Spin-controlled topological phase transition in non-Euclidean space

Modulation of topological phase transition has been pursued by researchers in both condensed matter and optics research fields, and has been realized in Euclidean systems, such as topological photonic crystals, topological metamaterials, and coupled resonator arrays. However, the spin-controlled topological phase transition in non-Euclidean space has not yet been explored. Here, we propose a non-Euclidean configuration based on Möbius rings, and we demonstrate the spin-controlled transition between the topological edge state and the bulk state. The Möbius ring, which is designed to have an 8π period, has a square cross section at the twist beginning and the length/width evolves adiabatically along the loop, accompanied by conversion from transverse electric to transverse magnetic modes resulting from the spin-locked effect. The 8π period Möbius rings are used to construct Su–Schrieffer–Heeger configuration, and the configuration can support the topological edge states excited by circularly polarized light, and meanwhile a transition from the topological edge state to the bulk state can be realized by controlling circular polarization. In addition, the spin-controlled topological phase transition in non-Euclidean space is feasible for both Hermitian and non-Hermitian cases in 2D systems. This work provides a new degree of polarization to control topological photonic states based on the spin of Möbius rings and opens a way to tune the topological phase in non-Euclidean space. Graphical abstract Supplementary Information The online version contains supplementary material available at 10.1007/s12200-024-00110-w.

Ⅵ. Band structure for Möbius ring as the link ring in both x and y direction

I. Breaking the degeneracy of TE and TM modes
In the main text, it mentions that even the spin-lock effect works in theory, the simulation results show that such a Möbius ring is still unable to achieve spin lock.We The two modes cannot be distinguished during the ring twist process.To break the degeneracy, we therefore design that the length and width of 8PMR change adiabatically during the twist process.
The results of a straight waveguide with length and width changing adiabatically and twist operation are shown in Figure S2(a).It shows that in this case, the TE mode and TM mode can be transformed into each other along the propagation direction.We also compare it with a regular waveguide without the length and width adiabatically changing shown in Figure S2(b).The results show that in this case, the modes cannot be transformed along the propagation direction.As a consequence, the adiabatically changing of length and width of 8PMR is necessary and essential.

Figure S1
The simulation results of degeneracy between TE and TM modes in a 8-π period Möbius ring without the length and width changing.There is no spin-lock effect.

II. Non-adiabatic-evolution Möbius ring
If the diameter of a Möbius ring is not large enough, the adiabatic process will be affected.In this case, the light-lock effect still works, but the locking light is not just the circular polarized light, there will be other direction polarized light locking.We

Figure S3
The linear polarized locked light in a non-adiabatic-evolution Möbius ring.
We therefore analyzed the influence of non-adiabatic process on the phases in Möbius ring.We found that an induced extra phase  factor could well describe this phenomenon.In this case, the Eq.2 can be transformed into Equation S1 as below. (S1) Then, the parameter  0 needs to meet the following condition, i.e., In this case, the locked light will then be the linearly polarized light as Figure S3 shows.
If the value of δ is other quantity, it is difficult to judge the specific polarized direction, and the locked light might be some elliptical polarized light.In the main text, we reckon the Möbius rings have undergone adiabatic evolution, therefore, they all have the spinlock property.

III. Calculation for Möbius ring as the link ring in SSH model
As Figure 2(a) shows in the main text, 8PMR is as a link ring in add-drop type micro rings (unit in SSH).Here, we demonstrate more concretely in Figure S4.In a unit, the coupling strength is equal as  1 , and the coupling strength is  2 between units.The dynamic phase factor of the half cycle accumulation in the link ring (Möbius) is also considered to be the same, represented by ', i.e., ' =  − '.For transmission matrix method (TMM), there can be four matrices to express the process.
Where α represents the intrinsic loss of light propagating half cycle in a link ring (Möbius ring), and the value is the same in the main ring (regular ring).The symbol Y represents the Möbius characteristics, i.e., the twist operation, and it can be written as �, it is derived from the coupling relationship in add-drop type micro rings.The coupling relationship with regular ring as the link ring in add-drop type micro rings is shown in Section Ⅴ, and the coupling relationship with Möbius ring as the link ring in add-drop type micro rings has the similar derivation process, but the coupling coefficient k and transmission coefficient t need to be expressed by matrices.
We can therefore obtain the expression of transmission matrix as From the above derivation, we can obtain the total matrix as The further schematic of Möbius ring as the link ring in SSH model.
Furthermore, by using the same transmission relationship, the eigenvalue distributions of SSH chains with finite length, as well as the corresponding bulk states and topological edge states can also be solved as shown in Figure 2(e)-2(h) in the main text.

IV. Calculation for Möbius ring as the link ring in CROW arrays
The coupling relationship with Möbius ring as the link ring in CROW arrays is similar to the that in SSH model, but there should be both X and Y directions coupling relationship for CROW arrays.As shown in Equation S4, the X-direction coupling relationship is the same as the SSH case because the Möbius ring is as the link ring along the X direction.The Y-direction coupling relationship could be simpler because the regular rings are as the link rings along the Y direction (This part can refer to Section Ⅴ).We therefore can obtain as follows: By using the TMM, we can then obtain the corresponding energy band structures.
Moreover, by using the similar transmission relationship, the projective band structures and the mode distributions can be obtained as shown in Figure 3 and Figure 4 in the main text.

V. Calculation for regular ring as the link ring in add-drop type micro rings
In section Ⅲ、Ⅳ, we have discussed the coupling relationship in SSH model and CROW arrays based on the Möbius ring as the link ring in the X direction.These derivations are on account of the coupling relationship of regular coupled resonant rings.
Here, we will briefly show the relationships and more information can refer to reference.
The racetrack shaped resonant ring structure is used here, so that the coupling region can be replaced by two coaxial straight waveguide coupling model.The coupling matrix between two resonant rings can be written as: where the  and  represent the transmittance and coupling coefficient in the coaxial waveguide coupler respectively.These two parameters also have the following constraints:  2 +  2 = 1.When the dielectric constant and wavelength in the ring are selected, the factor  in the matrix is only related to the gap size of the coupling region, (S8) and the factor   is the effective length of the region.By applying the coupling matrix to the add-drop type micro rings, the coupling relationship among   ,   ,   and  ℎℎ can be obtained as follows.
Where the intrinsic loss of light propagating half cycle in a link ring is α and the coupling region of the ring is symmetrical. represents the phase accumulation in the link ring.' represents the phase accumulation in the upper half (divided by the coupling region) of the link ring.Up till now, all the parameters in the transmission matrix become physical quantities that only depend on the optical frequency, and we can further fit the simulation results of an actual configuration with specific structural parameters, all coupling parameter values corresponding to the configuration can be obtained.

Ⅵ. Band structure for Möbius ring as the link ring in both x and y direction
In section Ⅵ, we have calculated the projective band structure for Möbius ring as the link ring in both x and y direction, and compared it to the case where Möbius ring as the link ring in x direction.As shown in Figure S5, when Möbius rings are the link ring in both x and y direction, the positions of bands of RCP and LCP will be changed, and there will still be spinlocked effect.
have simulated the 8-π period Möbius ring (8PMR) with the same sizes of ring's length and width.The results are shown in Figure S1.The input light is in a [1, −] form from the upper position of the ring, and four monitors are placed at the symmetrical positions, including the top, bottom, left and right respectively, of the ring.We select two monitors on the bottom (Figure S1(a)) and right positions (Figure S1(b)) of the ring.The bottom monitor demonstrates the anti-clockwise phase distributions along the propagation direction (Ex) and the right monitor demonstrates the clockwise phase distributions along the propagation direction (Ey).Due to the different spin directions of the light, this Möbius cannot play the spin lock function.This is caused by the TE mode and TM mode degeneration as the sizes of length and width in 8PMR are equal.
(a) The data obtained from the bottom monitor, showing the anti-clockwise phase distributions.(b) The data obtained from the right monitor, showing the clockwise phase distributions.

Figure S2
Figure S2The simulation results of breaking the TE and TM modes degeneracy.(a) have verified the light-lock phenomenon by simulation as shown in Figure S3.This Möbius ring has diameter size of 6 um, and the input light is in the [1, 0] form, which represents linear polarized light, from the right position of the ring.Due to the nonadiabatic evolution of the Möbius waveguide ring, the monitor shows that locked light is also linear polarized light instead of the circular polarized light.
we can simplify the original mode of [; ] as [1;  * (−1)  *  −  2  ].If the value of  is zero, the situation is the same as the description in the main text, which represents the adiabatic evolution occurring in a Möbius ring.If the value of δ is π, the supported mode becomes [1, (−1)  ], which represents the linear polarized light.