Introduction

Two prolific themes spanning across today’s solid-state, cold-atom and optical physics are the topology of waves in periodic potentials1,2,3,4,5,6,7 and localised coherent states, such as, e.g., solitons and vortices. The recent and ongoing work on the vortex and skyrmion matter and light8,9,10,11, soliton crystals12,13,14,15,16,17,18,19 and soliton gas20,21 points at the opportunity of using the localised structures of the coherent light and matter waves for designing new electromagnetic materials. Here, we present the proof-of-concept results revealing the topological properties of the periodic soliton sequences, i.e., soliton metacrystals, generated in optical microresonators.

For this work, we should distinguish the dissipative and conservative optical solitons and single out the former for their robustness and longevity22. Crystals of dissipative optical solitons are known to exist in Kerr ring microresonators14,15,16,17,18, fibre lasers23,24 and in the exciton-polariton resonators25. Methods of the on-demand positioning of the individual dissipative solitons, circling with the typical repetition rates between ten GHz and one THz, inside the ring resonators have also been successfully demonstrated. These methods included - pump modulation16, using bi-chromatic pump19, single pump frequency tuning26, and the excitation of the desired soliton ordering by applying a sequence of pulses with the repetition rate higher than the resonator free spectral range27. Though the soliton crystals in optical resonators are well established, their band structure and its topology remain an uncharted territory.

The focus of many prior studies of topological properties of nonlinear optical systems has been on the cases when taking the no-nonlinearity limit leaves a linear system prepossessing a specially designed linear periodic structure, see, e.g.,28,29,30,31,32 for topological solitons and33,34,35,36,37,38,39,40,41,42,43 for other topology vs nonlinearity interplay in the arrays of optical resonators and waveguides, and photonic crystals. The soliton crystals studied below require only a single optical resonator and are sustained by the pump-loss and dispersion-nonlinearity balances.

In this work, we demonstrate the soliton metacrystals consisting of the bound soliton pairs so that the soliton arrangement looks like the famous Su-Schrieffer-Heeger (SSH) lattice4. The relative positioning of the solitons within a unit cell is used as the control parameter helping to reveal topological and chiral properties. To prove the non-trivial topology of the metacrystal band structure, we develop a theory of their geometrical phase44,45, and also demonstrate the topological edge states4 in the metacrystals with defects. A feature of the multimode resonators is that the trains of the modelocked solitons they generate are periodic in space and time. We found that the spatio-temporal chirality of the soliton metacrystals leads to the distinct butterfly-like structure hidden in their optical spectra.

Results and discussion

Band structure of soliton metacrystals

Various practical and conceptual realizations of time crystals in condensed matter and optics46,47,48,49,50,51,52 and the cross-disciplinary applications of chirality53,54,55,56 are currently attracting the increasing attention. Here, we consider an optical ring microresonator14,15,16,17 which frequency spectrum is approximated by ωμ = ω0 + D1μ + D2μ2/2! + D3μ3/3!, where μ = 0, ± 1, ± 2, …  is the mode number, D1 is the linear repetition rate (inverse of the round-trip time), and D2,3 are the second and third order dispersion coefficients. ωμ typically belong to the infrared-to-visible range. The pump laser frequency ωp is tuned around ω0, so that δ = ω0 − ωp is the detuning parameter. The envelope, ϕ, of the multimode optical field inside the resonator is

$$\phi (\theta ,t)=\mathop{\sum}\limits_{\mu }{\phi }_{\mu }(t){e}^{i\mu \theta }=\phi (\theta +2\pi ,t),\,\theta =\vartheta -{\tilde{D}}_{1}\tau ,$$
(1)

where, ϕμ are the mode amplitudes, ϑ ∈ [0, 2π) is the polar angle, and \({\tilde{D}}_{1}\) is the nonlinear repetition rate.

A soliton crystal, with the spatial period 2π/K (K = 1, 2, … ) and the temporal one \(2\pi /K{\tilde{D}}_{1}\), is defined as a particular realisation of Eq. 1,

$$\phi (\theta ,t)={{\Phi }}(\theta )=\mathop{\sum}\limits_{l}{\phi }_{lK}{e}^{ilK\theta }={{\Phi }}\left(\theta +\frac{2\pi }{K}\right),$$
(2)

where the non-zero modes are separated by the step K, μ = lK (l = 0, ± 1, ± 2, … ). The corresponding optical spectrum of the soliton crystal is equidistant and is expressed as \({\tilde{\omega }}_{lK}={\omega }_{p}+lK{\tilde{D}}_{1}\). The envelope ϕ obeys the microresonator implementation of the Lugiato-Lefever model57,

$$i{\partial }_{t}\phi =\delta \phi +\mathop{\sum }\limits_{j=1}^{3}\frac{{d}_{j}}{j!}{(-i{\partial }_{\theta })}^{j}\phi -\gamma | \phi {| }^{2}\phi -\frac{i\kappa }{2}(\phi -h),$$
(3)

where \({d}_{1}={D}_{1}-{\tilde{D}}_{1}\), d2,3 = D2,3, κ is the linewidth, h2 is the pump power, and γ is the nonlinear parameter.

If a single soliton and its small excitations correspond to an atom, then a bound state of two solitons is a soliton-molecule58,59,60,61. A periodic arrangement of the soliton-molecules makes up what we call a metacrystal. The numerically computed family of such crystals with two solitons per unit cell, θ ∈ [0, 2π/K), and K = 24 is shown in Fig. 1. The metacrystal structure is SSH-like4, but none of the terms in Eq. 3 is modulated, i.e., our lattice is self-sustained by the dispersion-nonlinearity and pump-loss balances. When the inter-soliton distance equals half of the unit cell length, π/K, then the metacrystal (molecular crystal) with period 2π/K degenerates to the usual (atomic) soliton crystal14,15.

Fig. 1: Soliton metacrystals and dimerization choices.
figure 1

The middle panel shows how two solitons are positioned in the unit cell for varying separation distance, Sα. Two side panels show all 2K = 48 solitons along the ring circumference. The right plot corresponds to SA = SB, i.e., when the molecular crystal degenerates to the atomic one. The left plot shows the molecular metacrystal and illustrates the two dimerization choices applied in this work, SA + SB = 2π/K. The scaled dispersion and detuning parameters are d2/κ = 5 × 10−4, d3/κ = 5 × 10−7, and δ/κ = 30. The crystal repetition rates vary with Sα, but d1/κ remains close to − 0.01059. The dimensionless field amplitude, ψ, is expressed as γ∣ϕ∣2/κ = ∣ψ∣2. The dimensionless pump parameter is γh2/κ = 25. Crystals are computed with 512 points across the unit cell, θ ∈ [0, 2π/K).

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The difference of the interaction strengths between the left and right neighbours of a given soliton, in other words, the imbalance of the intra-cell and inter-cell coupling rates, is controlled by the separation distance Sα. There exist two choices of Sα, i.e., dimerization choices, corresponding to the same crystal, Sα = SA and Sα = SB = 2π/K − SA (see Fig. 1). In what follows, we take π/K ⩽ SA < 2π/K and 0 < SB ⩽ π/K, so that the choice of the unit-cell A, i.e., dimerization A, corresponds to the inter-cell coupling being stronger than the intra-cell one. SA,B = π/K is the degeneracy point with equal coupling rates.

Soliton metacrystals should modify the spectrum of small amplitude perturbations, i.e., elementary excitations, supported by the resonator. Physically, these excitations are slow modulations of the amplitudes and phases of the resonator modes having frequencies in the radio- to tera-hertz range. Using an analogy with the density waves in the solid state crystals, we term these excitations - phonons. While the nonlinear interactions between the crystal and phonon modes are imperative in what follows, phonon to phonon interactions are weak and should be disregarded.

The phonon band structure is computed using the Bloch theorem. This is achieved by taking a soliton crystal, Eq. 2, and its complex conjugate, and representing the phonon field by the cell-periodic two-component Bloch vectors, \({{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}(\theta )={{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}(\theta +2\pi /K)\),

$$\left[\begin{array}{c}\phi \\ {\phi }^{* }\end{array}\right]= \left[\begin{array}{c}{{{\Phi }}}^{(\alpha )}\\ {{{\Phi }}}^{(\alpha )* }\end{array}\right]+\mathop{\sum}\limits_{k,n}{e}^{t{\lambda }_{k,n}}\times \left[{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}{e}^{ik\theta -it{\beta }_{k,n}}\right.\\ \left.+\,{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{x}{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )* }{e}^{-ik\theta +it{\beta }_{k,n}}\right],{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{x}=\left[\begin{array}{cc}0&1\\ 1&0\end{array}\right].$$
(4)

Every Bloch vector is characterised by (i) Bloch momentum, k, which can be restricted to the first Brillouin zone, k = 1, 2, …, K, (ii) band index, n, and (iii) dimerization index, α = A, B. βk,n = βk+K,n are the phonon frequencies and λk,n = λk+K,n express the balance between the dissipation and parametric gain. The momentum varies discretely because of the ring geometry. The condition of the crystal stability, λk,n ⩽ 0, is satisfied for all solutions shown in Fig. 1. Below we use the ket notation for \({{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}=\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle\) and bra, \(\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|\), for its transpose and complex conjugate.

The Bloch states solve the eigenvalue problem

$${\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}{\widehat{H}}_{k}^{(\alpha )}\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle =({\beta }_{k,n}+i{\lambda }_{k,n}+i\frac{1}{2}\kappa )\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle ,{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}=\left[\begin{array}{cc}1&0\\ 0&{{{{{{{\rm{-}}}}}}}}1\end{array}\right],$$
(5)

where \({\widehat{H}}_{k}^{(\alpha )}\) is the system Hamiltonian, see Methods for details. The phonon spectrum, unlike the eigenstates, does not change if α = A is replaced with B. Fig. 2a shows the phonon spectra of the soliton crystals with SA = SB. The values of βk,n scale with the linewidth, κ, which, in the chip-integrated microresonators, varies in the range from tens to hundreds of MHz. The continuum of states starts around βk,n ≈ ± δ and is shaped like a typical band structure. Fig. 2b narrows the view to the top edge of the continuum, where one can see the band-crossing feature and bandgap (yellow shading). d3 ≠ 0 produces the small amplitude dispersive waves62 with the strongly tilted straight-line spectrum, see Fig. 2a.

Fig. 2: Phononic band structure in soliton metacrystals.
figure 2

Panels (a–c) show the spectrum vs momentum k varying across the two Brillouin zones. The number of Bloch states in one Brillouin zone is K = 24. Panels (b, c) zoom on the band structure around the top edge of the continuum, see βk,n/κ ≈ 30 in (a) Panel (d) demonstrates how the band structure changes with the inter-soliton distance Sα. The blue and yellow shading in (b–d) highlight the first and second bandgaps.

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When SA becomes ≠ SB, either way, the band crossing opens up to the bandgap, see the blue shading in Fig. 2c. At the same time, the yellow shaded SA = SB bandgap starts narrowing and comes to the near band-crossings for SA,B = π(1 ± 0.5)/K, see Fig. 2d. We recall that the textbook SSH chain with the nearest-neighbour coupling has only two bands that cross at the Brillouin zone edges when the inter and intra-cell coupling rates (r and \(r^{\prime}\)) are equal. The topological gap opens up for \(r\ne r^{\prime}\), and its width equals \(| r-r^{\prime} |\)4. Thus, the first two bands in the soliton crystal lattice behave very similarly to what happens in the SSH chain. To some extent, the similarity with the SSH model comes unexpectedly, since the spectrum of the Jacobian operator, \({\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}{\widehat{H}}_{k}^{(\alpha )}-i\frac{1}{2}\kappa \hat{{{{{{{{\bf{1}}}}}}}}}\), defines the phonon band structure in soliton crystals, while in the SSH model4, and for the non-interacting electrons in crystals45, this is the Hamiltonian spectrum.

Geometrical phase and Wannier-centres

If the soliton crystal is considered from the same point of view as the ionic crystal lattice shaping clouds of bound electrons in dielectrics, then the next natural step would be to understand the shapes of the phononic clouds existing around the solitons. This is achieved by developing the theory like the theory of polarization in crystalline dielectrics, where it has also provided a transparent interpretation of the geometrical (Zak) phase of an electron acquired after its passage over the Brillouin zone (BZ)45,63,64,65.

The k-space periodicity and gauge-freedom properties of the Bloch vectors are,

$${{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}={{{{{{{{\bf{u}}}}}}}}}_{k+K,n}^{(\alpha )}{e}^{iK\theta },\quad {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\to {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}{e}^{i{\xi }_{k}}.$$
(6)

Here, ξk are the phases, which are arbitrary across the first BZ and extendable beyond it with ξk = ξk+K. The discreteness of k requires special care when formulating the gauge invariant computational strategy for the geometric phase. Adopting the approach of64,65, we have derived the following equation for the geometric phase,

$${{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}={{{{{{{\rm{Im}}}}}}}}\ln \mathop{\prod }\limits_{k=1}^{K}\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{k+1,n}^{(\alpha )}\right\rangle ,$$
(7)

of phonons belonging to a specific band in the soliton crystals, see Methods. The simultaneous bra and ket occurrences for every k make Eq. 7 gauge invariant and applicable to any Bloch-function phases generated by the eigenvalue solver.

We check topological properties of phonons in soliton crystals numerically by observing the π-flipping of their geometric phase on dragging the system through the band crossing points. Fig. 3a, b show \({{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}\) vs Sα for n = 0, 1. The n = 0 phase flips from − π to zero at the point SA,B = π/K, where the n = 0 and n = 1 bands make the near-crossing, see Fig. 2d. The topology of the n = 1 band reflects on the co-existence of one crossing with the n = 0 band and two crossings with the n = 2 band, and, hence, its geometrical phase flips three times.

Fig. 3: Geometrical (Zak) phases and Wannier functions in soliton metacrystals.
figure 3

a, b Numerically computed Zak phases for the n = 0 (a) and n = 1 (b) bands. The phases flip by π at the band-crossing points, cf. Fig. 2d. c The change of the positions and relative strength of the peaks of the n = 0 Wannier function vs the soliton separation, Sα, and θ − θJ (restricted to the one and a half unit cell length to boost resolution). The black-orange colour-map shows \(| {V}_{J,0}^{(\alpha )}|\), see Eq. m.19. The magenta (α = B) and blue (α = A) vertical doted lines mark the values of \({{{{{{{{\mathcal{Z}}}}}}}}}_{0}^{(\alpha )}/K\) from (a). d shows three selected profiles of \(| {V}_{J,0}^{(\alpha )}|\) for Sα marked by the white horizontal lines in (c).

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If the number of cells is taken to infinity while their length is kept fixed at 2π/K, then k becomes continuous and, hence, ∂k is well defined. In this limit, the geometric phase acquires a more familiar form, \({{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}=\int\nolimits_{0}^{K}{{{{{{{{\mathcal{A}}}}}}}}}_{k,n}^{(\alpha )}dk\), where \({{{{{{{{\mathcal{A}}}}}}}}}_{k,n}^{(\alpha )}=i\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}{\partial }_{k}\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle\). \({{{{{{{{\mathcal{A}}}}}}}}}_{k,n}^{(\alpha )}\) differs from the Berry connection for the non-interacting electrons44,45 by the Pauli matrix \({\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\) before ∂k. Taking a unit cell with the coordinate θ = θJ and defining the corresponding Wannier function as \(\left|{{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right\rangle ={K}^{-1}{\sum }_{k}\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle {e}^{ik(\theta -{\theta }_{J})}\), we have further demonstrated that the geometric phase is directly proportional to the averaged coordinate of the phononic cloud centre in a given cell,

$$\frac{{{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}}{K}=\left\langle {{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}(\theta -{\theta }_{J})\left|{{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right\rangle .$$
(8)

With \({\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\to \hat{{{{{{{{\bf{1}}}}}}}}}\), the above would be the equation for the band- or Wannier-centre of the electron wave function in a crystal45,63,64,65, see Methods for more details. For our unit cell definition, see Fig. 1, the n = 0 phonon Wannier centres can be located either in the middle or at the boundary of a unit cell. Fig. 3c,d are explicit about the matching between \({{{{{{{{\mathcal{Z}}}}}}}}}_{0}^{(\alpha )}/K\) and the geometric centres of the Wannier functions. Recent papers demonstrating measurements of the geometric phases of cold atoms in optical lattices66 and light in waveguide arrays67,68 have applied various practical methods to detect positions of the respective wave-packets.

Metacrystals with defects: Edge states and chirality

For the SSH model, creating the lattice defect by removing one A-type unit cell, i.e., breaking two strong bonds in the chain, introduces the edge state inside the bandgap4. To study the bandgap states in the soliton metacrystals, we first extracted one pair of solitons and then tuned Sα. α = B corresponds to breaking the weak bond and α = A to the strong one, see Fig. 1. The corresponding phonon spectrum plotted as a function of Sα, see Fig. 4a, shows the near-crossing of the two bands at Sα = π/K, and the state inside the bandgap emerging for Sα either < π/K (B-type defect) or > π/K (A-type defect). The eigenvectors of the bandgap state are localised around the defect on either side from π/K, see Fig. 4b. For α = A, the state vector is localised on the edge solitons, while, for α = B, it is guided in the middle of the defect, see Fig. 4b. The data presentation for the SSH edge states similar to the style chosen in Fig. 4b,c can be found in50.

Fig. 4: Phonon edge states in soliton metacrystals with defects.
figure 4

a The phonon spectrum in and around the n = 0, 1 bandgap in the metacrystal with the defect made by removing one unit cell, i.e., 2-soliton defect. b The eigenfunctions of the bandgap states, \(| {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}{| }^{2}\), vs Sα and θ. The stems along top axes mark locations of the individual solitons in a crystal with defect for SA = 3π/2K. c is like (b), and (d) is like (a), but for the 7-soliton defect created by removing the three unit cells plus one soliton. The red diamonds in (a) and (d) mark the spectra of the edge states. Crystals with defects are computed with 16384 points across the resonator circumference, θ ∈ [0, 2π). \(\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle\) is computed using Eq. 5 with k = 0, K = 1, which corresponds to θ ∈ [0, 2π), see Methods for details.

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If the defect is made by the odd number of solitons, then shifting between the α = A and α = B dimerizations corresponds to moving the weak bond from one side of the defect to the other, see, e.g., the structure of the 7-soliton defect in Fig. 5a. Then, the edge state also migrates together with the weak bond, see Fig. 4c. With the defects larger than one unit cell, the bandgap contains not only the edge state but also other states guided inside the defect not at its edges, see the black doted lines inside the blue shading in Fig. 4d. The guidance mechanism for these modes is akin to the band-gap guidance in the hollow-core photonic crystal fibres69.

Fig. 5: Chirality of soliton metacrystals with defects.
figure 5

a, b show metacrystals with the differently arranged 7-soliton defects and K = 24. a, b geometries are called left and right, respectively. The missing soliton locations are marked with dashed lines. Note that the θ-interval shown is < 2π. c, d show the examples of spectra of the achiral and chiral defects. e shows the normalised frequency spectra of the crystals with the right and left defects, i.e., \({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(right)}}}}}}}}}\) and \({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(left)}}}}}}}}}\), minus the \({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(ac)}}}}}}}}}\) spectrum corresponding to the achiral geometry, Sα = π/K. Red dots correspond to the right defect, \(({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(right)}}}}}}}}}-{\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(ac)}}}}}}}}})/\kappa\), and blue dots to the left one, \(({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(left)}}}}}}}}}-{\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(ac)}}}}}}}}})/\kappa\). The values of μ shown are restricted to μ ∈ [360, 1200], with the step being 120.

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The defects consisting of the odd-number of solitons break the left-right symmetry of the metacrystals and thereby appear as an interesting object to study the spatio-temporal and spectral manifestations of chirality56. We consider a crystal to be chiral, providing that the mirror image of the solitons around the defects can not be made to coincide with the original soliton arrangement. This is true for any SA ≠ SB, while, the SA = SB = π/K case corresponds to the achiral defect. Indeed, the unpaired soliton can be removed either on the left or right side of the defect, and then the two crystals with the same Sα are connected by the chiral symmetry transformation. Tuning Sα and plotting the crystals around the defect edges makes the chirality obvious, see the tilts of the solitons at the defect edges in Fig. 5a, b.

The chiral pairs of the right and left defects have the spectra which are hard to distinguish. Therefore, we have just shown one of them in Fig. 5d. The spectrum of the ahiral defect is shown for comparison in Fig. 5c. The envelope of the large amplitude spectral lines match well with the spectra of the ideal crystals, while the small amplitude dense spectrum originates from the defect. To make the spectral signature of the chirality effect obvious, we have taken the frequency spectra of the left and right defects, i.e., \({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(left)}}}}}}}}}\) and \({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(right)}}}}}}}}}\), and subtracted from them the achiral spectrum, \({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(ac)}}}}}}}}}\). Here, \({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(..)}}}}}}}}}={\omega }_{p}+\mu {\tilde{D}}_{1}^{{{{{{{{\rm{(..)}}}}}}}}}\) is the mode-locked (dispersion free) frequency spectrum of the soliton crystal with defect and \({\tilde{D}}_{1}^{{{{{{{{\rm{(..)}}}}}}}}}\) is its numerically computed free spectral range. \({\tilde{D}}_{1}^{{{{{{{{\rm{(..)}}}}}}}}}\) is a function of Sα, pump and all the other parameters. We shall recall that the mode-locked spectrum for the ideal crystal, μ = lK, has been already defined after Eq. 5. Fig. 5e shows how \(({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(left)}}}}}}}}}-{\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(ac)}}}}}}}}})/\kappa\) and \(({\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(right)}}}}}}}}}-{\tilde{\omega }}_{\mu }^{{{{{{{{\rm{(ac)}}}}}}}}})/\kappa\) vary with Sα. The spectral differences make the butterfly pattern with the symmetry axis at Sα = π/K and, thereby, reveal the spatio-temporal chirality of the soliton metacrystals.

Conclusions

We have conjectured and confirmed numerically that the train of solitons in microresonators can be considered as a spatio-temporal topological metamaterial, i.e., soliton metacrystals. The optical spectra of such crystals correspond to the octave wide frequency combs. The elementary excitations, i.e., phonons, in metacrystals are the radio to terahertz modulations propagating across the spectrum of the resonator modes. We have computed the phononic Bloch states and their geometrical (Zak) phases. The Zak phase has also been expressed via the Wannier functions and demonstrated to have the meaning of the effective polarization of the phononic clouds around the solitons.

Zak phase changes in the steps of π with the tuning of the soliton separation, which proves the non-trivial topology of the metacrystal band structure. Further to this point, we have found the phononic edge states upon introducing the one unit cell and larger defects. We have demonstrated that the optical (photon) spectra of metacrystals with the left and right defects can be used to characterize chirality in time. Subtracting the optical spectrum of the achiral crystal from the chiral ones reveals the butterfly structure of frequencies, see Fig. 5e, similar to the butterfly wings illustrating chirality in nature.

Features of the proposed realisation of the SSH-like soliton lattice are in its spatio-temporal and dissipative nature and that it does not require spatial or temporal modulations of the material refractive index and pump laser and fabrication of the complex arrays of coupled resonators and waveguides. Further, we should note that one out of twenty microresonator soliton crystals illustrated in14 shows the SSH-like arrangement of soliton molecules with several embedded defects, see Fig. 3j in14. This suggests that the frequency combs with topological properties may have already been unintentionally captured in the experimental measurements. Our results intend to motivate further theory, experiments and applications so that this topic can grow and take its place in the interdisciplinary areas of topological and ultrafast physics. We are also forecasting applications of the soliton metacrystals and their topological states in classical and quantum information processing, where the microresonator solitons and topological photonics are already making an impressive start70,71,72,73,74.

Methods

In order to shorten notations we have dropped the band index, n, and the dimerization superscript, (α), in the first part of Methods. They are subsequently reintroduced in the second and third parts.

Bloch formalism

To analyse and solve Eq. 1 we set the ansatz consisting of the time independent solution which could be a crystal with or without defect, Φ(θ) = Φ(θ + 2π), plus the small amplitude phonon field, \(\tilde{\phi }(\theta ,t)\). Hence,

$$\phi ={{\Phi }}(\theta )+\tilde{\phi }(\theta ,t),\,\tilde{\phi }(\theta ,t)=\tilde{\phi }(\theta +2\pi ,t).$$
(m.1)

Linearising Eq. 1 for \(\tilde{\phi }\) yields

$$i({\partial }_{t}+\frac{1}{2}\kappa )\left[\begin{array}{c}\tilde{\phi }\\ {\tilde{\phi }}^{* }\end{array}\right]={\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}{\widehat{H}}_{0}\left[\begin{array}{c}\tilde{\phi }\\ {\tilde{\phi }}^{* }\end{array}\right],\theta \in [0,2\pi ),$$
(m.2)

where \({\widehat{H}}_{0}\) is specified in Eq. m.8.

We now assume that Φ(θ) has a period 2π/K, where K could also be one,

$${{\Phi }}(\theta )=\mathop{\sum}\limits_{l}{\phi }_{lK}{e}^{ilK\theta }={{\Phi }}\left(\theta +\frac{2\pi }{K}\right).$$
(m.3)

Equations for ϕlK have been solved using the Newton method. Parameters used are typical for Si3N4 resonators: D1/2π = 100GHz, D2/2π = 50kHz, D3/2π = 50Hz, γ/2π = 30MHz/W, κ/2π = 100MHz and laser power \({{{{{{{\mathcal{W}}}}}}}}=520\)mW, so that \({h}^{2}=({D}_{1}/2\pi \kappa ){{{{{{{\mathcal{W}}}}}}}}=83\)W, see Eqs. 1–3 and Ref. 57 for further parameter and model discussion. The scaled values of the parameters are given in the Fig. 1 caption. The approximate analytical expressions for the non-topological soliton crystals in the Lugiato-Lefever equation have been reported in, e.g.,75,76.

The phonon field is set as77,78

$$\tilde{\phi }=\mathop{\sum }\limits_{k=1}^{K}\left[{x}_{k}(t,\theta ){e}^{ik\theta }+{y}_{k}^{* }(t,\theta ){e}^{-ik\theta }\right],$$
(m.4)
$${x}_{k}={X}_{k}(\theta ){e}^{-it{\beta }_{k}+t{\lambda }_{k}},{y}_{k}={Y}_{k}(\theta ){e}^{-it{\beta }_{k}+t{\lambda }_{k}}.$$
(m.5)

Here, k is the Bloch angular momentum. Xk(θ) and Yk(θ) are the cell-periodic functions, i.e., have the same periodicity as Φ(θ), therefore their Fourier series are

$${X}_{k}(\theta )=\mathop{\sum}\limits_{l}{X}_{k,l}{e}^{ilK\theta },\quad {Y}_{k}(\theta )=\mathop{\sum}\limits_{l}{Y}_{k,l}{e}^{ilK\theta }.$$
(m.6)

Substituting Eq. m.4 into Eq. m.2 gives

$$i({\partial }_{t}+\frac{1}{2}\kappa )\left|{{{{{{{{\bf{Q}}}}}}}}}_{k}\right\rangle ={\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}{\widehat{H}}_{k}\left|{{{{{{{{\bf{Q}}}}}}}}}_{k}\right\rangle ,\,\left|{{{{{{{{\bf{Q}}}}}}}}}_{k}\right\rangle =\left[\begin{array}{c}{x}_{k}\\ {y}_{k}\end{array}\right],$$
(m.7)
$${\widehat{H}}_{k}=\left[\begin{array}{cc}{\widehat{D}}_{k}&-\gamma {{{\Phi }}}^{2}\\ -\gamma {{{\Phi }}}^{* 2}&{\widehat{D}}_{-k}^{* }\end{array}\right],\theta \in [0,2\pi /K),$$
(m.8)
$${\widehat{D}}_{k}=\delta -2\gamma | {{\Phi }}{| }^{2}+\mathop{\sum}\limits_{j\geqslant 1}\frac{{d}_{j}}{j!}{\left(k-i{\partial }_{\theta }\right)}^{j},$$

and then, applying Eq. m.5, we find

$$({\beta }_{k}+i{\lambda }_{k}+i\frac{1}{2}\kappa )\left|{{{{{{{{\bf{u}}}}}}}}}_{k}\right\rangle ={\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}{\widehat{H}}_{k}\left|{{{{{{{{\bf{u}}}}}}}}}_{k}\right\rangle ,\,\left|{{{{{{{{\bf{u}}}}}}}}}_{k}\right\rangle =\left[\begin{array}{c}{X}_{k}\\ {Y}_{k}\end{array}\right].$$
(m.9)

The above is Eq. 5 in the main text. One computational advantage of the Bloch formalism comes from the property that

$${\beta }_{k}={\beta }_{k+K},{\lambda }_{k}={\lambda }_{k+K},\left|{{{{{{{{\bf{u}}}}}}}}}_{k}\right\rangle =\left|{{{{{{{{\bf{u}}}}}}}}}_{k+K}\right\rangle {e}^{iK\theta },$$
(m.10)

and, therefore, k can be restricted to a single Brillouin zone, k = 1, 2, …, K. The other is that the spectral bandwidth \(\mu \in [-{\mu }_{\max },{\mu }_{\max }]\), see Eq. 1, is now achieved by taking \(l\in [-{l}_{\max },{l}_{\max }]\) with \({l}_{\max }={\mu }_{\max }/K\). For K starting from 2 and above, this allows to work with the proportionally smaller number of modes when Eq. m.9 is solved in the Fourier space.

Geometrical phase and gauge invariance

The Zak phase formalism that needs to be developed for the soliton metacrystals should reflect on two features - (A) nonlinear interaction between the phonon and crystal and (B) the discrete sampling of the states in the Brillouin zone. In order to make our derivations transparent, we first neglect (B) and, hence, take the limit of a large number of unit cells, so that k is continuous in the Brillouin zone. k is now assumed to vary adiabatically in time, k = k(t). The equation of motion, Eq. m.7, for the phonon wave function, \(\left|{{{{{{{{\bf{Q}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle\), is solved by the substitution44,45

$$\left|{{{{{{{{\bf{Q}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle ={f}_{k,n}^{(\alpha )}(t)\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle ,$$
(m.11)

where \({f}_{k,n}^{(\alpha )}\) is the function to be found and the phonons are assumed confined to the band n. For ∣ωk,n∣ ≫ ∣λk,n∣, κ, the left eigenvector of \({\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}{H}_{k}^{(\alpha )}\) is \(\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\)77. Substituting Eq. m.11 to Eq. m.8 and projecting leads to

$$i\frac{{{{{{{{\rm{d}}}}}}}}{f}_{k,n}^{(\alpha )}}{{f}_{k,n}^{(\alpha )}}\approx {\beta }_{k,n}{{{{{{{\rm{d}}}}}}}}t-{{{{{{{{\mathcal{A}}}}}}}}}_{k,n}^{(\alpha )}{{{{{{{\rm{d}}}}}}}}k,$$
(m.12)
$${{{{{{{{\mathcal{A}}}}}}}}}_{k,n}^{(\alpha )}=\frac{i}{{N}_{n}}\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}{\partial }_{k}\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle ,\,{N}_{n}=\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle ,$$

where Nn is the normalization constant.

The βk,n and \({{{{{{{{\mathcal{A}}}}}}}}}_{k,n}^{(\alpha )}\) terms in Eq. m.12 are the instantaneous dynamical and geometrical phases. Hence, the total geometric phase acquired during k(t) passing the first Brillouin zone (BZ) is

$${{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}=\int_{BZ}{{{{{{{{\mathcal{A}}}}}}}}}_{k,n}^{(\alpha )}{{{{{{{\rm{d}}}}}}}}k.$$
(m.13)

Our \({{{{{{{{\mathcal{A}}}}}}}}}_{k,n}^{(\alpha )}\) differs from the Berry connection emerging in the linear problems4,44,45, by the Pauli matrix \({\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\) before ∂k, which has been previously reported in, e.g., photonic crystals with parametric gain33 and LC-circuits34,35. The normalisation condition applied by us is Nn = ± 1, see, e.g.,8,77 for how similar norms appear in the theory of Bose condensates. \({\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\) also enters the Berry curvature of the Bogolyubov excitations of Bose condensates79. Thought in the example considered here Nn = 1, we have retained Nn for generality in what follows.

The phases of the Bloch functions are not uniquely defined and can be changed by the gauge transform, which, in the discrete BZ, takes the form

$${{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\to {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}{e}^{i{\xi }_{k}},\,{\xi }_{k}={\xi }_{k+K}.$$
(m.14)

For the k-continuous, \({{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}\) remains invariant on applying Eq. m.1445. When k is sampled discretely, a variety of available numerical schemes to deal with ∂k will not generally comply with the gauge invariance. Therefore, designing a practical algorithm to compute \({{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}\) is an important issue to consider.

This brings us to the point (B) and we assume that k is advancing in the unitary steps from 1 to K, and each step takes the time dt to make, then Eq. m.12 updates as

$$i\frac{{f}_{k+1,n}^{(\alpha )}-{f}_{k,n}^{(\alpha )}}{{f}_{k,n}^{(\alpha )}}\approx {\beta }_{k,n}{{{{{{{\rm{d}}}}}}}}t$$
(m.15)
$$-i\left(\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{k+1,n}^{(\alpha )}\right\rangle -\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle \right)/{N}_{n},$$

and, therefore, the total geometric phase is

$${{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}=-i\mathop{\sum }\limits_{k=1}^{K}\left(\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{k+1,n}^{(\alpha )}\right\rangle -{N}_{n}\right)/{N}_{n}.$$
(m.16)

Checking of Eq. m.16 reveals the discretization scheme induced violation of the gauge invariance. Indeed, Eq. m.16 is invariant on applying \(({{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )},{{{{{{{{\bf{u}}}}}}}}}_{k+1,n}^{(\alpha )})\to ({{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}{e}^{i{\xi }_{k}},{{{{{{{{\bf{u}}}}}}}}}_{k+1,n}^{(\alpha )}{e}^{i{\xi }_{k}})\) and not relative to Eq. m.14.

The recipe to restore the gauge invariance is given by the theory of polarization in crystalline solids64,65. We first take the exponent from Eq. m.16, and then use ex = 1 + x + … ,

$$\exp \{i{{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}\}= \mathop{\prod }\limits_{k=1}^{K}\exp \left\{\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{k+1,n}^{(\alpha )}\right\rangle \frac{1}{{N}_{n}}-1\right\}\\ \approx \mathop{\prod }\limits_{k=1}^{K}\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{k+1,n}^{(\alpha )}\right\rangle \frac{1}{{N}_{n}}.$$
(m.17)

Taking the logarithm gives the gauge invariant expression for the geometric phase of the soliton crystals,

$${{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}= \ {{{{{{{\rm{Im}}}}}}}}\ln \mathop{\prod }\limits_{k=1}^{K}\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{k+1,n}^{(\alpha )}\right\rangle \frac{1}{{N}_{n}}\\ = \ {{{{{{{\rm{Im}}}}}}}}\ln \left\{\frac{1}{{N}_{n}^{K}}\times \left\langle {{{{{{{{\bf{u}}}}}}}}}_{1,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{2,n}^{(\alpha )}\right\rangle \left\langle {{{{{{{{\bf{u}}}}}}}}}_{2,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{3,n}^{(\alpha )}\right\rangle \ldots \right.\\ \left.\ldots \left\langle {{{{{{{{\bf{u}}}}}}}}}_{K-1,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{K,n}^{(\alpha )}\right\rangle \left\langle {{{{{{{{\bf{u}}}}}}}}}_{K,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\left|{{{{{{{{\bf{u}}}}}}}}}_{1,n}^{(\alpha )}{e}^{-iK\theta }\right\rangle \right\}.$$
(m.18)

‘\(\ln\)’ is defined by its principal value, such that \({{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}\in [-2\pi ,0)\). For K even, \({N}_{n}^{K}=1\) and the above coincides with Eq. 7 in the main text.

Wannier formalism

The direct and inverse Bloch-to-Wannier transforms are defined as

$$\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle = \mathop{\sum}\limits_{J}\left|{{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right\rangle {e}^{-ik(\theta -{\theta }_{J})}=\left[\begin{array}{c}{X}_{k,n}^{(\alpha )}\\ {Y}_{k,n}^{(\alpha )}\end{array}\right],\\ \left|{{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right\rangle = \frac{1}{K}\mathop{\sum}\limits_{k}\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle {e}^{ik(\theta -{\theta }_{J})}=\left[\begin{array}{c}{W}_{J,n}^{(\alpha )}\\ {V}_{J,n}^{(\alpha )}\end{array}\right],$$
(m.19)

where \(\left|{{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right\rangle\) are Wannier functions, θJ = 2π(J − 1)/K and \(J\in {\mathbb{Z}}\) numbers the unit cells. Our gauge choice for the Bloch functions is specified by the condition Im\({X}_{k,n}^{(\alpha )}=0\) at the point θ = 3π/2K.

We now recall Eq. (m.9) and, then, transform Eq. m.13,

$${{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}=i{\int}_{BZ}dk\left\langle {{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}{\partial }_{k}\left|{{{{{{{{\bf{u}}}}}}}}}_{k,n}^{(\alpha )}\right\rangle$$
(m.20)
$$ =i{\int}_{BZ}dk\int\nolimits_{{\theta }_{J}-\pi /K}^{{\theta }_{J}+\pi /K}\left({X}_{k,n}^{(\alpha )* }{\partial }_{k}{X}_{k,n}^{(\alpha )}-{Y}_{k,n}^{(\alpha )* }{\partial }_{k}{Y}_{k,n}^{(\alpha )}\right)d\theta \\ =\frac{i}{K}{\int}_{BZ}dk\int\nolimits_{{\theta }_{J}-\pi }^{{\theta }_{J}+\pi }\left({X}_{k,n}^{(\alpha )* }{\partial }_{k}{X}_{k,n}^{(\alpha )}-{Y}_{k,n}^{(\alpha )* }{\partial }_{k}{Y}_{k,n}^{(\alpha )}\right)d\theta .$$

The last part of Eq. m.20 uses the fact that the Bloch functions are cell periodic and, hence, the integration can be extended to the entire ring.

If the ring is large, the problem can be well approximated by the straight-line infinite crystal lattice45 with the period 2π/K. Then, ∑k → ∫BZdk and expressing the Bloch functions via the Wannier ones allows to take the k-derivative inside Eq. m.20,

$${{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )} =\mathop{\sum}\limits_{J}\int\nolimits_{{\theta }_{J}-\pi }^{{\theta }_{J}+\pi }\left[| {W}_{J,n}^{(\alpha )}{| }^{2}-| {V}_{J,n}^{(\alpha )}{| }^{2}\right](\theta -{\theta }_{J})d\theta \\ =\mathop{\sum}\limits_{J}\left\langle {{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}(\theta -{\theta }_{J})\left|{{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right\rangle .$$
(m.21)

Wannier functions for all J coincide if plotted vs θ − θJ, therefore, Eq. m.21 simplifies to

$${{{{{{{{\mathcal{Z}}}}}}}}}_{n}^{(\alpha )}=K\left\langle {{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}(\theta -{\theta }_{J})\left|{{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right\rangle ,$$
(m.22)

see Eq. 8 in the main text. Thus, the geometric phase of soliton crystals is determined by the coordinate of the effective centre of the phonon cloud in the unit cell, i.e., by the Wannier-centre, \(\left\langle {{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right|{\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}(\theta -{\theta }_{J})\left|{{{{{{{{\bf{w}}}}}}}}}_{J,n}^{(\alpha )}\right\rangle\). For the ring geometry with the relatively large number of unit cells, as we are dealing with, Eq. m.22 remains a good approximation.