Quantum Generalized Hydrodynamics

Physical systems made of many interacting quantum particles can often be described by Euler hydrodynamic equations in the limit of long wavelengths and low frequencies. Recently such a classical hydrodynamic framework, now dubbed generalized hydrodynamics (GHD), was found for quantum integrable models in one spatial dimension. Despite its great predictive power, GHD, like any Euler hydrodynamic equation, misses important quantum effects, such as quantum fluctuations leading to nonzero equal-time correlations between fluid cells at different positions. Focusing on the one-dimensional gas of bosons with delta repulsion, and on states of zero entropy, for which quantum fluctuations are larger, we reconstruct such quantum effects by quantizing GHD. The resulting theory of quantum GHD can be viewed as a multicomponent Luttinger liquid theory, with a small set of effective parameters that are fixed by the thermodynamic Bethe ansatz. It describes quantum fluctuations of truly nonequilibrium systems where conventional Luttinger liquid theory fails.

Figure 1

(a) Zero-entropy GHD describes the motion of the Fermi contour ${\mathrm{\Gamma }}_t$, parametrized as in Eq. (3), which separates the regions in phase space where the Fermi factor $n(x,\theta )$ is one (orange) or zero (white) at a given time $t$. In any small interval $[x,x+dx]$ the fluid is in a state called split Fermi sea [69, 70, 71, 72] labeled by Fermi rapidities ${\theta }_1<{\theta }_2<\dots <{\theta }_{2q}$; the number of fluid components $q$ is a piecewise constant function of $x$ and $t$. (b) In this Letter the contour ${\mathrm{\Gamma }}_t$ is allowed to have quantum fluctuations around the classical solution to the zero-entropy GHD equations (4a) and (4b). The quantum fluctuations are captured by a chiral boson with density $\delta \widehat{\rho }(s)$ living along the contour.

Figure 2

Quantum quench from double to single well in the 1D Bose gas with delta repulsion. We compare the predictions of GHD and QGHD (orange curves) to time-dependent DMRG simulation for $N=10$ (light gray) and $N=20$ particles (dark gray). First row: Fermi contour evolved with GHD. Second row: Density profile predicted by GHD, compared with DMRG. Third and fourth row: Connected density-density correlator $⟨\widehat{\rho }(x)\widehat{\rho }(x_0)⟩$ predicted by QGHD and compared with DMRG, for two different positions $x_0$. Each row shows the corresponding quantity as a function of the spatial coordinate $x$, at different times, expressed as a fraction of the period $\tau$ (from $t=0$ in the first column to $t=0.3\tau$ in the last). For the DMRG simulation we work with particles on a lattice at very low density. The parameters are as follows: repulsion strength $\overline{g}=0.1$; $L=800$ lattice sites; number of particles $N=10$, 20; $\hslash =30/N$; prequench potential $V_0(x)=(x/L{)}^4-0.12(x/L{)}^2$; postquench potential $V(x)={\omega }^2{x}^2/2$ with $\omega =0.3/L$ (and period $\tau =2\pi /\omega$). The dimensionless Lieb parameter $\gamma =\overline{g}/\hslash \rho$ is of order 1, so we are far from both the Gross-Pitaevski limit and the Tonks-Girardeau limit.