Spin-Polarized Fractional Corner Charges and Their Photonic Realization
TTopological spin corner charge in the presence of spin-orbit coupling
Ran Gladstein Gladstone, Minwoo Jung, and Gennady Shvets School of applied and engineering physics, Cornell University ∗ Department of physics, Cornell University (Dated: February 26, 2021)We investigate several C second order topological insulators that are independently topologicalor trivial for two different spins, or pseudo-spins, using tight binding models and first principleselectromagnetic simulations in the presence of spin orbit coupling (SOC). When the insulator issecond order topological for both spins the corner charge is quantized to 0 and a topological phasetransition can occur due to SOC. However, when the insulator is second order topological for onlyone of the spins, the corner charge is quantized to 1/2 and no topological transition can manifest. The interplay of spin and topology has long been ofinterest to physicists [1–6]. Shortly after Chern insula-tors [1] had been discovered, researchers have extendedthe same idea to a system possessing spin 1 / / Z topological invariant, aspin Chern insulator possesses a Z topological invariantdue to time-reversal and inversion symmetries [11]. Theaddition of spin to the system has therefore changed itstopological classification.More recently, physicists have extended the conceptof topological insulators to higher order topological in-sulators (HOTIs) [12–14], and their photonic counter-parts (PHOTIs) [15–18], which are d dimensional topo-logical insulator with a bulk topological invariant pre-dicting topological states localized on some of its d − n dimensional terminations, where n ∈ N and d ≥ n ≥ C symmetry(rotation by 60 ◦ ). For spinless models, these topologi-cal crystalline insulators have been shown to support aHOTI phase with a quantized corner charge of 1 / / C symmetry (rotation by 180 ◦ ), which defines the cor-ner charge for the C models we inspect, we can definea new C operator that rotates the lattice for both spinssimultaneously, and show that the new system of twospins and SOC is invariant to that operator. Thus a newconserved corner charge arises; the total corner charge,which is the sum of the corner charges of the differentspins when no SOC is included. The difference betweenthe hybrid phases and the double topological phase isthen immediately apparent. The hybrid phases possesa quantized total corner charge of 1 / C symmetry, therebyallowing a Kekul´e distortion [28], which opens a bandgap at the Γ point of the Brillouin Zone. A couple ofexamples of such PhC designs are shown in Fig.1(b) and a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1: Spin degenerate photonic topological insulatorbased on Kekul´e distortion. (a) Top and side view ofPhC unit cell. (b) Band structure for first hybrid phase. a is the magnitude of the primitive vectors of the unitcell. r = r mid = 0 . a , r = r mid = 0 . a , g top = g bot = 0 . a . (c) Band structure for secondhybrid phase. r = r mid = 0 . a , r = r mid = 0 . a , g top = g bot = 0 . a . (d) Field profiles for the bandsbelow the band gap at the Γ point for the structures of(b),(c). Dashed blue box for TE-like modes and dashedred box for TM-like modes. Arrows point to the bandstructure each field profile belongs to.(c). When setting the gaps between the PEC elementsand plates, g top = g bot also possesses mid-plane mirrorsymmetry z → − z . When this is the case, the PhC’selectromagnetic modes can be classified as either TE-like or TM-like, which are orthogonal to each other [29].The TE-like modes are distinguished by having a non-vanishing H z component at the z = h/ E z component, asshow in Fig.1(d).It has been shown that for such PhCs the modesbelow and above the band gap at the Γ point haveprofiles that correspond to p x , p y and d xy , d x − y or-bitals [16, 30]. When the bands below the band gappossess p orbital profile, the PhC has an orbital Chernnumber of C orb = 0 [31], where the traditional role ofthe spin DOF is taken by the orbital DOF, and a cornercharge of Q c = 0 (mod 1) [26]. Changing the parametersof the PhC can cause a band inversion, switching theorbital profiles of the bands below and above the bandgap, such that the bands below possess d orbital pro-files [16, 30, 31]. When such a band inversion occurs, theorbital Chern number switches to C orb = 1 and cornercharge of Q c = 1 / Q c = 1 / Q c = 0 (mod 1). Since the polarizationsplay a similar role to spin, we can define a spin cornercharge Q ↑ / ↓ c where the superscript ↑ ( ↓ ) denotes the cor-ner charge associated with the spin-up (spin-down) de-gree of freedom. We note that to confirm the cornercharge for each polarization, one has to inspect the fieldprofiles of all the relevant bands below or above the bandgap, not just at the Γ point but also at the M and poten-tially the K points. In the supplementary information weverify the corner charges can be correctly predicted fromthe bulk band properties at the high symmetry points.In order to make theoretical predictions for PhCs withsimultaneous Kekul´e distortion for two polarizations weuse a nearest neighbor tight-binding (TB) model realspace Hamiltonian of a hexagonal lattice, with two spinor layer subspaces, each possessing its own Kekul´e tex-ture. We add Kane-Mele SOC by introducing next-nearest neighbor coupling with a hopping amplitude signthat depends on the spin and hopping direction (clock-wise or counter-clockwise) [1, 2]. The tight binding modelis therefore given by H = (cid:88) (cid:104) ij (cid:105) t ↑ in c † i ↑ c j ↑ + (cid:88) (cid:104) i (cid:48) j (cid:48) (cid:105) t ↑ out c † i (cid:48) ↑ c j (cid:48) ↑ + (cid:88) (cid:104) ij (cid:105) t ↓ in c † i ↓ c j ↓ + (cid:88) (cid:104) i (cid:48) j (cid:48) (cid:105) t ↓ out c † i (cid:48) ↓ c j (cid:48) ↓ + (cid:88) (cid:104)(cid:104) ij (cid:105)(cid:105) αβ it SOC ν ij s zαβ c † iα c jβ , (1)where the first (third) term describes the nearest neigh-bor intra unit cell hoppings of the spin up (down) elec-trons, the second (fourth) term describes the nearestneighbor inter unit cell hoppings of the spin up (down)electrons, and the fifth term describes the usual nextnearest neighbor Kane-Mele SOC [2]. If SOC is ne-glected, each spin is independent of the other and, forthis case, it has been shown that this model accuratelypredicts the band inversion discussed above [16, 31]. Foreach spin, when t in > t out the bulk phase is trivial andwhen t in < t out it is topological.The band structure of two such Hamiltonians areshown in Fig.2(a) and (b). In Fig.2(a) the solid linesshow the band structure for a Hamiltonian with topolog-ical Kekul´e textures for both spins, while Fig.2(b) solidFIG. 2: Bulk band structures of topological and hybridphases with and without SOC. (a) Bulk bands for t ↑ in = t ↓ in = 0 . t ↑ out = t ↓ out = 1 . t SOC = 0 in solidlines and with SOC t SOC = 0 . t ↑ in = t ↓ out = 0 . t ↑ out = t ↓ in = 1 . t SOC = 0 for solidlines and t SOC = 0 . r = 0 . a, r mid1 = 0 . a,r = 0 . a, r mid2 = 0 . a, h mid = 0 . a, g bot = 0 . a,g top = 0 . a . (d) Hybrid phase with SOC. Parameters:Same as Fig.1(c) except for g top = 0 . a .lines show the band structure when the Kekul´e textureis topological only for one of the spins. The addition ofKane-Mele SOC, shown with dashed lines, has a drasti-cally different effect on the different Hamiltonians. Whenthe Kekul´e texture is topological for both spins, the ad-dition of SOC can cause a band gap closure with a Diraccone and further increase of the SOC term causes theband gap to reopen [10], indicating a topological phasetransition. However, when only one of the spins has atopological Kekul´e texture, the addition of Kane-MeleSOC can never close the band gap. The reason for thisdifference is that the SOC only causes bands of the sameorbital type to repel each other. Therefore, for the topo-logical phase, where the d-orbital bands of both polariza-tions are degenerate below the band gap, they will repeleach other, but for the hybrid phases the bands of bothpolarizations have different orbital profiles and will notrepel each other.For a spinless C Hamiltonian, the corner charge isquantized due to C and C symmetries [26]. Since Kane-Mele SOC preserves both of these symmetries, there isstill a conserved corner charge even in its presence. How-ever, the conserved corner charge is now dependent onthe properties of all the bands below the band gap in-stead of the bands of only one of the spins, because thetwo spin subspaces are no longer independent. Since theaddition of small SOC compared to the nearest neigh- bor hopping amplitudes does not change the C and C eigenvalues of the bands at high symmetry points, theconserved corner charge in the presence of small SOC isthe sum of the corner charge of both spins when SOC, orthe total corner charge. For the case shown in Fig.2(a)the total corner charge is Q c = Q ↑ c + Q ↓ c = 0 (mod 1). Incontrast, for the case of Fig.2(b) the total corner chargeis Q c = Q ↑ c + Q ↓ c = 1 / z → − z symme-try, which we’ve established in previous works to emulateKane-Mele SOC in PhCs [23, 24]. We have designed thestructure to exhibit a closing the band gap with a Diraccone as in Fig.2(a). In Fig.2(d) we show the band struc-ture of the PhC from the right panel of Fig.1(b), whichis topological only for one of the polarizations. The ad-dition of SOC in this case slightly increases the band gapsize, in similar fashion to Fig.2(b).For the hybrid phases, we verify using a real spacetight binding simulation that corner modes exist at thezero energy at every 120 ◦ corner, as shown in Fig.3(a).Indeed, the addition of SOC to the hybrid phase, shownin Fig.3(a) right, does not affect the zero energy cornerstates. We repeat this in first principles electromagneticeigenmode simulations using COMSOL Multiphysics andshow (Fig.3(c)) that corner states manifest at a 120 ◦ cor-ner, and that SOC does not affect them (Fig.3(d)). Wenote that we used edge roughening [16] at the interfacebetween domains, which means we altered the radii ofthe cylinders along the interface in order to create a bandgap between the edge modes to reveal the existence of thecorner states. In the tight binding model this is done bymodulating the hopping terms along the interface.For the topological phase interfacing the trivial phasewe can see that in the absence of SOC we can still detectcorner states at the mid-gap (Fig.4(a),(c)), but as SOCis added the bulk band gap closes, eventually reopeningwith no corner states (Fig.4(b),(d)). The addition of suf-ficient SOC causes a first and second order topologicalphase transition to a spin-Hall phase, but unlike previ-ously known spin-Hall PhCs, the analytic model predictsthat the Kekul´e order is responsible for a folding of the K and K (cid:48) valleys into the Γ point, resulting in the arearrangement of the topologically protected kink statesfrom 2 per valley to 4 at the Γ point. This spin Hallphase is therefore expected to be a novel first order topo-logical phase, distinct from our previous works spin Hallphase [23, 24].In practice, one has to ensure a high degree of degen-eracy of all the relevant bulk bands below and above theFIG. 3: Corner spectra for hybrid phases interfacing at120 ◦ . (a) Left: eigenmodes for interface between twodomains with t ↑ in = t ↓ out = 1 . , t ↓ in = t ↑ out = 0 . t ↑ in = t ↓ out = 0 . , t ↓ in = t ↑ out = 1 . t SOC = 0 .
4. (b) Left: firstprinciples eigenmodes for interface between two hybridphases with no SOC. Corner mode | E | in top leftinsets and | H | in the bottom right insets, with colorbar marking the relative intensity. Left domainparameters are as in Fig.1(b) left panel and rightdomain are as in Fig.1(b) right panel. Edge rougheningused is r rough1 = 0 . r , r rough2 = 1 . r . Right: same asleft but with g top = 0 . a for the left domain and g bot = 0 . a, g top = 0 . a for the right domain.band gap to prevent significant scattering of the kinkstates [23]. To show that we indeed achieved a new firstorder topological phase in the photonic system, we in-spect the kink spectrum of a Kekul´e distorted PhC withSOC interfacing the trivial phase of Fig. 4(c), which hasno SOC (see Fig.5(b)). This allows us to achieve effec-tive spin degeneracy without having to design both bulkPhCs so that the band gaps overlap between them. A su-percell kink eigenmode simulation of this design reveals,as expected, two kink modes with linear dispersion cross-ing at the edge projected Γ point. Furthermore, this newphotonic spin Hall phase has an integer spin Chern num-bers, compared to our previous works [23, 24], which hadhalf integer Chern numbers. This allows us to achievetopologically protected kink states without relying on anadditional non-trivial bulk PhC.We recall that the total corner charge for the topolog-ical phase is 0, which serves as an example of a systemwith a trivial bulk invariant that nevertheless exhibitscorner states. Even more surprising, the corner statesremain at the mid-gap until the band gap closes, becauseone is symmetric to reflection while the other is antisym- FIG. 4: Corner spectra for hybrid phases interfacing at120 ◦ . (a) Left: eigenmodes for interface between twodomains with t ↑ in = t ↓ in = 1 . t ↓ out = t ↑ out = 0 . t ↑ in = t ↓ in = 0 . , t ↓ out = t ↑ out = 1 . t SOC = 0 . | E | in top left insets and | H | in the bottomright insets, with color bar marking the relativeintensity. Parameters for topological domain: r = 0 . a, r mid1 = 0 . a, r = 0 . a, r mid2 =0 . a, h mid = 0 . a, g top = g bot = 0 . a . Edgeroughening used is r rough1 = 0 . r , r rough2 = 1 . r . Right:same as left but g top = 0 . a for the topological domainand g bot = 0 . a for the trivial domain.metric. This means that even though there is no bulkinvariant predicting the robustness of the corner states,they are robust to SOC, much in the way that topologi-cal edge states cross the entire band gap until the bandgap is closed. The difference between the cases is thatthe first order invariant remains unchanged up until theband gap is closed and reopened, while the bulk cornercharge in this case switches to 0 immediately upon intro-duction of SOC of any strength. This example serves toshow how the addition of spin to a HOTI can actuallyrender its bulk invariant, and thus BBC, void of meaning.We therefore do not call the total conserved corner chargetopological. Perhaps there exists a yet undiscovered bulktopological invariant that predicts these differences andmaintains the BBC, but we do not know of such an in-variant as of the writing of this letter.Since our results apply to any C system with twoDOFs coupled to each other with coupling that preserves C and C symmetries, one isn’t limited to spinful con-densed matter systems if they try to achieve one of thephases discussed in our letter. While the topologicaland trivial phases may be relatively straightforward toFIG. 5: Kink spectrum of supercell simulation of spinHall phase with Kekul´e distortion interfacing a trivialphase with no SOC. b = √ a is the supercell periodicity.Kink modes in red and black solid lines and bulk modesin blue asterisks. Inset: Top view of supercall structure.Topological domain with SOC is in blue and trivialdomain in yellow. Parameters for topological domain: r = 0 . a, r mid1 = 0 . a, r = 0 . a, r mid2 =0 . a, h mid = 0 . a, g bot = 0 . a, g top = 0 . a . Fortrivial domain: r = 0 . a, r mid1 = 0 . a, r =0 . a, r mid2 = 0 . a, h mid = 0 . a, g bot = g top = 0 . a .Edge roughening used is r rough1 = 0 . r , r rough2 = 1 . r .achieve using a spinful crystalline system, that is not thecase for the hybrid phases, because the difference be-tween spin-dependent hopping amplitudes is rather large.However, the hybrid phases may be more approachableby using bilayer graphene (BLG) where each layer has adifferent Kekul´e texture.Our designs open a new direction for active control oftopological corner states, where active control of the mid-plane mirror symmetry can switch the topological cornerstates. For example, using the thermo-optical effect onecan apply an external electromagnetic field to changethe refractive index of some elements of the PhC [33],thereby breaking the mid-plane mirror symmetry, caus-ing a topological phase transition to a spin-Hall phase.Furthermore, our work also shows how robustness to suchsymmetry breaking can be achieved to produce devicesthat are immune to that type of fabrication faults with-out compromising the number of protected corner statesthe device supports on each corner.In summary, we’ve studied three different C PHOTIsand one trivial C with two independent Kekul´e texturesand shown they react in completely different ways to theaddition of SOC. We’ve explained the origin of these dif-ferences and shown the effect on their corner modes usingboth an analytic model and first principles electromag- netic simulations. 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