Uncomputably complex renormalisation group flows

• Published in: Nature communications Vol. 13; no. 1; p. 7618
• Main Authors: Watson, James D., Onorati, Emilio, Cubitt, Toby S.
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 09.12.2022; Nature Publishing Group; Nature Portfolio
• Subjects: 639/766/119/2795; 639/766/259; 639/766/483/481; 639/766/483/640; Critical point; Energy; Exact solutions; Feature extraction; Hilbert space; Humanities and Social Sciences; multidisciplinary; Phase diagrams; Physical properties; Physics; Science; Science (multidisciplinary)
• ISSN: 2041-1723; 2041-1723
• DOI: 10.1038/s41467-022-35179-4

Renormalisation group methods are among the most important techniques for analysing the physics of many-body systems: by iterating a renormalisation group map, which coarse-grains the description of a system and generates a flow in the parameter space, physical properties of interest can be extracted. However, recent work has shown that important physical features, such as the spectral gap and phase diagram, may be impossible to determine, even in principle. Following these insights, we construct a rigorous renormalisation group map for the original undecidable many-body system that appeared in the literature, which reveals a renormalisation group flow so complex that it cannot be predicted. We prove that each step of this map is computable, and that it converges to the correct fixed points, yet the resulting flow is uncomputable. This extreme form of unpredictability for renormalisation group flows had not been shown before and goes beyond the chaotic behaviour seen previously. Renormalisation group methods serve for finding analytic solutions, critical points and computing phase diagrams of many-body systems. Here the authors demonstrate that renormalisation group schemes can be constructed for undecidable many-body systems, giving rise to the types of renormalisation group flow which are strictly more unpredictable than chaotic flows.