Boundary overlap in the open XXZ spin chain

TL;DR

This work computes the overlaps between ground states of the open XXZ spin chain before and after a boundary quench (changing one boundary field $h_-$) in the gapped massive antiferromagnetic regime $\Delta>1$. Using the boundary algebraic Bethe Ansatz, Slavnov-type determinant formulas for scalar products, and Gaudin extraction to convert determinant ratios into a generalized Cauchy determinant, the authors derive a compact product expression for the overlap in the thermodynamic limit. The result depends on whether the ground-state Bethe roots are all real or include a boundary root, and on the parity of the chain length, yielding explicit $q$-Pochhammer and theta-function factors; all finite-size corrections are exponentially small in $L$. This first step toward boundary-quench dynamics opens the way to studying excited states and boundary-driven dynamics via integrability-based overlaps.

Abstract

In this paper we compute the overlaps of the ground states for the open spin chains after a change of one of the boundary magnetic fields. It can be considered as the first step toward the study of the boundary quench problem: behaviour of an open spin chain after an abrupt change of one boundary magnetic field.

Figures (5)

• Figure 1: contour $\Gamma_\epsilon$
• Figure 2: The overlap via exact diagonalisation for a chain of length $L=17$ compared to the ABA exact result at the thermodynamic limit obtained in section \ref{['sec-odd-L']}. Here $\zeta=1.8$, $h_+=-1$, $h_1^-=0$, and the value of the overlap $S(\{\lambda\},\{\mu\})$ is plotted for different values of $h_2^-$: when $h_2^- < -h_+=1$, we are in the case \ref{['sec-odd-case-1']} of section \ref{['sec-odd-L']}, and the overlap is given by \ref{['overlap-odd-1']}; when $h_2^- > -h_+=1$, we are in the case \ref{['sec-odd-case-2']} of section \ref{['sec-odd-L']} and the overlap vanishes, see \ref{['overlap-odd-2']}.
• Figure 3: The overlap via exact diagonalisation for a chain of length $L=18$ compared to the ABA exact result at the thermodynamic limit obtained in section \ref{['sec-even-L']}. Here $\zeta=1.5$, $h_+=2$, $h_1^-=-1$, and the value of the overlap $S(\{\lambda\},\{\mu\})$ is plotted for different values of $h_2^-$ for $h_2^-<h_\text{cr}^{(1)}$: we are here in the configuration \ref{['case-even-1-ii']} of the case \ref{['case-even-1']} from section \ref{['sec-even-L']}, and the overlap is given by \ref{['overlap-even-1']}.
• Figure 5: The overlap via exact diagonalization for different chain sizes compared to the exact result from ABA at the thermodynamic limit. Here, the values of the overlap are plotted for $\zeta=1.5$, $h_+=2$ and $h_1^-=-1$. They are also presented in a table to show the rapid (exponential) convergence of the numerical results to the analytic values. In this plot, we are in the exact same configuration as in figure \ref{['bolbol']}.
• Figure 6: The overlap via exact diagonalization for different chain sizes compared to the exact result from ABA at the thermodynamic limit. Here, the values of the overlap are plotted for $\zeta=1.8$, $h_+=0$ and $h_1^-=1$. They are also presented in a table to show the rapid (exponential) convergence of the numerical results to the analytic values. In this plot, we are in the same exact configuration as in figure \ref{['fig:plot2']}.