Electrons in metals interact via the Coulomb interaction. The size of this effect is not negligible: it is of order e 2 / a where a is the mean distance between electrons. For a metal this ≈ 3 eV, that is, comparable to E F . An approach to the problem would be to treat the interaction as a perturbation. However, the perturbation is strong and long-ranged, and this gives rise to many difficulties, such as divergences in finite orders of perturbation theory. The problems can be overcome by an application of the methods of quantum field theory to the problem, and is treated in books on many-body theory. We will not pursue this approach here, but try to get some of the central conclusions with physical reasoning. At first glance, we might be tempted to conclude that the sharp Fermi surface that we have discussed, and which is readily observed, should be wiped out by the interaction. The much smaller energy k B T does lead to smearing of the Fermi distribution; it is natural to conclude that this is true for interactions as well. This natural conclusion is not correct in three dimensions. The situation in one dimension is complicated, and the Fermi surface does disappear, leading to a state called the Luttinger liquid. In three dimensions the sharp Fermi surface survives the presence of interactions. This is an important phenomenon which demands explanation.
Electrons in metals interact via the Coulomb interaction. The size of this effect is not negligible: it is of order e 2 / a where a is the mean distance between electrons. For a metal this ≈ 3 eV, that is, comparable to E F . An approach to the problem would be to treat the interaction as a perturbation. However, the perturbation is strong and long-ranged, and this gives rise to many difficulties, such as divergences in finite orders of perturbation theory. The problems can be overcome by an application of the methods of quantum field theory to the problem, and is treated in books on many-body theory. We will not pursue this approach here, but try to get some of the central conclusions with physical reasoning. At first glance, we might be tempted to conclude that the sharp Fermi surface that we have discussed, and which is readily observed, should be wiped out by the interaction. The much smaller energy k B T does lead to smearing of the Fermi distribution; it is natural to conclude that this is true for interactions as well. This natural conclusion is not correct in three dimensions. The situation in one dimension is complicated, and the Fermi surface does disappear, leading to a state called the Luttinger liquid. In three dimensions the sharp Fermi surface survives the presence of interactions. This is an important phenomenon which demands explanation.
It is quite beyond the purpose of this book to give a comprehensive review of the modern theory of metals. I shall touch only on those topics which are related to the mainstream of this course: the problem of strong interactions. As I have noted before, strong interactions can appear as a result of renormalization, and from the aesthetic point of view this is, perhaps, the most interesting case. As we shall see in Part IV, such renormalizations are particularly strong in one-dimensional systems, where practically any interaction leads to quite dramatic effects. Therefore a one-dimensional electron gas neither undergoes a phase transition (this is forbidden due to the low dimensionality), nor behaves like a system of free electrons. On the contrary, in higher dimensions at low temperatures a system of electrons either undergoes a phase transition (superconductivity, magnetic ordering, etc.), or behaves like a free electron gas. When temperatures as low as several degrees became experimentally available, physicists discovered that an enormous amount of experimental data on normal metals can be described by the model where one neglects electron–electron interactions. This apparent miracle was explained by Landau who demonstrated that the interaction pattern drastically simplifies close to the Fermi surface. Provided the system does not undergo a symmetry breaking phase transition, all interactions except forward scattering effectively vanish on the Fermi surface and low-lying excitations carry quantum numbers of electrons. Therefore they are called quasi-particles . The meaning of the word quasi will become clear in a moment.
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
We especially need imagination in science. aria Mitchell Summary The many-body problem consists of two parts: the first is the non-interacting system in a materials-specific external potential; the second is the Coulomb interaction that makes the problem so hard to solve. The most straightforward idea is to use perturbation theory, with the Coulomb interaction as perturbation. This is conceptually simple, but it turns out to be difficult in practice, since the Coulomb interaction is often not small compared with typical energy differences, it is long-ranged and in the thermodynamic limit there is an infinite number of particles, contributing with an infinite number of mutual interaction processes. The present chapter outlines how one can deal with this problem. It contains an overview of facts that one can also find in many standard textbooks on the many-body problem, but that are useful to keep in mind in order to look at later chapters from a sound and well-established perspective. The many-body problem is a tough one, and it has many facets. Sorting it out is like putting together a huge puzzle. The eight introductory chapters of this book provide pieces of the puzzle, and ideas on what one might do about it. In the present chapter we choose to go in one of the possible directions, in order to arrive at something tangible. The chapter gives the general framework and the main ideas; specific approximations are the topic of Chs. 10–15. The idea is to start from an independent-particle problem and add the Coulomb interaction as a perturbation. This is not easy: first, the interaction is responsible for a rich variety of phenomena that are completely absent otherwise, such as the finite lifetime of quasiparticles, or additional structures in spectra due to the fact that a quasi-particle excitation may transfer its energy to other elementary excitations, for example plasmons. Second, because of the two-body Coulomb interaction, the problem scales badly with the number of electrons, and straightforward perturbation theory for the many-body hamiltonian with the Coulomb interaction as perturbation rapidly becomes intractable or even useless, especially in large systems. To get started, Sec. 9.1 recalls why things are not so easy. The following sections try to solve one problem after the other, starting from Sec. 9.2 where the Green's function is reformulated in a way that is appropriate for a perturbation expansion.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.
We use higher-dimensional bosonization to study the normal state of electrons in weakly coupled metallic chains interacting with long-range Coulomb forces. Particular attention is paid to the crossover between Luttinger and Fermi-liquid behavior as the interchain hopping ${t}_{\ensuremath{\perp}}$ is varied. Although in the physically interesting case of finite but small ${t}_{\ensuremath{\perp}}$ the quasiparticle residue does not vanish, the single-particle Green's function exhibits the signature of Luttinger-liquid behavior (i.e., anomalous scaling and spin-charge separation) in a large intermediate parameter regime. Using realistic parameters, we find that the scaling behavior in this regime is characterized by an anomalous dimension of the order of unity, as suggested by recent experiments on quasi-one-dimensional conductors. Our calculation also gives insights into the approximations inherent in higher-dimensional bosonization; in particular, we show that the replacement of a curved Fermi surface by a finite number M of flat patches can give rise to unphysical nesting singularities in the single-particle Green's function, which disappear only in the limit $M\ensuremath{\rightarrow}\ensuremath{\infty}$ or if curvature effects are included. We also compare our approach with other methods. This work extends our recent letter [Phys. Rev. Lett. 74, 2997 (1995)].
In most metals, the electrons behave as if they were independent of one another. Although each electron really does interact with the other electrons and phonons, the net effect is felt only as a kind of molasses through which the otherwise freely moving electron must slog. Thus one an usually treat metallic systems as a low-density gas of weakly interacting particles, accounting for the average interactions simply by assigning an effective mass to the electrons. Such a model is commonly called a Fermi liquid. In one-dimensional systems, however, the Coulomb forces between the electrons intervene more strongly and produce quite a different behavior.