Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect
The influence of the inhomogeneity on the macroscopic thermal pairwise entanglement for the system of coupled spins 1/2 (qubits) has been studied. The most important effect of the inhomogeneity on the thermal entanglement is in the new role of the external potential (magnetic field), which can pro...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1169572017-05-19T03:02:42Z Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect Zvyagin, A.A. Низкотемпеpатуpный магнетизм The influence of the inhomogeneity on the macroscopic thermal pairwise entanglement for the system of coupled spins 1/2 (qubits) has been studied. The most important effect of the inhomogeneity on the thermal entanglement is in the new role of the external potential (magnetic field), which can produce nonzero entanglement for qubits, situated not far from the inhomogeneity. 2012 Article Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect / A.A. Zvyagin // Физика низких температур. — 2012. — Т. 38, № 3. — С. 266-272. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 03.65.Ud, 03.67.Mn, 03.67.Bg, 75.10.Pq http://dspace.nbuv.gov.ua/handle/123456789/116957 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Zvyagin, A.A. Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect Физика низких температур |
| description |
The influence of the inhomogeneity on the macroscopic thermal pairwise entanglement for the system of
coupled spins 1/2 (qubits) has been studied. The most important effect of the inhomogeneity on the thermal entanglement
is in the new role of the external potential (magnetic field), which can produce nonzero entanglement
for qubits, situated not far from the inhomogeneity. |
| format |
Article |
| author |
Zvyagin, A.A. |
| author_facet |
Zvyagin, A.A. |
| author_sort |
Zvyagin, A.A. |
| title |
Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect |
| title_short |
Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect |
| title_full |
Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect |
| title_fullStr |
Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect |
| title_full_unstemmed |
Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect |
| title_sort |
macroscopic thermal entanglement in a spin chain caused by the magnetic field: inhomogeneity effect |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2012 |
| topic_facet |
Низкотемпеpатуpный магнетизм |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/116957 |
| citation_txt |
Macroscopic thermal entanglement in a spin chain
caused by the magnetic field: Inhomogeneity effect / A.A. Zvyagin // Физика низких температур. — 2012. — Т. 38, № 3. — С. 266-272. — Бібліогр.: 21 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT zvyaginaa macroscopicthermalentanglementinaspinchaincausedbythemagneticfieldinhomogeneityeffect |
| first_indexed |
2025-07-08T11:22:50Z |
| last_indexed |
2025-07-08T11:22:50Z |
| _version_ |
1837077650059821056 |
| fulltext |
© A.A. Zvyagin, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 3, pp. 266–272
Macroscopic thermal entanglement in a spin chain
caused by the magnetic field: Inhomogeneity effect
A.A. Zvyagin
Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Str., 38, D-01187, Dresden, Germany
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: zvyagin@ilt.kharkov.ua
Received September 15, 2011
The influence of the inhomogeneity on the macroscopic thermal pairwise entanglement for the system of
coupled spins 1/2 (qubits) has been studied. The most important effect of the inhomogeneity on the thermal en-
tanglement is in the new role of the external potential (magnetic field), which can produce nonzero entanglement
for qubits, situated not far from the inhomogeneity.
PACS: 03.65.Ud Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.);
03.67.Mn Entanglement measures, witnesses, and other characterizations;
03.67.Bg Entanglement production and manipulation (for entanglement in Bose–Einstein condensates);
75.10.Pq Spin chain models.
Keywords: thermal entanglement, spin chain, impurity.
Entanglement is the fundamental aspect for a quantum
many-body system, additional to correlations, which its'
classical counterpart possesses [1]. It shows the nonlocal
nature, in which the entangled system contains correlations
that cannot be attributed to its' subsystems alone. The main
subject of the theory of the quantum computation is the
two-level quantum system, a qubit, which plays the role of
the elementary cell containing information. It is the entang-
lement between qubits, which, due to the very heart of
quantum mechanics [2,3], permits to manipulate with ma-
ny qubits simultaneously, manifesting the main advantage
of a quantum computer, the quantum coherence [4]. Ma-
croscopic entanglement demonstrates that nonlocal corre-
lations persist even in the thermodynamic limit, where the
number of particles in the system (the number of qubits in
a quantum computer) tends to infinity. The majority of
theoretical results for entangled multi-qubit systems was
obtained for = 0T case [1]. However, physics of the real
world ever deals with nonzero temperatures, hence, it is
very important to study the thermal entanglement. The
thermal entanglement for spin systems was studied recent-
ly theoretically (however, mostly for few-spin systems, not
the macroscopic entanglement) [1,5]. Moreover, several
experimental groups recently reported the observation of
the thermal entanglement in real magnetic systems [6].
The aim of our work is to calculate analytically the cha-
racteristics of the macroscopic thermal entanglement for a
model system consisting of interacting two-level subsys-
tems (qubits) with an inhomogeneity. We attack the prob-
lem by the investigation of the thermal entanglement of
a semi-infinite chain with the interaction between neigh-
boring spins 1/2, which is coupled to an impurity, de-
scribed by two independent parameters. The general in-
terest to the entanglement in one-dimensional critical sys-
tems is due to universal predictions for the scaling of the
entanglement entropy [7], caused by the conformal inva-
riance [8]. A particulary important result, which the con-
sidered model permits us to obtain, is the influence of local
levels, caused by the impurity, on the thermal entang-
lement. This result is impossible to obtain within the boun-
dary conformal field theory, which is mostly used for the
studies of the entanglement of critical chains with impu-
rities, or Kondo impurities in metals [9,10]. It is worth
mentioning that the characteristics of the ground-state en-
tanglement were calculated numerically for similar models
for a finite number of qubits in Refs. 11 and 12, Ref. 13
studied the influence of a local level caused by the inho-
mogeneous field on the ground-state entanglement, and
Ref. 14 studied the thermal entanglement of a system con-
sisted of two qubits with an inhomogeneous field.
Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 3 267
In this paper we find exactly that the thermal entangle-
ment is very sensitive to the presence of the inhomogeneity
in a system of coupled qubits. The temperature range, at
which one can use the quantum macroscopic entanglement
between qubits is dependent on the distance to the inho-
mogeneity and on the applied field. We show that one can
govern that range for the thermal entanglement by apply-
ing an external homogeneous potential to the system (in
the case of the spin chain it is the external magnetic field).
Usually it is believed that such a field destroys the quan-
tum entanglement: Two antiferromagnetically coupled
spins 1/2 without the field are in the most entangled Bell
(singlet) state [2], while a large field transforms their wave
function to the one, which is the product of wave functions
of each spin, i.e., the entanglement between those spins be-
comes minimal for high values of the field. Surprisingly,
our exact results suggest that the field can cause the onset
of a nonzero macroscopic thermal entanglement, absent
without such a field.
The standard measure for the entanglement between
subsystems A and B of a quantum system is the von Neu-
mann entropy. For a subsystem A/B it is defined as
/ / /2= tr logA B A B A BS − ρ ρ , where /A Bρ is the reduced
density matrix for the subsystem A (B). If the subsystem
consists of only one qubit (spin 1/2), a simple measure for
the entanglement, related to the von Neumann entropy,
is the tangle [15], (1)= 4 detj jτ ρ , where (1) 0= (1/ 2) jjρ σ +
z z
j jS+ 〈 〉σ is the 2 2× matrix, 0
jσ is the identity matrix,
2z zSσ ≡ , and brackets denote averaging in the ground
state or at nonzero temperature, so that 2= (1/ 4) .z
j jSτ −〈 〉
The von Neumann entropy is related to the tangle as
(1) = [(1/ 2)(1 1 )],S h + −τ where 2( ) = (1 )logh x x x x− − − ×
2(1 ).log x× − However, the von Neumann entropy (and,
hence, tangle) is used only if the state of the total system
A+B is pure. To analyze the pairwise entanglement, which
provides more information about the quantum coherence,
we need to consider the subsystem consisting of two qubits
(two spins 1/2). The pairwise entanglement of two spins
1/2 situated at the sites n and m of the quantum system
can be studied using the reduced density matrix (2)
nmρ .
The latter can be expressed as a 4 4× matrix (2) =nmρ
,= n m n mS Sμ ν μ ν
μ ν 〈 〉σ ⊗σΣ , where , = 0, , ,x y zμ ν , and ⊗
denotes the direct product. The concurrence of the entan-
glement of two qubits (spins 1/2) is determined [16] as
1 2 3 4= max(0, )nmC μ −μ −μ −μ . Here 1,2,3,4μ (with 1μ
being the largest value) are the square roots of the eigen-
values of the matrix (2) (2)
nm nmρ ρ , where (2)
nmρ is the spin-
flipped matrix of (2)
nmρ , *(2) (2)= ( )y y y y
nm n m n mρ σ ⊗σ ρ σ ⊗σ .
Then the entanglement of formation is = ( )fE h x , where
2 1/2= (1/2) [1 ( /2)]nmx C+ − . fE is a monotonous function
of the concurrence with ( = 0) = 0f nmE C and ( =1) =f nmE C
= 1. This is why, for simplicity we study the concurrence
in what follows. For the uniaxial spin system in the ab-
sence of the antisymmetric interactions one can take
the advantage of the knowledge of the symmetries. The
concurrence can be calculated using the function
2 2 1/2= 2 max (0,| | [([1/4] ) (1/4)( ) ] ),nmC X Z M Mn m− + − +
where for the spin model we have = ,z
n nM S〈 〉 = ,z z
n mZ S S〈 〉
and = 2 = 2 = =x x y y
n m n m n m n mX S S S S S S S S+ − − +〈 〉 〈 〉 〈 〉 〈 〉 ( =nS±
= ;x y
n nS iS± = = 0x y y x
n m n mS S S S〈 〉 〈 〉 ).
To set the stage let us consider the Hamiltonian of the
inhomogeneous semi-infinite spin 1/2 chain [17]
1
1 1
=1 =1
= ( )
N N
yx x y z
n n n nn
n n
J S S S S H S
−
+ +− + − γ −∑ ∑H
0 1 00 1( ) ,y yx x zJ S S S S HS′ ′− + − γ (1)
where J ( J ′ ) describes the strength of spin–spin interac-
tions between neighboring spins in the host chain (between
the chain and impurity situated at the site 0), γ ( )′γ de-
notes the gyromagnetic ratio for the host (impurity),
H is the external magnetic field, and N is the number of
sites in the chain (we consider N odd; in the thermodyna-
mic limit we have N →∞ ). It is known [17] that ther-
modynamic properties of the host chain do not depend
on the sign of .J After the Jordan–Wigner transforma-
tion [18] †2 =1 2 ,z
n n nS a a− †
<= (1 2 ) ,Пn m n m m nS a a a+ − =nS−
† †
<= (1 2 ),Пn m n m ma a a− where na and †
na are Fermi
operators of creation and destruction, the above Hamilto-
nian is exactly transformed to the quadratic form
1
†† †
1 1
=1 =1
= ( )
2
N N
n n n n nn
n n
J a a a a H a a
−
+ +− + + γ −∑ ∑H
† † †
1 0 00 1 0( ) ( ) .
2 2
J Ha a a a Ha a N
′
′ ′− + + γ − γ + γ (2)
Notice that the fermionic form of the Hamiltonian can
(approximately) describe such inhomogeneous systems as
a linear chain of Josephson junctions, cavity QED systems,
linear ion traps, coupled quantum dots, etc., which were
proposed to model qubits in a quantum computer [19].
This quadratic form can be diagonalized by the unitary
transformation =n na u aλλ λΣ , = 0,1, , .n N… For the con-
sidered semi-infinite chain with the impurity in the ther-
modynamic limit we can exactly write the eigenfunctions
nuλ in the co-ordinate representation and eigenvalues λε .
There are two contributions. The first one is related to the
continuous spectrum (in the thermodynamic limit) of the
chain = cosk H J kε γ − , with the wave functions =k
nu
1/2 2= (2/ ) [2( cos ) sin sin (( 1) )]kA x k nk I n kπ + − − for 0,n ≠
1/2
0 = (2/ ) sin ,k
ku A I kπ where we denote 2= 4( cos )kA x k+ +
4 24 cos ( cos )I I k x k+ − + , = /I J J′ measures the re-
lative strength of the impurity-host coupling, =x
= ( 1) /H Jγ α − , and = /′α γ γ , which measures the rela-
tive difference between the local parameter of the impurity
and the host (their gyromagnetic ratios). Extended states,
associated with the continuous spectrum, are present, natu-
rally, in the homogeneous system, and the impurity renor-
malizes the parameters of extended states. This renormali-
zation can be taken into account using the boundary
A.A. Zvyagin
268 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 3
conformal field theory. On the other hand, for 2 2 2I x> ∓
local levels, caused by the impurity, are split off. These
impurity levels have the energies 1,2 = Hε γ −
2
1,2 1,2( 1)/2 ,J r r− + with the wave functions in the co-ordinate
representation 2 2 2 1/2
,1,2 1,2 1,2 1,2= [(1 ) / (1 [ 1] )] ,n
nu Ir r I r− + −
and 2 2 2 1/2
0,1,2 1,2 1,2= [(1 ) / (1 [ 1] )]u r I r− + − , where 1,2 =r
2 2 21/2= [ ( 1) ] / ( 1)x x I I± + − − . Wave functions of the lo-
cal states can be characterized by their localization lengths
2
1,2 1,2= ( 2 2 ) / ln ( )I x rξ −Θ − ∓ , where ( )aΘ is the Hea-
viside step function. Figure 1 shows that the localization
length depends on the applied field and can be very large.
Let us denote ,= n mX g , and = (1/ 2)n nM h− . The
tangle is equal to = 4 (1 )n n nh hτ − . Tangles for the impuri-
ty site and sites situated not very far from the impurity be-
have in a similar way, while for tangles for sites situated
very far from the impurity the low-temperature jump at the
critical value of the field is replaced by the kink, usual for
the homogeneous chain, i.e., the main difference for tan-
gles is in the ground (pure) state. It is illustrated in Fig. 2,
where the field behavior of the ground-state tangles is pre-
sented. Tangles in the vicinity of the impurity show jumps
in the field behavior at the critical value of the field
2 2
0 = /2 ( ).H I J Iγ α −α Jumps are caused by the local
level 1ε , which exists if the following inequality holds
2 > 2 /(1 )I ′γ + α . Tangles have jumps when arguments of
the Heaviside functions vanish. On the other hand, at
2 = 2 2I x+ we have 2
1 = 1r (no local level) and there is
no jump. The value of the jump exponentially decays with
the distance from the inhomogeneity, and tangles for sites
situated far from the impurity behave like the tangle, aver-
age over the chain (in that case we sum nh with respect to
n and divide the sum by N, 1
av 0= kh n dkπ−π ∫ , where
1= [1 exp ( / )]k kn T −+ ε and we use units, for which Boltz-
mann's constant is equal to 1): Its field behavior reveals a
kink at = / ,sH J γ characteristic for the second order
quantum phase transition to the spin-polarized state. Ob-
viously, in the spin-polarized state > sH H , or 0> ,H H
tangles are equal to zero. For comparison, in Fig. 3 we
present the field dependencies of the ground-state tangles
4τ and 5τ , i.e. for qubits, situated not very far from the
inhomogeneity. We can see that these tangles manifest
jumps at 0= ,H H however, the values of tangles (except
of at = 0H ) are much smaller than the ones near the im-
purity.
For any chain with interactions between only x and y
components of neighboring spins 1/2 (which Hamiltonian
can be transformed to the quadratic fermion form) we can
use the Wick theorem, which implies 2 = (1/ 2) nZ h− −
2
,2 2 | |m n m n mh h h g− + − for uniaxial systems without
antisymmetric interactions. The concurrence in this case is
400
300
200
100
–2
–1
0
1
2
2.0
1.5
1.0
0.5
0
I
x
�
1
Fig. 1. (Color online) Localization length 1ξ as the function of
the relative strength of the impurity-host coupling = /I J J′ and
the relative strength of the magnetic field, which affects the spin
of the impurity = ( ) / .x H J′γ − γ One can see that for some
values of the impurity-host stength and the field the correlation
length can be of order of 500 sites of the chain.
1.0
0.8
0.6
0.4
0.2
0
�
� ��� ��� ���
H
Fig. 2. (Color online) The ground-state tangles avτ (dotted line),
0τ (solid line), and 1τ (dashed line) as the function of the ap-
plied magnetic field H (we use units in which Planck's and
Boltzmann's constants, gyromagnetic ratio γ and the exchange
integral J in the chain are equal to 1 for = 2.2I and = 1.2α ).
0τ and 1τ manifest jumps at 2 2 1/2
0 = /2 ( ( ))H I J Iγ α − α
which take place in the region of the parameters for the impurity
[20] 2 > 2 / (1 ),I ′γ + α as the contribution of the local level. The
average tangle shows no low-temperature jump at 0H comparing
to the ones for the impurity site and near the impurity; instead
avτ manifests the kink (due to the second order quantum phase
transition) at the critical value = / ,sH J γ characteristic for the
homogeneous system.
Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 3 269
2
, ,= 2 max 0,| | ( | | )nm n m n m n mC g h h g⎛ ⎡− − ×⎜ ⎢⎣⎝
1/22
,( | | 1 ) .n m n m n mh h g h h ⎞⎤× − + − − ⎟⎦ ⎠
(3)
Expressions are simplified for the case of zero magnetic
field ( = = 1/ 2n mh h ), for which we obtain = 1nτ and
2
, ,= 2 max (0,| | | | 1 / 4)nm n m n mC g g+ − , i.e., the concur-
rence becomes nonzero for ,| | ( 2 1) / 2n mg ≤ − at = 0H .
From now on let us concentrate on the case of the pair-
wise entanglement between neighboring qubits (spins 1/2).
Using the values for the wave functions and the energy
spectra we obtain the impurity site
2 2
0
0
2= sink
k
n
h dk I k
A
π
+
π ∫
2 2
1,21,2
2 2
1,2 1,2
(1 ) ( 2 2 )
,
1 ( 1)
r n I x
I r
− Θ −
+
+ −
∑
∓
(4)
2
0,1
0
2= 2 sin ( cos )k
k
n
g dk I k x k
A
π
+ +
π ∫
2 2
1,21,2
1,2 2 2
1,2 1,2
(1 ) ( 2 2 )
,
1 ( 1)
r n I x
Ir
I r
− Θ −
+
+ −
∑
∓
(5)
where 1
1,2 1,2= [1 exp ( / )]n T −+ ε . For the characteristics
of sites with 0n ≠ we get
2 2
0
2= 4( cos ) sink
n
k
n
h dk x k kn
A
π
⎡ + +⎣π ∫
4 2 2sin ( ( 1)) 4 ( cos )sin sin ( ( 1))I k n I x k kn k n ⎤+ − − + − +⎦
2 2
1,2 1,22 2
1,2 2 2
1,2 1,2
(1 ) ( 2 2 )
,
1 ( 1)
n r n I x
I r
I r
− Θ −
+
+ −
∑
∓
(6)
2
, 1
0
2= 4( cos ) sink
n n
k
n
g dk x k kn
A
π
+ ⎡ + ×⎣π ∫
4sin ( ( 1)) sin ( ( 1)) sink n I k n kn× + + − −
2 22 ( cos )[sin sin ( ( 1))sin ( ( 1))]I x k kn k n k n ⎤− + + + − +⎦
2 2
1,2 1,22 2 1
1,2 2 2
1,2 1,2
(1 ) ( 2 2 )
.
1 ( 1)
n r n I x
I r
I r
+ − Θ −
+
+ −
∑
∓
(7)
We can average the concurrences for the total chain. In
this situation the contributions from local levels can be
neglected, and the average values are caused by only ex-
tended states: 1
av 0= cos kg k n dk
π−−π ∫ . Similar behavior
is revealed by qubits of the chain, situated very far from
the impurity site, at distances larger than the localization
lengths. Obviously, the average concurrence (and tangle)
does not depend on the impurity parameters: Those contri-
butions are of order of 1N − . The dependency of the aver-
age concurrence for the considered chain is shown in
Fig. 4.
We can see that the ground-state (and low-temperature)
dependence of the average tangle on the field differs from
the one at the impurity site: Instead of the jump, caused by
the local level, there is a kink, usual for the second order
0 0.5 1.0 1.5 2.0
H
0.10
0.08
0.06
0.04
0.02
0
�
Fig. 3. (Color online) The ground-state tangles 4τ (dashed line),
and 5τ (solid line) as the function of the applied magnetic field
H (parameters are the same as in Fig. 2). We can see the jump at
0,H however the value of tangles are much smaller than in the
vicinity of the inhomogeneity.
Fig. 4. (Color online) The average concurrence avC as the func-
tion of temperature T and the applied magnetic field H for =1.J
0.3
0.2
0.1
0
C
a
v
0
00.5
0.51.0
1.0
1.5 1.5
2.0
H
T
A.A. Zvyagin
270 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 3
quantum phase transition to the spin-polarized phase, cha-
racteristic for spin systems of such a symmetry.
Figures 5 and 6 present the temperature and magnetic
field dependencies of the concurrences 01C (near the im-
purity) and 45C (several lattice sites from the impurity).
One can see the drastic difference between the magnetic
field behavior of concurrences at the impurity site 12C
behaves similarly, as well as the average concurrence
(Fig. 4), and for sites, situated not very far from the im-
purity. The concurrence at the impurity site is maximal at
zero field, and decays with the growth of the field. Pay
attention that the concurrence at the impurity site is larger
than the average concurrence, while the one for the sites,
situated in several lattice sites from the impurity, is much
smaller. The critical temperature, at which the concurrence
becomes zero does not depend on H. Notice, however, that
such critical temperatures are different for the concur-
rences at the impurity, nor very far from the impurity, and
from the average over the chain (Fig. 4). Similar behavior
(except of the scale and the low-temperature region) is
manifested by average values of concurrences. In contrast,
concurrences for spins, situated not very far from the im-
purity site, reveal a very different magnetic field behavior
(cf. Fig. 6): They are zero at = 0,H and become nonzero
for large enough values of the field (of order of the coupl-
ing constant).
Hence, the pairwise thermal entanglement for neighbor-
ing qubits with inhomogeneity can be caused by the homo-
geneous external field. Such a behavior is very unusual. As
a rule, the homogeneous external magnetic field, which
does not violate the uniaxial symmetry in the spin-1/2
model of qubits is used for the initialization of qubit states
to the state with the lowest entanglement. However, our
results show that the same field can produce the macro-
scopic thermal entanglement for qubits, situated not very
far from the inhomogeneity.
We have also performed the systematic study of the
temperature dependencies of concurrences with a distance
from the inhomogeneity at the fixed value of the field
= 0H . Figures 7 and 8 present such a dependencies as a
0.75
0.50
0.25
0
0
0.5
1.0
1.5
4
3
2
1
T
H
C
01
Fig. 5. (Color online) The concurrence 01C at the impurity site
as the function of temperature T and the applied magnetic field
H. Parameters are the same as in Fig. 2. The concurrence be-
comes smaller with the growth of the field.
0.0015
0.0010
0.0005
0
C
45
0
0.5
1.0
1.5
0
4
3
2
1
TH
Fig. 6. (Color online) The concurrence 45C in a short distance
from the impurity as the function of temperature T and the ap-
plied magnetic field H. Parameters are the same as in Fig. 2. No-
tice much smaller scale for 45C comparing to 01.C The concur-
rence is zero at = 0H and becomes nonzero for large enough H.
Fig. 7. (Color online) The concurrence 01C at the impurity qubit
as the function of temperature T and the parameter of the inho-
mogeneity I at zero field = 0H .
0.8
0.6
0.4
0.2
C01
0
4
2
T
–4
–3
–2
–1 0
1
2 3
4
I 6
8
10
Macroscopic thermal entanglement in a spin chain caused by the magnetic field: Inhomogeneity effect
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 3 271
function of the temperature and the inhomogeneity para-
meter I for the pairwise concurrences at the impurity qu-
bit, 01C and for the next to the impurity qubit 12C .
We can see a very different behavior of the pairwise
concurrences as a function of the inhomogeneity parameter
for 01C and 12C : while the pairwise concurrence for the
impurity qubit is zero at small values of the impurity-host
coupling, the pairwise concurrence for the qubit, situated
next to the impurity, is nonzero for small values of the
coupling, and becomes zero for the large values. Such a
behavior is related to the different dependencies of consi-
dered concurrences on the coupling I, and, as the conse-
quence, on the localized levels, see Eqs. (3) and (4)–(7). At
the same value of the field, = 0,H the concurrences 23C ,
34C and 45C are zero in the same interval of parameters
T and I. For comparison, Fig. 9 shows the temperature
behavior of the pairwise concurrence at the impurity qubit
at = 0.5 /H J γ .
Our results can be applied for the infinite chain with an
impurity, by the formal substitution 2 22I I→ , | |n n→ ,
hence the studied effects are caused by the impurity itself,
and not by the free boundary of the considered above semi-
infinite chain.
In summary, we have studied the influence of the in-
homogeneity (in particular, of the local levels, caused by
that inhomogeneity) on the thermal macroscopic pairwise
entanglement for the system of coupled spins 1/2 (qubits)
with the interactions, which preserve the uniaxial symme-
try. The simple model has permitted to obtain exact analyt-
ic formulas for the characteristics of the macroscopic
thermal entanglement. Our analytic 0T ≠ results support
recent predictions based on numerical calculations mostly
for finite systems, that the spatial inhomogeneity essential-
ly affects the entanglement in real systems [10–14,21]. The
most important effect of the inhomogeneity on the macro-
scopic thermal entanglement is in the unexpected role of
the external magnetic field, which can produce nonzero en-
tanglement for qubits, situated not very far from the inho-
mogeneity.
I thank MPI PKS Dresden for kind hospitality. The sup-
port from the Institute of Chemistry of V.N. Karazin Khar-
kov National University is acknowledged.
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