# Analytical solution to 1D transverse Ising model

There has been a lot of work on the transverse Ising model, but when even limited to the 1D case, Monte Carlo simulations or Mean-field theory seem to have been the go-to approaches. So I wonder, has the 1D transverse Ising model ever been solved exactly?

With $\Gamma$ the applied external field, which we can set to one $\Gamma=1,$ and $J$ the spin-spin interaction strength, which we assume to be constant. Moreover we know the different spin components satisfy commutation relations of the type: $[S_i^x,S_i^z]\propto S_i^y.$ To clarify, by solving this model, I mean solving for the eigenvalues and eigenstates of the system, and specially, estimating the ferro-paramagnetic quantum phase transition based on $J.$
Note that the point of the phase transition can be inferred from the self-duality of the model and is at $\Gamma=J$ (depending on your normalization of $S$).