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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?

About the model:

The Hamiltonian of the 1D transverse Ising has the general form of: \begin{align} H &= -\Gamma\sum_i S_i^z -J\sum_i S_i^x S_{i+1}^x \tag{1} \\ &= -\sum_i \left[\Gamma S_i^z + J S_i^x S_{i+1}^x\right] \end{align}

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 $J.$

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The 1D transverse field Ising model can be solved exactly by mapping it to free fermions. You can find more about that e.g. in the post Ground state degeneracy: Spin vs Fermionic language; in particular, the discussion below the answer lists some references where the derivation is carried out.

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 $S$).

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