Differential equation for local magnetization in the boundary Ising model

We show that the local magnetization in the massive boundary Ising model on the half-plane with boundary magnetic field satisfies second order linear differential equation whose coefficients are expressed through Painleve function of the III kind.

💡 Research Summary

The paper addresses the problem of determining the exact spatial dependence of the local magnetization ⟨σ(x)⟩ in the massive Ising model defined on the half‑plane when a magnetic field h is applied at the boundary. While the bulk Ising model at criticality is well known to be described by conformal field theory, and its boundary versions without a bulk mass have been linked to Painlevé V or VI equations, the simultaneous presence of a bulk mass m and a boundary field h has resisted a closed‑form description. The authors overcome this difficulty by exploiting the fermionic representation of the Ising model (via the Majorana–Onsager transformation) and by treating the boundary as a mixed (Robin‑type) condition that incorporates the magnetic field linearly.

The central object of study is the one‑point function ⟨σ(x)⟩ = ⟨B|σ(x)|0⟩, where |0⟩ is the bulk vacuum and |B⟩ encodes the boundary state. By constructing the two‑point Green function G(x,y)=⟨B|ψ(x)ψ(y)|0⟩ for the massive Majorana fermion ψ and performing an operator‑product expansion of the spin field σ in terms of ψ, the authors express ⟨σ(x)⟩ in terms of G(x,x) and its derivatives. The fermionic Green function satisfies the massive Dirac equation together with the mixed boundary condition, which can be solved by a combination of Fourier transform and Wiener‑Hopf factorisation. The solution involves a special function τ(t) that is precisely the τ‑function of the Painlevé III equation.

From these ingredients the authors derive a second‑order linear differential equation for the magnetization:

 d²⟨σ⟩/dx² + p(x) d⟨σ⟩/dx + q(x)⟨σ⟩ = 0,

where the coefficient functions are expressed through τ(2mx) and its logarithmic derivatives:

 p(x) = d/dx ln τ(2mx),

 q(x) = ¼