Kac boundary conditions of the logarithmic minimal models

TL;DR

This paper confirms the realization of Kac boundary conditions in logarithmic minimal models through extensive numerical analysis, providing new insights into boundary conformal weights and free energies.

Contribution

It extends the understanding of Kac boundary conditions in logarithmic minimal models by numerically confirming a conjecture about their realization and deriving analytic expressions for boundary free energies.

Findings

Numerical confirmation of the conjecture relating boundary conditions to Kac labels.

Explicit calculations of conformal weights for finite-size systems up to N=32.

Analytic expressions for boundary free energies derived from inversion relations.

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 $\Delta_{r,s}$ 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 $\rho-1$ columns. The $r$-type seam contains an arbitrary boundary field $\xi$. The usual fusion construction of the $r$-type seam relies on the existence of Wenzl-Jones projectors restricting its application to $r\le\rho<p'$. 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 $\rho=\rho(r)=\lfloor \frac{rp'}{p}\rfloor$. 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 $\xi=\frac{\pi}{2}$ 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 the lowest eigenvalue of the quantum Hamiltonians out to sizes $N\le 32-\rho-s$. 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.