Correlation Inequalities for the Quantum XY Model

We show the positivity or negativity of truncated correlation functions in the quantum XY model with spin 1/2 (at any temperature) and spin 1 (in the ground state). These Griffiths–Ginibre inequalities of the second kind generalise an earlier result of Gallavotti.

setting, at least with S = 1 2 and S = 1. It follows that many truncated correlation functions take a fixed sign.
Let denote the (finite) set of sites that host the spins. The Hilbert space of the model is H = ⊗ x∈ C 2S+1 with S ∈ 1 2 N. Let S i , i = 1, 2, 3 denote usual spin operators on C 2S+1 ; that is, they satisfy the commutation relations [S 1 , S 2 ] = iS 3 , and other relations obtained by cyclic permutation of the indices 1, 2, 3. They also satisfy the identity (S 1 ) 2 +(S 2 ) 2 +(S 3 ) 2 = S(S + 1). Finally, let S i x = S i ⊗ 1 \{x} denote the spin operator at site x. We consider the hamiltonian Here, J i A is a nonnegative coupling constant for each subset of A ⊂ and each spin direction i ∈ {1, 2}. The expected value of an observable a (that is, an operator on H ) in the Gibbs state with hamiltonian H and at inverse temperature β > 0 is Clearly, other inequalities can be generated using spin symmetries. The corresponding inequalities for the classical XY model have been proposed in [5].
The proof of Theorem 1 can be found in Sect. 3. It is based on Ginibre's structure [3]. It is simpler than Gallavotti's, who used an ingenious approach based on the Trotter product formula, on a careful analysis of transition operators, and on Griffiths' inequalities for the classical Ising model [2]. Our proof allows us to go beyond pair interactions.
A consequence of Theorem 1 is the monotonicity of certain spin correlations with respect to the coupling constants: Corollary 2 Under the same assumptions as in the above theorem, we have for all A, B ⊂ that The first inequality states that correlations increase when the coupling constants increase (in the same spin direction). The second inequality is perhaps best understood classically; if the first component of the spins increases, the other components must decrease because the total spin is conserved. Corollary 2 follows immediately from Theorem 1 since We use this corollary in Sect. 2 to give a partial construction of infinite-volume Gibbs states.
The case of higher spins, S > 1 2 , is much more challenging, but we have obtained an inequality that is valid in the ground state of the S = 1 model. Recall that the states · and ·; · s , defined in The proof of this theorem can be found in Sect. 4. It uses Theorem 1.

Infinite Volume Limit of Correlation Functions
Infinite volume limits of Gibbs states are notoriously delicate issues; we show in this section that Theorem 1 (and Corollary 2) give partial but useful information: For Gibbs states "with + boundary conditions", the infinite volume limits of many correlation functions exist. Let us recall the notion of infinite volume limit. Let (t ) ⊂⊂Z d be a sequence of real or complex numbers, indexed by finite subsets of Z d . We say that t → t as along every sequence ( n ) of increasing finite subsets that tends to Z d . That is, the sequence satisfies n+1 ⊃ n , and, for any finite A ⊂⊂ Z d , there exists n A such that n ⊃ A for all n ≥ n A . We assume the interaction is finite-range: There exists R such that J i Let ∂ R = R \ be the exterior boundary of . We consider the hamiltonian H η R with field on the exterior boundary: Temperature does not play a rôle in this section so we set β = 1. The relevant (finite volume) Gibbs state is the linear functional that, to any operator a on H , assigns the value Traces are taken in H R (and a on H is identified with a ⊗ 1 ∂ R on H R ). We comment below on the relevance of this definition for Gibbs states. But first, we observe that the limit η → ∞ exists.

Proposition 4 For all operators a on H , the limit in (2.4) exists and is equal to
where traces are taken in H and Proof We can add a convenient constant to the hamiltonian without changing the corresponding Gibbs state, so we consider Tra exp where P + x is the projector onto the eigenstates of S 1 x with eigenvalue 1 2 . Writing P + A = x∈A P + x , we have Then, since the Trotter expansion converges uniformly in η, we have The challenge is to prove that a (+) converges as Z d , for any operator a on H with ⊂⊂ Z d (again, a on H is identified with a ⊗ 1 \ on H with ⊃ ). We can use the correlation inequalities to establish the existence of the infinite volume limit for certain operators a.  Finally, let us comment on the relevance of this Gibbs state with + boundary conditions. Consider the case of the isotropic XY model, where J 1 A = J 2 A for all A ⊂⊂ Z d . At low temperatures, the infinite volume state · (+) = lim Z d · (+) is expected to be extremal and to describe a system with spontaneous magnetisation in the direction 1 of the spins. One can apply rotations in the 1-2 plane to get all other (translation-invariant) extremal Gibbs states. Much work remains to be done to make this rigorous, but Theorem 5 seems to be a useful step.

The Case S = 1 2
We can define the spin operators as S i = 1 2 σ i , where the σ i s are the Pauli matrices It is convenient to work with the hamiltonian with interactions in the 1-3 spin directions, namely Following Ginibre [3], we introduce the product space H ⊗ H . Given an operator a on H , we consider the operators a + and a − on the product space, defined by 3) The Gibbs state in the product space is · = 1 Z ( ) 2 Tr · e −H ,+ , (3.4) where H ,+ = H ⊗ 1 + 1 ⊗ H . Without loss of generality, we set β = 1 in this section. We also need the Schwinger functions in the product space, namely ·; · s = 1 Z ( ) 2 Tr · e −s H ,+ · e −(1−s)H ,+ . (3.5)

Lemma 6 For all observables a, b on
It is enough to prove the second line. The right side is equal to The first two lines of the right side give 2 a; b s and the last two lines give 2 a b .
Next, a simple lemma with a useful formula.

Lemma 7 For all operators a, b on H , we have
The proof is straightforward algebra. Notice that both terms of the right side have positive factors. Now comes the key observation that leads to positive (and negative) correlations.

Lemma 8 There exists an orthonormal basis on
As a consequence, there exists an orthonormal basis of H ⊗H such that S 1 x,+ , S 1 x,− , S 3 x,+ , and −S 3 x,− have nonnegative matrix elements.
Proof of Theorem 1 for S = 1 2 We use Lemma 6 in order to get In order to make visible the sign of the right side, we expand the exponentials in Taylor with ε i , ε j ∈ {1, 3}. Further, all products ( S i x ) ± can be expanded using Lemma 7 in polynomials of S i x,± , still with positive coefficients. Finally, observe that the total number of operators S 3 x,− , x ∈ , is always even; then each S 3 x,− can be replaced by −S 3 x,− . We now have the trace of a polynomial, with positive coefficients, of matrices with nonnegative elements (by Lemma 8). This is positive.
The second inequality (with S 3 instead of S 2 ) is similar. The only difference is that ( S 3 x ) − gives a polynomial where the number of S 3 x,− is odd. Hence the negative sign.

The Case S = 1
This section is much more involved, and our result is sadly restricted to the ground state. Our strategy is inspired by the work of Nachtergaele on graphical representations of the Heisenberg model with large spins [6]. We consider a system where each site hosts a pair of spin 1 2 particles. The inequalities of Theorem 1 apply. By projecting onto the triplet subspaces, one gets a correspondence with the original spin 1 system. We prove that all ground states of the new model lie in the triplet subspace, so the inequality can be transferred. These steps are detailed in the rest of the section.
It is perhaps worth noticing that the tensor products in this section play a different rôle than those in Sect. 3.

The New Model
We introduce the new lattice˜ = × {1, 2}. The new Hilbert space is (4.1) Let R i be the following operator on C 2 ⊗ C 2 : Here, σ i are the Pauli matrices in C 2 as before. We denote R i x = R i ⊗1 \{x} the corresponding operator at site x ∈ . As before, we choose the interactions to be in the 1-3 spin directions; the hamiltonian onH is The coupling constants J i A are the same as those of the original model on H . The expected value of an observable a in the Gibbs state with hamiltonianH is where the normalisationZ ( ) is the partition functioñ We similarly define Schwinger functions ·; · ∼ s for s ∈ [0, 1]. It is useful to rewriteH as the hamiltonian of spin 1 2 particles on the extended lattice˜ . Given a subset X ⊂˜ , we denote suppX its natural projection onto , i.e. suppX = x ∈ : (x, 1) ∈ X or (x, 2) ∈ X . (4.6) We also denote D(˜ ) the family of subsets of˜ where each site of appears at most once. Notice that |D(˜ )| = 3 | | . Finally, let us introduce the coupling constants From these definitions, we can writeH using Pauli operators as (4.8)

Correspondence with the Spin 1 Model
The Hilbert space at a given site, C 2 ⊗C 2 , is the orthogonal sum of the triplet subspace (that is, the symmetric subspace, which is of dimension 3) and of the singlet subspace (of dimension 1). Let P triplet denote the projector onto the triplet subspace, and let P triplet = ⊗ x∈ P triplet .
We define a new Gibbs state, namely with partition function Z ( ) = Tr P triplet e −βH . In order to state the correspondence between the models with different spins, let V : C 3 → C 2 ⊗ C 2 denote an isometry such that

Proof of Theorem 3
We can assume that for any x ∈ , there exist i ∈ {1, 3} and A x such that J i A > 0the extension to the general case is straightforward. Since all ground states lie in the triplet subspace, we have for all subsets A, B ⊂ and for all s ∈ [0, 1], We used Theorem 1. We have obtained the first inequality of Theorem 3. The second inequality follows in the same way.