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arXiv:cond-mat/9406024 · Physical review. B, Condensed matter

Low-energy properties of fermions with singular interactions

B. L. Altshuler, L. B. Ioffe, Andrew J. Millis

Published Nov 15, 1994• 345 citations
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Abstract

We calculate the fermion Green function and particle-hole susceptibilities for a degenerate two-dimensional fermion system with a singular gauge interaction. We show that this is a strong-coupling problem, with no small parameter other than the fermion spin degeneracy N. We consider two interactions, one arising in the context of the t-J model and the other in the theory of half-filled Landau level. For the fermion self-energy we show that the qualitative behavior found in the leading order of perturbation theory is preserved to all orders in the interaction. The susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathit{Q}}$ at a general wave vector Q\ensuremath{\ne}2${\mathbf{p}}_{\mathbf{F}}$ retains the Fermi-liquid form. However, the 2${\mathit{p}}_{\mathit{F}}$ susceptibility ${\mathrm{\ensuremath{\chi}}}_{2\mathit{p}\mathit{F}}$ either diverges as T\ensuremath{\rightarrow}0 or remains finite but with nonanalytic wave-vector, frequency, and temperature dependence. We express our results in the language of recently discussed scaling theories, give the fixed-point action, and show that at this fixed point the fermion-gauge-field interaction is marginal in d=2, but irrelevant at low energies in d\ensuremath{\ge}2.

DOI
10.1103/physrevb.50.14048

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We calculate the fermion Green function and particle-hole susceptibilities for a degenerate two-dimensional fermion system with a singular gauge interaction. We show that this is a strong-coupling problem, with no small parameter other than the fermion spin degeneracy N. We consider two interactions, one arising in the context of the t-J model and the other in the theory of half-filled Landau level. For the fermion self-energy we show that the qualitative behavior found in the leading order of perturbation theory is preserved to all orders in the interaction. The susceptibility \ensuremathχQ{\mathrm{\ensuremath{\chi}}}_{\mathit{Q}}\ensuremathχQ​ at a general wave vector Q\ensuremath{\ne}2pF{\mathbf{p}}_{\mathbf{F}}pF​ retains the Fermi-liquid form. However, the 2pF{\mathit{p}}_{\mathit{F}}pF​ susceptibility \ensuremathχ2pF{\mathrm{\ensuremath{\chi}}}_{2\mathit{p}\mathit{F}}\ensuremathχ2pF​ either diverges as T\ensuremath{\rightarrow}0 or remains finite but with nonanalytic wave-vector, frequency, and temperature dependence. We express our results in the language of recently discussed scaling theories, give the fixed-point action, and show that at this fixed point the fermion-gauge-field interaction is marginal in d=2, but irrelevant at low energies in d\ensuremath{\ge}2.
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.
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.
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