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 hardcore 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.

Generalized-hydrodynamic approach to inhomogeneous quenches: correlations, entanglement and quantum effects

“…Note also, in connection to section 4, that evolution from such states would give zero entanglement entropy within the quasi-particle picture (which captures linear growth only). However, they still display a sublinear growth of entanglement entropy that can be exactly accessed with the method reviewed in the present section [277].…”

“…In particular, since in our quench both initial and trapping potentials are harmonic, the Wigner function is just an ellipse that rotates in time [277] (see figure 13(a)). This means that for any given position x, at any time t there are always no more than two Fermi points.…”

“…Even in the TG limit, and for evolutions from zero-entropy states, it is possible to find cases where the simplified description (130) does not hold. Indeed, it is sufficient to look at situations where more than two Fermi points occur [139,277]. This is the case, for instance, of a quench from a double-to a single-well potential, see figure 13(b) (note that this setup is also relevant from the experimental point of view, see, e.g.…”

“…The final result is exactly equation (134), where the flux Jacobian A ab can generically couple different chiral excitations. The Hamiltonian (134), together with the symplectic structure (133), defines the theory of generalized quantum hydrodynamics, which has been applied in both the free [277] and in the interacting [143] case to get results for correlations functions and entanglement entropy. Note that it takes the form of a multi-component, spatially inhomogeneous, time-dependent LL, but now, importantly, constructed on top of the exact profile (at the Euler scale) given by GHD, as derived from the full quantum initial model.…”

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. Nonetheless, they can easily appear in out-of-equilibrium settings: see for instance, inversion of population in pumped cavities [76] or quenches from the double(multi)-well potential [32,77]. So far, we have described our problem qualitatively.…”