Surface critical phenomena and scaling in the eight-vertex model.

Abstract

We give a physical interpretation of the entries of the reflection $K$ matrices of Baxter's eight-vertex model in terms of an Ising interaction at an open boundary. Although the model still defies an exact solution, we nevertheless obtain the exact surface free energy from a crossing-unitarity relation. The singular part of the surface energy is described by the critical exponents ${\ensuremath{\alpha}}_{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2\ensuremath{-}\ensuremath{\pi}/2\ensuremath{\mu}$ and ${\ensuremath{\alpha}}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1\ensuremath{-}\ensuremath{\pi}/\ensuremath{\mu}$, where \ensuremath{\mu} controls the strength of the four-spin interaction. These values reduce to the known Ising exponents at the decoupling point $\ensuremath{\mu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\pi}/2$ and confirm the scaling relations ${\ensuremath{\alpha}}_{s}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{b}+\ensuremath{\nu}$ and ${\ensuremath{\alpha}}_{1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{b}\ensuremath{-}1$.