Kac boundary conditions of the logarithmic minimal models

Abstract

We develop further the implementation and analysis of Kac boundary conditions in the general logarithmic minimal models ${\cal LM}(p,p')$ with $1\le p<p'$ and $p,p'$ coprime. Working in a strip geometry, we consider the $(r,s)$ boundary conditions, which are organized into infinitely extended Kac tables labeled by $r,s=1,2,3,...$. They are conjugate to Virasoro Kac representations with conformal dimensions $Δ_{r,s}$ 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 $ξ$. The usual fusion construction of the $r$-type seam relies on the existence of Wenzl-Jones projectors restricting its application to $r\leρ<p'$. This limitation was recently removed by Pearce, Rasmussen and Villani who further conjectured that the conformal boundary conditions labeled by $r$ are realized, in particular, for $ρ=ρ(r)=\lfloor \frac{rp'}{p}\rfloor$. 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 $ξ=\fracπ{2}$ if $[\fracρ{p'}]=0$ and $ξ=[\frac{ρp}{p'}]\fracπ{2}$ otherwise. For these boundary conditions, we obtain the Kac conformal weights $Δ_{r,s}$ by numerically extrapolating the finite-size corrections to the lowest eigenvalue of the quantum Hamiltonians out to sizes $N\le 32-ρ-s$. 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.