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Physical Review A

Chaotic renormalization-group trajectories

P.H. Damgaard, Guðmar Þorleifsson

Published Aug 1, 1991• 10 citations
Publisher

Abstract

Under certain conditions, the renormalization-group flow of models in statistical mechanics can change dramatically under just very small changes of given external parameters. This can typically occur close to bifurcations of fixed points, close to the complete disappearance of fixed points, or in regions where the renormalization-group flow becomes chaotic. We present some explicit examples of these phenomena for the case of a Lie group valued spin-model analyzed by means of a variational real-space renormalization group. By directly computing the free energy of these models around the parameter regions in which such nontrivial modifications of the renormalization-group flow occur, we can extract the physical consequences of these phenomena.

DOI
10.1103/physreva.44.2738

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Under certain conditions, the renormalization-group flow of models in statistical mechanics can change dramatically under just very small changes of given external parameters. This can typically occur close to bifurcations of fixed points, close to the complete disappearance of fixed points, or in regions where the renormalization-group flow becomes chaotic. We present some explicit examples of these phenomena for the case of a Lie group valued spin-model analyzed by means of a variational real-space renormalization group. By directly computing the free energy of these models around the parameter regions in which such nontrivial modifications of the renormalization-group flow occur, we can extract the physical consequences of these phenomena.
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The renormalization group (RG) in statistical physics focuses on ground-state properties of equilibrium systems. However, it is unclear how it should be generalized to nonunitary quantum dynamics caused by dissipation and measurement backaction, in which the notion of conserved energy is absent. Here, we extend the RG to cover nonunitary quantum dynamics governed by quantum operations. By performing coarse-graining in real time, we find that the competition between decoherence and coherent dynamics plays a decisive role in the behavior of the RG flow. In particular, we find that chaotic behavior without fixed points emerges in the RG flow when coherent dynamics is dominant, with the parity-time transition serving as a prototypical example. The measurement-induced parity-time transition belongs to the universality class of the one-dimensional Yang-Lee edge singularity, which serves as a guide for experimentally realizing imaginary fields in lattice spin systems with a quantum system.
In the previous chapter, the ferromagnetic Ising model provided a simple example of a phase diagram with an associated fixed point structure of the renormalization group flows. There were stable fixed points corresponding to low and high temperature phases, and a critical fixed point controlling the behaviour of critical Ising Systems. However, more realistic systems often have more complicated phase diagrams, and therefore a richer fixed point structure. In this chapter we study some of these examples, and show how, even with a rather qualitative description of renormalization group flows, it is possible to understand phase diagrams from the renormalization group viewpoint. More importantly, when more than one non-trivial fixed point is present, the question arises as to which is the dominant one in a particular region of the phase space. The renormalization group answers this question through the theory of cross-over behaviour . The existence of such phenomena, whereby different fixed points may influence the properties of the same system on different length scales, is totally absent in mean field treatments. Ising model with vacancies As a first example, consider a generalisation of the Ising model in which the spin variables s ( r ) may take the value 0 as well as ±1. This may be viewed as the classical version of a quantum spin-1 magnet, or as a lattice gas of magnetic particles, with | s ( r )| playing the role of the occupation number.
We study renormalization group flows in a space of observables computed by Monte Carlo simulations. As an example, we consider three-dimensional clock models, i.e., the XY spin model perturbed by a Z_{q} symmetric anisotropy field. For q=4, 5, 6, a scaling function with two relevant arguments describes all stages of the complex renormalization flow at the critical point and in the ordered phase, including the crossover from the U(1) Nambu-Goldstone fixed point to the ultimate Z_{q} symmetry-breaking fixed point. We expect our method to be useful in the context of quantum-critical points with inherent dangerously irrelevant operators that cannot be tuned away microscopically but whose renormalization flows can be analyzed as we do here for the clock models.
This chapter describes the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, the general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Phase transitions occur in macroscopic systems when one or more external parameters like temperature, pressure, electrical field, magnetic field, etc are varied. The chapter demonstrates the utility of the alternative formulations of thermodynamics and statistical mechanics in the context of phase transitions and shows how their use clarifies and unifies the theory of phase transitions. It deals with the basic thermodynamic description of a single component fluid. To understand different phases and associated transitions, one studies how the couplings flow in the parameter space under repeated Renormalization Group (RG) transformations. The flows around the fixed point are easily studied by linearizing the RG equations around the fixed points and determining the eigenvalues and eigenvectors of the linearized RG transformations.