Introduction Much of traditional condensed matter physics falls under the Fermi liquid paradigm. The notion goes back to Landau and implies that a liquid of fermions (such as 3 He or conduction electrons in a metal) can be treated as a system of essentially free particles [1]. Certain conditions apply: symmetries must remain unbroken and the energies probed should be "small." But with these provisos the only price to pay for removing the interactions is to keep track on how certain parameters (quasiparticle masses, lifetimes, etc.) renormalize. The Fermi-liquid picture has proven enormously successful and explains how we can get away with single-particle quantum mechanics when doing elementary solid state physics. Fortunately --- to keep us busy! --- experimentalists have found a number of lab systems that violate the standard Fermi liquid picture. (An outstanding example is the metallic phase of the high-T c superconductors.) The desi
Introduction Much of traditional condensed matter physics falls under the Fermi liquid paradigm. The notion goes back to Landau and implies that a liquid of fermions (such as 3 He or conduction electrons in a metal) can be treated as a system of essentially free particles [1]. Certain conditions apply: symmetries must remain unbroken and the energies probed should be "small." But with these provisos the only price to pay for removing the interactions is to keep track on how certain parameters (quasiparticle masses, lifetimes, etc.) renormalize. The Fermi-liquid picture has proven enormously successful and explains how we can get away with single-particle quantum mechanics when doing elementary solid state physics. Fortunately --- to keep us busy! --- experimentalists have found a number of lab systems that violate the standard Fermi liquid picture. (An outstanding example is the metallic phase of the high-T c superconductors.) The desi
The Fermi liquid is perhaps the most familiar quantum many-body state of solid state physics; we met it briefly in Section 16.2.2. It is the generic state of fermions at nonzero density, and is found in all metals. Its basic characteristics can already be understood in a simple free fermion picture. Noninteracting fermions occupy the lowest energy single-particle states, consistent with the exclusion principle. This leads to the fundamental concept of the Fermi surface: a surface in momentum space separating the occupied and empty single fermion states. The lowest energy excitations then consist of quasiparticle excitations which are particle-like outside the Fermi surface, and hole-like inside the Fermi surface. Landau's Fermi liquid theory is a careful justification for the stability of this simple picture in the presence of interactions between fermions. Just as we found in Chapters 5 and 7 for the quantum Ising and rotor models, interaction corrections modify the wavefunction of the quasiparticle and so introduce a quasiparticle residue A ; however, they do not destabilize the integrity of the quasiparticle, as we review in Section 18.1. The purpose of this chapter is to describe two paradigms of symmetry breaking quantum transitions in Fermi liquids. In the first class, studied in Section 18.2, the broken symmetry is related to the point-group symmetry of the crystal, while translational symmetry is preserved; consequently, the order parameter resides at zero wavevector. In the second class, studied in Section 18.3, the order parameter is at a finite wavevector, and so translational symmetry is also broken.
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.
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe. This book is a pedagogical and systematic introduction to the new concepts and quantum field theoretical methods in condensed matter physics. It discusses many basic notions in theoretical physics which underlie physical phenomena in nature, including a notion that unifies light and electrons. Topics covered include dissipative quantum systems, boson condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, and spin liquid and string-net condensation. Methods discussed include the path integral, Green's functions, mean-field theory, effective theory, renormalization group, bosonization in one- and higher dimensions, non-linear sigma-model, quantum gauge theory, dualities, projective construction, and exactly soluble models beyond one-dimension.
For pure fermionic systems, and in the absence of symmetry breaking (e.g. magnetism, superconductivity, …), two types of metallic phases are well established: (i) the “normal” Fermi liquid phase in d = 3 presented in Chapters 12 and 19 via the response and the Green functions techniques; and (ii) the “anomalous” Luttinger liquid phase in d = 1 originally formulated with the bosonization technique by which the fermionic operators are represented in terms of “bosonic” density operators (Luther and Peschel (1974); Haldane (1981)) with the exact solution (Mattis and Lieb (1965)) of the Tomonaga–Luttinger model (Tomonaga (1950); Luttinger (1963)) discussed inmany reviews and books (see for instance the book by Giamarchi (2004)). Here, to stress similarities and differences between the Fermi liquid and the Luttinger liquid, we will follow the alternative route of response and Green functions techniques (Dzyaloshinskii and Larkin (1973), Di Castro and Metzner (1991), Metzner et al. (1998)) and add a few comments on the renormalization group approach (Metzner and Di Castro (1993)). As we have seen in Chapters 12 and 19, the Fermi liquid is described asymptotically by the free low-lying single-particle excitations, i.e., the quasiparticles with a zero-kelvin discontinuous occupation number in momentum space n k (see Eq. (18.54)) at a well defined Fermi surface (k = k F ): where here k F> and k F< indicate the limit k → k F from above and below. This discontinuity still marks the Fermi surface in the interacting system. The finite reduction of the single-particle spectral weight z kF with respect to the Fermi gas (z kF = 1) is given by the finite (i.e. non-critical) “wavefunction renormalization” (see Eq. (18.94)). The presence of the discontinuity at the Fermi surface, together with the Pauli principle, compel the inverse quasiparticle lifetime to be τ −1 ≈ max( T 2 , ϵ 2 ), where _ is the deviation of the energy of the quasiparticle from the Fermi energy. The energy uncertainty ħτ −1 due to the finite lifetime of a quasiparticle near the Fermi surface is small compared to its energy ϵ , and the quasiparticle concept is well defined. All the momentum transferring scattering processes become asymptotically ineffective, as derived phenomenologically in Chapter 12 and confirmed microscopically in Chapter 19.
In this brief overview, I address some of the fundamental topics related to the physics of strongly correlated fermions, which have been also the subject of my research. After addressing the question why the field has a fundamental meaning, we next turn to the specific problems with their simple theoretical description. Those topics are: (i) the concept of spindependent masses of heavy particles and its subsequent experimental verification; (ii) the correlated metal–insulator transitions of the Mott–Hubbardtype; (iii) real space pairing and its subsequent experimental verification for high-temperature superconducting cuprates. I mention also the persistence of quantum spin and charge excitations in the superconducting phase. All those phenomena confirm the view that the strongly correlated fermionic \nsystems represent a new class of quantum liquids, with some properties quantitatively different than those of the Landau Fermi liquids. Namely, they can be classified as those that fall in between those of the anomalous Landau Fermi liquid and localized-magnetic-moment system.
Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.