Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium

Understanding the general principles underlying strongly interacting quantum states out of equilibrium is one of the most important tasks of current theoretical physics. With experiments accessing the intricate dynamics of many-body quantum systems, it is paramount to develop powerful methods that encode the emergent physics. Up to now, the strong dichotomy observed between integrable and nonintegrable evolutions made an overarching theory difficult to build, especially for transport phenomena where space-time profiles are drastically different. We present a novel framework for studying transport in integrable systems: hydrodynamics with infinitely many conservation laws. This bridges the conceptual gap between integrable and nonintegrable quantum dynamics, and gives powerful tools for accurate studies of space-time profiles. We apply it to the description of energy transport between heat baths, and provide a full description of the current-carrying nonequilibrium steady state and the transition regions in a family of models including the Lieb-Liniger model of interacting Bose gases, realized in experiments.

Popular Summary

When many particles interact, intricate phenomena often result that are hard to predict solely from knowledge of underlying physical laws. This fact is particularly true if quantum mechanics plays an important role in the interaction, for instance at very low temperatures. Surprisingly, in many cases, one can describe the time evolution of such complex quantum systems simply using the physics of classical fluids. This ability has been extremely useful in recent studies of quantum physics out of equilibrium. However, there is a class of systems where little is yet known concerning such emergent dynamics: integrable systems. Integrability—the presence of a large hidden symmetry—has already been observed in experiments of ultracold atoms to preclude ordinary thermalization. Here, we develop a new hydrodynamic theory that accounts for integrability in such ultracold atom systems and in many other systems. We use this theory to solve a long-standing problem in the nonequilibrium physics of integrable systems.

Our work focuses on far-from-equilibrium states achieved using heat baths. Assuming local entropy maximization, we derive, purely from the tenets of hydrodynamics, new dynamical equations representing the time evolution of the infinity of conserved densities and currents, applicable to a large integrability class including the Lieb-Liniger model realizable in experiments. Using these equations, we calculate exact currents and space-time profiles far from equilibrium that emerge after two independent baths are put into contact.

We expect that our formalism will pave the way for tackling a variety of transport problems in integrable models, including the release of one-dimensional Bose gases from confining traps and dynamics within a smooth potential landscapes.

Figure 1

The partitioning protocol. With ballistic transport, a current emerges after a transient period. Dotted lines represent different values of $\xi =x/t$. If a maximal velocity exists (e.g., due to the Lieb-Robinson bound), initial reservoirs are unaffected beyond it (light-cone effect). The steady state lies at $\xi =0$.

Figure 2

The functions $\mathcal{j}(\xi )$ (dots) and $\mathcal{e}(\xi )$ (squares) for ${\beta }_L=1$ and ${\beta }_R=30$ in the sinh-Gordon model.

Figure 3

The functions ${\beta }_L^2\mathcal{j}(\xi >0)$ for ${\beta }_R=30{\beta }_L$ and ${\beta }_L={10}^{-p}$, with $p=0$ (stars), 1 (triangles), 2 (inverted triangles), 3 (squares), and 4 (circles). The continuous bold line represents the conformal value ${\beta }_L^2\mathcal{j}(\xi )=\frac{\pi }{12}(1-\frac{1}{900})$ which, as expected, is reached at high temperatures. Dashed curves are interpolations.

Figure 4

Verification of the inequalities Eq. (3) in the sinh-Gordon model. Displayed are the functions ${\beta }_L^2{\mathcal{j}}^{\mathrm{sta}}$ (circles), ${\beta }_L^2({\mathcal{e}}^L-{\mathcal{e}}^R)/2$ (triangles), and ${\beta }_L^2({\mathcal{k}}^L-{\mathcal{k}}^R)/2$ (squares).

Figure 5

Energy current in the Lieb-Liniger model for low temperatures $\lambda =3$ and chemical potentials $\mu =3$ (circles), $\mu =6$ (squares), and $\mu =10$ (triangles). The CFT value $\frac{\pi }{12}(1-\frac{1}{25})$ (bold horizontal line) is reached for high values of $\mu$. By plotting the currents against $\xi /v_F$, we observe the collapse of the various curves, which becomes better as $\mu$ increases. The regions where plateaus emerge are roughly $\xi /v_F\in [-1,1]$ with $v_F\approx 2.5$, 2.9, 3.6.

Figure 6

Energy current in the Lieb-Liniger model for low temperatures, small coupling, and negative chemical potential (circles). The dashed curve represents the current [Eq. (46)] for the same temperatures and chemical potential.

Figure 7

Energy current in the Lieb-Liniger model for low temperatures, large coupling, and chemical potential $\mu =6$ (circles). Local stationary points occur at ${\alpha }_{L,R}=0$, that is, $\xi =\pm \sqrt{2\mu }=\pm 3.46$ (the Fermi velocity). The dashed curve represents the current [Eq. (47)] for the same temperatures and chemical potential, whose profile is not dissimilar to the plots shown in Fig. 5. As before, the bold horizontal line is the CFT value $\frac{\pi }{12}(1-\frac{1}{25})$. The agreement is extremely good.

Figure 8

A characteristic profile of the Lieb-Liniger particle density for $T_{L,R}\ll \mu$, $\lambda =3$, and $\mu =6$. The local maxima and minima are located around $\xi =\pm v_F$. The dashed curve is an interpolation.

Figure 9

A characteristic profile of the Lieb-Liniger particle current for $T_{L,R}\ll \mu$, $\lambda =3$, and $\mu =6$. The local maxima and minima are located around $\xi =\pm v_F$. The dashed curve is an interpolation.

Figure 10

Effective velocity in the sinh-Gordon model for $\xi =0$. Displayed are the effective velocity $v^{\mathrm{eff}}(\theta )$ (blue line) and the bare relativistic velocity $\tanh \theta$ (red line).