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On the Relationship Between Topological and Geometric Defects E-Mail: Sgriffin@Lbl.Gov and Nicola.Spaldin@Mat.Ethz.Ch Journal of Physics: Condensed Matter TOPICAL REVIEW Related content - Characteristics and controllability of On the relationship between topological and vortices in ferromagnetics, ferroelectrics, and multiferroics geometric defects Yue Zheng and W J Chen - Multiferroics of spin origin Yoshinori Tokura, Shinichiro Seki and To cite this article: Sinéad M Griffin and Nicola A Spaldin 2017 J. Phys.: Condens. Matter 29 343001 Naoto Nagaosa - Quantum spin ice: a search for gapless quantum spin liquids in pyrochlore magnets M J P Gingras and P A McClarty View the article online for updates and enhancements. This content was downloaded from IP address 128.3.150.53 on 12/09/2017 at 23:26 IOP Journal of Physics: Condensed Matter Journal of Physics: Condensed Matter J. Phys.: Condens. Matter J. Phys.: Condens. Matter 29 (2017) 343001 (8pp) https://doi.org/10.1088/1361-648X/aa7b5c 29 Topical Review 2017 On the relationship between topological © 2017 IOP Publishing Ltd and geometric defects JCOMEL Sinéad M Griffin1,2,4 and Nicola A Spaldin3,5 343001 1 Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States of America S M Griffin and N A Spaldin 2 Department of Physics, University of California Berkeley, Berkeley, CA 94720, United States of America 3 Materials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland On the relationship between topological and geometric defects E-mail: [email protected] and [email protected] Printed in the UK Received 15 March 2017, revised 16 June 2017 Accepted for publication 23 June 2017 Published 25 July 2017 CM Abstract 10.1088/1361-648X/aa7b5c The study of topology in solids is undergoing a renaissance following renewed interest in the properties of ferroic domain walls as well as recent discoveries regarding skyrmionic lattices. Each of these systems possess a property that is ‘protected’ in a symmetry sense, 1361-648X and is defined rigorously using a branch of mathematics known as topology. In this article we review the formal definition of topological defects as they are classified in terms of 34 homotopy theory, and discuss the precise symmetry-breaking conditions that lead to their formation. We distinguish topological defects from defects that arise from the details of the stacking or structure of the material but are not protected by symmetry, and we propose the term ‘geometric defects’ to describe the latter. We provide simple material examples of both topological and geometric defect types, and discuss the implications of the classification on the resulting material properties. Keywords: topological defect, geometric frustration, ferroic domain wall (Some figures may appear in colour only in the online journal) 1. Introduction and ferroelectric domain walls, both of these systems pos- sess a property that is ‘protected’ in a symmetry sense, and so Topological defects in materials have a long and rich history in can be rigorously defined by its corresponding mathematical the context of domain walls separating equivalent low-symmetry topological characteristics. At the same time, there has been states of different orientations in ferroics. Such defects are func- a flurry of exciting new discoveries regarding the properties tional entities in their own right, often showing exotic behavior of interfaces [5, 6], domain walls [7, 8], vortices [9], and spin such as sheet superconductivity in the ferroelastic twin walls of textures [10], particularly in ferroic and multiferroic systems, tungsten oxide, WO3 [1] or fast ionic transport along twin walls which it is tempting, although not always strictly correct, to in feldspar and perovskite structures [2]. discuss within a topological framework. Recently, interest in the concept of topology in condensed In this paper we review the distinction between topological matter has exploded following the discovery of skyrmionic defects as they are strictly classified in terms of mathematical magnetic lattices [3] and the experimental verification of the homotopy theory, and other defects arising from stacking or existence topological insulators [4]. Like some ferroelastic structural aspects of a material for which the term is formally inapplicable. In particular, we distinguish between topological 4 http://sites.google.com/site/sineadv0 defects resulting from precise symmetry-breaking conditions 5 http://theory.mat.ethz.ch classified by non-trivial homotopy groups, and defects which 1361-648X/17/343001+8$33.00 1 © 2017 IOP Publishing Ltd Printed in the UK J. Phys.: Condens. Matter 29 (2017) 343001 Topical Review are a consequence of the topography of the system. Since the latter often result from some type of geometric frustration, we refer to them in this work as geometric defects. Understanding the conditions for existence of topological or geometric defects in a system is essential for understanding its physical properties and potential applications. The occur- rence of topological defects in particular can prevent a system from reaching a single domain state, resulting instead in multi- ple ferroelectric or magnetic domains. These domains are sep- arated by domain walls, which can have novel properties and applications. An example is the conducting charged ferroelec- tric domain walls that are trapped in multiferroic hexagonal manganites by virtue of the topology [11]. In magnetic mat- erials with non-uniform spin textures, the topological ground Figure 1. Mexican-hat potential displaying the degenerate choice of ground states for spontaneous-symmetry breaking phase state is typically more energetically stable than a monodomain transitions. phase. As a result it can be controlled at a much lower energy cost, suggesting a plethora of applications in spin-based mem- have any orientation, giving an infinite manifold of degenerate ory and logic devices [12, 13]. Geometric defects, on the other ground states; in real materials, spin–orbit coupling term lifts hand, are associated with residual entropy which can lead to this rotational symmetry and causes a finite number of pre- exotic emergent phenomena such as the monopole-type exci- ferred easy axes. While many phase transitions in physics are tations observed in spin-ice systems [14]. caused by changes in temperature, the concept of spontaneous symmetry breaking is also applicable to changes driven by for example external electric or magnetic fields or pressure. 2. Topological defects The canonical spontaneous symmetry breaking phase trans ition is described by the ‘Mexican hat’ potential shown Topological defects were first defined rigorously by Tom in figure 1 with many examples existing in cosmology, high- Kibble in field theories in the context of cosmological phase energy physics and materials science [17]. The hat indicates transitions in the early universe [15, 16]. The mechanism a complex order-parameter function comprising an amplitude introduced by Kibble gives the symmetry requirements for the and a phase, with the horizontal distance from the peak of formation of topological defects during a phase transition, and the hat giving the magnitude of the order parameter, and the predicts—based on the particular details of the initial and final angle around the hat giving its phase. An example of a phase symmetries—what kinds of defects will form. In this sec- transition in condensed matter described by such a potential tion we describe the requirements of the Kibble mech anism, is superconductivity, for which the condensate wavefunction which determine whether a topological defect will form and is the order parameter. The non-superconducting (high-sym- how it will manifest. We then give three simple examples— metry) to superconducting (low-symmetry) phase transition misfit dislocations, skyrmions and ferroelectric domain spontaneously breaks the symmetry in such a way that the intersections—of topological defect formation in materials. allowed values of the order parameter can point anywhere on a circle (in field-theory language this is described as an S1 order-parameter space.) When the system drops from the 2.1. Spontaneous symmetry breaking high-symmetry peak of the hat to the low-symmetry brim, it The first requirement for topological defect formation within chooses a particular angle of phase, θ, from the infinite collec- tion of degenerate ground states with θ 0, 2π , losing its the Kibble mechanism is that the phase transition be spontane- ∈{ } ously symmetry breaking. A spontaneously symmetry break- initial U(1) symmetry in the process. Since all choices of θ ing phase transition is one that exhibits a symmetry change have the same energy, there exists a massless Goldstone boson from a higher to a lower symmetry state, which offers multiple running around the brim of the hat, which is a signature of the degenerate choices of the ground state. (Note that isomorphic initial higher symmetry of the system. phase transitions, such as the evaporation of a liquid to a gas, are not spontaneously symmetry breaking since the symmetry 2.2. Non-trivial homotopy does not change across the transition.) The development of the low symmetry state can be described by the onset of an Kibble’s second condition for topological defect formation order parameter, φ, which contains the relevant variables that is that the symmetry breaking is described by a non-trivial emerge during the transition. homotopy group [15]. Two topological spaces are homotopic A well-known example of a spontaneous symmetry break- if they can be deformed into each other by continuous trans- ing transition is the paramagnetic to ferromagnetic trans- itions such as twisting and pulling. The canonical case is the ition in magnets. Here the loss of time-reversal symmetry is topological equivalence of a coffee cup and a doughnut: a described by the onset of a magnetic order parameter, which coffee cup can be transformed continuously into a doughnut is often taken to be the magnetization. In the absence of aniso- by pushing the base of the coffee cup upwards until it meets tropy, the axis of magnetization in the ferromagnetic state can its rim, and then pulling outwards.
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