Continuous quantum phase transitions

TLDR

A quantum system can undergo a continuous phase transition at absolute zero as a Hamiltonian parameter is varied, and unlike classical finite‑temperature transitions, its dynamic and static critical behaviors are intimately intertwined. The authors aim to demonstrate that analyzing the path integral formulation of quantum statistical mechanics yields substantial insight into such transitions. This approach maps the quantum system onto a classical statistical mechanics model in which time acts as an additional spatial dimension. The mapping produces scaling forms for finite‑temperature behavior governed by finite‑size scaling theory and introduces a temperature‑dependent dephasing length that controls the crossover between quantum and classical fluctuations.

Abstract

A quantum system can undergo a continuous phase transition at the absolute zero of temperature as some parameter entering its Hamiltonian is varied. These transitions are particularly interesting for, in contrast to their classical finite temperature counterparts, their dynamic and static critical behaviors are intimately intertwined. We show that considerable insight is gained by considering the path integral description of the quantum statistical mechanics of such systems, which takes the form of the {\em classical} statistical mechanics of a system in which time appears as an extra dimension. In particular, this allows the deduction of scaling forms for the finite temperature behavior, which turns out to be described by the theory of finite size scaling. It also leads naturally to the notion of a temperature-dependent dephasing length that governs the crossover between quantum and classical fluctuations. We illustrate these ideas using Josephson junction arrays and with a set of recent experiments on phase transitions in systems exhibiting the quantum Hall effect.