Soft Self-Assembly of Weyl Materials for Light and Sound

Downloaded by guest on September 24, 2021 a www.pnas.org/cgi/doi/10.1073/pnas.1720828115 Korea; of Republic 02792, Seoul Fruchart Michel light sound for and materials Weyl of self-assembly Soft odfietetplgclqatte n ntr,rvasaduality a reveals turn, essential in is and symmetry quantities topological chiral the a define There, occur to lines (11–15). Weyl frequency and zero graphene) in at points Dirac here Weyl to symmetry-protected consider (similar mechanical points particu- we a contrast, require In that not symmetry. do points lar and Weyl frequency liquid finite in the at occur space that generically real Note in (10). disap- charge, pairs crystals opposite to of hedgehog–antihedgehog made point not to Weyl a but similar with space annihilate momentum they that unless in means pear moved point. which be crossing perturbations, can against the they robust near describes are eigenstates which points Bloch charge, Weyl Weyl the lin- topological a in a a such singularity Crucially, by exhibits the (7–9). characterized locally directions is structure all point band in crossing 3D band a ear where points, Weyl a and perturbations. or character certain vortex topological against a a robustness hence, with to them (similar confers the point which a in hedgehog), degenerate of eigenstates the existence Bloch of the the neighborhood by of configuration protected singular be How- as particular may exist. to (4–6). degeneracies unstable, parameters system such symmetry the and ever, of particular tuning rare fine a a generically require of they are presence degeneracies the Accidental by are which enforced exist, structure also not invariance). degeneracies band accidental translation a so-called (beyond exis- However, in symmetries the from additional degeneracies stem of and case: points tence the symmetric highly not at way. appear fact, possible mainly every in other each is, cross to This seem observer, casual a to wings butterflies’ several of 3). coloration (2, proper- structural optical the peculiar as in such resulting notably ties, gap most of band photonic which kinds a (1), host all description can a to such applies struc- by periodic encompassed it spatially tures are solids, crystals wavelength. in photonic example, given behav- phonons For the waves. a and understand electrons at to of developed and first ior direction was theory given band a While in propagate can T matter topological system. model a as approach self-assembly general copolymer our block illustrate using self-assembled We discovery. for and pipeline design material symmetry-driven such achieve a to how using describe and constraints points structure Weyl host self-assembled to satisfy a to that has constraints symmetry general of identify the combination we simulations, a numerical Using and methods invariance. group-theoretical in time-reversal even of channels, one-way presence of the existence opti- the bulk for allow their surface which topological states, In of with endowed selectivity are points. materials Weyl frequency response, Weyl cal and protected angular to topologically and addition bottom- with photonic a 3D crystals engineer present to we phononic self-assembly Here, on sound. based approach and up mesostruc- light design for to platform materials unparalleled crystals. tured an atomic offer of they symmetry such, 2017) the As 29, scale November mesoscopic review the that for phases at (received structured replicate 2018 highly into 5, self-assemble March can approved materials and Soft MA, Cambridge, University, Harvard Weitz, A. David by Edited 60637 IL Chicago, Chicago, of University The oet nttt,Lie nvriy edn20 A h Netherlands; The RA, 2300 Leiden University, Leiden Institute, Lorentz h ipeto uhtplgcldgnrce r so-called are degeneracies topological such of simplest The which bands, of set complicated a typically is structure band A epoaaino ae nsailyproi ei sde- is media periodic that frequencies the spatially determines which in theory, band waves by scribed of propagation he | a,1 metamaterials en-elJeon Seung-Yeol , c eateto aeil cec n niern,CrelUiest,Ihc,N 45;and 14850; NY Ithaca, University, Cornell Engineering, and Science Materials of Department | polymers b aynHur Kahyun , | colloids | semimetal b ai Cheianov Vadim , b etrfrCmuainlSine oe nttt fSineadTechnology, and Science of Institute Korea Science, Computational for Center ulse nieArl2 2018. 2, April online Published 1073/pnas.1720828115/-/DCSupplemental. at online information supporting contains article This 1 pc ru ftesrcue,adtesetao h fetv aitnasi available is Hamiltonians effective the the eigenvectors, of spectra numerical at the the Zenodo charges and of on the structures, representations the structures, of irreducible band group photonic the space the points, compute Weyl to the used of code The deposition: Data the under Published Submission. Direct PNAS a is article This interest. of conflict no U.W., declare V.C., authors K.H., The S.-Y.J., M.F., and paper. data; the and wrote analyzed V.V. U.W., V.V. and V.C., and K.H., M.F. S.-Y.J., M.F., research; research; performed V.V. designed V.V. and M.F. contributions: Author wider sur- a to and applicable is energy strategy elastic our phases However, between structured (61). highly competition tension face of Block materi- the variety soft light. from a of and arising into example self-assemble paradigmatic sound that a for als as materials used bottom-up are Weyl viable copolymers realize a provides to (60) show strategy self-assembly we matter article, this soft In how 54–59). 39, (28–32, approaches top-down 53). (48–51), (52, fields materials rotation gapped as active or or such with magnetic (44–47), way, contrast drives some channels external in sharp with one-way invariance in of time-reversal existence breaking is requires the This where materials, (38–43). the invariance topological at time-reversal broken appear when not even that states is samples, context) surface finite electronic arc of the response boundary in protected selective arcs topologically Weyl Fermi frequency of of (called and existence properties angular the peculiar their and as the mul- such by for way points, enabled the applications pave funda- may their tiple discoveries Beyond such (37). importance, plasma mental and magnetized as crystals (33–36) homogeneous well acoustic in as and phononic, (22–26) (27–32), the arsenide photonic in in tantalum matter semimetal condensed Weyl electronic so-called in observed experimentally so-called and (16–21). motions modes self-stress mechanical free zero-frequency between owo orsodnemyb drse.Eal rcatlrnzlieui.lor [email protected] Email: addressed. be [email protected]. may correspondence whom To otcuily on n ih ae a rpgt nthe on propagate imperfections. from can backscattering waves without directions. surface light of material and handful sound a crucially, in material Most can the sound inside or topological propagate light such only wavelengths, of certain at presence that, The means points. objects propaga- Weyl wave self- as of known for feature tion blueprint exotic an breaking. a with providing their materials by and assembled approach symmetries this crystal illustrate describe We tradition- to methods struc- group-theoretical used for such ally tailored of material job symmetries of a for the tures, arrangements self-assembly controlling symmetric soft requires the synthesis Harnessing scale crystals. larger atomic can a that on structures complex replicate into self-assemble materials Soft Significance l htncWy aeil eindu onwaebsdon based are now to up designed materials Weyl photonic All been have (7–9) equation Weyl the following Excitations a lihWiesner Ulrich , https://doi.org/10.5281/zenodo.1182581 NSlicense. PNAS c n icnoVitelli Vincenzo and , PNAS | o.115 vol. www.pnas.org/lookup/suppl/doi:10. d . eateto Physics, of Department | o 16 no. a,d,1 | E3655–E3664

PHYSICS range of self-assembled materials, because it is rooted in symme- a generic blueprint for mesostructured material design by self- try. In the same way as the arrangement of atoms into various assembly. crystalline structures is responsible for the diverse properties of natural materials, the self-assembly of soft mesoscopic structures Weyl Points and Symmetry Requirements with various space group symmetries provides an unparalleled The 3D band structure of an electronic system possessing Weyl platform to synthesize unique materials. points exhibits linear band crossings locally described by the Fully unleashing the potential of soft matter self-assembly in Hamiltonian (7–9) material design involves a constant interplay between the full- H (k) = qi vij σj , [1]

wave optical (or acoustic, etc.) equations of motion of the sys- where q = k − k0 is the wave vector relative to the Weyl point’s tem on one hand and its structural description in terms of free position k0, σj indicates the Pauli matrices, and vij is an invert- energy minimization subject to external fields and constraints on ible effective velocity matrix describing the band crossing at first the other hand. Those problems are generally not analytically order in q. While this description seems at first sight peculiar to tractable and require considerable computational power to be quantum mechanical systems, it is also applicable to all kinds of solved numerically for a wide range of parameters. Here, our waves, as we will see in the following with the example of light. goal is to design a bottom-up method to create Weyl materi- Crucially, such a Weyl point is characterized by an integer-valued als. While self-assembly is a global process taking place in real topological charge, which describes the singularity in the eigen- space, Weyl points exist in reciprocal space, as they are features states near the crossing point, and it can be expressed as (8, 9) of the band structure describing wave propagation in the sys- tem. Hence, we have to solve an inverse problem involving both C1 = sgn det(v). [2] descriptions. To shortcut this difficulty, we combine a minimum input of full-wave computations with a comprehensive symmetry Although the existence of such topologically protected Weyl analysis that determines analytically the desired symmetry break- points does not require a particular symmetry, a crucial inter- ing fields without performing heavy numerical simulations. play between such degeneracies and symmetries exists. Notably, This article is organized as follows. In the first section, we Weyl points cannot be obtained when both time-reversal sym- review the definition of a Weyl point and the properties of band metry and space inversion symmetry are present (8, 62), because structures with such singularities. This allows us to identify a first inversion symmetry requires a Weyl point located at point k set of symmetry constraints on our candidate systems. The sec- on the Brillouin zone to have a partner of opposite charge at ond section is devoted to the realization of a self-assembled block −k. Time-reversal symmetry requires a Weyl point located at copolymer structure that meets this set of minimal requirements, k to have a partner of the same charge at −k, which implies namely breaking inversion symmetry. We then move on to iden- that this topological charge must be zero and that no Weyl tify what (other) symmetries should be broken to obtain Weyl points exist. Hence, either time-reversal or inversion symmetry points and how to do so by applying suitable strains. We then (or both of them) has to be lifted to allow for Weyl points in confirm that the designed photonic structure indeed exhibits the band structure. In a time-reversal invariant system, Weyl Weyl points through full-wave computations of Maxwell equa- points come in pairs of points with identical charge, and the tions. Simulating the self-assembled structures with broken sym- simplest situation then consists of two such pairs with opposite metries is required to determine the quantitative features of the charges (Fig. 1B). band structure and most crucially, to show our method. How- A hallmark of Weyl materials is the existence of topological ever, to predict the existence of Weyl points, our framework surface states at the interface with a band gap material. At a only requires the band structure of the unmodified system with- plane interface, such as the one pictured in Fig. 1A, the trans- out symmetry-breaking alterations. This enables the extension lational invariance is preserved in two directions, and the surface of our design to other kinds of waves: from the full-wave band is described by a 2D surface Brillouin zone as represented in Fig. structures of the unperturbed dispersive photonic, phononic, and 1B. In addition to conical dispersions stemming from the projec- acoustic systems, we can predict that only the first two will exhibit tion of the bulk Weyl points, the surface band structure features Weyl points when altered and strained. The last section provides a manifold of arc surface states (represented in purple in Fig.

A BC

D

Fig. 1. Bulk Weyl points and arc surface states. (A) Sketch of an interface between a band gap material and a Weyl material. Arc surface states (purple) appear at this interface. (B) In a time-reversal invariant system, inversion symmetry has to be broken for Weyl points to exist. The simplest situation consists of four Weyl points with charges ±1 (red and blue, respectively) in the bulk Brillouin zone (bulk BZ). A plane interface preserves space periodicity in two directions and is hence described by a 2D surface Brillouin zone (surface BZ). Crucially, topological arc surface states (represented in purple) appear between Weyl points of opposite charge on the surface BZs. (C) The surface dispersion relation at the interface between a Weyl material and a gapped system features conical dispersions relations, which are the projections of the Weyl points. In addition, a manifold of topological arc surface states (light purple) appears. The intersection of this manifold with a plane of constant frequency (or energy) is sometimes called a Fermi arc in reference to the situation in electronic solid-state physics, where this plane is set at the Fermi energy. (D) The arc surface states may be observed by creating defects at the interface to couple them with incident waves.

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At observed (31). be isolation sample the could the frequency plug- into 28), by antenna a observed an were ging such (27, states surface require arc bands frequencies, not microwave do other states from face isolated 33). 31, spectrally (30, conserved partially) least broken (at momen- with be tangent system to the has a of tum in component a as as This strong invariance, 33). as time-reversal 31, not (30, however, to interface invari- is, the immune time-reversal robustness at propagation observed Although be wave 35). can unidirectional backscattering (33, preserved, sound is for ance (29– and light states for arc 58) both Such materials con- 32, 3D perturbations. origin in to topological observed their experimentally robustness were that certain fact a the them is fers states interests main surface the arc of of one connected). importance, are points fundamental Weyl their which Beyond (i.e., connec- states the surface the determines “Dirac of only tivity bulk topology a however, of points. invariant, projections gauge Weyl the the as connecting seen string” be can they pictorially, which at vectors present. wave an are of states is set surface locations the there protected surface representing topologically the points, arc cones, connecting Weyl Weyl the space the the excite of momentum to also in close curve may frequencies arc-shaped it all When interface, At structure. the band states. the hits by wave ruled beam trans- as or the the reflected material on either bulk Depending be the may in system. it mitted the vector, wave on its shone and frequency sound beam or monochromatic a light consider of us Let origin. topological of 1C) ∗ tcn oee,b rdce rmtebl hog h niedt fteBerry the of data entire the cases, through particular bulk In (31). the conditions from determined boundary predicted not (66). the be connection indeed on however, is depends states can, surface and it Fu–Kane– In topological topology (63–65). weak the zone the (or Brillouin of bulk numbers by shape the Chern of exact first surfaces the or weak planes contrast, by 2D but on defined points invariants) Weyl Mele the of charges the oepeiey h oooyo h adsrcuei ul hrceie o nyby only not characterized fully is structure band the of topology the precisely, More h rteapeo htnccytldslyn elpoints Weyl displaying crystal photonic a of example first The are they when useful most are points Weyl bulk the While states: bulk of topology the from arise states surface arc The ∗ steDrcsrn snot is string Dirac the As uldtiso cutcadpooi rsasaepoie in provided are crystals phononic In and Appendix. but acoustic networks. concreteness, on gyroidal for asym- details crystals two an photonic full the on inducing focus of we by properties following, obtained the elastic be the structure can in crystal metry a phononic with a left larly, are is we constant net- g complete, dielectric majority is the the process Fig. of where their in this rid tune pictorially get summarized After to is to 2. agent process is whole etching The step mild matrix. last work a between con- The of dielectric asymmetry radii. use different respective the optical of through an materials or depositing stants induce by to gyroidal either both them deposi- us between 90) allows (89, difference dielectric chemical networks or the 69) (87, (68, Crucially, minority material metal tion]. high-permittivity gyroidal by a the example, with of [for it alternating 88) one replace etch an to tech- selectively and struc- or standard networks to resulting self-assembly, gyroid us polymer The allow and double the niques blocks. blocks, asymmetric After B C gyroid. an of its of double called composed but composed is still blocks, then ture is A is matrix of of inversion composed the one space instance, be by For may image 83). labyrinths (82, interwoven gyroidal network are the majority chirality matrix opposite a of inside networks gyroid- self- distinct minority which chemically two terpolymers, shaped where triblock gyroid double of ABC a scheme, in use material shape assemble self-assembly to bottom-up the a choose a of on instead with advantage compatible we control take not direct To is scheme. has it synthesis one but cell, where struc- unit 28, milled the ref. the like in structures tures engineered is This in modifications. majority possible local without certainly of the rid get while to impossible almost blocks, A blocks. B the the of of composed say, is matrix composed, by described are minority is Both gyroidal inversion, space the by of by obtained described one is Hence, networks inversion. space through size where well- (86), triply is a which surface, sin(2 85), the gyroid 84, matrix isosurface of (67, a the by a surface one is approximated curvature between matrix inside constant the interface net- periodic interwoven and The minority networks are 83). two minority (82, chirality where network energy opposite structure, majority interface of the double-gyroid parame- of a works Flory–Huggins A minimization to the blocks constrained the leads and the rel- between B), B, energy the parame- interaction and and polymerization, system the A characterizing of of ter of degree set fractions two average well-chosen ative the together of a glued (typically For composed B ters and bonds. are A covalent by They by denoted material. blocks polymer soft immiscible self-assembling few. a a but name to (75–79), 81) colloids 80, patchy (61, and copolymers block anisotropic and of soft dispersions surfactants various 74), amphiphilic in (70–72), (73, crystals appear which liquid among generically as surfaces, Gyroids such com- materials, minimal surface. immiscible of gyroid variety the the a between is to links lead con- of can energy presence ponents surface the of by minimization other. strained the each circumstances, with such interactions In repulsive several have or two components where linked situations in self-assemble naturally gyroids space divides gyroid 2. labyrinth the Fig. interpenetrating in of the shown to structures surface con- corresponding The regions infinitely two (67). an into discovered Schoen is curvature gyroid Alan mean zero A by of (28). surface periodic system triply the nected reduce of to points symmetry strategic addi- at the drilled which deliberately in were gyroid, holes double tional a called structure symmetric highly (−x rcal,tersligsrcuehsivrinsmer htis that symmetry inversion has structure resulting the Crucially, Bdbokcplmr r h rhtpleapeo such of example archetypal the are copolymers diblock AB double that is science matter soft from fact remarkable A a π , h eodmnrt ewr sotie rmtefirst the from obtained is network minority second The . y −y cos(2π ) , −z x , ) ≥ y z and , sin(2 + ) t B and , z r esrdi nt fteui cell unit the of units in measured are g ε π air (x z cos(2π ) 1 = , y g , (x z PNAS usd fsc ein.Simi- regions. such of outside ε ) A , ≥ y x , for t = ) z hl t hrlpartner, chiral its while , | ) ≡ o.115 vol. g t (x sin(2 g , (with (−x y , π z | ) x , o 16 no. ≥ cos(2π ) −y 0 t ≤ A , −z , t | ε < B ) E3657 y √ ≥ + ) for SI 2) t .

PHYSICS A B

Fig. 2. Self-assembly process and effect of strain. (A) The self-assembly of triblock terpolymers leads to “colored” double gyroids, where the two minority networks (red and blue) are chemically distinct (68, 69). Starting from the self-assembled structure, a series of selective etching, partial dissolution, and backfilling steps leads to an asymmetric double gyroid made of high dielectric constant materials, which constitute a 3D photonic metacrystal. Crucially, the photonic band structure of such a system has a threefold degeneracy at the center of the Brillouin zone (the Γ point), which is represented in purple in B, Upper Right. This threefold degeneracy can be split into a set of Weyl points by an appropriate strain (in this case, pure shear) represented in red and blue (for Weyl points of charge 1 and −1, respectively) in B, Lower Right.

Effective Description of the Band Structure resort to numerical simulations. However, with minimal input To obtain Weyl points, the symmetry of the double gyroid must from a numerical full-wave solution complemented with the full be reduced further. Full-wave numerical simulations reveal that knowledge of symmetries in the problem, one can determine an the photonic band structure of a dielectric double gyroid has a effective Hamiltonian, which is sufficient for perturbative design threefold quadratic degeneracy at the Γ point (the center of the purposes. Similar considerations apply to other kinds of waves first Brillouin zone) (91). From the point of view of symmetries, propagating in periodic media (for example, elastic waves) (SI the threefold degeneracy is allowed by the existence of 3D irre- Appendix). ducible representations of the subgroup of symmetries leaving By reducing the full description of the system (contained in the the Γ point invariant, namely the irreducible representation T1g Maxwell operator) to the subspace spanned by a few relevant degrees of freedom, one obtains an effective Hamiltonian de- of the full octahedral group m3¯m (or Oh in Schoenflies nota- tion) (SI Appendix). This threefold degeneracy can be split into scribing a few bands in the vicinity of a (usually high-symmetry) k pairs of Weyl points by symmetry-breaking perturbations (27, 39) point 0 of the Brillouin zone. For example, the eigenstates k as represented in Figs. 2B and 3. The systematic description of involved in a degeneracy at 0 can then be used as a basis to H (q) a band structure near a high-symmetry point of the Brillouin describe the effective Hamiltonian operator as a matrix , q = k − k zone as well as the effect of symmetry-breaking perturbations where 0. Both the Maxwell operator and the effective can be obtained from group theory. This approach, known as the Hamiltonian operator are invariant with respect to the symme- g k method of invariants, originated within condensed matter physics tries of the group of the wave vector 0, defined as the subgroup k (92–96), but it also applies to photonic systems (97, 98) and more of symmetries that leave 0 invariant. The general idea of the the- generally, to all kinds of waves in periodic media. ory of invariants is that symmetries can be used to construct the For example, in the absence of charges and currents, Maxwell effective Hamiltonian matrix from scratch by combining a set of X 3 × 3 equations can be written in a convenient way as basis matrices (which form a basis of, say, the space of Hermitian matrices) and irreducible functions K(q) of the wave 2 2 2 " −1 # vector components (like qx + qy + qz ). The basis matrices rep- E  ε 0  0 i rot E  i∂t = , [3] resent operators in the basis of eigenstates at k0. As such, they H 0 µ −i rot 0 H change when a symmetry operation is applied. This is also the case in irreducible functions as the symmetry operation is applied where E and H are the electric and magnetic fields, respec- to the momentum vector. As the effective Hamiltonian operator tively, while  and µ are the spatially varying permittivity and is invariant when a symmetry is applied, it is possible to deter- permeability of the medium, respectively. In this form, the oper- mine all terms allowed by symmetry by selecting all combinations ator in square brackets, called the Maxwell operator, plays the of the form K(q)X , which are left invariant by the action of the role of a Hamiltonian† (1, 99, 100). This full-wave Maxwell symmetries. equation is usually impossible to solve analytically: one has to More precisely, if the eigenstates at k0 form an irreducible representation Γ, then the matrix representation H (q) of the effective Hamiltonian operator describing the corresponding bands will be covariant with respect to the symmetries. Namely, † For normal materials where permittivity and permeability are strictly positive, the D(g)H (g −1q)D(g)−1 = H (q) D Maxwell operator is Hermitian with respect to a relevant scalar product (1, 99). An , where is a representation of additional constraint stemming from the source-free equations has to be taken into Γ acting on the effective Hamiltonian by its adjoint action as a ∗ account, which commutes with the Maxwell operator. representation of Γ × Γ . The effective Hamiltonian can then be

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PHYSICS method to compute the space groups of the structures, based on Numerical Computation of Photonic Band Structures the open source spglib library (101), is detailed in SI Appendix]. To confirm the existence of Weyl points in the strained asym- Correspondingly, the point group at the Γ point is 222 (or D2 in metric double gyroid, we proceed to a full-wave computation of Schoenflies notation). As we shall see, the effective description the band structure using the well-established open source pack- of the band structure near the Γ point predicts the appearance of age MPB, which determines the fully vectorial eigenmodes of Weyl points in this situation. Maxwell equations with periodic boundary conditions (102). Lin- The effect of a reduction in symmetry on the effective Hamil- ear crossings between the fourth and fifth bands are observed tonian can be determined using subduction rules between the in the situation described in the previous section (in Fig. 4B, original symmetry group and its subgroup, which describe how the relevant bands are red and purple, and the Weyl points and the original irreducible representations combine into the new avoided crossing are marked by gray circles). In this case, the ones. In a system with lower symmetry, it is possible to combine difference between a nodal line and a set of Weyl points has to some X γ and Kδ in a way that was previously not allowed. In be searched on the Γ − P line. In the asymmetric double-gyroid our case, going from Oh to D2 allows various new terms in the structure, a local gap separates the fourth and fifth bands along effective Hamiltonian, which becomes H (k) = H0(k) + ∆H (k), this line, which closes in the inversion-symmetric structure. To where ensure that such crossings are indeed Weyl points, we compute their topological charge from the numerically computed eigen- 2 ∆H (k) = β± Λ± + γi ki Li + δ± K± Id + ζ± k Λ±, [7] modes (SI Appendix). We find that the topological charge of the crossing point on the Γ − H 0 axis is +1, while the charge of the † † 0 Λ+ = Λ + Λ , and Λ− = i(Λ − Λ ) while K+ = K and K− = K . crossing point on the Γ − N axis is −1 (the crossing points on Implicit summation over i = x, y, z and ± is assumed. (We the Γ − N 0 and Γ − H 0 axes have the same charge as their time- imposed, as an additional constraint, that time-reversal symme- reversal counterparts). try be preserved.) In this expression, β±, γi , δ±, and ζ± are An asymmetry only in either the dielectric constants or the generically nonvanishing free parameters, which depend on the gyroids’ thicknesses is sufficient to obtain photonic Weyl points. details of the system. The strained symmetric double gyroid can The effects of both perturbations are similar but not identical: also be described in such a way: the only difference is that all their combination may allow us to optimize for additional fea- symmetry groups now include inversion symmetry. Shear strain tures in the band structure (not necessarily topological; for exam- then reduces the point group at Γ from Oh to D2h, which imposes ple, avoiding frequency overlaps) (an example is in SI Appendix). γi = 0. Here, we focus on the Weyl points: the effect of the dielectric Particularly noteworthy in Eq. 7 are the constant terms with constant asymmetry on the local gap on the Γ − P line and on the prefactors β±, which allow the threefold degeneracy at Γ to positions of the Weyl points is shown in Fig. 5B (the effect of the be lifted. As such constant terms do not break inversion sym- gyroid thickness asymmetry can be found in SI Appendix). While metry, they cannot single-handedly lead to the appearance of the strain angle affects both the relative positions of the Weyl Weyl points. Instead, they split the threefold degeneracy into points and the gap on Γ − P (Fig. 5A), the relative positions of an entire nodal line of degeneracies, similar to the one pre- the Weyl points are almost not affected by the asymmetry. dicted in ref. 27, which is robust against (small) inversion- Additionally, both the dielectric and thickness asymmetries preserving perturbations. In contrast, a perturbation of the form gradually open a complete band gap between the second and Eq. 7 generically breaks inversion symmetry and produces Weyl third bands. Here, this effect is unwanted, as it reduces the band- points as observed in Fig. 3F (SI Appendix discusses typical width available for the Weyl points. It may, however, turn out spectra of the effective Hamiltonians with different symme- to be useful in other contexts. As the strain tends to reduce tries). Hence, we can predict that, in a well-chosen parame- the size of this band gap, we also obtain a 3D strain-tunable ter range, the strained asymmetric double gyroid will exhibit photonic band gap material (103, 104), with properties that can Weyl points. be adjusted through the dielectric and thickness asymmetries.

A

Fig. 4. Photonic band structures. Photonic band B structures of (A) the symmetric double gyroid and (B) the shear-strained asymmetric double gyroid. The threefold quadratic band crossing at the Γ point of the band structure of the unperturbed double gyroid is split into Weyl points on the Γ − N0 and Γ − H0 lines (in contrast, there is no crossing on the Γ − P line, which distinguishes the pair of Weyl points from a nodal line). The first eight bands of the band structures were computed with the MPB pack- 3 age (102) on a (64 × 3) grid, with (A) εA = εB = 16, tA = tB = 1.1, and θ = 0 and (B) εA = 20.5, εB = 11.5, tA = tB = 1.1, and θ = 0.3. Here, ω0 = 2πc/a, where c is the speed of light in vacuum.

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(34), frequency-dependent 35, crystals (33, a group- phononic crystals with dis- (ii) same (i) (i.e., tensor), shown: (37) dielectric the already media were photonic systems, points consider persive Weyl We waves. where photonic of examples kinds three on other to applies focused analysis theoretical we Sound and While Light for Materials Weyl Self-Assembled mechanical of measure strain-induced high-precision the a of properties. strain- achieve tracking such to optical of points combination an Weyl a with sen- envision methods strain can realize sensing we to Here, used (105). been have sors materials gap tunable Such both on uncertainty the of magnitude of order an observables. structural provides and This reduction case). strain numerical the the spurious of the Similarly, symmetry. from effects of distinguishable the not but are threshold, asymmetry this of below magnitude of ingless Appendix order (SI the errors to numerical correspond symmetry-breaking lines dashed by delimited regions rcate al. et Fruchart the on near appears points crossings Weyl figures. band the other of of in set one constant another of Γ kept artifact: position is an the parameter in is each jump which abrupt at The value the to w in represented points, angle Weyl strain the the of (A) features with characteristic of Evolution 5. Fig. ee h adsrcue r optdwt h e fprmtr in bands, parameters fifth of the set and Γ the fourth with the computed between are 4B structures Fig. band the Here, ˆ AB + − − = H P n h omlzdpstoso h w nqiaetWy points, Weyl inequivalent two the of positions normalized the and }, 0 w IAppendix (SI value this near lines na(32 a on + vlto ftemi etrso h htncbn structures. band photonic the of features main the of Evolution /kΓN C 0 k × and 3) 3 w ˆ w rd epo ohtemnmmo h oa gap local the of minimum the both plot We grid. ˆ − ± θ = hudbt aihat vanish both should n ( and w − /kΓH h ilcrcasymmetry dielectric the B) 0 Γ h rydse ie correspond lines dashed gray The k. on.Ters olw from follows rest The point. .Frtelclgp ih green light gap, local the For ). ∆f 45 Γ P θ .Tedt r o mean- not are data The ). = = min{|f wihi lal not clearly is (which 0 5 (k ) δε − Γ ≡ f − 4 (k ε N θ B | )| 0 = − and k 0.4 ε C, A ∈ . ai nmr,a h lsafeunypoie eghscale. length not a is provides frequency equations plasma Maxwell the of band as invariance the anymore, 6 scale gold, valid Fig. the of in frequency crystals, represented plasma tonic is made the which gyroid with of double metal structure a Drude consider a We of 107). (106, frequency-dependent a still tensor with is dielectric but a light equations of to Maxwell propagation leading by the described field, where elec- electromagnetic crystal the the photonic of dispersive with oscillations couple plasma density the tron frequency, acoustic high considered at dielectric the in frequencies. arise low reasonably to at seem least at not system, does but systems n optto r in are 48 computation and a with performed are tations In GPa. 81 obtained and namely are 109–111, epoxy refs. in lon- and values the steel tensor the for elastic in of assumed sound components values the of The from speeds epoxy. transverse in waves and transverse gitudinal of speed the is with computed case value the tabulated from the deviations significant no show results the and 0, stand- gold of frequency plasma ( double-gyroid the vacuum. with metallic in metal a ing Drude of a structure of made band structure photonic Dispersive (A) waves. ω In strained. when we points such, Weyl where As exhibit found. to is systems circles) such boundary gray (A expect by wall structures (highlighted hard band degeneracy with threefold phononic a and gyroid B), photonic double dispersive a the of In outside conditions. (C confined matrix. air elastic in epoxy sound an in embedded steel 6. Fig. C B A p /2π hnlgtpoaae nasrcuemd famtlo na in or metal a of made structure a in propagates light When C 44 epoxy c ' stesedo ih nvcu.W s h lsafeunyo gold, of frequency plasma the use We vacuum. in light of speed the is adsrcue fteuprubddul yodfrdifferent for gyroid double unperturbed the of structures Band 2.19 C, = ω 9Gaand GPa 1.59 × 0 = hnncbn tutr o neatcdul yodin gyroid double elastic an for structure band Phononic B) 10 2π 15 Γ/2π c z(0) and (108), Hz air where /a, IAppendix SI = ρ steel 5.79 = × × 70k m kg 7780 a c air ρ 48 10 ' . epoxy stesedo on nar l compu- All air. in sound of speed the is 12 × 0 m h osterm loss The nm. 500 C PNAS z(0) In (108). Hz IJ 8gi.Mr eal ntemodel the on details More grid. 48 = as 10k m kg 1180 −3 c t 2 | cutcbn tutr for structure band Acoustic ) , = C o.115 vol. 11 steel C 44 ndsesv pho- dispersive In A. 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PHYSICS The case of a unit cell of size a = 100 nm was previously con- sidered in ref. 68. At such scales, the threefold degeneracy may still be present but is overlapped by highly dense plasma bands, and it cannot be identified. Here, we consider a unit cell of size a = 500 nm. A threefold degeneracy appears near ω/ω0 ∼ 0.5. Inspection of the eigenvectors shows that the electric field trans- forms along the 3D irreducible representations T1u (more details are in SI Appendix). Similarly, in a phononic crystal, elastic waves propagate in a spatially periodic structure. Here, we consider a double gyroid made of steel embedded in an epoxy matrix, which couples elastically the two enantiomeric gyroids (110). As observed in Fig. 6 A and B, a threefold degeneracy is found near ω/ω0 ∼ 0.95. Inspection of the numerical eigenvectors shows that they also transform according to the 3D irreducible repre- sentation T1u (more details are in SI Appendix). According to our analysis, such threefold degeneracies will be split into Weyl points by inducing an asymmetry in the enantiomeric gyroid net- works and applying an appropriate strain in both the dispersive photonic and phononic systems. By contrast, we consider the case of an acoustic system, where sound propagates in air outside a double gyroid-shaped labyrinth. Here, no threefold degeneracy at Γ seems to appear in the band structure (at least below ω/ω0 ∼ 1.75) (Fig. 6C and SI Appendix) for the values of the parameters we considered. As a consequence, we do not expect Weyl points to appear under strain at those frequencies. Finally, the band structure of electrons constrained to move on gyroid-shaped nanostructured semiconductors displays multiple degeneracies (113, 114), which could also give rise to Weyl points under strain. In this situa- tion, however, one has to take into account the spin degrees of freedom of the electrons, which are also affected by the curva- ture, and therefore, we can draw no definitive conclusion from our analysis, which would have to be adapted to include double- group representations. The self-assembly of asymmetric double-gyroid structures has already been shown experimentally in block copolymers (69). Directed self-assembly can induce mechanical strains in the direction of growth (115), which according to our symmetry anal- ysis, would automatically lead to the appearance of Weyl points without the need of applying external perturbations. Moreover, gyroid-based systems appear to be unusually resistant to the appearance of cracks when strained (116–118), possibly as a result of their 3D cocontinuous structure, making them a partic- ularly good fit for our strain-based design. The size a of the unit cell of the structures obtained by block polymer self-assembly crucially depends on the blocks’ molar mass. With current exper- imental techniques, the accessible unit cell sizes range from a few nanometers to a hundred nanometers (83, 119). In pho- tonic crystals, this constraint on the unit cell size means that we can expect Weyl points to appear at wavelengths of order λopt ∼ a/0.5 ' 200 nm (or smaller), which are at UV wavelength. Depending on the materials used in the process, the light fre- quency may be high enough for the dielectric function not to Fig. 7. Symmetry-driven mesostructured material discovery pipeline. To be constant anymore, but as we have shown, Weyl points can obtain mesostructured materials with a set of desired properties, we sug- also occur in dispersive photonic crystals. While the direct obser- gest the following automated discovery pipeline. We start from a library of vation of a Weyl band structure at such frequencies is chal- self-assembled structures, which is scanned for candidates matching symme- lenging, such self-assembled photonic crystals could be used in try requirements for a set of target properties. This requires us to automat- X-ray/UV optics (for example, to realize Veselago lenses as pro- ically determine the space group of each structure: a script space group.py posed in ref. 42). does this job for structures represented as a skeletal graph (SI Appendix). To generate an optical response in the visible spectrum, it A best candidate for the initial structure is then selected, and its properties would be interesting to explore hierarchical self-assembly of are numerically computed. For example, we compute the band structure, gyroids using soft building blocks larger than standard polymeric from which the topological charges of the Weyl points (if any) are deter- monomers, such as superstructures formed by anisotropic col- mined by a script weyl charge.py. An effective description is then extracted loids (75, 78), or liquid crystalline phases (72). from the numerical data: here, we need to determine the irreducible rep- resentations of the numerical eigenvectors, a job performed by the script Symmetry-Driven Discovery of Self-Assembled Materials irreps.py (SI Appendix). The effective description then allows us to deter- mine which modifications should lead to the desired properties (for exam- Both the possible existence of a threefold degeneracy and its split- ple, through a symmetry reduction). Here, this step could also be automated ting into Weyl points are predicted by group theory. In this study, using https://github.com/greschd/kdotp-symmetry. Finally, the properties of we did not need to make an initial guess of a structure lead- the modified structure are numerically determined and compared with the ing to a threefold degeneracy at Γ, because we used the well- desired properties. In case of failure, a new initial structure is selected from known example of a double gyroid. Note, however, that symmetry the library, and the process is iterated.

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(2015) dislocations V to Vitelli bound AS, Meeussen modes Topological J, a Paulose (2015) in V 19. solitons Vitelli via BG, conduction Chen ge Nonlinear J, (2014) Paulose V Vitelli 18. N, lattices. Upadhyaya isostatic BG, in Chen modes ge boundary 17. Topological (2013) along TC stress Lubensky and CL, softness Localizing Kane (2017) V 16. Vitelli J, Paulose A, Souslov G, Baardink three 15. in lines weyl and S phonons OR, Topological Bilal (2016) 14. T Lubensky C, Kane O, semimetals. Stenull nodal topological of 13. analog Phonon in (2016) modes A Weyl Vishwanath Mechanical Y, Bahri (2016) HC, T Po Lubensky V, 12. Vitelli M, Falk BG, Chen DZ, Rocklin 11. 1 agB ta.(07 ietosraino oooia ufc-tt rsi photonic in arcs surface-state topological of observation Direct (2017) al. et B, Yang 31. 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