Universality classes 617which is simply a spin-flip transformation, the magnetization in a perfectly ordered
state changes sign. Note that the spin-flip transformation has the same effect as per-
forming aparity transformation, in which we let the spatial coordinatez→−z.^1
Consider next the effect of the transformation in eqn. (16.4.1) on the unperturbed
HamiltonianH 0 of our idealized magnetic model. The unperturbed (h= 0) Hamilto-
nian is
H 0 =−1
2
∑
<i,j>Jijσiσj. (16.4.2)If the spins in eqn. (16.4.2) are transformed according to eqn. (16.4.1), the Hamiltonian
becomes
H′ 0 =−1
2
∑
<i,j>Jijσ′iσ′j, (16.4.3)which has exactly the same form as eqn. (16.4.2). Thus, the transformation in eqn.
(16.4.1) preserves the form of the Hamiltonian. The HamiltonianH 0 is said to be
invariantunder a spin-flip transformation. This is not unexpected since the spin-flip
transformation is equivalent to a parity transformation, which is merely a different
choice of coordinates, and physical results should not depend on this choice. Thus, the
HamiltonianH 0 exhibitsparity invariance.
Readers having some familiarity with the concepts of group theory will recognize
that the spin-flip transformation, together with the trivial identity transformation
σ′i=σi, form a complete group of transformations with elements{ 1 ,− 1 }, a group
known asZ 2. The HamiltonianH 0 is invariant under both of the operations of this
group. The magnetization of an ordered state, on the other hand, is not. Based on
these notions, we introduce the concept of anorder parameter, which is needed to
define universality classes.
Suppose the unperturbed HamiltonianH 0 of a system is invariant with respect to
all of the transformations of a groupG. If two phases can be distinguished by a specific
thermodynamic average〈φ〉(either a classical phase space average or a quantum trace)
that is not invariant under one or more of the transformations ofG, then〈φ〉is called
anorder parameterfor the system. Because the magnetizationmis not invariant under
one of the transformations inZ 2 , it can serve as an order parameter for the Ising model
withH 0 given by eqn. (16.3.4).
The systems in a universality class are characterized by two parameters: (1) the
numberdof spatial dimensions in which the system exists, and (2) the dimensionnof
the order parameter. All systems possessing the same values ofdandnbelong to the
same universality class. Thus, comparing the gas–liquid system and theh= 0 Ising
model defined by eqn. (16.3.4), it should be clear why these two systems belong to
the same universality class. In both cases, the number of spatial dimensions isd= 3.
(In fact, these models can be defined in any number of dimensions.) In addition, the
dimension of the order parameter for each system isn= 1, since for both systems,
the order parameter (ρorm) is a simple scalar quantity. Consequently, the idealized
magnetic model can be used to determine the critical exponents ofthed= 3,n= 1
(^1) The general parity transformation is a complete reflection of all three spatial axes,r→−r.