Report on scipost_202102_00007v2
Daniel Spitz, Jürgen Berges, Markus K. Oberthaler, Anna Wienhard · 2021
Inspired by topological data analysis techniques, we introduce persistent homology observables and apply them in a geometric analysis of the dynamics of quantum field theories.As a prototype application, we consider data from a classical-statistical simulation of a two-dimensional Bose gas far from equilibrium.We discover a continuous spectrum of dynamical scaling exponents, which provides a refined classification of nonequilibrium self-similar phenomena.A possible explanation of the underlying processes is provided in terms of mixing strong wave turbulence and anomalous vortex kinetics components in point clouds.We find that the persistent homology scaling exponents are inherently linked to the geometry of the system, as the derivation of a packing relation reveals.The approach opens new ways of analyzing quantum manybody dynamics in terms of robust topological structures beyond standard field theoretic techniques.3.5 Scaling species and exponents mixing conjecture 15 4 Persistent homology observables and self-similarity 16 4.1 Persistent homology observables via functional summaries 16 4.2The asymptotic persistence pair distribution and geometric quantities 17 4.3 Self-similar scaling approach 18 4.3.1 Scaling ansatz to the asymptotic persistence pair distribution 19 4.3.2A heuristic packing relation 19 5 Exponent shifts, persistences and Betti number distributions 20 5.1 Amplitude redistribution-induced exponents shifts 20 5.2 Persistence distributions 22 5.3 Betti numbers as a consistency check 23 6 Conclusions 24 A The mathematics of persistent homology 26 A.1 Relevant notions from algebraic topology 26 A.2 The construction and structure of persistent homology groups 26 B The computational pipeline 27 C Packing relation from bounded total persistence 28 D Relating persistent homology exponents to correlation function exponents 29 E Details on the nonrelativistic Bose gas simulations 29 F Numerical convergence of persistent homology observables 30 G Numerical protocol to extract persistent homology scaling exponents 31 References 33