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arXiv:1207.1441 · Physical Review B

Thermally isolated Luttinger liquids with noisy Hamiltonians

Luca D’Alessio, Armin Rahmani

Published May 2, 2013• 9 citations
arXivHTMLPDF

Abstract

We study the dynamics of a quantum-coherent thermally isolated Luttinger liquid with noisy Luttinger parameter. To characterize the fluctuations of the absorbed energy in generic noise-driven systems, we first identify two types of energy moments, which can help tease apart the effects of classical (sample-to-sample) and quantum sources of fluctuations. One type of moment captures the total fluctuations due to both sources, while the other one captures the effect of the classical source only. We then demonstrate that, in the Luttinger liquid case, the two types of moments agree in the thermodynamic limit, indicating that the classical source dominates. In contrast to equilibrium thermodynamics, in this driven system the relative fluctuations of energy do not decay with the system size. Additionally, we study the deviations of equal-time correlation functions from their ground-state value, and find a simple scaling behavior.

DOI
10.1103/physrevb.87.174301

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We study the dynamics of a quantum-coherent thermally isolated Luttinger liquid with noisy Luttinger parameter. To characterize the fluctuations of the absorbed energy in generic noise-driven systems, we first identify two types of energy moments, which can help tease apart the effects of classical (sample-to-sample) and quantum sources of fluctuations. One type of moment captures the total fluctuations due to both sources, while the other one captures the effect of the classical source only. We then demonstrate that, in the Luttinger liquid case, the two types of moments agree in the thermodynamic limit, indicating that the classical source dominates. In contrast to equilibrium thermodynamics, in this driven system the relative fluctuations of energy do not decay with the system size. Additionally, we study the deviations of equal-time correlation functions from their ground-state value, and find a simple scaling behavior.
We develop the theory of hydrodynamic fluctuations for one-dimensional quantum liquids. At low temperatures the relaxation of interacting electrons in a single-channel wire separates into two classes: equilibration, which changes the numbers of right and left movers through backscattering and is exponentially slow, and thermalization, which redistributes energy and momentum at a fixed number of movers at a power-law rate. This separation of time scales gives the electron liquid a two-fluid description analogous to superfluid 4^44He, with the zero mode of the Luttinger liquid as the superfluid component and the gas of thermal excitations as the normal component. Using Onsager reciprocity and the fluctuation-dissipation theorem, we add Langevin fluxes to the two-fluid hydrodynamic equations, including a mutual-friction noise generated by rare backscattering events that has no counterpart in helium. We apply the theory to Coulomb-coupled quantum wires and calculate the dynamic structure factors, which contain a second-sound peak next to the plasmon doublet; the Coulomb drag resistivity, where intrawire equilibration acts as a bottleneck and turns the hydrodynamic drag into an activated temperature dependence; and the near-field heat transfer between the wires. Mode frequencies, spectral weights, linewidths, and the crossover scales of the transport coefficients are expressed through the thermodynamic properties of the liquid and a few dissipative coefficients, providing spectroscopic and transport probes of two-fluid behavior in one-dimensional quantum systems.
The concept of Tomonaga--Luttinger liquids (TLL) on the basis of the free-boson models is ubiquitous in theoretical descriptions of low-energy properties in one-dimensional quantum systems. In this work, we develop a squeezed-field path-integral description for gapless one-dimensional systems beyond the free-boson picture of the TLL paradigm. In the squeezed-field description, the parameter of the Bogoliubov transformation for the TL Hamiltonian becomes a dynamical squeezing field, and its fluctuations give rise to corrections to the free-boson results. We derive an effective nonlinear Lagrangian describing the dispersion relation of the squeezing field, and interactions between the excitations of the TLL and the squeezing modes. Using the effective Lagrangian, we analyze the imaginary-time correlation function of a vertex operator in the non-interacting limit. We show that a side-band branch emerges due to the fluctuation of the squeezing field, in addition to the standard branch of the free-boson model of the TLL paradigm. Furthermore, we perturbatively analyze the spectral function of the density fluctuations for an ultracold Bose gas in one dimension. We evaluate the renormalized values of the phase velocities and spectral weights of the TLL and side-band branches due to the interaction between the TLL and the squeezing modes. At zero temperature, the renormalized dispersion relations are linear in the momentum, but at nonzero temperatures, these acquire a nonlinear dependence on the momentum due to the thermal population of the excitation branches.
This Ph.D. thesis concerns a very active area of current research in theoretical physics, which is non-equilibrium statistical mechanics. More specifically, the focus will be on the study of the non-equilibrium dynamics and fluctuations in isolated many body systems driven out of equilibrium. We shall consider primarily quantum systems, albeit many of the theoretical techniques that will be used can be applied to both classical and quantum statistical systems. The novel contribution of this work is two-fold. First, we have developed new analytical techniques to study exactly the large-deviation statistics of physical quantities relevant for the description of systems driven out of equilibrium, like the work done in a quantum quench or the time-integrated current characterizing transport of matter, energy or charge etc, discussed in Part I of the thesis. Second, we have singled out a class of experimentally relevant nondisordered quantum systems showing a dramatically slow approach to thermal equilibrium and we have provided quantitative predictions for this behavior, which is discussed in Part II.
Abstract The interplay between dissipation and interactions could give rise to exotic phenomena. In this work, we apply the non-Hermitian (NH) linear response theory to investigate the density response of a spinless Luttinger liquid under local NH perturbations. We show that NH backward scattering results in interference between correlation functions arising from dissipation and Langevin noise terms, leading to fluctuations in the induced density rate of change as the Luttinger parameter K varies. For odd integer values of K, this rate vanishes, indicating immunity to dissipation. We demonstrate that the steady-state induced density is predominantly determined by Langevin noise, underscoring the necessity of including Langevin noise terms in NH Hamiltonians. For attractive interactions, the response to NH perturbations grows exponentially and exhibits a π/2\pi /2π/2 π / 2 phase shift, indicating stronger sensitivity to NH perturbations compared to Hermitian ones. Additionally, we show that NH forward scattering leads to a decay of the induced density. We refer to these peculiar behaviors of density fluctuations under NH perturbations as anomalous Friedel oscillations. Our findings offer new prospects for sensing and understanding dissipation and interactions.
The Luttinger liquid model, which describes interacting electrons in a single-channel quantum wire, is completely integrable in the absence of disorder and as such does not exhibit any relaxation to equilibrium. We consider relaxation processes induced by inelastic electron-electron interactions in a disordered Luttinger liquid, focusing on the equilibration rate and its essential differences from the electron-electron scattering rate as well as the rate of phase relaxation. In the first part of the paper, we review the basic concepts in a disordered Luttinger liquid at equilibrium. These include the elastic renormalization, dephasing, and interference-induced localization. In the second part, we formulate a conceptually important framework for systematically studying the nonequilibrium properties of a strongly correlated (non-Fermi) Luttinger liquid. We derive a coupled set of kinetic equations for the fermionic and bosonic distribution functions that describe the evolution of a nonequilibrium Luttinger liquid. Remarkably, the energy equilibration rate in the conducting disordered quantum wire (at sufficiently high temperature, when the localization effects are suppressed by dephasing) is shown to be on the order of the rate of elastic scattering off disorder, independent of the interaction constant and temperature.
Abstract Fluctuation-dissipation relations, i.e., the relation between two-time correlation and linear response functions, were successfully used to search for signs of equilibration and to identify effective temperatures in the non-equilibrium behavior of a number of macroscopic classical and quantum systems in contact with thermal baths. Among the most relevant cases in which the effective temperatures thus defined were shown to have a thermodynamic meaning one finds the stationary dynamics of driven super-cooled liquids and vortex glasses, and the relaxation of glasses. Whether and under which conditions an effective thermal behavior can be found in quantum isolated many-body systems after a global quench is a question of current interest. We propose to study the possible emergence of thermal behavior long after the quench by studying fluctuation-dissipation relations in which (possibly time- or frequency-dependent) parameters replace the equilibrium temperature. If thermalization within the Gibbs ensemble eventually occurs these parameters should be constant and equal for all pairs of observables in ‘partial’ or ‘mutual’ equilibrium. We analyze these relations in the paradigmatic quantum system, i.e., the quantum Ising chain, in the stationary regime after a quench of the transverse field. The lack of thermalization to a Gibbs ensemble becomes apparent within this approach.