Nonlinearity and Topology

Nonlinearity and Topology Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver Abstract The interplay of nonlinearity and topology results in many novel and emergent properties across a number of physical systems such as chiral magnets, nematic liquid crystals, Bose-Einstein condensates, photonics, high energy physics, etc. It also results in a wide variety of topological defects such as solitons, vortices, skyrmions, merons, hopfions, monopoles to name just a few. Interaction among and collision of these nontrivial defects itself is a topic of great interest. Curvature and underlying geometry also affect the shape, interaction and behavior of these defects. Such properties can be studied using techniques such as, e.g.the Bogomolnyi decomposition.Some applications of this interplay,e.g.innonreciprocalphotonicsas well as topological materials such as Dirac and Weyl semimetals, are also elucidated. Avadh Saxena Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA e-mail: [email protected] Panayotis G. Kevrekidis Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA e-mail: [email protected] Jesús Cuevas-Maraver Grupo de Fisica No Lineal, Departamento de Fisica Aplicada I, Universidad de Sevilla, Escuela Politécnica Superior, C/ Virgen de África, 7 41011 Sevilla, Spain Jesús Cuevas-Maraver Instituto de Matemáticas de la Universidad de Sevilla (IMUS) Edificio Celestino Mutis, Avda. Reina Mercedes s/n 41012 Sevilla, Spain e-mail: [email protected] 1 arXiv:2001.10038v2 [nlin.PS] 31 Jan 2020 2 Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver 1 Introduction The main context of this chapter is how topological effects in nonlinear systems give rise to a rich playground of excitations and properties. Topology, whether in real space or momentum space or more generally in a parameter space, is associated with certain system properties remaining unaltered under continuous deformation. It follows that during deformation neighboring points remain close to each other. Topology could be local, e.g. change in the lattice or network due to a defect, or global. The latter means attributes such as the genus (g) or Euler characteristic (χ) are overall or global features of a system. Boundary conditions play an important role (through χ), for example in finite carbon nanotubes or edge states in topological materials: quantum Hall systems, topological insulators [1], topological superconductors [2], Dirac and Weyl semimetals [3], etc. The latter are three dimensional analogs of graphene featuring gapless electronic excitations that are protected by topology and (time reversal, space inversion or other crystalline) symmetry [3]. Very recent experiments indicate that these materials, specifically Weyl semimetals, may also provide a realization of axions, very weakly interacting neutral particles in quantum field theory and potential candidates for dark matter, in condensed matter [4, 5]. In many physical systems and materials [6] there are point defects as well as extended or topological defects. The topological defects can significantly alter the physical properties and dynamics of the system. Apart from the celebrated solitonlike defects there is a whole slew of more elaborate ones that include skyrmions, merons, hopfions, monopoles, dislocations, disclinations among others. In this chapter we discuss how such defects arise in chiral magnets, nematic liquid crystals, Bose-Einstein condensates (BECs), etc. We also discuss the role of topology in the momentum space, particularly in the context of topological materials. The combination of nonlinearity and topology also provides a highly desirable functionality in photonics, namely nonreciprocity, which is quite important for a variety of photonic devices including optical isolators [7]. We thus provide examples of the interplay between nonlinearity and topology in photonics as well as condensed matter analogs. In addition, we illustrate the role of geometry and topology in determining spin textures via the so-called Bogomolnyi decomposition [8]. Finally, we discuss several open problems and future directions with regard to the role of topology in the presence of nonlinearity. 2 Topological defects in Nonlinear field theories We consider a variety of topological defects that arise in a number of field theories including the nonlinear σ-model [9, 10]. The Hamiltonian for the latter is given by H=∫(∇n)2d2x,n2=1,(1) Nonlinearity and Topology 3 where the unit vector nlives on a unit sphere. This model can support scalar soliton configurations under special conditions. However, under a scaling transformation x→λxand y→λythe Hamiltonian Hremains invariant and thus a soliton solution can be trivially scaled to a point because there is no length scale in the plane. We will return to this point later when we consider the nonlinear σ-model on curved manifolds. We note here that to describe disordered Weyl semimetals an anisotropic topological term can be analytically derived from the action of the nonlinear σmodel [11]. In the next section we consider a variety of topological defects such as skyrmions, merons and hopfions. Given their extensive interest and applicability, subsequently in Sec. 4, we consider vortices and vortex loops/rings. We then turn to different prototypical applications such as liquid crystals (and the emergence of skyrmions in them) in Sec. 5, as well as Bose-Einstein condensates in Sec. 6. After providing an example of a theoretical tool for the study of topology in curved manifolds via the Bogomolnyi decomposition (section 7), we present a broader perspective of the impact of topological ideas in Materials (Sec. 8), Optics (Sec. 9) and Acoustics and beyond (Sec. 10). Then in Sec. 11, we summarize our findings and present our Conclusions, as well as some directions for future work. 3 Skyrmions, Merons and Hopfions Beyond the well known solitons, there are more exotic topological defects such as vector field or spin textures called skyrmions [10], which can have topological charge of one (or two or even more). As shown in Fig. 1, in a skyrmion at the outer boundary all spins point up (red arrows) whereas at the center there is a spin pointing down (blue arrow). Therefore somewhere in the middle the spins have to lie in the plane (green arrows). Half skyrmions are also referred to as merons and have a topological charge of one half: the outer spins point up whereas the spin in the center lies in the plane. There are many other related topological defects such as sphalerons and bags (or lumps) known in high energy physics [9]. Similarly, three dimensional defects, e.g. vortex lines, vortex loops (rings) and knots appear in many physical systems, including, e.g. polymeric knots [12]. 3.1 Skyrmions in Chiral Magnets Before discussing magnets we note that beyond magnetic materials (e.g. ferroelectrics) skyrmions have been observed at interfaces [13]. In this case, the texture is given in terms of polar vectors or electric dipoles. Nanodots and nanocomposites can also stabilize skyrmions (due to boundary conditions) in polar materials [14]. The Hamiltonian for a chiral magnet (i.e. lacking spatial inversion symmetry in its crystal structure, e.g. MnSi) consists of the nonlinear σ-model plus the Zeeman 4 Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver BLOCH -and NEEL -type Skyrmions Bloch (MnSi)Neel (GaV4S8) Fig. 1 (a) Bloch and (b) Néel type skyrmions in a chiral magnet MnSi and a lacunar spinel GaV4S8, respectively. Below the skyrmion textures their radial cross sections show Bloch and Néel wall, respectively. Reproduced from [16]. ©2015 by the Nature Publishing Group term in addition to the Dzyaloshinskii-Moriya interaction [15] H=∫J(∇n)2+Dn· (∇ × n) − n·Bd2x,(2) where Jdenotes the (magnetic) exchange constant, Bis the external magnetic field in the last term in the Hamiltonian representing the Zeeman interaction and D represents the strength of the Dzyaloshinskii-Moriya interaction. The latter arises from the spin-orbit interaction at the microscopic level. Here nis a unit vector describing the direction of the magnetic moment. The topological charge associated with a skyrmion spin configuration is given by [15] Q=1 4π∫dr2n· (∂xn×∂yn)=±1.(3) For a metallic material when a conduction electron traverses across a skyrmion it gets spin polarized as a result of its interaction with the spin configuration of the skyrmion. In addition, the conduction electron is subjected to an effective electric and magnetic field [15], respectively given by E= ~ 2e[n· (∇n×∂tn)],(4) B= ~c 2e[n· (∂xn×∂yn)].(5) These emergent fields give rise to the topological and skyrmion Hall effects. Nonlinearity and Topology 5 Antiskyrmions: Mn1.4Pt0.9Pd0.1Sn A.K. Nayak et al. Nature 548, 561 (2017) Fig. 2 (a) Anti-skyrmions (center panels) in Mn1.4Pt0.9Pd0.1Sn and its comparison with the Bloch (left panels) and Néel skyrmions (right panels). The radial cross sections are shown below the spin textures. Schematics of the magnetic moments (green arrows) and the associated Lorentz deflections of transmitted electrons (red arrows) are shown in the middle row. The bottom row shows the corresponding simulated Lorentz Transmission Electron Microscopy (LTEM) patterns with dark and bright lobes. Reproduced from [17]. ©2017 by the Nature Publishing Group Depending on the material or system, skyrmions can have chirality (i.e. Bloch skyrmion as in MnSi) or no chirality (i.e. Néel skyrmion as in GaV4S8) as depicted in Fig. 1. They are so called because the radial cut of a Bloch skyrmion provides a magnetic Bloch domain wall (which is a strictly 3D structure) whereas a radial cut of the Néel skyrmion leads to a magnetic Néel domain wall (which is a 2D structure). Note that the spin configurations of the two types of skyrmions are topologically equivalent. Anti-skyrmions, which have structural characteristics of both the Bloch and Néel skyrmions, have also been observed using Lorentz Transmission Electron Microscopy (LTEM) [17] in tetragonal Heusler materials even above the room temperature, as shown in Fig. 2. We note here that spin-1 photonic skyrmions have also been described in the literature [18]. 3.2 Merons Half-skyrmions whose field covers only a hemisphere (i.e. topological charge 1/2) are known as merons. They have been observed (and modeled) in liquid crystals [19] and magnetic multi-layers [20]. If we add a magnetic anisotropy energy term 6 Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver Fig. 3 Comparison of skyrmion and meron latices in chiral magnets. A triangular skyrmion lattice arises from a spiral phase (top row) but a triangular meron lattice is at best metastable (bottom left). In contrast, a square meron lattice is stable (bottom right) but a square skyrmion lattice is at best metastable. On the other hand, in nematic liquid crystals a triangular meron lattice is stable but a triangular skyrmion lattice is unstable (not shown). Reproduced from [19] An2 zto the skyrmion Hamiltonian in Eq. (2), in the large anisotropy limit a skyrmion breaks into merons [21]. When A>0it is called the easy-plane anisotropy which is what we will consider here. The case of A<0is called the easy-axis anisotropy. As Ais increased the skyrmion size increases, particularly due to the expansion of the equatorial region of the skyrmion. At a certain large value of A, skyrmions become unstable and merons emerge. In chiral magnets a stable triangular lattice of skyrmions emerges from the spiral (or helical) phase as depicted in Fig. 3. In the spiral phase all spins are parallel and point in the same direction at one end. After a helical twist of 2πthey come back to the original parallel state, see Fig. 3(a). The square skyrmion lattice is at best metastable; however, a square meron lattice is allowed. On the contrary, in nematic liquid crystals the triangular skyrmion lattice is at best metastable but a triangular meron lattice is stable [19]. Merons have been experimentally observed at room temperature in a chiral-lattice magnet Co8Zn9Mn3, a material exhibiting in-plane magnetic anisotropy [22]. In this material a meron-antimeron square lattice emerges from the helical state of spins and then transforms into a triangular lattice of skyrmions when a magnetic field is applied, see Fig. 4. Interestingly, in analogy with the baryon model in high energy Nonlinearity and Topology 7 Experimental Observation of Merons X.Z. Yu et al., Nature 564, 95 (2018) Fig. 4 (a) - (d) Various experimentally observed meron and anti-meron spin structures in the chiral magnet Co8Zn9Mn3at room temperature. (e) Theoretically predicted and (f) LTEM observed meron/anti-meron square lattice. Reproduced from [22] physics, skyrmion bags have been observed both in chiral magnets and nematic liquid crystals [23]. These bags are multi-skyrmion configurations where a large skyrmion contains a variable number of antiskyrmions inside it, as depicted in Fig. 5. Merons can be compared and contrasted with magnetic vortices; there is a difference in their spin configurations. 3.3 Hopfions and Torons Three dimensional topological solitons (appearing in 3+1 dimensional scalar field theories) that can be characterized by the integer-valued Hopf invariant are known as hopfions [10]. Ludwig Faddeev proposed their existence in the 1970s. They represent one of the best known examples of knot solitons in field theory. In 1931 Heinz Hopf considered a link of two loops, thus paving the way for the linking number of circles as a topological invariant, i.e. the Hopf number. The topological charge or linking number of a hopfion is the homotopy group of the Hopf map π3(S2)=Z, where Zis the group of relative integers. The Hopf fibration is a topologically stable texture of a smooth, global configuration of a field. In effect, it is an interwoven structure of preimages. A preimage is defined as the 8 Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver Fig. 5 Skyrmion bags in chiral magnets (MnSi) and nematic liquid crystals. Top panels corresponds to experimental (optical micrographs) observations in MnSi whereas middle panels are simulations of the above states. Second and fourth panels in the bottom panel are the result of simulations for the state of the optical micrograph at their left. This figure has been adapted from the preprint version of [23] (arXiv:1806.02576v1). Only some of its panels were finally published in [23]. ©2019 by the Nature Publishing Group set of all points where a field orientation takes a specific value. The Hopf fibration has been observed in liquid crystals [24] and so are hopfions [25], see Fig. 6 and Fig. 7. Similarly, there are light controlled torons in liquid crystals [26, 27]. The toron is essentially a tube of double twist which is wrapped upon itself such that its boundary forms a torus. It contains two point defects, which can be manipulated to create a defect free structure topologically equivalent to a Hopf fibration. Finally, hopfions have also been considered in chiral magnets [28] and torus knots have been described as hopfions [29]. 3.4 Monopoles As such, free magnetic monopoles do not exist in nature but recent advances in condensed matter and atomic physics have demonstrated the existence of effective magnetic monopoles in artificial spin ice [30, 31], chiral magnets [32] and BECs [33, 34]. The latter are created in 87Rb atom condensates in a synthetic magnetic field. Note that a monopole-antimonopole pair is necessarily connected by a Dirac string. When two skyrmion tubes touch at a point it creates an effective magnetic monopole because the emergent magnetic field [see Eq. (5)] at that point is radially outward [32], see Fig. 8. Similarly, a moving heldgehog at the end of a skyrmion Nonlinearity and Topology 9 / I I --- ;' ........ 52 Fig. 6 Experimentally deduced hopfion texture in a liquid crystal (left panel) and a schematic of the hopfion as a knotted soliton (right panel). The hopfion is obtained by linking the circlelike preimages residing on nested tori in the material’s 3D space. The preimages correspond to color-coded points on S2. Reproduced from [25]. ©2017 by the Nature Publishing Group Fig. 7 Left panel: The texture of a toron (reproduced from [26]). ©2010 by the Nature Publishing Group). Second panel: Flow lines of the Hopf fibration. Third panel: The preimage surface of the Hopf fibration. Right panel: The experimental preimage of a Hopf fibration. Reproduced from [24]. ©2010 by the American Physical Society line (in a ferromagnetic nanowire) constitutes an emergent magnetic monopole [35], see Fig. 9. An experimentally observed and simulated monopole in a Bose-Einstein condensate is depicted in Fig. 10. 4 Vortices and Vortex Loops In many physical systems such as fluids, superconductors and magnets (and those modeled by the 3D Heisenberg model) vortices, vortex lines and vortex loops (rings) [37] are observed. Creation and dynamics of trefoil-like (and other) knotted vortices have been studied in water using specially shaped hydrofoils [38]. Since magnetic, superconducting and other vortices as well as their dynamics have been studied 16 Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver Fig. 12 The top panel shows the snapshots of the evolution of two vortex rings leapfrogging through each other. The bottom panel illustrates this type of motion in the radial-polar (r,z)plane on a co-moving reference frame. The solid lines correspond to the orbits predicted by ordinary differential equations for the radius and vertical position of the rings as a function of time. The dashed lines are the corresponding numerics of the full 3-dimensional NLS equation. Bottom panel is adapted from [121]. ©2016 by the American Institute of Physics Of increased interest recently has been not only this structure, but also its onecomponent counterpart, which in the relevant recent BEC literature is referred to as a hopfion state [127, 128, 129]. The latter consists of a VR and a VL in the same component with the axis of the VR constituting the line of vorticity of the VL. A stable hopfion state was found to exist both in the setup of [127] which, however, involved the rather elaborate realization of radially increasing nonlinear interactions and the purely dynamical exploration of [128] for condensates that are being rotated (and was suggested to be stable only for some intermediate rotation rate). Lastly, the study of quantum (vortex) knots is one that has only a relatively short history in the context of atomic BECs. To the best of our knowledge, the possibility of such complex topological structures was introduced in the work of [130] (see also [131]), illustrating how different torus knots T K,q, with co-prime Kand q, can be generated in the wavefunction of an atomic species. Subsequently the work of [132] seemed to put a full stop on the subject through the extensive simulation of 1458 vortex knots from the so-called “knot atlas” [133], and finding that the trapless Gross-Pitaevskii equation (GPE) could not support stable knots: all of the simulated knots would eventually untie into simpler patterns. Nevertheless, the recent work of [134, 135], both at the level of the Biot-Savart dynamical law (for the vortex knot motion), as well as at that of the full 3D GPE has, perhaps counter-intuitively, indicated that the trefoil knot can be a very long lived structure in the context of a trapped atomic BEC [136]. Nonlinearity and Topology 17 7 Topology and curved manifolds: Bogomolnyi decomposition We nowprovide anexample of howthetypesofconfigurations considered herein may be used to minimize the energy of a system of classical spins. The continuum limit of (classical) Heisenberg spins on a two-dimensional (planar or) curved manifold corresponds to the nonlinear σmodel. That is, the corresponding Hamiltonian His given by (1). If we impose homogeneous boundary conditions on the vector field nin the plane R2, i.e. limr→∞ n→n0, then we can compactify the plane into the surface of a sphere S2. This allows us to classify different configurations according to the homotopy class π2(S2)=Z, where Zis the group of relative integers [137, 138]. Topology does not directly help us to state anything about the energy of the field configuration but indirectly, by invoking the so-called Bogomolnyi inequalities [8], it enables us to establish energy bounds for configurations belonging to equivalent homotopy classes labeled by n∈Z. The inequality in the present case can be expressed as (∂in−ij ∂jn)2≥0,(7) whereby it follows that H≥∫n· (∂xn×∂yn)dxdy.(8) Thus, the minimum energy in each homotopy class is attained when ∂in=±ij ∂jn,(9) i.e. when these self-dual equations are satisfied by the field configurations. If we consider this model on a plane (R2), there is no characteristic length. As a result the nonlinear σmodel Hamiltonian can be scaled and thus all the nontrivial field configurations (satisfying homogeneous boundary conditions) can be scaled as well. This situation is drastically changed if there is a characteristic length scale, e.g. if the underlying manifold is curved. We will introduce a length scale in two ways: (i) first we will consider the nonlinear σmodel on a rigid cylinder and then (ii) we will also apply an axial magnetic field through the cylinder. In the first case the radius ρ0of the cylinder is the characteristic length whereas in the second case there is an additional length scale introduced by the magnetic field B. There are other ways of introducing a length scale, e.g. through magnetic anisotropy, ellipticity of the cylinder cross section, etc. but we will not consider these different cases here. We set the unit vector n=(sin θcos Φ,sin θsin Φ,cos θ)in terms of the colatitude θand the azimuthal angle Φ. Next, we write the Hamiltonian [137] in terms of cylindrical coordinates (ρ, z, φ) H=J∫∫cyl (∂zθ)2+sin2θ(∂zΦ)2+(∂φθ)2/ρ2 0+sin2θ(∂φΦ)2/ρ2 0ρ0dzdφ, (10) 18 Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver where Jdenotes the spin exchange interaction energy. In order to invoke topological considerationsletusimposehomogeneousboundaryconditions,i.e.limz→∓∞ ≡0[π] and limz→∓∞ dθ/dz =0. If we seek cylindrically symmetric solutions then Φ=φ and ∂θ/∂φ =0. Thus the Hamiltonian simplfies to H=2πρ0J∫∞ −∞ (∂zθ)2+sin2θ/ρ2 0dz .(11) The variation of this Hamiltonian (δH=0), i.e. the Euler-Lagrange equation turns out to be the celebrated sine-Gordon equation d2θ(z)/dz2=(1/2ρ2 0)sin 2θ , (12) with the well known kink solution θ(z)=arctan[exp(z/ρ0)]. It is depicted in Fig. 13(a). The energy for this configuration is H=8πJ, which is the minimum energy belonging to the first homotopy class. By invoking the technique used by Belavin and Polyakov [138], or equivalently the Bogomolnyi decomposition [8], we note that the solutions that correspond to the minimum energy in each homotopy class satisfy the first order self-dual equations ρ0∂xθ=±sin θ∂φΦ, ∂φθ=∓ρ0sin θ∂zΦ.(13) If we apply an external magnetic field (B) along the axis of the cylinder then the Hamiltonian is modified as [140] Hmag =J∫∫cyl (∇n)2dS −gµ∫∫cyl n·BdS ,(14) where gis the gyromagnetic ratio and µdenotes the magnetic moment. With homogeneous boundary conditions (θ=0as z→ ±∞) the Hamiltonian simplifies as Hmag =2J(2πρ0)∫∞ −∞ [θ2 z+{sin2θ/2ρ2 0+(1/ρ2 B)(1−cos θ)}]dz ,(15) where ρ2 B=2J/gµBis the magnetic length scale. Variation of this Hamiltonian leads to θzz =(1/2ρ2 0)sin 2θ+(1/ρ2 B)sin θ , (16) which is the double sine-Gordon equation. The corresponding 2π-kink solution that is consistent with the boundary conditions is given by θ(z)=2 arcsin 1 qcosh2(z/ξ)−(ξ2/ρ2 0)sinh2(z/ξ) ,(17) where the kink width ξ=ρ0ρB/(ρ2+ρ2 B)1/2is another characteristic length in the problem. This solution is depicted in Fig. 13(b). Nonlinearity and Topology 19 (a) (b) Fig. 13 (a) Heisenberg spins on a cylinder as a sine-Gordon π-soliton (reproduced from [139], ©1994 by Elsevier Science B.V.) and (b) in the presence of an axial magnetic field as a double sine-Gordon 2πsoliton (reproduced from [140]) Because of the homogeneous boundary conditions at the cylinder boundaries all the spins point in the same direction, see Fig. 13. Thus they can be compactified to a single spin and the spin configuration in Fig. 13(a) covers the unit sphere once, i.e. it is a skyrmion of topological charge 1. Similarly, the spin configuration in Fig. 13(b) covers the unit sphere twice, thus it is a skyrmion of topological charge 2. If the cylinder were semi-infinite, it will be topologically equivalent to a plane with a hole of radius ρ0. The spin configuration in this case will be a half-skyrmion (or a meron) [141]. If the cylinder is elastic (i.e. deformable) then the geometric frustration caused by the mismatch of the cylinder radius and kink width can be relieved by a pulselike deformation in the region of the magnetic kink [137, 140]. The Bogomolnyi technique is quite general and can be used in a broader context. Another application of the Bogomolnyi decomposition is in the calculation of elastic deformation energy of vesicles as a function of genus [142]. The latter has significance in the context of the Willmore conjecture [143]. 20 Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver 8 Topological Materials In conventional materials such as metals, insulators and semiconductors the nonrelativistic Schrödinger equation describes the energy dispersion of low-lying electronic excitations, ES=p2/2m∗, which is quadratic in the electron momentum p with effective mass m∗. However, over the past decade there is a growing class of materials, which are known as Dirac materials [144] or more generally topological materials, exhibiting linear electronic dispersion in their band structure. Examples include topological insulators [1, 145], topological superconductors [2], topological crystalline insulators [146] as well as Dirac semimetals and Weyl semimetals [3, 147]. One of the distinguishing features of these materials are Dirac points, where the (conduction and valence) bands touch each other at an isolated set of points. The corresponding band features are called Dirac cones. These points are topologically protected due to specific (time reversal, spatial inversion or crystalline) symmetries in that they are robust under perturbations. An example is graphene where the protection comes from the sublattice symmetry (of the underlying honeycomb lattice) and the energy dispersion of its electrons is linear in the momentum. Specifically, it is given by the relativistic Dirac equation: ED=cσ·p+mc2σ. Here σ=(σx, σy)denotes Pauli matrices and the speed of light cis replaced by the Fermi velocity vF. In d spatial dimensions (with c=1) the Dirac equation is written as (iγµ∂µ−m)ψ=0 [3], where µ=0,1, ..., dwith µ=0denoting time and the Dirac gamma matrices γµanticommute. In odd dimensions (d=1,3, ...) it can be simplified. In particular, for d=1one gets i∂tψ=(γ0γ1p+mγ0)ψwith momentum p=−i∂x. If we further consider massless (m=0) particles, we get the one-dimensional Weyl equation i∂tψ±=±pψ±. Thus we get simple linear dispersion E±=±prepresenting the right and left moving chiral particles or Weyl fermions. Topology seems to enhance the nonlinear response of topological materials. A recent important experimental technique is TFISH (Terahertz Field-Induced Second Harmonic Generation) which allows to understand the nonlinear response of topological Dirac and Weyl semimetals, e.g. TaAs [148]. In particular, second harmonic generation (SHG) is found to be enhanced in Weyl semimetals. The latter contain Weyl points (or nodes) in their electronic structure at which linearly dispersing, nondegenerate bands cross. They also exhibit Fermi arc surface states that are attached to the Weyl nodes in the bulk material (Fig. 14). In these materials there is a transition between the topological and non-topological (or trivial) phases which proceeds via a gapless state. When an appropriate nonlinearity is added to the usual (linear) Dirac equation and Weyl equation, a nonlinear Dirac (NLD) equation [149] and a nonlinear Weyl (NLW) equation [150] results, respectively. Their properties and attendant nonlinear (topological) excitations such as solitons and vortices [151] are quite different as compared to the nonlinear Schrödinger (NLS) equation. Currently, a significant amount of analytical and numerical effort is being devoted to understanding the stability and collision dynamics of solitons in NLD and NLW equations [152]. Nonlinearity and Topology 21 Fig. 14 The double Dirac cone structure of a Weyl semimetal with two Weyl nodes connected by a Fermi arc. Here EFdenotes the Fermi energy. These materials exhibit characteristic surface states which can be observed experimentally. Reproduced from [156]. ©2011 by the American Physical Society. In closing this section we note that in addition to chiral magnets, liquid crystals and BECs, there are many other materials including topological materials in which a variety of topological defects can form under right conditions. In particular, ferroelectrics [13], multiferroic materials, e.g. Cu2OSeO3[153] and magnetic shape memory alloys such as Ni2MnGa [154] can also support skyrmion-like topological excitations. There has been a recent observation of skyrmions in the heterostructures of a ferromagnet (Cr2Te3) and a topological insulator (Bi2Te3) [155]. It would be highly desirable to observe hopfions in such materials as well. Although we have not discussed dislocations separately here, they play an important role in determining the properties of topological materials [157]. 9 Nonreciprocal Topological Photonics Many of the modern photonic devices such as optical isolators and optical circulators are based on the principle of Lorentz reciprocity. It entails that in a (i) linear, (ii) time-independent material or medium with (iii) symmetric property (or constitutive optical) tensors, the received and transmitted fields are identical for both forward and time-reversed propagation directions [158]. However, in some cases reciprocity is deleterious, as, e.g. in self-echo in antennas. In addition, for many desired and emerging optical functionalities (e.g. optical circulators) it is important to break Lorentz reciprocity by relaxing any of the three conditions. Clearly, one way to obtain nonreciprocity is by way of introducing optical nonlinearity. The latter in conjunction with non-Hermitian photonics and topological photonics can significantly enhance nonreciprocity [7]. Examples of optical nonlinearities include the Kerr effect, two-photon absorption and the thermo-optic effect. An example of nonreciprocal topological photonic setup employing a nonlinear coupled resonator array is depicted in Fig. 15. 22 Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver Fig. 15 Nonlinear coupled ring lattice or resonator array for transmittance studies related to nonreciprocal topological photonics. (a) Transmittance Tthrough a linear lattice as a function of phase shift ξ. (b) Transmittance as a function of input power I through the nonlinear lattice. (c) Field intensity distribution (normalized) above the discontinuity (I>3). (d) Average phase shift as a function of I. Reproduced from [159]. Creative Commons Attribution License (CC BY) https: //creativecommons.org/ licenses/by/3.0/ One can create topologically nontrivial photonic band structures in analogy with the electronic band structure of topological materials discussed above. In particular, one can create topological edge states that are robust against perturbations or defects. One way to create such states is by forming an interface between a topologically nontrivial and a trivial optical material. Just like in electronic topological materials, photonic nontrivial topological bands cannot be deformed to trivial bands in an adiabaticway.Aphotonic realization of a two-dimensionalelectronic Cherninsulator uses a lattice of magnetized ferrite rods at microwave scale [160] in which the lattice edge state acts as an isolating waveguide. This structure leads to almost perfect forward transmission and exponentially suppressed backward transmission for frequencies in the photonic bandgap. There are two ways of realizing nonlinear topological photonic structures. One can consider nonlinear propagation dynamics in an otherwise linear topological photonic system. Here the nonlinearity locally alters the system properties. An example is that of waveguide arrays with evanescent coupling between neighboring waveguides. Another way is to use a probe beam to induce a phase transition in the dynamics of a linearized probe beam. Coupled optical resonator lattices provide a realization of this type where one gets quite strong nonlinear effects due to the resonant light confinement as compared to waveguide lattices. Finally, we note that a one dimensional nontrivial topological lattice can be modeled by the photonic Su-Schrieffer-Heeger (SSH) model [161, 162]. Nonlinearity and Topology 23 10 Topological modes in Acoustics and Beyond Admittedly, the study of topological insulators has drawn considerable interest in a variety of fields, among other reasons because of the ability of such media to feature transport that is immune to the presence of defects [1, 145]. One of the most recent venues for such studies has been in the area of mechanical and acoustic systems, where the topological properties can inspire the design of unconventional mechanical structures with unique elastic and vibrational properties [163, 164, 165, 166, 167]. This, in turn, can lead to significant new paradigms in the realm of energy harvesting, as well as in that of vibration isolation [168]. One of the canonical examples that is possible to realize in this mechanical setting is a direct analogue of a dimer in the form of an SSH model as has been suggested e.g. in [169]. This enables through its corresponding phononic band-gap structure the emergence of a zero-frequency topological mode. Finite (non-zero) frequency topological modes can also be achieved [170]. There have been numerous recent efforts in this direction of harnessing topological properties of suitable mechanical media to improve the propagation or storage of energy. These include, among others, the examination of edge solitons [171] and their ability for nonlinear conduction in topological mechanical insulators [172, 173], the study of nonlinear edge states that arise in phononic lattices [174], as well as the examination of topological band transitions in tunable phononic systems, under the variation of suitable (e.g. stiffness) parameters [175, 176]. Lastly, we touch upon the theme of topology optimization which is an important method used for many industrial and technological applications. For structural robustness and additive manufacturing, nonlinear topology optimization [177] and topology optimization for geometrically nonlinear structures [178] have been recently studied. In the former case a nonlinear elastic model of the materials is considered along with plasticity aspects in conjunction with invoking the von Mises (yield) criterion. In the latter case it is assumed that the structures under consideration experience large displacement but small strain. In particular, high-resolution topology optimized solutions are obtained for structures that are geometrically nonlinear. Invariably, the results are quite important for many engineering applications. 11 Conclusions and Future Work In this chapter, we have delineated the importance of topology in a variety of physical systems and discussed the ubiquity of topological defects such as skyrmions, merons, vortices, hopfions and monopoles in a number of distinct nonlinear systems, e.g. chiral magnets, liquid crystals, BEC, etc. We have also elucidated the pervasive role oftopology in nonlinear condensed matter and photonic, as wellasphononic systems. Althoughwedidnot discuss it here, topologyin soft matter is also quiteimportante.g. in topological colloids [179]. In addition, we illustrated the interplay of nonlinearity, topology and geometry by considering the nonlinear σmodel on simple curved 24 Avadh Saxena, Panayotis G. Kevrekidis, and Jesús Cuevas-Maraver manifolds. The resulting spin configurations are sine-Gordon or double sine-Gordon solitons. Despite the above significant progress over the last few years, there are several important open problems related to the interplay of nonlinearity and topology. Nonlinearity with “fragile topology” in a quantum system is a topic for future research. Fragile topology (as opposed to strong topology) refers to a set of quantum phenomena that endow materials or systems with unusual properties [180, 181, 182]. Examples include the misaligned layers of graphene [183] and “knotty” electronic quantum states in some topological materials. In the latter case electrons are restricted to move along certain directions. Understanding these states properly may require considerations other than K-theory [184], as discussed in e.g. [180]. Akin to the study of solitoncollisions and vortex interactions, it would be desirable to study interaction between and collision of different hopfions. This certainly is a challenging numerical problem. A study of one-dimensional NLD and NLW equations on (planar and space) curves and higher (two and three) dimensional such equations on curved manifolds will provide important insights into the interplay of topology, geometry and nonlinearity. In particular, the shape and dynamics of soliton and vortex solutions will be modified by the curved geometry. There are three fundamental (relativistic) fermions in nature: Dirac, Weyl (with zero mass) and Majorana (which are their own antiparticles and thus neutral). Similar to NLD and NLW equations there could exist a nonlinear Majorana equation; it would be intriguing to explore its soliton solutions and their dynamics. Majorana fermions also have potential applications in the growing field of (braiding-based) topological quantum computing [185], especially with fault tolerance. We note here that the nonlinear dynamics of Majorana modes has been studied using topological Josephson junctions [186]. Similarly, it was proposed that Majorana-like modes of light can also be realized in a one-dimensional array of nonlinear cavities [187]. During the last few years there have been studies of a class of nonlinear models that harbor kink solitons with non-exponential tails. Collision of such kinks with nonexponential tails, e.g. power-law [188] or super-exponential, in 1+1 dimensional field theories is an important open question. Recently the field of quantum time crystals [189] has emerged with many insights including topological considerations [190]. The role of nonlinearity in quantum time crystals with topological aspects remains an open field. Acknowledgements A.S. and P.G.K. acknowledge the support of the U.S. Department of Energy. Specifically, LANL is operated by Triad National Security, L.L.C. for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. 892333218NCA000001. J.C.M. thanks financial support from MAT2016-79866-R project (AEI/FEDER, UE). This material is based upon work supported by the National Science Foundation under Grant No. DMS-1809074 (P.G.K.). Finally, P.G.K. gratefully acknowledges the support of the Leverhulme Trust towards a visiting fellowship at the University of Oxford and the kind hospitality of the Mathematical Institute of the University of Oxford. Nonlinearity and Topology 25 References 1. M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010) 2. M. Sato, Y. Ando, Rep. Prog. Phys. 80, 076501 (2017) 3. N.P. Armitage, E.J. Mele, A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018) 4. J. Gooth, B. Bradlyn, S. Honnali, C. Schindler, N. Kumar, J. Noky, Y. 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