[Paper Review] Persistent homology analysis of deconfinement transition in effective Polyakov-line model

Takehiro Hirakida, Kouji Kashiwa+4 more|arXiv (Cornell University)|Oct 17, 2018

Topological and Geometric Data Analysis23 references4 citations

TL;DR

This study applies persistent homology to the effective Polyakov-line model on a lattice to analyze multiscale topological structures of center clusters in the confined and deconfined phases. By mapping local Polyakov-line values to three domains in the complex plane and analyzing their spatial distributions via alpha complex filtration, the authors identify a distinct topological transition: in the confined phase, all domains show similar hole birth/death times, while in the deconfined phase, the ${\cal Z}_0$ domain exhibits random-like clustering. The configuration-averaged ratio of hole birth to death times serves as a robust, nonlocal order parameter for the confinement-deconfinement transition, with significantly reduced statistical error compared to the Polyakov-line expectation value.

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.

Motivation & Objective

• To investigate the multiscale topological structure of center clusters in the effective Polyakov-line model using persistent homology.
• To identify topological signatures that distinguish the confined and deconfined phases of QCD-like systems.
• To construct a nonlocal order parameter based on persistent homology for the confinement-deconfinement transition.
• To compare the topological properties of spatial distributions in different Polyakov-line domains ($\mathcal{Z}_0$, $\mathcal{Z}_1$, $\mathcal{Z}_2$) across phases.
• To assess the statistical robustness and sensitivity of the persistent homology-based order parameter compared to the conventional Polyakov-line.

Proposed method

• Map local Polyakov-line values on a rectangular lattice to three domains in the complex plane: $\mathcal{Z}_0$, $\mathcal{Z}_1$, and $\mathcal{Z}_2$ based on phase angles.
• Apply persistent homology to spatial distributions of data points in each domain by constructing alpha complexes through filtration over increasing radii.
• Extract topological invariants—specifically, the birth and death times of topological holes (1-cycles) from the persistence diagram.
• Compute the configuration-averaged ratio of hole birth time to death time as a nonlocal order parameter.
• Compare results with a random configuration model at the same occupation rate to assess non-randomness.
• Use the ratio of birth to death times as a phase-sensitive indicator, sensitive to structural changes across the transition.

Experimental results

Research questions

• RQ1 How do the multiscale topological structures of center clusters differ between the confined and deconfined phases in the effective Polyakov-line model?
• RQ2 Can persistent homology detect a topological signature of the confinement-deconfinement transition that is not captured by the Polyakov-line expectation value?
• RQ3 How does the spatial distribution of data points in the $\mathcal{Z}_0$, $\mathcal{Z}_1$, and $\mathcal{Z}_2$ domains differ topologically across phases?
• RQ4 Is the configuration-averaged ratio of hole birth to death times a robust and sensitive nonlocal order parameter for the phase transition?
• RQ5 To what extent do the topological features in the data space reflect underlying physical phase structure beyond symmetry-based order parameters?

Key findings

• In the confined phase, the topological structure of holes in all three domains ($\mathcal{Z}_0$, $\mathcal{Z}_1$, $\mathcal{Z}_2$) exhibits similar birth and death times, indicating a common topological tendency.
• In the deconfined phase, the $\mathcal{Z}_0$ domain shows dense, random-like clustering with early-born, short-lived holes, while $\mathcal{Z}_1$ and $\mathcal{Z}_2$ domains exhibit late-born, short-lived structures.
• The configuration-averaged ratio of hole birth to death time shows a sharp transition between phases, with distinct behaviors in $\mathcal{Z}_0$ and $\mathcal{Z}_1,\mathcal{Z}_2$ domains in the deconfined phase.
• The persistent homology-based order parameter exhibits significantly smaller statistical error than the Polyakov-line expectation value, indicating higher sensitivity with fewer configurations.
• The topological structure in the $\mathcal{Z}_0$ domain of the deconfined phase is qualitatively similar to a random configuration with 98% occupation rate, suggesting loss of long-range topological order.
• The difference between the EPL model and the random configuration in the same occupation rate indicates the presence of more complex, nontrivial multiscale topological structures in the physical system.