The propagator of the finite XXZ spin-$\tfrac{1}{2}$ chain

Abstract

We derive contour integral formulas for the real space propagator of the spin-1 2 XXZ chain. The exact results are valid in any finite volume with periodic boundary conditions, and for any value of the anisotropy parameter. The integrals are on fixed contours, that are independent of the Bethe Ansatz solution of the model and the string hypothesis. The propagator is obtained by two different methods. First we compute it through the spectral sum of a deformed model, and as a by-product we also compute the propagator of the XXZ chain perturbed by a Dzyaloshinskii-Moriya interaction term. As a second way we also compute the propagator through a lattice path integral, which is evaluated exactly utilizing the so-called F -basis in the mirror (or quantum) channel. The final expressions are similar to the Yudson representation of the infinite volume propagator, with the volume entering as a parameter. As an application of the propagator we compute the Loschmidt amplitude for the quantum quench from a domain wall state.

Exact time evolution formulae in the XXZ spin chain with domain wall initial state

“…Another interesting project would be to make contact with exact results based on finite particle Bethe Ansatz (see [75] for the XXZ chain, [76,77] for the asymmetric exclusion process, and [71], for SSEP). The analogous real-time problem at ∆ = 1 is also very interesting, with connections to a classical Landau-Lifshitz equation [78][79][80] on the hydrodynamic side.…”

Non-Equilibrium Steady State of the Lieb-Liniger model: multiple-integral representation of the time evolved many-body wave-function

“…Next we evaluate the residue of the overlaps M s (λ; µ) at λ = µ from (30). As mentioned earlier, the N -dimensional pole of M s (λ; µ) at this point is due to the presence of the functions g(µ, λ) = ic/(µ − λ) in the matrix M N (µ; λ).…”

“…Our objective, in the present work, is to derive an exact multiple integral formula for K(µ; z; x, t) as a first step of this program. Similar multiple integral representations of many-body wave-functions in integrable models under special initial conditions have been also studied in [29,30], even though those works focus on the many-body propagator or Green's function which formally solves the time-dependent problem when the initial positions of the particles are given, while here we consider instead the case where the initial state is an eigenstate of the pre-quench Hamiltonian H 0 .…”

Exact time evolution formulae in the XXZ spin chain with domain wall initial state

“…Another interesting project would be to make contact with exact results based on finite particle Bethe ansatz (see [78,79] for the XXZ chain, [80,81] for the asymmetric exclusion process, and [74], for SSEP). The analogous real-time problem at Δ = 1 is also very interesting, with connections to a classical Landau-Lifshitz equation [82][83][84] on the hydrodynamic side.…”