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Advances In Physics

Quantum-coherent transport of a heavy particle in a fermionic bath

Achim Rosch

Published May 1, 1999• 35 citations
Publisher

Abstract

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.

DOI
10.1080/000187399243446

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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 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 $\ensuremath{\sigma}$, respectively. The mobility is shown to obey two different power laws below and above T∗{T}^{*}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.
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.
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.
Heavy fermion systems are explained on the basis of the Fermi liquid theory. The specific heat, magnetic susceptibility and electrical resistivity are discussed. Using Anderson's orthogonality theorem, we show that the Fermi liquid is nothing but a local spin singlet state at every site. Heavy fermion systems The Fermi liquid theory is independent of a model Hamiltonian, and can be applied to any system as long as the system remains a Fermi liquid. The theory tells us that even if the electron interaction becomes strong, physical quantities behave as those of the non-interacting Fermi gas. The difference between them with and without interaction is not qualitative but quantitative. As seen at the Mott transition in Hubbard systems, the effective mass of the electron increases near the transition point. As a system realizing such a large effective mass, a Fermi liquid system called the heavy electron system or the heavy fermion system attracts general interest. The heavy fermion systems are composed of the rare earth metals such as Ce and Yb, and actinide atoms such as U. The heavy fermions are nothing but the quasi-particles in the Fermi liquid theory. As a quasi-particle in a strongly correlated electron system, the heavy fermion is an important issue to be studied in the development of the Fermi liquid theory. The heavy fermion realized in f -electron systems is one of the heavy quasi-particles appearing near the Mott transition.
We consider two mutually interacting fermionic particle species on a one-dimensional lattice and study how the mass ratio $\ensuremath{\eta}$ between the two species affects the (equilibration) dynamics of the particles. Focusing on the regime of strong interactions and high temperatures, two well-studied points of reference are given by (i) the case of equal masses $\ensuremath{\eta}=1$, i.e., the standard Fermi-Hubbard chain, where initial nonequilibrium density distributions are known to decay, and (ii) the case of one particle species being infinitely heavy, $\ensuremath{\eta}=0$, leading to a localization of the lighter particles in an effective disorder potential. Given these two opposing cases, the dynamics in the case of intermediate mass ratios $0<\ensuremath{\eta}<1$ is of particular interest. To this end, we study the real-time dynamics of pure states featuring a sharp initial nonequilibrium density profile. Relying on the concept of dynamical quantum typicality, the resulting nonequilibrium dynamics can be related to equilibrium correlation functions. Summarizing our main results, we observe that diffusive transport occurs for moderate values of the mass imbalance and manifests itself in a Gaussian spreading of real-space density profiles and an exponential decay of density modes in momentum space. For stronger imbalances, we provide evidence that transport becomes anomalous on intermediate timescales, and in particular, our results are consistent with the absence of strict localization in the long-time limit for any $\ensuremath{\eta}>0$. Based on our numerical analysis, we provide an estimate for the ``lifetime'' of the effective localization as a function of $\ensuremath{\eta}$.