Kac boundary conditions of the logarithmic minimal models**This paper is dedicated to Jean-Bernard Zuber on the occassion of his retirement.

Abstract

We develop further the implementation and analysis of Kac boundary conditions in the general logarithmic minimal models with 1 ⩽ p < p′ and p, p′ coprime. Specifically, working in a strip geometry, we consider the (r, s) Kac boundary conditions. These boundary conditions are organized into infinitely extended Kac tables labeled by the Kac labels r, s = 1, 2, 3, …. They are conjugate to Virasoro Kac representations with conformal dimensions Δr, s given by the usual Kac formula. On a finite strip of width N, built from a square lattice, the associated integrable boundary conditions are constructed by acting on the vacuum (1, 1) boundary with an s-type seam of width s − 1 columns and an r-type seam of width ρ − 1 columns. The r-type seam contains an arbitrary boundary field ξ. While the usual fusion construction of the r-type seam relies on the existence of Wenzl–Jones projectors restricting its application to r ⩽ ρ < p′, this limitation was recently removed by Pearce et al who further conjectured that the conformal boundary conditions labeled by r are realized, in particular, for . In this paper, we confirm this conjecture by performing extensive numerics on the commuting double row transfer matrices and their associated quantum Hamiltonian chains. Letting [x] denote the fractional part, we fix the boundary field to the specialized values if and otherwise. For these boundary conditions, we obtain the Kac conformal weights Δr, s by numerically extrapolating the finite-size corrections to the lowest eigenvalue of the quantum Hamiltonians out to sizes N ⩽ 32 − ρ − s. Additionally, by solving local inversion relations, we obtain general analytic expressions for the boundary free energies allowing for more accurate estimates of the conformal data.