Extraordinary transition in the two-dimensional O(n) model

TL;DR

This paper investigates the extraordinary surface transition in the two-dimensional O(n) model for n<1, revealing unique surface critical behavior and spectral properties through conformal perturbation theory and transfer matrix analysis.

Contribution

It provides the first detailed analysis of the extraordinary transition in 2D O(n) models for n<1, highlighting differences from higher-dimensional cases and fixed boundary conditions.

Findings

Surface critical behavior differs from fixed boundary conditions.

All surface scaling dimensions match those of the ordinary transition.

Eigenvalues of the transfer matrix are reshuffled between sectors.

Abstract

The extraordinary transition which occurs in the two-dimensional O(n) model for $n<1$ at sufficiently enhanced surface couplings is studied by conformal perturbation theory about infinite coupling and by finite-size scaling of the spectrum of the transfer matrix of a simple lattice model. Unlike the case of $n\geq1$ in higher dimensions, the surface critical behaviour differs from that occurring when fixed boundary conditions are imposed. In fact, all the surface scaling dimensions are equal to those already found for the ordinary transition, with, however, an interesting reshuffling of the corresponding eigenvalues between different sectors of the transfer matrix.