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CRC Press eBooks

Bridging Phenomenological and Microscopic Aspects

V. Tripathi

Published Feb 2, 2018• 0 citations
Publisher

Abstract

This chapter provides a microscopic basis to the phenomenological understanding of Landau Fermi liquids. It examines the effect of electron density fluctuations on the Stoner instability, particularly to understand their effect on the mean-field condition for the instability. The temperature dependence coming from the distribution functions in the numerator has a characteristic scale of the order of the Fermi energy. Ward identities provide relations between the self-energy and the vertex functions, thus bridging the phenomenological and microscopic aspects. The microscopic justification of this assumption rests on Ward identities. The ideas of scaling theory, developed in the context of classical critical phenomena, can be generalized to the quantum case. Thus vertex corrections, through their relation to the Ward identity for the self-energy, enable to provide a microscopic justification of one of the key assumptions of Landau Fermi liquid theory.

DOI
10.1201/9781351121996-9

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This chapter provides a microscopic basis to the phenomenological understanding of Landau Fermi liquids. It examines the effect of electron density fluctuations on the Stoner instability, particularly to understand their effect on the mean-field condition for the instability. The temperature dependence coming from the distribution functions in the numerator has a characteristic scale of the order of the Fermi energy. Ward identities provide relations between the self-energy and the vertex functions, thus bridging the phenomenological and microscopic aspects. The microscopic justification of this assumption rests on Ward identities. The ideas of scaling theory, developed in the context of classical critical phenomena, can be generalized to the quantum case. Thus vertex corrections, through their relation to the Ward identity for the self-energy, enable to provide a microscopic justification of one of the key assumptions of Landau Fermi liquid theory.
In Landau's phenomenological Fermi-liquid theory (FLT), most physical quantities are derived from the knowledge of the energy variation δE[δn] corresponding to a change δn of the quasi-particle (QP) distribution function n≡{nkσ}. We show that the internal energy E[n] (or, more precisely, the thermodynamic potential Φ[n]), expressed as a function of the QP distribution n, can be interpreted as an effective potential (in the sense of field theory), which is obtained from the free energy by a Legendre transformation. This allows to obtain explicitly δΦ (or δE) starting from a microscopic Hamiltonian and to relate the Landau f function to the forward-scattering two-particle vertex without considering the collective modes as in the standard diagrammatic derivation of FLT. Out-of-equilibrium properties are obtained by extending the definition of the effective potential to space- and time-dependent configurations. Φ[n] is then a functional of the Wigner distribution function n≡{nkσ (r,t)}. It contains information about both the static and dynamic properties of the Fermi liquid. In particular, it yields the quantum Boltzmann equation satisfied by nkσ (r, t). Finally, we show how δΦ[δn] can be derived (in the static case) using a finite-temperature renormalization-group approach. In agreement with previous results based on this technique, we find that the Landau f function is defined by the fixed-point value of the Ω-limit of the forward-scattering two-particle vertex.
We consider the interaction-driven Mott transition at zero temperature from the viewpoint of microscopic Fermi liquid theory. To this end, we derive an exact expression for the Landau parameters within the dynamical mean-field theory (DMFT) approximation to the single-band Hubbard model. At the Mott transition, the symmetric and the antisymmetric Landau parameters diverge. The vanishing compressibility at the Mott transition directly implies the divergence of the forward-scattering amplitude in the charge sector, which connects the proximity of the Mott phase to a tendency toward phase separation. We verify the expected behavior of the Landau parameters in a DMFT application to the Hubbard model on the triangular lattice at finite temperature. Exact conservation laws and the Ward identity are crucial to capture vertex divergences related to the Mott transition. We furthermore generalize Leggett's formula for the static susceptibility of the Fermi liquid to the static fermion-boson response function. In the charge sector, the limits of small transferred momentum and frequency of this response function commute at the Mott transition.
Abstract This paper is a short review on the foundations and recent advances in the microscopic Fermi-liquid (FL) theory. We demonstrate that this theory is built on five identities, which follow from conservation of the total charge (particle number), spin, and momentum in a translationally and SU(2)-invariant FL. These identities allow one to express the effective mass and quasiparticle residue in terms of an exact vertex function and also impose constraints on the “quasiparticle” and “incoherent” (or “low-energy” and “high-energy”) contributions to the observable quantities. Such constraints forbid certain Pomeranchuk instabilities of a FL, e.g., towards phases with order parameters that coincide with charge and spin currents. We provide diagrammatic derivations of these constraints and of the general (Leggett) formula for the susceptibility in arbitrary angular momentum channel, and illustrate the general relations through simple examples treated in perturbation theory.
This paper is a short review on the foundations and recent advances in the microscopic Fermiliquid (FL) theory.We demonstrate that this theory is built on five identities, which follow from conservation of total charge (particle number), spin, and momentum in a translationally and SU (2)invariant FL.These identities allows one to express the effective mass and quasiparticle residue in terms of an exact vertex function and also impose constraints on the "quasiparticle" and "incoherent" (or "low-energy" and "high-energy") contributions to the observable quantities.Such constraints forbid certain Pomeranchuk instabilities of a FL, e.g., towards phases with order parameters that coincide with charge and spin currents.We provide diagrammatic derivations of these constraints and of the general (Leggett) formula for the susceptibility in arbitrary angular momentum channel, and illustrate the general relations through simple examples treated in the perturbation theory.It is our great pleasure to write this article for the special volume in celebration of the 85th birthday of Lev Petrovich Pitaevskii, who, in our view, is one the greatest physicist of his generation and the model of a scientist and citizen.His seminal volumes on Statistical Physics (with E. M. Lifshitz), 1 Physical Kinetics 2 (also with E. M. Lifshitz), and Quantum Electrodynamics 3 (with V. B. Berestetskii and E. M. Lifshitz), all parts of the Landau and Lifshitz Course on Theoretical Physics, as well as on Bose-Einstein condensation and superfluidity 4 (with S. Stringari) are not only used by every contemporary physicist but also, as we are positive, will serve the future generations of scientists from around the world.The paper we present for this volume is devoted to the microscopic theory of a Fermi liquid, which was pioneered by Lev Petrovich in the early 1960s.The general relations he obtained with Landau, which express the effective mass m * /m and the quasiparticle residue Z in terms of the vertex function (the Pitaevskii-Landau relations), set the gold standard for many-body theory.We hope that Lev Petrovich and other readers of this volume will find our summary of recent developments in this field interesting.
The general interaction between quasiparticles and particle-hole excitations in a neutral Fermi system at T=0T=0T=0 is investigated. Density and spin-density fluctuations and collective modes are treated on an equal basis. Use is made of Ward identities to relate the vertex function $\ensuremath{\Gamma}$ to the quasiparticle self-energy $\ensuremath{\Sigma}$, and logarithmic corrections to the self-energy are obtained. The coefficients of the logarithmic terms are calculated analytically in terms of known Landau parameters.