The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin
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The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin M. Correggi · A. Giuliani
Received: 24 July 2012 / Accepted: 6 September 2012 © Springer Science+Business Media, LLC 2012
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. Keywords Ferromagnetic Heisenberg model · Magnons · Spin waves
1 Introduction An important open problem in theoretical and mathematical physics is the proof of long range order in the three-dimensional (3D) quantum Heisenberg ferromagnet (FM) at low temperatures. While the existence of long range order at low temperatures in the classical Heisenberg model and in the quantum Heisenberg antiferromagnet can be proved by reflection positivity methods [11–13], the broken phase of the quantum ferromagnet eluded any rigorous treatment so far. From a heuristic point of view, a very useful and suggestive representation of the quantum FM is in terms of spin waves, an idea first introduced by Bloch in his seminal work [1, 2]. The spin waves are the lowest energy excitations, which give the dominant contribution to the free energy at low temperatures; they satisfy a Bose statistics and are in many respects the analogues of the phonons in crystals (see, e.g., [19] for a classical and comprehensive review). Bloch’s theory was later generalized in several directions by Herring and Kittel [15], Holstein and Primakoff [18], Dyson [9], and it was used, among other things, to compute M. Correggi () · A. Giuliani Dipartimento di Matematica, Università degli Studi Roma Tre, L.go S. Leonardo Murialdo 1, 00146 Rome, Italy e-mail: [email protected]
M. Correggi, A. Giuliani
a low temperature expansion for the spontaneous magnetization in zero external magnetic field: after a few erroneous attempts [22, 32, 34, 39, 40], Dyson’s result [10] was confirmed by a number of different methods [7, 20, 30, 31, 33, 35, 37, 41–43] and further extended, more recently, by the effective Lagrangian method [16, 17]. The conclusion is that at low energies the corrections to the simple Bloch’s theory coming from the interactions among spin waves are so small that for most practical purposes the linear theory is enough, both in the presence or in the absence of an external magnetic field. While physically Bloch’s theory is accepted and in good agreement with experiments, from a more mathematical point of view there is no confirmation of its correctness yet. It is fair to
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