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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214 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS.<br />
Because of the periodic boundary conditions this result is symmetric under the<br />
change j ↔ N − j. Also, we have ΓN = 1, which is to be expected because site<br />
N is equal to site 0, and < σ0σN >=< σ 2 0 >= 1.<br />
The expression is simplified when we divide the numer<strong>at</strong>or and denomin<strong>at</strong>or<br />
by the eigenvalue λ+ to the power N. Note th<strong>at</strong> λ+ > λ− > 0. The r<strong>at</strong>io of the<br />
two eigenvalues is simple, we have<br />
This gives<br />
λ−<br />
λ+<br />
= eβJ − e−βJ eβJ = tanh(βJ) (9.118)<br />
+ e−βJ Γj = tanhj (βJ) + tanh N−j (βJ)<br />
1 + tanh N (βJ)<br />
(9.119)<br />
This expression is complic<strong>at</strong>ed, but we have to remember th<strong>at</strong> we always<br />
need to invoke the thermodynamic limit after all calcul<strong>at</strong>ions have been done.<br />
Th<strong>at</strong> is the case here, we have our final result. If the temper<strong>at</strong>ure is non-zero,<br />
tanh(βJ) < 1 and in the limit N ≫ j only the lowest order terms survives. Note<br />
th<strong>at</strong> we have given explicit meaning to the idea of the thermodynamic limit, we<br />
require th<strong>at</strong> N is large compared to j! In th<strong>at</strong> case we can write<br />
Γj = tanh j (βJ) (9.120)<br />
This formula shows th<strong>at</strong> as a function of the distance j the spin correl<strong>at</strong>ion<br />
function decays as a power law. Hence it can be rel<strong>at</strong>ed to a correl<strong>at</strong>ion length<br />
via<br />
where the correl<strong>at</strong>ion length ξ is found by<br />
j<br />
−<br />
Γj = e ξ (9.121)<br />
ξ = − j<br />
= −[log tanh(βJ)]−1<br />
log(Γj)<br />
(9.122)<br />
As expected, we find th<strong>at</strong> the spin correl<strong>at</strong>ion decays exponentially. If Γj is<br />
close to one, spins tend to point in the same direction, and the magnetiz<strong>at</strong>ion<br />
is strongly correl<strong>at</strong>ed. The spin correl<strong>at</strong>ion length measures the distance over<br />
which spins are correl<strong>at</strong>ed and tend to point in the same direction. Note th<strong>at</strong><br />
this common direction can fluctu<strong>at</strong>e!<br />
The correl<strong>at</strong>ion length increases with decreasing temper<strong>at</strong>ure. We see th<strong>at</strong><br />
in the limit T → 0 the hyperbolic tangent approaches one and the logarithm<br />
gives zero. We can find the behavior by<br />
tanh(βJ) =<br />
1 − e−2βJ<br />
≈ 1 − 2e−2βJ<br />
1 + e−2βJ (9.123)<br />
and the logarithm of one plus a small term is approxim<strong>at</strong>ed by the small term.<br />
Th<strong>at</strong> gives