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A brief introduction to Luttinger liquids

Johannes Voit, Theoretische Physik, Universität Bayreuth, D- Bayreuth (germany

Published Jan 1, 2000• 11 citations
OpenAlex

Abstract

Abstract. I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant Kρ, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems. WHAT IS A LUTTINGER LIQUID ANYWAY? Ordinary, three-dimensional metals are described by Fermi liquid theory. Fermi liquid theory is about the importance of electron-electron interactions in metals. It states that there is a 1:1-correspondence between the low-energy excitations of a free Fermi gas, and those of an interacting electron liquid which are termed “quasiparticles” [1]. Roughly speaking, the combination of the Pauli principle with low excitation energy (e.g. T ≪ EF) and the large phase space available in 3D, produces

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A brief introduction to Luttinger liquids

Johannes Voit

I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant Kρ, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems.

arXiv:cond-mat/0005114•AIP conference proceedings•Jan 1, 2000•10 citations

One-dimensional Fermi liquids

Johannes Voit

We review the progress in the theory of one-dimensional (ID) 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 1D quantum 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 Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.

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Conservation laws and correlation functions in the Luttinger liquid

Walter Metzner, C. Di Castro

The low-energy properties of interacting Fermi systems are highly constrained by conservation laws. They generally simplify the structure of the underlying renormalization group by reducing the number of independent renormalization constants. In one dimension, all properties of normal metallic fixed points are uniquely determined by separate charge and spin conservation for states near the left and right Fermi points, respectively. We construct the general Luttinger-liquid theory of one-dimensional (1D) metals directly from these conservation laws. Luttinger-liquid parameters emerge naturally from the velocities associated with the conserved currents at the Luttinger-liquid fixed point. Instead of bosonization, one may thus use techniques familiar from Fermi-liquid theory, i.e., Feynman diagrams, equations of motion, and Ward identities. The choice of a technique comprising both Fermi- and Luttinger-liquid theory makes the similarities and differences of both theories particularly transparent, and sets the stage for constructing non-Fermi-liquid metallic fixed points in d>1. Several generic properties and asymptotic conservation laws of 2D non-Fermi-liquid metals are discussed.

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Colloquium: The spin-incoherent Luttinger liquid

Gregory A. Fiete

In contrast to the well-known Fermi-liquid theory of three dimensions, interacting one-dimensional and quasi-one-dimensional systems of fermions are described at low energy by an effective theory known as Luttinger liquid theory. This theory is expressed in terms of collective many-body excitations that show exotic behavior such as spin-charge separation. Luttinger liquid theory is commonly applied on the premise that ``low energy'' describes both the spin and charge sectors. However, when the interactions in the system are very strong, as they typically are at low particle densities, the ratio of spin to charge energy may become exponentially small. It is then possible at very low temperatures for the single-spin excitation energy to be low compared to the characteristic single excitation charge energy, but still high compared to the characteristic spin energy. This energy window of near ground-state charge degrees of freedom but highly thermally excited spin degrees of freedom is called a spin-incoherent Luttinger liquid. The spin-incoherent Luttinger liquid exhibits a higher degree of universality than the Luttinger liquid and its properties are qualitatively distinct. In this Colloquium some recent theoretical developments in the field are detailed and experimental indications of such a regime in gated semiconductor quantum wires are described.

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Electrons in one dimension and the Luttinger liquid

Gabriele F. Giuliani, Giovanni Vignale

Non-Fermi liquid behavior In this chapter we begin the study of electronic systems that are not Landau Fermi liquids. These systems are like totalitarian societies in which the behavior of the individual is subordinated to the needs of the organization: their low-energy excitations are collective, rather than single-particle-like. A strongly collective behavior is not at all unusual in condensed matter. For example, the low energy excitations of a crystal lattice are acoustic phonons, which are collective oscillations of the atoms about their equilibrium positions. However, such examples are usually associated with a broken symmetry (translational symmetry in this case): when it comes to homogeneous electron liquids, the familiar picture of Landau's quasiparticles is so ingrained that we tend to regard any departure from it as a surprising phenomenon. Nevertheless, departures from the normal Fermi liquid pattern can and do occur in two typical scenarios: In three- and two-dimensional systems in which the electron liquid is strongly correlated , i.e., when the order of magnitude of the coulomb interaction greatly exceeds the kinetic bandwidth, In quasi-one-dimensional systems, for any strength of the interaction. A trivial example of the first scenario is offered by the three-dimensional electron liquid at very low-density. In this case the electrons form a Wigner crystal and their collective behavior arises immediately from the loss of translational symmetry. Amuch more complex example is offered by the two-dimensional electron liquid at high magnetic field. Because the kinetic energy is effectively suppressed by the magnetic field, the structure of the groundstate and the low-lying excitations is entirely controlled by the coulomb interaction.

Cambridge University Press eBooks•Mar 31, 2005•0 citations

LUTTINGER LIQUIDS

Thierry Giamarchi

The bosonization technique is useful for solving the interacting problem and for extracting a certain number of properties such as power law behavior of the correlation functions. However, this derivation is, strictly speaking, based on a linearization of the spectrum close to the Fermi energy. One can thus wonder what will remain of the results of the previous chapter when it is impossible to stay confined close to the Fermi level, for example, when the interactions become strong. In order to answer this question, it is necessary to rederive the bosonization formulas, in a more phenomenological but more general way. This chapter demonstrates that a similar concept as Fermi liquids exists in one dimension. This concept, called Luttinger liquid, encompasses most of the physics of simple one-dimensional systems and will be the starting point for the study more complex situations. The concept of Luttinger liquids is explained using very low level techniques.

Oxford University Press eBooks•Dec 18, 2003•0 citations
Abstract. I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant Kρ, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems. WHAT IS A LUTTINGER LIQUID ANYWAY? Ordinary, three-dimensional metals are described by Fermi liquid theory. Fermi liquid theory is about the importance of electron-electron interactions in metals. It states that there is a 1:1-correspondence between the low-energy excitations of a free Fermi gas, and those of an interacting electron liquid which are termed “quasiparticles” [1]. Roughly speaking, the combination of the Pauli principle with low excitation energy (e.g. T ≪ EF) and the large phase space available in 3D, produces
I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant Kρ, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems.
We review the progress in the theory of one-dimensional (ID) 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 1D quantum 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 Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.
The low-energy properties of interacting Fermi systems are highly constrained by conservation laws. They generally simplify the structure of the underlying renormalization group by reducing the number of independent renormalization constants. In one dimension, all properties of normal metallic fixed points are uniquely determined by separate charge and spin conservation for states near the left and right Fermi points, respectively. We construct the general Luttinger-liquid theory of one-dimensional (1D) metals directly from these conservation laws. Luttinger-liquid parameters emerge naturally from the velocities associated with the conserved currents at the Luttinger-liquid fixed point. Instead of bosonization, one may thus use techniques familiar from Fermi-liquid theory, i.e., Feynman diagrams, equations of motion, and Ward identities. The choice of a technique comprising both Fermi- and Luttinger-liquid theory makes the similarities and differences of both theories particularly transparent, and sets the stage for constructing non-Fermi-liquid metallic fixed points in d>1. Several generic properties and asymptotic conservation laws of 2D non-Fermi-liquid metals are discussed.
In contrast to the well-known Fermi-liquid theory of three dimensions, interacting one-dimensional and quasi-one-dimensional systems of fermions are described at low energy by an effective theory known as Luttinger liquid theory. This theory is expressed in terms of collective many-body excitations that show exotic behavior such as spin-charge separation. Luttinger liquid theory is commonly applied on the premise that ``low energy'' describes both the spin and charge sectors. However, when the interactions in the system are very strong, as they typically are at low particle densities, the ratio of spin to charge energy may become exponentially small. It is then possible at very low temperatures for the single-spin excitation energy to be low compared to the characteristic single excitation charge energy, but still high compared to the characteristic spin energy. This energy window of near ground-state charge degrees of freedom but highly thermally excited spin degrees of freedom is called a spin-incoherent Luttinger liquid. The spin-incoherent Luttinger liquid exhibits a higher degree of universality than the Luttinger liquid and its properties are qualitatively distinct. In this Colloquium some recent theoretical developments in the field are detailed and experimental indications of such a regime in gated semiconductor quantum wires are described.
Non-Fermi liquid behavior In this chapter we begin the study of electronic systems that are not Landau Fermi liquids. These systems are like totalitarian societies in which the behavior of the individual is subordinated to the needs of the organization: their low-energy excitations are collective, rather than single-particle-like. A strongly collective behavior is not at all unusual in condensed matter. For example, the low energy excitations of a crystal lattice are acoustic phonons, which are collective oscillations of the atoms about their equilibrium positions. However, such examples are usually associated with a broken symmetry (translational symmetry in this case): when it comes to homogeneous electron liquids, the familiar picture of Landau's quasiparticles is so ingrained that we tend to regard any departure from it as a surprising phenomenon. Nevertheless, departures from the normal Fermi liquid pattern can and do occur in two typical scenarios: In three- and two-dimensional systems in which the electron liquid is strongly correlated , i.e., when the order of magnitude of the coulomb interaction greatly exceeds the kinetic bandwidth, In quasi-one-dimensional systems, for any strength of the interaction. A trivial example of the first scenario is offered by the three-dimensional electron liquid at very low-density. In this case the electrons form a Wigner crystal and their collective behavior arises immediately from the loss of translational symmetry. Amuch more complex example is offered by the two-dimensional electron liquid at high magnetic field. Because the kinetic energy is effectively suppressed by the magnetic field, the structure of the groundstate and the low-lying excitations is entirely controlled by the coulomb interaction.
The bosonization technique is useful for solving the interacting problem and for extracting a certain number of properties such as power law behavior of the correlation functions. However, this derivation is, strictly speaking, based on a linearization of the spectrum close to the Fermi energy. One can thus wonder what will remain of the results of the previous chapter when it is impossible to stay confined close to the Fermi level, for example, when the interactions become strong. In order to answer this question, it is necessary to rederive the bosonization formulas, in a more phenomenological but more general way. This chapter demonstrates that a similar concept as Fermi liquids exists in one dimension. This concept, called Luttinger liquid, encompasses most of the physics of simple one-dimensional systems and will be the starting point for the study more complex situations. The concept of Luttinger liquids is explained using very low level techniques.