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While it is generally said that there are no phase transitions in classical lattice systems in one spatial dimension, there are also exceptions to this rule. Rigorous proofs involve some fairly strong assumptions about the statistical weights, such as positivity. I wonder what the situation is for quantum spin chains at positive temperature. That is, under what assumptions can one prove that there are no phase transitions at positive temperature?

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For translationally-invariant finite-range lattice 1d Hamiltonians, the absence of phase transitions at positive temperatures has been proved by Araki:

H. Araki, "Gibbs states of a one-dimensional quantum lattice," Comm. Math. Phys. 14 (1969), 120-157.

Later he gave a different (and less computation-heavy) proof which also covers a wider class of Hamiltonians:

H. Araki, "On uniqueness of KMS states of one-dimensional quantum lattice systems", Comm. Math. Phys. 44 (1975), 1-7.

In particular, the 2nd proof covers the case without translation-invariance.

On the other hand, the 1d Ising model with long-range interactions is a famous counter-example. See two papers by F. Dyson and references therein:

F. Dyson, Comm. Math. Phys. 12 (1969), 91-107; Comm. Math. Phys. 21 (1971), 269-283.

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  • $\begingroup$ Are there any results regarding 1D finite sized systems at finite temperature? For example, are the poles in the free energy as as function of inverse temperature when extended to the whole complex plane? $\endgroup$ Aug 18, 2021 at 13:24

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