Mode Delocalization in Disordered Photonic Chern Insulator
Udvas Chattopadhyay, Sunil Mittal, Mohammad Hafezi, Y. D. Chong
MMode Delocalization in Disordered Photonic Chern Insulator
Udvas Chattopadhyay, Sunil Mittal, Mohammad Hafezi, and Y. D. Chong
1, 3, ∗ Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore 637371, Singapore Department of Electrical and Computer Engineering,The University of Maryland at College Park, College Park, MD 20742, USA Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore 637371, Singapore
In disordered two dimensional Chern insulators, a single bulk extended mode is predicted to existper band, up to a critical disorder strength; all the other bulk modes are localized. This behaviorcontrasts strongly with topologically trivial two-dimensional phases, whose modes all become local-ized in the presence of disorder. Using a tight-binding model of a realistic photonic Chern insulator,we show that delocalized bulk eigenstates can be observed in an experimentally realistic setting.This requires the selective use of resonator losses to suppress topological edge states, and acquiringsufficiently large ensemble sizes using variable resonator detunings.
I. INTRODUCTION
One of the most intriguing features of the IntegerQuantum Hall Effect is the extraordinary accuracy ofquantization in the Hall resistivity of about 1 part in 10 .Disorder plays an important role in this phenomenon;without disorder, the Integer Quantum Hall Effect’s cel-ebrated conductance quantization plateaus could notexist . As the seminal work of Anderson and co-workershas shown, the effects of disorder are strongly dependenton the spatial dimensionality . In one dimension, ar-bitrarily weak disorder localizes all states, whereas threedimensional systems host localized states at low ener-gies and extended states at high energies, separated bya mobility edge. In two dimensions (2D), the effects ofdisorder depend on the time-reversal and spin symme-tries of the system . For normal 2D materials, which arein the orthogonal symmetry class, all states are localizedby disorder, similar to the one dimensional case . Forthe unitary class, which includes Integer Quantum Hallsystems and other Chern insulators, localization occursvia a mechanism called “levitation and annihilation”: inthe limit of infinite system size, the introduction of dis-order causes all states to localize except for one state per(topologically nontrivial) band, which remains extended;with increasing disorder strength, extended states in ad-jacent bands can move towards each other and annihilate,producing a transition to a purely-localized phase .The inter-plateau longitudinal conductance peaks inthe Integer Quantum Hall Effect constitute the princi-pal experimental evidence for the special bulk extendedstates in 2D unitary disordered systems . In Chern in-sulators without Landau levels, there is thus far little ex-perimental evidence for these states, nor for theorized be-haviors such as levitation and annihilation, though manynumerical studies have been performed . In con-densed matter settings, such experiments are very chal-lenging due to the need to fabricate large samples withcontrolled amounts of disorder.This paper investigates the possibility of using pho-tonics to probe the localization behavior of 2D Chern insulators. Over the last decade, photonics has emergedas a versatile setting for realizing phenomena associatedwith band topology , including reflectionless edgetransport , topological pumping , spin and val-ley Hall edge states , Fermi arcs , and more. Topo-logical phenomena also hold promise for novel deviceapplications in photonics, such as highly-robust waveg-uides and delay lines , amplifiers , isolators , andlasers . There are several reasons to consider usingphotonic topological insulators to study the localizationproperties of topological phases. First, different disor-der configurations can be implemented on a single de-vice by means of optical, thermal, acoustic, or electricalpumps , which should simplify the acquisition ofensembles with many independent disorder realizations.Second, it is possible to excite any frequency in the bandor band gap via a number of available launching schemes.Third, field distributions can be observed by near-fieldimaging or other techniques, allowing for the accurateand direct determination of quantities such as localiza-tion lengths . Fourth, losses can be controllably in-corporated into photonic structures , which, as we shallsee, is helpful for distinguishing the experimental signa-tures of bulk delocalization from the effects of topologicaledge states. Although photonics has already been exten-sively employed for the experimental study of ordinaryAnderson localization , it has never been used to inves-tigate the peculiar localization properties of bulk statesin Chern insulators.The downside, however, is that photonic Chern insu-lators must be formed from deliberately structured pho-tonic media, such as photonic crystals, metamaterials,coupled resonators, or waveguide arrays . In some ofthese platforms, fabrication technologies are unable tocreate lattices that are sufficiently large, relative to theunit cell, for localization studies. Moreover, several plat-forms exhibit rather high radiative and material losses;while a loss-induced decay length of (say) several unitcells might be acceptable for the purposes of demonstrat-ing topological edge transport, it could complicate local-ization studies through the introduction of exponentiallydecreasing intensity profiles to states that are supposed a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 1: (a) Schematic of a finite lattice of coupled opticalresonators, composed of N x unit cells along x (length) and N y unit cells along y (width). A loss term iγ is added to thesites along the top and bottom edges. Light is injected intothe sites on the first column via uniformly excited couplingwaveguides (orange arrows); there are no coupling waveguideson the rightmost column, which serves as a closed edge. Inset:close-up view of one unit cell, showing the selected directionof circulation within the resonators. (b)–(c) Calculated bandstructures of the semi-infinite lattice (infinite length and fi-nite width, with losses omitted), for (b) the topological phase( M = 0) and (c) the trivial phase ( M = 4 J ). Edge states areshown in blue. In (b), the right-moving (left-moving) edgestate is localized on the bottom (top) edge. In (c), there areedge states localized on the bottom edge, but these do notspan the gap. to be delocalized.We focus on a type of 2D photonic Chern insulatorconsisting of a lattice of on-chip coupled ring resonators,which has recently been proposed and implemented .This system is amenable to theoretical analysis since itcan be accurately described by a tight-binding model .It has been experimentally realized using silicon photon-ics, featuring large lattice sizes of up to 15 × . Thelattice parameters can be altered by methods such as op-tical pumping , so disorder ensembles can be readilygenerated in each photonic lattice through spatially in-homogenous pumping, eliminating the need to fabricatemany different samples.Using tight-binding simulations, we show that thisplatform could be used to access the delocalization ofbulk Chern insulator states and the levitation and anni-hilation phenomenon. A clear experimental signature canbe achieved with a lattice size of about 50 ×
12, which isa modest increase relative to existing experiments , and far smaller than the lattices in previous numerical local-ization studies (which typically feature sample lengthsof 10 or more) . Silicon-on-insulator, whichis typically used in experiments involving coupled ringresonators , has a loss level of ∼ ∼ and we show that such a level of loss doesnot affect the key results. The topological edge statesof the Chern insulator tend to conflict with the experi-mental signature of the delocalized bulk states, but wefind that the former can be suppressed simply by addinglosses to the resonators along the lattice edges. Hence,the photonic lattice can provide a way to explore thelocalization behavior of bulk states in disordered topo-logical insulators, which have thus far resisted in-depthexperimental investigation. II. MODEL
We consider a photonic Chern insulator of a type thathas recently been proposed and implemented using sil-icon photonics . As shown in Fig. 1(a), the systemconsists of an bipartite square lattice of resonant “siterings” coupled to off-resonant “link rings”. The site ringsoccupy two sublattices, denoted by A and B , and thelink rings introduce nearest-neighbor and next-nearest-neighbor couplings between them. Light propagationwithin the lattice can be decomposed into two pseudospinsectors (corresponding to clockwise or counterclockwisecirculation in the site rings), which do not interact due tolocal momentum conservation at the inter-ring couplingregions . Within each sector, time reversal symme-try is effectively broken (however, the physical structureis time reversal symmetric and can be fabricated fromordinary dielectric materials).In the absence of disorder and losses, the system isdescribed by the following tight-binding Hamiltonian : H = ( (cid:15) + M ) n A + ( (cid:15) − M ) n B + V nn + V nnn , (1)where (cid:15) = 2 J cot( φ/
2) (2) n A = (cid:88) r a † r a r , n B = (cid:88) r b † r b r (3) V nn = W (cid:88) r (cid:2) a † r ( b r + b r + x + y ) + b † r ( a r − x + a r − y ) (cid:3) (4) V nnn = W (cid:88) r (cid:104) a † r (cid:88) ± ˆ a r ± y + b † r (cid:88) ± b r ± x (cid:105) (5) W = J exp( iφ/
4) csc( φ/ , W = J csc( φ/ . (6)Here, a r and b r are the annihilation operators for the A and B sublattices on the unit cell at position r (with r + x denoting the position one unit cell to the right, etc.), M is a sublattice-dependent resonator detuning for the siterings, and J and φ parameterize the couplings mediatedby the link rings. For φ (cid:54) = 0, time reversal symmetryis effectively broken. Each eigenvalue of H , denoted by δν , corresponds to the detuning of a photonic eigenmoderelative to a reference frequency.For a given nonzero φ , which depends on the detuningof the link rings relative to the site rings, the lattice sup-ports both a topological band insulator phase (a Cherninsulator) and a topologically trivial phase (a normal in-sulator), depending on the value of M/J , the relative de-tuning between the site ring sublattices . In the follow-ing, we take φ = π (i.e., link rings exactly anti-resonantwith the site rings), with M = 0 for the Chern insulatorand M = 4 J for the normal insulator. In Fig. 1(b)–(c), their bandstructures are plotted for a disorder-freequasi-one-dimensional geometry (i.e., a strip that is infi-nite in x , and finite in y with open boundary conditions).For the Chern insulator ( M = 0), the gap is spanned bychiral edge states that are localized to opposite edges,consistent with the Chern numbers of ± . For the normal insulator( M = 4 J ), there are no edge states spanning the gap.We consider rectangular lattices of length N x andwidth N y . As indicated in Fig. 1(a), light is injected uni-formly into the lattice via waveguides coupled to the siterings on the left edge (i.e., all site rings along that edgeare excited with equal intensity and phase). The lightreturning from the lattice bulk is assumed to outcouplethrough the same coupling waveguides.In an actual experiment, the intensities on individ-ual resonators can be determined from direct measure-ments of weak light scattering . The site intensitiesin column n can be calculated from the frequency-domainGreen’s function (which we will obtain using the methodof Kramer and McKinnon , which is based on the re-cursive Green’s function technique ). However, thecalculation has a subtle dependence on whether the ex-citations on column 1 are mutually coherent or incoher-ent. First, consider the incoherent case, in which differentsites bear no fixed phase relationship with one another.The total intensity on column n is Tr (cid:0) G † G (cid:1) , where G isthe Green’s function matrix beween sites on column 1and sites on column n . We can thus define T = 1 N y Tr (cid:0) G † G (cid:1) , (7)which is the standard definition of the transmittance asused in the electronic transport literature for determiningthe conductance of a sample . On the other hand, inphotonic experiments the inputs typically originate froma single laser source with strong spatial coherence. Inthat case, the effective transmittance in column n is T c = (cid:104) ψ |G † G| ψ (cid:105)(cid:104) ψ | ψ (cid:105) , (8)where | ψ (cid:105) is the input vector (e.g., [1 , , . . . ,
1] if all inputwaveguides have the same phase).Anderson-type disorder is introduced into the latticein the form of a random detuning on each site ring, drawn independently from a uniform distribution over[ − W/ , W/ W is a tunable disorder strengthparameter. In an actual photonic lattice, such disor-der is introduced in part by unavoidable fabrication im-perfections; a previous experiment found this to be onthe order of the coupling strength J (the basic “energy”scale of the tight-binding model) . Additional disor-der can be introduced through spatially inhomogenouspumping . We assume that an ensemble of inde-pendent disorder configurations can thus be achieved.We can then compute the disorder average (cid:104) log( T ) (cid:105) forthe incoherent input case. This is related to the localiza-tion length ξ by (cid:104) log( T ) (cid:105) ∝ − nξ . (9)Therefore, ξ can be extracted from a linear fit of (cid:104) log( T ) (cid:105) against n . For coherent inputs, we substitute T with T c in Eq. (9). We find numerically that although the valuesof (cid:104) log T (cid:105) and (cid:104) log T c (cid:105) are generally different, both casesyield the same fitted value of ξ , as shown in Appendix A.Hence, the use of spatially coherent inputs in a photonicexperiment is consistent with the standard definition ofthe localization length, based on Eq. (7), which is whatour subsequent numerical results are based on.We then define the normalized localization length ξ N ( δν, W ) = ξN y , (10)which depends implicitly on the operating frequency de-tuning δν , as well as the disorder strength W . In thelocalized regime, ξ N decreases with N y and vanishes inthe N y → ∞ limit. When extended states are present, ξ N increases with N y and diverges in the N y → ∞ limit.For the critical states at a mobility edge, ξ N approachesa finite constant in as N y → ∞ . Hence, for given δν and W , we can detect the presence of extended states byfinding how ξ N ( δν ) varies with N y .In numerical studies of bulk localization in 2D lattices,it has been conventional to take periodic boundary con-ditions along the upper and lower edges, so that the sam-ple forms an edgeless waveguide . The reason fordoing this is that Chern insulators and other 2D topologi-cal insulators host topological edge states that are robustagainst disorder and extended along the edge, which caninterfere with the signature of bulk localization of de-localization. However, since we aim to investigate thefeasibility of a realistic on-chip photonic experiment, itis not appropriate for us to impose periodic boundaryconditions. Instead, we introduce losses to the sites re-siding at the edge of the system, as shown by the grayoutlines in Fig. 1(a). The losses are modelled as an imag-inary contribution to the detuning, − iγ , where we take γ = J (later, we will also study the effects of smallerlosses on all the other sites). In experiments, losses canbe deliberately introduced by using adding lossy mate-rials or claddings or additional scattering defects to the FIG. 2: Normalized localization length ξ N versus relative detuning δν for samples with length N x = 50, different widths N y ,different disorder strengths W , and open boundary condition along y . A loss term − iγ with γ = J is added to the edge sites tosuppress the edge state. The ensemble size used is 500. Black arrows indicate the critical behavior ( ξ N increasing or constantwith N y ), as shown in the inset for W = 6. For large W , levitation and annihilation of the critical states is observed, and thebulk states are completely localized for W ≥ J . The band gap of the ordered bulk system is highlighted in yellow. resonators . It would be desirable to avoid signifi-cantly altering the real detuning of the edge resonatorswhile doing so; otherwise, the edge states may simply beshifted onto adjacent rows further into the bulk. III. RESULTS
Fig. 2 shows the normalized localization length ξ N ver-sus the source frequency detuning δν , for different disor-der strengths W and different lattice widths N y . Thelattices are in the Chern insulator phase ( M = 0), andhave fixed length N x = 50. At the upper and loweredges, we impose open (Dirichlet-like) boundary condi-tions. As mentioned, losses are added to the sites on theupper and lower edges to suppress the topological edgestate; the losses on the other resonators are assumed tobe negligible.For most values of δν , we find that ξ N decreases withincreasing N y . As explained in the previous section, thisindicates that the eigenmodes at these frequencies arelocalized by the disorder. However, in Fig. 2(a)–(d),representing disorder strengths up to W = 8 J , thereare two regions on each side of the band gap where ξ N is constant or increases with N y . This is evidence forthe bulk extended states believed to exist in disorderedChern insulators . In agreement with theoreti-cal predictions, there is only a narrow range of frequencieswithin each band where this phenomenon occurs. The signature of delocalization is more easily observ-able if the ensemble size is large. In Fig. 2, each datapoint is averaged from an ensemble of 500 independentdisorder realizations. It would be experimentally un-feasible to fabricate one physical sample for each dis-order realization, but it should be possible to imple-ment disorder dynamically via spatially inhomogenousand actively-switchable optical, thermal, acoustic, orelectrical pumps . This would allow numerous in-dependent disorder realizations to be generated with afew physical samples (corresponding to different valuesof N y ). Furthermore, according to our simulations, delo-calization may still be observable for ensemble sizes of aslow as 50.Fig. 3(a) shows the Chern number calculated for thelower band versus the disorder strength W . TheChern number is computed by summing the Berry fluxthrough each plaquette of the discretized Brillouin zonefollowing Fukui’s method , using a computationally ef-ficient coupling-matrix method to calculate the Berryflux . With increasing W , the Chern number is quan-tized to unity until W ≈ J , after which it decreasesto zero. This is consistent with the results shown inFig. 2, where the signature of the delocalized bulk mode,a property of the Chern insulator, disappears between W = 8 J and W = 10 J . In Fig. 3(b), we plot the range ofcritical frequencies (i.e., the frequency range over which ξ N does not decrease with N y ) versus W . As the sys-tem transitions from Chern insulator to normal insulator, FIG. 3: (a) Chern number of the lower band versus disorderstrength W for a 32 ×
32 lattice (averaged over 200 disorder re-alizations). (b) Critical energy δν c (where the extended stateis located) versus W for the realistic sample corresponding toFig. 2. Each shaded region indicates the detuning range overwhich ξ N does not decrease with N y , and the dots and solidlines indicate the mid-point of the range. (c) Normalized lo-calization length versus δν for the hermitian quasi-1D system( N x = 10000) with periodic boundary along y at a disorderstrength W = 8 J . (d) Critical energy δν c versus W for semiinfinite lattice corresponding to (c). Anderson localizationtransition by annihilation can be observed at W = 9 J . the critical frequencies shift toward one another, and for W (cid:38) J become impossible to distinguish over the sta-tistical noise. Hence, a photonic lattice can provide ev-idence for the long-standing theoretical prediction thattopologically insulating behavior is destroyed by strongdisorder through the levitation and annihilation of thedelocalized bulk states .In Fig. 3(c), we present simulation results for an ideal-ized and experimentally unrealistic lattice with periodicboundary conditions on the upper and lower edges, anda much greater length of N x = 10 . Delocalization isobserved at around the same frequencies as in Fig. 2(d).This shows that the results in Fig. 2, which were ob-tained for experimentally realistic open boundary condi-tions and much shorter length N x = 50, accurately cap-ture the frequencies at which the delocalized bulk statesare supposed to occur. In Fig. 3(d), we plot the range ofcritical frequencies versus disorder strength W using theidealized (periodic boundary conditions and large N x )lattice. It displays the same levitation and annihilationbehavior as in Fig. 3(b). The large error bar near the an-nihilation point ( W ≈ J ) is due to the fact that the twohumps seen in Fig. 3(c) merge into a single large humpnear the annihilation point, for the range of N y consid-ered here. In Fig. 3(b), this was not observed due to thesmaller lattice size and the effects of the open boundary FIG. 4: Plots of normalized localization length versus rela-tive detuning δν for different scenarios. (a) Chern insulator( M = 0) with disorder strength W = 6 J , with the losses onthe upper and lower edge sites omitted. Transmission nowpeaks in the bulk gap (yellow region) due to the topologicaledge states, and the signature of bulk delocalization cannotbe discerned. (b) Chern insulator ( M = 0) with disorderstrength W = 6 J , and with small losses − iγ (cid:48) on the non-edgesites. Here we take γ (cid:48) = 10 − J . As before, the sites on theupper and lower edges have loss − iJ , and all other param-eters are the same as in Fig. 2. Bulk delocalization can beobserved at δν ≈ − J . For clarity, only negative values of δν are plotted. (c)–(d) Normal insulator ( M = 4 J ) with disorderstrengths of (c) 4 J and (d) 6 J and length N x = 50. conditions.To verify that the loss on the edge sites is necessaryfor the observation of bulk delocalization, in Fig. 4(a)we plot ξ N versus δν with these losses omitted. In thiscase, high transmission is observed in the frequency rangecorresponding to the bulk gap, due to transport by thenow-unsuppressed topological edge states. The frequencyrange over which ξ N is constant or increasing with N y appears to occur near the center of the bulk gap, ratherthan within each band as in Fig. 2.In Fig. 4(b), we show that the delocalization signaturecan still be observed when there is weak but nonzerolosses on the other resonators. In real photonic struc-tures, some material and radiative loss is always present.Here, we assign each non-edge site a loss of − iγ (cid:48) , where γ (cid:48) = 10 − J . This level of losses is consistent what canbe achieved experimentally. For example, the silicon-on-insulator implementation of the model in Ref. 28 had aloss of γ (cid:48) ≈ . J and a silicon nitride platform canachieve a loss-level smaller by a factor of around 1 / γ (cid:48) = 10 − J should be achievable,and it should hence be possible to observe a frequency re-gion in which ξ N is approximately constant with N y .Finally, to verify the topological origin of the bulk ex-tended state, Fig. 4(c)–(d) shows the results for M = 4 J ,for which the lattice is in its normal insulator phase. Inthis case, there is no clear range of frequencies in which ξ N is constant or increasing with N y . IV. DISCUSSION
We have proposed an experimentally feasible way toprobe the localization behavior of disordered Chern in-sulators using a recently-developed photonic platform .Using a realistic tight-binding model , we showed thatit should be possible to observe the existence of ex-tended states in the bulk bands, a characteristic featureof disordered Chern insulators, as well as the levitationand annihilation of these extended states under increas-ing disorder . The required system sizes, disorderstrengths, and loss levels are all in the experimentallyaccessible range.There may be other ways to use photonic Chern in-sulators to probe the interplay of disorder and topolog-ical phases, such as level-spacing statistics . Apartfrom observing localization lengths from averaged inten-sity measurements, it may also be possible to probe thesystem using the Wigner time delay, which is insensitiveto losses and has previously been used to establish theextended nature of photonic topological edge states ;however, we have thus far been unable to find a clearsignature of bulk state delocalization in Wigner time de-lay statistics. It may also be interesting to explore theeffects of adding optical gain to such a system, to de-termine whether bulk extended modes could be observedthrough their promotion into lasing modes.Finally, although our proposal has focused on pho-tonic resonator lattices, similar ideas can be generalizedto other photonic lattices, and more broadly to otherbosonic systems, such as acoustic, electrical or mechani-cal lattices . Acknowledgments
We thank Prof. Caio Lewenkopf for helpful discus-sions. CYD and UC acknowledge support from the Sin-gapore Ministry of Education Tier 3 grant MOE2016-T3-1-006 and Tier 1 grant RG187/18. MF and SM ac-knowledge support from the US Air Force Office of Sci-entific Research (AFOSR) Multidisciplinary UniversityResearch Ini-tiative (MURI) grants FA95501610323 andFA95502010223, the US Office of Naval Research (ONR)MURI grant N00014-20-1-2325, and the National ScienceFoundation grant PHY1820938.
Appendix A: Coherent versus Incoherent Excitation
Let G ba denote the Green’s function matrix elementbetween site a on the input column 1 to site b on column n . If the input on site a is ψ (1) a , the complex wave am-plitude on site b is ψ ( n ) b = (cid:80) a G ba ψ (1) a , and the intensityon that site is I ( n ) b = (cid:12)(cid:12)(cid:12) ψ ( n ) b (cid:12)(cid:12)(cid:12) = (cid:88) ac G ba G ∗ bc ψ (in) a ψ (in) ∗ c . (A1)For uniform incoherent excitation (i.e., no fixed phaserelationship between different input sites), we take anaverage over an ensemble of input wave amplitudes withequal magnitude (cid:112) I in /N y and random phases. Averag-ing over the ensemble gives the mean intensity (cid:68) I ( n ) b (cid:69) = I in N y (cid:88) a | G ba | , (A2)Hence, the total intensity in column n is I ( n ) = (cid:88) b (cid:68) I ( n ) b (cid:69) = I in N y Tr (cid:0) G † n G n (cid:1) . (A3)Normalizing by the total input intensity I in yields thetransmittance formula given in Eq. (7). On the otherhand, if the input is spatially coherent, we do not performan average over the random phases, and the intensitydepends explicitly on the input vector, leading to Eq. (8).As shown in Fig. 5(a), (cid:104) log( T ) (cid:105) and (cid:104) log( T c ) (cid:105) exhibitthe same scaling with distance n . Hence, the same esti-mate for the localization length ξ N is obtained regardlessof whether T or T c is used in Eq. (9). As an example,Fig. 5(b) shows the plot of ξ N versus δν , similar to Fig. 2,for disorder strength W = 6 J . Estimating the localiza-tion length using either T (incoherent excitation) or T c (coherent excitation) gives essentially the same results. FIG. 5: (a) Plots of (cid:104) log( T ) (cid:105) and (cid:104) log( T c ) (cid:105) versus column in-dex n for samples of length N x = 50, width N y = 10, anddisorder strength W = 6 J , averaging over 500 disorder real-izations. The straight lines are linear least squares fits. Thefitted slopes are almost identical, so incoherent and coher-ent excitation give the same localization length estimate. (b)Normalized localization length ξ N extracted using T and T c ,plotted against relative detuning δν/J for disorder strength W = 6 J . All other parameters are the same as in Fig. 2. ∗ Electronic address: [email protected] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev.Lett. , 494 (1980). R. E. Prange and S. M. Girvin,
The Quantum Hall Effect (Springer-Verlag, 1990). K. I. Wysokinski, European Journal of Physics , 535(2000). P. W. Anderson, Phys. Rev. , 1492 (1958). E. Abrahams, P. W. Anderson, D. C. Licciardello, andT. V. Ramakrishnan, Phys. Rev. Lett. , 673 (1979). F. Evers and A. D. Mirlin, Rev. Mod. Phys. , 1355(2008). N. A. Garc´ıa-Mart´ınez, A. G. Grushin, T. Neupert,B. Valenzuela, and E. V. Castro, Phys. Rev. B , 245123(2013). A. MacKinnon and B. Kramer, Phys. Rev. Lett. , 1546(1981). A. MacKinnon and B. Kramer, Zeitschrift f¨ur Physik BCondensed Matter , 1 (1983). D. E. Khmelnitskii, Phys. Lett. A , 182 (1984). R. B. Laughlin, Phys. Rev. Lett. , 2304 (1984). S. N. Evangelou, Phys. Rev. Lett. , 2550 (1995). Y. Asada, K. Slevin, and T. Ohtsuki, Phys. Rev. Lett. ,256601 (2002). M. Onoda, Y. Avishai, and N. Nagaosa, Phys. Rev. Lett. , 076802 (2007). J. J. Mare˘s, J. Kri˘stofik, and P. Hub´ık, Phys. Rev. Lett. , 4699 (1999). P. T. Coleridge, in
Encyclopedia of Condensed MatterPhysics , edited by F. Bassani, G. L. Liedl, and P. Wyder(Elsevier, 2005), p. 248, ISBN 978-0-12-369401-0. Z. Xu, L. Sheng, D. Y. Xing, E. Prodan, and D. N. Sheng,Phys. Rev. B , 075115 (2012). E. V. Castro, M. P. L´opez-Sancho, and M. A. H. Vozme-diano, Phys. Rev. B , 085410 (2015). Z. Qiao, Y. Han, L. Zhang, K. Wang, X. Deng, H. Jiang,S. A. Yang, J. Wang, and Q. Niu, Phys. Rev. Lett. ,056802 (2016). T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi,L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilber-berg, et al., Rev. Mod. Phys. , 015006 (2019). M. Kim, Z. Jacob, and J. Rho, Light: Science & Applica-tions , 130 (2020). S. Raghu and F. D. M. Haldane, Phys. Rev. A , 033834(2008). Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljaˇci´c,Nature , 772 (2009). M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer,D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Sza-meit, Nature , 196 (2013). M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor,Nat. Phys. , 907 (2011). M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor,Nat. Photon. , 1001 (2013). D. Leykam, S. Mittal, M. Hafezi, and Y. D. Chong, Phys.Rev. Lett. , 023901 (2018). S. Mittal, V. V. Orre, D. Leykam, Y. D. Chong, andM. Hafezi, Phys. Rev. Lett. , 043201 (2019). Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zil-berberg, Phys. Rev. Lett. , 106402 (2012). W. Hu, J. C. Pillay, K. Wu, M. Pasek, P. P. Shum, and Y. D. Chong, Phys. Rev. X , 011012 (2015). W. Hu, H. Wang, P. P. Shum, and Y. D. Chong, Phys.Rev. B , 184306 (2017). S. Mittal, S. Ganeshan, J. Fan, A. Vaezi, and M. Hafezi,Nat. Photon. , 180 (2016). O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P.Chen, Y. E. Kraus, and M. C. Rechtsman, Nature , 59(2018). L.-H. Wu and X. Hu, Phys. Rev. Lett. , 223901 (2015). T. Ma and G. Shvets, New Journal of Physics , 025012(2016). L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljaˇci´c, Nat.Photon. , 294 (2013). V. Peano, M. Houde, F. Marquardt, and A. A. Clerk, Phys.Rev. X , 041026 (2016). X. Zhou, Y. Wang, D. Leykam, and Y. D. Chong, New J.Phys. , 095002 (2017). G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman,Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, andM. Segev, Science , eaar4003 (2018). M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren,M. Segev, D. N. Christodoulides, and M. Khajavikhan,Science , eaar4005 (2018). Y. Zeng, U. Chattopadhyay, B. Zhu, B. Qiang, J. Li,Y. Jin, L. Li, A. G. Davies, E. H. Linfield, B. Zhang, et al.,Nature , 246 (2020). P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys.Rev. B , 7837 (1996). R. El-Ganainy, M. Khajavikhan, D. N. Christodoulides,and S. K. Ozdemir, Comm. Phys. , 37 (2019). M. Segev, Y. Silberberg, and D. N. Christodoulides, Nat.Photon. , 197 (2013). C. Wang, Y. Su, Y. Avishai, Y. Meir, and X. R. Wang,Phys. Rev. Lett. , 096803 (2015). J. F. Bauters, M. J. R. Heck, D. John, D. Dai, M.-C. Tien,J. S. Barton, A. Leinse, R. G. Heideman, D. J. Blumenthal,and J. E. Bowers, Opt. Express , 3163 (2011). D. J. Thouless and S. Kirkpatrick, J. Phys. C: Solid StatePhys. , 235 (1981). C. H. Lewenkopf and E. R. Mucciolo, Journal of Com-putational Electronics , 203 (2013), ISSN 15698025,1304.3934. S.-H. Zhang, W. Yang, and K. Chang, Phys. Rev. B ,075421 (2017). K. Slevin, P. Markoˇs, and T. Ohtsuki, Phys. Rev. Lett. ,3594 (2001). G. Paulin and D. Carpentier, New J. Physics , 023026(2012). Y. Su, C. Wang, Y. Avishai, Y. Meir, and X. R. Wang,Scientific Reports , 33304 (2016). E. V. Castro, R. de Gail, M. P. L´opez-Sancho, and M. A. H.Vozmediano, Phys. Rev. B , 245414 (2016). F. Gao, Z. Gao, X. Shi, Z. Yang, X. Lin, H. Xu, J. D.Joannopoulos, M. Soljaˇci´c, H. Chen, L. Lu, et al., NatureCommunications , 11619 (2016). Y.-F. Zhang, Y.-Y. Yang, Y. Ju, L. Sheng, R. Shen, D.-N. Sheng, and D.-Y. Xing, Chinese Physics B , 117312(2013). T. Fukui, Y. Hatsugai, and H. Suzuki, Phys. Soc. Jpn. ,1674–1677 (2005). S. Mittal, J. Fan, S. Faez, A. Migdall, J. M. Taylor, and
M. Hafezi, Phys. Rev. Lett. , 087403 (2014). Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, andB. Zhang, Phys. Rev. Lett. , 114301 (2015). C. H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling, and R. Thomale, Communica-tions Physics , 39 (2018). R. S¨usstrunk and S. D. Huber, Science349