Renormalization Group Theory

Abstract

In this chapter, we discuss the renormalization-group (RG) approach to quantum field theory. As we will see, renormalization group theory is not only a very powerful technique for studying strongly-interacting problems, but also gives a beautiful conceptual framework for understanding many-body physics in general. The latter comes about because in practice we are often interested in determining the physics of a many-body system at the macroscopic level, i.e. at long wavelengths or at low momenta. As a result we need to eliminate, or integrate out, the microscopic degrees of freedom with high momenta to arrive at an effective quantum field theory for the long-wavelength physics. The Wilsonian renormalization group approach is a very elegant procedure to arrive at this goal. The approach is a transformation that maps an action, characterized by a certain set of coupling constants, to a new action where the values of the coupling constants have changed. This is achieved by performing two steps. First, an integration over the high-momentum degrees of freedom is carried out, where the effect of this integration is absorbed in the coupling constants of the action that are now said to flow. Second, a rescaling of all momenta and fields is performed to bring the relevant momenta of the action back to their original domain. By repeating these two steps over and over again, it is possible to arrive at highly nonperturbative approximations to the exact effective action.At a continuous phase transition the correlation length diverges, which implies that the critical fluctuations dominate at each length scale and that the system becomes scale invariant. This critical behavior is elegantly captured by the renormalization-group approach, where a critical system is described by a fixed point of the above two-step transformation. By studying the properties of these fixed points, it is possible to obtain accurate predictions for the critical exponents that characterize the nonanalytic behavior of various thermodynamic quantities near the critical point. In particular we find that the critical exponent ν, associated with the divergence of the correlation length, is in general not equal to 1/2 due to critical fluctuations that go beyond the Landau theory of Chap. 9. It has recently been possible to beautifully confirm this theoretical prediction with the use of ultracold atomic gases. Moreover, the renormalization-group approach also explains universality, which is the observation that very different microscopic actions give rise to exactly the same critical exponents. It turns out that these different microscopic actions then flow to the same fixed point, which is to a large extent solely determined by the dimensionality and the symmetries of the underlying theory. As a result, critical phenomena can be categorized in classes of models that share the same critical behavior. In condensed-matter physics, many phase transitions of interest fall into the XY universality class, such as the transition to the superfluid state in interacting atomic Bose gases and liquid 4He, and the transition to the superconducting state in an interacting Fermi gas. Therefore, we mainly focus on this universality class in the following.KeywordsRenormalization GroupCorrelation LengthFeynman DiagramQuantum Phase TransitionTricritical PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.