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Physical review. B, Condensed matter

Landau f function for the dilute Fermi gas in two dimensions

Jan R. Engelbrecht, Mohit Randeria, Lizeng Zhang

Published May 1, 1992• 64 citations
Publisher

Abstract

We study the Landau f function for a two-dimensional repulsive Fermi system in the dilute limit. We find the leading terms in the low-density expansion using second-order perturbation theory, following a method of Abrikosov and Khalatnikov. Further, using diagrammatic methods, we determine the essentially singular contribution coming from the two-hole bound state. The absence of a divergence in the f function, together with our earlier results on the quasiparticle lifetime, clearly demonstrates that the system is a Fermi liquid in the low-density regime.

DOI
10.1103/physrevb.45.10135

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"This thesis is divided into two parts. The first describes a study of the dilute Fermi gas with short range repulsive interactions in 2D, with special emphasis on questions related to the validity of Fermi liquid theory. We find that despite the existence of non-perturbative effects due to two-dimensionality, namely linearly dispersing excitations of ""bound"" holes, the system remains in a Fermi liquid state. The quasi-particles are both long lived and weakly interacting. We also provide a microscopic calculation of the Fermi liquid parameters which determine the response functions in the standard way."

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We study the Landau f function for a two-dimensional repulsive Fermi system in the dilute limit. We find the leading terms in the low-density expansion using second-order perturbation theory, following a method of Abrikosov and Khalatnikov. Further, using diagrammatic methods, we determine the essentially singular contribution coming from the two-hole bound state. The absence of a divergence in the f function, together with our earlier results on the quasiparticle lifetime, clearly demonstrates that the system is a Fermi liquid in the low-density regime.
We study the dilute-gas expansion for a two-dimensional Fermi system with arbitrary short-range repulsive interactions. In contrast with the three-dimensional case, we find an unusual pole in the vertex part in the particle-particle channel, for all center-of-mass momenta q2kF{\mathit{k}}_{\mathit{F}}kF​. We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2kF{\mathit{k}}_{\mathit{F}}kF​. We study the effect of these bound states on the single-particle self-energy and find that the quasiparticles are well defined. Thus in the low-density regime, there is no breakdown of Fermi-liquid theory. In the Appendix we discuss the two-particle phase shift in the dilute Fermi gas, its connection with bound states, and the analogy with the potential scattering phase shift.
The question of the breakdown of Fermi liquid theory in two dimensions is examined in the context of perturbation theory for a dilute interacting Fermi gas. The quasiparticle interaction function, fic;u;, is calculated for such a system. The interaction function, calculated to second order in terms of the dimensionless coupling constant, shows various singularities. The most divergent terms appear in the cross channel, but cancel out leaving a much weaker singularity in the limit of two moment a approaching each other (0 —f 0 ). As in the case of the three dimensional Fermi gas, the Cooper channel contains a logarithmic singularity in the limit 0 —f 7r. This singularity can be summed and is known to be harmless to the structure of Fermi liquid theory. A different feature in two dimensions is the existence of such a singularity for 0 —p 0. This feature needs further investigation. Calculations have also been extended to a polarized Fermi gas and the result is equivalent to the unpolarized case and does not show any additional features. In conclusion, the results do not indicate the presence of strong divergences which could cause the breakdown of Fermi liquid theory in two dimensions for a dilute interacting Fermi gas.
We examine the validity of Fermi-liquid theory in the low-density two-dimensional Hubbard model. From the ground-state energy we obtain the quasiparticle energy and the Landau interaction function. Then we prove that the quasiparticle-interaction function is finite and that the quasiparticle residue is nonzero for repulsive interactions. We also find, as a function of the excitation energy \ensuremath{\epsilon}, that the quasiparticle lifetime \ensuremath{\sim}1/${\mathrm{\ensuremath{\epsilon}}}^{2}$ln\ensuremath{\epsilon}. Hence, we conclude that Fermi-liquid theory holds in our model.
We study the Fermi liquid properties of a single component Fermi gas with p-wave interactions. In the weak repulsive limit, we obtain exact perturbative expansions for the ground state energy, the chemical potential, and the effective mass of the Landau quasiparticle up to second order in scattering volume a. We also calculated the corresponding Landau functions and Landau parameters and show that they satisfy the general Fermi liquid identities. Using the Landau transport equation, we show that undamped zero sound only appears in the second order in scattering volume, in contrast to the s-wave case.
This article gives a fairly detailed survey of some of the problems raised when the interaction energy fσσ'kk' between 2 fermionic quasiparticles (in 2 dimensions) is singular when ∣k‾−k‾′∣→0|\underline{k}-\underline{k}'|\to 0∣k​−k​′∣→0. Before dealing with singular interactions, it is shown how a non-singular fσσ'kk' leads to a 2-dimensional Fermi liquid theory, which is internally consistent, at least as far as its infrared properties are concerned. The quasiparticle properties are calculated in detail. The question of whether singular interactions arise for the dilute Fermi gas, with short-range repulsive interactions, is investigated perturbatively. One finds a weak singularity in fσσ'kk', when the dimensionality D = 2, but it does not destabilize the Fermi liquid. A more sophisticated analysis is then given, to all orders in the interaction, using the Lippman-Schwinger equation as well as a phase shift analysis for a finite box. The conclusion is that any breakdown of Fermi liquid theory must come from non-perturbative effects. An examination is then made of some of the consequences arising if a singular interaction is introduced — the form proposed by Anderson is used as an example. A hierarchy of singular terms arise in all quantities — this is shown for the self-energy, and also the 3–point and 4–point scattering functions. These may be summed in a perfectly consistent manner. Most attention is given to the particle-hole channel, since it appears to lead to results different from those of Anderson. Nevertheless it appears that it is possible to derive a sensible theory starting from a singular effective Hamiltonian — although Fermi Liquid theory breaks down, all fermionic quantities may be calculated consistently. Finally, the effect of a magnetic field (which cuts off the infrared divergences) is investigated, and the de Haas-van Alphen amplitude calculated, for such a singular Fermionic system.
"This thesis is divided into two parts. The first describes a study of the dilute Fermi gas with short range repulsive interactions in 2D, with special emphasis on questions related to the validity of Fermi liquid theory. We find that despite the existence of non-perturbative effects due to two-dimensionality, namely linearly dispersing excitations of ""bound"" holes, the system remains in a Fermi liquid state. The quasi-particles are both long lived and weakly interacting. We also provide a microscopic calculation of the Fermi liquid parameters which determine the response functions in the standard way."