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renormalization group

May 14, 2011

Renormalization group (RG) ideas are all the rage in physics these days, and one of my goals for graduate school is to become more familiar with this framework. Here are my thoughts so far.

The renormalization group is an extremely powerful tool that can be used to determine which microscopic aspects of a system are relevant to its macroscopic behavior. It exploits the crucial property that macroscopic systems are — for mathematical purposes — effectively infinite, and that there is a huge range of scales between the microscopic picture and the resolution at which finite-size effects start to manifest themselves.

The critical point

In its simplest application, RG also requires the system to be at a so-called critical point, which can be defined in the following way. Imagine that the system in question can be fully specified by a finite number of knobs or parameters

\mu = (\alpha,\beta,\gamma,\ldots)

and that as we tune one of the knobs \alpha there is a critical point \alpha_c where some extensive macroscopic observable \phi becomes multivalued. In other words, when we tune \alpha through \alpha_c in different copies of the system, individual copies end up with different values of \phi. Since \phi is extensive, we know it can be written as the average of some local quantity \phi(x)

\phi = \frac{1}{V} \int \phi(x) \, dV

where \phi(x) is called the order-parameter. It gives the local value of the multivalued macroscopic observable \phi at different points x within the system. Typically, we define \phi so that its numerical value in the single-valued region is zero.

Thus, with some precision we can now say that a critical point is a point in parameter space where the system average of some order parameter \phi becomes nonzero and multi-valued. Since the entire (effectively infinite) system chooses a particular value of \phi, this requires that the local order parameter \phi(x) is correlated over infinitely large regions. In other words, its correlation length \xi diverges:

\xi \to \infty as \alpha \to \alpha_c

More to come…

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