Bipartite fidelity of critical dense polymers

We investigate the bipartite fidelity Fd for a lattice model described by a logarithmic CFT: the model of critical dense polymers. We define this observable in terms of a partition function on the pants geometry, where d defects enter at the top of the pants lattice and exit in one of the legs. Using the correspondence with the XX spin chain, we obtain an exact closed-form expression for Fd and compute the leading terms in its 1/N asymptotic expansion as a function of x = NA/N, where N is the lattice width at the top of the pants and NA is the width of the leg where the defects exit. We find an agreement with the results of Stéphan and Dubail for rational CFTs, with the central charge and conformal weights specialised to c = −2 and ∆ = ∆1,d+1 = d(d−2)/8. We compute a second instance F2 of the bipartite fidelity for d = 2 by imposing a dierent rule for the connection of the defects. In the conformal setting, this choice corresponds to inserting two boundary condition changing fields of weight ∆ = 0 that are logarithmic instead of primary. We compute the asymptotic expansion in this case as well and find a simple additive correction compared to F2, of the form −2 log((1 + x)/(2√x)). We confirm this lattice result with a CFT derivation and find that this correction term is identical for all logarithmic theories, independently of c and ∆