How Important Is Quantum Mechanics

The question of how important quantum mechanics is for understanding continuous phase transitions has several facets. On the one hand, one may ask whether quantum mechanics is even needed to explain the existence and properties of the bulk phases separated by the transition. This question can be decided only on a case-by-case basis, and very often quantum mechanics is essential as, e.g., for the superconducting phase. On the other hand, one can ask how important quantum mechanics is for the behavior close to the critical point and thus for the determination of the universality class to which the transition belongs. It turns out that the latter question has a remarkably clear and simple answer Quantum mechanics does not play any role in determining the critical behavior if the transition occurs at a finite temperature it does play a role, however, at zero temperature. 

In classical statistical mechanics, static and dynamic behaviors decouple. Consider a classical Hamiltonian H pi,qi) = +Hpot (]i) consisting of 

The situation is different in quantum statistical mechanics. Here, the kinetic and potential parts of the Hamiltonian do not commute with each other. Consequently, the partition function Z = does not factorize, 

The homogeneity law, Eq. [5], for the free energy can be generalized easily to the quantum case (see, e.g.. Ref. 10). For the generic case of power-law dynamical scaling, it takes the form 

The quantum-to-classical mapping can be exploited for computational studies of quantum phase transitions. If one is only interested in finding the universal critical behavior at the quantum critical point (i.e., in the critical exponents) and not in nonuniversal quantities, it is often easier to perform a simulation of the equivalent classical system instead of the original quantum system. We will come back to this point later in the chapter. 

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