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Physical review. B, Solid state

Two-dimensional Fermi gas

Paul Bloom

Published Jul 1, 1975• 173 citations
Publisher

Abstract

A calculation is given of the equilibrium properties of a two-dimensional Fermi gas at 0 \ifmmode^\circ\else\textdegree\fi{}K. The ladder diagrams for the self-energy are summed, corresponding to a low-density expansion. We obtain the Fermi energy, the quasiparticle lifetime, the effective mass, the ground-state energy, and the discontinuity at the Fermi surface.

DOI
10.1103/physrevb.12.125

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Cambridge University Press eBooks•May 16, 2013•0 citations
A calculation is given of the equilibrium properties of a two-dimensional Fermi gas at 0 \ifmmode^\circ\else\textdegree\fi{}K. The ladder diagrams for the self-energy are summed, corresponding to a low-density expansion. We obtain the Fermi energy, the quasiparticle lifetime, the effective mass, the ground-state energy, and the discontinuity at the Fermi surface.
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 energy spectrum of the one-dimensional electron gas, solved by Luther and Emery for a particular attractive interaction strength UUU of the backward scattering across the Fermi surface, has been calculated for small deviations from this particular limit. The energy spectrum shows a gap at the Fermi energy, whose magnitude is very sensitive to the interaction UUU. The density of states shows a gap at zero temperature, which is smeared out by the quasiparticle lifetime for $T\ensuremath{\ne}0$.
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.
1. Introduce density distribution and momentum distribution of non-interacting Fermi gas. 2. Discuss the transport property of a Fermi gas through a quantum point contact. 3. Discuss Fermi surface as a key of a non-interacting Fermi gas and its generalization to a Fermi liquid. 4. Introduce the effective mass and the quasi-particle residue as two key parameters of the Fermi liquid. 5. Introduce the self-energy and the universal relations between the effective mass, the quasi-particle residue and the self-energy. 6. Discuss Fermi polaron as an example of the Fermi liquid, and show these universal relations with this example. 7. Discuss the divergence of the effective mass and vanishing of quasi-particle residue when Fermi liquid description fails. 8. Introduce the radio-frequency spectroscopy and how it measures the polaron energy and the quasi-particle residue. 9. Summarize different spectroscopy measurements in ultracold atomic systems.
The two-dimensional (2D) nature of the high-Tc materials has lead to renewed interest in the properties of fermions in two dimensions. Calculations of the corrections to the leading Fermi-liquid (FL) behaviour of the single-particle self-energy Σ(p, E), and the specific heat have shown that the differences between 2D and three-dimensional FLs arise from the difference in phase space between two and three dimensions. Here we investigate the imaginary part Σ″(p, ξp) of Σ(p, E) on shell at a low density which has a (p - pF)2 In (p - pf) dependence in two dimensions where pF is the Fermi wavenumber. We find that both particle-hole (ph) and particle-particle (pp) contributions are important in this limit so that in principle no class of diagrams can be neglected in the low-density limit in two dimensions, and one is effectively limited to second order in the interaction for reliable quantitative results. This is in contrast with three dimensions where the ph contribution becomes negligible compared with the pp contribution at low densities and only pp ladder diagrams survive. We argue, however, that the density dependence of Σ(p, E) is unchanged by high orders in perturbation theory and that in the limit pF → 0 the form of the dependence of Σ(p, E) on p - pF is given by the second-order contribution. We find that the (p - pF)2 In (p - pf) behaviour is limited to a values of p - pf Λ; pF so that the region over which FL behaviour is evident shrinks with decreasing density.
Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.