Persistent homology analysis of deconfinement transition in effective Polyakov-line model

Abstract

The persistent homology analysis is applied to the effective Polyakov-line model on a rectangular lattice to investigate the confinement-deconfinement nature.The lattice data are mapped onto the complex Polyakov-line plane without taking the spatial average and then the plane is divided into three domains. This study is based on previous studies for the clusters and the percolation properties in lattice QCD, but the mathematical method of the analyses are different. The spatial distribution of the data in the individual domain is analyzed by using the persistent homology to obtain information of the multiscale structure of center clusters. In the confined phase, the data in the three domains show the same topological tendency characterized by the birth and death times of the holes which are estimated via the filtration of the alpha complexes in the data space, but do not in the deconfined phase. By considering the configuration averaged ratio of the birth and death times of holes, we can construct the nonlocal order-parameter of the confinement-deconfinement transition from the multiscale topological properties of center clusters.

Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology

“…For discrete models they construct α-complexes on subsets of the lattice sites with the same spin. This is similar to the approach used by Hirakida et al in [21] who look at the effective Polyakov line model.…”

“…But recently, among other geometric and topological approaches [16][17][18][19], there has been an interest in using persistent homology, a tool from the new field of topological data analysis (TDA), to produce interpretable features which are inherently sensitive to topological objects. These can then be compared in their own right, or fed into a machine learning model [20][21][22][23][24][25].…”

“…The idea here is that a thermodynamic phase transition necessarily coincides with a change in the topology of the energy level set, although such a change does not turn out to be a sufficient condition [28]. In the present work however, we shall make use of a newer paradigm, investigated also in [20][21][22][23][24], which we call persistent homology as an observable. Given a sampled configuration of a model, we construct a sequence of geometric complexes based on that configuration.…”

Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology

“…But recently there has been an interest in using persistent homology, a tool from the new field of topological data analysis (TDA), to produce interpretable features which are inherently sensitive to topological objects. These can then be compared in their own right, or fed into a machine learning model [18][19][20][21][22][23].…”

“…This approach is based on the topology hypothesis for the origin of phase transitions [24,25] and is the approach used in [23]. In the present work however, we shall make use of a newer paradigm, investigated also in [18][19][20][21][22], which we call persistent homology as an observable. Given a sampled configuration of a model, we construct a sequence of geometric complexes based on that configuration.…”