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Physical review. B, Condensed matter

Noncanonical-transformation approach to the problem of an itinerant particle interacting with a Fermi sea

I. E. Perakis, Yia‐Chung Chang

Published Mar 15, 1993• 14 citations
Publisher

Abstract

We present a noncanonical-transformation solution to the problem of an itinerant particle interacting with a Fermi sea and apply the solution to the study of the excitonic effects on the optical spectra of a modulation-doped quantum wire. In the case of equal particle and Fermi electron mass, this provides the exact solution in one dimension. For unequal masses or higher dimensions, we provide the first term in the systematic expansion of the exact interacting eigenstates in terms of many-body correlations (${\mathit{e}}^{\mathit{S}}$ method). Our approximation conserves momentum, includes the three-body correlations between the itinerant particle and the electron and hole of one pair of Fermi sea excitations, and produces single Slater-determinant eigenstates. Numerical results were obtained for parameters relevant to modulation-doped quantum wires. They show strong dependence of the eigenstates on the total momentum of the system and reveal a new momentum scale of power-law behavior associated with the valence-subband width. For total momentum approaching the Fermi surface, this momentum scale diminishes to zero and large momentum excitations across the Fermi surface become important. In addition, for a total momentum approaching ${\mathit{k}}_{\mathit{F}}$ from above, a resonant structure in the wave function moving toward the Fermi surface is observed.

DOI
10.1103/physrevb.47.6573

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We present a noncanonical-transformation solution to the problem of an itinerant particle interacting with a Fermi sea and apply the solution to the study of the excitonic effects on the optical spectra of a modulation-doped quantum wire. In the case of equal particle and Fermi electron mass, this provides the exact solution in one dimension. For unequal masses or higher dimensions, we provide the first term in the systematic expansion of the exact interacting eigenstates in terms of many-body correlations (eS{\mathit{e}}^{\mathit{S}}eS method). Our approximation conserves momentum, includes the three-body correlations between the itinerant particle and the electron and hole of one pair of Fermi sea excitations, and produces single Slater-determinant eigenstates. Numerical results were obtained for parameters relevant to modulation-doped quantum wires. They show strong dependence of the eigenstates on the total momentum of the system and reveal a new momentum scale of power-law behavior associated with the valence-subband width. For total momentum approaching the Fermi surface, this momentum scale diminishes to zero and large momentum excitations across the Fermi surface become important. In addition, for a total momentum approaching kF{\mathit{k}}_{\mathit{F}}kF​ from above, a resonant structure in the wave function moving toward the Fermi surface is observed.
We study the motion of a particle in a degenerate Fermi sea in one dimension by calculating numerically the one-particle spectral function as well as a number of two-particle response functions. The problem is formulated in terms of the eigenstates of the static case, in which the particle is localized at a particular site. In spite of the Anderson orthogonality catastrophe, accurate results for small systems (up to 100 sites and electrons) are obtained with relatively few basis states, i.e., 0 and 1 particle-hole pair excitations. The mobile particle in one dimension does not display ``quasiparticle'' behavior, and most response functions show power-law threshold behavior as in the case of a localized particle, in complete agreement with exact results. We discuss practical applications to optical spectra and to the dynamical mobility of optically generated holes in doped semiconductor quantum wires.
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.
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The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) is momentum dependent. Using Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.
We consider the problem of a single particle interacting with NNN identical fermions, at zero temperature and in one dimension. We calculate the binding energy as well as the effective mass of the single particle. We use an approximate method developed in the three-dimensional case, where the Hilbert space for the excited states of the NNN fermions is restricted to have at most two particle-hole pairs. When the mass of the single particle is equal to the fermion mass, we find excellent agreement with the exact results of McGuire. When the mass of the single particle is infinite, we solve exactly the problem and find again excellent agreement between approximate results and exact ones. This overall agreement in one dimension gives a strong validation for the approximate method applied in three dimensions. Moreover it shows that our approximate treatment is excellent for the one-dimensional problem in the general case with respect to the mass of the single particle.
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