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arXiv:1605.02357 · Physical Review Letters

Dynamics of a Mobile Impurity in a One-Dimensional Bose Liquid

Aleksandra Petković, Zoran Ristivojević

Published Sep 1, 2016• 20 citations
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Abstract

We develop a microscopic theory of a quantum impurity propagating in a one-dimensional Bose liquid. As a result of scattering off thermally excited quasiparticles, the impurity experiences the friction. We find that, at low temperatures, the resulting force scales either as the fourth or the eighth power of temperature, depending on the system parameters. For temperatures higher than the chemical potential of the Bose liquid, the friction force is a linear function of temperature. Our approach enables us to find the friction force in the crossover region between the two limiting cases. In the integrable case, corresponding to the Yang-Gaudin model, the impurity becomes transparent for quasiparticles and thus the friction force is absent. Our results could be further generalized to study other kinetic phenomena.

DOI
10.1103/physrevlett.117.105301

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We develop a microscopic theory of a quantum impurity propagating in a one-dimensional Bose liquid. As a result of scattering off thermally excited quasiparticles, the impurity experiences the friction. We find that, at low temperatures, the resulting force scales either as the fourth or the eighth power of temperature, depending on the system parameters. For temperatures higher than the chemical potential of the Bose liquid, the friction force is a linear function of temperature. Our approach enables us to find the friction force in the crossover region between the two limiting cases. In the integrable case, corresponding to the Yang-Gaudin model, the impurity becomes transparent for quasiparticles and thus the friction force is absent. Our results could be further generalized to study other kinetic phenomena.
We study the motion of a slow quantum impurity in one-dimensional environments focusing on systems of strongly interacting bosons and weakly interacting fermions. While at zero temperature the impurity motion is frictionless, at low temperatures finite friction appears. The dominant process is the scattering of the impurity off two fermionic quasiparticles. We evaluate the friction force and show that, at low temperatures, it scales either as the fourth or the sixth power of temperature, depending on the system parameters. This is a result of the scattering of the impurity off two fermionic quasiparticles that are situated around different Fermi points. It is the dominant process at low temperatures. We also evaluate the contribution to the friction force originating from the scattering of the impurity off two fermionic quasiparticles that are situated around different Fermi points. It behaves as the tenth power of temperature.
We study the motion of a heavy impurity in a one-dimensional Bose gas. The impurity experiences the friction force due to scattering off thermally excited quasiparticles. We present detailed analysis of an arbitrarily strong impurity-boson coupling in a wide range of temperatures within a microscopic theory. Focusing mostly on weakly interacting bosons, we derive an analytical result for the friction force and uncover new regimes of the impurity dynamics. Particularly interesting is the low-temperature T^{2} dependence of the friction force obtained for a strongly coupled impurity, which should be contrasted with the expected T^{4} scaling. This new regime applies to systems of bosons with an arbitrary repulsion strength. We finally study the evolution of the impurity with a given initial momentum. We evaluate analytically its nonstationary momentum distribution function. The impurity relaxation towards the equilibrium is a realization of the Ornstein-Uhlenbeck process in momentum space.
We develop a systematic perturbation theory for the quasiparticle properties of a single impurity immersed in a Bose-Einstein condensate. Analytical results are derived for the impurity energy, effective mass, and residue to third order in the impurity-boson scattering length. The energy is shown to depend logarithmically on the scattering length to third order, whereas the residue and the effective mass are given by analytical power series. When the boson-boson scattering length equals the boson-impurity scattering length, the energy has the same structure as that of a weakly interacting Bose gas, including terms of the Lee-Huang-Yang and fourth order logarithmic form. Our results, which cannot be obtained within the canonical Fröhlich model of an impurity interacting with phonons, provide valuable benchmarks for many-body theories and for experiments.
We investigate the indirect interaction between two isolated impurities in a Luttinger liquid described by a microscopic lattice model. To treat the electron-electron interaction UUU, the functional renormalization group method is used. For comparison, we also study the U=0U=0U=0 case. We find that for a wide range of impurity parameters the impurity interaction V12{V}_{12}V12​ as a function of their separation rrr oscillates with decaying amplitude between being attractive and repulsive. For half-filling of the band and in a crossover regime between weak and strong impurities, the interaction becomes purely attractive. For U=0U=0U=0 and independent of the impurity strength, the amplitude of the interaction energy falls off as 1∕r1∕r1∕r. For U>0U>0U>0, the decay for small separations and weak to intermediate impurities is governed by a UUU dependent exponent larger than $\ensuremath{-}1$, which crosses over to $\ensuremath{-}1$ for large rrr. The crossover scale depends on the impurity strength and UUU. We present simple pictures which explain our results in the limits of weak and strong impurities. We finally also consider attractive interactions U<0U<0U<0.
We study the decay of Bogoliubov quasiparticles in one-dimensional Bose gases. Starting from the hydrodynamic Hamiltonian, we develop a microscopic theory that enables one to systematically study both the excitations and their decay. At zero temperature, the leading mechanism of decay of a quasiparticle is disintegration into three others. We find that low-energy quasiparticles (phonons) decay with the rate that scales with the seventh power of momentum, whereas the rate of decay of the high-energy quasiparticles does not depend on momentum. In addition, our approach allows us to study analytically the quasiparticle decay in the whole crossover region between the two limiting cases. When applied to integrable models, including the Lieb-Liniger model of bosons with contact repulsion, our theory confirms the absence of the decay of quasiparticle excitations. We account for two types of integrability-breaking perturbations that enable finite decay: three-body interaction between the bosons and two-body interaction of finite range.