Solvable Hydrodynamics of Quantum Integrable Systems

The conventional theory of hydrodynamics describes the evolution in time of chaotic many-particle systems from local to global equilibrium. In a quantum integrable system, local equilibrium is characterized by a local generalized Gibbs ensemble or equivalently a local distribution of pseudomomenta. We study time evolution from local equilibria in such models by solving a certain kinetic equation, the “Bethe-Boltzmann” equation satisfied by the local pseudomomentum density. Explicit comparison with density matrix renormalization group time evolution of a thermal expansion in the XXZ model shows that hydrodynamical predictions from smooth initial conditions can be remarkably accurate, even for small system sizes. Solutions are also obtained in the Lieb-Liniger model for free expansion into vacuum and collisions between clouds of particles, which model experiments on ultracold one-dimensional Bose gases.

Figure 1

Convergence of the method as $dt\to 0$ for an initial Gaussian temperature profile given by Eq. (9) with ${\beta }_0=2.0$, ${\beta }_M=0.1$, and $L=8$ in the XXZ spin chain at $\mathrm{\Delta }=\frac{1}{2}$. The numerical solution at time $t=20$ is rapidly converging as $dt$ is decreased, with $dt\sim 10$ being already quite accurate. Insets: Top: Relative error in total energy, showing energy conservation as $dt\to 0$. Bottom: Close-up of the main figure showing convergence.

Figure 2

Time evolution of energy density from various initial temperature profiles at $t=0$ for the XXZ spin chain at $\mathrm{\Delta }=\frac{1}{2}$. Left: High temperature Gaussian initial state. Middle: Low temperature Gaussian initial state. Right: Two-reservoir setup with temperatures ${\beta }_0=2$, ${\beta }_M=1$ connected through a $\tanh (x/L)$ interpolation with $L=8$. Inset: Energy current at $x=0$ showing the approach to a nonequilibrium steady state at long times.

Figure 3

Hydrodynamic evolutions in the Lieb-Liniger model with interaction strength $c=1$: the top panel depicts free expansion initial conditions ($dt=0.1$) and the lower panel models a collision between clouds of bosons with opposite initial momenta ($dt=0.05$).