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arXiv:2305.05396 · Physical review. E

Topological Kolmogorov complexity and the Berezinskii-Kosterlitz-Thouless mechanism

Vittorio Vitale, Tiago Mendes-Santos, Álex Rodríguez, Marcello Dalmonte

Published Mar 1, 2024• 4 citations

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 nonlocal 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 paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely 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, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.

DOI: 10.1103/physreve.109.034102

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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 nonlocal 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 paper, we show how topology and complexity are directly intertwined concepts in the context of classical statistical mechanics. We concretely 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, which displays scaling behavior. Our approach paves the way for an understanding of topological phenomena emergent from many-body interactions from the perspective of Kolmogorov complexity.

The fundamental concepts of phases of matter and thermal phase transitions constitute the cornerstone of our understanding of the physical universe. The historical development of the phase transition theory from Landau’s spontaneous symmetry breaking paradigm to modern topological phase transition theories represents a major milestone in the evolution of numerous scientific disciplines. From the perspective of emergent philosophy, the interplay of topological excitations leads to enriched physical phenomena. One prominent prototype is the Berezinskii-Kosterlitz-Thouless (BKT) phase transition, where unbinding of integer vortices occurs in the absence of spontaneous breaking of continuous U(1) symmetry. Using the state-of-the-art tensor network methods, we express the partition function of the two-dimensional XY-related system in terms of a product of one-dimensional transfer operators. From the singularities of the entanglement entropy of the one-dimensional transfer operator, we accurately determine the complete phase diagram of the partition function. This method provides new insights into the emergent phenomenon driven by topological excitations, and sheds new light on future studies of 2D systems with continuous symmetries.

The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.

The Berezinskii-Kosterlitz-Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models these are vortices: above a transition temperature

T_{\rm BKT}

vortices are free, below this transition temperature they get confined. In this work we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition, driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent

z

as the quantum Griffiths transition but have nothing to do with disorder.

The physics of the topological state of matter is the second revolution in quantum mechanics. We study the effect of interactions on the topological quantum phase transition and the quantum Berezinskii-Kosterlitz-Thouless (QBKT) transition in topological state of a quantum many-body condensed matter system. We predict a topological quantum phase transition from topological superconducting phase to an insulating phase for the interacting Kitaev chain. We observe interesting behaviour from the results of renormalization group study on the topological superconducting phase. We derive the renormalization group (RG) equation for QBKT through different routes with a few exact solutions along with the physical explanations, wherein we find the existence of two new important emergent phases apart from the two conventional phases of this model Hamiltonian. We also present results of a length-scale dependent study to predict asymptotic freedom like behaviour of the system. We do rigorous quantum field theoretical renormalization group calculations to solve this problem.

Quantum many-body systems are characterized by patterns of correlations defining highly nontrivial manifolds when interpreted as data structures.Physical properties of phases and phase transitions are typically retrieved via correlation functions, that are related to observable response functions.Recent experiments have demonstrated capabilities to fully characterize quantum many-body systems via wavefunction snapshots, opening new possibilities to analyze quantum phenomena.Here, we introduce a method to data mine the correlation structure of quantum partition functions via their path integral (or equivalently, stochastic series expansion) manifold.We characterize path-integral manifolds generated via state-of-the-art quantum Monte Carlo methods utilizing the intrinsic dimension (ID) and the variance of distances between nearest-neighbor (NN) configurations: the former is related to data-set complexity, while the latter is able to diagnose connectivity features of points in configuration space.We show how these properties feature universal patterns in the vicinity of quantum criticality, that reveal how data structures simplify systematically at quantum phase transitions.This is further reflected by the fact that both ID and variance of NN distances exhibit universal scaling behavior in the vicinity of second-order and Berezinskii-Kosterlitz-Thouless critical points.Finally, we show how non-Abelian symmetries dramatically influence quantum data sets, due to the nature of (noncommuting) conserved charges in the quantum case.Complementary to neural-network representations, our approach represents a first elementary step towards a systematic characterization of path-integral manifolds before any dimensional reduction is taken, that is informative about universal behavior and complexity, and can find immediate application to both experiments and Monte Carlo simulations.

The Berezinskii-Kosterlitz-Thouless (BKT) phase transition drives the unbinding of topological defects in many two-dimensional systems. In the two-dimensional Coulomb gas, it corresponds to an insulator-conductor transition driven by charge deconfinement. We investigate the global topological properties of this transition, both analytically and by numerical simulation, using a lattice-field description of the two-dimensional Coulomb gas on a torus. The BKT transition is shown to be an ergodicity breaking between the topological sectors of the electric field, which implies a definition of topological order in terms of broken ergodicity. The breakdown of local topological order at the BKT transition leads to the excitation of global topological defects in the electric field, corresponding to different topological sectors. The quantized nature of these classical excitations, and their strict suppression by ergodicity breaking in the low-temperature phase, afford striking global signatures of topological-sector fluctuations at the BKT transition. We discuss how these signatures could be detected in experiments on, for example, magnetic films and cold-atom systems.