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arXiv:cond-mat/9705065 · Physical review. B, Condensed matter

Single-particle properties of a two-dimensional Fermi liquid at finite frequencies and temperatures

Jungsoo Kim, Dermot Coffey

Published Jan 1, 1998• 3 citations
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Abstract

We review the leading momentum, frequency, and temperature dependences of the single-particle self-energy and the corresponding term in the entropy of a two-dimensional Fermi liquid (FL) with a free-particle spectrum. We calculate the corrections to these leading dependences for the paramagnon model and the electron gas and find that the leading dependences are limited to regions of energy and temperature which decrease with decreasing number density of fermions. This can make it difficult to identify the frequency- and temperature-dependent characteristics of a FL ground state in experimental quantities in low-density systems even when complications of band structure and other degrees of freedom are absent. This is an important consideration when the normal-state properties of the undoped cuprate superconductors are analyzed.

DOI
10.1103/physrevb.57.542

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We review the leading momentum, frequency, and temperature dependences of the single-particle self-energy and the corresponding term in the entropy of a two-dimensional Fermi liquid (FL) with a free-particle spectrum. We calculate the corrections to these leading dependences for the paramagnon model and the electron gas and find that the leading dependences are limited to regions of energy and temperature which decrease with decreasing number density of fermions. This can make it difficult to identify the frequency- and temperature-dependent characteristics of a FL ground state in experimental quantities in low-density systems even when complications of band structure and other degrees of freedom are absent. This is an important consideration when the normal-state properties of the undoped cuprate superconductors are analyzed.
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.
If the dynamical quasiparticle spectrum of a Fermi liquid (FL), ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{dy}}$, given by the poles of the single-particle propagator, is used in the expression for the entropy of a noninteracting Fermi gas, one finds that the result is different from that calculated directly from the thermodynamic potential. In particular, the coefficient of the T3{\mathit{T}}^{3}T3lnT terms, the leading correction to the linear T dependence for a three-dimensional (3D) FL, is overestimated when calculated from ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{dy}}$. The thermodynamic properties may be calculated correctly from a second quasiparticle spectrum, the statistical quasiparticle spectrum, ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{st}}$. In this paper I calculate the corrections to leading linear energy and temperature dependences for a 2D FL. I compare the difference between the two spectra in the 2D and 3D cases. Although these differences are qualitatively the same, they are quantitatively larger in the 2D case.
We study the finite-temperature Fermi-liquid to non-Fermi-liquid crossover in the two-dimensional Hubbard model for a range of dopings using the self-consistent ladder dual fermion method. We consider relatively high temperatures where by comparing the single-particle self-energy and the two-particle spin susceptibility we identify a suppression of the density of states near the Fermi level caused by a quasi-antiferromagnetic behavior that is itself characterized by a long, but finite, correlation length scale which we extract. We compute the single-particle self-energy, and the charge/spin susceptibilities for the entire doping range at relatively high temperature, and also for the half-filled weak-coupling case at relatively low temperatures. We perform fluctuation diagnostics to decompose the single-particle self-energy into scattering qqq-vector and bosonic frequency contributions in the spin channel. Within this framework we find that the key contributions to the single-particle self-energy that give non-Fermi-liquid character show complicated dependence on zero, positive, and negative frequency bosonic modes. We find that for a nominal choice of ${t}^{\ensuremath{'}}$ that electron- and hole-doped systems with similar Fermi-liquid character have distinct spin and charge susceptibility and distinct spin-fluctuation diagnostics spectra.
The temperature-dependent self-energy of the hole-doped two-dimensional Hubbard model is calculated perturbatively up to second order and used to find the spectral function for all points in the Brillouin zone. Single-particle dispersion, density of states, and Fermi surface topology are thus obtained and the appearance of the strong-coupling features studied. The renormalized quantities are temperature dependent due to an interplay between the electron correlation and topology of free-electron bands close to the Fermi energy.
A frequency- and momentum-renormalization-group acceleration together with an analytical approach is used to obtain the retarded Green's function in the self-consistent and conserving fluctuation-exchange (FLEX) approximation for the two-dimensional Hubbard model in the normal state and in the superconducting state. The analytical expressions for this approach are given. For band-fillings near half filling the self-energy in the normal state exhibits Fermi-liquid behaviour for, low temperatures and frequencies near the chemical potential, if the momentum is chosen near the Fermi-surface. However due to the presence of large many body effects the observed Fermi-liquid region near the chemical potential and near the Fermi-surface is very small. Results for the single particle density of states, the occupation number and the spectral function are presented. The superconducting state with symmetry is obtained for U = 2 to U = 6 and a (U, n)-phase diagram for the transition temperature Tc is presented. A maximum Tc/t of 0.0275 is obtained for U = 6 near half filling.
The rearrangement of single-particle degrees of freedom of a dilute two-dimensional electron gas in the vicinity of the quantum critical point is examined within a microscopic approach. It is shown that just beyond the critical point, the Landau state undergoes self-consistent rearrangement of the quasiparticle spectrum and momentum distribution. At very low temperatures, there emerges a multi-connected quasiparticle momentum distribution. With increasing temperature, two crossovers occur: a fermion condensate appears in the first and disappears in the second, giving way to universal non-Fermi-liquid behavior. Manifestations of these crossovers in thermodynamic properties of the electron gas are studied and characterized. The four quasiparticle phases predicted to exist in the vicinity of the critical point are collected in a schematic phase diagram.