Staggered long-range order for diluted quantum spin models

We study an annealed site diluted quantum XY model with spin $S\in \frac{1}{2}\mathbb{N}$. We find regions of the parameter space where, in spite of being a priori favourable for a densely occupied state, phases with staggered occupancy occur at low temperatures.


Introduction
Two quantum spin models (the XY model with spin 1 2 and the Heisenberg antiferromagnet) on the hypercubic lattice Z d (d ≥ 2) with the annealed site dilution are considered. The models are formulated in terms of the Hamiltonian (1.1) Here S i x 's are the standard spin-S operators acting on the site x and n x is the occupancy number indicating presence or absence (n x = 1 or n x = 0) of a particle at the site x. The parameters µ and κ are the chemical potential and the interaction parameter for the particle occupancy. The XY model with spin- 1 2 is obtained by the choosing (S , u) = ( 1 2 , 0) while S ≥ 1 2 and u = −1 yields the Heisenberg antiferromagnet.
Our main claim concerns the existence of a staggered long range order characterised by the presence of two distinct states (in the thermodynamic limit) with preferential occupation of either the even or the odd sublattice. Indeed, it will be proven that such states occur in a region of parameters µ and κ, at intermediate inverse temperatures, β.
The existence of such states can be viewed as a demonstration of an "effective entropic repulsion" caused by the interaction of quantum spins leading to an impactful restriction of the "available phase space volume". As a result, occupation of adjacent sites might turn out to be unfavourable-it results in an effective repulsion between particles in nearest neighbour sites and as a result eventually leads to a staggered order. It is easy to understand that this is the case for the annealed site diluted Potts model with large number of spin states q [3] where this effect is indeed caused by a pure entropic repulsion: two nearest neighbour occupied sites contribute the Boltzmann factor q + q(q − 1)e −β which is at low temperatures much smaller than the factor q 2 obtained from two next nearest neighbour spins that are free to take all possible spin values entirely independently. Actually, the same is true-even though less obvious-in the case of diluted models with classical continuous spins [4]. Our results constitute an extension of similar claims to a quantum situation.
To get a control on effective repulsion, we rely on a standard tool-the chessboard estimates which follow from reflection positivity. The classical references on this topic are [5,6,7,8,9] with a recent review [1]. For our case the treatment in [2] is especially useful. In particular, we use the setting from [2, Section 3.3] for an efficient formulation of the long range order in terms of coexistence of the corresponding infinite-volume KMS states.
While we are restricting ourselves only to the case u = 0, −1, the models with −1 < u < 0 are also covered by reflection positivity, hence results might be extended to this region. However, one would require bounds on the expectation of certain occupancy configurations (see Lemmas 3.4-3.7), that seem harder to achieve than in the cases u = 0, −1.
We introduce the models and state the main result in Section 2. The proof is deferred to Section 3.

Setting and Main Results
For a fixed even L ∈ N, we consider the torus T L = Z d /LZ d consisting of L d sites that can be identified with the set {−L/2 + 1, −L/2 + 2, . . . , L/2 − 1, L/2} d . On the torus T L we take the algebra A L of observables of all functions A : {0, 1} T L → M L where M L is s the C * -algebra of linear operators acting on the space ⊗ x∈T L C 2S +1 with S ∈ 1 2 N (complex (2S + 1) |T L |dimensional matrices).
A particular example of an observable is the Hamiltonian H L ∈ A L of the form (1.1) with the periodic boundary conditions (on the torus T L ), Here, E L is the set of all edges connecting nearest neighbour sites in the torus T L and (S 1 , S 2 , S 3 ) are the spin-S matrices. The Gibbs state on the torus is given by with Z L (β) = n Tr e −βH L . Infinite volume states of a quantum spin system are formulated in terms of KMS states, an analog of DLR states for classical systems. Let us briefly recall this notion in the form to be used in our situation. Here we follow closely the treatment from [2] which can be consulted for a more detailed discussion of KMS states in a setting similar to ours. Let A denote the C * algebra of quasilocal observables, where the overline denotes the norm-closure. We define the time evolution operators α (L) t acting on A ∈ A L and for any t ∈ R as α (L) It is well known that for a local operator A we can expand α (L) t (A) as a series of commutators, The map t → α (L) t extends to all t ∈ C and, as L → ∞, α (L) t converges in norm to an operator α t on A uniformly on compact subsets of C (one can consult the proof, for example, in [10] and see that the same proof structure works in this case). A state · β on A (a positive linear functional ( A β ≥ 0 if A ≥ 0) such that 1 β = 1) is called a KMS state (or is said to satisfy the KMS condition) with a Hamiltonian H at an inverse temperature β, if we have AB β = α −iβ (B)A β (2.6) for the above defined family of operators α t at imaginary values t = −iβ.
One can see that the Gibbs state (2.2) satisfies the KMS condition for the finite volume time evolution operator.
A special class of observables are classical events 1 F I obtained as a product of the identity I ∈ M L with the indicator 1 F of an occupation event F ⊂ {0, 1} T L . Often we will consider (classical) block events depending only on the occupation configuration on the block-cube of 2 d sites, C = {0, 1} d ⊂ T L . Namely, the events of the form E×{0, 1} T L \C where E ⊂ {0, 1} C . We will refer to these events directly as block events E and use a streamlined notation In particular, to characterise the long-range order states mentioned above, we introduce the block events G e = {n e } and G o = {n o } where n e and n o are the even and the odd staggered configurations on C: n e x = 1 iff x is an even site in C and n e x = 1 iff x is an odd site in C. Notice that the sets G e and G o are disjoint.
The main result for the quantum system with Hamiltonian (2.1) can now be stated as follows.
Theorem 2.1. Let u = −1 and S ≥ 1 2 or u = 0 and S = 1 2 . For each case there exists µ 0 > 0 and a function κ 0 (both depending on u, S , and d) that is positive on (0, µ 0 ) and such that for any µ > 0, κ < max(κ 0 (µ), 0), and any 0 < ε < 1 2 , there exists β 0 = β 0 (µ, κ, ǫ) such that for any β > β 0 there exist two distinct KMS states, · e β and · o β , that are staggered, The proof of this theorem is the content of Section 3. For the technical estimates, we will restrict ourselves to the two-dimensional case d = 2. The proof in higher dimensions employing the same methods is straigtforward but rather cumbersome. For d = 2 we construct the function κ 0 explicitly.
Notice that the claim is true for any negative κ. This is not so surprising, negative κ triggers antiferromagnetic staggered order at low temperatures. More interesting is the case, established by the theorem, when this happens for positive κ where it is a demonstration of an effective entropic repulsion stemming from the quantum spin.  1 Any such reflection (parallel P 1 and P 2 of distance L/2 in arbitrary half-integer position and orthogonal to any coordinate axis) will be called reflections through planes between the sites or simply reflections (we will not use the other reflections through planes on the sites that are useful for classical models).

Further, consider two subalgebras
is the set of all operators of the form A + ⊗ I acting on the subspace ⊗ x∈T + L C 2S +1 and I is the identity on the complemen- The reflection θ can be naturally elevated to an involution θ : acting on M + L in a properly parametrized basis as θ(A + ⊗ I) = I ⊗ A + and reflecting the configuration n, Finally, we say that a state · on A L is reflection positive with respect to θ if for any A, B ∈ A + L we have and Here, A denotes the complex conjugation of the matrix A. The standard consequence of the reflection positivity is the Cauchy-Schwartz inequality for any A, B ∈ A + L . In our situation of an annealed diluted quantum model, we are dealing with the state for any A ∈ A L and with the Hamiltonian H L ∈ A L of the form (2.1).
The standard proof of reflection positivity may be extended to this case. 1 Notice that on the torus, the reflection with respect to P 1 is identical with that with respect to P 2 (just notice that |x Lemma 3.1. The state · L, β is reflection positive for any θ through planes between the sites and any µ, κ ∈ R, β ≥ 0, and any u ≤ 0. consists of all terms of the Hamiltonian with (both) sites in T + L and D α θD α , with D α ∈ A + L indexed by α, are representing the terms containing the sites adjacent to both sides to the reflection plane.
Notice that to have a correct negative sign with the terms D α θD α , we need the condition u ≤ 0. Indeed, if {x, y} is an edge crossing the reflection plane the corresponding D α 's are 1 Recalling that in the standard basis iS 2 = iS 2 , we need u ≤ 0. Note that the term x,y n x n y (κ − S (S + 1)/S 2 ) is simply a constant times the identity for each n and can be bounded by ±d|T L | |κ − S (S + 1)/S 2 |, hence we can pull it out of the trace and the sum as a constant and ignore it, as we do for the remainder of the proof. For the claim (3.3) we need to show that for any A ∈ A + L . Adapting the standard proof, see e.g. [8, Theorem 2.1], by Trotter formula we get The needed claim will be verified once show that for all k.
Indeed, proceeding exactly in the same way as in the proof of Theorem 2.1 in [8], we can conclude that for each n, m ∈ {0, 1} T + L the operator F k (n, m) can be written as a sum of terms of the form completing thus the proof.
3.2. Chessboard estimates. Consider T L partitioned into (L/2) d disjoint 2 × 2 × · · · × 2 blocks C t ⊂ T L labeled by vectors t ∈ T L/2 with 2t denoting the position of their lower left corner. Clearly, C t = C + 2t with C 0 = C.
If t ∈ T L/2 with |t| = 1, we let θ t be the reflection with respect to the plane between C and C t corresponding to t. Further, if E is a block event, For other t's in T L/2 we define ϑ t (E) by a sequence of reflections (note that the result does not depend on the choice of sequence leading from C to C t .). If all coordinates of t are even this simply results in the translation by 2t.
Chessboard estimates are formulated in terms of a mean value of a homogenised pattern based on a block event E disseminated throughout the lattice, (3.10) If u ≤ 0, E 1 , ..., E m are block events, and t 1 , ..., t m ∈ T L/2 are distinct, we get, by a standard repeated use of reflection positivity, the chessboard estimates Note that we have chosen to split T L into 2 × 2 × · · · × 2 blocks with the bottom left corner of the basic block C at the origin (0, 0, . . . , 0). If we had instead replaced the basic block C by its shift C + e by the unit vector e = (1, 0, . . . , 0), the same estimate would hold with the new partition with all blocks shifted by e. We will use this fact in the sequel.
The proof of the useful property of subadditivity of the function q L, β for classical systems [1, Lemma 5.9] can be also directly extended to our case.
Proof. Using subadditivity of · L, β , we get Using now the chessboard estimate we get Let us introduce the set B of bad configurations, B = {0, 1} C \ (G e ∪ G o ), and use τ r to denote the shift by r ∈ T L . The proof of the existence of two distinct KMS states is based on the following lemma.

Lemma 3.3.
There exists a function κ 0 as stated in Theorem 2.1 such that for any ε > 0, µ > 0 and κ < κ 0 (µ) there exists β 0 such that for any β > β 0 , any L sufficiently large, and any distinct t 1 , t 2 ∈ T L , We claim that if β is a weak limit of L,M;β as L → ∞ and then M → ∞ then β is a KMS state at inverse temperature β invariant under translations by 2t for t ∈ T L . Indeed translation invariance comes from the spatial averaging in ρ L,M (E). As in [2] we need to show that β satisfies the KMS condition (2.6). For an observable A on the 'front' of the torus, T front L , we have If ε is small enough then the right-hand side of this inequality will be greater than 1/2, hence in the thermodynamic limit G e will dominate.
To prove Lemma 3.3 we use a version of Peierls' argument hinging on chessboard estimates.

Peierls argument.
For a given occupation configuration, consider the event τ 2t 1 (G e ) ∩ τ 2t 2 (G o ) that the blocks C t 1 and C t 2 have different staggered configurations described by G e and G o respectively. The idea is to show the existence of a contour separating the points t 1 and t 2 and to use chessboard estimates to show that occurence of such a countour is unprobable.
Consider the set of all blocks (labeled by) t ∈ T L/2 such that a translation of the even staggered configuration τ 2t (G e ) is occurring on it. Let ∆ ⊂ T L/2 be its connected component containing t 1 . Consider the component ∆ ⊂ T L/2 of ∆ c containing t 2 . The set of edges γ of the graph T L/2 between vertices of ∆ and its complement ∆ c is a minimal cutset of ∆. Informally, γ is a contour between ∆ with all its holes except the one containing t 2 filled up and the remaining component containing t 2 -a contour separating t 1 and t 2 . The standard fact is that the number of contours with a fixed number of edges |γ| = n separating two vertices t 1 and t 2 is bounded by c n with a suitable constant c.
Given a contour γ of length |γ| = n, there exists a coordinate direction such that there are at least n/d edges in γ aligned along this direction. Precisely half of them have their outer endpoint (the vertex in ∆) "on the left" of its inner endpoint, choosing (arbitrarily) the direction of the chosen coordinate axis (without loss of generality we can take for this the first coordinate axis) as e 1 , there are at least n/(2d) edges {t, t + e 1 } such that t ∈ ∆ and t + e 1 ∈ ∆. Now, the crucial claim is that with each contour we can associate at least 1/2 of the n/(2d) bad blocks (with a configuration from ϑ 2t (B)), all belonging to a given fixed partition: either to our original partition of T L labelled by T L/2 or to a new partition of T L with the basic block C shifted by a unit vector from T L in direction e 1 . Indeed, any block corresponding to an outer vertex t above is either bad or, if not, it has to be a translation τ 2t (G o ) of the odd staggered configuration (being the even staggered configuration would be in contradiction with the assumption that ∆ is a component of the set of blocks with even staggered configuration). However, then the block shifted by a unit vector in T L in direction e 1 features an odd staggered configuration on its left-hand half and an even staggered configurations on its right-hand half, i.e., a configuration that belongs to the properly shifted set B (here it is helpful that the set B is invariant with respect to the reflection through the middle plane of the block).
We use S (γ) to denote this collection of at least |γ|/(4d) bad blocks associated with contour γ. Given that, according to the construction above, all blocks from S (γ) belong to the same partition (either the original one or a shifted one), we can use the chessboard estimate based on the the corresponding partition to bound the probability that all blocks of a given set S (γ) are bad by As a result, assuming that q L, β (B) ≤ 1 (we will later show it can be made arbitrarily small), the expectation of the event τ 2t 1 (G e )∩τ 2t 2 (G o ) is bounded by Here, 2 |γ|/(2d)+1 is the bound on the number of sets S (γ) associated with the contour γ once the direction e 1 is chosen.
This leads to the final bound We now see that Lemma 3.3 will hold if q L, β (B) can be made arbitrarily small by tuning the parameters of the model correctly. Hence we turn our attention to this.
For the remaining technical part of this section we restrict ourselves to the two-dimensional case. We use Z (k) L (β) to denote the corresponding quantities for k ∈ {0, 1, . . . , 4}. For notational consistency we also denote the contribution of staggered configurations on T L as Z e L (β) and Z o L (β).

Lemma 3.4.
For any u, µ, κ ∈ R with |u| ≤ 1 we have Proof. It follows immediately from the observation that in these cases there are no interactions between spins at neighbouring sites.
Obtaining bounds for the remaining disseminated configurations is more difficult and will be done separately for the two considered types of quantum spin models. First we prove estimates for the antiferromagnetic case.

Lemma 3.5.
For u = −1 (the antiferromagnet) and any µ, κ ∈ R we have Proof. We present the proof for Z (4) L (β) and Z (3) L (β), the other two inequalities follow by a simpler application of the same method. First, using the unitary operator U = x∈T e L e iπS 2 x , we get For a site x 0 consider its four nearest neighbours x 1 , x 2 , x 3 , x 4 . The operator − x,y S x · S y can be written as as a sum of L 2 /2 operators of the form of B (4) x 0 := −S x 0 · 4 k=1 S x k summing over x 0 on even sublattice. According to [5,Theorem C.2], the largest eigenvalue of each operator B (4) x 0 is S (4S + 1). As a result, we get the bound Combining with the prefactor, the inequality follows.
For Z (3) L (β) we follow the same procedure combining, however, operators B (ℓ) x 0 := −S x 0 · ℓ k=1 S x k , with ℓ = 1, 2, 3, 4 neighbours of x 0 . Note that a dissemination of a block of three occupied and one unoccupied site throughout the lattice via reflections yields a pattern where 1/4 of the 2 × 2 blocks are empty and the remaining blocks are full, with the empty blocks evenly spaced throughout the lattice. Thus there are 5L 2 /4 edges with both endsites occupied and we can tile these edges by L 2 /8 copies of each of the operators B (ℓ) x 0 , ℓ = 1, 2, 3, 4. Observe that (4 + 3 + 2 + 1)L 2 /8 = 5L 2 /4 yields the correct number of edges. Nevertheless a tiling yielding the claimed bound uses L 2 /8 operators B (4) x 0 and L 2 /4 operators B (3) x 0 arranged in each 16 × 16 cell as shown below.
The inequality follows by using the claim [5, Theorem C.2] that the largest eigenvalue of the operator B (ℓ) x 0 is S (ℓS + 1). Collecting the terms we get the claimed bound.
The pattern of Z (2) L (β) consists of alternating double lines of occupied and unoccupied sites resulting in tiling with L 2 /4 operators of the form B (3) x 0 whose largest eigenvalue is S (3S + 1).
The bound for Z (1) L (β) is also straightforward with L 2 /4 edges and L 2 /8 of operators B (2) x 0 whose largest eigenvalue is S (2S + 1). Notice that tiling with L 2 /4 of operators B (1) x 0 would be also possible, but would lead to a bigger bound.
Explicit expressions for the function κ 0 are