Classical Ising chain in transverse field

The spin 1/2 Ising chain in transverse field is considered the prototypical system for quantum phase transitions. However, very little is apparently known in literature about its classical counterpart, not to be confused with the standard classical Ising model: while the latter is constructed from classical discrete variables, the model we consider is a chain of classical vectors of modulus 1, interacting via an Ising-like Hamiltonian. When an uniform field is applied perpendicular to the exchange interaction, both the quantum model and its classical counterpart get to be characterized by a critical field separating a ferromagnetically ordered state of minimal energy from a paramagnetic one, The properties of the classical model, and especially the behaviour of the correlation length, are investigated at low temperature around the critical field and compared with those of the quantum model, in order to single out the role played by quantum and classical fluctuations at finite temperature; the possibility to experimentally observe peculiar quantum critical effects in Ising spin chains is discussed.