646 Critical phenomena
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KKK123Fig. 16.18A renormalization group trajectory.defines the critical surface, which in this case, is a one-dimensionalcurve in theK 1 K 2
plane as illustrated in Fig. 16.19. Here, the black curve represents the critical “sur-
face” (curve), and the point at which the arrows meet is the critical fixed point. The
full coupling constant space represents the space of all physicalsystems containing
nearest-neighbor and next-nearest-neighbor couplings. If we wish to consider the sub-
set of systems with no next-nearest-neighbor coupling (K 2 = 0), the point at which
the lineK 2 = 0 intersects the critical surface defines the critical valueK 1 cand the
corresponding critical temperature, and it is an unstable fixed point of an RG trans-
formation withK 2 = 0. Similarly, if we consider a model for whichK 26 = 0, then the
point at which this line intersects the critical surface determines the critical value of
K 1 for such a model. In fact, for any of these models,K 1 clies on the critical surface
and iterates toward the critical fixed point under the full RG transformation. Thus,
we have an effective definition of a universality class: All models characterized by the
same critical fixed point belong to the same universality class and share the same
critical properties.