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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!

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    $\begingroup$ What do you mean by "the temperature"? All I have is a partition function/density operator $\exp(-\beta H)$. If I rescale $H \rightarrow \lambda H$ I can change $\beta$ to whatever I want, as you know from the Ising model. So do you mean something more concrete? If I normalize my imaginary time to have radius $1$, then the transformed partition look something like $\exp(-\beta H')$ (but $H'$ depends on $\beta$). Think about what an answer would look like. $\endgroup$ Jun 27, 2013 at 16:01
  • $\begingroup$ Consider the following: when a quantum system (QS) is mapped to the partition function of a classical one (CS), the integration imaginary 'time' period (or extra dimension) is inversely proportional to the temperature T (Feynman and Hibbs). Therefore the effect of T in the QS is to determine the path length in the CS, making the effect of T implicit in the CS. But as Ceperley mentions (Rev. Mod. Phys., Fig. 3 and 4), the T of the CS has no relation to that of the QS. So my question was can an effective T still be defined for the CS? $\endgroup$
    – MaviPranav
    Jun 28, 2013 at 6:38

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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.

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Temperature in the classical model is mapped to imaginary time in the quantum model. By analytic continuation, one can obtain the real-time evolution. The matrix elements of the time-evolution operator of the quantum model at zero temperature will get mapped to the matrix elements of the transfer matrix of the classical model at an appropriate temperature which depends on the time in the time-evolution operator.

This is my cursory understanding. Hope it helps.

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