Boundary energy and boundary states in integrable quantum field theories
A. LeClair, G. Mussardo, H. Saleur, S. Skorik
TL;DR
This work addresses the genuine Casimir problem in 1+1D integrable quantum field theories with boundaries, introducing an R-channel Thermodynamic Bethe Ansatz (TBA) for scalar theories and adapting the Destri–De Vega (DDV) approach for non-scalar cases such as sine-Gordon. The authors derive a closed-form expression for the boundary-ground-state energy and develop a formalism to compute boundary-state overlaps, normalizations, and scalar products, linking boundary data to the spectrum and partition functions. They verify ultraviolet limits against conformal field theory and Cardy’s results, and provide explicit finite-L partition functions for the critical Ising model with a boundary magnetic field, as well as results for boundary energies and S-matrices in $O(n)$ and minimal models. The work creates a unified framework connecting TBA, boundary CFT, and lattice regularizations to quantify boundary effects in integrable QFTs, with broad applications to Ising, sine-Gordon, and related models.
Abstract
We study the ground state energy of integrable $1+1$ quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new, ``R-channel TBA'', where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scalar products of boundary states, exact partition functions for the critical Ising model in a boundary magnetic field, and also results for the energy, excited states and boundary S-matrix of $O(n)$ and minimal models.