Editing Quantum phase transition (section)

==Classical description==

To understand quantum phase transitions, it is useful to contrast them to [[Phase transition|classical phase transitions]] (CPT) (also called thermal phase transitions).<ref>{{cite journal|last1=Jaeger|first1=Gregg|title=The Ehrenfest Classification of Phase Transitions: Introduction and Evolution|journal=Archive for History of Exact Sciences|date=1 May 1998|volume=53|issue=1|pages=51–81|doi=10.1007/s004070050021|s2cid=121525126}}</ref> A CPT describes a cusp in the thermodynamic properties of a system. It signals a reorganization of the particles; A typical example is the [[freezing]] transition of water describing the transition between liquid and solid. The classical phase transitions are driven by a competition between the [[energy]] of a system and the [[entropy]] of its thermal fluctuations. A classical system does not have entropy at zero temperature and therefore no phase transition can occur. Their order is determined by the first discontinuous derivative of a thermodynamic potential.

A phase transition from water to ice, for example, involves latent heat (a discontinuity of the [[internal energy]] <math>U</math>) and is of first order. A phase transition from a [[ferromagnet]] to a [[paramagnet]] is continuous and is of second order. (See [[phase transition]] for Ehrenfest's classification of phase transitions by the derivative of free energy which is discontinuous at the transition). These continuous transitions from an ordered to a disordered phase are described by an order parameter, which is zero in the disordered and nonzero in the ordered phase. For the aforementioned ferromagnetic transition, the order parameter would represent the total magnetization of the system.

Although the thermodynamic average of the order parameter is zero in the disordered state, its fluctuations can be nonzero and become long-ranged in the vicinity of the critical point, where their typical length scale ''ξ'' (correlation length) and typical fluctuation decay time scale ''τ<sub>c</sub>'' (correlation time) diverge:

:<math> \xi \propto |\epsilon |^{-\nu}\,\,= \left (\frac{|T-T_c|}{T_c}\right )^{-\nu} </math>

:<math> \tau_c \propto \xi^{z} \propto |\epsilon |^{-\nu z}, </math>

where

:<math>\epsilon = \frac{T-T_c}{T_c} </math>

is defined as the relative deviation from the critical temperature ''T<sub>c</sub>''. We call ''&nu;'' the ([[correlation length]]) ''[[critical exponent]]'' and ''z'' the ''dynamical critical exponent''. Critical behavior of nonzero temperature phase transitions is fully described by [[classical thermodynamics]]; [[quantum mechanics]] does not play any role even if the actual phases require a quantum mechanical description (e.g. [[superconductivity]]).