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One-Dimensional Fermi Liquids, Or Following Haldane “Luttinger Liquids” [3], Are the Main Subjects of This Review Article

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One-Dimensional Fermi Liquids, Or Following Haldane “Luttinger Liquids” [3], Are the Main Subjects of This Review Article One-dimensional Fermi liquids Johannes Voit Bayreuther Institut f¨ur Makromolek¨ulforschung (BIMF) and Theoretische Physik 1 Universit¨at Bayreuth D-95440 Bayreuth (Germany)1 and Institut Laue-Langevin F-38042 Grenoble (France) submitted to Reports on Progress in Physics on November 19, 1994 arXiv:cond-mat/9510014v1 29 Sep 1995 last revision and update on February 1, 2008 1Present address Abstract We review the progress in the theory of one-dimensional (1D) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle- hole bubble producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless one-dimensional quan- tum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a “Luttinger liquid” implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t J- or the extended Hub- − bard model. Also 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect. We discuss in detail various solutions of the Luttinger model which emphasize different aspects of the physics of 1D Fermi liquids. Correlation functions are calculated in detail using bosonization, and the relation of this method to other approaches is discussed. The correlation functions decay as non-universal power-laws, and scaling relations between their exponents are parameterized by the effective coupling constant. Charge-spin sepa- ration only shows up in dynamical correlations. The Luttinger liquid concept is developed from perturbations of the Luttinger model. Mainly specializing to the 1D Hubbard model, we review a variety of mappings for complicated models of interacting electrons onto Lut- tinger models, and thereby obtain their correlation functions. We also discuss the generic behaviour of systems not falling into the Luttinger liquid universality class because of gaps in their low-energy spectrum. The Mott transition provides an example for the tran- sition from Luttinger to non-Luttinger behaviour, and recent results on this problem are summarized. Coupling chains by interactions or tunneling allows transverse coherence to establish in the single- or two-particle dynamics, and drives the systems away from a Luttinger liquid. We discuss the influence of charge-spin separation and of the anomalous dimensions on the transverse dynamics of the electrons. The edge states in the quan- i tum Hall effect provide a realization of a modified, chiral Luttinger liquid whose detailed properties differ from those of the standard model. The article closes with a summary of experiments which can be interpreted in favour of Luttinger liquid-correlations in the “normal” state of quasi-1D organic conductors and superconductors, charge density wave systems, and semiconductors in the quantum Hall regime. ii Contents 1 Introduction 1 1.1 Motivation.................................... 1 1.2 Purposeandstructureofthisreview . .... 5 2 Fermi liquid theory and its failure in one dimension 8 2.1 TheFermiliquid ................................ 8 2.2 Breakdown of Fermi liquid theory in one dimension . ....... 10 3 The Luttinger model 14 3.1 Low-energy phenomenology in 1D – the Luttinger model . ....... 14 3.1.1 Ground state and elementary excitations of 1D fermions ...... 14 3.1.2 Tomonaga-LuttingerHamiltonian . .. 15 3.1.3 Symmetries and conservation laws . .. 16 3.2 BosonsolutionoftheLuttingermodel . .... 18 3.2.1 DiagonalizationofHamiltonian . .. 19 3.2.2 Bosonization............................... 24 3.3 Physical Properties of the Luttinger Model – Thermodynamics and Corre- lationFunctions................................. 27 3.3.1 Thermodynamicsandtransport . 28 3.3.2 Single- and two-particle correlation functions . ......... 30 3.4 Dynamical correlations: the spectral properties of Luttinger liquids . 36 3.5 Alternativemethods .............................. 38 3.5.1 Greenfunctionmethods . 38 3.5.2 Otherbosonicschemes . 42 3.6 Conformalfieldtheoryandbosonization . ..... 43 3.6.1 Conformal invariance at a critical point . ..... 43 3.6.2 TheGaussianmodel .......................... 50 4 The Luttinger Liquid 56 4.1 Theconjecture ................................. 56 4.2 Luttinger model with nonlinear dispersion – the emergence of higher har- monics...................................... 58 4.3 BackwardandUmklappscattering . .. 59 4.4 Latticemodels:Hubbard&Co. 64 iii 4.4.1 Models.................................. 64 4.4.2 BetheAnsatz .............................. 65 4.4.3 Low-energy properties of one-dimensional lattice models ...... 69 4.5 Electron-phonon interaction and impurity scattering . ............ 82 4.6 TransportinLuttingerliquids . .... 86 4.6.1 Electron-electronscattering . ... 86 4.6.2 Electron-impurity scattering . ... 88 4.6.3 Electron-phononscattering. .. 94 4.7 The notion of a Landau-Luttinger liquid . ..... 95 5 Alternatives to the Luttinger liquid: spin gaps, the Mott transition, and phase separation 98 5.1 Spingaps .................................... 98 5.2 TheMotttransition............................... 100 5.3 Phaseseparation ................................107 6 Extensions of the Luttinger Liquid 109 6.1 Multi-componentmodels . 110 6.2 Crossover to higher dimensions . 114 6.3 EdgestatesinthequantumHalleffect . 128 7 The normal state of quasi-one-dimensional metals – a Luttinger liquid?134 7.1 Organic conductors and superconductors . ......134 7.2 Inorganic charge density wave materials . .......138 7.3 Semiconductor heterostructures . .139 8 Summary 141 iv Chapter 1 Introduction 1.1 Motivation Strongly correlated fermions are an important problem in solid state physics. Over the last one or two decades, experiments on many classes of materials have provided evidence that strong correlations are a central ingredient for the understanding of their physical properties. Among them are the heavy fermion compounds, the high-Tc superconductors, a variety of intimately related organic metals, superconductors, and insulators, just to name a few. Also in normal metals, the interactions between the electrons are rather strong, although the correlations may be much weaker than in the systems mentioned before. The effective dimension of the electron gas plays an important role in correlating interacting fermions, and the materials listed are essentially three-(3D), two-(2D), and one-dimensional (1D), respectively. Correlations are also very important in semiconduc- tor heterostructures and quantum wires, being two- or one-dimensional, including the Quantum Hall regime. The theoretical description of strongly interacting electrons poses a formidable prob- lem. Exact solutions of specific models usually are impossible, exception made for certain one-dimensional models to be discussed later. Fortunately, such exact solutions are rarely required (and more rarely even practical) when comparing with experiment. Most mea- surements, in fact, only probe correlations on energy scales small compared to the Fermi energy EF so that only the low-energy sector of a given model is of importance. More- over, only at low energies can we hope to excite only a few degrees of freedom, for which a meaningful comparison to theoretical predictions can be attempted. Correlated fermions in three dimensions are a well studied problem. Their theo- retical description, by Fermi liquid theory, is approximate but well understood [1, 2]. It becomes an asymptotically exact solution for low energies and small wavevectors (E E , k k , T 0). The limitation to low energies is instrumental here → F | | → F → because, together with Fermi statistics, it implies that the phase space for excitations is severely restricted. In one dimension, there is a variety of exactly solvable models, which have been known for quite a time, but a deeper understanding of their mutual relation- ships and their relevance for describing the generic low-energy physics, close to the 1D 1 Fermi surface, has emerged only rather recently. These relations as well as the properties of such one-dimensional Fermi liquids, or following Haldane “Luttinger liquids” [3], are the main subjects of this review article. Here we shall use the terms “one-dimensional Fermi liquids” and “Luttinger liquids” synonymously, although, as we show below, Fermi liquid behaviour as it is established in 3D is not possible in 1D. Fermi liquid theory is based on (but not exhausted by) a picture of quasi-particles evolving out of the particles (holes) of a Fermi gas upon adiabatically switching on in- teractions [1, 2]. They are in one-to-one correspondence with the bare particles and, specifically, carry the same quantum
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