Quantum Field Theory of Critical and off Critical 2-D Ising Model with Boundary Magnetic Field

Masaki Oshikawa

Anatoly Konechny

Abstract Critical 2D Ising model with a boundary magnetic field is arguably the simplest QFT that interpolates between two non-trivial fixed points. We use the diagonalising Bogolyubov transformation for this model to investigate two quantities. Firstly we explicitly construct an RG interface operator that is a boundary condition changing operator linking the free boundary condition with the one with a boundary magnetic field. We investigate its properties and in particular show that in the limit of large magnetic field this operator becomes the dimension 1/16 primary field linking the free and fixed boundary conditions. Secondly we use Schrieffer-Wolff method to construct effective Hamiltonians both near the UV and IR fixed points.

R. Chatterjee

Ian K. Affleck

Using previous results from boundary conformal field theory and integrability, a phase diagram is derived for the two-dimensional Ising model at its bulk tri-critical point as a function of boundary magnetic field and boundary spin-coupling constant. A boundary tri-critical point separates phases where the spins on the boundary are ordered or disordered. In the latter range of coupling constant, there is a non-zero critical field where the magnetization is singular. In the former range, as the temperature is lowered, the boundary undergoes a first-order transition while the bulk simultaneously undergoes a second-order transition.

Jakob E. Björnberg

Eran Sela, Andrew K. Mitchell

We study the local magnetization in the 2D 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 β. This model is equivalent to the semi-infinite quantum critical 1D transverse field Ising model at temperature , with a symmetry-breaking field proportional to 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. The validity of our result as the continuum limit of the 1D lattice model is confirmed numerically, exploiting a modified Jordan–Wigner representation. Applications of the result are discussed.