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arXiv:2011.05750 · Physical review. B./Physical review. B

Spin diffusion and spin conductivity in the two-dimensional Hubbard model

Martin Ulaga, Jernej Mravlje, J. Kokalj

Published Apr 14, 2021• 6 citations
arXivHTMLPDF

Abstract

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 $T$ 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.

DOI
10.1103/physrevb.103.155123

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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.
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 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.
The spin-diffusion constant of the two-dimensional t-J model is calculated using an analytical approach at high temperatures and a recently developed numerical method based on the Lanczos technique combined with random sampling in the intermediate temperature regime. A simple relation, \ensuremath{\sigma}=Ds{\mathit{D}}_{\mathit{s}}Ds​\ensuremath{\chi}, between spin conductivity and spin diffusion is established and used to calculate the latter. In the high-temperature and low-doping limit, the calculated diffusion constant agrees with known results for the Heisenberg model. At small hole doping, Ds{\mathit{D}}_{\mathit{s}}Ds​ increases approximately linearly with doping, which leads us to an important conclusion that hopping processes enhance spin diffusion at high temperatures. At modest hole doping, \ensuremath{\delta}\ensuremath{\sim}0.25, diffusion exhibits a nonmonotonic temperature dependence, which indicates anomalous spin dynamics at small frequencies.
Dynamical properties are notoriously difficult to compute in numerical treatments of the Fermi-Hubbard model, especially in two spatial dimensions. However, they are essential in providing us with insight into some of the most important and less well-understood phases of the model, such as the pseudogap and strange metal phases at relatively high temperatures, or unconventional superconductivity at lower temperatures, away from the commensurate filling. Here, we use the numerical linked-cluster expansions to compute spin and charge optical conductivities of the model at different temperatures and strong interaction strengths via the exact real-time-dependent correlation functions of the current operators. We mitigate systematic errors associated with having a limited access to the long-time behavior of the correlators by introducing fits and allowing for non-zero Drude weights when appropriate. We compare our results to available data from optical lattice experiments and find that the Drude contributions can account for the theory-experiment gap in the DC spin conductivity of the model at half filling in the strong-coupling region. Our method helps paint a more complete picture of the conductivity in the two-dimensional Hubbard model and opens the door to studying dynamical properties of quantum lattice models in the thermodynamic limit.
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Studies relying on hydrodynamic theory and Kardar-Parisi-Zhang (KPZ) scaling have found that in the one-dimensional Hubbard model spin and charge transport are for all temperatures T > 0 anomalous superdiffusive at zero magnetic field, h = 0, and zero chemical potential, μ = 0, respectively. However, this contradicts recent exact results that at very low temperature charge transport rather is normal diffusive. In this Letter we identify the mechanisms that control the different types of temperature dependence of the h = 0 spin and μ = 0 charge transport and find that the latter is normal diffusive for all finite temperatures T > 0, in contrast to the hydrodynamic theory and KPZ scaling predictions.