© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physical review. B, Condensed matter

Path-integral method for a heavy particle moving in a periodic potential and screened by a light degenerate Fermi gas

Gergely T. Zimányi, Károly Vladár, A. Zawadowski

Published Aug 15, 1987• 25 citations

Abstract

We study a heavy particle moving in a periodic lattice of arbitrary dimension and interacting with a degenerate fermionic heat bath. We assume that the Fermi energy of the screening particles is large compared to the tight-binding bandwidth of the particle. We develop a path-integral method for the partition function of the heavy particle where we integrate out the electronic degrees of freedom. We perform also a summation over the paths between the screening-interaction points that makes the method capable of treating band motion and hopping. Following the method of Anderson and Yuval, we derive scaling equations by eliminating that part of the phase space of the light particles which is far from the Fermi energy. We find that the screening strength is not renormalized; the bandwidth of the heavy particle is, however, essentially reduced. That reduction coincides with the reduction in the two-site model. Extrapolating the result for larger values of the screening strength we find band motion and localization at zero temperature depending on the coupling strength; thus the behavior is of Ohmic type. We show that the results obtained can be combined with the classical kinetic equation to get the diffusion coefficient. We believe that the present work based on the path-integral method helps to clarify the connection between results obtained by different methods. The relation of the model to other models, especially to the Caldeira-Leggett model, is discussed in detail.

DOI: 10.1103/physrevb.36.3186

Related papers

Path integral of the two-dimensional Su-Schrieffer-Heeger model

Marco Zoli

The equilibrium thermodynamics of the two-dimensional Su-Schrieffer-Heeger model is derived by means of a path integral method that accounts for the variable range of the electronic hopping processes. While the lattice degrees of freedom are classical functions of time and are integrated out exactly, the electron particle paths are treated quantum mechanically. The free energy of the system and its temperature derivatives are computed by summing at any $T$ over the ensemble of relevant particle paths which mainly contribute to the total partition function. In the low $T$ regime, the heat capacity over $T$ ratio shows an upturn peculiar of a glassy-like behavior. This feature is more sizeable in the square lattice than in the linear chain as the overall hopping potential contribution to the total action is larger in higher dimensionality.

arXiv:cond-mat/0504259•Physical Review B•May 24, 2005•19 citations

Influence-functional theory for a heavy particle in a Fermi gas

Lutz Bönig, Kurt Schönhammer, W. Zwerger

We use Feynman's influence-functional theory to study the quantum dynamics of a heavy particle moving in a free Fermi gas with arbitrary average velocity. A semiclassical expansion yields a nonlinear Langevin equation with the exact friction coefficient as derived in an earlier publication. The fluctuations around a steady state far from equilibrium are due to a nonclassical state-dependent noise term and can be described by a diffusion constant. In the limit of zero average velocity, the Einstein relation is fulfilled for arbitrary temperatures. For finite velocities the diffusion around the steady state is different in longitudinal and transverse directions and can be expressed in terms of the transport cross section and a ``diffusion'' cross section. In the case where the frictional force exhibits a maximum as a function of velocity and thus an unstable branch for v>${\mathit{v}}_{\mathit{c}}$, the longitudinal diffusion constant diverges on approaching ${\mathit{v}}_{\mathit{c}}$ from below. Numerical results for the noise spectrum and the temperature and velocity dependence of the diffusion constants are presented for simple repulsive interaction potentials in one and three dimensions.

Physical review. B, Condensed matter•Jul 1, 1992•10 citations

Scaling theory of dynamic scattering centers and moving heavy particles in a fermionic environment

Károly Vladár

A dynamic scattering object and the slowly building screening fermion cloud around it are examined. The models previously studied for strong on-site scattering---x-ray absorption, the anisotropic Kondo problem, the Anderson model, etc.---have the common feature that all values of the bare scattering matrix commute with one another. Nozi\`eres and De Dominicis's path-integral method can be applied directly to these problems because the scatterings in different channels are independent. The aim of this paper is to extend the theory to those problems, in which the scattering matrix has noncommuting values as, e.g., in the case of a scattering center moving in real space. Integrating over the fermionic degrees of freedom, we get to the problem of a one-dimensional dielectric medium with moving domain walls. In contrast to the commutative case, more-than-two-particle interactions may arise between the domain walls. Renormalization is carried out by keeping the terms that are the most divergent in the short-time-cutoff parameter.

Physical review. B, Condensed matter•Jul 15, 1991•3 citations

Two-level system coupled to a boson mode

J. Florencio

The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/${\mathit{k}}_{\mathit{B}}$T. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.

Physical review. B, Condensed matter•Jul 1, 1990•4 citations

Renormalization of the periodic Anderson model: An alternative analytical approach to heavy-fermion behavior

A. Hübsch, K. W. Becker

In this paper a recently developed projector-based renormalization method (PRM) for many-particle Hamiltonians is applied to the periodic Anderson model with the aim to describe heavy-fermion behavior. In this method high-energetic excitation operators instead of high energetic states are eliminated. We arrive at an effective Hamiltonian for a quasifree system which consists of two noninteracting heavy-quasiparticle bands. The resulting renormalization equations for the parameters of the Hamiltonian are valid for large as well as small degeneracy ${\ensuremath{\nu}}_{f}$ of the angular momentum. An expansion in $1∕{\ensuremath{\nu}}_{f}$ is avoided. Within an additional approximation which adapts the idea of a fixed renormalized $f$ level ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$, we obtain coupled equations for ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$ and the averaged $f$ occupation $⟨{n}_{f}⟩$. These equations resemble to a certain extent those of the usual slave boson mean-field (SB) treatment. In particular, for large ${\ensuremath{\nu}}_{f}$ the results for the PRM and the SB approach agree perfectly whereas considerable differences are found for small ${\ensuremath{\nu}}_{f}$.

arXiv:cond-mat/0406653•Physical Review B•Apr 20, 2005•9 citations

Particle path correlations in a phonon bath

Marco Zoli

The path integral formalism is applied to derive the full partition function of a generalized Su-Schrieffer-Heeger Hamiltonian describing a particle motion in a bath of oscillators. The electronic correlations are computed versus temperature for some choices of oscillators energies. We study the perturbing effect of a time averaged particle path on the phonon subsystem deriving the relevant temperature dependent cumulant corrections to the harmonic partition function and free energy. The method has been applied to compute the total heat capacity up to room temeperature: a low temperature upturn in the heat capacity over temperature ratio points to a glassy like behavior ascribable to a time dependent electronic hopping with variable range in the linear chain.

arXiv:cond-mat/0308132•Journal of Physics Condensed Matter•Aug 29, 2003•1 citation

We study a heavy particle moving in a periodic lattice of arbitrary dimension and interacting with a degenerate fermionic heat bath. We assume that the Fermi energy of the screening particles is large compared to the tight-binding bandwidth of the particle. We develop a path-integral method for the partition function of the heavy particle where we integrate out the electronic degrees of freedom. We perform also a summation over the paths between the screening-interaction points that makes the method capable of treating band motion and hopping. Following the method of Anderson and Yuval, we derive scaling equations by eliminating that part of the phase space of the light particles which is far from the Fermi energy. We find that the screening strength is not renormalized; the bandwidth of the heavy particle is, however, essentially reduced. That reduction coincides with the reduction in the two-site model. Extrapolating the result for larger values of the screening strength we find band motion and localization at zero temperature depending on the coupling strength; thus the behavior is of Ohmic type. We show that the results obtained can be combined with the classical kinetic equation to get the diffusion coefficient. We believe that the present work based on the path-integral method helps to clarify the connection between results obtained by different methods. The relation of the model to other models, especially to the Caldeira-Leggett model, is discussed in detail.

The equilibrium thermodynamics of the two-dimensional Su-Schrieffer-Heeger model is derived by means of a path integral method that accounts for the variable range of the electronic hopping processes. While the lattice degrees of freedom are classical functions of time and are integrated out exactly, the electron particle paths are treated quantum mechanically. The free energy of the system and its temperature derivatives are computed by summing at any

T

over the ensemble of relevant particle paths which mainly contribute to the total partition function. In the low

T

regime, the heat capacity over

T

ratio shows an upturn peculiar of a glassy-like behavior. This feature is more sizeable in the square lattice than in the linear chain as the overall hopping potential contribution to the total action is larger in higher dimensionality.

We use Feynman's influence-functional theory to study the quantum dynamics of a heavy particle moving in a free Fermi gas with arbitrary average velocity. A semiclassical expansion yields a nonlinear Langevin equation with the exact friction coefficient as derived in an earlier publication. The fluctuations around a steady state far from equilibrium are due to a nonclassical state-dependent noise term and can be described by a diffusion constant. In the limit of zero average velocity, the Einstein relation is fulfilled for arbitrary temperatures. For finite velocities the diffusion around the steady state is different in longitudinal and transverse directions and can be expressed in terms of the transport cross section and a ``diffusion'' cross section. In the case where the frictional force exhibits a maximum as a function of velocity and thus an unstable branch for v>

{\mathit{v}}_{\mathit{c}}

, the longitudinal diffusion constant diverges on approaching

{\mathit{v}}_{\mathit{c}}

from below. Numerical results for the noise spectrum and the temperature and velocity dependence of the diffusion constants are presented for simple repulsive interaction potentials in one and three dimensions.

A dynamic scattering object and the slowly building screening fermion cloud around it are examined. The models previously studied for strong on-site scattering---x-ray absorption, the anisotropic Kondo problem, the Anderson model, etc.---have the common feature that all values of the bare scattering matrix commute with one another. Nozi\`eres and De Dominicis's path-integral method can be applied directly to these problems because the scatterings in different channels are independent. The aim of this paper is to extend the theory to those problems, in which the scattering matrix has noncommuting values as, e.g., in the case of a scattering center moving in real space. Integrating over the fermionic degrees of freedom, we get to the problem of a one-dimensional dielectric medium with moving domain walls. In contrast to the commutative case, more-than-two-particle interactions may arise between the domain walls. Renormalization is carried out by keeping the terms that are the most divergent in the short-time-cutoff parameter.

The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/

{\mathit{k}}_{\mathit{B}}

T. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.

In this paper a recently developed projector-based renormalization method (PRM) for many-particle Hamiltonians is applied to the periodic Anderson model with the aim to describe heavy-fermion behavior. In this method high-energetic excitation operators instead of high energetic states are eliminated. We arrive at an effective Hamiltonian for a quasifree system which consists of two noninteracting heavy-quasiparticle bands. The resulting renormalization equations for the parameters of the Hamiltonian are valid for large as well as small degeneracy ${\ensuremath{\nu}}_{f}$ of the angular momentum. An expansion in $1∕{\ensuremath{\nu}}_{f}$ is avoided. Within an additional approximation which adapts the idea of a fixed renormalized

f

level ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$, we obtain coupled equations for ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\epsilon}}}_{f}$ and the averaged

f

occupation

⟨{n}_{f}⟩

. These equations resemble to a certain extent those of the usual slave boson mean-field (SB) treatment. In particular, for large ${\ensuremath{\nu}}_{f}$ the results for the PRM and the SB approach agree perfectly whereas considerable differences are found for small ${\ensuremath{\nu}}_{f}$.

The path integral formalism is applied to derive the full partition function of a generalized Su-Schrieffer-Heeger Hamiltonian describing a particle motion in a bath of oscillators. The electronic correlations are computed versus temperature for some choices of oscillators energies. We study the perturbing effect of a time averaged particle path on the phonon subsystem deriving the relevant temperature dependent cumulant corrections to the harmonic partition function and free energy. The method has been applied to compute the total heat capacity up to room temeperature: a low temperature upturn in the heat capacity over temperature ratio points to a glassy like behavior ascribable to a time dependent electronic hopping with variable range in the linear chain.