Exponent relations at quantum phase transitions, with applications to metallic quantum ferromagnets

TL;DR

This paper derives and discusses scaling laws and critical exponent relations at quantum phase transitions, especially in metallic ferromagnets, highlighting the complexity due to multiple dynamical exponents and applying the theory to real materials.

Contribution

It introduces new scaling laws involving specific-heat exponents and extends classical finite-size scaling ideas to quantum first-order transitions, with applications to metallic quantum ferromagnets.

Findings

Multiple dynamical exponents at quantum transitions.

Discontinuous transitions in clean systems, continuous in disordered.

Rich crossover phenomena explained by fixed points.

Abstract

Relations between critical exponents, or scaling laws, at both continuous and discontinuous quantum phase transitions are derived and discussed. In general there are multiple dynamical exponents at these transitions, which complicates the scaling description. Some rigorous inequalities are derived, and the conditions needed for these inequalities to be equalities are discussed. New scaling laws involving the specific-heat exponents are derived and and contrasted with their counterparts at classical phase transitions. We also generalize the ideas of Fisher and Berker and others for applying (finite-size) scaling theory near a classical first-order transition to the quantum case. We then apply and illustrate all of these ideas by using the quantum ferromagnetic phase transition in metals as an explicit example. This transition is known to have multiple dynamical scaling exponents, and in general it is discontinuous in clean systems, but continuous in disordered ones. Furthermore, it displays many experimentally relevant crossover phenomena that can be described in terms of fixed points, originally discussed by Hertz, that ultimately become unstable asymptotically close to the transition and give way to the asymptotic fixed points. These fixed points provide a rich environment for illustrating the general scaling concepts and exponent relations. We also discuss the quantum-wing critical point at the tips of the tricritical wings associated with the discontinuous quantum ferromagnetic transition from a scaling point of view.