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Advances In Physics

The quantum criticality conundrum

R. B. Laughlin, Gilbert George Lonzarich, Philippe H. Monthoux, David Pines

Published Jun 1, 2001• 64 citations
Publisher

Abstract

It is well known theoretically that materials possessing distinct adjoining quantum phases at zero temperature can display behaviour that is universal and insensitive to microscopic details. This is the phenomenon of quantum criticality. Universality is, however, a central feature of the phases themselves and one essential to identifying them. Recent developments in the study of correlated electron materials suggest that quantum criticality is present in these materials and has been masking the true nature of their phases. This raises troubling questions about the relation between theory and experiment in physics, as well as in science generally.

DOI
10.1080/00018730110098534

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It is well known theoretically that materials possessing distinct adjoining quantum phases at zero temperature can display behaviour that is universal and insensitive to microscopic details. This is the phenomenon of quantum criticality. Universality is, however, a central feature of the phases themselves and one essential to identifying them. Recent developments in the study of correlated electron materials suggest that quantum criticality is present in these materials and has been masking the true nature of their phases. This raises troubling questions about the relation between theory and experiment in physics, as well as in science generally.
Critical points and their properties are one of the most active topics in condensed-matter physics.The activity in this field can be partially attributed to the abundance of theoretically and experimentally accessible phenomena in the vicinity of continuous phase transitions.In recent years quantum critical points have attracted a particular amount of interest.Although quantum phase transitions are observed only at absolute zero temperature, they also affect certain properties at nonzero temperatures, which can be studied experimentally.One remarkable feature of critical points is that in their vicinity macroscopic properties do not depend on the microscopic details of the sample and the underlying Hamiltonian.Therefore, the various transitions can be assigned to different universality classes, where all representatives of a class show exactly the same properties.Furthermore, the so-called scaling hypothesis implies that close to the critical point the observable quantities follow a universal scaling form.As a result, by properly rescaling all system-dependent parameters it is possible to collapse all data of a particular quantity onto a single curve, given by the respective universal scaling function.
In recent years, quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. These transitions, which are accessed at zero temperature by variation of a non-thermal control parameter, can influence the behaviour of electronic systems over a wide range of the phase diagram. Quantum phase transitions occur as a result of competing ground state phases. The cuprate superconductors which can be tuned from a Mott insulating to a d -wave superconducting phase by carrier doping are a paradigmatic example. This review introduces important concepts of phase transitions and discusses the interplay of quantum and classical fluctuations near criticality. The main part of the article is devoted to bulk quantum phase transitions in condensed matter systems. Several classes of transitions will be briefly reviewed, pointing out, e.g., conceptual differences between ordering transitions in metallic and insulating systems. An interesting separate class of transitions is boundary phase transitions where only degrees of freedom of a subsystem become critical; this will be illustrated in a few examples. The article is aimed at bridging the gap between high-level theoretical presentations and research papers specialized in certain classes of materials. It will give an overview on a variety of different quantum transitions, critically discuss open theoretical questions, and frequently make contact with recent experiments in condensed matter physics.
One of the most powerful and prized concepts in physics is that of universal behaviour. Universality allows us to make predictions for entire classes of systems without requiring knowledge of the microscopics, and can be found in classical and quantum systems in both equilibrium and in their dynamics. Often in many-body systems, this universal behaviour is found in regimes where effects at macroscopic length scales dominate over microscopic fluctuations, which is particularly true at a phase transition. In this thesis, we will address universality in quantum many-body physics, and its connection to the branch of mathematics known as catastrophe theory (CT). In CT, singularities in a theory take on several universal forms, known as catastrophes, which can be shown to manifest themselves in classical mechanical trajectories. We extend the concept of catastrophes to their wave variants, known as diffraction integrals, and identify how these universal features appear in many-body wavefunctions and observables. Specifically, in Chapter 2, we examine how the wavefunction of a δ\deltaδ-kicked Hamiltonian can be mapped exactly onto the Pearcey function, along with the effects of a phase transition on the diffraction. In Chapter 3, we examine the free-Fermion representation of the one-dimensional transverse-field Ising model in a similar vein, and identify the presence of catastrophes away from and through the critical point.
Abstract The theory of quantum critical phenomena is introduced to study some current many-body problems in condensed matter physics. Renormalization group concepts are applied to strongly correlated electronic materials which are close to a zero-temperature instability. These systems have enhanced effective masses and susceptibility. Scaling arguments yield the exponents which govern the critical behavior of these quantities in terms of the usual critical exponents associated with a zero-temperature phase transition. We show the existence of a new energy scale, related to the quantum nature of the many-body instability, which can be generally associated with the setting of Fermi-liquid behavior with decreasing temperature in three-dimensional strongly interacting electronic systems. The theory of quantum critical phenomena is used to investigate the Kondo lattice problem, which provides a model to describe heavy-fermion systems and to introduce a scaling theory of the Mott transition with special emphasis on charge fluctuation effects. However, this report is not a review on heavy fermions and Mott insulators. The microscopic theories of these systems are still controversial and present some of the most challenging and instigating problems in condensed matter physics. This state of affairs stimulated the author to review and extend the scaling approach. The scaling theory we develop provides a powerful tool, based on the notion of universality, to understand the physical properties of correlated systems beyond the mean-field level. This is illustrated by our treatment of the one-dimensional Hubbard model, where, although the Fermi-liquid fixed point does not survive the fluctuations, the scaling approach is still useful. Finally, we discuss briefly how disorder affect our results.