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 $J.$


1 Answer 1


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 $S$).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.