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arXiv:cond-mat/9409006 · International Journal of Modern Physics B

TWO PARTICLE CORRELATIONS AND ORTHOGONALITY CATASTROPHE IN INTERACTING FERMI SYSTEMS

Walter Metzner, Claudio Castellani

Published Jul 20, 1995• 3 citations
arXivHTMLPDF

Abstract

The wave function of two fermions, repulsively interacting in the presence of a Fermi sea, is evaluated in detail. We consider large but finite systems in order to obtain an unambiguous picture of the two-particle correlations. As recently pointed out by Anderson, in d≤2 dimensions the particles may be correlated even when situated on the Fermi surface. The “partial exclusion principle” for two particles with opposite spin on the same Fermi point is discussed, and related to results from the T-matrix approximation. Particles on different Fermi points are shown to be uncorrelated in d>1. Using the results for the two-particle correlations we find that the orthogonality effect induced by adding an extra particle to a (tentative) two-dimensional Fermi liquid is finite.

DOI
10.1142/s021797929500080x

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The wave function of two fermions, repulsively interacting in the presence of a Fermi sea, is evaluated in detail. We consider large but finite systems in order to obtain an unambiguous picture of the two-particle correlations. As recently pointed out by Anderson, in d≤2 dimensions the particles may be correlated even when situated on the Fermi surface. The “partial exclusion principle” for two particles with opposite spin on the same Fermi point is discussed, and related to results from the T-matrix approximation. Particles on different Fermi points are shown to be uncorrelated in d>1. Using the results for the two-particle correlations we find that the orthogonality effect induced by adding an extra particle to a (tentative) two-dimensional Fermi liquid is finite.
This chapter first studies the interaction effect of a Fermi sea on a single particle propagator and the related phenomenon of orthogonality catastrophe in X-ray spectrum. It then looks at a weakly interacting Fermi gas using Hartree–Fock approximation and introduces the notion of Fermi liquid. It discusses the concept and the appearance of quasiparticle in Fermi liquid. In addition to the standard quasiparticle description of Fermi liquid, the chapter also introduces a hydrodynamic description of Fermi liquid, which can be viewed as higher dimensional bosonization. Using the spin-density-wave state as an example, it then discusses how interaction can cause instability of Fermi liquid and generate a symmetry breaking phase transition to a spin ordered phase. Several theoretical methods, such as mean-field theory, variatonal approach, and RPA method are introduced within the interacting fermion systems. The low energy O(3) non-linear sigma model for the spin ordered phase is derived, where close attention has been paid to the possible topological terms in the non-linear sigma model. The effect of topological terms on the low energy dynamics of spin waves is discussed.
In the framework of exactly solvable quantum many-body systems we study models of interacting spin one-half Fermions in one dimension. The first part deals with systems of spin one-half Fermions which interact via repulsive contact interaction. A reformulation of the Bethe-Ansatz solvable many-body wave function is presented. This simplifies considerably the calculations for the highly imbalanced case, where very few particles of one species (minority Fermions) are present. For the other particle species (majority Fermions) the thermodynamic limit is taken. We assume the majority Fermions to be in the ground state such that their non-interacting momentum distribution is a Fermi-sea. Upon this we consider excitations where the particles of the minority species may occupy an arbitrary state within the Fermi-sea. In the case of only a single minority Fermion, the many-body wave function can be expressed as a determinant. This allows us to derive exact thermodynamic expressions for several expectation values as well as for the density-density correlation function. Moreover it is possible to find closed expressions for the single particle Green's function. All of the above mentioned quantities show a non-trivial dependence on the minority particle's momentum. In particular the Green's function in the Tonks-Girardeau regime of hardcore interaction is shown to undergo a transition from the one of impenetrable Bosons to that of free Fermions as the extra particle's momentum varies from the core to the edge of the Fermi-sea. This transition becomes manifest in an algebraic asymptotic decay of the Green's function. If two minority Fermions are present, the many-body wave function turns out to be more complicated. Nevertheless it is possible to derive exact expressions for the two and the three particle density-density correlation functions. Furthermore we calculate the system's total energy and based on that, identify terms which have a natural interpretation as effective interaction energy for the two minority Fermions in the presence of the Fermi-sea. The second part is devoted to the study of one-dimensional systems consisting of two fermionic particle species with different masses. We show that for specific kinds of interaction potentials and for certain relations between the masses and the coupling constants, the particle creation and annihilation operators of such a system can be constructed exactly.
Pairing lies at the heart of superfluidity in fermionic systems. Motivated by recent experiments in mesoscopic Fermi gases, we study up to six fermionic atoms with equal masses and equal populations in two different spin states, confined in a quasi-two-dimensional harmonic trap. We couple a stochastic variational approach with the use of an explicitly correlated Gaussian basis set, which enables us to obtain highly accurate energies and structural properties. Utilising two-dimensional two-body scattering theory with a finite-range Gaussian interaction potential, we tune the effective range to model realistic quasi-two-dimensional scattering. We calculate the excitation spectrum, pair correlation function, and number of pairs as a function of increasing attractive interaction strength. For up to six fermions in the ground state, we find that opposite spin and momentum pairing is maximised well below the Fermi surface in momentum space. By contrast, corresponding experiments on twelve fermions have found that pairing is maximal at the Fermi surface and strongly suppressed beneath [M. Holten et al., Nature 606, 287-291 (2022)]. This suggests that the Fermi sea — which acts to suppress pairing at low momenta via Pauli blocking — emerges in the transition from six to twelve particles.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.
In this paper we study a model of s-wave marginal Fermi liquid superconductor at two dimensions. Besides the usual Bardeen-Cooper-Schrieffer(BCS) point attractive interaction, the fermions in our system also interact {\em via} a long range repulsive interaction v(q)=gD/q2v(q)=g_{D}/q^2v(q)=gD​/q2. We find that although superconductivity is not destroyed by the long-ranged repulsion interaction (demonstrated by the existence of Meissner effect), nevertheless the BCS pairing function ΔBCSΔ_{BCS}ΔBCS​ is strongly suppressed resulting in power-law correlation ∼∣x⃗∣−2γ\sim|\vec{x}|^{-2γ}∼∣x∣−2γ. This observation is explained within a general picture of marginal Fermi liquid.