Abstract
We study the density-density correlation function of the 1D Lieb–Liniger model and obtain an exact expression for the small momentum limit of the static correlator in the thermodynamic limit. We achieve this by summing exactly over the relevant form factors of the density operator in the small momentum limit. The result is valid for any eigenstate, including thermal and non-thermal states. We also show that the small momentum limit of the dynamic structure factors obeys a generalized detailed balance relation valid for any equilibrium state.
Highlights
One of the most celebrated and experimentally relevant one-dimensional many-body interacting model is the one governed by the Lieb-Liniger Hamiltonian [1] N H =− ∂ 2 xi + 2c δ(xi − x j) (1)i=1 i> j for a system of N bosons with positions {xi}Ni=1 on a line of length L, interacting point-wise with coupling constant c
In this paper we report a new progress in this direction and obtain an exact expression for the small momentum limit of the static structure factor, the Fourier transform of the two-point density-density correlation function, with the density operator defined as ρ(x) = δ(x − x j)
We show that a generalization of the detailed balance to non-thermal equilibrium states is possible in the small momentum regime of density-density correlations
Summary
One of the most celebrated and experimentally relevant one-dimensional many-body interacting model is the one governed by the Lieb-Liniger Hamiltonian [1]. Some examples are the dynamics induced by an abrupt change of the coupling constant (quantum quench) [25] or by a quick and strong Bragg pulse [26] It was shown in [27] that for an integrable model as (1) the steady state of a non-equilibrium unitary time evolution (when this is reached) can be represented with a single eigenstate, fixed by the expectation values of the conserved quantities of the Hamiltonian on the initial state. This is called the GeneralizedGibbs-Ensemble (GGE) saddle-point state [28,29,30] and it determines the expectation values of all the local observables at equilibrium [31,32,33,34,35], and allows to reconstruct the time evolution towards it [36,37,38,39] These developments open up new challenging problems of computing a correlation function on an arbitrary eigenstate of the system. Notable examples are the ground state of the Lieb-Liniger model [1] and the split Fermi sea [41]
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