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Quantum Spin Systems at Positive Temperature

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Abstract

We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be used to prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature β and the magnitude of the quantum spins \(\mathcal{S}\) satisfy \(\beta\ll\sqrt\mathcal{S}\). From the quantum system we require that it is reflection positive and that it has a meaningful classical limit; the core technical estimate may be described as an extension of the Berezin-Lieb inequalities down to the level of matrix elements. The general theory is applied to prove phase transitions in various quantum spin systems with \(\mathcal{S}\gg1\). The most notable examples are the quantum orbital-compass model on \(\mathbb{Z}^2\) and the quantum 120-degree model on \(\mathbb{Z}^3\) which are shown to exhibit symmetry breaking at low-temperatures despite the infinite degeneracy of their (classical) ground state.

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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, 90095, U.S.A

    Marek Biskup, Lincoln Chayes & Shannon Starr

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  1. Marek Biskup
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  2. Lincoln Chayes
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  3. Shannon Starr
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Correspondence to Marek Biskup.

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Communicated by H. Spohn

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Biskup, M., Chayes, L. & Starr, S. Quantum Spin Systems at Positive Temperature. Commun. Math. Phys. 269, 611–657 (2007). https://doi.org/10.1007/s00220-006-0135-9

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