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Topics in Many-Body Physics

Jan R. Engelbrecht

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

"This thesis is divided into two parts. The first describes a study of the dilute Fermi gas with short range repulsive interactions in 2D, with special emphasis on questions related to the validity of Fermi liquid theory. We find that despite the existence of non-perturbative effects due to two-dimensionality, namely linearly dispersing excitations of ""bound"" holes, the system remains in a Fermi liquid state. The quasi-particles are both long lived and weakly interacting. We also provide a microscopic calculation of the Fermi liquid parameters which determine the response functions in the standard way."

Related papers

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

Failure of Fermi Liquid Theory in 2-D: A Signal from Perturbation Theory

G. Baskaran

We study the perturbative correction to the ground state energy eigenvalue of a 2-dimensional dilute fermi gas with weak short-range two body repulsion. From the structure of the energy shift we infer the presence of an induced two body long range repulsive interaction $ {\displaystyle {1\over{r^2}} }$ among the constituent electrons indicating a potential instability of the fermi liquid ground state.

arXiv:cond-mat/9404023•Apr 11, 1994•0 citations

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

New results for interacting Fermi systems: the 2D Hubbard model at low density and the two coupled chain problem

Prof Michele Fabrizio

A central problem in the theory of strongly correlated fermion systems, concerns the nature of their low-lying excitations. It is generally accepted that, except for special cases with symmetry breaking, three dimensional (3D) interacting Fermi systems, are well described in terms of weakly correlated quasi-particles with quantum numbers in one-toone correspondence with those identifying the excitations of a non-interacting Fermi gas. This is the basic assumption underlying the Landau theory of Fermi liquids (FL), which was originally proposed as a phenomenological description of strongly interacting fermions by Landau in early 1956[1), an then established microscopically by Nozieres and Luttinger in 1962[2]. Quite a different scenario occurs in lD interacting Fermi systems. Apart from specific models developing gaps in the excitation spectrum, most fermion systems in one dimension with repulsive interactions, do have low-lying gapless excitations, like ordinary Fermi liquids, but totally different from the quasi-particles predicted by Landau theory. The low-energy behavior of most of these lD interacting fermion systems with gapless linear excitations, can be understood in terms of few model dependent constants, which parametrize all the long-wavelength properties...

•Nov 2, 1992•0 citations

Three-component Fermi gases in quasi-two-dimensional geometries

T. Kirk

This thesis concerns quasi-two-dimensional (quasi-2D) ultracold Fermi systems with short range interactions. Here, quasi-2D geometries are those that lie between two- and three-dimensions. We study few- and many-body systems composed of two or three distinguishable species of fermions. Of principle interest are the bound states that form in these systems and how these bound states are affected by the quasi-2D geometries. In the first research chapter, we focus on the two-dimensional (2D) regime of quasi- 2D. We begin by developing an effective 2D model of this 2D-like geometry. We use this model to analyse a Fermi system consisting of three fermionic species with equal interactions between all distinguishable pairs. Starting with the few-body problem, we show that there is always a three-body bound state (trimer) in the effective 2D model and that these 2D trimers are stable. This is notably different from the extremely unstable ‘Efimov’ trimers found in three-dimensional (3D) systems. Using a low-density expansion and a variational approach, we investigate the fate of the 2D trimers as the system’s density is increased from the three-body limit to the many-body limit. We find that remnants of trimers can persist in the form of strong three-body correlations in the many-body system. In the second research chapter, we analyse a realistic experimental system of three distinguishable fermions confined to quasi-2D. We develop a model that goes beyond the previous chapter in two regards. Firstly, we allow for unequal interactions between the pairs of fermions. Secondly, we use a full quasi-2D model that is not specific to the 2D-like regime. With this general model, we use the scattering and confinement parameters from experiments on ultracold 6Li atoms to calculate the trimer spectrum across the dimensional crossover from 2D to 3D. We show that one state smoothly evolves from an unstable 3D-like trimer to a 2D-like trimer as the attractive interactions are decreased. We furthermore compute the trimer wave function and quantify the stability of the trimer with respect to three-body recombination by determining the probability that three fermions approach each other at short distances. Our results indicate that the lifetime of the trimer can be enhanced by at least an order of magnitude in the quasi-2D geometry. This opens the door to realising long-lifetime trimers in ultracold gas experiments and furthermore, the many-body state of trimers predicted in the preceding chapter. vii In the final research chapter, we examine an alternative approach to accessing quasi-2D geometries by using a system of coupled 2D planes. We study the two-component Fermi gas in this geometry, as a preliminary to studying the three-component gas. We find that the interplanar coupling can be tuned to drive a dimensional crossover from 2D to 3D. We study the effect of this dimensional crossover on the two- and many-body systems. In general, we find that two-body correlations are strengthened by weakening the coupling between 2D planes. In the many-body system, we use mean-field theory to find that the critical temperature of pairing is increased by weakening the coupling between 2D planes. This represents a first step towards a more general representation of quasi-2D geometries, namely: A system of coupled quasi-2D planes.

•Jul 28, 2018•0 citations

Fermi liquid of two-dimensional polar molecules

Zhen-Kai Lu, Gora V. Shlyapnikov

We study Fermi-liquid properties of a weakly interacting two-dimensional gas of single-component fermionic polar molecules with dipole moments $d$ oriented perpendicularly to the plane of their translational motion. This geometry allows the minimization of inelastic losses due to chemical reactions for reactive molecules and, at the same time, provides a possibility of a clear description of many-body (beyond mean-field) effects. The long-range character of the dipole-dipole repulsive interaction between the molecules, which scales as $1/{r}^{3}$ at large distances $r$, makes the problem drastically different from the well-known problem of the two-species Fermi gas with repulsive contact interspecies interaction. We solve the low-energy scattering problem and develop a many-body perturbation theory beyond the mean field. The theory relies on the presence of a small parameter ${k}_{F}{r}_{*}$, where ${k}_{F}$ is the Fermi momentum and ${r}_{*}=m{d}^{2}/{\ensuremath{\hbar}}^{2}$ is the dipole-dipole length, with $m$ being the molecule mass. We obtain thermodynamic quantities as a series of expansion up to the second order in ${k}_{F}{r}_{*}$ and argue that many-body corrections to the ground-state energy can be identified in experiments with ultracold molecules, as it has been recently done for ultracold fermionic atoms. Moreover, we show that only many-body effects provide the existence of zero sound and calculate the sound velocity.

arXiv:1111.7114•Physical Review A•Feb 13, 2012•39 citations
"This thesis is divided into two parts. The first describes a study of the dilute Fermi gas with short range repulsive interactions in 2D, with special emphasis on questions related to the validity of Fermi liquid theory. We find that despite the existence of non-perturbative effects due to two-dimensionality, namely linearly dispersing excitations of ""bound"" holes, the system remains in a Fermi liquid state. The quasi-particles are both long lived and weakly interacting. We also provide a microscopic calculation of the Fermi liquid parameters which determine the response functions in the standard way."
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.
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 q2kF{\mathit{k}}_{\mathit{F}}kF​, 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}''(kF{\mathit{k}}_{\mathit{F}}kF​,\ensuremath{\omega}) is obtained. The 2D case is also contrasted with a similar calculation in 1D.
A central problem in the theory of strongly correlated fermion systems, concerns the nature of their low-lying excitations. It is generally accepted that, except for special cases with symmetry breaking, three dimensional (3D) interacting Fermi systems, are well described in terms of weakly correlated quasi-particles with quantum numbers in one-toone correspondence with those identifying the excitations of a non-interacting Fermi gas. This is the basic assumption underlying the Landau theory of Fermi liquids (FL), which was originally proposed as a phenomenological description of strongly interacting fermions by Landau in early 1956[1), an then established microscopically by Nozieres and Luttinger in 1962[2]. Quite a different scenario occurs in lD interacting Fermi systems. Apart from specific models developing gaps in the excitation spectrum, most fermion systems in one dimension with repulsive interactions, do have low-lying gapless excitations, like ordinary Fermi liquids, but totally different from the quasi-particles predicted by Landau theory. The low-energy behavior of most of these lD interacting fermion systems with gapless linear excitations, can be understood in terms of few model dependent constants, which parametrize all the long-wavelength properties...
This thesis concerns quasi-two-dimensional (quasi-2D) ultracold Fermi systems with short range interactions. Here, quasi-2D geometries are those that lie between two- and three-dimensions. We study few- and many-body systems composed of two or three distinguishable species of fermions. Of principle interest are the bound states that form in these systems and how these bound states are affected by the quasi-2D geometries. In the first research chapter, we focus on the two-dimensional (2D) regime of quasi- 2D. We begin by developing an effective 2D model of this 2D-like geometry. We use this model to analyse a Fermi system consisting of three fermionic species with equal interactions between all distinguishable pairs. Starting with the few-body problem, we show that there is always a three-body bound state (trimer) in the effective 2D model and that these 2D trimers are stable. This is notably different from the extremely unstable ‘Efimov’ trimers found in three-dimensional (3D) systems. Using a low-density expansion and a variational approach, we investigate the fate of the 2D trimers as the system’s density is increased from the three-body limit to the many-body limit. We find that remnants of trimers can persist in the form of strong three-body correlations in the many-body system. In the second research chapter, we analyse a realistic experimental system of three distinguishable fermions confined to quasi-2D. We develop a model that goes beyond the previous chapter in two regards. Firstly, we allow for unequal interactions between the pairs of fermions. Secondly, we use a full quasi-2D model that is not specific to the 2D-like regime. With this general model, we use the scattering and confinement parameters from experiments on ultracold 6Li atoms to calculate the trimer spectrum across the dimensional crossover from 2D to 3D. We show that one state smoothly evolves from an unstable 3D-like trimer to a 2D-like trimer as the attractive interactions are decreased. We furthermore compute the trimer wave function and quantify the stability of the trimer with respect to three-body recombination by determining the probability that three fermions approach each other at short distances. Our results indicate that the lifetime of the trimer can be enhanced by at least an order of magnitude in the quasi-2D geometry. This opens the door to realising long-lifetime trimers in ultracold gas experiments and furthermore, the many-body state of trimers predicted in the preceding chapter. vii In the final research chapter, we examine an alternative approach to accessing quasi-2D geometries by using a system of coupled 2D planes. We study the two-component Fermi gas in this geometry, as a preliminary to studying the three-component gas. We find that the interplanar coupling can be tuned to drive a dimensional crossover from 2D to 3D. We study the effect of this dimensional crossover on the two- and many-body systems. In general, we find that two-body correlations are strengthened by weakening the coupling between 2D planes. In the many-body system, we use mean-field theory to find that the critical temperature of pairing is increased by weakening the coupling between 2D planes. This represents a first step towards a more general representation of quasi-2D geometries, namely: A system of coupled quasi-2D planes.
We study Fermi-liquid properties of a weakly interacting two-dimensional gas of single-component fermionic polar molecules with dipole moments ddd oriented perpendicularly to the plane of their translational motion. This geometry allows the minimization of inelastic losses due to chemical reactions for reactive molecules and, at the same time, provides a possibility of a clear description of many-body (beyond mean-field) effects. The long-range character of the dipole-dipole repulsive interaction between the molecules, which scales as 1/r31/{r}^{3}1/r3 at large distances rrr, makes the problem drastically different from the well-known problem of the two-species Fermi gas with repulsive contact interspecies interaction. We solve the low-energy scattering problem and develop a many-body perturbation theory beyond the mean field. The theory relies on the presence of a small parameter kFr∗{k}_{F}{r}_{*}kF​r∗​, where kF{k}_{F}kF​ is the Fermi momentum and ${r}_{*}=m{d}^{2}/{\ensuremath{\hbar}}^{2}$ is the dipole-dipole length, with mmm being the molecule mass. We obtain thermodynamic quantities as a series of expansion up to the second order in kFr∗{k}_{F}{r}_{*}kF​r∗​ and argue that many-body corrections to the ground-state energy can be identified in experiments with ultracold molecules, as it has been recently done for ultracold fermionic atoms. Moreover, we show that only many-body effects provide the existence of zero sound and calculate the sound velocity.