An Ising model in a magnetic field with a boundary

TL;DR

This paper derives boundary reflection matrices for dilute A_L lattice models, including the Ising model in a magnetic field, and calculates critical surface exponents directly from the model.

Contribution

It provides the first direct calculation of the Ising surface magnetic exponent without relying on scaling relations.

Findings

Derived diagonal reflection matrices for dilute A_L models.

Calculated surface free energy and critical exponents.

Obtained the Ising surface magnetic exponent _s = -15/7 for L=3.

Abstract

We obtain the diagonal reflection matrices for a recently introduced family of dilute ${\rm A}_L$ lattice models in which the ${\rm A}_3$ model can be viewed as an Ising model in a magnetic field. We calculate the surface free energy from the crossing-unitarity relation and thus directly obtain the critical magnetic surface exponent $\delta_s$ for $L$ odd and surface specific heat exponent for $L$ even in each of the various regimes. For $L=3$ in the appropriate regime we obtain the Ising exponent $\delta_s = -\frac{15}{7}$, which is the first determination of this exponent without the use of scaling relations.