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arXiv:2610.37301

Hydrodynamic fluctuations in Luttinger liquids

Alex Levchenko

Published Oct 27, 2026• 0 citations
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Abstract

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 $^4$He, 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.

DOI
10.48550/arxiv.2610.37301

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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.
Luttinger liquid theory of one-dimensional quantum systems ignores exponentially weak backscattering of particles. This endows Luttinger liquids with superfluid properties. The corresponding two-fluid hydrodynamic description available at present applies only to Galilean-invariant systems, whereas most experimental realizations of one-dimensional quantum liquids lack Galilean invariance. Here we develop the two-fluid theory of such quantum liquids. In the low-frequency limit the theory reduces to single-fluid hydrodynamics. However, the absence of Galilean invariance brings about three new transport coefficients. We obtain expressions for these coefficients in terms of the backscattering rate.
We review some of the recent results on equilibration of one-dimensional quantum liquids. The low-energy properties of these systems are described by the Luttinger liquid theory, in which the excitations are bosonic quasiparticles. At low temperatures, the relaxation of the gas of excitations toward full equilibrium is exponentially slow. In electronic Luttinger liquids, these relaxation processes involve backscattering of electrons and give rise to interesting corrections to the transport properties of one-dimensional conductors. We focus on the phenomenological theory of the equilibration of a quantum liquid and obtain an expression for the relaxation rate in terms of the excitation spectrum.
We develop a theory of Coulomb drag in ultraclean double layers with strongly correlated carriers. In the regime where the equilibration length of the electron liquid is shorter than the interlayer spacing the main contribution to the Coulomb drag arises from hydrodynamic density fluctuations. The latter consist of plasmons driven by fluctuating longitudinal stresses, and diffusive modes caused by temperature fluctuations and thermal expansion of the electron liquid. We express the drag resistivity in terms of the kinetic coefficients of the electron fluid. Our results are nonperturbative in interaction strength and do not assume Fermi-liquid behavior of the electron liquid.
We consider the effect of the curvature in fermionic dispersion on the observable properties of Luttinger liquid (LL). We use the bosonization technique where the curvature is irrelevant perturbation, describing the decay of LL bosons (plasmon modes). When possible, we establish the correspondence between the bosonization and the fermionic approach. We analyze modifications in density correlation functions due to curvature at finite temperatures, TTT. The most important application of our approach is the analysis of the Coulomb drag by small momentum transfer between two LLs, which is only possible due to curvature. Analyzing the ac transconductivity in the one-dimensional drag setup, we confirm the results by Pustilnik et al. for TTT dependence of drag resistivity, ${R}_{12}\ensuremath{\sim}{T}^{2}$ at high and ${R}_{12}\ensuremath{\sim}{T}^{5}$ at low temperatures. The bosonization allows for treating both intra- and interwire electron-electron interactions in all orders, and we calculate exact prefactors in the low-TTT drag regime. The crossover temperature between the two regimes is ${T}_{1}\ensuremath{\sim}{E}_{F}\ensuremath{\Delta}$, with $\ensuremath{\Delta}$ relative difference in plasmon velocities. We show that $\ensuremath{\Delta}\ensuremath{\ne}0$ even for identical wires, due to lifting of degeneracy by interwire interaction, U12{U}_{12}U12​, leading to crossover from ${R}_{12}\ensuremath{\sim}{U}_{12}^{2}{T}^{2}$ to ${R}_{12}\ensuremath{\sim}{T}^{5}∕{U}_{12}$ at $T\ensuremath{\sim}{U}_{12}$.
At low temperatures, elementary excitations of a one-dimensional quantum liquid form a gas that can move as a whole with respect to the center of mass of the system. This internal motion attenuates at exponentially long time scales. As a result, in a broad range of frequencies, the liquid is described by two-fluid hydrodynamics, and the system supports two sound modes. The physical nature of the two sounds depends on whether the particles forming the quantum liquid have a spin degree of freedom. For particles with spin, the modes are analogous to the first and second sound modes in superfluid 4He^{4}\mathrm{He}4He, which are the waves of density and entropy, respectively. When dissipative processes are taken into account, we find that at low frequencies, the second sound is transformed into heat diffusion, while the first sound mode remains weakly damped and becomes the ordinary sound. In a spinless liquid, the entropy and density oscillations are strongly coupled and the resulting sound modes are hybrids of the first and second sound. As the frequency is lowered and dissipation processes become important, the crossover to a single-fluid regime occurs in two steps. First the hybrid modes transform into predominantly density and entropy waves, similar to the first and second sound, and then the density wave transforms to the ordinary sound and the entropy wave becomes a heat-diffusion mode. Finally, we account for the dissipation due to viscosity and intrinsic thermal conductivity of the gas of excitations, which controls attenuation of the sound modes at high frequencies.
The natural excitations of an interacting one-dimensional system at low energy are the hydrodynamic modes of a Luttinger liquid, protected by the Lorentz invariance of the linear dispersion. We show that beyond low energies, where the quadratic dispersion reduces the symmetry to Galilean, the main character of the many-body excitations changes into a hierarchy: calculations of dynamic correlation functions for fermions (without spin) show that the spectral weights of the excitations are proportional to powers of R2/L2{\mathcal{R}}^{2}/{L}^{2}R2/L2, where R\mathcal{R}R is a length-scale related to interactions and LLL is the system length. Thus only small numbers of excitations carry the principal spectral power in representative regions on the energy-momentum planes. We have analyzed the spectral function in detail and have shown that the first-level (strongest) excitations form a mode with parabolic dispersion, like that of a renormalized single particle. The second-level excitations produce a singular power-law line shape to the first-level mode and multiple power laws at the spectral edge. We have illustrated a crossover to a Luttinger liquid at low energy by calculating the local density of states through all energy scales: from linear to nonlinear, and to above the chemical potential energies. In order to test this model, we have carried out experiments to measure the momentum-resolved tunneling of electrons (fermions with spin) from/to a wire formed within a GaAs heterostructure. We observe a well-resolved spin-charge separation at low energy with appreciable interaction strength and only a parabolic dispersion of the first-level mode at higher energies. We find a structure resembling the second-level excitations, which dies away rapidly at high momentum in line with the theoretical predictions here.