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arXiv:1503.04175 · Physical Review B

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

T. R. Kirkpatrick, D. Belitz

Published Jun 5, 2015• 60 citations
arXivHTMLPDF

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. Scaling laws involving the specific-heat exponents that are specific to quantum phase transitions are derived 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.

DOI
10.1103/physrevb.91.214407

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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. Scaling laws involving the specific-heat exponents that are specific to quantum phase transitions are derived 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.
We employ scaling arguments and optimal fluctuation theory to establish a general relation between quantum Griffiths singularities and the Harris criterion for quantum phase transitions in disordered systems. If a clean critical point violates the Harris criterion, it is destabilized by weak disorder. At the same time, the Griffiths dynamical exponent z' diverges upon approaching the transition, suggesting unconventional critical behavior. In contrast, if the Harris criterion is fulfilled, power-law Griffiths singularities can coexist with clean critical behavior, but z' saturates at a finite value. We present applications of our theory to a variety of systems including quantum spin chains, classical reaction-diffusion systems and metallic magnets, and we discuss modifications for transitions above the upper critical dimension. Based on these results we propose a unified classification of phase transitions in disordered systems.
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We show that the critical behavior of disordered ferromagnets can be understood in terms of an effective, translational invariant Hamiltonian with renormalized coupling parameters which exhibit a tricritical as well as a critical point. The stable critical fixed point describes the critical behavior of weakly disordered systems, whereas the tricritical fixed point is responsible for the critical behavior of strongly disordered systems at least in the experimentally accessible range of temperature. This necessitates a crossover theory to describe the crossover between these two fixed points and to calculate critical properties as a function of concentration and temperature. In this paper we study the symmetric phase. Susceptibility and specific heat are calculated in scaling form using nonlinear scaling fields to verify that disordered ferromagnets exhibit universal critical behavior. The corresponding effective exponents exhibit nonmonotonous concentration- and temperature-dependent behavior. According to our theory, Fisher-renormalized tricritical behavior dominates in the experimentally accessible temperature range which leads to extreme values of the effective exponents ${\ensuremath{\gamma}}_{\mathrm{eff}}$=2 and ${\ensuremath{\alpha}}_{\mathrm{eff}}$=-1 in the limit of strong dilution. However, in accord with the Harris criterion we verify that the asymptotic behavior (T\ensuremath{\rightarrow}Tc{T}_{c}Tc​) is not changed by dilution if the critical exponent of the specific heat of the pure system ${\ensuremath{\alpha}}_{\mathrm{pure}<0}$, independent of the value of the concentration.
The phase-ordering dynamics that results from domain coarsening is considered for itinerant quantum ferromagnets. The fluctuation effects that invalidate the Hertz theory of the quantum phase transition also affect the phase ordering. For a quench into the ordered phase, there appears a transient regime where the domain growth follows a different power law than in the classical case, and for asymptotically long times, the prefactor of the t1∕2{t}^{1∕2}t1∕2 growth law has an anomalous magnetization dependence. A quench to the quantum critical point results in a growth law that is not a power-law function of time. Both phenomenological scaling arguments and renormalization-group arguments are given to derive these results, and estimates of experimentally relevant length and time scales are presented.
Quantum Phase Transitions Quantum phase transitions, in contrast to temperature-driven critical phenomena, occur due to a competition between different parameters describing the basic interactions of the system. Their specific feature is the quantum character of the critical fluctuations. This implies, through the uncertainty principle, that energy fluctuations and time are coupled. Then at zero temperature time plays a crucial and fundamental role, the static properties being entangled with the dynamics (Continentino, 1994a; Sondhi et al ., 1997; Sachdev, 1999). In this book we are mainly interested in quantum phase transitions that occur in electronic many-body systems and how scaling concepts can be useful to understand their properties close to these transitions (Continentino, 1994a). Even though a similar approach has been used in the case of interacting bosons (Fisher et al ., 1989) the fermionic problem has its own idiosyncrasies and difficulties. For example, there is no natural order parameter associated with the localisation transition in the electronic case, while bosons at zero temperature are either localised or superfluid so that the superfluid order parameter can be used to distinguish between both phases. If the results of the study of quantum phase transitions were restricted to zero temperature, this would be an interesting but purely academic area of research. What is really exciting about this subject is the effect of quantum critical points (QCP) in the finite temperature phase diagram of actual physical systems, even far away from the QCP (Freitas, 2015). As we will show, there is a special line in this phase diagram, the quantum critical trajectory , where the temperature dependence of the thermodynamic and transport properties is governed by the quantum critical exponents, i.e. those associated with the QCP. This and the observed crossover effects induced by temperature in the non-critical side of the phase diagram of a material with a QCP are sufficient to make the study of quantum phase transitions an inevitable subject. We start this chapter by introducing the scaling theory of quantum critical phenomena, since scaling concepts are used throughout this book.
At a quantum critical point the ground state of a many-particle quantum system undergoes a phase transition when some control parameters such as pressure, magnetic field, or chemical composition is varied. The universal behaviour characteristic for such quantum phase transitions often affects a wide temperature range and gives rise to novel material properties. This chapter gives a short overview over five lectures held during the Les Houches summer school ‘Quantum theory from small to large scales’ in 2010. After an introductory chapter, field driven magnetic quantum phase transitions of insulators are used as an example to discuss some of the concepts underlying quantum phase transitions. Both experimentally and theoretically, quantum phase transitions in metals are much less understood compared to insulating systems. After a brief review of the standard approach to describe those systems, the importance of multiple time scales and associated multiple critical exponents z are discussed. Finally, emergent gauge theories close to critical points are investigated. As an example, it discusses why a gauge theory describes the (classical) phase transitions of a nematic, if topological defects are suppressed.