Correlation function (statistical mechanics): Difference between revisions

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==The connection between phase transitions and correlation functions==

[[File:Ferromagnetic correlation functions around Tc.svg|thumb|left|alt=The caption is very descriptive.|Equal-time correlation functions, <math>C(r,\tau =0)</math>, as a function of radius for a ferromagnetic spin system above, at, and below at its critical temperature, <math>T_ C</math>. Above <math>T_ C</math>, <math>C(r,\tau =0)</math> exhibits a combined exponential and power-law dependence on distance: <math>C (r,\tau = 0)\propto r^{-\vartheta} e^{-r/\xi (T)} </math>. The power-law dependence dominates at distances short relative to the correlation length, <math>\xi</math>, while the exponential dependence dominates at distances large relative to <math>\xi</math>. At <math>T_ C</math>, the correlation length diverges, <math>\xi (T_C)=\infty</math>, resulting in solely power-law behaviourbehavior: <math>C(r,\tau =0) \propto r^{-(d-2+\eta)}</math>. <math>T_ C</math> is distinguished by the extreme non-locality of the spatial correlations between microscopic values of the relevant order parameter without long-range order. Below <math>T_ C</math>, the spins exhibit spontaneous ordering but finite correlation length again. Continuous order-disorder transitions can be understood as the process of the correlation length, <math>\xi</math>, transitioning from being finite in the low-temperature, ordered state, to infinite at the critical point, and then again finite in a high-temperature, disordered state.]]

Continuous phase transitions, such as order-disorder transitions in metallic alloys and ferromagnetic-paramagnetic transitions, involve a transition from an ordered to a disordered state. In terms of correlation functions, the equal-time correlation function is non-zero for all lattice points below the critical temperature, and is non-negligible for only a fairly small radius above the critical temperature. As the phase transition is continuous, the length over which the microscopic variables are correlated, <math>\xi</math>, must transition continuously from being infinite to finite when the material is heated through its critical temperature. This gives rise to a power-law dependence of the correlation function as a function of distance at the critical point. This is shown in the figure in the left for the case of a ferromagnetic material, with the quantitative details listed in the section on magnetism.

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The alignment that would naturally arise as a result of the interaction between spins is destroyed by thermal effects. At high temperatures exponentially-decaying correlations are observed with increasing distance, with the correlation function being given asymptotically by

:<math>G (r) \approx \frac{1}{r^{\vartheta}}\exp{\left(-\frac{r}{\xi}\right)}\,,</math>

where r is the distance between spins, and d is the dimension of the system, and <math>\vartheta</math> is an exponent, whose value depends on whether the system is in the disordered phase (i.e. above the critical point), or in the ordered phase (i.e. below the critical point). At high temperatures, the correlation decays to zero exponentially with the distance between the spins. The same exponential decay as a function of radial distance is also observed below <math>T_c</math>, but with the limit at large distances being the mean magnetization <math>\langle M^2 \rangle</math>. Precisely at the critical point, an algebraic behaviourbehavior is seen

:<math>G (r) \approx \frac{1}{r^{(d-2+\eta)}}\,,</math>

where <math>\eta</math> is a [[critical exponent]], which does not have any simple relation with the non-critical exponent <math>\vartheta</math> introduced above.