Sine-Gordon quantum field theory on the half-line with quantum boundary degrees of freedom

TL;DR

This paper explores the quantum sine-Gordon model on a half-line with dynamical boundary degrees of freedom, demonstrating quantum integrability, constructing non-local charges, and proposing reflection matrices with duality properties.

Contribution

It extends classical boundary conditions to the quantum level, explicitly constructs non-local conserved charges, and conjectures reflection matrices for solitons and bound states.

Findings

Classical boundary conditions are preserved at quantum level.

Explicit construction of non-local conserved charges.

Proposed soliton and bound state reflection matrices.

Abstract

The sine-Gordon model on the half-line with a dynamical boundary introduced by Delius and one of the authors is considered at quantum level. Classical boundary conditions associated with classical integrability are shown to be preserved at quantum level too. Non-local conserved charges are constructed explicitly in terms of the field and boundary operators. We solve the intertwining equation associated with a certain coideal subalgebra of $U_q(\hat{sl_2})$ generated by these non-local charges. The corresponding solution is shown to satisfy quantum boundary Yang-Baxter equations. Up to an exact relation between the quantization length of the boundary quantum mechanical system and the sine-Gordon coupling constant, we conjecture the soliton/antisoliton reflection matrix and boundstates reflection matrices. The structure of the boundary state is then considered, and shown to be divided in two sectors. Also, depending on the sine-Gordon coupling constant a finite set of boundary bound states are identified. Taking the analytic continuation of the coupling, the corresponding boundary sinh-Gordon model is briefly discussed. In particular, the particle reflection factor enjoys weak-strong coupling duality.