 Classical Ising chain as a quantum particle

Hubbard-Stratonovich decoupling of an exponentially coupled Ising chain gives a scalar-field action with a nearest-neighbor gradient term and $U(m)=\tanh (\kappa /2)m^2/J-\log [2\cosh (2m+h)]$. Its continuum partition function is the Euclidean path integral of the displayed quantum particle. At zero field the potential can have two wells, but a finite barrier still permits quantum tunnelling and gives a unique symmetric ground state. The excitation gap produces a finite classical correlation length, so there is no finite-temperature spontaneous magnetization for fixed positive $\kappa$. One classical spatial coordinate becomes imaginary time; a $d$-dimensional classical model corresponds to a quantum model with $d-1$ spatial dimensions.