Singular Fermi Liquids C. M. Varmaa,b, Z. Nussinovb, and Wim van Saarloosb aBell Laboratories, Lucent Technologies, Murray Hill, NJ 07974, U.S.A. 1 bInstituut–Lorentz, Universiteit Leiden, Postbus 9506, 2300 RA Leiden, The Netherlands Abstract An introductory survey of the theoretical ideas and calculations and the ex- perimental results which depart from Landau Fermi-liquids is presented. Common themes and possible routes to the singularities leading to the breakdown of Landau Fermi liquids are categorized following an elementary discussion of the theory. Sol- uble examples of Singular Fermi liquids include models of impurities in metals with special symmetries and one-dimensional interacting fermions. A review of these is followed by a discussion of Singular Fermi liquids in a wide variety of experimental situations and theoretical models. These include the effects of low-energy collective fluctuations, gauge fields due either to symmetries in the hamiltonian or possible dynamically generated symmetries, fluctuations around quantum critical points, the normal state of high temperature superconductors and the two-dimensional metallic state. For the last three systems, the principal experimental results are summarized and the outstanding theoretical issues highlighted. Contents 1 Introduction 4 1.1 Aim and scope of this paper 4 1.2 Outline of the paper 7 arXiv:cond-mat/0103393v1 [cond-mat.str-el] 19 Mar 2001 2 Landau’s Fermi-liquid 8 2.1 Essentials of Landau Fermi-liquids 8 2.2 Landau Fermi-liquid and the wave function renormalization Z 11 2.3 Understanding microscopically why Fermi-liquid Theory works 17 1 Present and permanent address Preprint submitted to Elsevier Preprint 1 February 2008 2.4 Principles of the Microscopic Derivation of Landau Theory 22 2.5 Modern derivations 27 2.6 Routes to Breakdown of Landau Theory 29 3 Local Fermi-Liquids & Local Singular Fermi-Liquids 34 3.1 The Kondo Problem 34 3.2 Fermi-liquid Phenemenology for the Kondo problem 38 3.3 Ferromagnetic Kondo problem and the anisotropic Kondo problem 39 3.4 Orthogonality Catastrophe 40 3.5 X-ray Edge singularities 41 3.6 A Spinless Model with Finite Range Interactions 43 3.7 A model for Mixed-Valence Impurity 45 3.8 Multi-channel Kondo problem 47 3.9 The Two-Kondo-Impurities Problem 51 4 SFL behavior for interacting fermions in one dimension 55 4.1 The One Dimensional Electron Gas 57 4.2 The Tomonaga-Luttinger Model 59 4.3 One-particle Spectral functions 62 4.4 Thermodynamics 63 4.5 Correlation Functions 64 4.6 The Luther-Emery Model 65 4.7 Spin charge separation 67 4.8 Spin-charge Separation in more than one-dimension? 69 4.9 Recoil and the Orthogonality Catastrophe in 1d and higher 72 4.10 Coupled One-dimensional Chains 76 4.11 Experimental observations of one-dimensional Luttinger liquid behavior 76 2 5 Singular Fermi-liquid behavior due to gauge fields 79 5.1 SFL behavior due to coupling to the electromagnetic field 79 5.2 Generalized gauge theories 82 6 Quantum Critical Points in fermionic systems 84 6.1 Quantum critical points in ferromagnets, antiferromagnets, and charge density waves 85 6.2 Quantum critical scaling 88 6.3 Experimental Examples of SFL due to Quantum-Criticality: Open Theoretical Problems 96 6.4 Special complications in heavy fermion physics 101 6.5 Effects of impurities on Quantum Critical Points 103 7 The High-Tc Problem in the Copper-Oxide Based Compounds 104 7.1 Some basic features of the high-Tc materials 105 7.2 Marginal Fermi Liquid behavior of the normal state 108 7.3 General Requirements on a Microscopic Theory 115 7.4 Microscopic Theory 117 8 The Metallic State in Two-Dimensions 123 8.1 The two-dimensional Electron Gas 124 8.2 Noninteracting Disordered Electrons: Scaling Theory of Localization 126 8.3 Interactions in Disordered Electrons 130 8.4 Finkelstein Theory 135 8.5 Compressibility, Screening length and a Mechanism for Metal- Insulator Transition 137 8.6 Experiments 138 8.7 Discussion of the Experiments in light of the theory of Interacting Disordered Electrons 144 8.8 Phase Diagram and Concluding Remarks 150 3 9 Acknowledgements 152 References 152 1 Introduction 1.1 Aim and scope of this paper In the last two decades a variety of metals have been discovered which dis- play thermodynamic and transport properties at low temperatures which are fundamentally different from those of the usual metallic systems which are well described by the Landau Fermi-liquid theory. They have often been re- ferred to as Non-Fermi-liquids. A fundamental characteristic of such systems is that the low-energy properties in a wide range of their phase diagram are dominated by singularities as a function of energy and temperature. Since these problems still relate to a liquid state of fermions and since it is not a good practice to name things after what they are not, we prefer to call them Singular Fermi-liquids (SFL). The basic notions of Fermi-liquid theory have actually been with us at an intuitive level since the time of Sommerfeld: He showed that the linear low temperature specific heat behavior of metals as well as their asymptotic low temperature resisitivity and optical conductivity could be understood by as- suming that the electrons in a metal could be thought of as a gas of non- interacting fermions, i.e., in terms of quantum mechanical particles which do not have any direct interaction but which do obey Fermi statistics. Mean- while Pauli calculated that the paramagnetic susceptibility of non-interacting electrons is independent of temperature, also in accord with experiments in metals. At the same time it was understood, at least since the work of Bloch and Wigner, that the interaction energies of the electrons in the metallic range of densities are not small compared to the kinetic energy. The rationalization for the qualitative success of the non-interacting model was provided in a mas- terly pair of papers by Landau [144,145] who initially was concerned with the properties of liquid 3He. This work introduced a new way of thinking about the properties of interacting systems which is a cornerstone of our understanding of condensed matter physics. The notion of quasiparticles and elementary ex- citations and the methodology of asking useful questions about the low-energy excitations of the system based on concepts of symmetry, without worrying about the myriad unnecessary details, is epitomized in Landau’s phenomen- logical theory of Fermi-liquids. The microscopic derivation of the theory was also soon developed. 4 Our perspective on Fermi-liquids has changed significantly in the last two decades or so. This is due both to changes in our theoretical perspective, and due to the experimental developments: on the experimental side, new materi- als have been found which exhibit Fermi-liquid behavior in the temperature dependence of their low temperature properties with the coefficients often a factor of order 103 different from the non-interacting electron values. These observations dramatically illustrate the power and range of validity of the Fermi-liquid ideas. On the other hand, new materials have been discovered whose properties are qualitatively different from the predictions of Fermi- liquid theory (FLT). The most prominently discussed of these materials are the normal phase of high-temperature superconducting materials for a range of compositions near their highest Tc. Almost every idea discussed in this review has been used to understand the high-Tc problem, but there is no consensus yet on the solution. It has of course been known for a long time that FLT breaks down in the fluctuation regime of classical phase transitions. This breakdown happens in a more substantial region of the phase diagram around the quantum critical point (QCP) where the transition temperature tends to zero as a function of some parameter, see Fig. 1. This phenomenon has been extensively investi- gated for a wide variety of magnetic transitions in metals where the transition temperature can be tuned through application of pressure or by varying the electronic density through alloying. Heavy fermion with their close competi- tion between states of magnetic order with localized moments and itinerant states due to Kondo-effects appear particularly prone to such QCP’s. Equally interesting are questions having to do with the change in properties due to impurities in systems which are near a QCP in the pure limit. The density-density correlations of itinerant disordered electrons at long wave- lengths and low energies must have a diffusive form. In two-dimensions this leads to logarithmic singularities in the effective interactions when the inter- actions are treated perturbatively. The problem of finding the ground state and low-lying excitations in this situation is unsolved. On the experimental side the discovery of the metal-insulator transition in two dimensions and the unusual properties observed in the metallic state makes this an important problem to resolve. The one-dimensional electron gas reveals logarithmic singularities in the effec- tive interactions even in a second-order perturbation calculation. A variety of mathematical techniques have been used to solve a whole class of interacting one-dimensional problems and one now knows the essentials of the correlation functions even in the most general case. An important issue is whether and how this knowledge can be used in higher dimensions. The solution of the Kondo problem and the realization that its low temper- 5 Fig. 1. Schematic phase diagram near a Quantum Critical Point. The parameter along the x-axis can be quite general, like the pressure or a ratio of coupling con- stants. Whenever the critical temperature vanishes, a QCP, indicated with a dot in the figure, is encountered.