Quantum spin systems at positive temperature

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 $\beta$ and the magnitude of the quantum spins $\CalS$ satisfy $β\ll\sqrt\CalS$. 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 $\CalS\gg1$. 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.

47 pages, version to appear in CMP (style files included)

References in corpus (7)

• Orbital order in classical models of transition-metal compounds
• Directional Ordering of Fluctuations in a Two-dimensional Compass Model
• Provable first-order transitions for liquid crystal and lattice gauge models with continuous symmetries
• Orbital ordering in transition-metal compounds: I. The 120-degree model
• Order by disorder, without order, in a two-dimensional spin system with O(2) symmetry
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Cited by in corpus (12)

• Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems
• Phase coexistence of gradient Gibbs states
• Dilution Effects in Two-dimensional Quantum Orbital System
• On Quantum Markov Chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order three
• Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley tree
• On Quantum Markov Chains on Cayley tree III: Ising model
• Phase Transitions for quantum Ising model with competing XY -interactions on a Cayley tree
• Existence of Néel order in the S=1 bilinear-biquadratic Heisenberg model via random loops
• True nature of long-range order in a plaquette orbital model
• Classical and quantum KMS states on spin lattice systems
• Quantum Vorticity at positive temperature for spin systems with continuous symmetry
• The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems