Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
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204 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.<br />
or<br />
χ =<br />
Tr S0βMe −β(H−hM)<br />
Tr e −β(H−hM)<br />
This can be transformed to<br />
−<br />
Tr S0e −β(H−hM) TrβM e −β(H−hM)<br />
Tr e −β(H−hM) 2<br />
(9.60)<br />
χ = β <br />
{〈S0Si〉 − 〈S0〉〈Si〉} (9.61)<br />
i<br />
χ = β <br />
〈S0 − m〉〈Si − m〉 (9.62)<br />
i<br />
which clearly shows th<strong>at</strong> χ is directly rel<strong>at</strong>ed to the fluctu<strong>at</strong>ions in the spin<br />
variables. Here we used 〈S0〉 = 〈Si〉 = m.<br />
If fluctu<strong>at</strong>ions are uncorrel<strong>at</strong>ed we have < AB >=< A >< B > which would<br />
seem to imply th<strong>at</strong> for uncorrel<strong>at</strong>ed fluctu<strong>at</strong>ions χ = 0. Th<strong>at</strong> is not correct,<br />
however. Fluctu<strong>at</strong>ions on the same site are always correl<strong>at</strong>ed, and in th<strong>at</strong> case<br />
we would get χ =< (S0 − m) 2 >, which is always positive indeed.<br />
It is customary to define a spin-correl<strong>at</strong>ion function Γi by<br />
Γi(T ) = 〈S0Si〉T − 〈S0〉T 〈Si〉T<br />
(9.63)<br />
If the spins <strong>at</strong> site 0 and site i are uncorrel<strong>at</strong>ed, Γi = 0. This spin-correl<strong>at</strong>ion<br />
function can also be expressed in terms of real coordin<strong>at</strong>es r, as long as we<br />
understand th<strong>at</strong> the value Γ(r, T ) is actually an average over a number of <strong>at</strong>omic<br />
sites in a volume ∆V which is small compared to the total volume of the crystal.<br />
In th<strong>at</strong> case we find<br />
<br />
χ = β d 3 rΓ(r, T ) (9.64)<br />
If for large values of r the spin-correl<strong>at</strong>ion function is proportional to r −p the<br />
integral is well-defined only if p > 3. Therefore, <strong>at</strong> Tc where the spin-correl<strong>at</strong>ion<br />
function diverges we need to have p 3. At temper<strong>at</strong>ures away from the critical<br />
temper<strong>at</strong>ure the spin correl<strong>at</strong>ion function does not diverge. If the correl<strong>at</strong>ion<br />
function would always be a power law, Γ ∝ r −p(T ) , this would imply p = 3 <strong>at</strong> the<br />
critical temper<strong>at</strong>ure, since p(T ) > 3 away from the critical point. Th<strong>at</strong> is not<br />
correct, however, because we also have an exponential factor, which disappears<br />
<strong>at</strong> the critical temper<strong>at</strong>ure. In general we write Γ ∝ r −p e −α(T )r and we use<br />
α(T ) > 0 away from the critical point and hence α(Tc) = 0. Since these are the<br />
only two possibilities and since we know from experiment th<strong>at</strong> the first one is<br />
wrong, we find th<strong>at</strong> correl<strong>at</strong>ions have to die out exponentially. The length scale<br />
corresponding to this exponential decay is called the correl<strong>at</strong>ion length, and <strong>at</strong><br />
the critical point the correl<strong>at</strong>ion length diverges and becomes infinite.<br />
If we have no correl<strong>at</strong>ions we find χ = β(< S 2 0 > −m 2 ) = β(1 − m 2 ). Here<br />
we know < S 2 0 >= 0 because the value of the spin variable is ±1 and hence