Large-Scale Description of Interacting One-Dimensional Bose Gases: Generalized Hydrodynamics Supersedes Conventional Hydrodynamics

The theory of generalized hydrodynamics (GHD) was recently developed as a new tool for the study of inhomogeneous time evolution in many-body interacting systems with infinitely many conserved charges. In this Letter, we show that it supersedes the widely used conventional hydrodynamics (CHD) of one-dimensional Bose gases. We illustrate this by studying “nonlinear sound waves” emanating from initial density accumulations in the Lieb-Liniger model. We show that, at zero temperature and in the absence of shocks, GHD reduces to CHD, thus for the first time justifying its use from purely hydrodynamic principles. We show that sharp profiles, which appear in finite times in CHD, immediately dissolve into a higher hierarchy of reductions of GHD, with no sustained shock. CHD thereon fails to capture the correct hydrodynamics. We establish the correct hydrodynamic equations, which are finite-dimensional reductions of GHD characterized by multiple, disjoint Fermi seas. We further verify that at nonzero temperature, CHD fails at all nonzero times. Finally, we numerically confirm the emergence of hydrodynamics at zero temperature by comparing its predictions with a full quantum simulation performed using the NRG-TSA-abacus algorithm. The analysis is performed in the full interaction range, and is not restricted to either weak- or strong-repulsion regimes.

Figure 1

(a) Left: density profile of LL gas suddenly released from a Gaussian potential $V(x)=-5e^{-[(x/50){]}^2}-1$. Right: Corresponding Fermi points ${\theta }_j^{\pm }(x)$. Initially, there are only $k=2$ Fermi points, but after the shock at $t\simeq 37$, there is a region where the red curve is multivalued, corresponding to $k=4$ Fermi points. (b) Same setup at finite temperature: the initial state is obtained from LDA at temperature $T=1$. After finite time, CHD quantitatively differs from GHD. Moreover, at a later time $t\simeq 35$, CHD has a shock (see the Supplemental Material [31]); in contrast, GHD has no shock.

Figure 2

The density profile under the evolution with $2m=1$, $c=1$, $U=0.03$, $a=1/576$, and a choice of ${\mu }_{\infty }$ such that there are $N=48$ particles in the ground state. We use periodic boundary conditions. Points show NRG-TSA-abacus data, full line the GHD simulation, and dotted line a linear sound wave approximation.