432 13 Monte Carlo Methods in Statistical Physics
and the susceptibility
χ(T)∼|TC−T|−γ,
withα= 0 andγ=− 7 / 4. Another important quantity is the correlation length, which is ex-
pected to be of the order of the lattice spacing forTis close toTC. Because the spins become
more and more correlated asTapproachesTC, the correlation length increases as we get
closer to the critical temperature. The discontinuous behavior of the correlationξnearTCis
ξ(T)∼|TC−T|−ν. (13.4)A second-order phase transition is characterized by a correlation length which spans the
whole system. The correlation length is typically of the order of some few interatomic dis-
tances. The fact that a system like the Ising model, whose energy is described by the inter-
action between neighboring spins only, can yield correlation lengths of macroscopic size at a
critical point is still a feature which is not properly understood. Stated differently, how can
the spins propagate their correlations so extensively whenwe approach the critical point, in
particular since the interaction acts only between nearestspins? Below we will compute the
correlation length via the spin-sin correlation function for the one-dimensional Ising model.
In our actual calculations of the two-dimensional Ising model, we are however always lim-
ited to a finite lattice andξwill be proportional with the size of the lattice at the critical point.
Through finite size scaling relations [75,78–80] it is possible to relate the behavior at finite
lattices with the results for an infinitely large lattice. The critical temperature scales then as
TC(L)−TC(L=∞)∝aL−^1 /ν, (13.5)withaa constant andνdefined in Eq. (13.4). The correlation length for a finite lattice size
can then be shown to be proportional to
ξ(T)∝L∼|TC−T|−ν.and if we setT=TCone can obtain the following relations for the magnetization, energy and
susceptibility forT≤TC
〈M(T)〉∼(T−TC)β∝L−β/ν,
CV(T)∼|TC−T|−γ∝Lα/ν,
and
χ(T)∼|TC−T|−α∝Lγ/ν.
13.4.2Critical Exponents and Phase Transitions from Mean-field Models
In order to understand the above critical exponents, we willderive some of the above relations
using what is called mean-field theory.
In studies of phase transitions we are interested in minimizing the free energy by varying
the average magnetisation, which is the order parameter forthe Ising model. The magnetiza-
tion disappears atTC.
Here we use mean field theory to model the free energyF. In mean field theory the local
magnetisation is a treated as a constant and all effects fromfluctuations are neglected. Stated
differently, we reduce a complicated system of many interacting spins to a set of equations
for each spin. Each spin sees now a mean field which is set up by the surrounding spins. We
neglect therefore the role of spin-spin correlations. A wayto achieve this is to rewrite the