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 $k\text{−}\mathrm{\Lambda }$ 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.

Figure 1

Three types of roots of the Lieb-Wu equations: (left) real $k$, (middle) $\mathrm{\Lambda }$ string, and (right) $k\text{−}\mathrm{\Lambda }$ string. Examples of $\mathrm{\Lambda }$ string for $a=4$ and $k\text{−}\mathrm{\Lambda }$ string for $a=-3$ are shown. Black dots in each complex plane represent roots. Those with circle are doubly degenerate corresponding to $k$ and $\pi -k$, which have the same value of $sink$. These three are different types of quasiparticles in the Hubbard model, and their charge and spin are listed at the bottom. $e(<0)$ is the electron charge.

Figure 2

Schematic picture of partitioning protocol. The initial state consists of two thermal equilibrium states; they are disconnected until $t=0$ and joined at the origin $x=0$ after that. The left and right states are prepared with the control parameters such as inverse temperature ${\beta }_{\mathrm{L}\left(\mathrm{R}\right)}$, chemical potential ${\mu }_{\mathrm{L}\left(\mathrm{R}\right)}$, and magnetic field $B_{\mathrm{L}\left(\mathrm{R}\right)}$. Local distributions of quasiparticles at $(x,t)$ depend only on the ray $\xi =x/t$. Outside the light cones $\xi \le V_{\mathrm{L}}$ or $\xi \ge V_{\mathrm{R}}$, the distributions are unchanged from those in the initial left or right thermal equilibrium state, respectively.

Figure 3

Sketch of how to determine the values of the filling function ${\vartheta }_a(w,\xi )$. The curve $\xi ={\stackrel{\circ }{v}}_a(w,\xi )$ divides the $\xi \text{−}w$ space into two parts, and the values of the light cones ${\xi }_a^{-}$ and ${\xi }_a^{+}$ are determined as the leftmost and rightmost points, respectively, on the curve. In the left (right) area, the filling function is determined as ${\vartheta }_a(w,\xi )={\vartheta }_a^{\mathrm{L}\left(\mathrm{R}\right)}\left(w\right)$.

Figure 4

Chemical potential dependence of (a) particle density and (b) energy density in the initial left equilibrium state. Five curves correspond to different temperatures $0.5\le {\beta }_{\mathrm{L}}\le 5$.

Figure 5

Profiles of particle and energy densities and their currents: (a) $n\left(\xi \right)$, (b) $j_n\left(\xi \right)$, (c) $e\left(\xi \right)$, and (d) $j_e\left(\xi \right)$, as functions of the ray $\xi =x/t$. The initial condition is $({\beta }_{\mathrm{L}},{\mu }_{\mathrm{L}})=(0.5,0)$. Black lines show the total values of each quantity with the cutoff $a_c=48$. Orange lines are contributions of scattering states ($a=0$). Light blue lines are the sum of contributions of bound states ($-47\le a\le -1)$. The other lines are contributions of quasiparticles with $-5\le a\le -1$.

Figure 6

Effects of varying the temperature $0.5\le {\beta }_{\mathrm{L}}\le 2$ on quasiparticle contributions in (a) $n\left(\xi \right)$, (b) $j_n\left(\xi \right)$, (c) $e\left(\xi \right)$, and (d) $j_e\left(\xi \right)$. The initial chemical potential is fixed at ${\mu }_{\mathrm{L}}=0$. For each temperature, the total value of density or current and the sum of bound state contributions are separately plotted.

Figure 7

Clogged region at finite temperatures $0.5\le {\beta }_{\mathrm{L}}\le 5$. The cutoff is $a_c=48$, and ${\xi }_{-47}^{-}$ is an approximate value of ${\xi }_{-\infty }^{-}$. The initial chemical potential is fixed at ${\mu }_{\mathrm{L}}=0$. The blue line is a fitting function ${\xi }_{-47}^{-}-{\xi }_0^{-}=a{b}^{-{\beta }_{\mathrm{L}}}$, where $a=1.82$ and $b=5.76$. (Inset) Semilogarithmic plot of ${\beta }_{\mathrm{L}}$ dependence of the width.

Figure 8

Extrapolation for checking if $n\left(\xi \right)=1$ in the limit of the cutoff $a_c\to \infty$. The initial condition is $({\beta }_{\mathrm{L}},{\mu }_{\mathrm{L}})=(1,0)$. (a) Analysis for those $\xi$ points in or near the clogged region: ${\xi }_l=-3+6l/95(15\le l\le 24)$. Symbols show the numerical data, and their linear fittings are plotted by dashed lines. (b) Magnified plot of the lower part in (a). Light cones ${\xi }_a^{-}$ in this $\xi$ range are also marked roughly at the place corresponding to their values. Notice ${\xi }_0<\cdots <{\xi }_{24}<{\xi }_a^{-}$ for all $a>0$.

Figure 9

Convergence check of energy density in the limit of the cutoff $a_c\to \infty$. (a) Analysis for those $\xi$ points in or near the clogged region: ${\xi }_l=-3+6l/95(15\le l\le 20)$. Symbols show the numerical results, and their linear fittings are plotted by dashed lines. The initial condition is $({\beta }_{\mathrm{L}},{\mu }_{\mathrm{L}})=(1,0)$. (b) Singularity of energy density. Symbols are extrapolated values determined in (a). The dashed line is a fitting function $a\sqrt{\xi -b}$, where $a=1.21$ and $b=-2.02$.

Figure 10

Effects of varying the chemical potential $-4\le {\mu }_L\le 0$ on quasiparticle contributions in (a) $n\left(\xi \right)$, (b) $j_n\left(\xi \right)$, (c) $e\left(\xi \right)$, and (d) $j_e\left(\xi \right)$. The initial temperature is fixed at ${\beta }_{\mathrm{L}}=1$. For each ${\mu }_{\mathrm{L}}$, the total value of density or current and the sum of bound state contributions are separately plotted.

Figure 11

Relation of particle density current $j_n\left(\xi \right)$ and energy current $j_e\left(\xi \right)$ plotted for $\xi ={\xi }_l=-3+6l/95,(l=0,1,\cdots ,95)$. The four panels correspond to different values of the initial temperature: ${\beta }_{\mathrm{L}}=$ (a) 0.5, (b) 1, (c) 1.5, and (d) 2. Nine data sets for $-10\le {\mu }_{\mathrm{L}}\le 0$ are plotted in each panel. The points $|{\xi }_l|\le 2$ are within some light cones, and the corresponding filling functions differ from those in the initial states.

Figure 12

Stationary values of two currents [(a), (c), and (e)] $j_n(\xi =0)$ and [(b), (d), and (f)] $j_e(\xi =0)$, plotted as a function of $\mathrm{\Delta }n=n^{\mathrm{L}}-n^{\mathrm{R}}=n^{\mathrm{L}}$. (a) and (b) are the numerical data for the Hubbard model with $u=2$. (c) and (d) are the results of the noninteracting systems ($u=0$) calculated by the GHD theory. Note that the half filling case ($\mathrm{\Delta }n=1$) is special, and $j_n\left(0\right)=2/\pi$ independent of ${\beta }_{\mathrm{L}}$. (e) and (f) show the comparison in the small $\mathrm{\Delta }n$ range with the noninteracting systems. Symbols show the data for $u=2$, while lines are the results for $u=0$.

Figure 13

Chemical potential dependence of ratio of the two stationary currents $j_e\left(0\right)$ and $j_n\left(0\right)$. Five curves correspond to different temperatures $0.5\le {\beta }_{\mathrm{L}}\le 5$.