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We develop further the implementation and analysis of Kac boundary conditionsin 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 tableslabeled by $r,s=1,2,3,...$. They are conjugate to Virasoro Kac representationswith conformal dimensions $\Delta_{r,s}$ given by the usual Kac formula. On afinite strip of width $N$, built from a square lattice, the associatedintegrable 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 ofwidth $\rho-1$ columns. The $r$-type seam contains an arbitrary boundary field$\xi$. The usual fusion construction of the $r$-type seam relies on theexistence of Wenzl-Jones projectors restricting its application to$r\le\rho<p'$. This limitation was recently removed by Pearce, Rasmussen andVillani who further conjectured that the conformal boundary conditions labeledby $r$ are realized, in particular, for $\rho=\rho(r)=\lfloor\frac{rp'}{p}\rfloor$. In this paper, we confirm this conjecture by performingextensive numerics on the commuting double row transfer matrices and theirassociated quantum Hamiltonian chains. Letting $[x]$ denote the fractionalpart, we fix the boundary field to the specialized values $\xi=\frac{\pi}{2}$if $[\frac{\rho}{p'}]=0$ and $\xi=[\frac{\rho p}{p'}]\frac{\pi}{2}$ otherwise.For these boundary conditions, we obtain the Kac conformal weights$\Delta_{r,s}$ by numerically extrapolating the finite-size corrections to thelowest eigenvalue of the quantum Hamiltonians out to sizes $N\le 32-\rho-s$.Additionally, by solving local inversion relations, we obtain general analyticexpressions for the boundary free energies allowing for more accurate estimatesof the conformal data.

Kac boundary conditions of the logarithmic minimal models