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arXiv:1901.05478 · Physical Review Letters

Thermal Transport in One-Dimensional Electronic Fluids

Rickmoy Samanta, Ivan V. Protopopov, A. D. Mirlin, Dmitri B. Gutman

Published May 20, 2019• 14 citations
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Abstract

We study thermal conductivity for one-dimensional electronic fluids. The many-body Hilbert space is partitioned into bosonic and fermionic sectors that carry the thermal current in parallel. For times shorter than the bosonic umklapp time, the momenta of Bose and Fermi components are separately conserved, giving rise to the ballistic heat propagation and imaginary heat conductivity proportional to T/iω. The real part of thermal conductivity is controlled by decay processes of fermionic and bosonic excitations, leading to several regimes in frequency dependence. At lowest frequencies or longest length scales, the thermal transport is dominated by Lévy flights of low-momentum bosons that lead to a fractional scaling, ω^{-1/3} and L^{1/3}, of heat conductivity with the frequency ω and system size L, respectively.

DOI
10.1103/physrevlett.122.206801

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We study thermal conductivity for one-dimensional electronic fluids. The many-body Hilbert space is partitioned into bosonic and fermionic sectors that carry the thermal current in parallel. For times shorter than the bosonic umklapp time, the momenta of Bose and Fermi components are separately conserved, giving rise to the ballistic heat propagation and imaginary heat conductivity proportional to T/iω. The real part of thermal conductivity is controlled by decay processes of fermionic and bosonic excitations, leading to several regimes in frequency dependence. At lowest frequencies or longest length scales, the thermal transport is dominated by Lévy flights of low-momentum bosons that lead to a fractional scaling, ω^{-1/3} and L^{1/3}, of heat conductivity with the frequency ω and system size L, respectively.
We study thermal conductivity in one-dimensional electronic fluids combining kinetic [R. Samanta, I. V. Protopopov, A. D. Mirlin, and D. B. Gutman, Thermal transport in one-dimensional electronic fluid, Phys. Rev. Lett. 122, 206801 (2019)] and hydrodynamic [I. V. Protopopov, R. Samanta, A. D. Mirlin, and D. B. Gutman, Anomalous hydrodynamics in one-dimensional electronic fluid, Phys. Rev. Lett. 126, 256801 (2021)] theories. The kinetic approach is developed by partitioning the Hilbert space into bosonic and fermionic sectors. We focus on the regime where the long-living thermal excitations are fermions and compute thermal conductivity. From the kinetic theory standpoint, the fermionic part of thermal conductivity is normal, while the bosonic one is anomalous, that scales as ω–1/3 and thus dominates in the infrared limit. The multi-mode hydrodynamic theory is obtained by projecting the fermionic kinetic equation on the zero modes of its collision integral. On a bare level, both theories agree and the thermal conductivity computed in hydrodynamic theory matches the result of the kinetic equation. The interaction between hydrodynamic modes leads to renormalization and consequently to anomalous scaling of the transport coefficients. In a four-mode regime, all modes are ballistic and the anomaly manifests itself in Kardar-Parisi-Zhang-like broadening with asymmetric power-law tails. “Heads” and “tails” of the pulses contribute equally to thermal conductivity, leading to ω–1/3 scaling of heat conductivity. In the three-mode regime, the system is in the universality class of a classical viscous fluid [Herbert Spohn, Nonlinear fluctuating hydrodynamics for anharmonic chains, J. Stat. Phys. 154, 1191 (2014); O. Narayan and S. Ramaswamy, Anomalous heat conduction in one-dimensional momentum-conserving systems, Phys. Rev. Lett. 89, 200601 (2002)].
The frequency and temperature dependence of the conductivity of a one-dimensional fermion system with attractive interactions is studied by using a renormalization-group technique. At half filling the real part of the conductivity has both a \ensuremath{\delta}(\ensuremath{\omega}) part and a divergent frequency behavior ${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ at finite frequencies, where \ensuremath{\nu} is a nonuniversal exponent depending on the interactoins. For the particular case of the attractive Hubbard model, logarithmic corrections appear and the conductivity behaves as 1/[\ensuremath{\omega} log102{\mathrm{log}}_{10}^{2}log102​(\ensuremath{\omega})], plus a \ensuremath{\delta}(\ensuremath{\omega}) part. Away from half filling the conductivity has a \ensuremath{\delta}(\ensuremath{\omega}) part and a gap up to a critical frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, where ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ is proportional to the doping with a prefactor depending on the interactions. The results obtained for the fermion model can be straightforwardly extended to the conductivity of an interacting one-dimensional boson model.
We study heat transport in a gas of one-dimensional fermions in the presence of a small temperature gradient. At temperatures well below the Fermi energy there are two types of relaxation processes in this system, with dramatically different relaxation rates. As a result, in addition to the usual thermal conductivity, one can introduce the thermal conductivity of the gas of elementary excitations, which quantifies the dissipation in the system in the broad range of frequencies between the two relaxation rates. We develop a microscopic theory of these transport coefficients in the limit of weak interactions between the fermions.
We study the thermal conductivity of the one-dimensional Fermi-Hubbard model at a finite temperature using a density matrix renormalization group approach. The integrability of this model gives rise to ballistic thermal transport. We calculate the temperature dependence of the thermal Drude weight at half filling for various interaction strengths. The finite-frequency contributions originating from the fact that the energy current is not a conserved quantity are investigated as well. We report evidence that breaking the integrability through a nearest-neighbor interaction leads to vanishing Drude weights and diffusive energy transport. Moreover, we demonstrate that energy spreads ballistically in local quenches with initially inhomogeneous energy density profiles in the integrable case. We discuss the relevance of our results for thermalization in ultracold quantum-gas experiments and for transport measurements with quasi-one-dimensional materials.
The impact of lattice vibrations on the electric conductivity has been considered a settled issue since the proof of the Migdal theorem. The precondition for its validity is a higher velocity of electronic than phononic excitation modes. In some new materials, e. g. weakly coupled spin chains and heavy fermion systems, this precondition is not fulfilled. The heat conductivity of such systems is calculated by means of perturbation theory in this work. Another subject is the interplay of Umklapp scattering and weak disorder in spin chains and one-dimensional correlated metals. Hidden conservation laws lead to a strong coupling between heat current and charge or spin current, respectively. The transport properties are calculated perturbatively within the hydrodynamic memory matrix formalism, which gives a lower bound to the conductivies. To calculate the Lorenz number of one-dimensional metals, we model within this formalism also the influence of phonons, which in physical systems contribute substantially to the heat current. The introduction of ultracold quantum gases in optical lattices in the recent years provides a powerful tool for the modeling of many systems to solid state physics, in particular systems out of equilibrium. The relatively slow dynamics of the cold atoms allows for time resolved imaging of the atoms and thus opens up new perspectives. In the last part of this thesis, a recent experiment, which measures the expansion of a correlated atom cloud in an optical lattice, is compared to a lattice Boltzmann simulation. Depending on the local density, the problem is shown to be described by the crossover from diffusive to ballistic transport.
We study the localization and decay properties as well as the thermal conductance of one-dimensional plasmons. Our model contains a Luttinger-liquid part with spatially random plasmon velocity and interaction parameter as well as a nonlinearity that is cubic in density. The scaling of the decay rate of plasmons is obtained in several regimes. At sufficiently high frequencies, it describes the inelastic lifetime of localized plasmon excitations that crosses over to the clean result with lowering frequencies. For higher frequencies, we analyze implications of many-body-localization effects that lead to a suppression of the decay rate. We find that the thermal conductance depends in a nontrivial fashion on the system size LLL. Specifically, it scales as ${L}^{\ensuremath{-}1/2}$ for sufficiently short wires and crosses over to ${L}^{\ensuremath{-}2/3}$ scaling for longer wires.