Logarithmic Minimal Models with Robin Boundary Conditions

TL;DR

This paper studies logarithmic minimal models with Robin boundary conditions, analytically and numerically determining boundary free energies and conformal spectra, revealing half-integer Kac labels for boundary operators.

Contribution

It introduces a construction of Robin boundary conditions in logarithmic minimal models using fusion and boundary seams, and computes their boundary free energies and spectra.

Findings

Boundary free energies are analytically calculated.

Conformal weights correspond to half-integer Kac labels.

Numerical analysis confirms the spectrum matches theoretical predictions.

Abstract

We consider general logarithmic minimal models ${\cal LM}(p,p')$, with $p,p'$ 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 $r\in{\Bbb Z}$ and $s\in{\Bbb N}$. 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 $\xi=-\tfrac{\lambda}{2}$ where $\lambda=\frac{(p'-p)\pi}{2p'}$ 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 $w=\big\lfloor\frac{|r|p'}{p}\big\rfloor$, $r\in{\Bbb Z}$ with the inverse $r=(-1)^{N+w+d}\big\lceil \frac{p w}{p'}\big\rceil$, we find that the conformal weights take the values $\Delta^{p,p'}_{r,s-\frac12}$ where $\Delta^{p,p'}_{r,s}$ 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}$}$.