Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions
The emptiness formation probability (EFP) that the first n spins of the chain are all aligned downwards is studied for the one-dimensional spin- 1/2 XX chain system with both critical and non-critical regimes with three spin and uniform long-range interactions. By using the emptiness formation proba...
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| Cite this: | Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions / P. Lou, J.Y. Lee // Condensed Matter Physics. — 2007. — Т. 10, № 3(51). — С. 425-432. — Бібліогр.: 22 назв. — англ. |
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Lou, P. Lee, J.Y. 2017-05-31T04:53:56Z 2017-05-31T04:53:56Z 2007 Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions / P. Lou, J.Y. Lee // Condensed Matter Physics. — 2007. — Т. 10, № 3(51). — С. 425-432. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 75.10.Jm, 75.40.Cx, 75.40.-s, 73.43.Nq DOI:10.5488/CMP.10.3.425 https://nasplib.isofts.kiev.ua/handle/123456789/118712 The emptiness formation probability (EFP) that the first n spins of the chain are all aligned downwards is studied for the one-dimensional spin- 1/2 XX chain system with both critical and non-critical regimes with three spin and uniform long-range interactions. By using the emptiness formation probability P(n), the block-block entanglement Sn and the magnetization Mz, the properties of the system are also studied. It is worthy of note that when the quantum phase transition from the phase III to the phase II occurs, the Emptiness Formation Probability P(n) of the system decreases, but the block-block entanglement Sn increases. Дослiджується ймовiрнiсть формування порожнини, тобто ймовiрнiсть того, що n перших спiнiв у ланцюжку є спрямованi вниз, для одновимiрної спiн- 1/2 XX ланцюжкової системи з триспiновою i однорiдною далекосяжною взаємодiями як у критичному, так i у некритичному режимах. З використанням ймовiрностi формування порожнини P(n), блок-блок сплутаностi Sn i намагнiченостi Mz дослiджуються також властивостi такої спiнової системи. Важливо вiдзначити, що коли вiдбувається квантовий фазовий перехiд з фази III у фазу II, ймовiрнiсть формування порожнини системи P(n) зменшується, тодi як блок-блок сплутанiсть Sn зростає. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF–2006–311–C00082). This work was also supported by the Korea Research Foundation and The Korean Federation of Science and Technology Societies Grant funded by Korean Government (MOEHRD, Basic Research Promotion Fund). en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions Ймовiрнiсть формування порожнини для спiн-1/2 XX ланцюжка з триспiновою та однорiдною далекосяжною взаємодiями Article published earlier |
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| title |
Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions |
| spellingShingle |
Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions Lou, P. Lee, J.Y. |
| title_short |
Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions |
| title_full |
Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions |
| title_fullStr |
Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions |
| title_full_unstemmed |
Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions |
| title_sort |
emptiness formation probability for the spin-1/2 xx chain with three spin and uniform long-range interactions |
| author |
Lou, P. Lee, J.Y. |
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Lou, P. Lee, J.Y. |
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2007 |
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English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
| title_alt |
Ймовiрнiсть формування порожнини для спiн-1/2 XX ланцюжка з триспiновою та однорiдною далекосяжною взаємодiями |
| description |
The emptiness formation probability (EFP) that the first n spins of the chain are all aligned downwards is studied for the one-dimensional spin- 1/2 XX chain system with both critical and non-critical regimes with three spin and uniform long-range interactions. By using the emptiness formation probability P(n), the block-block entanglement Sn and the magnetization Mz, the properties of the system are also studied. It is worthy of note
that when the quantum phase transition from the phase III to the phase II occurs, the Emptiness Formation Probability P(n) of the system decreases, but the block-block entanglement Sn increases.
Дослiджується ймовiрнiсть формування порожнини, тобто ймовiрнiсть того, що n перших спiнiв у ланцюжку є спрямованi вниз, для одновимiрної спiн- 1/2 XX ланцюжкової системи з триспiновою i однорiдною далекосяжною взаємодiями як у критичному, так i у некритичному режимах. З використанням ймовiрностi формування порожнини P(n), блок-блок сплутаностi Sn i намагнiченостi Mz дослiджуються також властивостi такої спiнової системи. Важливо вiдзначити, що коли вiдбувається квантовий фазовий перехiд з фази III у фазу II, ймовiрнiсть формування порожнини системи P(n) зменшується, тодi як блок-блок сплутанiсть Sn зростає.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118712 |
| citation_txt |
Emptiness formation probability for the spin-1/2 XX chain with three spin and uniform long-range interactions / P. Lou, J.Y. Lee // Condensed Matter Physics. — 2007. — Т. 10, № 3(51). — С. 425-432. — Бібліогр.: 22 назв. — англ. |
| work_keys_str_mv |
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| first_indexed |
2025-11-27T09:23:47Z |
| last_indexed |
2025-11-27T09:23:47Z |
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| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 3(51), pp. 425–432
Emptiness formation probability for the spin-1
2
XX chain
with three spin and uniform long-range interactions
P.Lou1,2, J.Y.Lee1
1 Department of Chemistry, Sungkyunkwan University, Suwon 440–746, Korea
2 Department of Physics, Anhui University, Hefei 230039, Anhui, P.R. China
Received March 29, 2007
The emptiness formation probability (EFP) that the first n spins of the chain are all aligned downwards is
studied for the one-dimensional spin- 1
2
XX chain system with both critical and non-critical regimes with three
spin and uniform long-range interactions. By using the emptiness formation probability P (n), the block-block
entanglement Sn and the magnetization Mz , the properties of the system are also studied. It is worthy of note
that when the quantum phase transition from the phase III to the phase II occurs, the Emptiness Formation
Probability P (n) of the system decreases, but the block-block entanglement Sn increases.
Key words: emptiness formation probability, spin- 1
2
XX chain, three spin and uniform long-range interactions
PACS: 75.10.Jm, 75.40.Cx, 75.40.-s, 73.43.Nq
1. Introduction
One of the most important issues emerging from the studies of condensed matter physics in
recent years is the concept of quantum criticality [1]. A quantum critical point marks a zero-
temperature phase transition between different ground states of a many body system as a result
of changes in parameters of the underlying Hamiltonian. It was previously stated at the quantum
critical point that there is no characteristic length or energy scale, and the system shows power-
law spatial correlations and gapless excitations [2]. Moreover, quantum phase transitions can occur
between phases that are not characterized by any local order parameter or symmetry breaking
description, but that have different quantum order, which is one of the most exciting and novel
streams of research in theoretical condensed matter [3].
Recently, quantum phase transitions have been studied in the context of the theory of entan-
glement where the points of quantum phase transitions are believed to be connected to the points
of extremality for the entanglement or its derivatives [4–9]. There are two widely used methods of
characterizing entanglement in quantum phase transitions of spin chains. The first method describes
the entanglement between two spins in the chain with the quantity called concurrence [10,11]. The
other method measures entanglement of a block of spins with the rest of the chain with the von
Neumann entropy (block-block entanglement), when the chain is in its ground state [12–17].
Recently we studied the block-block entanglement and quantum phase transition of the system
[17], based on the spin-1
2
XX chain model with three-spin and uniform long-range interactions [18].
We found that a generic finite order quantum phase transition in the chain can be consistently
identified and characterized by studying the behavior of the entanglement of the block with the
rest of the chain and its derivatives with respect to the parameter that controls the transition.
However, the very important quantity that characterizes a quantum spin system in the quantum
spin-liquid phase, which is the so-called EFP [19,20], i.e, the probability to find a ferromagnetic
string of the length “n” in the spin liquid ground state, has not been studied for such systems so
far. The motivation to study the EFP for this system is to explore if the EFP can supply new
important information, just as the block-block entanglement, to gain a deeper understanding of
the quantum states of system at zero temperature.
c© P.Lou, J.Y.Lee 425
P.Lou, J.Y.Lee
In this paper, we study the probability of the formation of a ferromagnetic string in the ground
state of the spin- 1
2
XX chain model with three-spin and uniform long-range interactions [18].
We show that a generic finite order quantum phase transition in the system can be consistently
identified and characterized by studying the EFP behavior and its derivatives with respect to the
parameter that controls the transition. In section II, we present the model and theory that we
employed. The EFP behavior is studied in section III. In section IV, we give the summary.
2. Model and theory
The exactly solvable quantum spin model considered here is the spin-1
2
XX chain with three-spin
and uniform long-range interactions, which is defined by the following Hamiltonian [18],
H = −
N
∑
l=1
[
(Sx
l Sx
l+1 + Sy
l Sy
l+1
) + α(Sx
l−1S
z
l Sy
l+1
− Sy
l−1
Sz
l Sx
l+1)
]
− λ
N
N
∑
l,k=1
Sz
l Sz
k , (1)
where Si
l (i = x, y, z) are spin operators of S = 1/2 on site l associated with the periodic boundary
conditions (Si
N+1 = Si
1), N is the total number of spins, α is the dimensionless parameter charac-
terizing the three-spin coupling constant (in unit of the nearest-neighbor exchange coupling), and λ
is the dimensionless parameter characterizing the uniform long-range interactions. This model can
be solved by using the Jordan-Wigner transformation [17,18,21,22] and the Hubbard- Stratonovich
transformation, and all the physical quantities can be calculated exactly [18]. The ground state
energy of the system is given by
e0(α, λ) = − 1
N
∑
k
[
cos(k) − α
2
sin(2k)
]
Θ [εk(Mz)] − λ (Mz)
2
, (2)
where Θ(x) = 1 when x > 0 and vanishes otherwise, and
εk(Mz) = cos(k) − α
2
sin(2k) + 2λMz. (3)
The magnetization (Mz) is given by
Mz =
1
N
∑
k
Θ [εk(Mz)] − 1
2
. (4)
Using equations (2), (3), and (4), the magnetization Mz and its contour map can be calculated
with parameters λ and α, and the result is shown in figure 1 [17]. It is clearly seen that there
Figure 1. The magnetization M
z and its contour map changes with parameter λ and α.
426
Emptiness formation probability for the spin- 1
2
XX chain
are four phases. Phases I and II are paramagnetic. But phase II breaks the chiral symmetry [21].
Phases III and IV are ferromagnetic. In phase III (called ferromagnetic δ phase), the magnetic
moment Mz < 1
2
, while in phase IV (called ferromagnetic γ), Mz ≡ 1
2
. There are two tricritical
points, one separates the paramagnetic, ferromagnetic δ, and paramagnetic chiral phases, and the
other separates the paramagnetic, ferromagnetic δ, and ferromagnetic γ phases. The quantum
phase transition from phase I to phase IV is of the first order, which results from the uniform long-
range interactions. The quantum phase transition from phase III to phase II, as well as from phase
IV to phase III, is also of the first order, but it is caused mainly by the three-spin interactions.
The quantum phase transition from phase I to phase III is also caused mainly by the three-spin
interactions, but it is of the second-order.
The EFP at zero temperature is defined to be the probability of formation of a ferromagnetic
string, i.e., the probability that the first n spins of the chain are all aligned downwards in the spin
liquid ground state |0〉 [19,20], and it can be expressed as
P (n) =
〈
0
∣
∣
∣
∣
∣
∣
n
∏
j=1
(
Sz
j +
1
2
)
∣
∣
∣
∣
∣
∣
0
〉
. (5)
By performing Jordan-Wigner transformation [22],
Sx
l =
1
2
l−1
∏
j=1
(1 − 2c†jcj)(c
†
l + cl),
Sy
l =
1
2i
l−1
∏
j=1
(c†l − cl)(1 − 2c†jcj),
Sz
l = c†l cl −
1
2
, (6)
and it is easy to show that equations (5) becomes
P (n) =
〈
0
∣
∣
∣
∣
∣
∣
n
∏
j=1
c†jcj
∣
∣
∣
∣
∣
∣
0
〉
= det
[〈
0|c†l cm|0
〉]n
l,m=1
, (7)
where we have used Wick’s theorem. The determinant like equation (7) is referred to as a Toeplitz
determinant. Therefore, in our model, the determinant P (n) is
P (n) =
∣
∣
∣
∣
∣
∣
∣
∣
f1,1 f1,2 ... f1,n
f2,1 f2,2 ... f2,n
... ... ... ...
fn,1 fn,2 ... fn,n
∣
∣
∣
∣
∣
∣
∣
∣
, (8)
in which the coefficients fm,n for an infinite chain are calculated by
fm,n = 〈0|c+
mcn|0〉 =
1
2π
∫
exp(−ik(n − m))dkΘ(εk(Mz)). (9)
It is noted that the correlation matrix Fn of the determinant P (n) is
Fn =
f1,1 f1,2 ... f1,n
f2,1 f2,2 ... f2,n
... ... ... ...
fn,1 fn,2 ... fn,n
, (10)
which is the correlation matrix of the reduced density matrix (ρL) of n contiguous spins [17]. Then,
the von Neumann entropy of a block of n contiguous spins, describing entanglement of the block
427
P.Lou, J.Y.Lee
with the rest of the chain, takes the form [17]
Sn =
n
∑
j=1
[− (1 − λj) log2 (1 − λn) − λj log2 λj ] , (11)
where λj is the j-th eigenvalue of the correlation matrix Fn. Then, the EFP of the system is
given by
P (n) =
n
∏
j=1
λj , (12)
i.e, the EFP of the system is the product of the eigenvalues of the correlation matrix Fn.
For example, the entanglement between one spin and the rest part of the system can be char-
acterized by the von Neumann entropy,
S1 = −(λ1) log(λ1) − (1 − λ1) log(1 − λ1), (13)
with
λ1 = f1,1 =
1
2
+ Mz. (14)
The EFP of the system is
P (1) =
1
2
+ Mz. (15)
The entanglement between two contiguous spins and the rest part of the system is charac-
terized by,
S2 = −(λ1) log(λ1) − (1 − λ1) log(1 − λ1) − (λ2) log(λ2) − (1 − λ2) log(1 − λ2), (16)
in which
λ1 = f1,1 − f1,2 =
1
2
+ Mz − f1,2,
λ2 = f1,1 + f1,2 =
1
2
+ Mz + f1,2 . (17)
Then, the EFP of the system is
P (2) =
(
1
2
+ Mz
)2
− (f1,2)
2. (18)
It is important to note that both the EFP and block-block entanglement of the system are
directly related to the eigenvalues of the correlation matrix Fn. Therefore, one can expect that
the quantum phase transitions of the system can be well identified by the EFP P (n), just like the
block-block entanglement [17].
3. Results and discussion
We first investigate the λ = 0 case. In this case, there are two phases in the system. The ground
state of the new phase is a chiral state Ok > 0 with gapless excitations [21]. It is separated by the
critical point αc = 1 from the ground state of Ok = 0 gapless phase. For the two phases, Mz = 0
and
fl,l+m =
1
mπ
sin
(
m
π
2
)
, α < 1,
1
2mπ
[1 − (−1)
m
] sin
(
m arcsin
(
1
α
))
, α > 1,
(19)
428
Emptiness formation probability for the spin- 1
2
XX chain
and
fl,l =
1
2
, (20)
for all α. Then, the results of S1, S2, S3 and S4 are given as follows, respectively [17].
S1 = 1, (21)
S2 =
2 −
[
(1 − 2
π
) log2(1 − 2
π
) + (1 + 2
π
) log2(1 + 2
π
)
]
, α < 1,
2 −
[
(1 − 2
πα
) log2(1 − 2
πα
) + (1 + 2
πα
) log2(1 + 2
πα
)
]
, α > 1,
(22)
and
S3 =
3 −
(
1 − 2
√
2
π
)
log2
(
1 − 2
√
2
π
)
−
(
1 +
2
√
2
π
)
log2
(
1 +
2
√
2
π
)
, α < 1,
3 −
(
1 − 2
√
2
πα
)
log2
(
1 − 2
√
2
πα
)
−
(
1 +
2
√
2
πα
)
log2
(
1 +
2
√
2
πα
)
, α > 1.
(23)
It is noted that dS1/dα = 0, but
dS2
dα
=
{
0, α < 1,
2
πα2 log2
(
1+ 2
πα
1− 2
πα
)
, α > 1,
(24)
and
dS3
dα
=
0, α < 1,
2
√
2
πα2 log2
(
1+
2
√
2
πα
1− 2
√
2
πα
)
, α > 1.
(25)
While, the EFP of the system are
P (1) =
1
2
, (26)
P (2) =
1
4
−
(
1
π
)2
, α < 1,
1
4
−
(
1
π
)2 ( 1
α
)2
, α > 1,
(27)
and
P (3) =
1
8
−
(
1
π
)2
, α < 1,
1
8
−
(
1
π
)2 ( 1
α
)2
, α > 1.
(28)
It is also noted that dP (1)/dα = 0, but
dP (2)
dα
=
0, α < 1,
2( 1
π
)2( 1
α
)3, α > 1,
(29)
429
P.Lou, J.Y.Lee
and
dP (3)
dα
=
0, α < 1,
2( 1
π
)2( 1
α
)3, α > 1.
(30)
It is displayed that at αc = 1, the behavior of EFP P (n) is similar to that of the block-block
entanglement Sn, i.e, the first derivative of the EFP, dP (n)/dα shows discontinuities, while P (n)
is continuous. The discontinuity in dP (n)/dα at αc = 1 does indicate the second-order quantum
phase transitions of the present case. Thus, there is one-to-one correspondence between quantum
phase transitions and the non-analyticity property of the EFP P (n).
When α → ∞, from the equation (19) we get
fl,l+m = fl+m,l = 0, (31)
for m > 1. Thus
P (n) =
(
1
2
)n
. (32)
Figure 2. The Emptiness Formation Probabi-
lity P (n) and its contour map changes with
parameter λ and α for n = 2.
Figure 3. The block-block entanglement Sn
and its contour map changes with parameter
λ and α for n = 10.
The numerical results for the EFP P (n) and its contour map with the variation of the pa-
rameters λ and α for n = 2 are displayed in figure 2. The numerical results for the block-block
entanglement Sn and its contour map changes with parameters λ and α for n = 10 are displayed
in figure 3. Comparing figure 2 with figure 1 and figure 3, it is noted that the phase diagrams are
consistent with each other. This means that there is one-to-one correspondence between quantum
phase transitions and the non-analyticity property of the EFP P (n) for the present model as well.
We have farther studied the properties of phases I, II, III and IV by using the combination
of the EFP P (n), the block-block entanglement Sn, and the magnetization Mz. Firstly, from
the magnetization Mz aspect angle, phases I and II are paramagnetic. Phases III and IV are
ferromagnetic phases. The Mz value is zero in both phases I and II. But, there is difference in
between phases III and IV. In phase III (called ferromagnetic δ phase), the magnetic moment Mz <
1
2
, while in phase IV (called ferromagnetic γ), Mz ≡ 1
2
in the long-range ordered ferromagnetic
phase. Thus, the magnetization Mz can be used to identify phases III and IV, but cannot be used
to characterize phases I and II. Secondly, the block-block entanglement Sn can be used to identify
phases I, II, III and IV, which may be seen from figure 3. The EFP P (n), just as the block-block
430
Emptiness formation probability for the spin- 1
2
XX chain
entanglement Sn, can also be used to distinguish the different phases of system as shown in figure 2.
For example, when the system is in the phase IV, since Mz ≡ 1
2
, and we have
fl,l = 1, (33)
and
fl,l+m = fl+m,l = 0. (34)
Thus
P (n) = 1. (35)
This implies that the probability to find a ferromagnetic string of the length “n” in the ground
state of phase IV is always 1, i.e, the long-range ordered ferromagnetic phase. While
Sn = 0. (36)
This indicates that the ground state of phase IV is ordered phase, and has no entanglement and
spin fluctuation.
It is noted that the behavior of the system occurs in between the phase II and III. If the quantum
phase transition takes place from the phase III to phase II, the EFP of the system decreases, but
the block-block entanglement Sn increases. That is to say, the phase III is more ordered(short-
range ordered), while the phase II is more entangled, and more spin fluctuating due to the quantum
frustration of the system.
4. Summary
In summary, we have explored the Emptiness Formation Probability P (n), the block-block en-
tanglement Sn, the magnetization Mz, and quantum phase transition, based on an exactly solvable
model of the spin- 1
2
XX Heisenberg chain with three-spin and uniform long-range interactions. We
find that the quantum phase transition is well characterized, just as the block-block entanglement
[17], by the Emptiness Formation Probability of the system. The evolution of the system shows a
number of characteristic features in the Emptiness Formation Probability P (n), the block-block
entanglement Sn, and the magnetization Mz. The EFP extends and deepens our previous under-
standing of quantum phase transitions.
Acknowledgements
This work was supported by the Korea Research Foundation Grant funded by the Korean
Government (MOEHRD) (KRF–2006–311–C00082). This work was also supported by the Korea
Research Foundation and The Korean Federation of Science and Technology Societies Grant funded
by Korean Government (MOEHRD, Basic Research Promotion Fund).
References
1. Sachdev S., Quantum Phase Transitions. Cambridge University Press, Cambridge, 1999.
2. Pachos J. K., Carollo A. C. M., Phil. Trans. R. Soc. A, 2006, 364, 3463.
3. Wen X-G., Quantum Field Theory of Many-Body Systems. Oxford University Press, 2004.
4. Chen Y., Zanardi P., Wang Z.D., Zhang F.C., New Journal of Physics, 2006, 8, 97.
5. Gu S.-J., Tian G.-S., Lin H.-Q., New Journal of Physics, 2006, 8, 61.
6. Wu L.-A., Sarandy M.S., Lidar D. A., Phys. Rev. Lett, 2004, 93, 250404.
7. Latorre J.I., Rico E., Vidal G., Quantum Inf. Comput, 2004, 4, 48.
8. Yang M.-F., Phys. Rev. A, 2005, 71, 030302(R).
9. Gu S.-J., Deng S.-S., Li Y.-Q., Lin H.-Q., Phys. Rev. Lett, 2004, 93, 86402.
10. Osterloh A., Amico L., Falci G., Fazio R., Nature (London), 2002, 416, 608.
431
P.Lou, J.Y.Lee
11. Osborne T. J., Nielsen M. A., Phys. Rev. A, 2002, 66, 032110.
12. Vidal G., Latorre J. I., Rico E., Kitaev A., Phys. Rev. Lett, 2003, 90, 227902.
13. Jin B. Q., Korepin V. E., J. Stat. Phys, 2004, 116, 79.
14. Fannes M., Haegemann B., Mosonyi M., J. Math. Phys, 2003, 44, 6005.
15. Keating J. P., Mezzadri F., Commun. Math. Phys, 2004, 252, 543.
16. Popkov V., Salerno M., Phys. Rev. A, 2005, 71, 012301.
17. Lou P., Lee J. Y., Phys. Rev. B, 2006, 74, 134402.
18. Lou P., Phys. Rev. B, 2005, 72, 064435.
19. Korepin V. E., Izergin A. G., Essler F. H. L., Uglov D. B., Phys. Lett. A, 1994, 190, 182.
20. Franchini F., Abanov A.G., J. Phys. A, 2005, 38, 5069.
21. Lou P., Wu W.-C., Chang M.-C., Phys. Rev. B, 2004, 70, 064405.
22. Jordan P., Wigner E., Z. Physik, 1928, 47, 631.
Ймовiрнiсть формування порожнини для спiн-1
2
XX ланцюжка
з триспiновою та однорiдною далекосяжною взаємодiями
П.Лу1,2, Ж.Й.Лi2
1 Факультет хiмiї, Унiверситет Сунгкянкван, Сувон, Корея
2 Факультет фiзики, унiверситет Ангуi, Гефей Ангуi, Китайська Народна Республiка
Отримано 29 березня 2007 р.
Дослiджується ймовiрнiсть формування порожнини, тобто ймовiрнiсть того, що n перших спiнiв у
ланцюжку є спрямованi вниз, для одновимiрної спiн- 1
2
XX ланцюжкової системи з триспiновою i
однорiдною далекосяжною взаємодiями як у критичному, так i у некритичному режимах. З вико-
ристанням ймовiрностi формування порожнини P (n), блок-блок сплутаностi Sn i намагнiченостi Mz
дослiджуються також властивостi такої спiнової системи. Важливо вiдзначити, що коли вiдбувається
квантовий фазовий перехiд з фази III у фазу II, ймовiрнiсть формування порожнини системи P (n)
зменшується, тодi як блок-блок сплутанiсть Sn зростає.
Ключовi слова: ймовiрнiсть формування порожнини, спiн- 1
2
XX ланцюжок, триспiновi i однорiднi
далекосяжнi взаємодiї
PACS: 75.10.Jm, 75.40.Cx, 75.40.-s, 73.43.Nq
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