© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Philosophical Magazine B

Two-dimensional Fermi liquids at low densities: The single-particle self-energy

Jungsoo Kim, Dermot Coffey

Published Nov 1, 1996• 5 citations

Abstract

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.

DOI: 10.1080/01418639608240350

Related papers

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

Jungsoo Kim, Dermot Coffey

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.

arXiv:cond-mat/9705065•Physical review. B, Condensed matter•Jan 1, 1998•3 citations

Quasiparticle spectra and the calculation of thermodynamics for a two-dimensional Fermi liquid

Dermot Coffey

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 ${\mathit{T}}^{3}$lnT 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.

Physical review. B, Condensed matter•May 15, 1995•1 citation

New collective mode and corrections to Fermi-liquid theory in two dimensions

Jan R. Engelbrecht, Mohit Randeria

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 q2${\mathit{k}}_{\mathit{F}}$, 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}''(${\mathit{k}}_{\mathit{F}}$,\ensuremath{\omega}) is obtained. The 2D case is also contrasted with a similar calculation in 1D.

Physical Review Letters•Aug 20, 1990•121 citations

Self-energy calculations in the Hubbard model

Christopher J. Rhodes, R L Jacobs

The authors have calculated the electronic self-energy of the two-dimensional Hubbard model as a function of frequency and momentum for various band fillings. They present this up to second order in the interaction parameter and also up to infinite order for ladder diagrams in the two-particle scattering channel. The authors have carried out direct k-space integrations using a triangle method. For a half-filled perfectly nested band, the imaginary part, lm Sigma (k f , omega ), has a linear- omega dependence for nested Fermi surfaces which is consistent with the marginal Fermi liquid phenomenology for high-T c superconductors. At, and near, half-filling there is a strong low-energy peak in lm Sigma (2) (k, omega ) when k is off the Fermi surface. The physical origin of this peak is discussed. The peak is enhanced in the ladder approximation. The corresponding structure in Re Sigma (2) (k, omega ) leads to multiple solutions of the Dyson equation for a wide range of values of the parameters. This leads to an appealing explanation of observed structure in some experimental angle-resolved photoemission spectra of high-T c materials. The authors also find anti-bound states split off from the band in the two-particle scattering ladders consistent with the suggestion of Anderson.

Journal of Physics Condensed Matter•Aug 2, 1993•4 citations

Analytic Properties of the Self-Energy of the Hubbard Model in One and Two Dimensions

V. Zlatić, Berislav Horvatić

A new method of calculating the properties of correlated electrons propagting on D -dimensional lattices with N D sites is used to study the analytic properties of the Hubbard model in one and two dimensions. The single-particle self-energy is evaluated in the weak-coupling limit around the Fermi energy, ω F , and momentum, p F , at T = 0. For D = 1, the imaginary part of the self-energy, around p F and ω F , behaves in the limit as Im Σ p F (ω F + ω) ~ |ω| for |ω| ~ 1/ N . For D = 2 at half-filling and along the (1,0) direction in the Brillouin zone, the dominate contribution to the self-energy around ω F is estimated as Im Σ p F (ω F + ω) ~ |ω| ln|ω|. For one and two dimensional lattices, the excitation spectra of the U = 0 and U → 0 models become qualitatively different as N → ∞ and can not be simply connected by the perturbation theory.

Physica Scripta•Jan 1, 1991•8 citations

Low-density repulsive Fermi gas in two dimensions: Bound-pair excitations and Fermi-liquid behavior

Jan R. Engelbrecht, Mohit Randeria

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 q2${\mathit{k}}_{\mathit{F}}$. We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2${\mathit{k}}_{\mathit{F}}$. 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.

Physical review. B, Condensed matter•Jun 1, 1992•67 citations

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.

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.

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

{\mathit{T}}^{3}

lnT 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 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 q2

{\mathit{k}}_{\mathit{F}}

, 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}''(

{\mathit{k}}_{\mathit{F}}

,\ensuremath{\omega}) is obtained. The 2D case is also contrasted with a similar calculation in 1D.

The authors have calculated the electronic self-energy of the two-dimensional Hubbard model as a function of frequency and momentum for various band fillings. They present this up to second order in the interaction parameter and also up to infinite order for ladder diagrams in the two-particle scattering channel. The authors have carried out direct k-space integrations using a triangle method. For a half-filled perfectly nested band, the imaginary part, lm Sigma (k f , omega ), has a linear- omega dependence for nested Fermi surfaces which is consistent with the marginal Fermi liquid phenomenology for high-T c superconductors. At, and near, half-filling there is a strong low-energy peak in lm Sigma (2) (k, omega ) when k is off the Fermi surface. The physical origin of this peak is discussed. The peak is enhanced in the ladder approximation. The corresponding structure in Re Sigma (2) (k, omega ) leads to multiple solutions of the Dyson equation for a wide range of values of the parameters. This leads to an appealing explanation of observed structure in some experimental angle-resolved photoemission spectra of high-T c materials. The authors also find anti-bound states split off from the band in the two-particle scattering ladders consistent with the suggestion of Anderson.

A new method of calculating the properties of correlated electrons propagting on D -dimensional lattices with N D sites is used to study the analytic properties of the Hubbard model in one and two dimensions. The single-particle self-energy is evaluated in the weak-coupling limit around the Fermi energy, ω F , and momentum, p F , at T = 0. For D = 1, the imaginary part of the self-energy, around p F and ω F , behaves in the limit as Im Σ p F (ω F + ω) ~ |ω| for |ω| ~ 1/ N . For D = 2 at half-filling and along the (1,0) direction in the Brillouin zone, the dominate contribution to the self-energy around ω F is estimated as Im Σ p F (ω F + ω) ~ |ω| ln|ω|. For one and two dimensional lattices, the excitation spectra of the U = 0 and U → 0 models become qualitatively different as N → ∞ and can not be simply connected by the perturbation theory.

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 q2

{\mathit{k}}_{\mathit{F}}

. We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2

{\mathit{k}}_{\mathit{F}}

. 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.