Finite temperature quantum condensations in the space of states: General Proof

TL;DR

This paper extends the concept of quantum phase transitions as condensations in the space of states to finite temperatures, providing a rigorous proof and identifying universal features of the critical surface.

Contribution

It offers a formal proof for finite temperature quantum condensations and phase transitions based on Hilbert space partitioning and free energy crossing, generalizing previous zero-temperature results.

Findings

Finite temperature quantum phase transitions occur when free energies of subspaces cross.

The critical surface exhibits universal features at high and low temperatures.

The proof uses an exact probabilistic representation of quantum dynamics.

Abstract

We formalize and prove the extension to finite temperature of a class of quantum phase transitions, acting as condensations in the space of states, recently introduced and discussed at zero temperature~(Ostilli and Presilla 2021 \textit{J. Phys. A: Math. Theor.} \textbf{54} 055005). In details, we find that if, for a quantum system at canonical thermal equilibrium, one can find a partition of its Hilbert space $\mathcal{H}$ into two subspaces, $\mathcal{H}_\mathrm{cond}$ and $\mathcal{H}_\mathrm{norm}$, such that, in the thermodynamic limit, $\dim \mathcal{H}_\mathrm{cond}/ \dim \mathcal{H} \to 0$ and the free energies of the system restricted to these subspaces cross each other for some value of the Hamiltonian parameters, then, the system undergoes a first-order quantum phase transition driven by those parameters. The proof is based on an exact probabilistic representation of quantum dynamics at an imaginary time identified with the inverse temperature of the system. We also show that the critical surface has universal features at high and low temperatures.