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arXiv:1407.4710 · Physical Review A

Temperature-driven crossover in the Lieb-Liniger model

Andreas Klümper, Ovidiu I. Pâţu

Published Nov 21, 2014• 12 citations
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Abstract

The large-distance behavior of the density-density correlation function in the Lieb-Liniger model at finite temperature is investigated by means of the recently derived nonlinear integral equations characterizing the correlation lengths. We present extensive numerical results covering all the physical regimes from weak to strong interaction and all temperatures. We find that the leading term of the asymptotic expansion becomes oscillatory at a critical temperature which decreases with the strength of the interaction. As we approach the Tonks-Girardeau limit the asymptotic behavior becomes more complex with a double crossover of the largest and next-largest correlation lengths. The crossovers exist only for intermediate couplings and vanish for $\ensuremath{\gamma}=0$ and $\ensuremath{\infty}$.

DOI
10.1103/physreva.90.053626

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The large-distance behavior of the density-density correlation function in the Lieb-Liniger model at finite temperature is investigated by means of the recently derived nonlinear integral equations characterizing the correlation lengths. We present extensive numerical results covering all the physical regimes from weak to strong interaction and all temperatures. We find that the leading term of the asymptotic expansion becomes oscillatory at a critical temperature which decreases with the strength of the interaction. As we approach the Tonks-Girardeau limit the asymptotic behavior becomes more complex with a double crossover of the largest and next-largest correlation lengths. The crossovers exist only for intermediate couplings and vanish for \ensuremathγ=0\ensuremath{\gamma}=0\ensuremathγ=0 and \ensuremath∞\ensuremath{\infty}\ensuremath∞.
This PhD thesis is devoted to the study of the long-time and large-distance asymptotic behaviour of the dynamical field--field correlation function of the Lieb--Liniger model in the limit of infinite repulsion (the one-dimensional impenetrable Bose gas). Extending previous works we consider a large class of non-thermal equilibrium conditions in addition to the case of thermal equilibrium that had been studied before. Our starting point is an exact representation of the field--field correlation function as the Fredholm determinant of an integrable integral operators. The integrable operator depends parametrically on time t, distance x and on the filling fraction, which is a function that characterizes the thermal or non-thermal equilibrium conditions. Our work is based on the rigorous asymptotic analysis, using Riemann--Hilbert techniques, of a more general integral operator that depends on two more functional parameters. The long-time and large-distance behaviour of the field--field correlation function is derived for two classes of filling fractions. These classes are characterized by the number of poles on the real axis (a generalization of Fermi points) that, together with the unique saddle point, contribute to the asymptotic expansion. For each class, we derive the long-time and large-distance asymptotic behaviour as a series in x^(-1/2) as x and t go to infinity for a fixed ratio x / t. We provide explicit closed-form expressions for the leading and sub-leading terms, logarithmic corrections, and overall constants in terms of special functions and simple integrals. For the impenetrable Bose gas in thermal equilibrium, we verify the derived asymptotic expansions by comparing them with the existing results in the literature and with numerical data. Our analysis is based solely on the analytic properties of the functions parametrizing the kernel of the integrable integral operator. This generalized approach will enable future applications to the long-time and large-distance asymptotic analysis of the Lieb--Liniger model with finite coupling constant.
We address the problem of calculating finite-temperature response functions of an experimentally relevant low-dimensional, strongly correlated system: the integrable one-dimensional Bose gas with a repulsive $\ensuremath{\delta}$-function interaction (the Lieb-Liniger model). Focusing on the dynamical density-density function, we present a Bethe ansatz-based method allowing for its accurate evaluation in finite but large systems, over broad ranges of momenta, frequencies, temperatures, and interaction parameters which are difficult to access using other methods. This allows us to quantify the reshaping of the zero-temperature critical behavior by thermal fluctuations, in experimentally accessible regimes.
We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.
We study the local correlations in the super Tonks-Girardeau gas, a highly excited, strongly correlated state obtained in quasi-one-dimensional Bose gases by tuning the scattering length to large negative values using a confinement-induced resonance. Exploiting a connection with a relativistic field theory, we obtain results for the two-body and three-body local correlators at zero and finite temperature. At zero temperature, our result for the three-body correlator agrees with the extension of the results of Cheianov et al. [Phys. Rev. A 73, 051604(R) (2006)], obtained for the ground state of the repulsive Lieb-Liniger gas, to the super-Tonks-Girardeau state. At finite temperature, we obtain that the three-body correlator has a weak dependence on temperature up to the degeneracy temperature TD{T}_{\mathrm{D}}TD​. We also find that, for temperatures larger than TD{T}_{\mathrm{D}}TD​, the values of the three-body correlator for the super-Tonks-Girardeau gas and the corresponding repulsive Lieb-Liniger gas are rather similar, even for relatively small couplings.