Local Magnetization in the Boundary Ising Chain at Finite Temperature

TL;DR

This paper derives an analytical scaling function for local magnetization in the boundary Ising model at criticality and finite temperature, confirming its continuum limit through numerical methods and discussing potential applications.

Contribution

It provides the full analytical scaling function for boundary magnetization at criticality, refining previous results and confirming the continuum limit numerically.

Findings

Analytical scaling function for local magnetization derived

Numerical confirmation of continuum limit achieved

Results applicable to boundary critical phenomena

Abstract

We study the local magnetization in the 2-D Ising model at its critical temperature on a semi-infinite cylinder geometry, and with a nonzero magnetic field $h$ applied at the circular boundary of circumference $\beta$. This model is equivalent to the semi-infinite quantum critical 1-D transverse field Ising model at temperature $T \propto \beta^{-1}$, with a symmetry-breaking field $\propto h$ applied at the point boundary. Using conformal field theory methods we obtain the full scaling function for the local magnetization analytically in the continuum limit, thereby refining the previous results of Leclair, Lesage and Saleur in Ref. \onlinecite{Leclair}. The validity of our result as the continuum limit of the 1-D lattice model is confirmed numerically, exploiting a modified Jordan-Wigner representation. Applications of the result are discussed.