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Correction of the expression for the correlation function in the main text to make it consistent with figure caption Tag: Reverted |
Undid revision 1283758967 by 158.47.243.116 (talk) These appear to be different equations. Note that the dimension d is mentioned in the following paragraph, so the original equation is supposed to depend on d. |
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Even in a magnetically disordered phase, spins at different positions are correlated, i.e., if the distance r is very small (compared to some length scale <math>\xi</math>), the interaction between the spins will cause them to be correlated.
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>C (r) \approx \frac{1}{r^{\vartheta}}\exp{\left(-\frac{r}{
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 behavior is seen
:<math>C (r) \approx \frac{1}{r^{(d-2+\eta)}}\,,</math>
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