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Physical review. B, Condensed matter

Motion of a particle in a Fermi sea: One dimension

Joachim F. Mueller, Andrei E. Ruckenstein, Stefan Schmitt-Rink

Published Apr 15, 1992• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.45.8902

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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.
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.
Using the Bethe-ansatz analysis as reformulated by Edwards, we calculate the spectral properties of a particle interacting with a bath of fermions in one dimension for the case of equal particle-fermion masses. These are directly related to singularities apparent in optical experiments in one-dimensional systems. The orthogonality catastrophe for the case of an infinite particle mass survives in the limit of equal masses. We find that the exponent \ensuremath{\beta} of the quasiparticle weight Z\ensuremath{\simeq}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$ is different for the two cases and proportional to their respective phase shifts at the Fermi surface; we present a simple physical argument for this difference. We also show that these exponents describe the low-energy behavior of the spectral function for repulsive as well as attractive interactions.
We consider a mobile impurity coupled to an ideal Fermi gas in one spatial dimension through an attractive contact interaction. We calculate the quasi-particle residue ZZZ exactly, based on Bethe Ansatz and diagrammatic Monte Carlo methods, and with varational Ansatz up to one particle-hole excitation of the Fermi sea. We find that the exact quasi-particle residue vanishes in the thermodynamic limit as a power law in the number of particles, consistent with the Luttinger-liquid paradigm and the breakdown of Fermi-liquid theory. The variational Ansatz, however, predicts a finite value of ZZZ, even in the thermodynamic limit. We also study how the presence of the impurity affects the density of the spin-up sea by calculating the pair correlation function. Subtracting the homogeneous background and integrating over all distances gives the charge QQQ. This charge turns out to grow continuously from 0 at zero coupling to 1 in the strong-coupling limit. The varational Ansatz predicts Q=0Q=0Q=0 at all couplings. So, although the variational Ansatz has been shown to be remarkably accurate for the energy and the effective mass, it fails even qualitatively when predicting ZZZ and the pair correlation function in the thermodynamic limit.
We study a ground-state ansatz for the single-hole-doped $t\text{\ensuremath{-}}J$ model in two dimensions via a variational Monte Carlo method. Such a single-hole wave function possesses finite angular momenta generated by hidden spin currents, which give rise to a novel ground-state degeneracy in agreement with recent exact diagonalization (ED) and density matrix renormalization group (DMRG) results. We further show that the wave function can be decomposed into a quasiparticle component and an incoherent momentum distribution in excellent agreement with the DMRG results up to an $8\ifmmode\times\else\texttimes\fi{}8$ lattice. Such a two-component structure indicates the breakdown of Landau's one-to-one correspondence principle, and in particular, the quasiparticle spectral weight vanishes by a power law in the large sample size limit. By contrast, turning off the phase string induced by the hole hopping in the so-called $\ensuremath{\sigma}\ifmmode\cdot\else\textperiodcentered\fi{}t\text{\ensuremath{-}}J$ model, a conventional Bloch-wave wave function with a finite quasiparticle spectral weight can be recovered, also in agreement with the ED and DMRG results. The present study shows that a singular effect already takes place in the single-hole-doped Mott insulator, by which the bare hole is turned into a non-Landau quasiparticle with translational-symmetry breaking. Generalizations to pairing and finite doping are briefly discussed.
The localization properties of quasiparticle pairs above a frozen Fermi sea in the two- and three-dimensional Anderson model with Hubbard interaction are investigated. The dependence of the width Γ of the local spectral density on the pair excitation energy ε2, the disorder W, the interaction strength U and the linear system size L is determined. The obtained results allow to emphasize an interaction-induced pair delocalization effect even when the quasiparticles remain individually localized. The corresponding mobility edge is further estimated and the disorder versus interaction phase diagram is determined. Further investigations focusing on the spectral properties of the system and the transition from Poisson to Wigner-Dyson statistics corroborate these conclusions. The relevance of this effect with respect to the one-quasiparticle lifetime is finally discussed.
The spectral functions of one hole in the t-J and one-band Hubbard models are calculated using exact diagonalization techniques on small lattices. Results for the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model are also presented. For the t-J model we found that there is a quasiparticle at the bottom of the hole spectrum with an energy well approximated by Eh{\mathit{E}}_{\mathit{h}}Eh​=-3.17+2.83J0.73{\mathit{J}}^{0.73}J0.73 (for 0.1\ensuremath{\le}J\ensuremath{\le}1.0, t=1) on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The rest of the spectrum is not incoherent: We identified at least two other peaks following a similar power-law behavior with J. We speculate that the J dependence of the results can be explained by a model where the hole is trapped in a confining potential as in the Ising limit. The bandwidth of the hole is linear in J in the region 0.1\ensuremath{\le}J\ensuremath{\le}0.4 although a power-law behavior is not excluded. The spectral weight of the quasiparticle grows like J0.5{\mathit{J}}^{0.5}J0.5 in the same region. We present new analytical results in the large J/t limit to understand the motion of the hole: In perturbation theory it can be shown that the momentum of the hole at large J/t is k=(\ensuremath{\pi},\ensuremath{\pi}) changing to k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) at intermediate J/t in agreement with numerical and spin-waves results. We show analytically and numerically that the bandwidth of the quasiparticle is of order t in the large J/t limit. This result corresponds to a spin-liquid state. The one-hole spectral function of the Hubbard model is obtained for lattices with 8 and 10 sites. A quasiparticle is also observed in this case. The bandwidth and the relation with the t-J model are discussed and a comparison with recent Monte Carlo results is made. We also review and extend previous results for the ground-state properties of the t-J model.