Generalized Hydrodynamic Approach To Charge And Energy Currents In The One-Dimensional Hubbard Model

We have studied nonequilibrium dynamics of the one-dimensional Hubbard model using the generalized hydrodynamic theory. We mainly investigated the spatiotemporal profile of particle density (equivalent to charge density), energy density, and their currents using the partitioning protocol; the initial state consists of two semi-infinite different thermal equilibrium states joined at the origin. In this protocol, a transient region where currents flow appears around the origin, and this region expands its size linearly with time. We examined how density and current profiles depend on initial conditions. Inside the transient region, we have found a clogged region where charge current is zero but nonvanishing energy current flows. This phenomenon is similar to spin-charge separation in Tomonaga-Luttinger liquids, in which spin and charge excitations propagate with different velocities. This region appears when one of the initial states has half-filled electron density, and it is located adjacent to this initial state. We have proved analytically the existence of the clogged region in the infinite temperature case of the half-filled initial state. The existence is confirmed also for finite temperatures by numerical calculations of generalized hydrodynamics. A similar analytical proof is also given for a clogged region of spin current when magnetic field is applied to one and only one of the two initial states. A universal proportionality of charge and spin currents is also proved for a special region, for general initial conditions of electron density and magnetic field. To examine the clogged region of charge current, we have calculated the current contributions of different types of quasiparticles in the Hubbard model. It is found that the charge current component carried by scattering states (i.e., Fermi liquid type) is canceled completely in the clogged region by the counter flows carried by bound-state quasiparticles (called k-A strings). Their contributions are not canceled for energy current, which is also proved analytically for special cases, and these different behaviors are the origin of the clogged region. Except for the clogged region, charge and energy densities are in good proportion to each other, and their ratio depends on the initial conditions. This proportionality, in nonequilibrium dynamics, is reminiscent of Wiedemann-Franz law in thermal equilibrium, which states current proportionality determined by temperature. The long-time stationary values of charge and energy currents were also studied with varying initial conditions. We have compared the results with the values of noninteracting systems and discussed the effects of electron correlations. The ratio of these stationary currents is also analyzed. We have found that the temperature dependence of the ratio is strongly suppressed by electron correlations and even reversed.