This review discusses instabilities of the Fermi-liquid state of conduction electrons in metals with particular emphasis on magnetic quantum critical points. Both the existing theoretical concepts and experimental data on selected materials are presented; with the aim of assessing the validity of presently available theory. After briefly recalling the fundamentals of Fermi-liquid theory, the local Fermi-liquid state in quantum impurity models and their lattice versions is described. Next, the scaling concepts applicable to quantum phase transitions are presented. The Hertz-Millis-Moriya theory of quantum phase transitions is described in detail. The breakdown of the latter is analyzed in several examples. In the final part experimental data on heavy-fermion materials and transition-metal alloys are reviewed and confronted with existing theory.

http://export.arxiv.org/pdf/cond-mat/0606317

Fermi-liquid instabilities at magnetic quantum phase transitions

Intermetallic compounds containing $f$-electron elements display a wealth of superconducting phases, which are prime candidates for unconventional pairing with complex order parameter symmetries. For instance, superconductivity has been found at the border of magnetic order as well as deep within ferromagnetically and antiferromagnetically ordered states, suggesting that magnetism may promote rather than destroy superconductivity. Superconducting phases near valence transitions or in the vicinity of magnetopolar order are candidates for new superconductive pairing interactions such as fluctuations of the conduction electron density or the crystal electric field, respectively. The experimental status of the study of the superconducting phases of $f$-electron compounds is reviewed.

Superconducting phases of f -electron compounds

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.

https://iopscience.iop.org/article/10.1088/0034-4885/66/12/R01/pdf

Quantum phase transitions

In several metals, including URhGe, superconductivity has recently been observed to appear and coexist with ferromagnetism at temperatures well below that at which the ferromagnetic state forms. However, the material characteristics leading to such a state of coexistence have not yet been fully elucidated. We report that in URhGe there is a magnetic transition where the direction of the spin axis changes when a magnetic field of 12 tesla is applied parallel to the crystal b axis. We also report that a second pocket of superconductivity occurs at low temperature for a range of fields enveloping this magnetic transition, well above the field of 2 tesla at which superconductivity is first destroyed. Our findings strongly suggest that excitations in which the spins rotate stimulate superconductivity in the neighborhood of a quantum phase transition under high magnetic field.

https://science.sciencemag.org/content/sci/309/5739/1343.full.pdf

Magnetic Field-Induced Superconductivity in the Ferromagnet URhGe

This review discusses a paradigm that has become of increasing importance in the theory of quantum phase transitions, namely, the coupling of the order-parameter fluctuations to other soft modes and the resulting impossibility of constructing a simple Landau-Ginzburg-Wilson theory in terms of the order parameter only. The soft modes in question are manifestations of generic scale invariance, i.e., the appearance of long-range order in whole regions in the phase diagram. The concept of generic scale invariance and its influence on critical behavior is explained using various examples, both classical and quantum mechanical. The peculiarities of quantum phase transitions are discussed, with emphasis on the fact that they are more susceptible to the effects of generic scale invariance than their classical counterparts. Explicit examples include the quantum ferromagnetic transition in metals, with or without quenched disorder; the metal-superconductor transition at zero temperature; and the quantum antiferromagnetic transition. Analogies with classical phase transitions in liquid crystals and classical fluids are pointed out, and a unifying conceptual framework is developed for all transitions that are influenced by generic scale invariance.

/pdf/how-generic-scale-invariance-influences-quantum-and-2vto41mf6z.pdf

How Generic Scale Invariance Influences Quantum and Classical Phase Transitions

It is shown that the critical temperature for spin-triplet, $p$-wave superconductivity mediated by spin fluctuations is generically much higher in a Heisenberg ferromagnetic phase than in a paramagnetic one, due to the coupling of the magnons to the longitudinal magnetic susceptibility. Together with the tendency of the low-temperature ferromagnetic transition in very clean Heisenberg magnets to be of first order, this qualitatively explains the phase diagram recently observed in ${\mathrm{UGe}}_{2}$.

/pdf/strong-enhancement-of-superconducting-t-c-in-ferromagnetic-4kn1ftrrum.pdf

Strong enhancement of superconducting T(c) in ferromagnetic phases.

General conditions are formulated that allow us to determine which quantum phase transitions in itinerant electron systems can be described by a local Landau-Ginzburg-Wilson (LGW) theory solely in terms of the order parameter. A crucial question is the degree to which the order parameter fluctuations couple to other soft modes. Three general classes of zero-wave-number order parameters, in the particle-hole spin-singlet and spin-triplet channels and in the particle-particle channel, respectively, are considered. It is shown that the particle-hole spin-singlet class does allow for a local LGW theory, while the other two classes do not. The implications of this result for the critical behavior at various quantum phase transitions are discussed, as is the connection with nonanalyticities in the wave-number dependence of order-parameter susceptibilities in the disordered phase.

https://arxiv.org/pdf/cond-mat/0109547.pdf

Local versus nonlocal order-parameter field theories for quantum phase transitions

We investigate the quantum phase transition of itinerant electrons from a paramagnet to a state which displays long-period helical structures due to a Dzyaloshinskii instability of the ferromagnetic state. In particular, we study how the self-generated effective long-range interaction recently identified in itinerant quantum ferromagnets is cut off by the helical ordering. We find that for a sufficiently strong Dzyaloshinskii instability the helimagnetic quantum phase transition is of second order with mean-field exponents. In contrast, for a weak Dzyaloshinskii instability the transition is analogous to that in itinerant quantum ferromagnets, i.e., it is of first order, as has been observed in MnSi.

http://arxiv.org/pdf/cond-mat/0009441.pdf

Quantum phase transition of itinerant helimagnets

The behaviour of the conductivity and the density of states, as well as the phase relaxation time, of disordered itinerant electrons across a quantum ferromagnetic transition is discussed. It is shown that critical fluctuations lead to anomalies in the temperature and energy dependence of the conductivity and the tunnelling density of states, respectively, that are stronger than the usual weak-localisation anomalies in a disordered Fermi liquid. This can be used as an experimental probe of the quantum critical behaviour. The energy dependence of the phase relaxation time at criticality is shown to be that of a marginal Fermi liquid.

Thomas Vojta

Papers

Strong enhancement of superconducting T(c) in ferromagnetic phases.

Local versus nonlocal order-parameter field theories for quantum phase transitions

Quantum phase transition of itinerant helimagnets

Transport Anomalies and Marginal Fermi-Liquid Effects at a Quantum Critical Point

An unexpected difference between regular and random order of updates in damage spreading simulations