Logarithmic minimal models with Robin boundary conditions

Abstract

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 ∈ Z and s ∈ 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 ξ = − λ 2 where λ = (p ′ −p)π 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 = 0 and restricting to the ground state sequences w = |r|p ′ p , r ∈ Z with the inverse r = (−1) N +w+d pw p ′ , we find that the conformal weights take the values ∆ p,p ′ r,s− 1 2 where ∆ 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.

Conformal partition functions of critical percolation fromD3thermodynamic Bethe Ansatz equations

“…However, the methods of this paper should extend to the more general Kac modules with highest weight ∆ r,s , which are realised on the lattice by including a seam on the boundary [37,63,64]. These methods should also extend to boundary conditions described by the one-and two-boundary Temperley-Lieb algebra [38,39,115,116]. In all these cases, it is expected that the transfer matrix eigenvalues will satisfy the same universal Y -system, encoded by the Dynkin diagram D p ′ .…”

“…The value d SLE path = 7 4 was proved by Beffara [55]. The incorporation of critical dense polymers LM (1,2) [38,[56][57][58][59][60] and critical percolation LM (2,3) into the framework of the family of logarithmic minimal models LM(p, p ′ ) [37,39,41,[61][62][63][64] establishes that these models are Yang-Baxter integrable. The transfer matrices of the logarithmic minimal models are built from so called transfer tangles of the planar Temperley-Lieb algebra [14,15], which we respectively denote by D(u) and T (u) for the boundary and the periodic cases.…”

Finite-size corrections for universal boundary entropy in bond percolation

“…The case F 2n (−1), n odd, was conjectured in [46] and proved in [47]. In the current setting it follows directly from Kummer's theorem [48] for the hypergeometric series (45). The top and bottom coefficients are known to be lim…”

“…It is conceivable that the special role of r ∈ N may be accounted for by the representation theory of the one-boundary Temperley-Lieb algebra, according to which the boundary conditions corresponding to the BCC operator φ r,r are expressible within the usual TL algebra in terms of a Jones-Wenzl projector that symmetrises the first physical strand with r − 1 extra ghost strands [11]. (See also [44] for an equivalent description in terms of boundary integrability, still for r ∈ N.) We also note that half-integer Kac labels of BCC operators are ubiquitous in the CFT of loop models [34], and have appeared recently in the boundary integrability framework as well [45]. is given by…”

“…The second is a result from Bailey [48] for the hypergeometric series (45), and follows from Kummer's theorem after an Euler transformation. Note that the first result is related to the second by the property F (x) = x n+1 F (1/x).…”