Abstract
We compute the boundary entropy for bond percolation on the square lattice in the presence of a boundary loop weight, and prove explicit and exact expressions on a strip and on a cylinder of size LL. For the cylinder we provide a rigorous asymptotic analysis which allows for the computation of finite-size corrections to arbitrary order. For the strip we provide exact expressions that have been verified using high-precision numerical analysis. Our rigorous and exact results corroborate an argument based on conformal field theory, in particular concerning universal logarithmic corrections for the case of the strip due to the presence of corners in the geometry. We furthermore observe a crossover at a special value of the boundary loop weight.
Highlights
When one perturbs a conformal boundary condition (CBC) by a relevant operator, it flows towards another CBC under the renormalisation group flow. These CBCs and their flows can be characterised by their boundary entropy SB which is defined in the conformal field theory (CFT) via the scalar product of a boundary state |B〉 with the ground state |0〉 of the conformal Hamiltonian: SB = − log〈B|0〉
Where An is the number of n×n alternating sign matrices (ASMs), AV2n+1 is the number of vertically symmetric ASMs of size 2n + 1, and C2n is the number of cyclically symmetric transpose complement plane partitions of size 2n
We have computed the overlap of the ground state of the Temperley-Lieb loop model with bulk loop weight β = 1 with that of the product state of small arcs, or deformed dimerised state
Summary
A 2D classical statistical mechanical model can be viewed as a 1+1D model evolving in imaginary time. These CBCs and their flows can be characterised by their boundary entropy SB which is defined in the CFT via the scalar product of a boundary state |B〉 with the ground state |0〉 of the conformal Hamiltonian: SB = − log〈B|0〉 These numbers are universal and have been computed analytically for many CFTs and for many different CBCs. In the context of CBCs relevant for loop models [11], several such analytical computations were presented in [12, 13]. In this paper we will focus on the TL loop model and provide several mathematically rigorous results on the computation of these scalar products on a lattice of size L and for the case where the bulk loop weight β = 1 At this value of the loop weight the model is well known to be equivalent to bond percolation [14]. We will be precise in terms of mathematical statements, writing “conjecture” for statements we are very confident about being true but for which a rigorous mathematical proof is lacking
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