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Validity of the Spin-Wave Approximation for the Free Energy of the Heisenberg Ferromagnet

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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.

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, L.go S. Leonardo Murialdo 1, 00146, Rome, Italy

    Michele Correggi & Alessandro Giuliani

  2. Institute of Science and Technology Austria, Am Campus 1, 3400, Klosterneuburg, Austria

    Robert Seiringer

Authors
  1. Michele Correggi
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  2. Alessandro Giuliani
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  3. Robert Seiringer
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Correspondence to Alessandro Giuliani.

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Communicated by H. Spohn

Copyright © 2015 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

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Correggi, M., Giuliani, A. & Seiringer, R. Validity of the Spin-Wave Approximation for the Free Energy of the Heisenberg Ferromagnet. Commun. Math. Phys. 339, 279–307 (2015). https://doi.org/10.1007/s00220-015-2402-0

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