Global properties of the Renormalization Group flow

We now want to show qualitatively that the global behaviour of the RG flow determines the phase diagram of the system. Since as we have already stated previously every fixed point of the RG flow can correspond to a phase of the system, it is very important to classify the fixed points of a possible RG flow. This classification can be done using the codimension and the correlation length of the point. We shall now see some results.

  • Fixed points with and are called sinks (since they have no relevant directions, so all the points in their neighbourhoods flow towards them), and correspond to stable bulk phases of the system; the nature of the coupling constants at the sink characterizes the relative phase. For example, a three-dimensional Ising model with nearest-neighbour interaction and in an external field turns out to have two sinks at and (note that considering a coupling constant is equivalent to considering as such), which correspond to the fact that, for all temperatures, in a positive (negative) external field the system has a positive (negative) magnetization
  • There are two types of fixed points with , called discontinuity and continuity fixed points. In the case of the Ising model, both have (but in other models we can have also fixed points with and . The former correspond to first order transitions where one of the order parameters changes abruptly. For example, in the same case as before the line of points flows under the RG towards the discontinuity point . The latter represents a phase of the system, generally a disordered one like the paramagnetic phase of the Ising model; in this case the line flows towards the continuity point . These points are generally not really interesting
  • Fixed points with can describe either points of multiple phase coexistence (if ) or multicritical points (if ). In the simplest case, i.e. , such fixed points correspond to triple points if or critical points if . In each case, a useful way to interpret the presence of two relevant directions is that these represent the two variables that must be tuned in order to place the system in the appropriate point (for example in the case of a magnet we must set and )
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