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arXiv:2008.06522 · Physical review. B./Physical review. B

Generalized hydrodynamics study of the one-dimensional Hubbard model: Stationary clogging and proportionality of spin, charge, and energy currents

Yuji Nozawa, Hirokazu Tsunetsugu

Published Jan 19, 2021• 25 citations
arXivHTMLPDF

Abstract

In our previous work [Nozawa and Tsunetsugu, Phys. Rev. B 101, 035121 (2020)], we studied the quench dynamics in the one-dimensional Hubbard model based on the generalized hydrodynamics theory for a partitioning protocol and showed the presence of a clogging phenomenon. Clogging is a phenomenon where vanishing charge current coexists with nonzero energy current, and we found it for the initial conditions that the left half of the system is prepared to be half filling at high temperatures with the right half being empty. Clogging occurs at all the sites in the left half and lasts for a time proportional to its distance from the connection point. In this paper, we use various different initial conditions and discuss two issues. The first issue is the possibility of clogging in a stationary state. When the electron density in the right half is initially set nonzero, we found that the left half-filled part expands for various sets of parameters in the initial condition. This means that the clogging phenomenon occurs at all the sites in the long-time stationary state, and we also discuss its origin. In addition, stationary clogging is accompanied by a back current, namely, particle density current flows towards the high-density region. We also found that spin clogging occurs for some initial conditions, i.e., the vanishing spin current coexists with nonzero energy current. The second issue is the proportionality of spin and charge currents. We found two spatiotemporal regions where the current ratio is fixed to a nonzero constant. We numerically studied how the current ratio depends on various initial conditions. We also studied the ratio of charge and energy currents.

DOI
10.1103/physrevb.103.035130

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In our previous work [Nozawa and Tsunetsugu, Phys. Rev. B 101, 035121 (2020)], we studied the quench dynamics in the one-dimensional Hubbard model based on the generalized hydrodynamics theory for a partitioning protocol and showed the presence of a clogging phenomenon. Clogging is a phenomenon where vanishing charge current coexists with nonzero energy current, and we found it for the initial conditions that the left half of the system is prepared to be half filling at high temperatures with the right half being empty. Clogging occurs at all the sites in the left half and lasts for a time proportional to its distance from the connection point. In this paper, we use various different initial conditions and discuss two issues. The first issue is the possibility of clogging in a stationary state. When the electron density in the right half is initially set nonzero, we found that the left half-filled part expands for various sets of parameters in the initial condition. This means that the clogging phenomenon occurs at all the sites in the long-time stationary state, and we also discuss its origin. In addition, stationary clogging is accompanied by a back current, namely, particle density current flows towards the high-density region. We also found that spin clogging occurs for some initial conditions, i.e., the vanishing spin current coexists with nonzero energy current. The second issue is the proportionality of spin and charge currents. We found two spatiotemporal regions where the current ratio is fixed to a nonzero constant. We numerically studied how the current ratio depends on various initial conditions. We also studied the ratio of charge and energy currents.
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 $k\text{\ensuremath{-}}\mathrm{\ensuremath{\Lambda}}$ 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.
We study the real-time and real-space dynamics of charge in the one-dimensional Hubbard model in the limit of high temperatures. To this end, we prepare pure initial states with sharply peaked density profiles and calculate the time evolution of these nonequilibrium states, by using numerical forward-propagation approaches to chains as long as 20 sites. For a class of typical states, we find excellent agreement with linear-response theory and unveil the existence of remarkably clean charge diffusion in the regime of strong particle-particle interactions. We additionally demonstrate that, in the half-filling sector, this diffusive behavior does not depend on certain details of our initial conditions, i.e., it occurs for five different realizations with random and nonrandom internal degrees of freedom, single and double occupation of the central site, and displacement of spin-up and spin-down particles.
We investigate the dynamical properties of the one-dimensional Hubbard model using the quantum Monte Carlo method and the numerical analytic continuation method based on the maximum entropy principle. Results for the spin, charge and current excitation spectrum are presented for various values of the coupling and the band filling for the system of 24 sites with the periodic boundary condition. We find that the total spectral density of the spin excitation exhibits a gapless structure irrespective of the coupling or the band filling. The charge excitation spectrum, on the other hand, clearly shows the gap formation at the half filled band and shifts its weight toward the higher energy region with increase of the coupling. This charge excitation gap, however, disappears with the hole doping. We also discuss the momentum dependence of the spectral functions.
The persistent current and spin-charge separation are studied in a one-dimensional (1D) Hubbard ring with side sites connected to the even sites of the ring. At half filling, the persistent current is always paramagnetic and has a period of flux quantum. Away from half filling, as the onsite repulsion UUU increases, the sign of the current may oscillate depending on the total number of electrons Ne{N}_{e}Ne​ and the hopping ttt between the ring site and the side site. For small hopping ttt, the number of electrons ${N}_{e}^{\ensuremath{'}}$ in the ring sites plays an important role in determining the sign of the current. In contrast to the 1D Hubbard model, a weak or a moderate UUU can strongly enhance the persistent current while the hopping ttt always suppresses the current. The study on the transmittance of the ring with two connected leads indicates that in the large UUU and small ttt case, the transmittance varies with a period $1∕{N}_{e}^{\ensuremath{'}}$ of a flux quantum, demonstrating a spin-charge separation in this system.
Abstract We consider the extended Hubbard model on a two-dimensional square lattice at half-filling. The model is investigated using the strong coupling diagram technique. We sum infinite series of ladder diagrams allowing for full-scale charge and spin fluctuations and the actual short-range antiferromagnetic order for nonzero temperatures. In agreement with earlier results, we find the first-order phase transition in the charge subsystem occurring at v = v c ≳ U/4 with v and U the intersite and on-site Coulomb repulsion constants. The transition reveals itself in an abrupt sign change of a sharp maximum in the zero-frequency charge susceptibility at the corner of the Brillouin. States arising at the transition have alternating deviations of electron occupations from the mean value on neighboring sites. Due to fluctuations, these alternating occupation deviations have short-range order. For the considered parameters, such behavior is found for U ≲ 5t with t the hopping constant. For the insulating case U ≳ 6t, in which the transition is not observed, we find a continuous growth of the Mott gap with v. The evolution of the electron density of states with increasing v is also considered.
We consider charge and spin transport in the one-dimensional Hubbard model at infinite temperature, half-filling, and zero magnetization. Implementing matrix-product-operator simulations of the nonequilibrium steady states of boundary-driven open Hubbard chains for up to 100 sites we find clear numerical evidence of diffusive transport for weak driving and any (nonzero and finite) value of the interaction UUU. In addition, nonequilibrium steady states are characterized by nonzero spin-up spin-down density-density correlations. For maximal driving, on the other hand, we find an insulating behavior and cosine-shaped density profiles.