Table of contents

`With a heavy heart, I have been converted to the idea that Fermi - Dirac, not Einstein - Bose is the correct statistics. I wish to write a short note on its application to paramagnetism.'

W Pauli (letter to Schrödinger, December 1926) [1].

Recent advances in materials science have re-opened a great debate about the nature of metals. For almost forty years, `Landau - Fermi liquid theory' has provided the mainstay of our understanding of the metallic state. While Fermi liquid theory provides an astonishingly successful description of conventional metals, it has become increasingly clear that the behaviour of many complex materials, such as the cuprate superconductors, certain f-electron materials and low-dimensional conductors, lie outside the its well-explored confines, suggesting that fundamentally novel and unexpected kinds of metallic behaviour occur in nature.

As Pauli's letter shows above, even the Fermi liquid began life as a subject of controversy. Fermi liquid theory is now well established; the important question is how to go beyond it. What is the essential issue? The theorist will tell you that we are really looking for new kinds of metallic fixed points; the experimentalist, that we are interested in discovering, characterizing and understanding new kinds of metals. The past few years have seen tremendous advances on both fronts. Theoretically, we have discovered and established the properties of a number of classes (or `fixed points') describing non-Fermi-liquid behaviour. Some examples include:

• The one-dimensional Luttinger liquid, in which the spin and charge excitations of the electronic fluid decouple into new quasiparticles called `holons' and `spinons' [2].

• A conventional metal at a zero-temperature critical point. At such a `quantum critical' point, fluctuations become sufficiently long-ranged in space and time that the interaction they induce is singular enough to destabilize the Fermi liquid ground state [3,4].

• Non-trivial impurity models, such as the two-channel Kondo model [5] - [8].

• The Chern - Simons theory of the half-filled Landau level and the closely related gauge theory of the `t-J' model [9,10].

On the experimental side, many classes of materials with properties apparently inconsistent with Fermi liquid theory have been discovered, including:

• High-temperature superconductors.

• Quasi one- and two-dimensional conductors.

• Many metallic alloys containing elements such as Ce or U with partially filled f-shells.

• Edge states of quantized Hall systems.

This is only a partial list of recent advances; furthermore, it is likely that there are new kinds of non-Fermi liquid behaviour yet to be discovered. This proof by example that non-Fermi liquid behaviour exists has led to a new sub-field of physics and many lively debates. On the theoretical side, many of the examples known involve rather special situations (e.g. low dimensionality, criticality, high symmetry); pressing questions arise relating to the possibility of extending this anomalous behaviour to more general models. On the experimental side, the connection between the realized non-Fermi liquid materials and the various theoretical models is by no means clear. When a real material with anomalous properties is encountered, the resulting debate centres on whether it falls into one of the pre-existing categories: is it a more conventional system in disguise (due to real-world effects such as disorder), or is it a new kind of non-Fermi liquid fixed point, as yet uncharacterized [11] - [13], [15]?

Over a six month period in the spring of 1996, a group convened at the Institute for Theoretical Physics in Santa Barbara to study `Non-Fermi Liquid Physics'. The programme was conceived as a way of bringing diverse theoretical and experimental efforts in this area together, framing the debate and identifying the open questions in the field. The International Conference on Non-Fermi Liquid Behaviour in Metals was held in June 1996 to conclude the programme. This volume contains a collection of papers that were written by conference or programme participants. In its preparation, we have been mindful of the youthfulness of this field. We feel it is vital to try to represent the fervor of scientific debate and discussion; thus we have tried to select articles which polemically or pedagogically describe possible resolutions of problems at the forefront of this field of research. Issues addressed in subsequent pages include:

• Quantum critical phenomena. The existing theories seem to account for the properties of transition metal ferromagnets [12], yet the various heavy-fermion systems on the brink of antiferromagnetism, such as [13], [14] and [12,15], show a marked deviation from the theoretical predictions. The dependence of the resistivity of over two decades is particularly dramatic in this respect. Could it be that disorder is affecting the nature of the magnetic fixed point? Both theoretical [17,18] and experimental [19] investigations of the interplay between disorder and quantum spin fluctuations are featured here. Another exciting possibility is that the interplay of the Kondo effect with the quantum critical behaviour produces a new kind of fixed point [12]. A related perspective is offered by Klein's group [16] who argue that even the classical critical behaviour of the resistivity of `bad metal' systems is not understood.

• The wide variety of non-Fermi liquid behaviour in f-electron materials reviewed by Maple and coworkers [11,20,21] continues to fascinate. Both optical [22] and neutron data [23] highlight the presence of anomalous spin and transport properties in several of these compounds. Here the debate takes many forms. One new idea aired at the meeting [24,25] is that many of the observed properties might be understood as a consequence of a broad distribution of Kondo temperatures. These authors recognize, however, that this can only be part of the story. Others have argued for a more intrinsic origin of the non-Fermi liquid behaviour, such as the quadrupolar Kondo effect or some other single-ion origin [26,27], or perhaps a wholly new kind of lattice non-Fermi liquid behaviour [28].

• Nowhere is the debate about possible non-Fermi liquid behaviour more fierce than in the discussion of the normal state of the cuprate superconductors. Experimentalists brought many new results to bear on this discussion at this meeting, including new insights into the nature of the spin-gap in under-doped cuprates [29] and the anomalous temperature dependence of Hall current relaxation time [30]. Each of these measurements spurs its own debate. Is the spin gap a reflection of novel spin pairing, as discussed by Anderson [31] or is it a reflection of a more conventional kind of Cooper pairing (with long-range order suppressed by fluctuation effects)? Alternatively, could it arise through gapping of the Fermi surface by antiferromagnetic fluctuations [32]? Likewise, is the presence of two relaxation rates seen in the optical Hall and conductivity measurements [30] a reflection of severe Fermi-surface anisotropy in the scattering rates [32], or does it arise from the formation of a new kind of fluid, where the Hall and electric currents relax in intrinsically different fashions [33,34]?

• Experimental manifestations of one-dimensional non-Fermi liquid behaviour, such as spin - charge decoupling, are still very much sought after as discussed by Allen and coworkers [35]. One area of active debate is the effect of interchain coupling on one-dimensional conductors, and whether spin - charge decoupling is robust against these effects [33]. Another area of interest is whether non-Fermi liquid behaviour can give rise to new kinds of correlation, such as odd-frequency pairing [37].

• Half-filled Landau level. Although the Fermi surface of composite fermions was initially characterized as a non-Fermi liquid, there is a growing school of thought that argues that the physical response functions of the composite Fermi surface may be described by a modified version of Landau - Fermi liquid theory, whereby the quasiparticles are subject to a singular `gauge' interaction. This discussion is reviewed by Simon [36].

Above all, we hope that this volume will be a source book for the field, offering something to interested readers, future students and participants in non-Fermi liquid physics. In closing, we would like to thank the staff of the ITP for their gracious assistance in organizing the non-Fermi liquids programme, and in particular, acknowledge the support of the National Science Foundation (grant No PHY94-07194), which provided the funds that made the whole event possible.

Piers Coleman (Rutgers University) Brian Maple (University of California) Andrew Millis (John Hopkins University) The Editors

The normal states of magnetic metals with vanishing Curie or Néel temperatures are investigated by means of measurements of the temperature and pressure dependence of the resistivity in the stoichiometric d and f compounds and . The results for the nearly ferromagnetic d metals may be described over a wide range in temperature and pressure in terms of a quantitative model of a marginal Fermi liquid based on dispersive spin-fluctuation spectra inferred from inelastic neutron scattering data. The behaviour of the antiferromagnetic f metals is also unconventional, but in a way which cannot yet be readily categorized. Near the critical pressure where , displays a resistivity of the form over nearly two decades in temperature before condensing into a short-coherence-length superconducting state below 0.4 K. The isoelectronic and isostructural compound , with a lattice cell volume slightly smaller than that of , remains normal down to 20 mK and shows a resistivity of the form over a decade below several kelvin at ambient pressure. These findings for the Ce systems have not yet been described consistently in terms of an extension of the model developed for the ferromagnetic d metals.

In heavy-fermion systems with 4f or 5f atoms (such as Ce or U) the competition between the on-site moment compensation by the Kondo effect and the long-range RKKY interaction between localized magnetic moments leads to the possibility of either a non-magnetic or a magnetically ordered ground state. However, even in the case of no long-range magnetic order as exemplified by , short-range dynamic intersite correlations are observed. Yet, the thermodynamic and transport properties of this alloy at very low temperatures T resemble those of a Fermi liquid (FL). Upon alloying with Au, long-range incommensurate antiferromagnetism is observed in for x > 0.1. For x = 0.1 where , the specific heat C depends on T as , the magnetic susceptibility as , and the T-dependent part of the electrical resistivity as . This is in marked contrast to the FL behaviour . It is suggested that low-energy spin excitations are at the origin of these non-Fermi-liquid (NFL) anomalies which occur at a zero-temperature quantum phase transition. Large magnetic fields B restore FL behaviour. The low-T range of FL behaviour in C and extends towards higher T with increasing B, with the crossover temperature varying roughly linearly with B. Apart from changing the Au concentration x, the magnetic - non-magnetic transition can be tuned by applying pressure p to antiferromagnetic samples with x > 0.1. For x = 0.3, at and NFL behaviour is observed in the specific heat for this critical pressure. For x = 0.2, where we likewise observe a logarithmic divergence of C/T and for p = 6.9 kbar we recover FL behaviour. Finally, we report on a remarkable `universality' of C/T in the system with M = Au, Pd, Pt: regardless of how is reached in this system (alloying with different elements M, varying concentration, or applying pressure), the C/T versus ln T curves are practically identical. Possible origins of NFL behaviour are discussed.

The quantum ferromagnetic transition of itinerant electrons is considered. We give a paedagogical review of recent results which show that zero-temperature soft modes that are commonly neglected invalidate the standard Landau - Ginzburg - Wilson description of this transition. If these modes are taken into account, then the resulting order parameter field theory is non-local in space and time. Nevertheless, both for disordered and for clean systems the critical behaviour has been exactly determined for spatial dimensions d > 2 and d > 1, respectively. The critical exponents characterizing the paramagnetic-to-ferromagnetic transition are dimensionality-dependent and substantially different both from mean-field critical exponents and from the classical Heisenberg exponents that characterize the transition at finite temperatures. Our results should be easily observable, particularly those for the disordered case, and experiments to check our predictions are proposed.

Recent work on the zero-temperature phases and phase transitions of strongly random electronic systems is reviewed. The transition between the spin glass and quantum paramagnet is examined, for both metallic and insulating systems. Insight gained from the solution of infinite-range models leads to a quantum field theory for the transition between a metallic quantum paramagnetic and a metallic spin glass. The finite-temperature phase diagram is described and crossover functions are computed in mean-field theory. A study of fluctuations about mean field leads to the formulation of scaling hypotheses.

High-pressure studies of transport, magnetic and thermal properties of a heavy-fermion antiferromagnet, , are reviewed. At ambient pressure, the Ce ions in the three nonequivalent sites in this compound are found to be very close to trivalent from -XANES (x-ray absorption near-edge structure) spectra and magnetic susceptibility measurements. With increasing pressure, (1.8 K at P = 0) of is suppressed and vanishes near . Non-Fermi-liquid (NFL) behaviour appears around 0.4 GPa in both the specific heat and AC magnetic susceptibility: and ). Above 0.62 GPa, the normal Fermi liquid state recovers as indicated by the T independence of and the dependence of magnetic resistivity. The variation of with pressure is analysed in terms of two models: the impurity Kondo model with three Kondo temperatures and the self-consistent renormalization (SCR) theory of spin fluctuations (SFs). We find that the crossover in is better described by the SCR theory. The characteristic SF temperature increases by a factor of 20 for 0.33 GPa , yielding a large Grüneisen parameter around 0.4 GPa.

We explore the behaviour of the dipolar-coupled Ising magnet, , in a transverse magnetic field. The transverse field, applied perpendicular to the Ising axis, introduces quantum channels for relaxation, thereby continuously depressing the spin-ordering temperature to zero. We compare the classical (thermally driven) and the quantum (transverse-field-driven) transitions for both the pure ferromagnet, , and the spin glass, , and we discuss the implications of these results for the T = 0 disordered ferromagnet (x = 0.5). Finally, we contrast these high-resolution studies of model quantum transitions in insulating magnets with the quantum critical behaviour of the highly correlated Mott - Hubbard metals and .

Certain chemically substituted Ce and U compounds have low-temperature physical properties that exhibit non-Fermi-liquid (NFL) characteristics and apparently constitute a new class of strongly correlated f-electron materials. The NFL behaviour takes the form of weak power law or logarithmic divergences in the temperature dependence of the physical properties that scale with a characteristic temperature , which, in some systems, can be identified with the Kondo temperature . These systems have complex temperature T - chemical substituent composition x phase diagrams, which contain regions displaying the Kondo effect, NFL behaviour, spin glass freezing, magnetic order, quadrupolar order, and, sometimes, even superconductivity. Possible origins of the NFL behaviour include a multichannel Kondo effect and fluctuations of an order parameter in the vicinity of a second-order phase transition at T = 0 K. Recent experiments on the systems and are reviewed. In the and systems, the low-temperature physical properties in the NFL regime scale with the U concentration and , suggesting that single-ion effects are responsible for the NFL behaviour.

The system exhibits an unconventional Kondo effect with non-Fermi-liquid characteristics at low temperatures for . Measurements of the low-temperature electrical resistivity and magnetization of high-purity samples with have been made as part of an effort to determine whether the non-Fermi-liquid behaviour persists to low U concentrations, i.e., whether it is a single-ion effect. Irreversible behaviour in the magnetization, reminiscent of spin glass freezing, is observed for samples with , while long-range antiferromagnetic ordering has previously been established by neutron diffraction measurements for a sample with x = 0.45. Magnetic relaxation, ac magnetic susceptibility, and specific heat measurements were performed to investigate this unusual magnetic ordering.

The transport, calorimetric and magnetic properties of have been investigated. For we have observed all low-temperature asymptotics predicted to date for the quadrupolar Kondo effect (QKE). Namely, this compound demonstrates a logarithmic increase of C/T where C is the specific heat, square-root temperature dependences of the resistivity and susceptibility, absence of magnetoresistivity as well as deviating to positive values of the non-linear susceptibility when temperature decreases. We have found that the Hall effect for the QKE is quenched in low magnetic fields. The anomalous properties of are compared with corresponding characteristics of and for x > 0.1.

We review the compiled measurements of the imaginary part of the dynamical magnetic susceptibility , static susceptibility , electrical resistivity and specific heat C(T) in the uranium intermetallics (x = 1, 1.5). We assess the temperature- and energy-dependences predicted by single-ion and disorder-dominated models and compare these results to experiments. For temperatures T and excitation energies in the range , T < 150 K, our analysis suggests that the dynamics of isolated uranium ions are responsible for the observed temperature and frequency scaling, although inter-ion interactions may become important at lower temperatures and frequencies.

We present a pedagogical and critical overview of the two-channel Kondo model and its possible relevance to a number of non-Fermi-liquid alloys and compounds. We survey the properties of the model, how a magnetic two-channel Kondo effect might obtain for ions in metals, and a quadrupolar Kondo effect for ions in metals. We suggest that the incoherent metal behaviour of the two-channel Kondo-lattice model may be useful in understanding the unusual normal-state resistivity of and speculate that the residual resistivity and entropy of the two-channel lattice paramagnetic phase might be removed by either antiferromagnetic (or antiferroquadrupolar) ordering or by a superconducting transition to an odd-frequency pairing state.

Evidence has emerged from nuclear magnetic resonance (NMR) and muon spin-rotation experiments that non-Fermi-liquid (NFL) behaviour in heavy-fermion alloys can in some cases be due to an inhomogeneous distribution of Kondo temperatures arising from disorder in the random alloy. Such `Kondo disorder' implies a broad distribution of the heavy-fermion spin polarization, which is reflected in the widths of NMR and lines. A simple model for the shape and width of the distribution of accounts for the temperature and field dependence of the bulk susceptibility in the NFL alloys , x = 1.0 and 1.5, and then agrees with NMR values of the width of the susceptibility distribution with no further adjustable parameters. Comparison of NMR and estimates of indicates that the susceptibility is disordered on an atomic length scale. In contrast, lines in are too narrow for Kondo disorder to account for NFL properties unless, as seems unlikely, the correlation length is long in this alloy. Similarly, small low-temperature NMR linewidths make it unlikely that Kondo disorder is the origin of NFL behaviour in .

We present a general model of disorder in Kondo alloys that, under certain conditions, leads to non-Fermi-liquid behaviour. The central underlying idea is the presence of a distribution of local Kondo temperature scales. If this distribution is broad enough, such that there are sites with arbitrarily low Kondo temperatures, a non-Fermi-liquid phase is formed. We analyse thermodynamics and transport in this approach and show it is consistent with a number of Kondo alloys. We also compare the predictions of this model with the measured dynamical magnetic response of these systems.

We report the results of an optical study of the alloy over a broad frequency range from 15 to . We have evaluated the frequency dependence of the scattering relaxation rate at various temperatures and find it to increase linearly with decreasing frequency and temperature. We interpret this as a manifestation of a non-Fermi-liquid behaviour of this alloy.

We present results of low-temperature calorimetric and resistive measurements on the isostructural heavy-fermion compounds and . `Non-Fermi-liquid' effects are established which suggest the nearness of an antiferromagnetic quantum critical point (QCP) in both systems. The observed deviations from the properties of a Landau Fermi liquid (FL) may be related to anomalous energy dependences of both the quasiparticle mass and the quasiparticle - quasiparticle scattering cross section. For , a moderately heavy FL can be recovered by application of moderate values of either magnetic field or hydrostatic pressure. For p = 1.7 GPa a novel, non-superconducting, phase transition has been discovered at .

The Fermi surfaces and single-particle spectral functions of several low-dimensional materials have been measured as part of an effort to assess the occurrence of non-Fermi-liquid behaviour in non-cuprate materials.

The substitution of transition metals for Cu in the heavy-fermion compound leads to unusual low-temperature properties. For example, the substitution of Pd suppresses long-range magnetic order and results in non-Fermi-liquid behaviour. In this report, we compare properties and phase diagrams for and based upon electrical resistivity and dc magnetic susceptibility measurements. In the Pt system, we find a rapid suppression of magnetic ordering similar to that seen in the Pd system. The electrical resistivity and dc magnetic susceptibility of exhibit low-temperature behaviour for suggestive of the non-Fermi-liquid behaviour found in for . The electrical resistivity of for reaches a concentration-dependent minimum before increasing at lower temperatures. At higher Pt concentrations , the electrical resistivity develops a broad peak at low temperatures, as in , suggesting short-range magnetic order although irreversibility was not seen in the magnetic susceptibility. The similarities of and differences between the two systems are discussed. The possibility of non-Fermi-liquid behaviour in is considered.

I summarize recent work on non-Fermi liquids within a certain generalized Anderson impurity model as well as in the large-dimensionality (D) limit of the two-band extended Hubbard model. The competition between local charge and spin fluctuations leads either to a Fermi liquid with renormalized quasiparticle excitations, or to non-Fermi liquids with spin - charge separation. These results provide new insights into the phenomenological similarities and differences between different correlated metals. While presenting these results, I outline a general strategy of local approach to non-Fermi liquids in correlated electron systems.

We observe that the appearance of two transport relaxation times in the various transport coefficients of cuprate metals may be understood in terms of scattering processes that discriminate between currents that are even, or odd under the charge-conjugation operator. We develop a transport equation that illustrates these ideas and discuss its experimental and theoretical consequences.

We study the temperature crossovers seen in the magnetic and transport properties of cuprates using a nearly antiferromagnetic Fermi-liquid model (NAFLM). We distinguish between underdoped and overdoped systems on the basis of their low-frequency magnetic behaviour and so classify the optimally doped cuprates as a special case of the underdoped cuprates. For the overdoped cuprates, we find, in agreement with earlier work, mean-field z = 2 behaviour of the magnetic variables associated with the fact that the damping rate of their spin fluctuations is essentially independent of temperature, while the resistivity exhibits a crossover from Fermi-liquid behaviour at low temperature to linear-in-T behaviour above a certain temperature . We demonstrate that above the proximity of the quasiparticle Fermi surface to the magnetic Brillouin zone boundary brings about the measured linear-in-T resistivity. For the underdoped cuprates we argue that the sequence of crossovers identified by Barzykin and Pines in the low-frequency magnetic behaviour (from mean-field z = 2 behaviour at high temperatures, , to non-universal z = 1 scaling behaviour at intermediate temperatures, , to pseudogap behaviour below ) reflects the development in the electronic structure of a precursor to a spin-density-wave state. This development begins at with a thermal evolution of the quasiparticle spectral weight which brings about temperature-dependent spin damping and ends at where the Fermi surface has lost pieces near corners of the magnetic Brillouin zone. For the resistivity is linear in T because this change in spectral weight does not affect the resistivity significantly; below vertex corrections act to bring about the measured downturn in and approximately quadratic-in-T resistivity for .

Far-infrared magnetotransmission measurements of normal-state thin films are discussed. The magneto-optical signals observed in circularly polarized light are consistent with the ac Hall effect of holes with a reduced scattering rate and an enhanced mass compared to the zero-field transport parameters. The results agree with a simple high-frequency extension of models which have been developed to explain the anomalous dc Hall effect in the normal state of high- superconductors in terms of two scattering rates. The Hall scattering rate , where and K. These experiments provide new tests of the Fermi-liquid versus non-Fermi-liquid nature of these materials.

We report on a study of the electromagnetic response of three different families of high- superconductors that in combination allowed us to cover the whole doping range from under- to overdoped. The discussion is focused on the ab-plane charge dynamics in the pseudogap state which is realized in underdoped materials below a characteristic temperature , a temperature that can significantly exceed the superconducting transition temperature . We explore the evolution of the pseudogap response by changing the doping level, by varying the temperature from above to below , or by introducing impurities in the underdoped compounds. We employ a memory function analysis of the ab-plane optical data that allows us to observe the effect of the pseudogap most clearly. We compare the infrared data with other experimental results, including the c-axis optical response, dc transport, and angle-resolved photoemission.

We discuss some generalities about the spin gap in cuprate superconductors and, in detail, how it arises from the interlayer picture. It can be thought of as spinon (uncharged) pairing, which occurs independently at each point of the 2D Fermi surface because of the momentum selection rule on interlayer superexchange and pair tunnelling interactions. Some predictions can be made.

We propose that in highly anisotropic, strongly correlated materials a novel breakdown of Fermi-liquid theory can result: the motion of the electrons in one or more directions may become entirely incoherent due to interaction effects even in the limit of zero temperature and an arbitrarily pure system. This paper presents an introduction to our arguments as to why quantum coherence is relevant to hopping between liquids of strongly interacting electrons and how it should be lost for sufficiently strong interactions, even though the hopping may remain relevant in a renormalization group sense. In this case a state with confined coherence obtains, even though the electrons themselves are not confined to individual planes in the low-energy limit. This proposal offers a natural resolution of certain apparent contradictions in the experimentally determined properties of the high-temperature superconductors and can uniquely explain the striking magnetoresistance anomalies observed for the organic superconductor .

is an itinerant ferromagnet with and a `bad metal' in the limit of . While the magnetic properties of in the paramagnetic phase, near the ferromagnetic phase transition and at low temperatures are normal and consistent with its being a strong itinerant ferromagnet, the transport properties (resistivity and magnetoresistance) sharply deviate from that of good metallic ferromagnets. We conjecture that the distinct transport behaviour of is related to its being a `bad metal' in the limit, and discuss the possible relevance of our results to the unusual transport properties of other `bad metals' such as high-temperature superconductors.

The fermionic Chern - Simons approach has had remarkable success in the description of quantum Hall states at even-denominator filling fractions . In this paper we review a number of recent works concerned with modelling this state as a Landau - Silin Fermi liquid. We will then focus on one particular problem with constructing such a Landau theory that becomes apparent in the limit of high magnetic field, or equivalently the limit of small electron band mass . In this limit, the static response of electrons to a spatially varying magnetic field is largely determined by kinetic energy considerations. We then remedy this problem by attaching an orbital magnetization to each fermion to separate the current into magnetization and transport contributions, associated with the cyclotron and guiding centre motions respectively. This leads us to a description of the state as a Fermi liquid of magnetized composite fermions which correctly predicts the -dependence of the static and dynamic response in the limit . As an aside, we derive a sum rule for the Fermi liquid coefficients for the Chern - Simons Fermi liquid. This paper is intended to be readable by people who may not be completely familiar with this field.