I'm trying to understand the connections between quantum models in d dimensions and classical models in (d+1) dimensions within two, possibly related, contexts:
(i) in path integral monte carlo, the Trotter-Suzuki decomposition gives an equivalence between a d-dimensional quantum spin model at a given temperature and a d+1 dimensional classical Ising model (e.g: Progress of Theoretical Physics, Vol. 56, No. 5, November 1976), and
(ii) in quantum critical systems, where the zero temperature maps to one extra additional infinite dimension of a classical system.
Now as Suzuki emphasizes in the above paper, critical properties of a d-dimensional quantum system need not be the same as that of a d+1 dimensional classical system.
Therefore, is my understanding correct that the temperature in the mapped classical system need not be the temperature in the original quantum system? If not, what is the temperature in the classical model?
Thanks!
Best Answer
The mapping between the quantum and the classical system is formal, but as you say, we can usually interpret a quantum phase transition of a $d$ dimensional quantum system (that is, a phase transition at zero temperature) as a (classical) phase transition in a $d+1$ dimensional classical system. The temperature of the quantum system maps to the size of the $d+1$th dimension of the classical system. Furthermore, a quantum phase transition is at zero temperature, and therefore is driven by a non-thermal control parameter (an external pressure, a magnetic field, etc.). Usually, we can modelize the effect of this control parameter by the parameter $r_0$ in front of the quadratic term in the action, which changes sign at the transition: $r_0 \phi^2$.
Here enter the confusion: in classical systems, one also assume that the parameter $r_0$ drives the transition (changes sign), and one usually assumes that $r_0\propto (T-T_c)$, where $T$ is the temperature of the classical system, and $T_c$ is the (meanfield) critical temperature. But this has nothing to do with the quantum-classical mapping, and it is just specific to stat-mech.