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arXiv:2312.04959 · Physical review. D/Physical review. D.

Detecting defect dynamics in relativistic field theories far from equilibrium using topological data analysis

Viktoria Noel, Daniel Spitz

Published Mar 11, 2024• 7 citations
arXivHTMLPDF

Abstract

We study nonequilibrium dynamics of relativistic N -component scalar field theories in Minkowski space-time in a classical statistical regime, where typical occupation numbers of modes are much larger than unity. In this strongly correlated system far from equilibrium, the role of different phenomena such as nonlinear wave propagation and defect dynamics remains to be clarified. We employ persistent homology to infer topological features of the nonequilibrium many-body system for different numbers of field components N via a hierarchy of cubical complexes. Specifically, we show that the persistent homology of local energy density fluctuations can give rise to signatures of self-similar scaling associated with topological defects, distinct from the scaling behavior of nonlinear wave modes. This contributes to the systematic understanding of the role of topological defects for far-from-equilibrium time evolutions of nonlinear many-body systems. Published by the American Physical Society 2024

DOI
10.1103/physrevd.109.056011

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We study nonequilibrium dynamics of relativistic N -component scalar field theories in Minkowski space-time in a classical statistical regime, where typical occupation numbers of modes are much larger than unity. In this strongly correlated system far from equilibrium, the role of different phenomena such as nonlinear wave propagation and defect dynamics remains to be clarified. We employ persistent homology to infer topological features of the nonequilibrium many-body system for different numbers of field components N via a hierarchy of cubical complexes. Specifically, we show that the persistent homology of local energy density fluctuations can give rise to signatures of self-similar scaling associated with topological defects, distinct from the scaling behavior of nonlinear wave modes. This contributes to the systematic understanding of the role of topological defects for far-from-equilibrium time evolutions of nonlinear many-body systems. Published by the American Physical Society 2024
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 many-body dynamics in terms of robust topological structures beyond standard field theoretic techniques.
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 simulated data 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 universal phenomena. A possible explanation of the underlying processes is provided in terms of mixing wave turbulence and 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 many-body dynamics in terms of robust topological structures beyond standard field theoretic techniques.
This thesis motivates and examines the use of methods from topological data analysis in detecting and analysing topological features relevant to models from sta-tistical physics and particle physics.In statistical physics, we use persistent homology as an observable of three dif-ferent variants of the two-dimensional XY model in order to identify relevant topo-logical features and study their relation to the phase transitions undergone by each model. We examine models with the classical XY action, a topological lattice action, and an action with an additional nematic term. In particular, we introduce a new way of computing the persistent homology of lattice spin model configurations and demonstrate its use in detecting topological defects called vortices. By considering the fluctuations in the output of logistic regression and k-nearest neighbours mod-els trained on persistence images, we develop a methodology to extract estimates of the critical temperature and the critical exponent of the correlation length. We put particular emphasis on finite-size scaling behaviour and producing estimates with quantifiable error. For each model we successfully identify its phase transition(s) and are able to get an accurate determination of the critical temperatures and critical exponents of the correlation length.In particle physics, we investigate the use of persistent homology as a means to detect and quantitatively describe center vortices in SU(2) lattice gauge theory in a gauge-invariant manner. The sensitivity of our method to vortices in the deconfined phase is confirmed by using twisted boundary conditions which inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a k-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical β and critical exponent of correlation length for the deconfinement phase transition. We also use persistent homology to study the stability of vortices under gradient flow and the classification of different vortex surface geometries.
In recent years, there have been important advances in understanding the far-from-equilibrium dynamics in different physical systems. In ultra-relativistic heavy-ion collisions, the combination of different methods led to the development of a weak-coupling description of the early-time dynamics. The numerical observation of a classical universal attractor played a crucial role for this. Such attractors, also known as non-thermal fixed points (NTFPs), have been now predicted for different scalar and gauge theories. An important universal NTFP emerges in scalar theories modeling ultra-cold atoms, inflation or dark matter, and its scaling properties have been recently observed in an ultra-cold atom experiment. In this proceeding, recent progress in selected topics of the far-from-equilibrium evolution in these systems will be discussed. A new method to extract the spectral function numerically is a particularly promising tool to better understand their microscopic properties.
The Kibble Zurek mechanism in a relativistic ${\ensuremath{\phi}}^{4}$ scalar field theory in D=(1+1)D=(1+1)D=(1+1) is studied using uniform matrix product states. The equal time two point function in momentum space G2(k){G}_{2}(k)G2​(k) is approximated as the system is driven through a quantum phase transition at a variety of different quench rates ${\ensuremath{\tau}}_{Q}$. We focus on looking for signatures of topological defect formation in the system and demonstrate the consistency of the picture that the two point function G2(k){G}_{2}(k)G2​(k) displays two characteristic scales, the defect density nnn and the kink width dK{d}_{K}dK​. Consequently, G2(k){G}_{2}(k)G2​(k) provides a clear signature for the formation of defects and a well defined measure of the defect density in the system. These results provide a benchmark for the use of tensor networks as powerful nonperturbative nonequilibrium methods for relativistic quantum field theory, providing a promising technique for the future study of high energy physics and cosmology.
We investigate universal behavior of isolated many-body systems far from equilibrium, which is relevant for a wide range of applications from ultracold quantum gases to high-energy particle physics. The universality is based on the existence of nonthermal fixed points, which represent nonequilibrium attractor solutions with self-similar scaling behavior. The corresponding dynamic universality classes turn out to be remarkably large, encompassing both relativistic as well as nonrelativistic quantum and classical systems. For the examples of nonrelativistic (Gross-Pitaevskii) and relativistic scalar field theory with quartic self-interactions, we demonstrate that infrared scaling exponents as well as scaling functions agree. We perform two independent nonperturbative calculations, first by using classical-statistical lattice simulation techniques and second by applying a vertex-resummed kinetic theory. The latter extends kinetic descriptions to the nonperturbative regime of overoccupied modes. Our results open new perspectives to learn from experiments with cold atoms aspects about the dynamics during the early stages of our universe.