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arXiv:cond-mat/9801015 · Physical Review Letters

Giant Mass and Anomalous Mobility of Particles in a Fermionic System

Achim Rosch, Thilo Kopp

Published May 25, 1998• 32 citations
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Abstract

We calculate the mobility of a heavy particle coupled to a Fermi sea within a nonperturbative approach valid at all temperatures. The interplay of particle recoil and of strong-coupling effects, leading to the orthogonality catastrophe for an infinitely heavy particle, is carefully taken into account. We find two novel types of strong-coupling effects: a new low-energy scale ${T}^{*}$ and a giant mass renormalization in the case of either near-resonant scattering or a large transport cross section $\ensuremath{\sigma}$, respectively. The mobility is shown to obey two different power laws below and above ${T}^{*}$. For $\ensuremath{\sigma}\ensuremath{\gg}{\ensuremath{\lambda}}_{f}^{2}$, where ${\ensuremath{\lambda}}_{f}$ is the Fermi wavelength, an exponentially large effective mass suppresses the mobility.

DOI
10.1103/physrevlett.80.4705

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We calculate the mobility of a heavy particle coupled to a Fermi sea within a nonperturbative approach valid at all temperatures. The interplay of particle recoil and of strong-coupling effects, leading to the orthogonality catastrophe for an infinitely heavy particle, is carefully taken into account. We find two novel types of strong-coupling effects: a new low-energy scale T∗{T}^{*}T∗ and a giant mass renormalization in the case of either near-resonant scattering or a large transport cross section \ensuremathσ\ensuremath{\sigma}\ensuremathσ, respectively. The mobility is shown to obey two different power laws below and above T∗{T}^{*}T∗. For \ensuremathσ\ensuremath≫\ensuremathλf2\ensuremath{\sigma}\ensuremath{\gg}{\ensuremath{\lambda}}_{f}^{2}\ensuremathσ\ensuremath≫\ensuremathλf2​, where \ensuremathλf{\ensuremath{\lambda}}_{f}\ensuremathλf​ is the Fermi wavelength, an exponentially large effective mass suppresses the mobility.
In this review, a detailed discussion of the behaviour of a heavy particle interacting with a Fermi sea is given. Particular emphasis is put on the issue of how strong correlations influence coherence and transport of the particle. First, we investigate the question of whether the heavy particle is a well defined quasiparticle at low temperatures. While in one dimension ( D = 1) and at a van Hove singularity in D = 2 the coherence of the particle is destroyed, the quasiparticle weight is finite in higher dimensions. The most important transport quantity is the diffusion constant or mobility of the heavy particle. We are able to describe both the well known high-temperature properties and the cross-over to the lowest temperatures in a unified approximation scheme based on a self-consistent evaluation of an effective action. Two strong-correlation effects of independent origin are discussed. The first arises if the scattering of the fermions from the heavy particle is nearly resonant, that is, if one of the scattering phase shifts δ is close to π/2. In this regime an anomalous exponent is observed in the temperature dependence of the mobility μ(T). In D = 3, for instance, the mobility is proportional to T-3/2 rather than to T-2. The second effect is a giant mass renormalization in the case of a large particle. In this situation, the low-temperature effective mass M* increases up to an exponentially large value, M* exp[c(r/λF)3], where r is the effective radius of the particle, λF the Fermi wavelength and c a non-universal constant of order one.
We investigate the low temperature behavior of a heavy particle in a fermionic bath. Two competing effects dominate the physics: a shakeup of the Fermi sea (orthogonality catastrophe) destroys the coherence, while the recoil of the particle tends to restore it by reducing the available phase space for scattering events. An effective action is considered which exactly implements these two aspects. In one dimension the coherence of the particle is destroyed---the propagator decays algebraically for long times. In two and higher dimensions coherent propagation survives.
We study the dynamics of a heavy particle of mass M moving in a one-dimensional repulsively interacting Fermi gas. The Fermi gas is described using the Luttinger model and bosonization. By transforming to a frame comoving with the heavy particle, we map the model onto a generalized ``quantum impurity problem.'' A renormalization-group calculation reveals a crossover from strong to weak coupling upon scaling down in temperature. Above the crossover temperature scale T∗{\mathit{T}}^{\mathrm{*}}T∗=(m/M)EF{\mathit{E}}_{\mathit{F}}EF​, the particle's mobility \ensuremath{\mu} is found to be (roughly) temperature independent and proportional to the dimensionless conductance g, characterizing the one-dimensional Luttinger liquid. Here m(\ensuremath{\ll}M) is the fermion mass, and EF{\mathit{E}}_{\mathit{F}}EF​ the Fermi energy. Below T∗{\mathit{T}}^{\mathrm{*}}T∗, in the weak-coupling regime, the mobility grows and diverges as \ensuremath{\mu}(T)\ensuremath{\sim}${\mathit{T}}^{\mathrm{\ensuremath{-}}4}$ in the T\ensuremath{\rightarrow}0 limit. \textcopyright{} 1996 The American Physical Society.
In this report we discuss the dynamical response of heavy quantum impurities immersed in a Fermi gas at zero and at finite temperature. Studying both the frequency and the time domain allows one to identify interaction regimes that are characterized by distinct many-body dynamics. From this theoretical study a picture emerges in which impurity dynamics is universal on essentially all time scales, and where the high-frequency few-body response is related to the long-time dynamics of the Anderson orthogonality catastrophe by Tan relations. Our theoretical description relies on different and complementary approaches: functional determinants give an exact numerical solution for time- and frequency-resolved responses, bosonization provides accurate analytical expressions at low temperatures, and the theory of Toeplitz determinants allows one to analytically predict response up to high temperatures. Using these approaches we predict the thermal decoherence rate of the fermionic system and prove that within the considered model the fastest rate of long-time decoherence is given by [Formula: see text]. We show that Feshbach resonances in cold atomic systems give access to new interaction regimes where quantum effects can prevail even in the thermal regime of many-body dynamics. The key signature of this phenomenon is a crossover between different exponential decay rates of the real-time Ramsey signal. It is shown that the physics of the orthogonality catastrophe is experimentally observable up to temperatures [Formula: see text] where it leaves its fingerprint in a power-law temperature dependence of thermal spectral weight and we review how this phenomenon is related to the physics of heavy ions in liquid [Formula: see text]He and the formation of Fermi polarons. The presented results are in excellent agreement with recent experiments on LiK mixtures, and we predict several new phenomena that can be tested using currently available experimental technology.
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.
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.