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Register a new userExact out-of-equilibrium steady states in the semiclassical limit of the interacting Bose gas
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Added by
Giuseppe Del Vecchio Del Vecchio
Publication date
2020
fields
Physics
and research's language is
English
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We study the out-of-equilibrium properties of a classical integrable non-relativistic theory, with a time evolution initially prepared with a finite energy density in the thermodynamic limit. The theory considered here is the Non-Linear Schrodinger equation which describes the dynamics of the one-dimensional interacting Bose gas in the regime of high occupation numbers. The main emphasis is on the determination of the late-time Generalised Gibbs Ensemble (GGE), which can be efficiently semi-numerically computed on arbitrary initial states, completely solving the famous quench problem in the classical regime. We take advantage of known results in the quantum model and the semiclassical limit to achieve new exact results for the momenta of the density operator on arbitrary GGEs, which we successfully compare with ab-initio numerical simulations. Furthermore, we determine the whole probability distribution of the density operator (full counting statistics), whose exact expression is still out of reach in the quantum model.
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The exact solution of the 1D interacting mixed Bose-Fermi gas is used to calculate ground-state properties both for finite systems and in the thermodynamic limit. The quasimomentum distribution, ground-state energy and generalized velocities are obtained as functions of the interaction strength both for polarized and non-polarized fermions. We do not observe any demixing instability of the system for repulsive interactions.
We extend the notion of the Eigenstate Thermalization Hypothesis (ETH) to Open Quantum Systems governed by the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) Master Equation. We present evidence that the eigenstates of non-equilibrium steady state (NESS) density matrices obey a generalization of ETH in boundary-driven systems when the bulk Hamiltonian is non-integrable, just as eigenstates of Gibbs density matrices are conjectured to do in equilibrium. This generalized ETH, which we call NESS-ETH, can be used to obtain representative pure states that reproduce the expectation values of few-body operators in the NESS. The density matrices of these representative pure states can be further interpreted as weak solutions of the GKLS Master Equation. Additionally, we explore the validity and breakdown of NESS-ETH in the presence of symmetries, integrability and many-body localization in the bulk Hamiltonian.
We investigate the low temperature behaviour of the integrable 1D two-component spinor Bose gas using the thermodynamic Bethe ansatz. We find that for strong coupling the characteristics of the thermodynamics at low temperatures are quantitatively affected by the spin ferromagnetic states, which are described by an effective ferromagnetic Heisenberg chain. The free energy, specific heat, susceptibility and local pair correlation function are calculated for various physical regimes in terms of temperature and interaction strength. These thermodynamic properties reveal spin effects which are significantly different than those of the spinless Bose gas. The zero-field susceptibility for finite strong repulsion exceeds that of a free spin paramagnet. The critical exponents of the specific heat $c_v sim T^{1/2}$ and the susceptibility $chi sim T^{-2}$ are indicative of the ferromagnetic signature of the two-component spinor Bose gas. Our analytic results are consistent with general arguments by Eisenberg and Lieb for polarized spinor bosons.
Aiming at studying the emergence of Non-Equilibrium Steady States (NESS) in quantum integrable models by means of an exact analytical method, we focus on the Tonks-Girardeau or hard-core boson limit of the Lieb-Liniger model. We consider the abrupt expansion of a gas from one half to the entire confining box, a prototypical case of inhomogeneous quench, also known as geometric quench. Based on the exact calculation of quench overlaps, we develop an analytical method for the derivation of the NESS by rigorously treating the thermodynamic and large time and distance limit. Our method is based on complex analysis tools for the derivation of the asymptotics of the many-body wavefunction, does not make essential use of the effectively non-interacting character of the hard-core boson gas and is sufficiently robust for generalisation to the genuinely interacting case.
We discuss recent results on the relation between the strongly interacting one-dimensional Bose gas and a gas of ideal particles obeying nonmutual generalized exclusion statistics (GES). The thermodynamic properties considered include the statistical profiles, the specific heat and local pair correlations. In the strong coupling limit $gamma to infty$, the Tonks-Girardeau gas, the equivalence is with Fermi statistics. The deviation from Fermi statistics during boson fermionization for finite but large interaction strength $gamma$ is described by the relation $alpha approx 1 - 2/gamma$, where $alpha$ is a measure of the GES. This gives a quantitative description of the fermionization process. In this sense the recent experimental measurement of local pair correlations in a 1D Bose gas of $^{87}$Rb atoms also provides a measure of the deviation of the GES parameter $alpha$ away from the pure Fermi statistics value $alpha=1$. Other thermodynamic properties, such as the distribution profiles and the specific heat, are also sensitive to the statistics. They also thus provide a way of exploring fractional statistics in the strongly interacting 1D Bose gas.
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