Overlaps after a boundary quantum quench in the integrable XXZ spin chain
Charbel Abetian
The aim of this thesis is to provide an exact analytical approach for computing overlaps between eigenstates of the open XXZ chain before and after a boundary quench, that is, an abrupt change of one of the boundary magnetic fields. These overlaps constitute a fundamental building block for the description of the out-of-equilibrium dynamics of quantum systems and play a crucial role in understanding their post-quench evolution. We focus on the open XXZ spin-½ chain with parallel boundary magnetic fields in the massive antiferromagnetic regime, one of the simplest (non-trivially interacting) integrable models with non-periodic boundary conditions. This system can be interpreted as a quantum chain interacting with its environment through boundary degrees of freedom. The Quantum Inverse Scattering Method (QISM) and the Algebraic Bethe Ansatz (ABA) provide a strong theoretical framework for studying such quantities. The goal of our analysis is to derive, within this formalism, exact expressions for the overlaps in the thermodynamic limit. Using Slavnov’s determinant formula for scalar products of Bethe states, we first derive a determinant representation for the overlaps between pre- and post-quench eigenstates. By employing what we refer to as the Gaudin extraction method, we show that we can extract from this (typically encountered) ratio of determinants, a Cauchy-like determinant, suitable for taking the thermodynamic limit. We show that the overlap between the gapped ground states (i.e., states whose Bethe root configurations contain no holes) before and after the quench can be written in a product form, valid up to exponentially small corrections in the system size. This allows us to obtain a closed analytical expression for the overlap in the infinite-size limit, that depends only on the physical parameters of the model, namely, the anisotropy parameter and the boundary magnetic fields. We further verify this formula through direct comparison with numerical diagonalization results. We extend this analysis to a broader class of (ground/excited) states characterized by the presence of a finite number of holes in their Bethe root distributions. We show that the Gaudin extraction method yields once again a Cauchy-type structure, now, perturbed by a matrix of rank equal to the number of holes. In this case, the overlap factorizes into a prefactor that can be written in a product form, similar to the case of the ground state, multiplied by a finite determinant whose size is equal to the number of holes n. In the thermodynamic limit, we obtain an explicit expression for the overlap that depends on the anisotropy, the boundary magnetic fields, and the hole positions. The results presented here lay the groundwork for a broader analytical study of boundary-driven quenches in integrable quantum systems through ABA techniques.
Jury : P. Baseilhac, O. Castro-Alvaredo, J. De Nardis, N. Kitanine (co-directeur de thèse), K. Kozlowski, V. Pasquier, V. Terras (directrice de thèse)