# Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet

Correggi M, Giuliani A, Seiringer R. 2015. Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet. Communications in Mathematical Physics. 339(1), 279–307.

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http://arxiv.org/abs/1312.7873
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*Journal Article*|

*Published*|

*English*

**Scopus indexed**

Author

Correggi, Michele;
Giuliani, Alessandro;
Seiringer, Robert

^{ISTA}^{}Department

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.

Publishing Year

Date Published

2015-06-23

Journal Title

Communications in Mathematical Physics

Volume

339

Issue

1

Page

279 - 307

IST-REx-ID

### Cite this

Correggi M, Giuliani A, Seiringer R. Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet.

*Communications in Mathematical Physics*. 2015;339(1):279-307. doi:10.1007/s00220-015-2402-0Correggi, M., Giuliani, A., & Seiringer, R. (2015). Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet.

*Communications in Mathematical Physics*. Springer. https://doi.org/10.1007/s00220-015-2402-0Correggi, Michele, Alessandro Giuliani, and Robert Seiringer. “Validity of the Spin-Wave Approximation for the Free Energy of the Heisenberg Ferromagnet.”

*Communications in Mathematical Physics*. Springer, 2015. https://doi.org/10.1007/s00220-015-2402-0.M. Correggi, A. Giuliani, and R. Seiringer, “Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet,”

*Communications in Mathematical Physics*, vol. 339, no. 1. Springer, pp. 279–307, 2015.Correggi, Michele, et al. “Validity of the Spin-Wave Approximation for the Free Energy of the Heisenberg Ferromagnet.”

*Communications in Mathematical Physics*, vol. 339, no. 1, Springer, 2015, pp. 279–307, doi:10.1007/s00220-015-2402-0.**All files available under the following license(s):**

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