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.Comment: 62 pages, 29 figs, review article for Rev. Mod. Phys; (v2) discussion extended, refs added; (v3) shortened; final version as publishe

This paper is concerned with the weak-moment magnetism in heavy-fermion materials and its relation to the non-Fermi liquid physics observed near the transition to the Fermi liquid. We explore the hypothesis that the primary fluctuations responsible for the non-Fermi liquid physics are those associated with the destruction of the large Fermi surface of the Fermi liquid. Magnetism is suggested to be a low-energy instability of the resulting small Fermi surface state. A concrete realization of this picture is provided by a fractionalized Fermi liquid state which has a small Fermi surface of conduction electrons, but also has other exotic excitations with interactions described by a gauge theory in its deconfined phase. Of particular interest is a three-dimensional fractionalized Fermi liquid with a spinon Fermi surface and a U(1) gauge structure. A direct second-order transition from this state to the conventional Fermi liquid is possible and involves a jump in the electron Fermi surface volume. The critical point displays non-Fermi liquid behavior. A magnetic phase may develop from a spin density wave instability of the spinon Fermi surface. This exotic magnetic metal may have a weak ordered moment although the local moments do not participate in the Fermi surface. Experimental signatures of this phase and implications for heavy-fermion systems are discussed.Comment: 20 pages, 8 figures; (v2) includes expanded discussion and solution of quantum Boltzmann equatio

Abstract. 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 behavior 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 are 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 on 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.

In spatial dimensions d ≥ 2, Kondo lattice models of conduction and local moment electrons can exhibit a fractionalized, non-magnetic state (FL * ) with a Fermi surface of sharp electron-like quasiparticles, enclosing a volume quantized by (ρa − 1)(mod 2), with ρa the mean number of all electrons per unit cell of the ground state. Such states have fractionalized excitations linked to the deconfined phase of a gauge theory. Confinement leads to a conventional Fermi liquid state, with a Fermi volume quantized by ρa(mod 2), and an intermediate superconducting state for the Z2 gauge case. The FL * state permits a second order metamagnetic transition in an applied magnetic field.The physics of the heavy fermion metals, intermetallic compounds containing localized spin moments on d or f orbitals and additional bands of conduction electrons, has been of central interest in the theory of correlated electron systems for several decades [1][2][3]. These systems are conveniently modelled by the much studied Kondo lattice Hamiltonian, in which there are exchange interactions between the local moments and the conduction electrons, and possibly additional exchange couplings between the local moments themselves. To be specific, one popular Hamiltonian to which our results apply is:Here the local moments are S = 1/2 spin S j , and the conduction electrons c jσ (σ =↑↓) hop on the sites j, j ′ of some regular lattice in d spatial dimensions with amplitude t(j, j ′ ), J K > 0 are the Kondo exchanges ( τ are the Pauli matrices), and explicit short-range Heisenberg exchanges, J H , between the local moments have been introduced for theoretical convenience. A chemical potential for the c σ fermions which fixes their mean number at ρ c per unit cell of the ground state is implied. We have not included any direct couplings between the conduction electrons as these are assumed to be well accounted by innocuous Fermi liquid renormalizations. For simplicity, we restrict our attention here to nonmagnetic states, in which there is no average static moment on any site ( S j = 0), and the spin rotation invariance of the Hamiltonian is preserved: the S j moments have been 'screened', either by the c σ conduction electrons, or by their mutual interactions (there is a natural extension of our results to magnetic states). It is widely accepted [1,[3][4][5][6][7] that such a ground state of H is a conventional Fermi liquid (FL) with a Fermi surface of 'heavy' quasiparticles, enclosing a volume, V F L determined by the Luttinger theorem:Hereis a phase space factor, v 0 is the volume of the unit cell of the ground state, ρ a = n ℓ + ρ c is the mean number of all electrons per volume v 0 , and n ℓ (an integer) is the number of local moments per volume v 0 . Note that ρ c,a need not be integers, and the (mod 2) in (2) allows neglect of fully filled bands. In d = 1, (2) has been established rigorously by Yamanaka et al. [5]. In general d, a non-perturbative argument for (2), assuming that the ground state is a Fermi liquid, has been provided by Oshikaw...

This article will give an overview on both theoretical and experimental developments concerning states with lattice symmetry breaking in the cuprate high-temperature superconductors. Recent experiments have provided evidence for states with broken rotation as well as translation symmetry, and will be discussed in terms of nematic and stripe physics. Of particular importance here are results obtained using the techniques of neutron and x-ray scattering and scanning tunneling spectroscopy. Ideas on the origin of lattice-symmetry-broken states will be reviewed, and effective models accounting for various experimentally observed phenomena will be summarized. These include both weak-coupling and strong-coupling approaches, with a discussion on their distinctions and connections. The collected experimental data indicate that the tendency toward uni-directional stripe-like ordering is common to underdoped cuprates, but becomes weaker with increasing number of adjacent CuO 2 layers.