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

Abstract

We derive contour integral formulas for the real space propagator of the spin-$\frac{1}{2}$ 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.

44 pages, v2: a minor comment added, v3: major revision (we added a second method, and also an application of the results), v4: minor modifications, references and acknowledgment added

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• Extreme boundary conditions and random tilings
• Integral formulas of ASEP and $q$-TAZRP on a ring
• Exact time evolution formulae in the XXZ spin chain with domain wall initial state
• Bethe Ansatz, Quantum Circuits, and the F-basis
• Integral formula for the propagator of the one-dimensional Hubbard model