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arXiv:2307.05320 · Physical review. B./Physical review. B

Thermoelectric effect on diffusion in the two-dimensional Hubbard model

Martin Ulaga, Jernej Mravlje, J. Kokalj

Published Oct 12, 2023• 1 citation
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Abstract

We study charge and heat transport in the square lattice Hubbard model at strong coupling using the finite-temperature Lanczos method. We construct the diffusion matrix and estimate the effect of thermoelectric terms on diffusive and hydrodynamic time evolution. The thermoelectric terms prevent the interpretation of the diffusion in terms of a single time scale. We discuss our results in relation to cold-atom experiments and measurements of heat conductivity based on the measurements of heat diffusion.

DOI
10.1103/physrevb.108.155118

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We study charge and heat transport in the square lattice Hubbard model at strong coupling using the finite-temperature Lanczos method. We construct the diffusion matrix and estimate the effect of thermoelectric terms on diffusive and hydrodynamic time evolution. The thermoelectric terms prevent the interpretation of the diffusion in terms of a single time scale. We discuss our results in relation to cold-atom experiments and measurements of heat conductivity based on the measurements of heat diffusion.
We study the electronic thermal conductivity ${\ensuremath{\kappa}}_{\text{el}}$ and the thermal diffusion constant DQ,el{D}_{\text{Q,el}}DQ,el​ in the square lattice Hubbard model using the finite-temperature Lanczos method. We make use of the Nernst-Einstein relation for thermal transport and connect the strong nonmonotonous temperature dependence of ${\ensuremath{\kappa}}_{\text{el}}$ to that of DQ,el{D}_{\text{Q,el}}DQ,el​ and the electronic specific heat cel{c}_{\text{el}}cel​. We present also the results for the Heisenberg model on a square lattice and ladder geometries. We study the effects of doping and consider the doped case also with the dynamical mean-field theory. We show that ${\ensuremath{\kappa}}_{\text{el}}$ is below the corresponding Mott-Ioffe-Regel value in almost all calculated regimes, while the mean free path is typically above or close to lattice spacing. We discuss the opposite effect of quasiparticle renormalization on charge and heat diffusion constants. We calculate the Lorenz ratio and show that it differs from the Sommerfeld value. We discuss our results in relation to experiments on cuprates. Additionally, we calculate the thermal conductivity of overdoped cuprates within the anisotropic marginal Fermi liquid phenomenological approach.
We study the spin diffusion and spin conductivity in the square lattice Hubbard model by using the finite-temperature Lanczos method. We show that the spin diffusion behaves differently from the charge diffusion and has a nonmonotonic TTT dependence. This is due to a progressive liberation of charges that contribute to spin transport and enhance it beyond that active at low temperature due to the Heisenberg exchange. We further show that going away from half filling and zero magnetization increases the spin diffusion, but that the increase is insufficient to reconcile the difference between the model calculations and the recent measurements on cold atoms.
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.
Recent experiments on cold atoms in optical lattices allow for a quantitative comparison of the measurements to the conductivity calculations in the square lattice Hubbard model. However, the available calculations do not give consistent results, and the question of the exact solution for the conductivity in the Hubbard model remained open. In this Letter, we employ several complementary state-of-the-art numerical methods to disentangle various contributions to conductivity and identify the best available result to be compared to experiment. We find that, at relevant (high) temperatures, the self-energy is practically local, yet the vertex corrections remain rather important, contrary to expectations. The finite-size effects are small even at the lattice size 4×4, and the corresponding Lanczos diagonalization result is, therefore, close to the exact result in the thermodynamic limit.
Recent experimental results suggest that a particular hydrodynamic theory describes charge fluctuations at long wavelengths in the square-lattice Hubbard model. Due to the continuity equation, the correlation functions for the charge and the current are directly connected: the parameters of the effective hydrodynamic model thus determine the optical conductivity. Here we investigate the validity of the proposed hydrodynamic theory in the full range of parameters of the Hubbard model. In the noninteracting case, there is no effective hydrodynamics, and the charge fluctuations present a rich variety of nonuniversal behaviors. At weak coupling, the optical conductivity is consistent with the hydrodynamic theory: at low frequency one observes a Lorentzian-shaped Drude peak, but the high-frequency asymptotics is necessarily different; the high-temperature limit for the product of the two hydrodynamic model parameters is also in agreement with numerical data. At strong coupling, we find that a generalization of the proposed hydrodynamic law is consistent with our quantum Monte Carlo, as well as the finite-temperature Lanczos results from literature. Most importantly, the temperature dependence of the hydrodynamic parameters as well as the dc resistivity are found to be very similar in the weak- and the strong-coupling regimes.
Strong interactions in many-body quantum systems complicate the interpretation of charge transport in such materials. To shed light on this problem, we study transport in a clean quantum system: ultracold lithium-6 in a two-dimensional optical lattice, a testing ground for strong interaction physics in the Fermi-Hubbard model. We determine the diffusion constant by measuring the relaxation of an imposed density modulation and modeling its decay hydrodynamically. The diffusion constant is converted to a resistivity by using the Nernst-Einstein relation. That resistivity exhibits a linear temperature dependence and shows no evidence of saturation, two characteristic signatures of a bad metal. The techniques we developed in this study may be applied to measurements of other transport quantities, including the optical conductivity and thermopower.