The free energy in a class of quantum spin systems and interchange processes

Abstract

We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin S = 1 2 the model is the Heisenberg ferromagnet, for general spin S ∈ 1 2 N it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by Tóth and by Penrose when S = 1 2 ). The critical temperature is shown to coincide (as a function of S) with that of the q = 2S + 1 state classical Potts model, and the phase transition is discontinuous when S ≥ 1.

Quantum Spins and Random Loops on the Complete Graph

“…We note that the limiting quantities agree with the corresponding expectations with respect to the Poisson-Dirichlet distributions; more precisely PD(2), for u = 1, and PD(1), for u < 1. Indeed, setting θ = 2 in (3.2), we find that 7) while setting θ = 1 yields Next, we explain how to derive this loop model from quantum spin systems. This will show that Theorem 3.1 is equivalent to Theorem 2.1.…”

“…, x 2S+1 ) of φ β under the condition i x i = 1 and x 1 ≥ x 2 ≥ · · · ≥ x 2S+1 . It was understood and proven by Björnberg, see [7,Theorem 4.2], that the answer involves the critical parameter…”

“…Specifically we will make use of tools developed by Alon, Berestycki and Kozma [2,5]. A similar approach has been followed in [7] in a calculation of the free energy and of the critical point of the model. In this section, we will also use the connection between representations of S n and symmetric polynomials.…”

“…It has been shown by Berestycki and Kozma in [5] that 13) where d λ is the dimension of the irreducible representation of S n with character χ λ (·) and r(λ) = χ λ ((1, 2))/d λ is the character ratio at a transposition. Furthermore, it has been shown in [7] that (6.14) where ε 1 (λ, n) → 0, uniformly in λ n, with (λ) ≤ θ. Note, moreover, that 1 n logf 0 (λ) =: ε 2 (λ, n) has the same property.…”

“…For the second part, let δ > 0 be arbitrary and let ε > 0 be such that x − x < ε implies |F (x) − F (x )| < δ. Applying the first part with A(k, n) = F (k/n) + ε 2 (k, n) − F (x ) we get In [7] it was proved that (for θ ≥ 3, that is S ≥ 1) φ β (·) is maximised at x 1 = x 2 = · · · = x θ = 1 θ when β < β c , and at some point satisfying x 1 > x 2 when β ≥ β c . Here we provide the following additional information about the maximiser.…”