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Register a new userBeyond Fermi-Liquid Theory: the $k$-Fermi liquids
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Tai-Kai Ng
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We study in this paper the general properties of a many body system of fermions in arbitrary dimensions assuming that the {em momentum} of individual fermions are good quantum numbers of the system. We call these systems $k$-Fermi liquids. We show how Fermi liquid, Luttinger liquid (or Fermi liquid with exclusion statistics) and spin-charge separation arises from this framework. Two exactly solvable $k$-Fermi liquid models with spin-charge separation are discussed as examples.
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An introductory survey of the theoretical ideas and calculations and the experimental 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. Soluble examples of Singular Fermi liquids (often called Non-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.
We propose in this paper an effective low-energy theory for interacting fermion systems which supports exclusion statistics. The theory can be viewed as an extension of Landau Fermi liquid theory where besides quasi-particle energy $xi_{mathbf{k}}$, the kinetic momentum $mathbf{k}$ of quasi-particles depends also on quasi-particle occupation numbers as a result of momentum ($k$)-dependent current-current interaction. The dependence of kinetic momentum on quasi-particles excitations leads to change in density of states and exclusion statistics. The properties of this new Fermi liquid state is studied where we show that the state (which we call $U(1)$-Fermi liquid state) has Fermi-liquid like properties except that the quasi-particles are {em not} adiabatically connected to bare fermions in the system and the state may not satisfy Luttinger theorem.
A system with charge conservation and lattice translation symmetry has a well-defined filling $ u$, which is a real number representing the average charge per unit cell. We show that if $ u$ is fractional (i.e. not an integer), this imposes very strong constraints on the low-energy theory of the system and give a framework to understand such constraints in great generality, vastly generalizing the Luttinger and Lieb-Schultz-Mattis theorems. The most powerful constraint comes about if $ u$ is continuously tunable (i.e. the system is charge-compressible), in which case we show that the low-energy theory must have a very large emergent symmetry group -- larger than any compact Lie group. An example is the Fermi surface of a Fermi liquid, where the charge at every point on the Fermi surface is conserved. We expect that in many, if not all, cases, even exotic non-Fermi liquids will have the same emergent symmetry group as a Fermi liquid, even though they could have very different dynamics. We call a system with this property an ersatz Fermi liquid. We show that ersatz Fermi liquids share a number of properties in common with Fermi liquids, including Luttingers theorem (which is thus extended to a large class of non-Fermi liquids) and periodic quantum oscillations in the response to an applied magnetic field. We also establis
We present the detailed formalism of the extremely correlated Fermi liquid theory, developed for treating the physics of the t-J model. We start from the exact Schwinger equation of motion for the Greens function for projected electrons, and develop a systematic expansion in a parameter lambda, relating to the double occupancy. The resulting Greens function has a canonical part arising from an effective Hamiltonian of the auxiliary electrons, and a caparison part, playing the role of a frequency dependent adaptive spectral weight. This adaptive weight balances the requirement at low omega, of the invariance of the Fermi volume, and at high omega, of decaying as c_0/(i omega), with a correlation depleted c_0 <1. The effective Hamiltonian H_{eff} describing the auxiliary Fermions is given a natural interpretation with an effective interaction V_{eff} containing both the exchange J(ij), and the hopping parameters t(ij). It is made Hermitian by adding suitable terms that ultimately vanish, in the symmetrized theory developed in this paper. Simple but important shift invariances of the t-J model are noted with respect to translating its parameters uniformly. These play a crucial role in constraining the form of V_{eff} and also provide checks for further approximations. The auxiliary and physical Greens function satisfy two sum rules, and the Lagrange multipliers for these are identified. A complete set of expressions for the Greens functions to second order in lambda is given, satisfying various invariances. A systematic iterative procedure for higher order approximations is detailed. A superconducting instability of the theory is noted at the simplest level with a high transition temperature.
We describe an analytical theory investigating the regime of validity of the Fermi liquid theory in interacting, via the long-range Coulomb coupling, two-dimensional Fermi systems comparing it with with the corresponding 3D systems. We find that the 2D Fermi liquid theory and 2D quasiparticles are robust up to high energies and temperatures of the order of Fermi energy above the Fermi surface, very similar to the corresponding three-dimensional situation. We calculate the phase diagram in the frequency-temperature space separating the collisionless ballistic regime and the collision-dominated hydrodynamic regime for 2D and 3D interacting electron systems. We also provide the temperature corrections up to third order for the renormalized effective mass, and comment on the validity of 2D Wiedemann-Franz law and 2D Kadawoki-Woods relation.
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