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Physical Review Letters

Effect of singular interaction terms on two-dimensional Fermi liquids

Philip C. E. Stamp

Published Apr 1, 1992• 31 citations
Publisher

Abstract

The structure of perturbation theory for an interacting two-dimensional Fermi system is analyzed, in the case where singular terms exist in the quasiparticle interactions. These give a hierarchy of singular terms in the one-particle self-energy. The dominant terms are self-consistently summed, to give a quasiparticle pole vanishing as Z(\ensuremath{\omega})\ensuremath{\sim}(\ensuremath{\omega}/${\mathrm{\ensuremath{\omega}}}_{0}$${)}_{0}^{\mathrm{\ensuremath{\rho}},\mathrm{\ensuremath{\delta}}2}$/${\mathrm{\ensuremath{\Elzxh}}}^{2}$N(0)${\mathrm{\ensuremath{\omega}}}_{0}$ near the Fermi energy (signaling a breakdown of Fermi-liquid theory), and a form for the scattering amplitude which is consistent with the assumed form for the quasiparticle interaction.

DOI
10.1103/physrevlett.68.2180

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The structure of perturbation theory for an interacting two-dimensional Fermi system is analyzed, in the case where singular terms exist in the quasiparticle interactions. These give a hierarchy of singular terms in the one-particle self-energy. The dominant terms are self-consistently summed, to give a quasiparticle pole vanishing as Z(\ensuremath{\omega})\ensuremath{\sim}(\ensuremath{\omega}/{\mathrm{\ensuremath{\omega}}}_{0}$${)}_{0}^{\mathrm{\ensuremath{\rho}},\mathrm{\ensuremath{\delta}}2}/\ensuremath\Elzxh2{\mathrm{\ensuremath{\Elzxh}}}^{2}\ensuremath\Elzxh2N(0)\ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​ near the Fermi energy (signaling a breakdown of Fermi-liquid theory), and a form for the scattering amplitude which is consistent with the assumed form for the quasiparticle interaction.
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.
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 consider a system of interacting fermions in two dimensions beyond the second-order perturbation theory in the interaction. It is shown that the mass-shell singularities in the self-energy, arising already at the second order of the perturbation theory, manifest a nonperturbative effect: an interaction with the zero-sound mode. Resumming the perturbation theory for a weak, short-range interaction and accounting for a finite curvature of the fermion spectrum, we eliminate the singularities and obtain the results for the quasiparticle self-energy and the spectral function to all orders in the interaction with the zero-sound mode. A threshold for emission of zero-sound waves leads a nonmonotonic variation of the self-energy with energy (or momentum) near the mass shell. Consequently, the spectral function has a kinklike feature. We also study in detail a nonanalytic temperature dependence of the specific heat $C(T)\ensuremath{\propto}{T}^{2}$. It turns out that although the interaction with the collective mode results in an enhancement of the fermion self-energy, this interaction does not affect the nonanalytic term in C(T)C(T)C(T) due to a subtle cancellation between the contributions from the real and imaginary parts of the self-energy. For a short-range and weak interaction, this implies that the second-order perturbation theory suffices to determine the nonanalytic part of C(T)C(T)C(T). We also obtain a general form of the nonanalytic term in C(T)C(T)C(T), valid for the case of a generic Fermi liquid, i.e., beyond the perturbation theory.
Landau-Fermi liquid theory is conventionally believed to hold whenever the interacting single-particle density of states develops a $\ensuremath{\delta}$-like component at the Fermi surface, which is associated with quasiparticles. Here we show that a microscopic justification can be actually achieved under more general circumstances, even in the case where coherent quasiparticles are totally missing and the interacting single-particle density of states vanishes at the chemical potential as a consequence of a pole singularity in the self-energy.
We study a two-dimensional Fermi gas with an arbitrary short-range repulsive interaction. We show that the vertex part has an unusual singularity in the particle-particle channel for all momenta q2kF{\mathit{k}}_{\mathit{F}}kF​, which is not found in higher dimensions. We interpret this as a collective mode representing a bound excitation of two holes. This mode, however, does not lead to an instability, or to a breakdown of Fermi-liquid theory in the low-density limit. The resulting Fermi-surface phase shift vanishes, and a \ensuremath{\Vert}\ensuremath{\omega}${\mathrm{\ensuremath{\Vert}}}^{5/2}$ correction to \ensuremath{\Sigma}''(kF{\mathit{k}}_{\mathit{F}}kF​,\ensuremath{\omega}) is obtained. The 2D case is also contrasted with a similar calculation in 1D.
We analyze the form of the fermionic propagator for two-dimensional fermions interacting with massless overdamped bosons. Examples include a nematic and Ising ferromagnetic quantum-critical points and fermions at a half-filled Landau level. Fermi-liquid behavior in these systems is broken at criticality by a singular self-energy, but the Fermi surface remains well defined. These are strong-coupling problems with no expansion parameter other than the number of fermionic species, NNN. The two known limits, N⪢1N⪢1N⪢1 and N=0N=0N=0, show qualitatively different behavior of the fermionic propagator $G({ϵ}_{k},\ensuremath{\omega})$. In the first limit, $G({ϵ}_{k},\ensuremath{\omega})$ has a pole at some ϵk{ϵ}_{k}ϵk​; in the other it is analytic. We analyze the crossover between the two limits. We show that the pole survives for all NNN, but at small NNN it only exists in a range O(N2)O({N}^{2})O(N2) near the mass shell. At larger distances from the mass shell, the system evolves and $G({ϵ}_{k},\ensuremath{\omega})$ becomes regular. At N=0N=0N=0, the range where the pole exists collapses and $G({ϵ}_{k},\ensuremath{\omega})$ becomes regular everywhere.