Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors
Using the explicit factorized formulas for matrix elements (formfactors) of the spin operators between the eigenvectors of the Hamiltonian of a finite quantum XY-chain in a transverse field, the spontaneous magnetization for σ^x and σ^y is re-derived in a simple way.
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nasplib_isofts_kiev_ua-123456789-132932025-02-09T15:40:50Z Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors Спонтанна намагніченість квантового XY-ланцюжка з формфакторів для скінченного ланцюжка Iorgov, N.Z. Загальні питання теоретичної фізики Using the explicit factorized formulas for matrix elements (formfactors) of the spin operators between the eigenvectors of the Hamiltonian of a finite quantum XY-chain in a transverse field, the spontaneous magnetization for σ^x and σ^y is re-derived in a simple way. Використовуючи факторизованi формули для матричних елементiв (формфакторiв) спiнових операторiв мiж власними векторами гамiльтонiана скiнченного квантового XY-ланцюжка в поперечному полi, дано простий вивiд формули для спонтанної намагнiченостi σ^x та σ^y. This article is based on the talk given at the Bogolyubov Kyiv conference “Modern Problems of Theoretical and Mathematical Physics”, September 15-18, 2009, Kyiv, Ukraine. The work was supported by the Program of Fundamental Research of the Physics and Astronomy Division of the NAS of Ukraine, the Ukrainian FRSF grants 28.2/083 and 29.1/028, by French-Ukrainian program “Dnipro” M17-2009 and the joint project PICS of CNRS and NAS of Ukraine. 2010 Article Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors / N.Z. Iorgov // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 116-120. — Бібліогр.: 18 назв. — англ. 2071-0194 PACS 75.10Jm, 75.10.Pq, 05.50+q, 02.30Ik https://nasplib.isofts.kiev.ua/handle/123456789/13293 en application/pdf Відділення фізики і астрономії НАН України |
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Загальні питання теоретичної фізики Загальні питання теоретичної фізики Iorgov, N.Z. Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors |
| description |
Using the explicit factorized formulas for matrix elements (formfactors) of the spin operators between the eigenvectors of the Hamiltonian of a finite quantum XY-chain in a transverse field, the spontaneous magnetization for σ^x and σ^y is re-derived in a simple way. |
| format |
Article |
| author |
Iorgov, N.Z. |
| author_facet |
Iorgov, N.Z. |
| author_sort |
Iorgov, N.Z. |
| title |
Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors |
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Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors |
| title_full |
Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors |
| title_fullStr |
Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors |
| title_full_unstemmed |
Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors |
| title_sort |
spontaneous magnetization of quantum xy-chain from finite chain form-factors |
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Відділення фізики і астрономії НАН України |
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2010 |
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Загальні питання теоретичної фізики |
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https://nasplib.isofts.kiev.ua/handle/123456789/13293 |
| citation_txt |
Spontaneous Magnetization of Quantum XY-chain from Finite Chain Form-factors / N.Z. Iorgov // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 116-120. — Бібліогр.: 18 назв. — англ. |
| work_keys_str_mv |
AT iorgovnz spontaneousmagnetizationofquantumxychainfromfinitechainformfactors AT iorgovnz spontannanamagníčenístʹkvantovogoxylancûžkazformfaktorívdlâskínčennogolancûžka |
| first_indexed |
2025-11-27T12:55:30Z |
| last_indexed |
2025-11-27T12:55:30Z |
| _version_ |
1849948251110768640 |
| fulltext |
N.Z. IORGOV
SPONTANEOUS MAGNETIZATION OF QUANTUM
XY-CHAIN FROM FINITE CHAIN FORM-FACTORS
N.Z. IORGOV
Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03680, Ukraine; e-mail: iorgov@ bitp. kiev. ua )
PACS 75.10Jm, 75.10.Pq,
05.50+q, 02.30Ik
c©2010
Using the explicit factorized formulas for matrix elements (form-
factors) of the spin operators between the eigenvectors of the Hamil-
tonian of a finite quantum XY-chain in a transverse field, the spon-
taneous magnetization for σx and σy is re-derived in a simple way.
1. Introduction
The quantum XY-chain is one of the simplest models
which is rich enough from the point of view of physics
and, at the same time, admits a strict mathematical
analysis. The study of this model was started in [1],
where it was rewritten in terms of fermionic operators
by means of the Jordan–Wigner transformation. Now
this relation is a standard mean to study different prop-
erties (the spectrum of Hamiltonian [1, 2], correlation
functions [3–5], emptiness formation probability [6], and
entanglement entropy [7–9]) of the XY-chain. Although
the Hamiltonian of the model is equivalent to the Hamil-
tonian of a free fermion system, the spin operators σx
and σy are expressed in terms of fermionic operators in
a non-local way. This non-locality leads to non-zero av-
erages 〈σx〉 and 〈σy〉 (spontaneous magnetization) in the
ferromagnetic phase of the model in the thermodynamic
limit.
In [10], we propose an alternative way to study corre-
lation functions of the XY-model: we derive the formu-
las for matrix elements of the spin operators σx and σy
between the eigenvectors of the Hamiltonian of a finite
quantum XY-chain in a transverse field. These formu-
las allow one to obtain at least formal expressions for
the multipoint multitime correlation functions at a fi-
nite temperature. In this paper, as an application of
the formulas for form-factors, the value of spontaneous
magnetization for σx and σy is re-derived in a simple
way.
In Section 2 we recall the definition of a finite quantum
XY-chain in a transverse field, its phase diagram, and
eigenvalues of the Hamiltonian and give general com-
ments on matrix elements of the spin operators between
the eigenvectors of the Hamiltonian. Section 3 is de-
voted to the description of a relation between the model
of quantum XY-chain and the Ising model on a 2D lat-
tice. In Section 4, we present formulas for matrix el-
ements (form-factors) of the spin operators σx and σy
between the eigenvectors of the Hamiltonian of a finite
quantum XY-chain derived in [10]. In Section 5, these
formulas are rewritten for the case of an infinite-length
chain. Here, we also re-derive the value of spontaneous
magnetization for σx and σy.
2. The Finite Quantum XY-chain in a
Transverse Field
The Hamiltonian of the XY-chain of length n in a trans-
verse field h is [1, 2]
H = −1
2
n∑
k=1
(
1 + κ
2
σx
kσ
x
k+1 +
1− κ
2
σy
kσ
y
k+1 + hσz
k
)
,
(1)
where σik are Pauli matrices, and κ is the anisotropy. In
the case κ = 0, we get an XX-chain (isotropic case). The
value κ = 1 corresponds to the quantum Ising chain in a
transverse field. In what follows, we restrict ourselves to
the case 0 < κ ≤ 1 and suppose the periodic boundary
condition σik = σik+n.
Now consider the values of h. Due to the relationship
of the XY chain and the 2D Ising model which will be
discussed below, the coupling constant h plays the role
of a temperature-like variable. The value h > 1 cor-
responds to the paramagnetic (disordered) phase. The
value 0 ≤ h < 1 corresponds to the ferromagnetic
(ordered) phase. At h = 1, there is a second-order
phase transition. If 0 ≤ h < (1 − κ2)1/2, it is an os-
cillatory region (because of the oscillatory behavior of
the two-point correlation function). Another peculiar-
ity related to this region is the following. At fixed κ,
0 < κ ≤ 1, in the region where (1 − κ2)1/2 < h < 1,
the NS-vacuum energy is lower than the R-vacuum en-
ergy (asymptotically, if n → ∞, they coincide). In the
116 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
SPONTANEOUS MAGNETIZATION OF QUANTUM XY-CHAIN
region 0 ≤ h < (1 − κ2)1/2, there are intersections at
special values of h of these vacuum levels even at finite
n. The number of these intersections grows with n. For
a detailed analysis of the oscillatory region, see [3, 11].
In this paper, we consider the ferromagnetic phase
which corresponds to 0 ≤ h < 1 in order to obtain a
non-zero value of spontaneous magnetization.
Using the Jordan–Wigner and Bogoliubov transforma-
tions, the Hamiltonian H of the XY-chain can be rewrit-
ten as the Hamiltonian of a system of free fermions and
can be diagonalized [1, 2]. The relation between the en-
ergies ε and the momenta q of fermionic excitations is
ε(q) =
(
(h− cos q)2 + κ2 sin2 q
)1/2
, q 6= 0, π , (2)
ε(0) = 1− h, ε(π) = h+ 1 .
The Hamiltonian H commutes with the operator V =
σz
1σ
z
2 · · ·σz
n. Since V2 = 1, the eigenvectors are separated
into two sectors with respect to the eigenvalue of V =
σz
1σ
z
2 · · ·σz
n with specific sets of possible momenta (E is
the energy of state, i.e. the eigenvalue of H):
– NS-sector: V→ +1⇒ “half-integer” momenta
q ∈ NS =
{
2π
n
(j + 1/2)
}
⇒ E = −1
2
∑
q∈NS
±ε(q).
Each −ε(q) in the expression for the energy corre-
sponds to a fermionic excitation with momentum
q. The number of excitations is even.
– R-sector: V→ −1⇒ “integer” momenta
q ∈ R =
{
2π
n
j
}
⇒ E = −1
2
∑
q∈R
±ε(q). (3)
The number of excitations is even. In the param-
agnetic phase (h > 1), the energy of the fermionic
excitation ε(0) becomes negative. In this case, we
define ε(0) = h−1 together with the swapping be-
tween the absence/presence of the excitation with
zero momentum. In other words, although the an-
alytical expressions for the energies E in terms of h
and κ are the same in both phases, due to the re-
definition of ε(0) in the case of h > 1, the number
of excitations in the paramagnetic/ferromagnetic
phase is odd/even.
We will denote the eigenvectors by the values of the
excited momenta q corresponding to −ε(q) in the expres-
sion for the energy E .
Formally, in order to calculate any correlation func-
tion for the XY-chain, it is sufficient to insert a reso-
lution of the identity operator as a sum of projectors
to eigenspaces of the Hamiltonian between spin opera-
tors. It is the so-called Lehmann expansion. Then the
problem is reduced to the problem of finding the matrix
elements of the spin operators σx
k, σ
y
k , and σz
k between
eigenstates of the Hamiltonian H.
– Matrix elements of σz
k:
The operator σz
k commutes with V = σz
1σ
z
2 · · ·σz
n.
Therefore, the action of σz
k does not change the
sector. In fact, the operator σz
k can be presented as
a bilinear combination of the operators of creation
and annihilation of fermionic excitations. Thus,
the matrix elements of σz
k between the eigenvectors
of H can be easily calculated (most of them are 0).
We will not consider such matrix elements in this
paper.
– Matrix elements of σx
k and σy
k:
The operators σx
k and σy
k anticommute with V =
σz
1σ
z
2 · · ·σz
n. Therefore, their action changes the
sector. The operators σx
k and σy
k cannot be pre-
sented in terms of fermionic operators in a local
way. All the matrix elements of them between the
eigenvectors of H from different sectors are non-
zero!
The idea of the derivation [10] of form-factors for σx
k
and σy
k of a quantum finite XY-chain was to use the rela-
tions between three models: the model of quantum XY-
chain in a transverse field, the Ising model on a 2D lat-
tice and the N = 2 Baxter–Bazhanov–Stroganov (BBS)
model. The relation between the first and second mod-
els was observed in [12], the relation between the second
and third models was found in [13]. The latter relation
together with the results on the separation of variables
for the BBS model allowed one to prove [14] the formulas
for the matrix elements of a spin operator of the Ising
model found in [15, 16]. In [10], by using these relations
between the models, we transferred the formulas for the
form-factors of the N = 2 BBS model to the model of
quantum XY-chain. A summarizing overview of the re-
sults on the separation of variables of the BBS model is
given in [17].
In the following sections, we describe the relation be-
tween the model of quantum XY-chain in a transverse
field and the Ising model on a 2D lattice. The parame-
ters of the models are (h,κ) and (Kx,Ky), respectively.
Then we present the formulas for the matrix elements
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 117
N.Z. IORGOV
of the spin operators σx and σy between eigenvectors of
the Hamiltonian of a finite quantum XY-chain derived
in [10] and take the thermodynamic limit of these for-
mulas. The obtained formulas allow us to re-derive the
value of spontaneous magnetization for σx and σy.
3. Relation between a Quantum XY-chain and
the Ising Model on a Lattice
The row-to-row transfer-matrix of the two-dimensional
Ising model with parameters Kx, Ky can be chosen as
tIs = T
1/2
1 T2T
1/2
1 , (4)
where
T1 = exp
(
n∑
k=1
K∗y σ
z
k
)
, T2 = exp
(
n∑
k=1
Kx σ
x
kσ
x
k+1
)
. (5)
The spin configurations of the rows are chosen to be
labeled by the eigenvectors of the operators σx
k, and the
parameter K∗y is dual to Ky, i.e. tanhKy = exp(−2K∗y ).
In [12], M. Suzuki observed that if we choose Kx and
K∗y such that
tanh 2Kx =
√
1− κ2
h
, cosh 2K∗y =
1
κ
, (6)
then Hamiltonian (1) of a XY-chain will commute with
the transfer-matrix of the 2D Ising model (4), and these
two operators have a common set of eigenvectors.
The dispersion relation for the fermions of the 2D Ising
model with energies γ(p) and momenta p is
cosh γ(p) =
(tx + t−1
x )(ty + t−1
y )
2(t−1
x − tx)
−
t−1
y − ty
t−1
x − tx
cos p , (7)
tx = tanhKx, ty = tanhKy. We also have a relation
between ε(p) given by (2) and γ(p):
sinh γ(p) =
√
1− κ2
κ
√
κ2 + h2 − 1
ε(p) . (8)
Relation (8) between the energies of fermionic excita-
tions of these two models seems to be new. The exis-
tence of such a relation is surprising, because the com-
mutativity of Hamiltonian (1) of the XY-chain and the
transfer-matrix (4) of the 2D Ising model does not imply
a priori any relation between their eigenvalues.
4. Formula for the Matrix Elements
We use the Bugrij–Lisovyy formula [15, 16] for the ma-
trix element of a spin operator between the eigenvectors
|Φ0〉Is = |q1, q2, . . . , qK〉NS
Is and |Φ1〉Is = |p1, p2, . . . , pL〉RIs
of the transfer matrix (4) for the finite 2D Ising model
ΞΦ0,Φ1 = |Is〈Φ0|σx
m|Φ1〉Is|2 =
= ξ ξT
(
ty − t−1
y
tx − t−1
x
)(K−L)2/2 ∏
1≤k≤K
1≤l≤L
sinh2 γ(qk)+γ(pl)
2
sin2 qk−pl
2
×
×
K∏
k=1
∏NS
q 6=qk
sinh γ(qk)+γ(q)
2
n
∏R
p sinh γ(qk)+γ(p)
2
L∏
l=1
∏R
p 6=pl
sinh γ(pl)+γ(p)
2
n
∏NS
q sinh γ(pl)+γ(q)
2
×
×
K∏
k<k′
sin2 qk−qk′
2
sinh2 γ(qk)+γ(qk′ )
2
L∏
l<l′
sin2 pl−pl′
2
sinh2 γ(pl)+γ(pl′ )
2
, (9)
ξ4 = 1− (sinh 2Kx sinh 2Ky)−2 =
1− h2
κ2
, (10)
ξ4T =
∏NS
q
∏R
p sinh2 γ(q)+γ(p)
2∏NS
q,q′ sinh γ(q)+γ(q′)
2
∏R
p,p′ sinh γ(p)+γ(p′)
2
, (11)
ty − t−1
y
tx − t−1
x
=
1− κ2
κ
√
κ2 + h2 − 1
, (12)
where we used (6) to write equivalent expressions in
terms of different parameters.
We use the following main result of paper [10]: the
matrix elements of spin operators between the eigenvec-
tors |Φ0〉XY = |q1, q2, . . . , qK〉NS
XY from the NS-sector and
|Φ1〉XY = |p1, p2, . . . , pL〉RXY from the R-sector of Hamil-
tonian (1) of the XY-chain are
|XY〈Φ0|σx
m|Φ1〉XY|2 =
=
κ
2(1 + κ)
(
C
1/2
Φ0,Φ1
+ C
−1/2
Φ0,Φ1
)2
ΞΦ0,Φ1 , (13)
|XY〈Φ0|σy
m|Φ1〉XY|2 =
=
κ
2(1− κ)
(
C
1/2
Φ0,Φ1
− C−1/2
Φ0,Φ1
)2
ΞΦ0,Φ1 , (14)
118 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
SPONTANEOUS MAGNETIZATION OF QUANTUM XY-CHAIN
where ΞΦ0,Φ1 is given by (9) and
CΦ0,Φ1 =
∏
p∈R e
γ(p)/2∏
q∈NS e
γ(q)/2
∏K
k=1 e
γ(qk)∏L
l=1 e
γ(pl)
. (15)
In the case of the quantum Ising chain (κ = 1), the
formula for the matrix element of the spin operator σx
k
derived in [14] can be expressed in terms of the energies
of excitations ε(q). In the case of a general XY-chain,
we were not able to find an analogous explicit formula
and need to use relation (8).
5. Asymptotics of Form-Factors in the Limit of
Infinite Chain and Spontaneous
Magnetization
In this section, we analyze the asymptotics of different
parts of form-factors in the thermodynamic limit (the
length n→∞) of the XY-chain. They can be obtained
from the integral representations for form-factors at fi-
nite n [16, 18]. We slightly change the method. Our
derivation is based on the following formulas valid for
arbitrary |λ| < 1 and |λ| > 1, respectively:
lim
n→∞
log
∏R
p (λ− eγ(p))∏NS
p (λ− eγ(p))
= 0 . (16)
lim
n→∞
log
∏NS
p (λ− e−γ(p))∏R
p (λ− e−γ(p))
= 0 . (17)
At λ = 0 and λ = eγ(q), they give, respectively,
Λ−1 =
1
2
(∑
q
NS
γ(q)−
∑
p
R
γ(p)
)
→ 0 ,
eη(q) =
∏NS
p
(
1− e−γ(q)−γ(p)
)∏R
p
(
1− e−γ(q)−γ(p)
) → 1 .
In turn, these two formulas yield∏NS
p sinh γ(q)+γ(p)
2∏R
p sinh γ(q)+γ(p)
2
→ 1 .
Using it twice for ξT (see (11)) for fixed q and p, respec-
tively, and taking into account that the left-hand sides
of (16) and (17) vanish exponentionally in n (see the
derivation below), we get∏NS
q
∏R
p sinh γ(q)+γ(p)
2∏NS
q,q′ sinh γ(q)+γ(q′)
2
,
∏NS
q
∏R
p sinh γ(q)+γ(p)
2∏R
p,p′ sinh γ(p)+γ(p′)
2
→ 1.
Therefore, ξT → 1 in the thermodynamic limit.
Finally, in the limit of the infinite XY-chain, formulas
(13), (14) and (9) become
|XY〈Φ0|σx
m|Φ1〉XY|2 =
= ΞΦ0,Φ1
2κ
1 + κ
cosh2
∑K
k=1 γ(qk)−
∑L
l=1 γ(pl)
2
, (18)
|XY〈Φ0|σy
m|Φ1〉XY|2 =
= ΞΦ0,Φ1
2κ
1− κ
sinh2
∑K
k=1 γ(qk)−
∑L
l=1 γ(pl)
2
, (19)
ΞΦ0,Φ1 = ξ
(
ty − t−1
y
tx − t−1
x
)(K−L)2/2 ∏
1≤k≤K
1≤l≤L
sinh2 γ(qk)+γ(pl)
2
sin2 qk−pl
2
×
×
K∏
k=1
1
n sinh γ(qk)
L∏
l=1
1
n sinh γ(pl)
×
×
K∏
k<k′
sin2 qk−qk′
2
sinh2 γ(qk)+γ(qk′ )
2
L∏
l<l′
sin2 pl−pl′
2
sinh2 γ(pl)+γ(pl′ )
2
. (20)
Formulas (18) and (19) at K = L = 0 allow us to
re-obtain the formulas for the spontaneous magnetiza-
tion found in [12]. Indeed, for the quantum XY-chain
in the ferromagnetic phase (0 ≤ h < 1) and in the
thermodynamic limit n → ∞ (when the energies of
|Φ0〉XY = |vac〉NS and |Φ1〉XY = |vac〉R asymptotically
coincide, giving the degeneration of the ground state),
the spontaneous magnetization is
〈σx,y〉XY = XY〈Φ0|σx,y|Φ1〉XY ,
〈σx〉XY =
√
2
(
κ2(1− h2)
(1 + κ)4
)1/8
, 〈σy〉XY = 0 .
At the end of this section, we give the derivation of
(16) and (17). It repeats the derivation of the asymp-
totics of η(q) in [18]. We use the function
T 2(z) =
∏R
q (cosh γ̄(p)− cos q)∏NS
q (cosh γ̄(p)− cos q)
, z = e−ip ,
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 119
N.Z. IORGOV
where γ̄(p) is defined as γ(p) in (7) but with the in-
terchange tx ↔ ty (i.e. γ̄(p) is the energy of fermionic
excitations corresponding to the evolution in the trans-
verse direction on a 2D Ising lattice). The evaluation of
the products over q gives
T (z) = tanh(nγ̄(p)/2) . (21)
On the other hand, due to the relation
cosh γ̄(p)− cos q =
tx − t−1
x
ty − t−1
y
(cosh γ(q)− cos p),
we have
log T (z) =
1
2
log
( ∏R
q (z − eγ(q))(z − e−γ(q))∏NS
q (z − eγ(q))(z − e−γ(q))
)
.
At |λ| > 1,
1
iπ
∮
|z|=1
dz log T (z)
z − λ
= − 1
iπ
∮
|z|=1
dz log(z− λ)
T ′(z)
T (z)
=
= log
∏NS
q (λ− e−γ(q))∏R
q (λ− e−γ(q))
,
where we have integrated by parts and taken the con-
tribution of the simple poles of T ′(z)/T (z) into account.
Similarly, this integral at |λ| < 1 is
1
iπ
∮
|z|=1
dz log T (z)
z − λ
= log
∏R
q (λ− eγ(q))∏NS
q (λ− eγ(q))
,
where we also took the simple pole at z = λ into account.
Due to (21), we have log T (z)→ 0 if n→∞. This proves
(16) and (17) in both cases of λ.
This article is based on the talk given at the Bo-
golyubov Kyiv conference “Modern Problems of Theoret-
ical and Mathematical Physics”, September 15-18, 2009,
Kyiv, Ukraine. The work was supported by the Program
of Fundamental Research of the Physics and Astronomy
Division of the NAS of Ukraine, the Ukrainian FRSF
grants 28.2/083 and 29.1/028, by French-Ukrainian pro-
gram “Dnipro” M17-2009 and the joint project PICS of
CNRS and NAS of Ukraine.
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Received 24.09.09
СПОНТАННА НАМАГНIЧЕНIСТЬ КВАНТОВОГО
XY-ЛАНЦЮЖКА З ФОРМФАКТОРIВ
ДЛЯ СКIНЧЕННОГО ЛАНЦЮЖКА
М.З. Iоргов
Р е з ю м е
Використовуючи факторизованi формули для матричних еле-
ментiв (формфакторiв) спiнових операторiв мiж власними век-
торами гамiльтонiана скiнченного квантового XY-ланцюжка в
поперечному полi, дано простий вивiд формули для спонтанної
намагнiченостi σx та σy.
120 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
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