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arXiv:2308.00486 · Physical Review A

Nonequilibrium dynamics in one-dimensional strongly interacting two-component gases

Ovidiu I. Pâţu

Published Nov 1, 2023• 3 citations
arXivHTMLPDF

Abstract

The derivation of determinant representations for the space-, time-, and temperature-dependent correlation functions of the impenetrable Gaudin-Yang model in the presence of a trapping potential is presented. These representations are valid in both equilibrium and nonequilibrium scenarios like the ones initiated by a sudden change of the confinement potential. In the equal-time case our results are shown to be equivalent to a multicomponent generalization of Lenard's formula from which Painlev\'e transcendent representations for the correlators can be obtained in the case of harmonic trapping and Dirichlet and Neumann boundary conditions. For a system in the quantum Newton's cradle setup the determinant representations allow for an exact numerical investigation of the dynamics and even hydrodynamization, which is outside the reach of generalized hydrodynamics or other approximate methods. In the case of a sudden change in the trap's frequency, we predict a many-body bounce effect, not present in the evolution of the density profile, which causes a nontrivial periodic narrowing of the momentum distribution with amplitude depending on the statistics of the particles.

DOI
10.1103/physreva.108.053304

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The derivation of determinant representations for the space-, time-, and temperature-dependent correlation functions of the impenetrable Gaudin-Yang model in the presence of a trapping potential is presented. These representations are valid in both equilibrium and nonequilibrium scenarios like the ones initiated by a sudden change of the confinement potential. In the equal-time case our results are shown to be equivalent to a multicomponent generalization of Lenard's formula from which Painlev'e transcendent representations for the correlators can be obtained in the case of harmonic trapping and Dirichlet and Neumann boundary conditions. For a system in the quantum Newton's cradle setup the determinant representations allow for an exact numerical investigation of the dynamics and even hydrodynamization, which is outside the reach of generalized hydrodynamics or other approximate methods. In the case of a sudden change in the trap's frequency, we predict a many-body bounce effect, not present in the evolution of the density profile, which causes a nontrivial periodic narrowing of the momentum distribution with amplitude depending on the statistics of the particles.
Analytical results on the correlation functions of strongly correlated many-body systems are rare in the literature and their importance cannot be overstated. We present determinant representations for the space-, time-, and temperature-dependent correlation functions of the strongly interacting one-dimensional Hubbard model in the presence of an external trapping potential. These representations are exact and valid in both equilibrium and nonequilibrium scenarios like the ones initiated by a sudden change of the confinement potential. In addition, they can be implemented numerically very easily, significantly outperforming other numerical approaches. As applications of our results we investigate the single-particle spectral functions of systems with harmonic trapping and show that dynamical quasicondensation occurs for both fermionic and bosonic spin-1/21/21/2 systems released from a Mott insulator state.
We address the problem of calculating the correlation functions of one-dimensional two-component gases with strong repulsive contact interactions. The model considered in this paper describes particles with fractional statistics and in appropriate limits reduces to the Gaudin-Yang model or the spinor Bose gas. In the case of impenetrable particles we derive a Fredholm determinant representation for the temperature-, time-, and space-dependent correlation functions which is very easy to implement numerically and constitutes the starting point for the analytical investigation of the asymptotics. Making use of this determinant representation and the solution of an associated Riemann-Hilbert problem we derive the low-energy asymptotics of the correlators in the spin-incoherent regime characterized by near ground-state charge degrees of freedom but a highly thermally disordered spin sector. The asymptotics present features reminiscent of spin-charge separation with the spin part exponentially decaying in space separation and oscillating with a period proportional to the statistics parameter, while the charge part presents scaling with anomalous exponents which cannot be described by any unitary conformal field theory. The momentum distribution and the Fourier transform of the dynamical Green's function are asymmetrical for arbitrary statistics, a direct consequence of the broken space-reversal symmetry. Due to the exponential decay the momentum distribution n(k)n(k)n(k) at zero temperature does not present algebraic singularities but the tails obey the universal decay ${lim}_{k\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}n(k)\ensuremath{\sim}C/{k}^{4}$ with the amplitude CCC given by Tan's contact. As a function of the statistics parameter the contact is a monotonic function reaching its minimum for the fermionic system and the maximum for the bosonic system.
One of quantum physics' fundamental, but largely unsolved, problems is the computation of the correlation functions in many-body systems. We address this problem in the case of one-dimensional spinor gases with repulsive contact interactions in the presence of a confining potential. We take advantage of the fact that in the strong-coupling limit, the wave function factorizes with the charge degrees of freedom expressed as a Slater determinant of spinless fermions and the spin sector described by a spin chain of Sutherland type with exchange coefficients that depend only on the trapping potential. This factorization is also present in the expressions for the correlation functions. Still, analytical and numerical investigations were hindered by the fact that the local exchange coefficients and the charge component of the correlators are expressed as $N\ensuremath{-}1$ multidimensional integrals, with NNN the number of particles, which are notoriously hard to compute using conventional methods. We introduce a different approach to calculating these integrals that is extremely simple, scales polynomially with the number of particles, and is several orders of magnitude faster than the previous methods reported in the literature. This allows us to investigate the static and dynamic properties, temperature dependence, and nonequilibrium dynamics for systems with a larger number of particles than previously considered and discover new phenomena. We show that contrary to natural expectations, the momentum distribution of strongly interacting trapped spinor gases becomes narrower as we increase the temperature and derive simple determinant representations for the correlators in the spin incoherent regime that is valid for both equilibrium and nonequilibrium situations. For homogeneous systems of impenetrable particles at zero temperature, we analytically compute the large distance asymptotics of the correlators, finding that the leading term of the asymptotic expansion is proportional to ${e}^{\ensuremath{-}x{k}_{F}ln\ensuremath{\kappa}/\ensuremath{\pi}}$, with $\ensuremath{\kappa}$ the number of components of the system, which precludes any singularity of the momentum distribution.
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.
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.