QuantumPhysics.dvi

If both ground states contribute to the partition function, thenthe total magnetization

will get wiped out, and the system will remain in a disordered phase. When the volume, or

N, is finite, this will always be the case. But whenN → ∞, it is possible for the system

to get stuck in one ground state or the other. The reason this onlyhappens for infinite

N is that it would then take an infinite number of spin flips to transition between the|+〉

and|−〉states, and this may get energetically impossible. When the system gets stuck in

one of its ground states,M(0) 6 = 0 and we have spontaneous magnetization, familiar from

ferromagnetism below the Curie temperature. The operation of spin reversal, which is a

symmetry of the Hamiltonian forB= 0 is then NOT a symmetry of the physical system

any more, as a definite non-zero value ofM(0) is not invariant underR. The symmetryR

is said to bespontaneously broken, and the system is then in an ordered phase, close to one

of its ground states. We have already shown that, for the 1-dimensional Ising model, this

phenomenon does not take place.

The 2-dimensional Ising model, however, does exhibit an ordered phase below a critical

temperatureTc. This is known since the model was solved exactly by Lars Onsager in 1944,

and the critical temperature is known analytically,

sh(2Jβc) = 1

1

βc

=kBTc= 2J× 1. 134542 (8.89)

The corresponding magnetization was computed by C.N. Yang,

M(0) =

(

1 −

1

sh^4 (2Jβ)

) 1 / 8

T < Tc

M(0) = 0 T > Tc (8.90)

Note that asTրTc, the expressionM(0) vanishes and joins continuously with theT > Tc

result. The phase transition atT=Tcis actually second order.

Whether the 3-dimensional Ising model allows for an exact solution isone of the great

outstanding problems of statistical mechanics. Proposals have been made that the model

behaves as a theory of free fermionic random surfaces, but the details have never been

conclusive. Numerical studies of the model have shown however that it also admits a phase

transition between ordered (low temperature) and disordered (high temperature) phases.