Non-Abelian Frame Charge Flow in Photonic Media
Item Type Article
Authors Wang, Dongyang;Wu, Ying;Zhang, Z. Q.;Chan, C. T.
Citation Wang, D., Wu, Y., Zhang, Z. Q., & Chan, C. T. (2023). Non-Abelian Frame Charge Flow in Photonic Media. Physical Review X, 13(2).
https://doi.org/10.1103/physrevx.13.021024 Eprint version Publisher's Version/PDF
DOI 10.1103/physrevx.13.021024
Publisher American Physical Society (APS)
Journal Physical Review X
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Link to Item http://hdl.handle.net/10754/691757
Supplementary Material for
Non-Abelian frame charge flow in photonic media
Dongyang Wang,1 Ying Wu,2 Z. Q. Zhang,1 C. T. Chan1
1. Department of Physics and Center for Metamaterials Research, Hong Kong University of Science and Technology, Hong Kong, China.
2. Division of Computer, Electrical and Mathematical Science and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia.
*Correspondence to: [email protected]
1. Introduction to the formulation of non-Abelian frame charges
Following the method proposed in Refs. 22 and 24 of main text, we here briefly review the mathematical formulation of multiband non-Abelian generalized quaternion charges that support our calculations.
For a PT symmetric N-dimensional Hamiltonian, the real eigenvectors define a frame. The rotation of such eigen-frame along a closed path L is characterized by integrating the path- ordered Berry-Wilczek-Zee (BWZ) connection over all bands as
𝑅(L) = 𝑒𝑥𝑝̅̅̅̅̅ (∮ 𝐴L(𝑘) ∙ 𝑑𝑘) , (1)
with [𝐴L(𝑘)]𝑎𝑗𝑖 = ⟨𝑢𝑘𝑖|𝜕𝑘𝑎|𝑢𝑘𝑗⟩, and i, j ∈ [1, 2, …, N].
The N-band BWZ connection can be decomposed into SO(N) basis. The generalized quaternion charges are calculated by lifting the connection basis of SO(N) to Spin(N) and formulated as,
𝑞L= 𝑅̅(L) = 𝑒𝑥𝑝̅̅̅̅̅ (∮ 𝐴̅̅̅(𝑘) ∙ 𝑑𝑘) ,L (2)
where the lifted BWZ connection matrix is denoted as 𝐴̅̅̅(𝑘). L
For a N-band Hamiltonian, the SO(N) basis can be constructed according to (𝐿𝑖𝑗)
𝑎𝑏= −𝛿𝑖𝑎𝛿𝑗𝑏+ 𝛿𝑖𝑏𝛿𝑗𝑎, (3)
and the Spin(N) basis are,
𝑡𝑖𝑗 = −1
4[Γ𝑖, Γ𝑗], (4)
where Гi are anti-commutating Gamma matrices in N dimensions.
The π rotation along a first homotopy (π1) loop can be formulated as
𝑒𝑗−1 = 𝑒𝜋𝑡1𝑗 = 2𝑡1𝑗, (5)
and more general descriptions of the rotations are taken as 𝑔𝑗 = {𝑒1, 𝑗 = 1
𝑒𝑗−1𝑒𝑗, 𝑗 ≥ 2 . (6)
The discrete group 𝑃̅𝑁 describing the topology of PT symmetric nodal lines is formed of elements as 𝑞 = [+1, −1, ±𝑔1, ±𝑔2, … ± 𝑔1𝑔2, ±𝑔1𝑔3, … , ±𝑔1𝑔2… 𝑔𝑁−1].
The group elements q = ±gi represent the ±π rotation of the ith and the (i+1)th eigenvectors of a Hamiltonian, which characterize the nodal line in momentum space that is formed between the ith and (i+1)th bands. The element of q = “+1” corresponds to 4nπ rotation of a co-plane spanned by two eigenvectors, q = “-1” corresponds to the (2n+1)2π rotation.
2. Homotopy loop characterization of the frame charges around nodal line crossing The elementary frame charge of q = ±gi in generalized quaternion group corresponds to a π1
loop encircling the linear degeneracy node formed between the ith and (i+1)th bands in the momentum space. By assigning the charges q = ±gi as directed arrows on the nodal lines, we can more clearly present the evolution of frame charges along nodal lines and explain their behaviours at the crossing point of nodal lines discussed in Fig. 1(d) of main text. To be more specific, the colour of arrow is set in accordance with the degeneracy lines, which represents the indices i/(i+1) of the bands forming the nodal line or the subscript “i” in charge q = ±gi. The arrow direction represents the “±” sign of charge q = ±gi, which is the “sense of rotation”
of the frame. We show below on how to characterize the crossed nodal lines through π1 loops or the merging of them.
For example, regarding the crossed nodal lines shown in Fig. S1(a) without braidings (but a third band is assumed to exist for multiband discussion), the green loop starting from a base point in momentum space encircles a branch of nodal line, and when we examine the eigen- frame rotation along the loop, we will retrieve a generalized quaternion charge of q = gi (frame rotation of π) in accordance with the assigned arrow on nodal line. Similarly, we can trace the eigen-frame rotation along the orange loop, which will also result into the charge of q = gi. However, as shown in Fig. S1(b) with orange loop (dashed), if the loop direction is reversed (compared to that of Fig. S1(a)) the sign of encircled charge becomes opposite as q = -gi., since that the frame rotation is also reversed due to the flipped loop.
One important property of the π1 homotopy loops is that they are allowed to adiabatically transform as long as no singularity or nodal lines are crossed, and the encircled topological charge remains unchanged. As such, two π1 loops starting and ending at the same base point are allowed to merge through adiabatic transformation, which corresponds to the frame charge multiplication.
Given the above information, we then consider the multiplication of frame charges by encircling two branches of nodal lines together. In Fig. S1(c), we show the merged loop of the two original loops in Fig. S1(b) (green and orange), and by deforming the loops by a small
amount, we can visually see that the merged loop now actually encircles two branches of nodal lines together (e.g., lower left one and upper right one). Such multiplication is noted as q = gi · (-gi) = +1 (corresponding to frame rotation of π - π = 0).
For easier understanding, the “+1” charge is manifested as the encircling of two “opposite arrows”. To illustrate the meaning of “opposite”, we can adiabatically transform the encircled nodal line branches in Fig. S1(c) into Fig. S1(d) without changing any topology, and we can easily see that one arrow on nodal line points towards the crossing point and the other arrow points outwards, which makes them “opposite”.
Now we consider the encircling of two arrows of “same direction”. In Fig. S1(e), the two loops both encircle a nodal line and carries a frame charge of q = gi. In Fig. S1(f), we show the merged loop encircling both nodal lines, that makes the multiplication of q = gi · gi = -1 (frame rotation of π + π = 2π). It can be noticed that the two arrows on encircled nodal lines both point towards the crossing point and are noted as “same direction”. The terminology of “opposite” and “same direction” arrows are used in main text for easier identifying the “+1” and “-1” frame charges.
We have thus shown that for the crossed nodal lines configuration shown in Fig. S1(a-f), frame charges of q = ± gi, +1, -1 can each be retrieved.
Let us now discuss the case of braiding on such crossing nodal lines. Regarding adjacent bands, we note that the charge multiplication is anticommutative, e.g., gi · gi+1 = -gi+1 · gi, which introduces the non-Abelian braiding between nodal lines. We present a pictorial explanation of non-Abelian braiding in Fig. S1(g), where we consider a configuration with only one blue nodal line for simplicity, and the red nodal line (e.g., formed between (i+1)th and (i+2)th bands) is “in front of” (from the viewpoint of the basepoint) the blue one (e.g., formed between ith and (i+1)th bands). We show three π1 loops encircling frame charges of q = gi+1, gi, and -gi+1, respectively (The loops are slightly opened for the eye to see which line is being encircled). The combination of such three green loops can be intuitively understood by adiabatically dragging the two marked points in purple along the grey dashed lines, which results into the combined loop in right panel of Fig. S1(g). As can be noticed, the merged loop encircles only the blue nodal line of upper right now. The merging of loops indicates a charge multiplication of q = gi+1 · gi · (-gi+1) = (-gi · gi+1)· (-gi+1) = -gi · [gi+1 · (-gi+1)] = -gi. Importantly, we can find that the resulting charge of q = -gi is with a minus sign compared to the charge of q = gi before braiding. The extra minus sign leads to the change of arrow direction on the upper right blue
nodal line in Fig. S1(g) (compared to the original configuration in Fig. S1(a-f)), and the two arrows on blue nodal lines are both pointing towards the braiding point.
Given the message above, we consider the braiding of full nodal lines configuration in Fig.
S1(h), it can be easily inferred that the braiding with red nodal line results into two possible arrow configurations on the blue nodal lines as shown, which make the junction point either a source or a sink of arrows. Take the source configuration as example, we show in Fig. S1(i) and (j) that the eigen-frame rotation will correspond to “-1” charge by tracing along both the two green loops, since that they are each encircling a pair of same direction arrows. This defines the “double -1” charge at the braiding point.
3. Numerical calculation of Zak phase
To account for the boundary modes of the photonic crystal, the Zak phases are calculated numerically after solving the eigenmodes of the proposed photonic crystal unit and retrieving the periodic part of Bloch modes for each band. Specifically, Wilson loops are constructed as straight lines along kx,y,z directions, respectively, which are closed by taking periodic gauge.
The Zak phases are formulated as
𝜙𝑍𝑎𝑘= −𝑖 log⟨𝑈1|𝑈2⟩⟨𝑈2|𝑈3⟩ ⋯ ⟨𝑈𝑛−1|𝑈𝑛⟩ , (7)
with 𝑈𝑛 = 𝑈1𝑒−𝑖𝐺𝑟 (G is the reciprocal lattice vector in BZ).
The calculated results are shown in Fig. S2. In Fig. S2(a), the nodal structure projection on ky
– kz plane is shown, the Zak phase ϕx is calculated to predict the surface modes on the y – z surface of the photonic crystal sample. In Fig. S2(b), the Zak phase ϕx distribution is examined along Г – Y – Z (avoiding the degeneracy positions) for the 1st band, and the Zak phase goes through a jump while crossing the blue nodal line. The projection of blue nodal line thus separates the surface BZ into two regions in Fig. S2(a), with the shaded region indicating π value and predicting the appearance of surface mode. However, as shown in Fig. 5(e) of the main text, there is no energy gap between projection of 1st and 2nd band in the ky – kz plane, thus no surface mode is observed on the y – z surface of the sample.
In Fig. S2(c), we show the nodal structure projection on the kx – kz plane. The Zak phase ϕy is calculated along Г – Z – X for the 1st band as shown in Fig. S2(d), and the blue nodal line again separates the surface BZ into two regions, the region with π Zak phase is shaded in blue in Fig.
S2(c). The measured surface modes on the x – z surface of photonic crystal are provided in note 4 below.
In Fig. S2(e), the projection of nodal structure on the kx – ky plane is shown. To explain the experimentally observed surface mode induced by the nodal ring in Fig. 6 of main text, we should consider the total contribution from the 1st and 2nd bands as shown in Fig. S2(f), where the unit cell selection (shown as inset) is in accordance with the experimental configuration, and the Zak phase ϕz is examined along Г – Y – X. We show the region with accumulated Zak phase of ϕz = π in red colour in Fig. S2(e), which is consistent with the experimental observation.
It should be mentioned that, when the resonators are gradually tuned as same length of L1 = L2, the shaded regions in Fig. S2(a) and (c) both gradually expand to fill the full surface BZ. The Zak phases thus predict surface modes on all the side surfaces of the uniaxial photonic crystal.
Another information that can be inferred from the Zak phase is the sign change on frame charges that are carried by periodic nodal line partners in neighbor Brillouin zones, as has been recently proposed by Ref. 34 of main text, where the relative sign depends on the Zak phase difference between participating bands as
𝑞𝑁𝐿= (−1)𝑚𝑞𝑁𝐿′, (8)
and m is the number of band pairs that carry different Zak phases.
We conduct the Zak phase calculation for a truncation of unit cell with mirror symmetry in z- direction, so that the Zak phase is quantized as shown in Fig. S2(g). We can then find from the results in Fig. S2 (b), (d) and (g) (inside the nodal ring projection) that no Zak phase difference exists for 1st and 2nd bands. As such, the arrows on nodal lines # 3,4,7,8 (or the frame charge q
= ±g1 carried) are the same as that of nodal lines #3’, 4’, 7’, 8’, such analysis is consistent with the numerical calculation in Appendix C 2.
4. Observation of surface modes on the x – z surface of biaxial crystal
The Zak phase calculation in note 3 predicts the surface modes on the x – z surface of the biaxial photonic crystal. We present here in Fig. S3 the numerically calculated and experimentally measured results of such surface states. The projected bands on kx – kz plane are shown at different kz-cut positions in the first and third rows. The surface modes are marked with magenta curve in the calculation results. Experimentally measured results are shown in the second and fourth rows, where the surface mode positions are indicated with dashed lines.
The light cones for air and substrate are also shown as blue and red colors, respectively.
5. Frame charge source identified at the Г point of an acoustic model
To verify that the topological features proposed in this manuscript can be extended to other multiband systems, we here provide the calculation for an acoustic continuum model. The Hamiltonian reads as
𝐻 = (
𝑣𝑇2𝑘2 (𝑣𝐿2− 𝑣𝑇2)𝑘𝑥𝑘𝑦 (𝑣𝐿2− 𝑣𝑇2)𝑘𝑥𝑘𝑧 (𝑣𝐿2− 𝑣𝑇2)𝑘𝑥𝑘𝑦 𝑣𝑇2𝑘2 (𝑣𝐿2− 𝑣𝑇2)𝑘𝑦𝑘𝑧 (𝑣𝐿2 − 𝑣𝑇2)𝑘𝑥𝑘𝑧 (𝑣𝐿2− 𝑣𝑇2)𝑘𝑦𝑘𝑧 𝑣𝑇2𝑘2
) , (9)
where 𝑘 = √𝑘𝑥2+ 𝑘𝑦2+ 𝑘𝑧2, vL and vT are the velocities of longitudinal and transverse modes.
The frame charges are calculated based on this Hamiltonian and the results are shown in the Fig. S4 below, where the blue nodal lines are two-fold degeneracies, and red lines are three- fold degeneracies. The arrow configurations show that the phononic Г point can also be identified with a source character, proving that the proposed non-Abelian frame charge features in this manuscript can be extended to phononic systems.
Supplementary Figures
Fig. S1. Homotopy characterization of crossing nodal lines. (a) Two nodal lines cross at a joint point and form into four branches, where the frame charges do not accumulate, two arrows flow in and then flow out. Each of the two π1 loops encircling the nodal lines possesses a charge of q = gi. (b) The direction of orange loop is inverted as dashed line and encircles a charge of q = -gi. The two loops can then merge together. (c) The combined path is shown, which gives charge multiplication of gi · (-gi) = 1. (d) The new π1 loop encircled nodal lines can adiabatically transform, which illustrates that the “+1” charge corresponds to a pair of “opposite” arrows. (e) The π1 loops encircling two “same direction” arrows are shown. (f) The two π1 loops join as a new one corresponding to the charge multiplication of gi · gi = -1, which corresponds to a pair of arrows with “same direction”. (g) The braiding between red and blue nodal lines. The combined green loop in left panel is a result of the multiplication of charges (encircled by original π1 loops before merging) as q = gi+1 · gi · (-gi+1), which is topologically equivalent to the loop in the right panel with q = -gi. (h) The braiding between the red nodal line and the two blue nodal lines (four branches) changes the direction of two arrows, making the crossing point a source or sink of arrows. (i-j). Two ways of achieving the “-1” charge are shown, which define a “double -1” charge.
Fig. S2. Zak phase in biaxial photonic crystal. (a) The nodal structure projection in the ky – kz plane, shaded region is with Zak phase of π for 1st band. (b) Zak phase ϕx examined along Г – Y – Z for (a), π value predicts surface mode within the projection gap between 1st and 2nd bands. (c) The nodal structure projection in the kx – kz plane, shaded region is with Zak phase of π for 1st band. (d) Calculated Zak phase ϕy along Г – Z – X. Within the projection gap between 1st and 2nd bands, surface mode is allowed to appear in the region with π value. (e) The nodal structure projection in the kx – ky plane, shaded region is with Zak phase of π with contribution from both the 1st and 2nd bands. (f) For the experimental measurement configuration (shown as inset), the accumulated Zak phase ϕz of 1st and 2nd bands is calculated, which predict the surface modes to appear in the projection gap between 2nd and 3rd bands, the region is shaded in red in (e). (g) Calculated Zak phase ϕz for the unit cell (shown as inset) truncated with mirror symmetry.
Fig. S3. Experimentally measured surface states on the x – z surface of biaxial photonic crystal. Simulation results are shown in the 1st and 3rd rows for the band projection in the kx – kz plane. Fixed kz values are adopted as 0.3 to 0.8 π/c. Surface modes are indicated with magenta colour. Experimentally measured band projections are shown in the 2nd and 4th rows, bulk band dispersions are plotted on top as white curves, predicted surface mode positions are indicated with dashed lines.
Fig. S4. Frame charges calculated for an acoustic continuum model. (a-l) The frame charges corresponding to the π1 loops in green.
