Quantum generalized hydrodynamics of the Tonks–Girardeau gas: density fluctuations and entanglement entropy

Abstract

We apply the theory of Quantum Generalized Hydrodynamics (QGHD) introduced in [Phys. Rev. Lett. 124, 140603 (2020)] to derive asymptotically exact results for the density fluctuations and the entanglement entropy of a one-dimensional trapped Bose gas in the Tonks-Girardeau (TG) or hard- core limit, after a trap quench from a double well to a single well. On the analytical side, the quadratic nature of the theory of QGHD is complemented with the emerging conformal invariance at the TG point to fix the universal part of those quantities. Moreover, the well-known mapping of hard-core bosons to free fermions, allows to use a generalized form of the Fisher-Hartwig conjecture to fix the non-trivial spacetime dependence of the ultraviolet cutoff in the entanglement entropy. The free nature of the TG gas also allows for more accurate results on the numerical side, where a higher number of particles as compared to the interacting case can be simulated. The agreement between analytical and numerical predictions is extremely good. For the density fluctuations, however, one has to average out large Friedel oscillations present in the numerics to recover such agreement.

Entanglement Hamiltonian during a domain wall melting in the free Fermi chain

“…To our best knowledge, a similar result is only known for the dynamics of a driven Tonks-Girardeau gas in harmonic traps [16]. A natural extension of the proposed method for generic quench settings is to join it with quantum generalised hydrodynamics to trace backward in time the quantum correlations, similarly to what recently done for the entanglement entropies and spectrum [21][22][23][24][25]. An interesting application of our result would be to discretise the field theoretical result (52) to engineer both numerically and experimentally the hydrodynamic entanglement Hamiltonian of the domain wall melting, on the lines of Refs.…”

“…A more general result for split Fermi seas can be found in Refs. [21,24,25]. Plugging this expression in Eq.…”

“…Roughly in parallel, the dynamical problem was also studied with the goal of completing such asymptotic approach with a quantum hydrodynamic theory. To date, thanks to the re-quantisation of the semi-classical evolution established by the generalised hydrodynamics [19,20], it has been possible to obtain promising results for the large-scale dynamics of the entanglement entropy [21][22][23][24][25] and to reproduce the numerical data obtained for the microscopic models with an impressive precision. Although some aspects of this emerging quantum hydrodynamic theory still remain to be clarified [26], it is a matter of fact that it provides asymptotically exact results for the entanglement dynamics, extending our analytical capabilities far beyond the current computational and conceptual limits set by standard lattice formulations.…”

Interplay between transport and quantum coherences in free fermionic systems

“…However, these states are not usually among the most natural choices as we will see with several examples, in particular at equilibrium. However, we note that they can easily appear in out-of-equilibrium settings: see for instance, inversion of population in pumped cavities [59] or quenches from the double(multi)-well potential [19,60].…”

Phase-space stochastic quantum hydrodynamics for interacting Bose gases

“…Including these higher order conserved quantities in its description allows GHD to simulate scenarios where CHD would otherwise suffer from the notorious gradient catastrophe (or derivative discontinu-ity) problem, and provides a more accurate description at finite-temperatures [3]. Standard GHD is capable of simulating ultra-cold Bose gases at any interaction strength, and since its discovery it has been extended to address (among other scenarios) near-integrable systems [6,21] and to include quantum fluctuations [22,23]. Yet, to date, these descriptions remain restricted to longwavelength excitations and are unable to capture quantum interference and other important short-wavelength phenomena such as the ones present in dispersive quantum shock waves [13,14,[24][25][26][27][28][29][30][31][32].…”