The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin

Abstract

We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if S→∞ and the inverse temperature β→0 in such a way that βS stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder.

Appendix: Two Technical Bounds

In this appendix we prove Propositions 2.2 and 2.3.

Proof of Proposition 2.2

Let us assume for definiteness that Λ ₁={x∈ℤ³:x _i =1,…,ℓ}. We rewrite $H_{\varLambda _{1}}^{N}$ by applying the Fourier transform , where $\phi_{\mathbf{k}}(\mathbf{x})=\prod_{i=1}^{3}\phi_{k_{i}}(x_{i})$,

(A.1)

and the set of momenta $\varLambda^{*}_{1,N}$ is

(A.2)

Then a straightforward computation shows that, if $\varepsilon (\mathbf{k})=\sum_{i=1}^{3}(1-\cos k_{i})$,

(A.3)

Now, for every k≠0, one has ε(k)≥c ₀ ℓ ⁻², for a suitable c ₀>0. Therefore,

(A.4)

where in the last inequality we used Plancherel’s identity and the definition of total spin operator , from which on . Now, the expression in square brackets in the r.h.s. of Eq. (A.4) can be bounded from below by Sℓ ³(S−S _T /ℓ ³), so that

(A.5)

Therefore, for all S _T such that $S - \ell^{-3}S_{T} > 6c_{0}^{-1}\ell^{2}$, we get , which proves the proposition with $c : = \frac{1}{2}J c_{0}$. □

Proof of Proposition 2.3

We start by investigating the part of $\mathcal{K}^{N}_{\varLambda _{1}}$ associated with the square roots in the first term in (1.7). Using the fact that $A_{\mathbf{x},\mathbf{y}}:= 1 - \sqrt{1 - \frac{\hat {n}_{\mathbf{x}}}{2S}} \sqrt{1 - \frac{\hat{n}_{\mathbf{y}}}{2S} }$ is a non-negative operator on the bosonic Hilbert space of interest, we get:

(A.6)

where in the first line we used the Cauchy–Schwarz inequality, while to go from the first to the second line we used the simple fact that $1-\sqrt{1-x}\le x$, ∀x∈[0,1]. Finally, the remaining term in $\mathcal{K}^{N}_{\varLambda _{1}}$ can be bounded trivially as $J\vert \sum_{{ \langle\mathbf{x}, \mathbf{y} \rangle}\subset \varLambda _{1}} n_{\mathbf{x}}n_{\mathbf{y}}\vert \le(\mbox{const.}) ( \sum_{\mathbf{x}\in\varLambda_{1}} \hat{n}_{\mathbf {x}} )^{2}$, which implies the desired estimate. □