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Kac boundary conditions of the logarithmic minimal models

Profile image of Paul PearcePaul Pearce

2015, Journal of Statistical Mechanics: Theory and Experiment

https://doi.org/10.1088/1742-5468/2015/01/P01018
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Abstract

We develop further the implementation and analysis of Kac boundary conditions in the general logarithmic minimal models LM(p, p ′ ) 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 ∆ 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. 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, Rasmussen and Villani who further conjectured that the conformal boundary conditions labeled by r are realized, in particular, for ρ = ρ(r) = ⌊ rp ′ p ⌋. 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 = π 2 if [ ρ p ′ ] = 0 and ξ = [ ρp p ′ ] π 2 otherwise. For these boundary conditions, we obtain the Kac conformal weights ∆ r,s by numerically extrapolating the finite-size corrections to the lowest eigenvalue of the quantum Hamiltonians out to sizes N ≤ 32ρs. 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.

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