The $(1+1)d$ transverse-field Ising chain is closely related to the $(2+0)d$ Ising model. In particular, the $(2+0)d$ classical Ising model has a transfer matrix that can be written suggestively as $e^{K_z \sum_i \sigma^z_{i} \sigma^z_{i+1}} e^{\bar{K}_x \sum_i \sigma^x_{i}}$; in the extremely anisotropic limit of $K_z, \bar{K}_x \to 0$, this reduces to an exponential of $-\sum_i\sigma^z_{i} \sigma^z_{i+1} + \frac{\bar{K}_x}{K_z} \sigma^x_{i}$, the Hamiltonian of the transverse-field Ising chain.
I will call the $(2+0)d$ classical Ising model the classical counterpart to the $(1+1)d$ transverse-field Ising chain.
Does the Heisenberg magnet of $$H = -J\sum_i \sigma^x_{i} \sigma^x_{i+1}+\sigma^y_{i} \sigma^y_{i+1}+\sigma^z_{i} \sigma^z_{i+1}$$ have a $(2+0)d$ classical counterpart in the sense above involving a transfer matrix?
