Logarithmic Minimal Models with Robin Boundary Conditions

Abstract

We consider general logarithmic minimal models $\mathcal{LM}(p,p^{\prime })$, with $p,p^{\prime }$ coprime, on a strip of $N$ columns with the $(r,s)$ Robin boundary conditions introduced by Pearce, Rasmussen and Tipunin. The associated conformal boundary conditions are labelled by the Kac labels $r\in \mathbb{Z}$ and $s\in \mathbb{N}$. The Robin vacuum boundary condition, labelled by $(r,s\!-\!\frac{1}{2})=(0,\mbox{$\textstyle \frac{1}{2}$})$, is given as a linear combination of Neumann and Dirichlet boundary conditions. The general $(r,s)$ Robin boundary conditions are constructed, using fusion, by acting on the Robin vacuum boundary with an $(r,s)$-type seam consisting of an $r$-type seam of width $w$ columns and an $s$-type seam of width $d=s-1$ columns. The $r$-type seam admits an arbitrary boundary field which we fix to the special value $\xi =-\frac{\lambda }{2}$ where $\lambda =\frac{(p^{\prime }-p)\pi }{2p^{\prime }}$ is the crossing parameter. The $s$-type boundary introduces $d$ defects into the bulk. We consider the associated quantum Hamiltonians and calculate analytically the boundary free energies of the $(r,s)$ Robin boundary conditions. Using finite-size corrections and sequence extrapolation out to system sizes $N+w+d\le 26$, the conformal spectrum of boundary operators is accessible by numerical diagonalization of the Hamiltonians. Fixing the parity of $N$ for $r\ne 0$ and restricting to the ground state sequences $w=\lfloor \frac{|r|p^{\prime }}{p}\rfloor$, $r\in \mathbb{Z}$ with the inverse $r=(-1{)}^{N+w+d}\lceil \frac{pw}{p^{\prime }}\rceil$, we find that the conformal weights take the values ${\Delta }_{r,s-\frac{1}{2}}^{p,p^{\prime }}$ where ${\Delta }_{r,s}^{p,p^{\prime }}$ is given by the usual Kac formula. The $(r,s)$ Robin boundary conditions are thus conjugate to scaling operators with half-integer values for the Kac label $s-\mbox{$\textstyle \frac{1}{2}$}$.

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