Three Order Parameters in Quantum XZ Spin-Oscillator Models with Gibbsian Ground States

Quantum models on the hyper-cubic d-dimensional lattice of spin-1/2 particles interacting with linear oscillators are shown to have three ferromagnetic ground state order parameters. Two order parameters coincide with the magnetization in the first and third directions and the third one is a magnetization in a continuous oscillator variable. The proofs use a generalized Peierls argument and two Griffiths inequalities. The class of spin-oscillator Hamiltonians considered manifest maximal ordering in their ground states. The models have relevance for hydrogen-bond ferroelectrics. The simplest of these is proven to have a unique Gibbsian ground state.


Introduction
In this paper we consider quantum lattice models of oscillators interacting with spins whose variables are indexed by the sites of a hyper-cube Λ with the finite number of sites |Λ| in the hyper-cubic lattice Z d . Interaction is considered to be short-range and translation invariant. The corresponding Hamiltonian H Λ is expressed in terms of the oscillators variables q Λ = (q x , x ∈ Λ) ∈ R |Λ| and spin 1 2 Pauli matrices S l Λ = (S l x , x ∈ Λ, l = 1, 3), defined in the tensor product of the 2 |Λ| -dimensional Euclidean space and the space of square integrable functions L 2 Λ = (⊗E 2 ) |Λ| ⊗ L 2 (R |Λ| ), For products of operators, functions and variables we use the following notation: The scalar product in (⊗E 2 ) |Λ| , L 2 Λ will be denoted by (·, ·) 0 , (·, ·), respectively. The Schwartz space of test functions on R n will be denoted by S(R n ).
We require that the Hamiltonian is well defined and bounded from below on (⊗E 2 ) |Λ| ⊗ C ∞ 0 (R |Λ| ), i.e. the tensor product of the 2 |Λ| -dimensional Euclidean space and the space of infinitely differentiable functions with compact supports. The ground state average for an observable (operator) F is determined by where Z Λ is a partition function. For partial cases of F we have where the integration is performed over R |Λ| andq [A] is the operator of multiplication by q x . We will employ the orthonormal basis ψ 0 Λ (s Λ ) of the Euclidean space (⊗E 2 ) |Λ| , diagonalizing S 3 Λ , which is chosen in the following way: ψ 0 Λ (s Λ ) = ⊗ x∈Λ ψ 0 (s x ), s x = ±1, ψ 0 (1) = (1, 0), ψ 0 (−1) = (0, 1), S 1 ψ 0 (s) = ψ 0 (−s), S 3 ψ 0 (s) = sψ 0 (s). For F ∈ L 2 Λ we have the following decomposition where the summation is performed over the |Λ|-fold Cartesian product (−1, 1) |Λ| of the set (−1, 1). The Hamiltonians in (1.1) are employed in hydrogen-bond ferroelectric crystal models, considered in [1,2,3], and describe interaction between heavy ions (oscillators with constant frequency) and protons (spins). The second term with J A = 0, |A| ≥ 2 corresponds to the energy of protons, tunneling along hydrogen bonds from one well to another, and J x is associated with the tunneling frequency. The last term in the expression for our Hamiltonian describes many-body interaction between protons (J 1 (A) is the intensity of the |A|-body interaction).
A rigorous analysis of a mean-field version of the Hamiltonian in (1. uniformly in lattice sites proportional to |Λ| −1 was carried out in [3] in the framework of the Bogolyubov approximating Hamiltonian method [4] and occurrence of spin and oscillator orderings (the corresponding order parameters are non-zero) for two-body interaction between protons at non-zero temperatures was proved. To establish such orderings for ground states without the mean-field limit in a general case is an important task for a theory. The oscillator and spin orderings are established if one proves the existence of ferromagnetic oscillator and spin long-range orders (lro's) in ground states for the corresponding Hamiltonians. This means that the ground state averages q xqy Λ , S j x S j y Λ , j = 1, 3 are bounded uniformly in Λ from below by positive numbers . Occurrence of the ferromagnetic lro's implies the existence of the spin order parameters (magnetizations in the first and third directions) M l Λ = |Λ| −1 x∈Λ S l x , l = 1, 3, and the oscillator order parameter Q Λ = |Λ| −1 x∈Λ q x in the thermodynamic limit (Λ → Z d ) since the ground state averages of their squares are uniformly bounded in Λ from below by a positive number.
In this paper we find functions V Λ , depending on oscillator variables, for which ground states or eigenstates Ψ Λ of the Hamiltonians in (1.1) are Gibbsian with the linear in q Λ spin-oscillator quasi-potential energy U is the ground state of the free oscillator Hamiltonian, that is the first term in the right-hand side of (1.1) with η = 0. We prove the maximal ordering in the corresponding systems (provided some simple conditions on U 0 and φ x are satisfied): magnetizations M l Λ , l = 1, 3, Q Λ are non-zero. No other ground states are known with such the property.
Gibbsian ground states were introduced by Kirkwood and Thomas in [5] in XZ spin-1 2 models with Hamiltonians (linear in S 1 ) that include the spin part of (1.1), i.e. the second and third terms, with only J x = 0 and the periodic boundary condition (this boundary condition is not essential). Matsui in [6,7] enlarged a class of spin-1 2 XZ-type models in which Gibbsian states exist. The method was further developed by Datta and Kennedy in [8]. An application of classical spins systems for constructing of quantum states was given in [9].
In [10] we showed how to find V Λ for a given Gibbsian ground state and established existence of lro's in S 1 and S 3 for a wide class of the spin-1 2 XZ models (see also [11,12]). This reference contains the most simple proofs of the existence of lro's in ground states of quantum many-body systems. A reader may find a review of the results concerning several quantum orders in it (see also [13,14,15]).
The ground state in (1.3) can be represented in the following equivalent form From (1.5), orthonormality of the basis (see the beginning of the third section) and the equalities ψ 2 0 (q)dq = 1, ψ 2 0 (q)qdq = 0 it follows that where σ x (s Λ ) = s x and Equalities in (1.6) reduce a calculation of averages in our quantum systems to a calculation of averages indexed by a star in Ising models. For a short-range ferromagnetic potential energy U * the first Griffiths inequality holds: σ [A] * Λ ≥ 0. As a result the following statement (principle) is true.
2) and ferromagnetic lro occur in the Ising model with the ferromagnetic potential energy U * given by (1.5), that is σ x σ y * Λ > 0 uniformly in Λ, then ferromagnetic lro occurs in oscillator variables and S 3 in the quantum spin-oscillator system with the Hamiltonian (1.1) and ground state Ψ Λ in (1.3).
Usefulness of Gibbsian ground states is explained by comparative simplicity of a proof of existence of lro. Our results show that Gibbsian ground states are expected to appear in many quantum spin-oscillator systems with non-trivial interactions.
Our paper is organized as follows. In Section 2 we formulate our main results in a lemma and theorem. In next sections we prove them.

Main result
We establish that Gibbsian ground states exist for J A depending on q Λ if . V Λ determines an unbounded operator. Negative J A generate positive functions V Λ and this enables us to prove the following lemma.
, is an eigenfunction of the Hamiltonian (1.1) with the zero eigenvalue and is its (unique) ground Remark 1. If the functions J A are only negative then Ψ Λ is the ground state of the self-adjoint extension of the Hamiltonian preserving positive definiteness.
The most simple translation invariant short-range U 0 is ferromagnetic II. Let the positive constants C, where |A| = 2. Then ground state ferromagnetic lro in S 1 occurs in arbitrary dimension d.
This equality follows from the equalities In the simplest case the conditions of item II of Theorem 1 can be checked without difficulty.
Note that if one uses the Pauli matrices with 1 2 instead of the unity as matrix elements then V Λ should be changed by adding to W A the number −|A| ln 2.
If one chooses the anti-ferromagnetic U 0 in (2.1), specifically, where x, y are nearest neighbors, then it can be easily proved that the spin lro in the third direction will be anti-ferromagnetic, generating a staggered magnetization (spins at the even and odd sublattices take different values). The interesting and important property of the Hamiltonians with V Λ given by (2.1) is that they are simply related to generators of stationary Markovian processes (see, also, [16]). We believe that it is possible to apply the same mathematical technique for proving existence of order and phase transitions in equilibrium quantum systems and non-equilibrium stochastic systems (see [17]).

Proof of Lemma 1
For our purpose it is convenient to pass to a new representation. It is determined by the Hilbert space of sequences of functions F (q Λ ; s Λ ), s x = ±1, which are found in the expansion of the vector F ∈ L 2 Λ mentioned at the beginning of the introduction, with the scalar product where the integration is performed over R |Λ| . Here we took into account the orthonormality of the basis, i.e. the equality where δ s,s ′ is the Kronecker delta. Let Here we took into account that differentiation commutes with T Λ . Our Hamiltonian is rewritten as follows The remarkable fact is that the symmetric operator P A and the harmonic operator h x have both common eigenvector Ψ Λ with the zero eigenvalue (see Remark 3 in the end of the paper). Note that the space of ground states of the operator h x (eigenfunctions with the zero eigenvalue) is 2 |Λ| -fold degenerate since S 3 x is diagonal and the Laplacian is translation invariant. From (1.4) and the definition of T x (φ) if follows that T −1 x (φ)Ψ Λ is equal to ψ 0 (q x ) multiplied by a function independent of q x The proof that Ψ Λ is an eigenvector with the zero eigenvalue of P A is inspired by our previous paper [10]. For simplicity we will omit q Λ in the expression for U in (1.3). Taking into consideration the equalities we obtain Here we changed signs of the spin variables s A in the first term in the sum in s Λ . Positive definiteness of the Hamiltonian follows from the following proposition. The scalar product in the Hilbert space L 2 Λ is given by (F 1 , F 2 ) = (F 1 (q Λ ), F 2 (q Λ )) dq Λ , where the integration is performed over R |Λ| . We have to show that −(P A F (q Λ ), F (q Λ )) 0 ≥ 0. Let us define the operator It is not difficult to check on the basis ψ 0 Λ that where I is the unit operator. Here we used the following equality For the operator P + A we have It is convenient to introduce the new scalar product The operator P + A is symmetric with respect to the new scalar product since It is not difficult to check that Here we took into account that the function under the sign of exponent is invariant under changing signs of the spin variables s A . From (3.1), (3.2) it follows that −P A ≥ 0.

From Proposition 3 it follows that H
The fact that the Hamiltonian is essentially self-adjoint is derived from the following proposition (see example X.9.3 in [18]). Proof . The following two inequalities are valid where µ is a real number, |y| = max j |y| j , || · || is the scalar product in the Hilbert space of square integrable functions. We tacitly assume that x 2 means the operator of multiplication of a squared variable. The first inequality follows from the inequalities (∂ j = ∂ ∂x j ) Here we took into account that V is positive and [∂ j , x k ] = ∂ j x k − x k ∂ j = δ j,k . As a result ||(y, x)ψ|| ≤ a||(−∆ + V + µ 2 x 2 )ψ|| + b||ψ||, a = µ −2 ǫ, b = n 2 + |y| 2 4ǫ .
For µ = 0 number a can be arbitrary small and from the Kato-Rellich theorem [18] it follows that the essential domain of −∆ + V + µ 2 x 2 + (y, x) coincides with the essential domain of −∆+V +µ 2 x 2 . But the last one coincides with S(R n )∩D(V ) [18, Theorem X.59]. Inequality (3.3) is sufficient, also, for the proof of the proposition in the case µ = y = 0 (this is explained in Example X.3 in [18]).
Since S 3 Λ is a diagonal operator on (⊗C 2 ) |Λ| the operator is the direct sum of 2 |Λ| copies of the minus |Λ|-dimensional Laplacian plus the three functions coinciding with the three functions in Proposition 4. From Proposition 4 it follows that this operator is essentially self-adjoint on the set ( which is the essential set for H Λ . This implies that H Λ is positive definite on its domain D(H Λ ) and Ψ Λ is its ground state.
Proof of uniqueness. We have to establish that the symmetric semigroup P t Λ , generated by −H Λ , maps non-negative functions into (strictly) positive functions (increases positivity) and this will imply that the ground state is unique (see Theorem XIII.44 in [18]). We will establish this property with the help of a perturbation expansion. The kernel of the semigroup P t 1 , generated by h Λ + V Λ , is expressed in terms of the Feynman-Kac (FK) formula [18,19] is the conditional Wiener measure and w Λ (t) is the sequence of continuous paths. The semigroup P t is represented as a perturbation series in powers of V 0 This series is convergent in the uniform operator norm [20] since V 0 is a bounded operator. Its perturbation expansion is given by where τ n+1 = t . We now use the following simple inequality x then it follows from the positivity of the the kernel P t 1 and V 0 that Here we utilized the semigroup property ofP t , the inequalities e a ≥ e −|a| , V 0 ≥ V − , Now, it is easily proved as in [10] that the matrix V − is irreducible. As a result there exists a positive integer n such that and that V n − , has positive non-diagonal elements [21,22]. Hence the kernel in the left-hand side of (3.4) is positive.

Order parameters
In this section we will prove Theorem 1, i.e. occurrence of different types of lro in the considered quantum systems. In the proof we will rely on the following basic theorem. where ϕ(A) ≥ 0, s x = ±1. Let, also, the uniform bound ϕ(x, y) ≥φ > 0 for nearest neighbors x, y hold and ϕ(A) = 0 for A with odd number of sites. Then for a sufficiently largeφβ > 1 and the dimension d ≥ 2 there is the ferromagnetic lro, that is, for the Gibbsian two point spin average the uniform in Λ bound holds where σ x (s Λ ) = s x and the magnetization (an order parameter) The proof of this theorem is based on an application of the generalized Peierls principle (argument). It will be given in the end of this section (see, also, [17,26]. The next theorem is the consequence of the basic theorem. Proof of item I of Theorem 1. Condition (1.2) shows that where J 2 (x ∪ A) = J 0 (x; A) and J 2 (A) = 0 for odd |A|. The last term is equal to where s ∅ = 1 and A 1 ∆A 2 = (A 1 ∪ A 2 )\(A 1 ∩ A 2 ). Due to translation invariance of interaction the first term is bounded by |Λ| A J 2 0 (0; A), where the summation is performed over Z d , and this expression is finite since the interaction is short-range. Hence subtracting a finite constant proportional to |Λ| from U * one sees that the result admits representation (4.1) with positive J * instead of ϕ such that J * (A) = 0 for odd |A| and for nearest neighbors x, y the inequality J * (x, y) ≥ (2η 2 µ + α)J. The basic theorem and Proposition 1 imply occurrence of ferromagnetic lro in S 3 and oscillator lro.
Proof of Theorem 2. Let χ ± x = 1 2 (1 ± σ x ) then one obtains Since the systems are invariant under the transformation of changing signs of spins the third and the second terms in the right-hand side of last equality are equal to zero and Hence if then the ferromagnetic lro occurs, i.e.
where a is independent of Λ. If one succeeds in proving that there exists a positive function E 0 (β) and positive constants a, a ′ independent of Λ such that and proves that E 0 is increasing at infinity then (4.2), (4.3) will hold for a sufficiently large inverse temperature β. The Peierls principle reduces the derivation of (4.4) to the derivation of the contour bound.
Peierls principle. Let the contour bound hold Proof of contour bound. Bricmont and Fontain derived the contour bound for the spin systems with the potential energy (4.1) with the help of the second Griffiths [23] and Jensen inequalities [24] (see also [25,26]) where dµ is a probability measure on a measurable space. Their proof starts form the inequality As a result (β ′ =φβ) x,y ∈Γ where ·, · Λ[Γ] is the average corresponding to the potential energy In the last line we applied the Jensen inequality. From the second Griffiths inequality it follows that the average σ x σ y Λ[Γ] is a monotone increasing function in ϕ A . So, in the potential energy determining this average we can put ϕ A = 0, except A = x, y and leave the coefficientφ in front of the bilinear nearest-neighbor pair potential in (4.1) without increasing the average. This leads to That is, Here we used in the denominator the inequality e −2 −1 β ′ ≤ e 2 −1 β ′ . Obviously, E tends to infinity if β ′ tends to infinity. This implies (4.5).

Discussion
We showed that in the considered lattice spin-boson models with J A ≤ 0 ground states are Gibbsian and the ground state averages for special observables are reduced to averages in classical Ising models. This means that existence of ground states order parameters is connected with existence of order parameters in the associated Ising models and that a breakdown of symmetries in the quantum systems is determined by a breakdown of symmetries in Ising models. We considered the free boundary conditions implying that for the cases of the perturbation V Λ , considered in the two theorems, the ground state averages of q x , S 3 x are zero, that is q x Λ = 0, S 3 x Λ = 0 if the associated Ising potential energy is an even function. In order to make such the averages non-zero (explicit symmetry breaking) one has to introduce special boundary conditions (quasi-averages) which have to single out pure Gibbsian states in the associated Ising models. It is known [28] that for the two-dimensional ferromagnetic Ising nearest-neighbor model there are two boundary conditions which generate pure states and that every other state is a convex linear combination of these two states. A discussion of a construction of ground states in lattice spin and fermion quantum systems with an explicit symmetry breaking a reader may find in [29].
The property of the ground state Ψ Λ to be a ground state with the zero eigenvalue of a local Hamiltonian H x was found earlier for special isotropic anti-ferromagnetic Heisenberg chains with valence bond ground state in [27].