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arXiv:2209.00700 · Physical review. B./Physical review. B

Generalized hydrodynamics of the repulsive spin- 12 Fermi gas

Stefano Scopa, Pasquale Calabrese, Lorenzo Piroli

Published Oct 31, 2022• 15 citations
arXivHTMLPDF

Abstract

We study nonhomogeneous quantum quenches in a one-dimensional gas of repulsive spin-$1/2$ fermions, as described by the integrable Yang-Gaudin model. By means of generalized hydrodynamics (GHD), we analyze in detail the real-time evolution following a sudden change of the confining potential. We consider in particular release protocols and trap quenches, including a version of the quantum Newton's cradle. At zero temperature, we employ a simplified phase-space hydrodynamic picture to characterize the dynamics of the particle- and spin-density profiles. Away from zero temperatures, we perform a thorough numerical study of the GHD equations, and provide quantitative predictions for different values of the temperature, external magnetic field, and chemical potential. We highlight the qualitative features arising due to the multicomponent nature of the elementary excitations, discussing in particular effects of spin-charge separation and dynamical polarization.

DOI
10.1103/physrevb.106.134314

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We study nonhomogeneous quantum quenches in a one-dimensional gas of repulsive spin-1/21/21/2 fermions, as described by the integrable Yang-Gaudin model. By means of generalized hydrodynamics (GHD), we analyze in detail the real-time evolution following a sudden change of the confining potential. We consider in particular release protocols and trap quenches, including a version of the quantum Newton's cradle. At zero temperature, we employ a simplified phase-space hydrodynamic picture to characterize the dynamics of the particle- and spin-density profiles. Away from zero temperatures, we perform a thorough numerical study of the GHD equations, and provide quantitative predictions for different values of the temperature, external magnetic field, and chemical potential. We highlight the qualitative features arising due to the multicomponent nature of the elementary excitations, discussing in particular effects of spin-charge separation and dynamical polarization.
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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. It offers a full and efficient framework to capture the long-wavelengths nonequilibrium dynamics in experimentally realizable one-dimensional systems, like Bose gases with delta repulsion. In this letter, we study the nonlinear sound waves emanating from initial density accumulations in the zero-temperature Bose gas. We numerically confirm the emergence of GHD by comparing its predictions with a full quantum simulation performed using the NRG-TSA-ABACUS algorithm. We prove that at zero temperature, GHD is equivalent to conventional Galilean hydrodynamics with nontrivial equations of state. We further predict, at long wavelengths and large densities, the appearance of sharp profiles in finite times, that immediately dissolve (no sustained shocks). The analysis is performed in the full interaction range, and is not restricted to either weak- or strong-repulsion regimes.
Generalized hydrodynamics (GHD) is a recent theoretical approach that is becoming a go-to tool for characterizing out-of-equilibrium phenomena in integrable and near-integrable quantum many-body systems. Here, we benchmark its performance against an array of alternative theoretical methods, for an interacting one-dimensional Bose gas described by the Lieb-Liniger model. In particular, we study the evolution of both a localized density bump and dip, along with a quantum Newton's cradle setup, for various interaction strengths and initial equilibrium temperatures. We find that GHD generally performs very well at sufficiently high temperatures or strong interactions. For low temperatures and weak interactions, we highlight situations where GHD, while not capturing interference phenomena on short lengthscales, can describe a coarse-grained behaviour based on convolution averaging that mimics finite imaging resolution in ultracold atom experiments. In a quantum Newton's cradle setup based on a double-well to single-well trap quench, we find that GHD with diffusive corrections demonstrates excellent agreement with the predictions of a classical field approach.
Following the removal of axial confinement the momentum distribution of a Tonks-Girardeau gas approaches that of a system of noninteracting spinless fermions in the initial harmonic trap. This phenomenon, called dynamical fermionization, has been experimentally confirmed in the case of the Lieb-Liniger model and theoretically predicted in the case of multicomponent systems at zero temperature. We prove analytically that for all spinor gases with strong repulsive contact interactions at finite temperature the momentum distribution after release from the trap asymptotically approaches that of a system of spinless fermions at the same temperature but with a renormalized chemical potential which depends on the number of components of the spinor system. In the case of the Gaudin-Yang model we check numerically our analytical predictions using the results obtained from a nonequilibrium generalization of Lenard's formula describing the time evolution of the field-field correlators.
By exploring a phase-space hydrodynamics description of one-dimensional free Fermi gas, we discuss how systems settle down to steady states described by the generalized Gibbs ensembles through quantum quenches. We investigate time evolutions of the fermions which are trapped in external potentials or a circle for a variety of initial conditions and quench protocols. We analytically compute local observables such as particle density and show that they always exhibit power-law relaxation at late times. We find a simple rule which determines the power law exponent. Our findings are, in principle, observable in experiments in a one-dimensional free Fermi gas or Tonk's gas (Bose gas with infinite repulsion).
Building on the Bethe ansatz and generalized hydrodynamics, we rigorously establish universal relations between Drude weights governing charge, spin, and energy transport and thermodynamic properties for the one-dimensional repulsive two-component continuum Fermi gas - the paradigmatic integrable Yang-Gaudin model. We analytically derive thermodynamic expressions for charge Drude weights: DnnD_{nn}Dnn​, DnmD_{nm}Dnm​, DneD_{ne}Dne​ (responses of particle, magnetization and energy currents to a chemical potential gradient) equal the particle density, magnetization, and temperature times entropy density, respectively. These relations constitute a hallmark of ballistic transport, and can be generalized to other 1D continuum integrable systems. Furthermore, we investigate spin Drude weight (DmmD_{mm}Dmm​, DmeD_{me}Dme​) at zero and low temperatures. They characterize the magnetization and energy current responses to a magnetic field gradient, revealing an essential spin-charge coupling feature in quantum transport. We find that the Drude weights DnmD_{nm}Dnm​, DmmD_{mm}Dmm​, and DmeD_{me}Dme​ across the μ-Hμ\text{-}Hμ-H plane remarkably map out the zero-temperature phase boundary. This work fills a critical gap in the understanding of transport within integrable quantum gases and furnishes a direct theoretical foundation for future ultracold-atom experiments.