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Register a new userQuantum quench dynamics of the attractive one-dimensional Bose gas via the coordinate Bethe ansatz
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We use the coordinate Bethe ansatz to study the Lieb-Liniger model of a one-dimensional gas of bosons on a finite-sized ring interacting via an attractive delta-function potential. We calculate zero-temperature correlation functions for seven particles in the vicinity of the crossover to a localized solitonic state and study the dynamics of a system of four particles quenched to attractive interactions from the ideal-gas ground state. We determine the time evolution of correlation functions, as well as their temporal averages, and discuss the role of bound states in shaping the postquench correlations and relaxation dynamics.
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We use the coordinate Bethe ansatz to exactly calculate matrix elements between eigenstates of the Lieb-Liniger model of one-dimensional bosons interacting via a two-body delta-potential. We investigate the static correlation functions of the zero-temperature ground state and their dependence on interaction strength, and analyze the effects of system size in the crossover from few-body to mesoscopic regimes for up to seven particles. We also obtain time-dependent nonequilibrium correlation functions for five particles following quenches of the interaction strength from two distinct initial states. One quench is from the non-interacting ground state and the other from a correlated ground state near the strongly interacting Tonks-Girardeau regime. The final interaction strength and conserved energy are chosen to be the same for both quenches. The integrability of the model highly constrains its dynamics, and we demonstrate that the time-averaged correlation functions following quenches from these two distinct initial conditions are both nonthermal and moreover distinct from one another.
The discrete polymer model with random Boltzmann weights with homogeneous inverse gamma distribution, introduced by Seppalainen, is studied in the case of a polymer with one fixed and one free end. The model with two fixed ends has been integrated by Thiery and Le Doussal, using coordinate Bethe Ansatz techniques and an analytic-continuation prescription. The probability distribution of the free energy has been obtained through the replica method, even though the moments of the partition sum do not exist at all orders due to the fat tail in the distribution of Boltzmann weights. To extend this approach to the polymer with one free end, we argue that the contribution to the partition sums in the thermodynamic limit is localised on parity-invariant string states. This situation is analogous to the case of the continuum polymer with one free end, related to the Kardar--Parisi--Zhang equation with flat boundary conditions and solved by Le Doussal and Calabrese. The expansion of the generating function of the partition sum in terms of numbers of strings can also be transposed to the log-gamma polymer model, with the induced Fredholm determinant structure. We derive the large-time limit of the rescaled cumulative distribution function, and relate it to the GOE Tracy--Widom distribution. The derivation is conjectural in the sense that it assumes completeness of a family of string states (and expressions of their norms already used in the fixed-end problem) and extends heuristically the order of moments of the partition sum to the complex plane.
We investigate the response of a one-dimensional Bose gas to a slow increase of its interaction strength. We focus on the rich dynamics of equal-time single-particle correlations treating the Lieb-Liniger model within a bosonization approach and the Bose-Hubbard model using the time-dependent density-matrix renormalization group method. For short distances, correlations follow a power-law with distance with an exponent given by the adiabatic approximation. In contrast, for long distances, correlations decay algebraically with an exponent understood within the sudden quench approximation. This long distance regime is separated from an intermediate distance one by a generalized Lieb-Robinson criterion. At long times, in this intermediate regime, bosonization predicts that single-particle correlations decay following a stretched exponential. This latter regime is unconventional as, for one-dimensional interacting systems, the decay of single-particle correlations is usually algebraic within the Luttinger liquid picture. We develop here an intuitive understanding for the propagation of correlations, in terms of a generalized light-cone, applicable to a large variety of systems and quench forms.
We investigate the lowest scattering state of one-dimensional Bose gas with attractive interactions trapped in a hard wall trap. By solving the Bethe ansatz equation numerically we determine the full energy spectrum and the exact wave function for different attractive interaction parameters. The resultant density distribution, momentum distribution, reduced one body density matrix and two body correlation show that the decreased attractive interaction induces rich density profiles and specific correlation properties in the weakly attractive Bose gas.
We extend the exact periodic Bethe Ansatz solution for one-dimensional bosons and fermions with delta-interaction and arbitrary internal degrees of freedom to the case of hard wall boundary conditions. We give an analysis of the ground state properties of fermionic systems with two internal degrees of freedom, including expansions of the ground state energy in the weak and strong coupling limits in the repulsive and attractive regimes.
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