〉B).
For finite inter-pair coupling J ′ 6= 0 or at elevated tempera-
tures the A-B pair can not be described by a pure state. In the
case of mixed states describing the subsystem A-B the concur-
rence may be calculated from the reduced density matrix ρAB
given in the standard basis |s〉i|s′〉j6. Concurrence can be fur-
ther expressed in terms of spin-spin correlation functions2,27,
where for systems that are axially symmetric in the spin space
the concurrence may conveniently be given in a simple closed
form37, which for the thermal equilibrium case simplifies fur-
ther,
CAB = 2max(0, |〈S+AS
〉〈P ↓
〉). (2)
Here S+i = (S
† = Sxi + ıS
i is the spin raising operator
for dot i and P ↑i =
(1 + 2Szi ), P
(1 − 2Szi ) are the
projection operators onto the state | ↑ 〉i or | ↓ 〉i, respectively.
We consider the concurrence at fixed temperature, therefore
the expectation values in the concurrence formula Eq. (2) are
evaluated as
〈O〉 =
〈n|O|n〉e−βEn , (3)
whereZ =
−βEn is the partition function, β = 1/T , and
{|n〉} is a complete set of states of the system. Note that due
to the equilibrium and symmetries of the system, several spin-
spin correlation functions vanish, 〈S+
〉 = 0, for example.
Figure 2: (Color online) (a) Zero-temperature concurrence CAB as a function of J
′/J and B/J . Different regimes are characterized by
particular ground state functions |φn〉 defined in Appendix A. (b) T/J = 0.1 results for CAB. (c) Next nearest concurrence CAC for T = 0,
and (d) for T/J = 0.1. Dashed lines separate CAB(C) > 0 from CAB(C) = 0.
In vanishing magnetic field, where the SU(2) symmetry is
restored, the concurrence formula Eq. (2) simplifies further
and is completely determined by only one38 spin invariant
〈SA · SB〉,
CAB = max(0,−2〈SA · SB〉 −
). (4)
The concurrence may be expected to be significant whenever
enhanced spin-spin correlations indicate A-B singlet forma-
tion.
B. Analytical results
There are several known results related to the model Eq. (1).
In the special case of J ′ = 0, for example, the tetramer con-
sists of two decoupled spin dimers with concurrence CAB
(or the corresponding thermal entanglement) as derived in
Refs. 20,21. Entanglement of a qubit pair described by the
related XXZ Heisenberg model with Dzyaloshinskii-Moriya
anisotropic interaction can be also obtained analytically22.
Hamiltonian H4 with additional four-spin exchange interac-
tion but in the absence of magnetic field was considered re-
cently in the various limiting cases39.
Tetramer model Eq. (1) considered here is exactly solvable
and in Appendix A we present the corresponding eigenvec-
tors and eigenenergies. The concurrence CAB is for this case
determined from Eq. (2) with
〉 = 1
− e3j/2/2− ej/2
eb + e−b
+ e−j/2+4j
/6 + e−j/2+2j
eb + e−b
+ e−j/2−2j
eb/4 + 1/3 + e−b/4
, (5)
where j = βJ , j′ = βJ ′, b = βB, and with
〈P ↑↓
〉 = 1
+ e−j/2+4j
+ e−j/2+2j
1 + e±b
+ e−j/2−2j
1/6 + e±b/2 + e±2b
. (6)
Z = e3j/2 + 2ej/2
eb + 1 + e−b
+ e−j/2+4j
+ e−j/2+2j
eb + 1 + e−b
+ e−j/2−2j
e2b + eb + 1 + e−b + e−2b
is the partition function.
Alternatively, one can define and analyze also the entangle-
ment between spins at sites A and C and the corresponding
concurrence CAC can be expressed from Eq. (2) by applying
Figure 3: (Color online) (a) Temperature and magnetic field depen-
dence of CAB for J
′/J = 0.4 and (b) J ′ = J . Dashed lines separate
CAB > 0 from CAB = 0.
additional correlators with replaced B→C,
− e−j/2+4j
− e−j/2+2j
eb + e−b
+ e−j/2−2j
eb/4 + 1/3 + e−b/4
, (8)
〈P ↑↓
〉 = 1
e3j/2/4 + ej/2
1/2 + e±b
+ e−j/2+4j
/12 + e−j/2+2j
e±b/2
+ e−j/2−2j
1/6 + e±b/2 + e±2b
. (9)
The line 2J ′ = J represents a particularly interesting spe-
cial case where two dimers are coupled symmetrically form-
ing a regular tetrahedron. An important property of this sys-
tem is the (geometrical) frustration of, e.g., qubits C-A-B.
Such a frustration is the driving force of the quantum phase
transition found in the Shastry-Sutherland model and is the
reason for similarity of the results for two coupled dimers and
a large planar lattice studied in the next Section.
C. Examples
In the low temperature limit the concurrence is determined
by the ground state properties while transitions between var-
ious regimes are determined solely by crossings of eigenen-
ergies, which depend on two parameters (J ′/J,B/J). There
are 5 distinct regimes for CAB shown in Fig. 2(a): (i) com-
pletely entangled dimers (singlets A-B and C-D, state |φ1〉
from Appendix A), CAB = 1; (ii) for B > J and smaller
J ′/J the concurrence is zero because the energy of the state
consisted of a product of fully polarized A-B and C-D triplets,
|φ12〉, is the lowest energy in this regime; (iii) concurrence
is zero also for J ′ > J/2 and low B/J , with the ground
state |φ2〉. There are two regimes corresponding to 12 step in
CAB where the ground state is either (iv) any linear combina-
tion of degenerate states |φ6,7〉, i.e., simultaneous A-B singlet
(triplet) and C-D triplet (singlet) for J ′ < J/2, or (v) state
|φ5〉 at J ′ > J/2 and larger B. Qubits A-C are due to special
topology never fully entangled, and the corresponding CAC is
presented in Fig. 2(c). In the limit of J ′ ≫ J the tetramer
corresponds to a Heisenberg model ring consisted of 4 spins
and in this case qubit A is due to tetramer symmetry equally
entangled to both neighbors (C and D), thus CAC =
At elevated temperatures the concurrence is smeared out as
shown in Figs. 2(b,d). Note the dip separating the two dif-
ferent regimes with CAB =
, seen also in the CAC =
case. This dip clearly separates different regimes discussed in
the previous T = 0 limit and signals a proximity of a disen-
tangled excited state. For sufficiently high temperatures van-
ishing concurrence is expected23. The critical temperature Tc
denoted by a dashed line is set by the magnetic exchange scale
J , since at higher temperatures local singlets are broken irre-
spectively of the magnetic field.
A rather unexpected result is shown in Fig. 3(a) where at
B & 2J and low temperatures the concurrence slightly in-
creases with increasing temperature due to the contribution of
excited A-B singlet components that are absent in the ground
state. Similar behavior is found for J ′ ∼ 0 around B ∼ J ,
which is equivalent to the case of a single qubit dimer20,21
(not shown here). There is no distinctive feature in tempera-
ture and magnetic field dependence of CAB when J
′ > J/2
and a typical results is shown in Fig. 3(b) for J ′ > J .
III. PLANAR ARRAY OF QUBIT PAIRS: THE
SHASTRY-SUTHERLAND LATTICE
A. Preliminaries
The central point of this paper is the analysis of pair en-
tanglement for the case of a larger number of coupled qubit
pairs. In the following it will be shown that the results corre-
sponding to tetramers considered in the previous Section can
be very helpful for better understanding pair-entanglement of
N > 4 qubits. There are several possible generalizations of
coupled dimers and one of the simplest in two dimensions is
the Shastry-Sutherland lattice shown in Fig. 1(b). Neighbor-
ing sites A-B are connected with exchange interaction J and
next-neighbors with J ′. The corresponding hamiltonian for
N/2 dimers (N sites) is given with
HN = J
Si · Sj + J ′
Si · Sj −B
Szi . (10)
Periodic boundary conditions are used. For the special case
N = 4 the model reduces to Eq. (1) where due to periodic
boundary conditions sites A-C (and other equivalent pairs) are
doubly connected, therefore a factor of 2 in Eq. (1), as men-
tioned in Sec. II.
The Shastry-Sutherland model (SSM) was initially pro-
posed as a toy model possessing an exact dimerized eigenstate
known as a valence bond crystal26. Recently, the model has
experienced a sudden revival of interest by the discovery of
the two-dimensional spin-liquid compound SrCu2(BO3)2
40,41
since it is believed that magnetic properties of this compound
are reasonably well described by the SSM42. In fact, several
generalizations of the SSM have been introduced to account
better for recent high-resolution measurements revealing the
magnetic fine structure of SrCu2(BO3)2
42,43,44,45. Soon af-
ter the discovery of the SrCu2(BO3)2 system, the SSM thus
became a focal point of theoretical investigations in the field
of frustrated AFM spin systems, particularly low-dimensional
quantum spin systems where quantum fluctuations lead to
magnetically disordered ground states (spin liquids) with a
spin gap in the excitation spectrum.
The SSM is a two-dimensional frustrated antiferromagnet
with a unique spin-rotation invariant exchange topology that
leads in the limit J ≫ J ′ to an exact gapped dimerized ground
state with localized spin singlets on the dimer bonds (dimer
phase). In the opposite limit, J ≪ J ′, the model becomes
ordinary AFM Heisenberg model with a long-range Néel or-
der and a gapless spectrum (Néel phase). While two of the
phases are known, there are still open questions regarding the
existence and the nature of the intermediate phases. Several
possible scenarios have been proposed, e.g.: either a direct
transition between the two states occurs at the quantum criti-
cal point near J ′/J ∼ 0.746,47, or a transition via an interme-
diate phase that exists somewhere in the range of J ′/J > 0.6
and J ′/J < 0.948. Although different theoretical approaches
have been applied, a true nature of the intermediate phase (if
any) has still not been settled. As will be evident later on, our
exact-diagonalization results support the first scenario.
The SSM phase diagram reveals interesting behavior also
for varying external magnetic field. In particular, experiments
on SrCu2(BO3)2 in strong magnetic fields show formation of
magnetization plateaus41,49, which are believed to be a con-
sequence of repulsive interaction between almost localized
spin triplets. Several theoretical approaches support the idea
that most of these plateaus are readily explained within the
(bare) SSM46,50,51. Recent variational treatment based on en-
tangled spin pairs revealed new insight into various phases of
the SSM48.
Although extensively studied, the zero-temperature phase
diagram of the SSM remains elusive. This lack of reliable
solutions is even more pronounced when considering thermal
fluctuations in SSM as only few methods allow for the inclu-
0 0.2 0.4 0.6 0.8 1
T/J=0.1-2 <S
> - 1/2.
-2 <S
> - 1/2
AFM limit
dimer limit
Figure 4: (Color online) Results for the Shastry-Sutherland lattice
with N = 20 sites and periodic boundary conditions. Presented are
renormalized spin-spin correlation functions −2〈SA · SB,C〉 − 12 as
a function of J ′/J and for various temperatures. Asterisk indicates
critical J ′c which roughly separates the dimer and Néel phase.
sion of finite temperatures in frustrated spin systems. In this
respect, the calculation of thermal entanglement between the
spin pairs would also provide a new insight into the complex-
ity of the SSM.
B. Numerical method
We use the low-temperature Lanczos method52 (LTLM), an
extension of the finite-temperature Lanczos method53 (FTLM)
for the calculation of static correlation functions at low tem-
peratures. Both methods are nonperturbative, based on the
Lanczos procedure of exact diagonalization and random sam-
pling over different initial wave functions. A main advan-
tage of LTLM is that it accurately connects zero- and finite-
temperature regimes with rather small numerical effort in
comparison to FTLM. On the other hand, while FTLM is lim-
ited in reaching arbitrary low temperatures on finite systems,
it proves to be computationally more efficient at higher tem-
peratures. A combination of both methods therefore provides
reliable results in a wide temperature regime with moderate
computational effort. We note that FTLM was in the past suc-
cessfully used in obtaining thermodynamic as well as dynamic
properties of different models with correlated electrons as are:
the t-J model,53 the Hubbard model,54 as well as the SSM
model.43,45
In comparison with the conventional Quantum Monte Carlo
(QMC) methods LTLM possesses the following advantages:
(i) it does not suffer from the minus-sign problem that usually
hampers QMC calculations of many-electron as well as frus-
trated spin systems, (ii) the method continuously connects the
zero- and finite-temperature regimes, (iii) it incorporates as
well as takes the advantage of the symmetries of the prob-
lem, and (iv) it yields results of dynamic properties in the
real time in contrast to QMC calculations where imaginary-
time Green’s function is obtained. The LTLM (FTLM) is on
the other hand limited to small lattices which usually leads
to sizable finite-size effects. To account for these, we ap-
Figure 5: (Color online) (a) Zero-temperature concurrence CAB for a 20-site cluster for various J
′/J and B/J . Shaded area represents the
regime of fully entangled dimers, CAB = 1. (b) The corresponding results for T/J = 0.1. (c) Next nearest concurrence CAC for T = 0, and
(d) for T/J = 0.1. Note qualitative and even quantitative similarity with the tetramer results, Fig. 2. Dashed lines separate CAB(C) > 0 from
CAB(C) = 0.
plied LTLM to different square lattices with N = 8, 16 and
20 sites using periodic boundary conditions (we note that
next-larger system, N = 32, was too large to be handled
numerically). Another drawback of the LTLM (FTLM) is
the difficulty of the Lanczos procedure to resolve degener-
ate eigenstates that emerge also in the SSM. In practice, this
manifests itself in severe statistical fluctuations of the cal-
culated amplitude for T → 0 since in this regime only a
few (degenerate) eigenstates contribute to thermal average.
The simplest way to overcome this is to take a larger num-
ber of random samples R ≫ 1, which, however, requires
a longer CPU time. We have, in this regard, also included
a small portion of anisotropy in the SSM (in the form of
the anisotropic interdimer Dzyaloshinskii-Moriya interaction
{AC}(S
C − S
C), D
z/J ∼ 0.01), which slightly
splits the doubly degenerate single-triplet levels. In this way,
R ∼ 30 per Sz sector was enough for all calculated curves
to converge within ∼ 1% for T/J < 1. Here, the number
of Lanczos iterations M = 100 was used along with the full
reorthogonalization of Lanczos vectors at each step.
C. Entanglement
Entanglement in the absence of magnetic field is most
prominently reflected in spin-spin correlation functions, e.g.,
〈SA ·SB〉 and 〈SA ·SC〉. In zero temperature limit due to quan-
tum phase transition at J ′c these correlations change sign. In
Fig. 4 are presented renormalized spin-spin correlation func-
tions (for positive values identical to concurrence) as a func-
tion of J ′/J : (i) CAB > 0 in dimer phase and (ii) CAC > 0
in the Néel phase. Critical J ′c is indicated by asterisk. The re-
sults for N = 16 are qualitatively and quantitatively similar to
the N = 20 case presented here. At finite temperatures spin
correlations are smeared out as shown in Fig. 4 for various T .
Limiting Heisenberg case, J ′ → ∞, is discussed in more de-
tail in the next Section. J ′ = 0 case corresponds to the single
dimer limit21 and Sec. II.
Complete phase diagram of the SSM at T = 0 but with fi-
nite magnetic field can be classified in terms of concurrence
instead of spin correlations. In Fig. 5(a)CAB is presented as
a function of (J ′/J,B/J) as in the case of a single tetramer,
Fig. 2(a). Presented results correspond to the N = 20 case,
while N = 16 system exhibits very similar structure (not
shown here). N = 8 and N = 4 cases are qualitatively sim-
ilar, the main difference being the value of critical J ′c which
increases with N . Remarkable similarity between all these
cases can be interpreted by local physics in the regime of fi-
nite spin gap, J ′ < J ′c. Qubit pairs are there completely entan-
gled, CAB = 1, and CAB ∼ 12 for magnetic field larger than
the spin gap, but B < J+2J ′. For even larger B concurrence
approaches zero, similar to the N = 4 case. Concurrence is
Figure 6: (Color online) Temperature and magnetic field dependence
of CAB for J
′/J = 0.4 and N = 20. Note the similarity with
the corresponding tetramer results, Fig. 3(a). Dashed lines separate
CAB > 0 from CAB = 0.
zero also for J ′ > J ′c, except along the B ∼ 4J ′ line where
weak finite concurrence could be the finite size effect. Similar
results are found also for N = 16, 8 cases, and are most pro-
nounced in the N = 4 case. At finite temperature the structure
of concurrence is smeared out [Fig. 5(b)] similar to Fig. 2(b).
Concurrence CAC corresponding to next-nearest neighbors
is, complementary to CAB, increased in the Néel phase of the
diagram, Fig. 5(c). The similarity with N = 4, Fig. 2(c) is
somewhat surprising because in this regime long-range cor-
relations corresponding to the gapless spectrum of AFM-like
physics are expected to change also short range correlations.
The only quantitative difference compared to N = 4 is the
maximum value of CAC ∼ 0.3 instead of 0.5 (beside the crit-
ical value J ′c discussed in the previous paragraph). Concur-
rence is very small for B > J + 2J ′. At finite temperatures
fine fluctuations in the concurrence structure are smeared out,
Fig. 5(d).
Temperature and magnetic field dependence of CAB in the
dimer phase is presented in Fig. 6 for fixed J ′/J = 0.4. Sim-
ilarity with the corresponding N = 4 tetramer case, Fig. 3(a),
is astonishing and is again the consequence of local physics in
the presence of a finite spin gap. Finite size effects (in com-
parison with N = 16 and N = 8 cases) are very small (not
shown). Dashed line represents the borderline of the CAB = 0
region: critical Tc ≈ 0.75J valid for B/J . 3, that is in this
regime nearly independent of B, is slightly larger than in the
single tetramer case where its insensitivity to B is even more
pronounced.
IV. HEISENBERG LIMIT
The concurrence corresponding to next-nearest neighbors
in SSM, CAC, is non zero in the Néel phase for J
′ > J ′c.
Typical result for concurrence in this regime (for fixed J ′/J =
1) in terms of temperature and magnetic field is presented in
Fig. 7(a). At zero temperature the concurrence is zero for B >
4J ′ [compare with Fig. 2(c) and Fig. 5(c)].
Figure 7: (Color online) (a) Next nearest neighbor concurrence CAC
for J ′ = J . (b) Heisenberg lattice result as a special case of the
SSM, J = 0. Shaded region represents CAC = 0. In the line shaded
region (low finite temperature and large magnetic field) our numer-
ical results set only the upper limit CAC < 5 · 10
−4. Dashed lines
separate CAC > 0 from CAC = 0.
In the limit J = 0 the model simplifies to the AFM Heisen-
berg model on a square lattice of N sites,
HAC = J
Si · Sj −B
Szi . (11)
Several results for this model have already been presented for
very small clusters55,56,57, however the temperature and mag-
netic field dependence of the concurrence for systems with
sufficiently large number of states and approaching thermo-
dynamic limit has not been presented so far.
In Fig. 7(b) we further presented temperature and magnetic
field dependence of concurrence for the Heisenberg model for
N = 20 (results for N = 16 are quantitatively similar, but not
shown here). Temperature and magnetic field dependence of
CAC exhibits peculiar semi-island shape where at fixed value
of B the concurrence increases with increasing temperature.
This effect is to some extent seen in all cases and is the con-
sequence of exciting local singlet states, which do not appear
in the ground state. At T → 0 finite steps with increasing B
correspond to gradual transition from the singlet ground state
to totally polarized state with total spin S = 10 and vanish-
ing concurrence. This is in more detail presented in Fig. 8(a)
for various N = 4, 8, 16, 20. At B = 0 and for N = 20
we get CAC = 0.19. It is interesting to compare this results
0 0.05 0.1
/2N=8
1/6+2N
-3/2C
Figure 8: (Color online) (a) Zero-temperature concurrence CAC in
the Heisenberg limit as a function of B/J ′ and for various N = 4,
8, 16, 20. Sections with different total spin values are addition-
ally labeled. (b) Finite-size scaling of concurrence in the absence of
magnetic field. Full line represents the fit corresponding to Ref. 58,
CAC ≈
+ 2N−3/2.
with the known finite-size analysis scaling for the ground state
energy of the Heisenberg model58. The same scaling gives
CAC ≈ 16 + 2N
−3/2. Our finite-size scaling, Fig. 8(b), is in
perfect agreement with this result for N → ∞ at T = 0 and
B = 0.
In the opposite limit of high magnetic fields, the vanishing
concurrence CAC = 0, is observed for B above the critical
value Bc = 4J
′ for all system sizes shown in Fig. 8(a). This
result can be deduced also analytically. Since in a fully po-
larized state CAC = 0, this Bc actually denotes a transition
from S1 = N/2 − 1 to S0 = N/2 ferromagnetic ground
state with energy E0 = N(J
′ − B)/2. The energy of the
one-magnon excitation above the ferromagnetic ground state
is given by the spin wave theory, which is in this case exact, as
E1 = E0 −J ′(2− cos kxa− cos kya)+B, where (kx, ky) is
the magnon wave vector and a denotes the lattice spacing. Ev-
idently, a transition to a fully polarized state occurs precisely
at Bc = 4J
′ at (π/a, π/a) point in the one-magnon Brillouin
zone.
V. SUMMARY
The aim of this paper was to analyze and understand
how concurrence (and related entanglement) of qubit pairs
(dimers) is affected by their mutual magnetic interactions. In
particular, we were interested in a planar array of qubit dimers
described by the Shastry-Sutherland model. This model is
suitable due to very robust ground state composed of entan-
gled qubit pairs which breaks down by increasing the in-
terdimer coupling. It is interesting to study both, the en-
tanglement between nearest and between next-nearest spins
(qubits) at finite temperature and magnetic field. The results
are based on numerical calculations using low-temperature
Lanczos methods on lattices of 4, 8, 16 and 20 sites with pe-
riodic boundary conditions.
A comprehensive analysis of concurrence for various pa-
rameters revealed two general conclusions:
(1) For a weak coupling between qubit dimers, J ′ < J ′c,
qubit pairs are locally entangled in accordance with the local
nature of the dimer phase. This is due to a finite singlet-triplet
gap (spin gap) in the excitation spectrum that is a consequence
of strong geometrical frustration in magnetic couplings. The
regime of fully entangled neighbors perfectly coincides with
the regime of finite spin gap as presented in Fig. 9. Calcu-
lated lines for various system sizes N in Fig. 9(a) denote re-
gions (shaded for N = 20) in the (J ′/J,B/J) plane where
CAB = 1 at T = 0. In the lower panel [Fig. 9(b)] the lines
represent the energy gap E1 − EGS between the first excited
state with energy E1 with total spin projection S
z = 1 and
the ground state with energy EGS and total spin projection
Sz = 0, calculated for B = 0. For J ′ < J ′c (full lines)
E1 − EGS corresponds to the value of the spin gap. With an
increasing magnetic field the spin gap closes (shaded region
for N = 20) and eventually vanishes at the CAB = 1 border
line. Shaded regions in Figs. 9(a),(b) therefore coincide. Note
also that the results for N = 16 and 20 sites differ mainly in
J ′c.
As a consequence of finite spin gap and local character of
correlations it is an interesting observation that even N = 4
results as a function of temperature and magnetic field quali-
tatively correctly reproduce N = 20 results in the regime of
J ′ < J ′c. The main quantitative difference is in a renormal-
ized value of J ′c = J/2 for N = 4, as is evident from the
comparison of Figs.2,3 and Figs.5,6. This similarity of the re-
sults appears very useful due to the fact that concurrence for
tetrahedron-like systems (N = 4) is given analytically (Sec.
(2) In the opposite, strong interdimer coupling regime, J ′ >
J ′c, the excitation spectrum is gapless and the concurrence be-
tween next-nearest qubits, CAC, exhibits a similar behavior as
in the antiferromagnetic Heisenberg model J/J ′ → 0. Our
B = 0 results coincide with the known result extrapolated to
the thermodynamic limit CAC ≈ 16 . In finite magnetic field
and T = 0 the concurrence vanishes at Bc = 4J
′ when the
system becomes fully polarized (ground state with the total
spin S = N/2). However, at elevated temperatures the con-
currence increases due to excited singlet states and eventually
drops to zero at temperatures above Tc ≈ J ′.
We can conclude with the observation that our analysis
of concurrence and related entanglement between qubit pairs
was also found to be a very useful measure for classify-
ing various phases of the Shastry-Sutherland model. As our
numerical method is based on relatively small clusters, we
were unable to unambiguously determine possible interme-
diate phases of the model in the regime J ′ ∼ J ′c, but we be-
lieve that concurrence will prove to be a useful probe for the
classification of various phases also in this regime using alter-
native approaches. However, we were able to sweep through
all other dominant regimes of the parameters including finite
temperature and magnetic field.
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
singlet - triplet gap
Figure 9: (Color online) (a) Zero-temperature CAB = 1 region in the
plane (J ′/J, B/J) for various N . (b) The corresponding spin gap
at B = 0 (the energy of the lowest total Sz = 1 state relative to the
ground state energy).
VI. ACKNOWLEDGMENTS
The authors acknowledge J. Mravlje for useful discussions
and the support from the Slovenian Research Agency under
Contract No. P1-0044.
Appendix A: EIGENENERGIES AND EIGENVECTORS FOR
PERIODICALLY COUPLED TWO QUBIT DIMERS
Consider two qubit dimers coupled into a tetramer and de-
scribed with the Hamiltonian Eq. (1) and Fig. 1(a). The model
is exactly solvable in the separate {S, Sz} subspaces corre-
sponding to different values of the total spin S and its z com-
ponent Sz . Following the abbreviations for singlet and triplet
states on nearest-neighbor (dimer) sites i and j,
|sij〉 =
|↑i↓j − ↓i↑j〉,
|t0ij〉 =
|↑i↓j + ↓i↑j〉,
|t+ij〉 = |↑i↑j〉,
|t−ij〉 = |↓i↓j〉, (A1)
the resulting eigenstates |φk〉 and eigenenergies Ek corre-
sponding to the hamiltonian Eq. (1) are:
S = 0, Sz = 0 :
|φ1〉 = |sAB〉|sCD〉,
E1 = −3J/2, (A2)
|φ2〉 =
− |t0AB〉|t0CD〉+ |t+AB〉|t
〉+ |t−
E2 = J/2− 4J ′. (A3)
S = 1, Sz = −1 :
|φ3〉 = |sAB〉|t−CD〉,
|φ4〉 = |t−AB〉|sCD〉,
E3,4 = −J/2−B, (A4)
|φ5〉 =
|t0AB〉|t
〉 − |t−
〉|t0CD〉
E5 = J/2− 2J ′ −B. (A5)
S = 1, Sz = 0 :
|φ6〉 = |sAB〉|t0CD〉,
|φ7〉 = |t0AB〉|sCD〉,
E6,7 = −J/2, (A6)
|φ8〉 =
〉 − |t−
E8 = J/2− 2J ′. (A7)
S = 1, Sz = 1 :
|φ9〉 = |sAB〉|t+CD〉,
|φ10〉 = |t+AB〉|sCD〉,
E9,10 = −J/2 +B, (A8)
|φ11〉 =
− |t0AB〉|t+CD〉+ |t
〉|t0CD〉
E11 = J/2− 2J ′ +B. (A9)
S = 2, Sz = −2 :
|φ12〉 = |t−AB〉|t
E12 = J/2 + 2J
′ − 2B. (A10)
S = 2, Sz = −1 :
|φ13〉 =
|t0AB〉|t−CD〉+ |t
〉|t0CD〉
E13 = J/2 + 2J
′ −B. (A11)
S = 2, Sz = 0 :
|φ14〉 =
2|t0AB〉|t0CD〉+ |t+AB〉|t
〉+ |t−
E14 = J/2 + 2J
′. (A12)
S = 2, Sz = 1 :
|φ15〉 =
|t0AB〉|t+CD〉+ |t
〉|t0CD〉
E15 = J/2 + 2J
′ +B. (A13)
S = 2, Sz = 2 :
|φ16〉 = |t+AB〉|t
E16 = J/2 + 2J
′ + 2B. (A14)
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http://arxiv.org/abs/cond-mat/9612126
http://arxiv.org/abs/cond-mat/0603444
http://arxiv.org/abs/quant-ph/0011036

  We show that temperature and magnetic field properties of the entanglement
between spins on the two-dimensional Shastry-Sutherland lattice can be
qualitatively described by analytical results for a qubit tetramer. Exact
diagonalization of clusters with up to 20 sites reveals that the regime of
fully entangled neighboring pairs coincides with the regime of finite spin gap
in the spectrum. Additionally, the results for the regime of vanishing spin gap
are discussed and related to the Heisenberg limit of the model.

Thermal entanglement of qubit pairs on the Shastry-Sutherland lattice
S. El Shawish
J. Stefan Institute, Ljubljana, Slovenia
A. Ramšak and J. Bonča
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia and
J. Stefan Institute, Ljubljana, Slovenia
(Dated: 2 April 2007)
We show that temperature and magnetic field properties of the entanglement between spins on the two-
dimensional Shastry-Sutherland lattice can be qualitatively described by analytical results for a qubit tetramer.
Exact diagonalization of clusters with up to 20 sites reveals that the regime of fully entangled neighboring pairs
coincides with the regime of finite spin gap in the spectrum. Additionally, the results for the regime of vanishing
spin gap are discussed and related to the Heisenberg limit of the model.
PACS numbers: 75.10.Jm, 03.65.Yz, 03.67.Mn
I. INTRODUCTION
In any physical system with subsystems in interaction, indi-
vidual parts of the system are to some extent entangled, even
if they are far apart, as realized already at the beginning of
modern quantum mechanics sixty years ago. Today it has be-
come appreciated that the ability to establish entanglement be-
tween quantum particles in a controlled manner is a crucial in-
gredient of any quantum information processing system1. On
the other hand, it turned out that the analysis of appropriately
quantified entanglement between parts of the system can also
be a very useful tool in the study of many body phenomena,
as is, e.g., the behavior of correlated systems in the vicinity of
crossovers between various regimes or even points of quantum
phase transition2.
Quantum entanglement of two distinguishable particles in a
pure state can be quantified through von Neuman entropy3,4,5.
Entanglement between two spin- 1
particles – qubit pair – can
be considered a physical resource, an essential ingredient of
algorithms suitable for quantum computation. For a pair of
subsystems A and B, each occupied by a single electron, an
appropriate entanglement measure is the entanglement of for-
mation, which can be quantified from the Wootters formula6.
In general, electron-qubits have the potential for even richer
variety of entanglement measure choices due to both their
charge and spin degrees of freedom. When entanglement is
quantified in systems of indistinguishable particles, the mea-
sure must account for the effect of exchange and it must ade-
quately deal with multiple occupancy states7,8,9,10,11,12. A typ-
ical example is the analysis of entanglement in lattice fermion
models (the Hubbard model, e.g.) where double occupancy
plays an essential role11.
In realistic hardware designed for quantum information
processing, several criteria for qubits must be fulfilled13: the
existence of multiple identifiable qubits, the ability to initial-
ize and manipulate qubits, small decoherence, and the ability
to measure qubits, i.e., to determine the outcome of compu-
tation. It seems that among several proposals for experimen-
tal realizations of such quantum information processing sys-
tems the criteria for scalable qubits can be met in solid state
structures consisting of coupled quantum dots14,15. Due to the
ability to precisely control the number of electrons in such
structures16, the entanglement has become experimentally ac-
cessible quantity. In particular, recent experiments on semi-
conductor double quantum dot devices have shown the evi-
dence of spin entangled states in GaAs based heterostuctures17
and it was shown that vertical-lateral double quantum dots
may be useful for achieving two-electron spin entanglement18.
It was also demonstrated recently that in double quantum dot
systems coherent qubit manipulation and projective readout is
possible19.
Qubit pairs to be used for quantum information processing
must be to a high degree isolated from their environment, oth-
erwise small decoherence requirement from the DiVincenzo’s
checklist can not be fulfilled. The entanglement, e.g., between
two antiferromagnetically coupled spins in contact with ther-
mal bath, is decreased at elevated temperatures and external
magnetic field20,21,22, and will inevitably vanish at some fi-
nite temperature23. Entanglement of a pair of electrons that
are confined in a double quantum dot is collapsed due to the
Kondo effect at low temperatures and for a very weak tunnel-
ing to the leads. At temperatures below the Kondo temper-
ature a spin-singlet state is formed between a confined elec-
tron and conduction electrons in the leads24. For other open
systems there are many possible sources of decoherence or
phase-breaking, for example coupling to phonon degrees of
freedom25.
The main purpose of the present paper is to analyze the ro-
bustness of the entanglement of spin qubit pairs in a planar
lattice of spins (qubits) with respect to frustration in magnetic
couplings, elevated temperatures as well as due to increasing
external magnetic field. The paper is organized as follows.
Sec. II introduces the model for two coupled qubit pairs –
qubit tetramer – and presents exact results for temperature and
magnetic field dependence of the entanglement between near-
est and next-nearest-neighboring spins in a tetrahedron topol-
ogy. In Sec. III the model is extended to infinite lattice of
qubit pairs described by the Shastry-Sutherland model26. This
model is convenient firstly, because of the existence of sta-
ble spin-singlet pairs in the ground state in the limit of weak
coupling between the qubit pairs, and secondly, due to a rel-
atively good understanding of the physics of the model in the
http://arxiv.org/abs/0704.0172v1
thermodynamic limit. Entanglement properties of the Shastry-
Sutherland model were so far not considered quantitatively.
Neverteless, several results concerning the role of entangle-
ment at a phase transition in other low-dimensional spin lattice
systems2,27,28,29,30,31,32, as well as in fermionic systems33,34,35
have been reported recently. Near a quantum phase transi-
tion in some cases entanglement even proves to be more ef-
ficient precursor of the transition compared to standard spin-
spin correlations35,36. In Sec. IV we discuss entanglement be-
tween nearest neighbors in the Heisenberg model, represent-
ing a limiting case of the Shastry-Sutherland model. Results
are summarized in Sec. V and some technical details are given
in Appendix A.
II. THERMAL ENTANGLEMENT OF A QUBIT
TETRAMER IN MAGNETIC FIELD
Consider first a double quantum dot composed of two adja-
cent quantum dots weakly coupled via a controllable electron-
hopping integral. By adjusting a global back-gate voltage,
precisely two electrons can be confined to the dots. The inter-
dot tunneling matrix element t determines the effective anti-
ferromagnetic (AFM) superexchange interaction J ∼ 4t2/U ,
where U is the scale of Coulomb interaction between two
electrons confined on the same dot. There are several possi-
ble configurations of coupling between such double quantum
dots. One of the simplest specific designs is shown schemat-
ically in Fig. 1(a): four qubits at vertices of a tetrahedron. In
addition to the coupling A-B, by appropriate arrangements of
gate electrodes the tunneling between A-C and A-D can as
well be switched on.
We consider here the case where J/U ≪ 1, thus double oc-
cupancy of individual dot is negligible and appropriate Hilbert
space is spanned by two dimers (qubit pairs): spins at sites
A-B and C-D are coupled by effective AFM Heisenberg mag-
netic exchange J and at sites A-C, B-C, A-D, B-D by J ′. The
corresponding hamiltonian of such a pair of dimers is given as
H4 = J(SA · SB + SC · SD) + (1)
+ 2J ′(SA · SC + SB · SC + SA · SD + SB · SD)−
− B(SzA + SzB + SzC + SzD),
where Si =
σi is spin operator corresponding to the site i
and B is external homogeneous magnetic field in the direction
of the z-axis. Factor 2 in Eq. (1) is introduced for convenience
– such a parameterization represents the simplest case of finite
Shastry-Sutherland lattice with periodic boundary conditions
studied in Sec. III.
A. Concurrence
We focus here on the entanglement properties of two cou-
pled qubit dimers. The entanglement of a pair of spin
qubits A and B may be defined through concurrence3, C =
2|α↑↑α↓↓ − α↑↓α↓↑|, if the system is in a pure state |ΨAB〉 =
ss′ αss′ |s〉A|s′〉B, where |s〉i corresponds to the basis | ↑ 〉i,
Figure 1: (Color online) (a) Two coupled qubit pairs (dimers) in tetra-
hedral topology. (b) Shastry-Sutherland lattice as realized, e.g., in the
SrCu2(BO3)2 compound.
| ↓ 〉i. Concurrence varies from C = 0 for an unentan-
gled state (for example | ↑ 〉A| ↑ 〉B) to C = 1 for com-
pletely entangled Bell states3 1√
(| ↑ 〉A| ↑ 〉B ± | ↓ 〉A| ↓ 〉B)
or 1√
(|↑ 〉A|↓ 〉B ± |↓ 〉A|↑ 〉B).
For finite inter-pair coupling J ′ 6= 0 or at elevated tempera-
tures the A-B pair can not be described by a pure state. In the
case of mixed states describing the subsystem A-B the concur-
rence may be calculated from the reduced density matrix ρAB
given in the standard basis |s〉i|s′〉j6. Concurrence can be fur-
ther expressed in terms of spin-spin correlation functions2,27,
where for systems that are axially symmetric in the spin space
the concurrence may conveniently be given in a simple closed
form37, which for the thermal equilibrium case simplifies fur-
ther,
CAB = 2max(0, |〈S+AS
〉〈P ↓
〉). (2)
Here S+i = (S
† = Sxi + ıS
i is the spin raising operator
for dot i and P ↑i =
(1 + 2Szi ), P
(1 − 2Szi ) are the
projection operators onto the state | ↑ 〉i or | ↓ 〉i, respectively.
We consider the concurrence at fixed temperature, therefore
the expectation values in the concurrence formula Eq. (2) are
evaluated as
〈O〉 =
〈n|O|n〉e−βEn , (3)
whereZ =
−βEn is the partition function, β = 1/T , and
{|n〉} is a complete set of states of the system. Note that due
to the equilibrium and symmetries of the system, several spin-
spin correlation functions vanish, 〈S+
〉 = 0, for example.
Figure 2: (Color online) (a) Zero-temperature concurrence CAB as a function of J
′/J and B/J . Different regimes are characterized by
particular ground state functions |φn〉 defined in Appendix A. (b) T/J = 0.1 results for CAB. (c) Next nearest concurrence CAC for T = 0,
and (d) for T/J = 0.1. Dashed lines separate CAB(C) > 0 from CAB(C) = 0.
In vanishing magnetic field, where the SU(2) symmetry is
restored, the concurrence formula Eq. (2) simplifies further
and is completely determined by only one38 spin invariant
〈SA · SB〉,
CAB = max(0,−2〈SA · SB〉 −
). (4)
The concurrence may be expected to be significant whenever
enhanced spin-spin correlations indicate A-B singlet forma-
tion.
B. Analytical results
There are several known results related to the model Eq. (1).
In the special case of J ′ = 0, for example, the tetramer con-
sists of two decoupled spin dimers with concurrence CAB
(or the corresponding thermal entanglement) as derived in
Refs. 20,21. Entanglement of a qubit pair described by the
related XXZ Heisenberg model with Dzyaloshinskii-Moriya
anisotropic interaction can be also obtained analytically22.
Hamiltonian H4 with additional four-spin exchange interac-
tion but in the absence of magnetic field was considered re-
cently in the various limiting cases39.
Tetramer model Eq. (1) considered here is exactly solvable
and in Appendix A we present the corresponding eigenvec-
tors and eigenenergies. The concurrence CAB is for this case
determined from Eq. (2) with
〉 = 1
− e3j/2/2− ej/2
eb + e−b
+ e−j/2+4j
/6 + e−j/2+2j
eb + e−b
+ e−j/2−2j
eb/4 + 1/3 + e−b/4
, (5)
where j = βJ , j′ = βJ ′, b = βB, and with
〈P ↑↓
〉 = 1
+ e−j/2+4j
+ e−j/2+2j
1 + e±b
+ e−j/2−2j
1/6 + e±b/2 + e±2b
. (6)
Z = e3j/2 + 2ej/2
eb + 1 + e−b
+ e−j/2+4j
+ e−j/2+2j
eb + 1 + e−b
+ e−j/2−2j
e2b + eb + 1 + e−b + e−2b
is the partition function.
Alternatively, one can define and analyze also the entangle-
ment between spins at sites A and C and the corresponding
concurrence CAC can be expressed from Eq. (2) by applying
Figure 3: (Color online) (a) Temperature and magnetic field depen-
dence of CAB for J
′/J = 0.4 and (b) J ′ = J . Dashed lines separate
CAB > 0 from CAB = 0.
additional correlators with replaced B→C,
− e−j/2+4j
− e−j/2+2j
eb + e−b
+ e−j/2−2j
eb/4 + 1/3 + e−b/4
, (8)
〈P ↑↓
〉 = 1
e3j/2/4 + ej/2
1/2 + e±b
+ e−j/2+4j
/12 + e−j/2+2j
e±b/2
+ e−j/2−2j
1/6 + e±b/2 + e±2b
. (9)
The line 2J ′ = J represents a particularly interesting spe-
cial case where two dimers are coupled symmetrically form-
ing a regular tetrahedron. An important property of this sys-
tem is the (geometrical) frustration of, e.g., qubits C-A-B.
Such a frustration is the driving force of the quantum phase
transition found in the Shastry-Sutherland model and is the
reason for similarity of the results for two coupled dimers and
a large planar lattice studied in the next Section.
C. Examples
In the low temperature limit the concurrence is determined
by the ground state properties while transitions between var-
ious regimes are determined solely by crossings of eigenen-
ergies, which depend on two parameters (J ′/J,B/J). There
are 5 distinct regimes for CAB shown in Fig. 2(a): (i) com-
pletely entangled dimers (singlets A-B and C-D, state |φ1〉
from Appendix A), CAB = 1; (ii) for B > J and smaller
J ′/J the concurrence is zero because the energy of the state
consisted of a product of fully polarized A-B and C-D triplets,
|φ12〉, is the lowest energy in this regime; (iii) concurrence
is zero also for J ′ > J/2 and low B/J , with the ground
state |φ2〉. There are two regimes corresponding to 12 step in
CAB where the ground state is either (iv) any linear combina-
tion of degenerate states |φ6,7〉, i.e., simultaneous A-B singlet
(triplet) and C-D triplet (singlet) for J ′ < J/2, or (v) state
|φ5〉 at J ′ > J/2 and larger B. Qubits A-C are due to special
topology never fully entangled, and the corresponding CAC is
presented in Fig. 2(c). In the limit of J ′ ≫ J the tetramer
corresponds to a Heisenberg model ring consisted of 4 spins
and in this case qubit A is due to tetramer symmetry equally
entangled to both neighbors (C and D), thus CAC =
At elevated temperatures the concurrence is smeared out as
shown in Figs. 2(b,d). Note the dip separating the two dif-
ferent regimes with CAB =
, seen also in the CAC =
case. This dip clearly separates different regimes discussed in
the previous T = 0 limit and signals a proximity of a disen-
tangled excited state. For sufficiently high temperatures van-
ishing concurrence is expected23. The critical temperature Tc
denoted by a dashed line is set by the magnetic exchange scale
J , since at higher temperatures local singlets are broken irre-
spectively of the magnetic field.
A rather unexpected result is shown in Fig. 3(a) where at
B & 2J and low temperatures the concurrence slightly in-
creases with increasing temperature due to the contribution of
excited A-B singlet components that are absent in the ground
state. Similar behavior is found for J ′ ∼ 0 around B ∼ J ,
which is equivalent to the case of a single qubit dimer20,21
(not shown here). There is no distinctive feature in tempera-
ture and magnetic field dependence of CAB when J
′ > J/2
and a typical results is shown in Fig. 3(b) for J ′ > J .
III. PLANAR ARRAY OF QUBIT PAIRS: THE
SHASTRY-SUTHERLAND LATTICE
A. Preliminaries
The central point of this paper is the analysis of pair en-
tanglement for the case of a larger number of coupled qubit
pairs. In the following it will be shown that the results corre-
sponding to tetramers considered in the previous Section can
be very helpful for better understanding pair-entanglement of
N > 4 qubits. There are several possible generalizations of
coupled dimers and one of the simplest in two dimensions is
the Shastry-Sutherland lattice shown in Fig. 1(b). Neighbor-
ing sites A-B are connected with exchange interaction J and
next-neighbors with J ′. The corresponding hamiltonian for
N/2 dimers (N sites) is given with
HN = J
Si · Sj + J ′
Si · Sj −B
Szi . (10)
Periodic boundary conditions are used. For the special case
N = 4 the model reduces to Eq. (1) where due to periodic
boundary conditions sites A-C (and other equivalent pairs) are
doubly connected, therefore a factor of 2 in Eq. (1), as men-
tioned in Sec. II.
The Shastry-Sutherland model (SSM) was initially pro-
posed as a toy model possessing an exact dimerized eigenstate
known as a valence bond crystal26. Recently, the model has
experienced a sudden revival of interest by the discovery of
the two-dimensional spin-liquid compound SrCu2(BO3)2
40,41
since it is believed that magnetic properties of this compound
are reasonably well described by the SSM42. In fact, several
generalizations of the SSM have been introduced to account
better for recent high-resolution measurements revealing the
magnetic fine structure of SrCu2(BO3)2
42,43,44,45. Soon af-
ter the discovery of the SrCu2(BO3)2 system, the SSM thus
became a focal point of theoretical investigations in the field
of frustrated AFM spin systems, particularly low-dimensional
quantum spin systems where quantum fluctuations lead to
magnetically disordered ground states (spin liquids) with a
spin gap in the excitation spectrum.
The SSM is a two-dimensional frustrated antiferromagnet
with a unique spin-rotation invariant exchange topology that
leads in the limit J ≫ J ′ to an exact gapped dimerized ground
state with localized spin singlets on the dimer bonds (dimer
phase). In the opposite limit, J ≪ J ′, the model becomes
ordinary AFM Heisenberg model with a long-range Néel or-
der and a gapless spectrum (Néel phase). While two of the
phases are known, there are still open questions regarding the
existence and the nature of the intermediate phases. Several
possible scenarios have been proposed, e.g.: either a direct
transition between the two states occurs at the quantum criti-
cal point near J ′/J ∼ 0.746,47, or a transition via an interme-
diate phase that exists somewhere in the range of J ′/J > 0.6
and J ′/J < 0.948. Although different theoretical approaches
have been applied, a true nature of the intermediate phase (if
any) has still not been settled. As will be evident later on, our
exact-diagonalization results support the first scenario.
The SSM phase diagram reveals interesting behavior also
for varying external magnetic field. In particular, experiments
on SrCu2(BO3)2 in strong magnetic fields show formation of
magnetization plateaus41,49, which are believed to be a con-
sequence of repulsive interaction between almost localized
spin triplets. Several theoretical approaches support the idea
that most of these plateaus are readily explained within the
(bare) SSM46,50,51. Recent variational treatment based on en-
tangled spin pairs revealed new insight into various phases of
the SSM48.
Although extensively studied, the zero-temperature phase
diagram of the SSM remains elusive. This lack of reliable
solutions is even more pronounced when considering thermal
fluctuations in SSM as only few methods allow for the inclu-
0 0.2 0.4 0.6 0.8 1
T/J=0.1-2 <S
> - 1/2.
-2 <S
> - 1/2
AFM limit
dimer limit
Figure 4: (Color online) Results for the Shastry-Sutherland lattice
with N = 20 sites and periodic boundary conditions. Presented are
renormalized spin-spin correlation functions −2〈SA · SB,C〉 − 12 as
a function of J ′/J and for various temperatures. Asterisk indicates
critical J ′c which roughly separates the dimer and Néel phase.
sion of finite temperatures in frustrated spin systems. In this
respect, the calculation of thermal entanglement between the
spin pairs would also provide a new insight into the complex-
ity of the SSM.
B. Numerical method
We use the low-temperature Lanczos method52 (LTLM), an
extension of the finite-temperature Lanczos method53 (FTLM)
for the calculation of static correlation functions at low tem-
peratures. Both methods are nonperturbative, based on the
Lanczos procedure of exact diagonalization and random sam-
pling over different initial wave functions. A main advan-
tage of LTLM is that it accurately connects zero- and finite-
temperature regimes with rather small numerical effort in
comparison to FTLM. On the other hand, while FTLM is lim-
ited in reaching arbitrary low temperatures on finite systems,
it proves to be computationally more efficient at higher tem-
peratures. A combination of both methods therefore provides
reliable results in a wide temperature regime with moderate
computational effort. We note that FTLM was in the past suc-
cessfully used in obtaining thermodynamic as well as dynamic
properties of different models with correlated electrons as are:
the t-J model,53 the Hubbard model,54 as well as the SSM
model.43,45
In comparison with the conventional Quantum Monte Carlo
(QMC) methods LTLM possesses the following advantages:
(i) it does not suffer from the minus-sign problem that usually
hampers QMC calculations of many-electron as well as frus-
trated spin systems, (ii) the method continuously connects the
zero- and finite-temperature regimes, (iii) it incorporates as
well as takes the advantage of the symmetries of the prob-
lem, and (iv) it yields results of dynamic properties in the
real time in contrast to QMC calculations where imaginary-
time Green’s function is obtained. The LTLM (FTLM) is on
the other hand limited to small lattices which usually leads
to sizable finite-size effects. To account for these, we ap-
Figure 5: (Color online) (a) Zero-temperature concurrence CAB for a 20-site cluster for various J
′/J and B/J . Shaded area represents the
regime of fully entangled dimers, CAB = 1. (b) The corresponding results for T/J = 0.1. (c) Next nearest concurrence CAC for T = 0, and
(d) for T/J = 0.1. Note qualitative and even quantitative similarity with the tetramer results, Fig. 2. Dashed lines separate CAB(C) > 0 from
CAB(C) = 0.
plied LTLM to different square lattices with N = 8, 16 and
20 sites using periodic boundary conditions (we note that
next-larger system, N = 32, was too large to be handled
numerically). Another drawback of the LTLM (FTLM) is
the difficulty of the Lanczos procedure to resolve degener-
ate eigenstates that emerge also in the SSM. In practice, this
manifests itself in severe statistical fluctuations of the cal-
culated amplitude for T → 0 since in this regime only a
few (degenerate) eigenstates contribute to thermal average.
The simplest way to overcome this is to take a larger num-
ber of random samples R ≫ 1, which, however, requires
a longer CPU time. We have, in this regard, also included
a small portion of anisotropy in the SSM (in the form of
the anisotropic interdimer Dzyaloshinskii-Moriya interaction
{AC}(S
C − S
C), D
z/J ∼ 0.01), which slightly
splits the doubly degenerate single-triplet levels. In this way,
R ∼ 30 per Sz sector was enough for all calculated curves
to converge within ∼ 1% for T/J < 1. Here, the number
of Lanczos iterations M = 100 was used along with the full
reorthogonalization of Lanczos vectors at each step.
C. Entanglement
Entanglement in the absence of magnetic field is most
prominently reflected in spin-spin correlation functions, e.g.,
〈SA ·SB〉 and 〈SA ·SC〉. In zero temperature limit due to quan-
tum phase transition at J ′c these correlations change sign. In
Fig. 4 are presented renormalized spin-spin correlation func-
tions (for positive values identical to concurrence) as a func-
tion of J ′/J : (i) CAB > 0 in dimer phase and (ii) CAC > 0
in the Néel phase. Critical J ′c is indicated by asterisk. The re-
sults for N = 16 are qualitatively and quantitatively similar to
the N = 20 case presented here. At finite temperatures spin
correlations are smeared out as shown in Fig. 4 for various T .
Limiting Heisenberg case, J ′ → ∞, is discussed in more de-
tail in the next Section. J ′ = 0 case corresponds to the single
dimer limit21 and Sec. II.
Complete phase diagram of the SSM at T = 0 but with fi-
nite magnetic field can be classified in terms of concurrence
instead of spin correlations. In Fig. 5(a)CAB is presented as
a function of (J ′/J,B/J) as in the case of a single tetramer,
Fig. 2(a). Presented results correspond to the N = 20 case,
while N = 16 system exhibits very similar structure (not
shown here). N = 8 and N = 4 cases are qualitatively sim-
ilar, the main difference being the value of critical J ′c which
increases with N . Remarkable similarity between all these
cases can be interpreted by local physics in the regime of fi-
nite spin gap, J ′ < J ′c. Qubit pairs are there completely entan-
gled, CAB = 1, and CAB ∼ 12 for magnetic field larger than
the spin gap, but B < J+2J ′. For even larger B concurrence
approaches zero, similar to the N = 4 case. Concurrence is
Figure 6: (Color online) Temperature and magnetic field dependence
of CAB for J
′/J = 0.4 and N = 20. Note the similarity with
the corresponding tetramer results, Fig. 3(a). Dashed lines separate
CAB > 0 from CAB = 0.
zero also for J ′ > J ′c, except along the B ∼ 4J ′ line where
weak finite concurrence could be the finite size effect. Similar
results are found also for N = 16, 8 cases, and are most pro-
nounced in the N = 4 case. At finite temperature the structure
of concurrence is smeared out [Fig. 5(b)] similar to Fig. 2(b).
Concurrence CAC corresponding to next-nearest neighbors
is, complementary to CAB, increased in the Néel phase of the
diagram, Fig. 5(c). The similarity with N = 4, Fig. 2(c) is
somewhat surprising because in this regime long-range cor-
relations corresponding to the gapless spectrum of AFM-like
physics are expected to change also short range correlations.
The only quantitative difference compared to N = 4 is the
maximum value of CAC ∼ 0.3 instead of 0.5 (beside the crit-
ical value J ′c discussed in the previous paragraph). Concur-
rence is very small for B > J + 2J ′. At finite temperatures
fine fluctuations in the concurrence structure are smeared out,
Fig. 5(d).
Temperature and magnetic field dependence of CAB in the
dimer phase is presented in Fig. 6 for fixed J ′/J = 0.4. Sim-
ilarity with the corresponding N = 4 tetramer case, Fig. 3(a),
is astonishing and is again the consequence of local physics in
the presence of a finite spin gap. Finite size effects (in com-
parison with N = 16 and N = 8 cases) are very small (not
shown). Dashed line represents the borderline of the CAB = 0
region: critical Tc ≈ 0.75J valid for B/J . 3, that is in this
regime nearly independent of B, is slightly larger than in the
single tetramer case where its insensitivity to B is even more
pronounced.
IV. HEISENBERG LIMIT
The concurrence corresponding to next-nearest neighbors
in SSM, CAC, is non zero in the Néel phase for J
′ > J ′c.
Typical result for concurrence in this regime (for fixed J ′/J =
1) in terms of temperature and magnetic field is presented in
Fig. 7(a). At zero temperature the concurrence is zero for B >
4J ′ [compare with Fig. 2(c) and Fig. 5(c)].
Figure 7: (Color online) (a) Next nearest neighbor concurrence CAC
for J ′ = J . (b) Heisenberg lattice result as a special case of the
SSM, J = 0. Shaded region represents CAC = 0. In the line shaded
region (low finite temperature and large magnetic field) our numer-
ical results set only the upper limit CAC < 5 · 10
−4. Dashed lines
separate CAC > 0 from CAC = 0.
In the limit J = 0 the model simplifies to the AFM Heisen-
berg model on a square lattice of N sites,
HAC = J
Si · Sj −B
Szi . (11)
Several results for this model have already been presented for
very small clusters55,56,57, however the temperature and mag-
netic field dependence of the concurrence for systems with
sufficiently large number of states and approaching thermo-
dynamic limit has not been presented so far.
In Fig. 7(b) we further presented temperature and magnetic
field dependence of concurrence for the Heisenberg model for
N = 20 (results for N = 16 are quantitatively similar, but not
shown here). Temperature and magnetic field dependence of
CAC exhibits peculiar semi-island shape where at fixed value
of B the concurrence increases with increasing temperature.
This effect is to some extent seen in all cases and is the con-
sequence of exciting local singlet states, which do not appear
in the ground state. At T → 0 finite steps with increasing B
correspond to gradual transition from the singlet ground state
to totally polarized state with total spin S = 10 and vanish-
ing concurrence. This is in more detail presented in Fig. 8(a)
for various N = 4, 8, 16, 20. At B = 0 and for N = 20
we get CAC = 0.19. It is interesting to compare this results
0 0.05 0.1
/2N=8
1/6+2N
-3/2C
Figure 8: (Color online) (a) Zero-temperature concurrence CAC in
the Heisenberg limit as a function of B/J ′ and for various N = 4,
8, 16, 20. Sections with different total spin values are addition-
ally labeled. (b) Finite-size scaling of concurrence in the absence of
magnetic field. Full line represents the fit corresponding to Ref. 58,
CAC ≈
+ 2N−3/2.
with the known finite-size analysis scaling for the ground state
energy of the Heisenberg model58. The same scaling gives
CAC ≈ 16 + 2N
−3/2. Our finite-size scaling, Fig. 8(b), is in
perfect agreement with this result for N → ∞ at T = 0 and
B = 0.
In the opposite limit of high magnetic fields, the vanishing
concurrence CAC = 0, is observed for B above the critical
value Bc = 4J
′ for all system sizes shown in Fig. 8(a). This
result can be deduced also analytically. Since in a fully po-
larized state CAC = 0, this Bc actually denotes a transition
from S1 = N/2 − 1 to S0 = N/2 ferromagnetic ground
state with energy E0 = N(J
′ − B)/2. The energy of the
one-magnon excitation above the ferromagnetic ground state
is given by the spin wave theory, which is in this case exact, as
E1 = E0 −J ′(2− cos kxa− cos kya)+B, where (kx, ky) is
the magnon wave vector and a denotes the lattice spacing. Ev-
idently, a transition to a fully polarized state occurs precisely
at Bc = 4J
′ at (π/a, π/a) point in the one-magnon Brillouin
zone.
V. SUMMARY
The aim of this paper was to analyze and understand
how concurrence (and related entanglement) of qubit pairs
(dimers) is affected by their mutual magnetic interactions. In
particular, we were interested in a planar array of qubit dimers
described by the Shastry-Sutherland model. This model is
suitable due to very robust ground state composed of entan-
gled qubit pairs which breaks down by increasing the in-
terdimer coupling. It is interesting to study both, the en-
tanglement between nearest and between next-nearest spins
(qubits) at finite temperature and magnetic field. The results
are based on numerical calculations using low-temperature
Lanczos methods on lattices of 4, 8, 16 and 20 sites with pe-
riodic boundary conditions.
A comprehensive analysis of concurrence for various pa-
rameters revealed two general conclusions:
(1) For a weak coupling between qubit dimers, J ′ < J ′c,
qubit pairs are locally entangled in accordance with the local
nature of the dimer phase. This is due to a finite singlet-triplet
gap (spin gap) in the excitation spectrum that is a consequence
of strong geometrical frustration in magnetic couplings. The
regime of fully entangled neighbors perfectly coincides with
the regime of finite spin gap as presented in Fig. 9. Calcu-
lated lines for various system sizes N in Fig. 9(a) denote re-
gions (shaded for N = 20) in the (J ′/J,B/J) plane where
CAB = 1 at T = 0. In the lower panel [Fig. 9(b)] the lines
represent the energy gap E1 − EGS between the first excited
state with energy E1 with total spin projection S
z = 1 and
the ground state with energy EGS and total spin projection
Sz = 0, calculated for B = 0. For J ′ < J ′c (full lines)
E1 − EGS corresponds to the value of the spin gap. With an
increasing magnetic field the spin gap closes (shaded region
for N = 20) and eventually vanishes at the CAB = 1 border
line. Shaded regions in Figs. 9(a),(b) therefore coincide. Note
also that the results for N = 16 and 20 sites differ mainly in
J ′c.
As a consequence of finite spin gap and local character of
correlations it is an interesting observation that even N = 4
results as a function of temperature and magnetic field quali-
tatively correctly reproduce N = 20 results in the regime of
J ′ < J ′c. The main quantitative difference is in a renormal-
ized value of J ′c = J/2 for N = 4, as is evident from the
comparison of Figs.2,3 and Figs.5,6. This similarity of the re-
sults appears very useful due to the fact that concurrence for
tetrahedron-like systems (N = 4) is given analytically (Sec.
(2) In the opposite, strong interdimer coupling regime, J ′ >
J ′c, the excitation spectrum is gapless and the concurrence be-
tween next-nearest qubits, CAC, exhibits a similar behavior as
in the antiferromagnetic Heisenberg model J/J ′ → 0. Our
B = 0 results coincide with the known result extrapolated to
the thermodynamic limit CAC ≈ 16 . In finite magnetic field
and T = 0 the concurrence vanishes at Bc = 4J
′ when the
system becomes fully polarized (ground state with the total
spin S = N/2). However, at elevated temperatures the con-
currence increases due to excited singlet states and eventually
drops to zero at temperatures above Tc ≈ J ′.
We can conclude with the observation that our analysis
of concurrence and related entanglement between qubit pairs
was also found to be a very useful measure for classify-
ing various phases of the Shastry-Sutherland model. As our
numerical method is based on relatively small clusters, we
were unable to unambiguously determine possible interme-
diate phases of the model in the regime J ′ ∼ J ′c, but we be-
lieve that concurrence will prove to be a useful probe for the
classification of various phases also in this regime using alter-
native approaches. However, we were able to sweep through
all other dominant regimes of the parameters including finite
temperature and magnetic field.
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
singlet - triplet gap
Figure 9: (Color online) (a) Zero-temperature CAB = 1 region in the
plane (J ′/J, B/J) for various N . (b) The corresponding spin gap
at B = 0 (the energy of the lowest total Sz = 1 state relative to the
ground state energy).
VI. ACKNOWLEDGMENTS
The authors acknowledge J. Mravlje for useful discussions
and the support from the Slovenian Research Agency under
Contract No. P1-0044.
Appendix A: EIGENENERGIES AND EIGENVECTORS FOR
PERIODICALLY COUPLED TWO QUBIT DIMERS
Consider two qubit dimers coupled into a tetramer and de-
scribed with the Hamiltonian Eq. (1) and Fig. 1(a). The model
is exactly solvable in the separate {S, Sz} subspaces corre-
sponding to different values of the total spin S and its z com-
ponent Sz . Following the abbreviations for singlet and triplet
states on nearest-neighbor (dimer) sites i and j,
|sij〉 =
|↑i↓j − ↓i↑j〉,
|t0ij〉 =
|↑i↓j + ↓i↑j〉,
|t+ij〉 = |↑i↑j〉,
|t−ij〉 = |↓i↓j〉, (A1)
the resulting eigenstates |φk〉 and eigenenergies Ek corre-
sponding to the hamiltonian Eq. (1) are:
S = 0, Sz = 0 :
|φ1〉 = |sAB〉|sCD〉,
E1 = −3J/2, (A2)
|φ2〉 =
− |t0AB〉|t0CD〉+ |t+AB〉|t
〉+ |t−
E2 = J/2− 4J ′. (A3)
S = 1, Sz = −1 :
|φ3〉 = |sAB〉|t−CD〉,
|φ4〉 = |t−AB〉|sCD〉,
E3,4 = −J/2−B, (A4)
|φ5〉 =
|t0AB〉|t
〉 − |t−
〉|t0CD〉
E5 = J/2− 2J ′ −B. (A5)
S = 1, Sz = 0 :
|φ6〉 = |sAB〉|t0CD〉,
|φ7〉 = |t0AB〉|sCD〉,
E6,7 = −J/2, (A6)
|φ8〉 =
〉 − |t−
E8 = J/2− 2J ′. (A7)
S = 1, Sz = 1 :
|φ9〉 = |sAB〉|t+CD〉,
|φ10〉 = |t+AB〉|sCD〉,
E9,10 = −J/2 +B, (A8)
|φ11〉 =
− |t0AB〉|t+CD〉+ |t
〉|t0CD〉
E11 = J/2− 2J ′ +B. (A9)
S = 2, Sz = −2 :
|φ12〉 = |t−AB〉|t
E12 = J/2 + 2J
′ − 2B. (A10)
S = 2, Sz = −1 :
|φ13〉 =
|t0AB〉|t−CD〉+ |t
〉|t0CD〉
E13 = J/2 + 2J
′ −B. (A11)
S = 2, Sz = 0 :
|φ14〉 =
2|t0AB〉|t0CD〉+ |t+AB〉|t
〉+ |t−
E14 = J/2 + 2J
′. (A12)
S = 2, Sz = 1 :
|φ15〉 =
|t0AB〉|t+CD〉+ |t
〉|t0CD〉
E15 = J/2 + 2J
′ +B. (A13)
S = 2, Sz = 2 :
|φ16〉 = |t+AB〉|t
E16 = J/2 + 2J
′ + 2B. (A14)
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http://arxiv.org/abs/cond-mat/9612126
http://arxiv.org/abs/cond-mat/0603444
http://arxiv.org/abs/quant-ph/0011036
