Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States
A class of general spin 1/2 lattice models on hyper-cubic lattice Zd, whose Hamiltonians are sums of two functions depending on the Pauli matrices S¹, S² and S³, respectively, are found, which have Gibbsian eigen (ground) states and two order parameters for two spin components x, z simultaneously fo...
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2006
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Цитувати: | Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States / W. Skrypnik // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1464422019-02-10T01:25:05Z Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States Skrypnik, W. A class of general spin 1/2 lattice models on hyper-cubic lattice Zd, whose Hamiltonians are sums of two functions depending on the Pauli matrices S¹, S² and S³, respectively, are found, which have Gibbsian eigen (ground) states and two order parameters for two spin components x, z simultaneously for large values of the parameter α playing the role of the inverse temperature. It is shown that the ferromagnetic order in x direction exists for all dimensions d ≥ 1 for a wide class of considered models (a proof is remarkably simple). 2006 Article Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States / W. Skrypnik // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 8 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 82B10; 82B20; 82B26 http://dspace.nbuv.gov.ua/handle/123456789/146442 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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A class of general spin 1/2 lattice models on hyper-cubic lattice Zd, whose Hamiltonians are sums of two functions depending on the Pauli matrices S¹, S² and S³, respectively, are found, which have Gibbsian eigen (ground) states and two order parameters for two spin components x, z simultaneously for large values of the parameter α playing the role of the inverse temperature. It is shown that the ferromagnetic order in x direction exists for all dimensions d ≥ 1 for a wide class of considered models (a proof is remarkably simple). |
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Article |
| author |
Skrypnik, W. |
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Skrypnik, W. Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Skrypnik, W. |
| author_sort |
Skrypnik, W. |
| title |
Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States |
| title_short |
Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States |
| title_full |
Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States |
| title_fullStr |
Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States |
| title_full_unstemmed |
Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States |
| title_sort |
order parameters in xxz-type spin 1/2 quantum models with gibbsian ground states |
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Інститут математики НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/146442 |
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Order Parameters in XXZ-Type Spin 1/2 Quantum Models with Gibbsian Ground States / W. Skrypnik // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 8 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT skrypnikw orderparametersinxxztypespin12quantummodelswithgibbsiangroundstates |
| first_indexed |
2025-07-10T23:59:03Z |
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2025-07-10T23:59:03Z |
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1837306429832167424 |
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Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 011, 6 pages
Order Parameters in XXZ-Type Spin 1
2
Quantum Models with Gibbsian Ground States
Wolodymyr SKRYPNIK
Institute of Mathematics, 3 Tereshchenkivs’ka Str., Kyiv 4, 01601 Ukraine
E-mail: skrypnik@imath.kiev.ua
Received October 19, 2005, in final form January 16, 2006; Published online January 24, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper011/
Abstract. A class of general spin 1
2 lattice models on hyper-cubic lattice Zd, whose Hamilto-
nians are sums of two functions depending on the Pauli matrices S1, S2 and S3, respectively,
are found, which have Gibbsian eigen (ground) states and two order parameters for two spin
components x, z simultaneously for large values of the parameter α playing the role of the
inverse temperature. It is shown that the ferromagnetic order in x direction exists for all
dimensions d ≥ 1 for a wide class of considered models (a proof is remarkably simple).
Key words: Gibbsian eigen (ground) states; quantum spin models
2000 Mathematics Subject Classification: 82B10; 82B20; 82B26
1 Introduction and main result
The existence of several long-range orders (lro’s) and order parameters in quantum many-body
systems is an important problem which is the first step towards a description of their phase
diagrams.
In our previous paper [1] we found a class of quantum spin 1
2 XZ-type systems on the hyper-
cubic lattice Zd with a Gibbsian ground state, characterized by the classical spin potential
energy U0(sΛ), in which two lro’s can occur for the spin operators S1 and S3 in a dimension
greater than one. In such the systems there is always the ferromagnetic lro for S1 even for d = 1
if a simple condition for U0 holds. The Hamiltonians, determined as symmetric matrices in the
2|Λ| dimensional complex Hilbert space C2|Λ| with the Euclidean scalar product (·, ·), were given
by
HΛ =
∑
A⊂Λ,|A|>0
JAPA, JA ≤ 0, PA = S1
[A] − e−
α
2
WA(S3
Λ), S1
[A] =
∏
x∈A
S1
x,
WA(S3
Λ) = U0
(
S3A
Λ
)
− U0
(
S3
Λ
)
, S3A
Λ =
(
S3
Λ\A,−S
3
A
)
. (1)
JA are real numbers, S1
x, S3
x, x ∈ Zd are the ‘unity’ Pauli matrices, Λ ⊂ Zd is a hypercube with
a finite cardinality |Λ| (number of sites). S1 is diagonal and S3
1,1 = −S3
2,2 = 1. S1 has zero
diagonal elements and S1
1,2 = S1
2,1 = 1. The matrices at different sites commute. Sl
[A] is the
abbreviated notation for the tensor product of the matrices Sl
x, x ∈ A and the unity matrices Ix,
x ∈ Λ\A.
The Gibbsian non-normalized state ΨΛ is given by
ΨΛ =
∑
sΛ
e−
α
2
U0(sΛ)Ψ0
Λ(sΛ), α ∈ R+, (2)
mailto:skrypnik@imath.kiev.ua
http://www.emis.de/journals/SIGMA/2006/Paper011/
2 W. Skrypnik
where the summation is performed over (×(−1, 1))|Λ|,
Ψ0
Λ(sΛ) = ⊗x∈Λψ0(sx), ψ0(1) = (1, 0), ψ0(−1) = (0, 1).
These systems differ from the XZ spin 1
2 systems, which admit Gibbsian ground states con-
sidered in [2]. The potential energy of the associated classical Gibbsian system, which generates
the ground state, is found there in the form of a perturbation expansion in a small parameter
(an analog of α). The authors proved that there is the ferromagnetic lro for S3 in the ground
state in some of their ferromagnetic systems. Our proof of the S1-lro is a simplified analog of
their proof. Uniqueness of a Gibbsian translation invariant ground state is established in the
thermodynamic limit for general XZ models with a sufficiently strong magnetic field in [3].
In [4] the classical Gibbsian states are identified with ground states of quantum Potts models.
The structure of the considered Hamiltonians are close to the Hamiltonians of XZ spin systems
which are represented as a sum of a diagonal and non-diagonal parts.
In this paper we consider the Hamiltonians
HΛ = H0Λ + VΛ, H0Λ =
∑
A, A′ ⊆ Λ,
A ∩A′ = ∅
φA,A′S1
[A]S
2
[A′], (3)
where φA,A′ are real valued coefficients, S2 is the second Pauli matrix with the zero diagonal
elements such that S2
1,2 = −S2
2,1 = −i and VΛ depends on S3
Λ. We find the expression for VΛ
which guarantees that ΨΛ given by (2) is the eigen (ground) state. This result is a generalization
of our previous result since with the help of (6) we reduce our Hamiltonian to the Hamiltonian (1)
with JA depending in S3
Λ. Our result is summarized in the following theorem
Theorem. Let VΛ be given by
VΛ = −
∑
A⊆Λ
JA
(
S3
A
)
e−
α
2
WA(S3
Λ), JA
(
S3
A
)
=
∑
A′⊆A
(−i)|A′|φA\A′,A′S3
[A′]. (4)
Then
I. ΨΛ is an eigenfunction of the Hamiltonian (3);
II. ΨΛ is its ground state if φA,A′ = 0 for odd |A′| and JA ≤ 0;
III. lro for S1 occurs in the eigenstate ΨΛ if lim
Λ→Zd
WA(sΛ) exists for |A| = 2 and is uni-
formly bounded. Moreover, 〈S1
[A]〉Λ ≥ a > 0, where a is a constant independent of Λ if
lim
Λ→Zd
WA(sΛ) exists and is uniformly bounded.
IV. lro occurs for S3 in the eigenstate ΨΛ if lro occurs in the classical spin system with the
potential energy U0.
If H0Λ coincides with the Hamiltonian of the XX Heisenberg model
H0Λ =
∑
x,y∈Λ
φx,y
(
S1
xS
1
y + S2
xS
2
y
)
,
then it can be shown without difficulty, utilizing the equality
(
S3
)2 = I, that for the following
choice U0(sΛ) =
∑
x∈Λ
uxsx the matrix VΛ in (4) is given by
VΛ =
∑
x,y∈Λ
φx,y
[
S3
xS
3
y coshα(ux − uy)− (S3
x − S3
y) sinh(α(ux − uy))− cosh(α(ux − uy))
]
.
Order Parameters in XXZ-Type Spin 1
2 Quantum Models with Gibbsian Ground States 3
If one puts φx,y = φx−y, φx = φ|x| = 0, |x| 6= 1, φ1 = J , q = eα and
ux = x1 + · · ·+ xd, x =
(
x1, . . . , xd
)
then the following Hamiltonian of the XXZ Heisenberg model is derived
HΛ = J
∑
〈x,y〉∈Λ
[
S1
xS
1
y + S2
xS
2
y +
q + q−1
2
S3
xS
3
y
]
− 2J
∑
〈x<y〉∈Λ
[
q − q−1
2
(S3
x − S3
y) +
q + q−1
2
]
,
where the summations are performed over nearest neighbor pairs and “<” means lexicographic
order. It is remarkable that the term linear in S3
x contributes only on the boundary of Λ. These
Hamiltonians coincide with the Hamiltonians proposed in [5, 6]. A reader may find out that
the Gibbsian ground states of these Hamiltonians are not unique. Gibbsian ground states for
a partial case of our Hamiltonians were considered in [7].
If FA depends on S3
A, S1
A then its expectation value in a state ΨΛ is given by
〈FA〉Λ = (ΨΛ,ΨΛ)−1
(
ΨΛ, FA
(
S1
A, S
3
A
)
ΨΛ
)
,
where (·, ·) is the Euclidean scalar product on C2|Λ|. Ferromagnetic lro for Sl occurs if
〈Sl
xS
l
y〉Λ ≥ al > 0, (5)
where the constants al are independent of Λ. It implies that the magnetization M l
Λ is an order
parameter in the thermodynamic limit since
〈(M l
Λ)2〉Λ ≥ al > 0, M l
Λ = |Λ|−1
∑
x∈Λ
Sl
x.
Besides, the inequality in the statement III of the Theorem for |A| = 1 implies that
〈M1
Λ〉Λ ≥ a > 0.
It is well known that in the classical Ising model with a ferromagnetic short-range potential
energy U0, generated by the nearest neighbor bilinear pair potential, there is the ferromag-
netic lro. Hence our quantum systems for such U0 admit two order parameters M l
Λ, l = 1, 3 in
the thermodynamic limit for sufficiently large α since for such U0 the condition in the statements
III–IV of the Theorem is true. For small values of α the magnetization in the third direction
vanishes for short-range pair interaction potentials.
The last statement of the theorem follows without difficulty since S3 is a diagonal matrix and
the ground state expectation value 〈FA〉Λ of a function FA depending on S3
A equals the classical
Gibbsian expectation value of the same function depending on the classical spins sA correspon-
ding to the potential energy U0(sΛ) and the “inverse temperature” α. The orthogonality of the
basis
(Ψ0
Λ(sΛ),Ψ0
Λ(s′Λ)) =
∏
x∈Λ
δsx,s′
x
,
where δs,s′ is the Kronecker symbol, has to be applied for proving that.
The proofs of the second statement of the Theorem is based on the proof that HΛ is positive
definite. Its condition implies that the semigroup generated by HΛ has positive matrix elements.
As a result the operator
H+
Λ = e
α
2
U0(S3
Λ)HΛe
−α
2
U0(S3
Λ).
4 W. Skrypnik
generates a Markovian process with a stationary state. We show that it is symmetric and positive
definite in the Euclidean scalar product with the operator weight e−αU0(S3
Λ).
For sufficiently small α and a pair simple ferromagnetic interaction with φA,A′ = 0 for |A′| 6= 0
lro at non-zero low temperature occurs for S1 [8].
2 Proof of Theorem
It easy to check that
S2 = −iS3S1 = iS1S3. (6)
From this equality the following equalities are derived
S2
[A′] = (−i)|A′|S3
[A′]S
1
[A′],
H0Λ =
∑
A, A′ ⊆ Λ,
A ∩A′ = ∅
(−i)|A′|φA,A′S1
[A]S
3
[A′]S
1
[A′] =
∑
A, A′ ⊆ Λ,
A ∩A′ = ∅
(−i)|A′|φA,A′S3
[A′]S
1
[A∪A′].
As a result
H0Λ =
∑
A⊆Λ
JA(S3
A)S1
[A]. (7)
(7) and (4) lead to the following expression for the Hamiltonian (3)
HΛ =
∑
A⊂Λ, |A|>0
JA(S3
A)PA. (8)
Proof of I. S1 flips spins:
S1
[A]Ψ
0
Λ(sΛ) = Ψ0
Λ(sA
Λ) = Ψ0
Λ(sΛ\A,−sA), S3
xΨ0
Λ(sΛ) = sxΨ0
Λ(sΛ).
These identities lead to
JA
(
S3
A
)
PAΨΛ =
∑
sΛ
(
JA(−sA)Ψ0
Λ(sΛ\A,−sA)− JA(sA)e−
α
2
WA(sΛ)Ψ0
Λ(sΛ)
)
e−
α
2
U0(sΛ).
From the definition of WA and after changing signs of the spin variables sA in the first term
in the sum it follows that
JA
(
S3
A
)
PAΨΛ =
∑
sΛ
[
JA(−sA)Ψ0
Λ(sΛ\A,−sA)e−
α
2
U0(sΛ) − JA(sA)Ψ0
Λ(sΛ)e−
α
2
U0(sA
Λ )
]
=
∑
sΛ
JA(sA)
(
e−
α
2
U0(sA
Λ ) − e−
α
2
U0(sA
Λ )
)
Ψ0
Λ(sΛ) = 0.
That is, every term in the sum for HΛΨΛ in (8) is equal to zero. This proves the statement.
Proof of II. It is necessary to prove that the Hamiltonian is positive-definite. For that purpose
we’ll use the operator H+
Λ introduced in the introduction. It is not difficult to check on the
basis Ψ0
Λ that
H+
Λ =
∑
A⊆Λ
JA
(
S3
A
)
e−
α
2
WA(S3
Λ)
(
S1
[A] − I
)
,
Order Parameters in XXZ-Type Spin 1
2 Quantum Models with Gibbsian Ground States 5
where I is the unity operator. If
F =
∑
sΛ
F (sΛ)Ψ0
Λ(sΛ), H+
ΛF =
∑
sΛ
(H+
ΛF )(sΛ)Ψ0
Λ(sΛ)
then, taking into account that JA(sA) is en even function in sx, we obtain
(H+
ΛF )(sΛ) = −
∑
A⊆Λ
JA(sA)e−
α
2
WA(sΛ)
(
F (sΛ)− F
(
sA
Λ
))
.
H+
Λ is symmetric with respect to the new scalar product
(F, F ′)U0 =
(
e−αU0(S3
Λ)F, F ′).
The check is given by
(H+
ΛF, F
′)U0 =
(
e−αU0(S3
Λ)H+
ΛF, F
′) =
∑
A⊆Λ
(
JA(S3
A)e−
α
2
[U0(S3
Λ)+U0(S3A
Λ )](S1
[A] − I)F, F ′)
=
∑
A⊆Λ
(
JA(S3
A)e−
α
2
[U0(S3
Λ)+U0(S3A
Λ )]F, (S1
[A] − I)F ′) = (F,H+
ΛF
′)U0 .
Here we used the equalities
e−
α
2
U0(S3
Λ)S1
[A] = S1
[A]e
−α
2
U0(S3A
Λ ), e−
α
2
U0(S3A
Λ )S1
[A] = S1
[A]e
−α
2
U0(S3
Λ)
and the fact that S1
[A] commutes with JA(S3
A). From the definitions it follows that
(H+
ΛF, F
′)U0 =
(
HΛe
−α
2
U0(S3
Λ)F, e−
α
2
U0(S3
Λ)F ′). (9)
H+
Λ is positive definite. This is a consequence of the relations
(H+
ΛF, F )U0 = −
∑
A⊆Λ
∑
sΛ
JA(sA)e−
α
2
[U0(sΛ)+U0(sA
Λ )]
(
F (sΛ)− F
(
sA
Λ
))
F (sΛ)
= −1
2
∑
A⊆Λ
∑
sΛ
JA(sA)e−
α
2
[U0(sΛ)+U0(sA
Λ )]
(
F (sΛ)− F
(
sA
Λ
))2 ≥ 0. (10)
Here we took into account that the exponential weight in the sum is invariant under changing
signs of spin variables sA. From (9), (10) it follows that HΛ is, also, positive definite. Statement
is proved.
Proof of III. We have to prove (5) for l = 1. From orthogonality of the basis it follows that
ZΛ = (ΨΛ,ΨΛ) =
∑
sΛ
e−αU0(sΛ)
and
〈S1
[A]〉Λ = Z−1
Λ
∑
sΛ
e−αU0(sΛ)e−
α
2
WA(sΛ) ≥ inf
sΛ,A
e−
α
2
WA(sΛ) ≥ e
−α
2
max
sΛ,A
|WA(sΛ)|
.
This proves the statement.
6 W. Skrypnik
3 Discussion
The proposed perturbations VΛ of the initial Hamiltonian H0Λ seem complicated. But it is not
always so. If the initial Hamiltonian coincides with the Hamiltonian of the XX Heisenberg
model, then in some cases, mentioned in the introduction, the perturbation is very simple and
produces an anisotropic quadratic in S3
x, x ∈ Λ term and a boundary term linear in S3
x, x ∈ Λ
only. A reduction of VΛ to a simpler form for a quadratic in S3
x, x ∈ Λ function U0 can be
found in [1]. There is an interesting problem to find out all the cases of the initial Hamiltonians,
commuting with the total spin in the third direction, φA,A′ and U0 leading to simple anisotropic
generalized XXZ models.
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1 Introduction and main result
2 Proof of Theorem
3 Discussion
|