Critical temperature of Heisenberg models on regular trees, via random loops

We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-$\frac12$ XXZ model and the spin-1 nematic model. Our formula is conjectured to be valid for large-dimensional cubic lattices. Our method of proof uses a probabilistic representation in terms of random loops.


Introduction and main result
The main goal of this study is to predict an expression for the critical temperature of a family of quantum spin systems on the cubic lattice Z ν that holds asymptotically for large dimension ν. More precisely, we propose the first two terms in the expansion in powers of ν −1 . The family of quantum spin systems includes the spin 1 2 ferromagnetic and antiferromagnetic Heisenberg models and the XXZ model. We also consider spin 1 quantum nematic systems. Our results are expected to be exact but they are not rigorous on Z ν . In fact we do not perform calculations with the cubic lattice but we consider the model on regular trees with d descendants; we obtain the first two terms of the critical inverse temperature in powers of d −1 . For trees our computations are completely rigorous. We conjecture that our expression applies to Z ν when taking d = 2ν − 1.
1.1. Random-loop model. Our method is based on using a random loop representation, which we now describe. The relevant model of random loops may be defined for arbitrary finite graphs, here we consider mainly trees. Let T denote an infinite rooted tree where each vertex has d ≥ 2 offspring, and write ρ for its root. We sometimes refer to the number of offspring of vertex as its outdegree. For m ≥ 0 let T m denote the subtree of T consisting of the first m generations (ρ being generation zero). Write V m and E m for the vertex-and edge sets of T m .
Let P m (⋅) denote a probability measure governing a collection ω = (ω xy ∶ xy ∈ E m ) of independent Poisson processes on the interval [0, 1], indexed by the edge-set E m , each having rate β (the inverse-temperature). We refer to realizations of ω as a collections of links, and to ω xy as the links supported by the edge xy. Thus, disjoint sub-intervals I, J ⊆ [0, 1] independently receive uniformly placed links, their number being Poisson-distributed with mean β I and β J , respectively. We write E m [⋅] for expectation under P m (⋅).
A given link is assigned to be a cross with probability u, otherwise a double-bar, independently between different links. The collection of links then decomposes T m × [0, 1] into a collection of disjoint loops in a natural way. Rather than giving a formal definition here, we refer to Fig. 1. A formal definition may be found e.g. in [17,Sect. 2.1].
The total number of loops is denoted = (ω). We actually work with a weighted version of P m (⋅), denoted P (θ) m (⋅) with a positive parameter θ. This is the probability measure whose Figure 1. Random loops coming from a configuration ω of crosses and bars, in the case when the underlying graph is a line with seven vertices. To each vertex corresponds a vertical line segment which is a copy of the interval [0, 1]. On following a loop one reverses direction when traversing a doublebar, maintains direction when traversing a cross, and proceeds periodically in the vertical direction. In this example there are (ω) = 4 loops. .

Note that P
(1) m = P m . All loops are small when β is small, this may be shown e.g. as in [9, Thm. 6.1]. But it is expected that there exists β c , that depends on the parameter θ and the outdegree d, such that a given points lies in an infinite loop with positive probability for β > β c . Our main result is a formula for β c ; it is asymptotic in the outdegree d → ∞, namely and we can prove that there are infinite loops for β > β c in the vicinity of β c . For a more precise statement, see Theorem 1.1 below. The first study of this model on trees is due to Angel [2], who established the presence of long loops for a range of parameters β when d ≥ 4; he only considered the case u = 1 and θ = 1. Angel's results were extended by Hammond [10,11]; he gave a precise characterisation of the critical parameter β c for large enough d. The formula (1.1) was established in [6] in the case θ = 1, our study following a suggestion of Hammond. Very recently Hammond and Hegde [12] proved that the formula (1.1) for θ = 1 truly identifies the critical point, not only in the local sense considered here and in [6]; their results hold for large d. Another extension to θ ≠ 1 has independently been proposed by Betz where S (i) x , i = 1, 2, 3 denotes the ith spin operator at site x ∈ Λ. Here, ∆ ∈ [−1, 1] is a parameter.
As was progressively understood in [15,1,17], this quantum system is represented by the model of random loops with θ = 2 and u = 1 2 (1 + ∆). Indeed, the quantum two-point correlation function is given by loop correlations, where {x ↔ y} is the event that (x, 0) and (y, 0) belong to the same loop. It follows that magnetic long-range order is related to the occurrence of large loops. On Z 3 , the critical inverse temperature has been computed numerically; it was found that For large ν, the lattice Z ν behaves like a tree of outdegree d = 2ν − 1. Our formula (1.1) gives In the case of spin-1 systems, the Hilbert space is H Λ = ⊗ x∈Λ C 3 and the hamiltonian is

5)
See [17]. The phase diagram of this model was determined in [7]. For 0 < u < 1 the system displays nematic long-range order at low temperatures (if d ≥ 3; also in the ground state when d = 2). This was rigorously proved in [14,17]. The corresponding loop model has parameter θ = 3, and the same u as in (1.5). Loop correlations are related to nematic long-range order, namely ⟨A x A y ⟩ = 2 9 P (θ=3) Λ (x ↔ y), (1.6) with A x = (S x ) 2 − 2 3 . We are not aware of numerical calculations of the critical inverse temperature β c for this model on Z 3 . With θ = 3 and d = 2ν − 1, the formula (1.1) gives In the rest of this paper we deal only with the probabilistic model of random loops defined above, and we allow θ to be any (fixed) positive real number. Our main result is that, as the distance between x and y goes to ∞, the two-point function vanishes or stays positive, according to whether β is smaller or larger than β c given above. Let us say that a loop visits a vertex x of T m if the loop contains a point (x, t) for some t ∈ [0, 1]. Motivated by (1.2) and (1.6) we consider m -probability that (ρ, 0) belongs to a loop which visits some vertex in generation m in T m .
Throughout this paper we work with β of the form for some fixed but arbitrary α 0 > 0. All error terms O(⋅), o(⋅) and constants may depend on α 0 but are otherwise uniform in α.
Theorem 1.1. Consider β of the form (1.7), and write Let us remark that for θ = 1 the result was shown in our previous work [6]. The arguments presented here are strengthened versions of those arguments. The basic strategy is to establish recursion inequalities for the sequence σ m , see Prop. 2.1. These are obtained by analyzing the local configuration around the root ρ, in particular we identify two events A 1 and A 2 which together contribute most of the probability in the regime we consider (d → ∞ and β as in (1.7)).

Proof of the main result
The indicator function of an event A will be written 1I A or 1I{A}. The partition function for the loop model on T m is written For given m ≥ 1 and ε > 0 we definẽ . (A priori we need not have σ m ≤ σ m−1 since they are computed using different measures.) In this section we will prove the following recursion-inequalities.
Our main result follows easily: Proof of Thm 1.1. First suppose α < α * . For d large enough the factor in square brackets in (2.3) is strictly smaller than 1. This easily gives that σ m decays to 0 exponentially fast. Now suppose α > α * . Clearly σ 0 = 1, and it is not hard to see that there exists a constant c 1 > 0 such that σ 1 ≥ c 1 for all d. This implies thatσ 1 = ε d if ε < c 1 . If also ε < 2(α − α * ) and d is large enough then (2.2) and induction on m give that σ m ≥σ m = ε d for all m ≥ 1.
Before turning to the proof of Prop. 2.1, let us describe some of the main ideas and also what new input is required compared to our previous work [6] on the case θ = 1. For the lower bound (2.2) we will estimate the probability of certain local configurations near ρ which guarantee that ρ is connected to generation m if certain of its children (or grandchildren) are. For the upper bound we similarly estimate P (θ) m (ρ ↔ m) in terms of the probability that certain of ρ's children (or grandchildren) are blocked from generation m. When θ ≠ 1, the configurations in the subtrees rooted at the children of ρ are not independent of the local configuration adjacent to ρ. Thus we must deal carefully with the factor θ (ω) and how it behaves in the local configurations which we consider. This involves obtaining estimates for the partition function Z m in terms of the partition function Z m−1 in the smaller tree, which is where the number z m in (2.1) becomes relevant.
As was the case in [6], the hardest part is the upper bound (2.3). This is because we must rule out connections due to 'lower order events' ((A 1 ∪ A 2 ) c in the notation below) where the loop structure is too complicated to handle directly. The main technical advance compared to [6] started with a simplification of the argument used there to deal with this difficulty. Having this simpler version allowed us to deal also with the correlations caused by the factor θ (ω) , see Prop. 2.4.

Preliminary calculations.
Let us first introduce some notations and prove some facts that will be used for establishing both bounds in Prop 2.1.
Write A 1 for the event that, for each child x of ρ, there is at most one link between ρ and x. Write A 2 for the event that: (i) there is a unique child x of ρ with exactly 2 links between ρ and x, (ii) for all siblings x ′ of x there is at most one link between ρ and x ′ , and (iii) for all children y of x there is at most one link between x and y. See Fig. 2.
).  To see this, one may imagine that the k links to ρ are put in last, one at a time. Each such link then merges some loop in the corresponding subtree with a loop visiting ρ. (This uses the tree-structure of the underlying graph, which implies that there can be no connections between ρ and the subtree until the link is put in.) It follows that and hence (recalling z m from (2.1)) Similarly, since the k children with links would need to be blocked from reaching distance m − 1, we also have For the event A 2 , we decompose it as , according as the 2 links from ρ to x are different sorts (crosses/double-bars) or the same (Fig 3). If we look at the restriction of Figure 3. Illustration of the possibilities for ω ρx on the event A 2 . On A same 2 there are two loops, one of which contains (ρ, 0); on A mix 2 only one. The latter is thus more advantageous for long connections. The random variable X has mean 2 3 .
ω to the link ρx only (i.e., at ω ρx ) then it has two loops on A same 2 and a single loop on A mix 2 . Let us number the children of x together with the children of ρ excepting x by i = 1, . . . , 2d−1. Then we have that where k denotes the total number of 1-links at ρ and at x. To see this one may again imagine that the 1-links are placed last, one at a time. If k = 0 then (2.8) holds due to our observation about A mix 2 and A same 2 above, if k > 0 then each link we place merges two previously disjoint loops.
Let Λ denote the loop in ω ρx containing (ρ, 0), and let Λ ρ = Λ ∩ ({ρ} × [0, 1]) and Λ x = Λ∩({x}×[0, 1]) denote the parts of Λ at ρ and at x, respectively. For B ρ m to happen, children of ρ which link to Λ need to be blocked from distance m − 1 and children of x which link to Λ need to be blocked from distance m − 2; the remaining children of ρ and x do not need to be blocked. In particular, on A mix 2 all children which link to either ρ or x need to be blocked. Write A mix 2 (x, k 0 , k 1 ) for the event that (i) ρx supports one link of each sort, (ii) among the remaining children of ρ exactly k 0 support 1 link and the rest 0, and (iii) among the children of x exactly k 1 support 1 link and the rest 0. Using (2.8) with k = k 0 + k 1 and a calculation similar to (2.7) we get For the case of A same 2 we may start with a similar decomposition, where A same 2 (x, k 0 , k 1 ) is defined as A mix 2 (x, k 0 , k 1 ) except for requiring the two links supported by ρx to be of the same sort instead. Here we may then further consider the number j 0 ∈ {0, . . . , k 0 } of links with an endpoint in Λ ρ as well as the number j 1 ∈ {0, . . . , k 1 } of links with an endpoint in Λ x . As mentioned above, these links need to be blocked, but the remaining do not. Recalling that the locations of links are uniform on [0, 1] this means that we obtain a factor Λ ρ (respectively Λ x ) for each of these j 0 (respectively, j 1 ) links, and hence (2.10) Here we have simply written E[⋅] for E m [⋅ A 2 ], this expectation is over the choice of crosses or double-bars and over the lengths Λ ρ and Λ x only. We note here that the joint expectations of Λ ρ and Λ x may be computed explicitly. Indeed, as illustrated in Fig. 3, there is a random variable X such that Λ ρ and Λ x have respective lengths X and 1 − X in the case of two crosses; X and X in the case of two double-bars; and Λ ρ = Λ x = X = 1 in the case of a mixture. One may check 1 3 . At this point, let us mention the following asymptotics, which will be useful several times: if σ = O(d −1 ) and x ∈ R then we have . (2.12) For the last step we used (2.11) to first order, and that (2.13)

Stochastic domination.
In some estimates we will want to approximate the complicated measure P (θ) m (⋅), which involves counting loops, by some simpler measure. For this we use stochastic domination. Let us define β + = (βθ) ∨ (β θ). Also let us define E + m in the same way as E m but with β replaced by β + ; thus the links form independent Poisson processes with rate β + . We say that an event A is increasing if it cannot be destroyed by adding more links; examples of increasing events include A c 1 and (A 1 ∪ A 2 ) c where A 1 and A 2 are as defined above. Stochastic domination tells us that (2.14) Proof of (2.14). We apply [8, Thm. 1.1]. Note that P (θ) m ≪ P + m and the density f (ω) = where ω denotes the number of links. Let ω ′ be obtained from ω by adding a 1 The conditional distribution of X equals that of the length of the segment between two uniform independent points on a circle (with circumference 1) which contains a given point.
single link. This link either splits a loop, merges two loops, or does not change the number of loops, hence The result follows since all three possible values are ≤ 1.
An immediate consequence of (2.14) is that there is some constant c > 0 such that We now deduce some information about the asymptotic behaviour of the numbers Proposition 2.2. There is a constant C and there are functions ε (2.19) and , this satisfies (2.18) by (2.15). From the expressions (2.6) and (2.12) we have Hence, using the asymptotics (2.11) and (2.13), for a function ε (4) (d) not depending on m but otherwise satisfying the bounds (2.18). Using the induction hypothesis we get where we have simply bounded the remaining difference involving the event (A 1 ∪ A 2 ) c from below by 0. Consider first the terms involving A 1 . From (2.6) and (2.7), bounding σ m−1 ≥ σ m−1 , and using the asymptotics (2.11) as well as the estimates Prop. 2.2 on z m we get Now consider the terms involving A 2 . Using that ζ m−1 , ζ m−2 ≤ 1 −σ m−1 , as well as the asymptotics (2.11) to order d −1 , we deduce from (2.9) that and from (2.10) that Here we used the properties of Λ ρ and Λ x stated below (2.10) (eee also Fig. 3). Using also (2.12) and (2.21) we get where r is defined in (2.17). Putting this together in (2.22) gives Since α * = 1 + q − r this gives (2.2).

Proof of the upper bound (2.3).
Write Σ ρ m for the complement of B ρ m , so that . The following will be proved at the end of this section: Proposition 2.4. For all d large enough there is a constant C such that Before proving this we show how to deduce (2.3). We have by taking the difference of the expressions (2.6) and (2.7) that In the last step we used the concavity of the function Similarly using (2.9) and concavity of f (x, y) Hence, bounding also σ m−1 and σ m−2 by their maximum, we have that where r = r(θ, u) was defined in (2.17). In the above, all O(d −2 ) terms are uniform in m. Since 1 + q − r = α * we see that (2.3) follows once we prove Prop. 2.4. In the following argument we will examine the subtreeŤ of T m which contains the root and is spanned by edges supporting at least two links. InŤ , the loop-structure is very complicated and we will not attempt to keep track of it. Instead we use thatŤ is likely to be small, and that a loop exiting it must do so across an edge supporting exactly one link, which is a simpler situation to analyze. Roughly speaking, the enforcement of the event A c 1 ∩ A c 2 will give rise to the factor d −2 , and the requirement that the loop exitsŤ will give a factor σ m−k for some k ≥ 1, which can then be bounded in terms of σ m−1 ∨ σ m−2 . The details are quite technical.
Proof of Prop. 2.4. We begin by definingŤ carefully: we letŤ be the (random) subtree of T m containing (1) the root ρ (2) any vertex in generation 1 with ≥ 2 links to ρ, (3) in general, any vertex in generation k with ≥ 2 links to some vertex ofŤ in generation k − 1.
Note that A c 1 ∩ A c 2 is precisely the event thatŤ has at least two edges. Let V k (Ť ) denote the set of vertices inŤ in generation k. For x a vertex ofŤ , x ∈ V m (Ť ), let d x denote its number of descendants not inŤ . Thus x has d x outgoing edges carrying only 0 or 1 links of ω. For 0 ≤ k ≤ m − 1 we let E k denote the set of outgoing edges from generation k (to generation k + 1) which carry precisely 1 link.
Note that if the loop of (ρ, 0) reaches generation m then either it reaches generation m withinŤ , or it passes some link of ∪ m−1 k=0 E k . Let us by convention set σ −1 = 1 and E m = V m (Ť ) . We claim that . (2.29) Intuitively, this is because if the loop exitsŤ through some edge in E k , then it has distance m−k −1 left to go to reach the m th generation of T m . A detailed justification of (2.29) requires dealing with the dependencies caused by the factor θ . To do this, let us introduce the following notation. First, letω denote the restriction of ω toŤ . Next, let ∂ +Ť denote the set of vertices y ∈ T m ∖Ť whose parent belongs toŤ , and write ω y for the restriction of ω to the subtree rooted at y. For simplicity, in the rest of this proof we simply write E for E m . We will make use of the fact that, givenω, the random collections (E j ) m−1 j=0 and (ω y ) y∈∂ +Ť are conditionally independent under E. This implies that for three functions F 1 (ω), F 2 (ω, (E j ) m−1 j=0 ), F 3 (ω, (ω y ) y∈∂ +Ť ) (2.30) we have E F 1 (ω)F 2 (ω, (E j ) m−1 j=0 )F 3 (ω, (ω y ) y∈∂ +Ť ) = E F 1 (ω)E[F 2 (ω, (E j ) m−1 j=0 ) ω]E[F 3 (ω, (ω y ) y∈∂ +Ť ) ω] . (2.31) Note that we have the decomposition (similar to (2.5)) whereˇ denotes the number of loops in the configurationω, and is a factorization into three functions as in (2.30). Turning to (2.29), by considering the possibilities that eitherŤ reaches generation m (meaning V m (Ť ) ≠ ∅) or that loop of (ρ, 0) passes some edge e ∈ ∪ m−1 k=0 E k , we have