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Cambridge University Press eBooks

Topological Defects

Aron J. Beekman, Louk Rademaker, Jasper van Wezel

Published Sep 3, 2026• 0 citations
Publisher

Abstract

In this chapter, we introduce topological defects as excitations complementary to the elementary Nambu–Goldstone modes and relate their classification to the presence of broken symmetries. We show that the proliferation of topological defects can lead to the suppression of long-range correlations and the emergence of algebraic long-range order. The Berezinskii–Kosterlitz–Thouless phase transition is discussed both generally and in the specific example of the one-dimensional Ising model. In the same context, the Kramers–Wannier duality between topological defects and elementary modes is introduced.

DOI
10.1017/9781009575782.010

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In this chapter, we introduce topological defects as excitations complementary to the elementary Nambu–Goldstone modes and relate their classification to the presence of broken symmetries. We show that the proliferation of topological defects can lead to the suppression of long-range correlations and the emergence of algebraic long-range order. The Berezinskii–Kosterlitz–Thouless phase transition is discussed both generally and in the specific example of the one-dimensional Ising model. In the same context, the Kramers–Wannier duality between topological defects and elementary modes is introduced.
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.
Abstract Statistical-mechanical systems often involve discrete elementary degrees of freedom such as spins in the Ising model. Field theories, on the other hand, have continuous fields, defined over the whole space-time or part of it, as fundamental degrees of freedom. These two seemingly different descriptions of physical phenomena can be related close to the critical point. The present chapter summarizes how the description by continuous fields emerges from discrete degrees of freedom in a more systematic manner than in previous chapters. The phenomenological Landau-Ginzburg approach, based on the concept of order parameter, is expanded to generate effective field theories. The important roles of symmetry and topology are also elucidated in some detail. Also shown are some of the important consequences of having a broken-symmetry phase, such as long-range order, the emergence of Nambu-Goldstone modes when the symmetry involved is continuous, and topological defects.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is an archetypal example of a topological phase transition, which is driven by the proliferation of vortices. In this Letter, we analyze the persistence of the BKT transition in the XY model under the influence of long-range algebraically decaying interactions of the form ∼1/r^{2+σ}. The model hosts a magnetized low temperature phase for sufficiently small σ. Crucially, in the presence of long-range interactions, spin waves renormalize the interaction between vortices, which stabilizes the BKT transition. As a result, we find that there is no direct transition from the magnetized to the disordered phase and that the BKT transition persists for arbitrary long-range exponents, which is distinct from previous results. The key methodological advance is the inclusion of coupling between spin wave and topological excitations. We do this by developing Landau-Peierls-type arguments and using higher-order renormalization group calculations, obtaining compatible results. We emphasize that Landau-Peierls-type arguments are a powerful tool for analyzing continuous spin models. We discuss the relevance of our findings for current Rydberg atom experiments, and highlight the importance of long-range couplings for other types of topological defects.
Condensed matter is the area of physics which deals with the phases of matter in condensed form with an aim to understand the properties of these phases and the transition between them based on the known laws of physics. This is usually done by figuring out some characteristic properties of the phase which sets it apart from the others, e.g., symmetry, topological order, entanglement, etc. The most common way of characterizing the transition between these phases is the broken symmetry. In this classification method, one finds the symmetry which phase breaks spontaneously during the transition and develops a local long-range order. This is the famous Landau’s symmetry breaking paradigm [ 1 ]. Landau’s symmetry breaking theory is a phenomenological theory which describes the transition from a symmetric phase (high-temperature phases) to a less symmetric/broken symmetry phase (phases at low temperature). A zero-temperature phase transition is also possible (by tuning the parameters of the Hamiltonian) and is known as quantum phase transition [ 2 ]. Although there were other paradigms of phase transition available, like Anderson localization [ 3 ], Berezinsky—Kosterlitz—Thouless transition [ 4 ] (broken symmetry but no long-range order), etc., Landau’s spontaneous symmetry breaking dominated over the others.
We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in 222 dimensions, with electromagnetic duality for finite gauge theories in 333 dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.