Model reduction of large scale systems, the renormalization group, and exotic critical phenomena

Uwe Claus Täuber

Originally, the term ‘dynamic critical phenomena’ was coined for time-dependent properties near second-order phase transitions in thermal equilibrium . The kinetics of phase transitions in magnets, at the gas–liquid transition, and at the normal- to superfluid phase transition in helium 4 were among the prominent examples investigated already in the 1960s. The dynamic scaling hypothesis , generalizing the scaling ansatz for the static correlation function and introducing an additional dynamic critical exponent, successfully described a variety of these experiments. Yet only the development of the systematic renormalization group (RG) approach for critical phenomena in the subsequent decade provided a solid conceptual foundation for phenomenological scaling theories. Supplemented with exact solutions for certain idealized model systems, and guided by invaluable input from computer simulations in addition to experimental data, the renormalization group now provides a general framework to explore not only the static and dynamic properties near a critical point, but also the large-scale and low-frequency response in stable thermodynamic phases. Scaling concepts and the renormalization group have also been successfully applied to phase transitions at zero temperature driven by quantum rather than thermal fluctuations. It is to be hoped that RG methods may help to classify the strikingly rich phenomena encountered in far-from-equilibrium systems as well. Recent advances in studies of simple reaction-diffusion systems, active to absorbing state phase transitions, driven lattice gases, and scaling properties of moving interfaces and growing surfaces, among others, appear promising in this respect.

David A. Lavis, Reimer Kühn, Roman Frigg

Abstract This paper presents an in-depth analysis of the anatomy of both thermodynamics and statistical mechanics, together with the relationships between their constituent parts. Based on this analysis, using the renormalization group and finite-size scaling, we give a definition of a large but finite system and argue that phase transitions are represented correctly, as incipient singularities in such systems. We describe the role of the thermodynamic limit. And we explore the implications of this picture of critical phenomena for the questions of reduction and emergence.

Franz Wegner

Kenneth Geddes Wilson

Yoshinori Sakamoto, Hisamitsu Mukaida, Chigak Itoi

We study the critical behavior of a random field $\mathrm{O}(N)$ spin model with a second-rank random anisotropy term in spatial dimensions $4<d<6$, by means of the replica method and the $1∕N$ expansion. We obtain a replica-symmetric solution of the saddle-point equation, and we find the phase transition obeying dimensional reduction. We study the stability of the replica-symmetric saddle point against the fluctuation induced by the second-rank random anisotropy. We show that the eigenvalue of the Hessian at the replica-symmetric saddle point is strictly positive. Therefore, this saddle point is stable and the dimensional reduction holds in the $1∕N$ expansion. To check the consistency with the functional renormalization group method, we obtain all fixed points of the renormalization group in the large $N$ limit and discuss their stability. We find that the analytic fixed point yielding the dimensional reduction is practically singly unstable in a coupling constant space of the given model with large $N$. Thus, we conclude that the dimensional reduction holds for sufficiently large $N$.