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Oxford University Press eBooks

The renormalization group

Tom Lancaster, Stephen J. Blundell

Published Apr 17, 2014• 0 citations
Publisher

Abstract

The renormalization group is introduced in this chapter. Examples treated include asymptotic freedom, Anderson localization, and the Kosterlitz–Thouless transition.
DOI
10.1093/acprof:oso/9780199699322.003.0035

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The physics of the topological state of matter is the second revolution in quantum mechanics. We study the effect of interactions on the topological quantum phase transition and the quantum Berezinskii-Kosterlitz-Thouless (QBKT) transition in topological state of a quantum many-body condensed matter system. We predict a topological quantum phase transition from topological superconducting phase to an insulating phase for the interacting Kitaev chain. We observe interesting behaviour from the results of renormalization group study on the topological superconducting phase. We derive the renormalization group (RG) equation for QBKT through different routes with a few exact solutions along with the physical explanations, wherein we find the existence of two new important emergent phases apart from the two conventional phases of this model Hamiltonian. We also present results of a length-scale dependent study to predict asymptotic freedom like behaviour of the system. We do rigorous quantum field theoretical renormalization group calculations to solve this problem.
The basic idea of the renormalization group is introduced and illustrative examples are presented. Emphasis is put on the application to the theory of critical phenomena. This article is prepared for pedagogical purposes. It is written at a level that a second-year graduate student in physical sciences can understand. No previous knowledge of critical phenomena or field theory is needed. We make no attempt to survey the field or cover a wide range of subjects. On the contrary, we limit the scope to the most basic aspects. We choose to elaborate at length to make the basic idea clear and the definitions precise, and to go through the examples very carefully. We feel that once these basic aspect are understood, there will be no difficulty in confronting the rapidly expanding literature on this subject.
We investigate the general features of the renormalization-group flow at the Berezinskii-Kosterlitz-Thouless (BKT) transition, providing a thorough quantitative description of the asymptotc critical behavior, including the multiplicative and subleading logarithmic corrections. For this purpose, we consider the RG flow of the sine-Gordon model around the renormalizable point which describes the BKT transition. We reduce the corresponding $\ensuremath{\beta}$ functions to a universal canonical form, valid to all perturbative orders. Then we determine the asymptotic solutions of the RG equations in various critical regimes: the infinite-volume critical behavior in the disordered phase, the finite-size scaling limit for homogeneous systems of finite size, and the trap-size scaling limit occurring in two-dimensional bosonic particle systems trapped by an external space-dependent potential.