Geometry of Bend: Singular Lines and Defects in Twist-Bend Nematics

We describe the geometry of bend distortions in twist-bend nematic liquid crystals in terms of their fundamental degeneracies, which we call $\beta$ lines. These represent a new class of line-like topological defect. We use them to construct and characterise novel structures, including grain boundary and focal conic smectic-like defects, Skyrmions, and knotted merons. We analyse their local geometry and global structure, showing that their intersection with any surface is twice the Skyrmion number. Finally, we demonstrate how arbitrary knots and links can be created and describe them in terms of merons, giving a new geometric perspective on the fractionalisation of Skyrmions.

Fresh perspectives invariably accompany the discovery of a new phase: The recent discovery of the twist-bend nematic phase [1][2][3] invites fresh consideration of nematic geometry and topology. The twist-bend nematic is a fluid mesophase in which the nematic orientation exhibits a heliconical modulation with nanoscale pitch and modest cone angle [4]. It occurs in compounds with a bent core architecture (banana molecules) and is characterised by a preferred state of non-zero bend distortion [5,6]. Thus the geometry of bend is a natural vehicle for describing the structural degeneracies and defects of the twist-bend nematic that equally applies quite generally to any material with orientational order.
Geometric elastic distortions pervade soft matter physics [7], providing a common conceptual framework for understanding many different materials as well as numerous methods -including boundary conditions, substrate topography and surface curvature -for designing or controlling properties and functionality [8][9][10][11][12][13][14][15][16]. Geometric methods also relate to topological properties through the Gauss-Bonnet theorem and Berry phase physics, so that geometric degeneracies possess both elastic and topological significance, giving them greater potential for material control. A common feature of many materials are structural degeneracies along lines or curves, with examples including flux lines in superconductors [17], fluid vortices [18], nodal lines in optical beams [19], C lines in electromagnetic fields [20], defect lines in liquid crystals [21] and umbilic lines in general [22]. In many instances these lines are fundamental to the organisation and properties of the entire material, simultaneously characterising it and offering a mechanism for controlling and engineering specific responses.
In this Letter, we introduce a new line-like geometric degeneracy associated to zeros of the bend in a unit vector field, that we call β lines. These lines occur in all materials with vector or orientational order, such as liquid crystals and ferromagnets, but have added significance when there is an energetic preference for non-zero bend, and in such materials β lines are a new type of topological defect. We give constructions of both smectic-like and Skyrmionic textures in twist-bend nematics and characterise them in terms of their β lines. We provide a description of the local structure of generic β lines and show that their signed intersection number with a surface gives a Skyrmion count. Finally, we show how complex three-dimensional textures encoding knotted β lines may be constructed, analogous to the 'heliknotons' recently created experimentally in cholesterics [23], and characterise them in terms of merons.
Orientational order is described by a unit vector n, called the director. Nematic symmetry, n ∼ −n, corresponds to alignment that is line-like, rather than vectorial. The bend b = (n · ∇)n = −n × (∇ × n) is the curvature of the director integral curves; it is a globally defined vector whose sign does not reverse under n → −n. As n is a unit vector the bend is everywhere orthogonal to it, b · n = 0. Thus, although bend is a vector field in threedimensional space, it is atypical, having only two degrees of freedom. In particular, the set of points where it vanishes -geometrically, the set of inflectional points in the integral curves of n -is one-dimensional, and forms a collection of fundamental curves in the material that are characteristic of it; we call them β lines.
Our model system for exploring the significance of β lines is the twist-bend nematic. It may be described by a Frank free energy with negative bend elastic constant [5], or by coupling the bend of the nematic director to a vector polarisation, p, coming from the 'banana' shape of the constituent molecules, with a free energy [6] where K is a Frank elastic constant, λ is a coupling constant, C is an elastic constant for the polarisation, and U sets the scale of the bulk ordering energy. This favours the heliconical director n = cos θ e z + sin θ(cos qz e x + sin qz e y ), with the conical angle θ and helical wavevector q determined by the elastic moduli [4,6]. The integral curves of the director are helices of constant curvature and torsion; the bend b = q sin θ cos θ(− sin qz e x + cos qz e y ) has constant magnitude and rotates at the arXiv:2002.00916v1 [cond-mat.soft] 3 Feb 2020 same rate as the director. On scales large compared to the helical pitch (2π/q) the twist-bend phase has the same elastic energy as a smectic [24,25] and exhibits all the features, textures and defects of a bone-fide smectic, despite there being no mass-density wave. These smectic-like defects are all associated with β lines; we remark that they are revealed by the director field and the degeneracies of its bend despite many of the textures we consider being nullhomotopic and hence invisible to the traditional homotopy theory methods.
We consider first screw dislocations in the helical integral curves of the twist-bend ground state, shown in Figs. 1(a,b). On a circle enclosing the dislocation the phase of the helices winds by 2πs, where the integer s is the strength of the screw dislocation -in the examples shown s = ±1. The bend winds by exactly the same amount, thereby guaranteeing the existence of a β line. These textures may be explicitly realised using n = cos θ e z + sin θ cos φ e x + sin φ e y , where φ = qz + s arctan(y/x) and the cone angle θ varies smoothly from its far field preferred value to vanish on the z-axis. As θ vanishes, the helical integral curves degenerate to a straight line along the z-axis which, having no curvature, is a β line. Using instead φ = qz +s arctan(z/x) yields an edge dislocation, as shown in Fig. 1(c). The β line does not coincide exactly with the dislocation (y-axis) but is displaced slightly to one side at the position of the hyperbolic point where ∇φ is zero [26]. Here, as we cross the β line, the integral helices pass through an inflectional configuration in the manner described in [27].
These examples can be set in more general context as follows: Let φ be a smectic phase field with N the smectic-A director, i.e. ∇φ/|∇φ| away from the singularities in φ, and e 1 , e 2 an orthonormal basis for the planes orthogonal to N that is parallel transported along it, (∇ N e 1 ) · e 2 = 0. Then set where the cone angle θ again vanishes along the singu-larities in φ. Away from the core regions the bend is b ≈ |∇φ| sin θ cos θ − sin φ e 1 + cos φ e 2 and winds by the same amount as the smectic phase field, again implying the presence of β lines. We show in Fig. 1(d) examples of this construction that realise grain boundaries (TGB phases) [28][29][30] and parabolic focal conic domains [31,32], but any smectic construction can be converted into a twist-bend texture in this way. β lines also occur in vortex structures in the director field -along the axes of double twist cylinders or the cores of Skyrmions. A canonical example is the double twist profile n = cos qr e z + sin qr e φ shown in Fig. 2(a), for which the bend is radial, b = − 1 r sin 2 qr e r , and vanishes linearly along the axis with winding number +1. It is instructive to compare this defect to the +1 screw dislocation of Fig. 1(a). In the double twist cylinder we have another family of integral helices degenerating to a straight line, however each helix encircles the β line, in contrast to the screw dislocation where they do not, and the texture is quasi two-dimensional, with no periodic modulation along the z-axis as in the screw disclocation. This obervation establishes that these two β lines are topologically distinct, in the sense that one cannot convert one into the other without creating additional β lines.
Skyrmions are non-singular field configurations carrying a topological charge Q = 1 4π n · ∂ x n × ∂ y n dxdy, that corresponds to an element of π 2 (RP 2 ) ∼ = Z. They are (meta)stable states in cholesterics and in ferromagnets with Dzyaloshinski-Moriya interaction [33,40]. The analogous helical director structures suggest it is possible Skyrmions also arise in twist-bend nematics. In Fig. 2(b) we show an isolated Skyrmion embedded in a heliconical background, obtained by numerical relaxation of (1); it is (meta)stable in simulation. The texture contains two β lines, the first with structure resembling the double twist cylinder shown in Fig. 2(a), the second a helix with pitch equal to that of the heliconical far field director. These β lines are a topological necessity and count the Skyrmion charge Q; as we shall show below, there are two β lines per Skyrmion. Fig. 2(c) shows a lattice of Skyrmions, in which the radial symmetry of the β lines is broken to hexagonal by the lattice. A full Skyrmion phase diagram, analogous to that constructed for cholesterics [35], would be of clear interest, although it is not the focus of this work; here, we simply note that we have confirmed (meta)stability for the heliconical far field angle θ ∈ [0.1, 0.5], U/C ∈ [0.1, 0.5], in simulations perfomed using periodic boundary conditions with box height chosen to match one pitch length (2π rotation) of the twist-bend director. The stability we have seen suggests that twist-bend Skyrmions could be directly nucleated by adapting techniques used in cholesteric cells or in magnetic systems.
Thus far we have discussed β lines in the context of experimentally relevant structures in the twist-bend nematic; we now turn to a more general description of their geometric structure and topological significance. In our simple examples, the director n is either colinear with the β line tangent t, as in Figs. 1(a, b) and Fig. 2(a), or orthogonal to it as in Fig. 1(c). However generically neither is the case, and n and t make some intermediate angle. Points where they are orthogonal have codimension one and are called Legendrian (see for example [36]); we remark that this can happen in two distinct ways, via either a saddle-node bifurcation or a Hopf bifurcation [37]. Points of colinearity are codimension two and do not occur except in situations of high symmetry. A local description of a generic point on a β line can be developed by introducing adapted coordinates n ≈ n x e x +n y e y +e z and expanding in a Taylor series, retaining only terms that contribute at linear order to the bend: Here ∇ ⊥ n = ∂xnx ∂ynx ∂xny ∂yny denotes the 2×2 matrix of orthogonal gradients of the director [22], and ∂ z ∇ ⊥ n is its rate of change along the local director; [s x , s y ] controls the angle between t and n. The winding number in the xy-plane is ±1 according to the sign of det (∇ ⊥ n| 0 ) 2 + ∂ z ∇ ⊥ n| 0 . When the derivatives ∂ z ∇ ⊥ n 0 are negligible this reduces to (det ∇ ⊥ n| 0 ) 2 and the winding is always +1, so that the different profiles of β lines are controlled crucially by the parallel derivatives of the orthogonal director gradients. We now describe the global structure beginning with a canonical orientation of β lines via the operator ∇b. Along the β lines there are two canonical frames; one comes from the curve and consists of its tangent vector t and normal plane ν; the other comes from the director n and its normal plane ξ. On the β line (∇b)·n = 0, so ∇b is a linear map into ξ with kernel t. For normal displacements away from the line, ∇b : ν → ξ is an isomorphism. We orient the β line such that this isomorphism preserves orientation, giving a canonical choice of β line tangent t. Note that this orientation reverses upon the replacement n → −n, which corresponds precisely to the change in sign of point defects (or Skyrmion charge) under the same replacement [38]. On the complement of the β lines there is the Frenet-Serret framing of the director integral curves. The β lines are singularities of this framing. We write b = κ e 1 , with κ the curvature of the integral curves, and e 2 = n × e 1 . This framing yields a connection 1-form for the plane field ξ, denoted ω = (∇e 1 ) · e 2 . The component of ω along the director is the torsion τ = ω(n) = (∇ n e 1 ) · e 2 , while the vector dual to it is the local pitch axis of the heliconical twist-bend state. For example, the smectic-based twist-bend director (2) has connection 1-form ω = cos θ dφ; the torsion is τ ≈ q cos 2 θ and the pitch axis is along ∇φ. Topological information is conveyed by the associated curvature 2-form Ω; again for (2) this is Ω = − sin θ dθ ∧ dφ = −1 2 ijk n i dn j ∧ dn k . Given a surface S, the β lines intersect it in a set of points p i and by the Gauss-Bonnet-Chern theorem where Int pi (β, S) denotes the signed intersection number at point p i of an oriented β line with an oriented surface S. For Skyrmion textures this total intersection number is 2Q, giving two β lines per Skyrmion as seen in Fig. 2. With these concepts in place, we now discuss fully three-dimensional twist-bend textures: it is possible to embed an arbitrary knotted or linked set of β lines into a heliconical background, via an extension of our constructions for screw and edge dislocations. Given any knot or link K, the director n = cos θ e z + sin θ cos φ K e x + sin φ K e y , where φ K = qz + 1 2 ω K , with ω K the solid angle function for K [39], embeds a helical winding of the director integral curves around a tubular neighbourhood of K; as before, the cone angle θ should be made to vary from its far field value to vanish along K. The phase winding in the helical integral curves guarantees the existence of a β line. Examples for the figure-eight knot and Hopf link are shown in Fig. 3.
The director texture is that of a meron tube extruded along K, Fig. 3(a-d). A meron is a fractionalisation of a Skyrmion that carries half the topological charge [40,41].
β lines provide a natural geometric perspective on this fractionalisation: since each Skyrmion comprises two β lines, a single β line represents half a Skyrmion, i.e. a meron. In terms of the heliconical phase field, φ K , these meron tubes are edge dislocations where heliconical layers terminate, Fig. 3(c, d). Exactly these structures were recently created experimentally in cholesteric cells and shown to form highly controllable and responsive knotted solitons [23]. In that experiment, links of 'escape up' and 'escape down' meron tubes combined to give non-zero Hopf invariant. For the twist-bend nematic phase, the small conical angle (θ ≈ 25 • [3]) gives an energetic preference to 'escape up' merons over 'escape down', whereas in cholesterics (θ = π/2) the two types of meron are degenerate. Even with only 'escape up' merons, where the Hopf invariant is trivial, the linking is still a relevant quantity with distinct textures for different values of the linking number, Lk(β i , β j ). In Fig. 3(e) we show an example for the Hopf link with linking numbers ±1 where the layer structure through the middle of the link is different in the two cases. The triviality, or otherwise, of the Hopf invariant can also be seen just from the β lines and the formula H = i SL(β i ) + i =j Lk(β i , β j ), familiar from helicity and abelian Chern-Simons theory [42]. The self-linking number, SL(β), is defined as follows: consider the total rotation B e 2 · de 1 of the Frenet-Serret frame about the director along any push-off B giving a zeroframing for the β line. Part of this rotation is an intrinsic Berry phase γ, equal to the area on the unit sphere bound by the curve traced out by n along B. The difference γ − B e 2 · de 1 = 2π SL(β) defines the self-linking.
We have given an initial description of geometric degeneracies in the bend of a vector field, which we call β lines, and their connection to topological features, including smectic singularities, Skyrmions and merons. We have couched the majority of the discussion around the twist-bend nematic phase, in which the β lines are novel topological defects, however the same structures arise in any orientationally ordered material. As one example in a different setting, active materials with extensile activity exhibit a bend driven instability in (three-dimensional) active nematics and cholesterics [43][44][45] and so naturally exist in states with non-zero bend distortion. The geometric degeneracies we have introduced here will also arise there and may provide a means for their analysis. This work was supported by the UK EPSRC through Grant No. EP/L015374/1. JB supported by a Warwick IAS Early Career Fellowship.