Topological Kolmogorov complexity and the Berezinskii-Kosterlitz-Thouless mechanism

TL;DR

This paper explores how topological features influence the complexity of classical many-body systems, revealing a connection between topology, phase transitions, and Kolmogorov complexity, especially in the XY model at the BKT transition.

Contribution

It introduces a theory linking topology and Kolmogorov complexity in classical statistical mechanics and demonstrates how complexity depends on topology in the XY model at the BKT transition.

Findings

Complexity is topology-dependent in the XY model at the BKT transition.

Intrinsic dimension remains topology-independent for Ising and Heisenberg models.

Complexity scales with genus in the BKT phase for g-tori.

Abstract

Topology plays a fundamental role in our understanding of many-body physics, from vortices and solitons in classical field theory, to phases and excitations in quantum matter. Topological phenomena are intimately connected to the distribution of information content - that, differently from ordinary matter, is now governed by non-local degrees of freedom. However, a precise characterization of how topological effects govern the complexity of a many-body state - i.e., its partition function - is presently unclear. In this work, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. In concrete, we present a theory that shows how the Kolmogorov complexity of a classical partition function sampling carries unique, distinctive features depending on the presence of topological excitations in the system. We confront two-dimensional Ising, Heisenberg, and XY models on several topologies and study the corresponding samplings as high-dimensional manifolds in configuration space, quantifying their complexity via the intrinsic dimension. While for the Ising and Heisenberg models the intrinsic dimension is independent of the real-space topology, for the XY model it depends crucially on temperature: across the Berezkinskii-Kosterlitz-Thouless (BKT) transition, complexity becomes topology dependent. In the BKT phase, it displays a characteristic dependence on the homology of the real-space manifold, and, for $g$-torii, it follows a scaling that is solely genus dependent. We argue that this behavior is intimately connected to the emergence of an order parameter in data space, the conditional connectivity, that displays scaling behavior.