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arXiv:1412.6693 · Physical Review B

Brownian scattering of a spinon in a Luttinger liquid

Maria-Theresa Rieder, Alex Levchenko, Tobias Micklitz

Published Dec 29, 2014• 1 citation
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Abstract

We consider strongly interacting one-dimensional electron liquids where elementary excitations carry either spin or charge. At small temperatures a spinon created at the bottom of its band scatters off low-energy spin and charge excitations and follows the diffusive motion of a Brownian particle in momentum space. We calculate the mobility characterizing these processes and show that the resulting diffusion coefficient of the spinon is parametrically enhanced at low temperatures compared to that of a mobile impurity in a spinless Luttinger liquid. We briefly discuss that this hints at the relevance of spin in the process of equilibration of strongly interacting one-dimensional electrons, and comment on implications for transport in clean single-channel quantum wires.

DOI
10.1103/physrevb.90.245434

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We consider strongly interacting one-dimensional electron liquids where elementary excitations carry either spin or charge. At small temperatures a spinon created at the bottom of its band scatters off low-energy spin and charge excitations and follows the diffusive motion of a Brownian particle in momentum space. We calculate the mobility characterizing these processes and show that the resulting diffusion coefficient of the spinon is parametrically enhanced at low temperatures compared to that of a mobile impurity in a spinless Luttinger liquid. We briefly discuss that this hints at the relevance of spin in the process of equilibration of strongly interacting one-dimensional electrons, and comment on implications for transport in clean single-channel quantum wires.
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We study the transport properties of interacting electrons in a disordered quantum wire within the framework of the Luttinger liquid model. The conductivity at finite temperature is nonzero only because of inelastic electron-electron scattering. We demonstrate that the notion of weak localization is applicable to the strongly correlated one-dimensional electron system. We calculate the relevant dephasing rate, which for spinless electrons is governed by the interplay of electron-electron interaction and disorder, thus vanishing in the clean limit.
Luttinger liquid theory accounts for the low-energy boson excitations of one-dimensional quantum liquids, but disregards the high-energy excitations. The most important high-energy excitations are holes, which have infinite lifetime at zero temperature. At finite temperatures, they can be scattered by thermally excited bosons. We describe the interaction of a hole with the bosons by treating it as a mobile impurity in a Luttinger liquid. This approach enables us to evaluate the scattering probability at arbitrary interaction strength. In general, the result is expressed in terms of the hole spectrum, its dependence on the density and momentum of the fluid, and the parameters of the Luttinger liquid Hamiltonian. In the special case of Galilean invariant systems, the scattering probability is expressed in terms of only the hole spectrum and its dependence on the fluid density. We apply our results to the problem of equilibration of one-dimensional quantum liquids.
We study the mobility of a particle coupled to a one-dimensional interacting fermionic system, namely, a Luttinger liquid. We bosonize the Luttinger liquid and find the effective interaction between the particle and the bosonic system. We show that the dynamics of this system is completely equivalent to the acoustic polaron problem where the interaction has purely electronic origin. This problem has a zero-mode excitation, or soliton, in the strong-coupling limit, which corresponds to the the formation of a polarization cloud due to the fermion-fermion interaction around the particle. We find that, due to the scattering of the residual bosonic modes, the soliton has a finite mobility and diffusion coefficient at finite temperatures that depend on the fermion-fermion interaction. We show that at low temperatures the mobility and the diffusion coefficient are proportional to ${\mathit{T}}^{\mathrm{\ensuremath{-}}4}$ and T5{\mathit{T}}^{5}T5, respectively, and at high temperatures the mobility vanishes as ${\mathit{T}}^{\mathrm{\ensuremath{-}}1}$ while the diffusion increases as T.
In one-dimensional electronic systems with strong repulsive interactions, charge excitations propagate much faster than spin excitations. Such systems therefore have an intermediate temperature range [termed the ``spin-incoherent Luttinger liquid'' (SILL) regime] where charge excitations are ``cold'' (i.e., have low entropy) whereas spin excitations are ``hot.'' We explore the effects of charge-sector disorder in the SILL regime in the absence of external sources of equilibration. We argue that the disorder localizes all charge-sector excitations; however, spin excitations are protected against full localization, and act as a heat bath facilitating charge and energy transport on asymptotically long time scales. The charge, spin, and energy conductivities are widely separated from one another. The dominant carriers of energy in much of the SILL regime are neither charge nor spin excitations, but neutral ``phonon'' modes, which undergo an unconventional form of hopping transport that we discuss. We comment on the applicability of these ideas to experiments and numerical simulations.