Consider the Ising field theory, a conformal field theory in 2 dimensions which corresponds to the minimal model $\mathcal{M}_{4,3}$ and it's perturbations by the relevant operators $\epsilon, \sigma$ corresponding to the energy density (not to be confused with the stress-energy tensor) and the spin respectively, given schematically by the following Hamiltonian
$$H=\mathcal{H}_{\text{Ising}}+m_f\int\epsilon(x) d^2x+h\int\sigma(x)d^2x.$$
Interestingly, the model is integrable in the directions $m_f=0$ and $h=0$. The operator $\epsilon$ is even under fermion $\mathbb{Z}_2$ symmetry and $\sigma$ is odd. I would like to understand the spontaneous symmetry breaking structure of phase space for small values of the couplings, if it exists. To explain what I mean, assume that initially $m_f>0$ and the magnetic field has a very small non-zero value. As I raise the value of the magnetic field is there a point where I can induce a first order transition or is there not? Can I describe it by some sort of Landau type Hamiltonian?
I am aware of the results in the spin lattice version of the Ising model, where there is such a first order transition when crossing the line $h=0$ for $T<T_c$, however I am not entirely certain that I can use these results to argue for the phase diagram of the continuum theory above (that is described by deformations around the critical point $T=T_c$). I know however that the mass the fermion gets for small departures from the critical point is proportional to the distance from the critical point $m_f\propto|1-T/T_c|$ so it would seem that in the lattice model, you induce a first order transition the moment you turn on the magnetic field.
What gives? Is the answer trivial and exactly the same as in the lattice model? Is the situation different for a negative mass parameter $m_f$? How can I connect the dots together?
Best Answer
The answer is exactly the same as what you get in the lattice model. The field theory is supposed to accurately describe the long-distance, low-energy behavior of the lattice model, so it better reproduces the phase diagram around the critical point. The sign of the mass determines which side of the transition you are in, and once you are on the ferromagnetic side, and with the magnetic field turned on, passing through $h=0$ crosses a first-order transition.
Also, once you add the magnetic field term the mapping to fermions no longer works: the $Z_2$ symmetry that is necessary for the bosonization/fermionization map to work is broken explicitly. Put it in another way, the $\sigma$ operator is a highly non-local operator in the fermionic representation.