Authors: M. Correggi, A. Giuliani, R. Seiringer
Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet
Abstract: We consider the quantum ferromagnetic Heisenberg model in three dimensions, for all spins S≥1/2. We rigorously prove the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy at low temperatures. Our proof comes with explicit, constructive upper and lower bounds on the error term. It uses in an essential way the bosonic formulation of the model in terms of the Holstein-Primakoff representation. In this language, the model describes interacting bosons with a hard-core on-site repulsion and a nearest-neighbor attraction. This attractive interaction makes the lower bound on the free energy particularly tricky: the key idea there is to prove a differential inequality for the two-particle density, which is thereby shown to be smaller than the probability density of a suitably weighted two-particle random process on the lattice.
Keywords: Quantum Heisenberg ferromagnet, 3D quantum spins, spin-wave theory, localization bounds, Holstein-Primakoff representation, random walk representation, functional inequalitites.
Dipartimento di Matematica, Sapienza Università di Roma
P.le Aldo Moro 5, 00185, Rome, Italy
E-mail address: michele DOT correggi AT gmail DOT com
Dipartimento di Matematica, Universit`a degli Studi Roma Tre
L.go S. Leonardo Murialdo 1, 00146, Rome, Italy
E-mail address: giuliani AT mat DOT uniroma3 DOT it
Am Campus 1, 3400 Klosterneuburg, Austria
E-mail address: robert DOT seiringer AT ist DOT ac DOT at