# Ising model phase transition when external magnetic field is non-zero

In the Ising model, when the external magnetic field is absent, it has a second order phase transition, because there is a discontinuity in the magnetic susceptibility vs Temperature, T (and magnetic susceptibility is the second derivative of the free energy).

When the external magnetic field is non-zero, the magnetic susceptibility vs T is now continuous. So, is there still a phase transition when the external magnetic field is non-zero? If so, what order is it?

No there is no thermodynamic phase transition when $$h\neq 0$$. This was first proved by Lee and Yang in 1952 in a famous paper. What they proved is that the free energy is an analytic function of the magnetic field in the whole complex plane except possibly along the imaginary axis. In particular, the only possible real value of $$h$$ at which analyticity might fail is $$h=0$$. There have been many further improvements since then.
• The free energy is analytic in $$h$$ for all values of $$(\beta,h)$$ ouside the segment $$\{h=0, \beta\geq\beta_c\}$$.
• The free energy is analytic in $$\beta$$ for all values of $$(\beta,h)$$ except at the point $$(\beta_c,0)$$.
• The free energy has an essential singularity (as a function of $$h$$) at $$h=0$$ when $$\beta\gg 1$$ (expected to be true for all $$\beta>\beta_c$$).