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Progress of Theoretical Physics

Fermi Liquid Theory of Linear Antiferromagnetic Chains. II

知司 山田

Published Apr 1, 1972• 1 citation
Publisher

Abstract

Anisotropic antiferromagnetic Heisenberg linear chains are studied by the method developed by Yang and Yang and Lieb, with the help of the Landau's Fermi liquid theory. Exact residual interactions between quasi-particles (holes) at Fermi surface are obtained. With these interactions “zero sound” velocity is calculated and found to coincide with the quasi-particle velocity at Fermi surface.

DOI
10.1143/ptp.47.1140

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Anisotropic antiferromagnetic Heisenberg linear chains are studied by the method developed by Yang and Yang and Lieb, with the help of the Landau's Fermi liquid theory. Exact residual interactions between quasi-particles (holes) at Fermi surface are obtained. With these interactions “zero sound” velocity is calculated and found to coincide with the quasi-particle velocity at Fermi surface.
We study the Fermi liquid properties of the cold atomic dipolar Fermi gases with the explicit dipolar anisotropy using perturbative approaches. Due to the explicit dipolar anisotropy, Fermi surfaces exhibit distortions of the ${d}_{{r}^{2}\ensuremath{-}3{z}^{2}}$ type in three dimensions and of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ type in two dimensions. The fermion self-energy, effective mass, and Fermi velocity develop the same anisotropy at the Hartree-Fock level proportional to the interaction strength. The Landau interaction parameters in the isotropic Fermi liquids become the tridiagonal Landau interaction matrices in the dipolar Fermi liquids which renormalize thermodynamic susceptibilities. With large dipolar interaction strength, the Fermi surface collapses along directions perpendicular to the dipole orientation. The dynamic collective zero sound modes exhibit an anisotropic dispersion with the largest sound velocity propagating along the polar directions. Similarly, the longitudinal ppp-wave channel spin mode becomes a propagating mode with an anisotropic dispersion in multicomponent dipolar systems.
I show that when non-linearities are taken into account the Landau theory of Fermi liquids predicts the existence of hyperbolic waves in fermionic systems. The zero sound is described by a infinite set of coupled non-linear partial differential equations, one for each harmonic of oscillation of the Fermi surface. When just a few harmonics are included it is possible to describe the dynamics of the first sound in a very simple way. These results lead to the interesting experimental possibility of having non-linear effects in quantum liquids. In particular I discuss the problem of generation of second harmonic in a Fermi system.
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.
We bosonize the low energy excitations of Fermi liquids in any number of dimensions in the limit of long wavelengths. The bosons are a coherent superposition of electron-hole pairs and are related with the displacements of the Fermi surface in some arbitrary direction. A coherent-state path integral for the bosonized theory is derived and it is shown to represent histories of the shape of the Fermi surface. The Landau theory of Fermi liquids can be obtained from the formalism in the absence of nesting of the Fermi surface and singular interactions. We show that the Landau equation for sound waves is exact in the semiclassical approximation for the bosons.
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.