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Physical Review Letters

Low-energy properties of two-dimensional fermions with long-range current-current interactions

Dmitri V. Khveshchenko, Philip C. E. Stamp

Published Sep 27, 1993• 80 citations
Publisher

Abstract

We calculate the one-particle Green function of 2D fermions interacting via a long-range transverse gauge field. Its asymptotic low-energy behavior is found within the eikonal expansion which consistently sums the infrared divergent terms given by ``maximally crossed diagrams.'' Instead of power law corrections to Fermi liquid theory we observe a much stronger singularity which implies a more radical breakdown of Fermi liquid theory than the usual orthogonality catastrophe.

DOI
10.1103/physrevlett.71.2118

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Physical review. B, Condensed matter•Oct 15, 1993•44 citations
We calculate the one-particle Green function of 2D fermions interacting via a long-range transverse gauge field. Its asymptotic low-energy behavior is found within the eikonal expansion which consistently sums the infrared divergent terms given by ``maximally crossed diagrams.'' Instead of power law corrections to Fermi liquid theory we observe a much stronger singularity which implies a more radical breakdown of Fermi liquid theory than the usual orthogonality catastrophe.
We study the behavior of response functions of two-dimensional (2D) fermions interacting via a long-range transverse gauge field in the eikonal approximation. We observe that an exponentially vanishing wave-function renormalization prevents divergences in the density-density correlation function and the pairing susceptibility. The wave-function renormalization also shows up in a suppression of the dHvA effect. Elaborating on an observation recently made by Ioffe, Lidsky, and Altschuler, we also infer an effective bosonic description which makes it possible to reproduce our eikonal results in terms of free 2D bosons.
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}2pF{\mathbf{p}}_{\mathbf{F}}pF​ retains the Fermi-liquid form. However, the 2pF{\mathit{p}}_{\mathit{F}}pF​ 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.
We calculate the Green's function, particle-hole susceptibility and Cooper pair susceptibility of a 2D electron gas with long range current-current interactions, using Haldane bosonization scheme. We find Luttinger Liquid properties of weakly decaying power-law tails in the spectral function and a Kohn anomaly in the 2$k\sb{F}$ particle-hole susceptibility. At finite temperature, the system exhibits more radical departure from Fermi Liquid. The specific heat was determined being linearly proportional to temperature. The strong correlations of spins and holes are demonstrated in the 2D t-J model of spin-boson and hole-fermion, through the instability against charge ordering we find from our charge susceptibility calculation of bubble and ladder diagrams. We calculate the linear-T dependence of resistivity from the hole self-energy to the second order. The excitation spectra of holon and spinon found from the self-energy calculations agree with the mean-field results to that order.
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 examine the low-energy behavior of two-dimensional electrons with current-current interactions. The interactions are long range, cannot be screened, and result in anomalies in perturbation theory. In the Hartree-Fock theory, we find an instability of the Landau Fermi-liquid state resulting in ``fermion-condensate'' states with a high degeneracy of levels at the Fermi energy. This degeneracy may be removed to give a Fermi-liquid state with vanishing Fermi velocity. As an example, we investigate further the Fermi-liquid state with the quasiparticle spectrum \ensuremath{\epsilon}(p)\ensuremath{\propto}(p-pF{\mathit{p}}_{\mathit{F}}pF​)3/2{)}^{3/2})3/2 proposed by Halperin et al. Application of the recent Haldane scheme of bosonization in higher dimensions to this state leads to coupling between bosons, which has the form required for a breakdown of the Landau Fermi-liquid behavior. A generalized form of the Luttinger liquid is obtained.