Logarithmic minimal models with Robin boundary conditions

We consider general logarithmic minimal models 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. On the lattice, these models are Yang-Baxter integrable loop models that are described algebraically by the one-boundary Temperley-Lieb algebra. The (r, s) Robin boundary conditions are a class of integrable boundary conditions satisfying the boundary Yang-Baxter equations which allow loop segments to either reflect or terminate on the boundary. The associated conformal boundary conditions are organized into infinitely extended Kac tables labelled by the Kac labels r is an element of Z and s is an element of N. The Robin vacuum boundary condition, labelled by (r, s - 1/2) = (0, 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 = -lambda/2 where lambda = (p' - p)pi/p' is the crossing parameter. The s-type boundary introduces d defects into the bulk. We consider the commuting double-row transfer matrices and their 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 <= 26, the conformal spectrum of boundary operators is accessible by numerical diagonalization of the Hamiltonians. Fixing the parity of N for r not equal 0 and restricting to the ground state sequences w = [vertical bar r vertical bar p'/p], r is an element of Z with the inverse r =(-1)(N+w+d)[pw/p'], we find that the conformal weights take the values Delta(r,s-1/2) (p,p') 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 - 1/2. Level degeneracies support the conjecture that the characters of the associated (reducible or irreducible) representations are given by Virasoro Verma characters.