  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}$}$.
