A 1d lattice realization of chiral fermions with a non-Hermitian Hamiltonian
AA 1d lattice realization of chiral fermions with a non-Hermitian Hamiltonian
Wenjie Xi, ∗ Wei-Zhu Yi, ∗ Yong-Shi Wu,
2, 1 and Wei-Qiang Chen
1, 3, † Institute for Quantum Science and Engineering and Department of Physics,Southern University of Science and Technology, Shenzhen 518055, China Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China Shenzhen Key Laboratory of for Advanced Quantum Functional Materials and Devices,Southern University of Science and Technology, Shenzhen 518055, China
According to the famous Nielsen-Ninomiya theorem, one can not put a 1d left moving chiralfermion on a 1d lattice with a local hermitian translation-invariant Hamiltonian. This is closelyrelated to the quantum anomaly of the chiral fermion. In this paper, we propose that the Nielsen-Ninomiya theorem can be circumvented by allowing non-hermiticity. We construct a 1d local non-hermitian model with a complex spectrum, where the imaginary part of the energy corresponds tothe inverse lifetime of the fermion. We show that despite our non-hermitian Hamiltonian respectsthe chiral symmetry, the model actually has correct quantum anomaly, including both chiral andgravitational anomaly. We present several evidences in various approaches.
The chiral fermions are the massless relativisticfermions that respects the chiral symmetry, the invari-ance of the action under chiral rotations. Since the chiralfermions can be described by the Weyl equation, they arealso called Weyl fermions. The chiral fermions has defi-nite chirality χ = ±
1, which corresponds to right-handedor left-handed in 3d case and right-moving or left-movingin 1d case. The chiral fermions and chiral symmetries arefundamentally important in high energy physics. For ex-ample, only left-handed quarks and leptons are invovledin weak interaction in the electroweak theory. And inquantum chromodynamics (QCD), since the masses ofup, down and strange quarks are much smaller than theenergy scale, the chiral symmetry and its spontaneousbreaking also plays very important roles. In condensedmatter physics, the Weyl fermions can emerge from var-ious systems, such as Weyl semimetals , B phase of su-perfluids He-III , and chiral edges of two-dimensionaltime-reversal symmetry broken topological phases , aslow-energy excitations and have attracted a lot of atten-tion.It has been known for a long time that one can not puta single chiral fermion on lattice with same spacetime di-mension, as stated by the Nielsen-Ninomiya theorem thatany local, Hermitian and translation-invariant lattice ac-tion in even-dimensional spacetime must contain an equalnumber of left- and right-handed chiral fermions . Thisis also referred as the fermion doubling problem. In thecondensed matter systems mentioned above, the chiralfermions either come in pairs or live only on the edge ofone higher dimensional system. People realize that thefermion doubling problem is closely related to the quan-tum anomaly of the field theory. It has been said that thequantum anomaly of a theory indicates it has a nontriv-ial one higher dimensional bulk, and hence can only berealized with a one higher dimensional lattice model .Recently, the study of non-hermitian systems has at-tracted a lot of attention . The non-hermiticity pro-vides a different approach for the lattice realization ofa chiral theory; for example, the non-hermitian latticemodel of the chiral Chern-Simons theory has been pro- posed by Nagata and Wu . Here we will show that a 1dlattice model can be constructed for a chiral fermion in1+1 dimensions with the help of non-hermiticity, despitethe existence of quantum anomaly. Technically, this isbecause one may circumvent the Nielsen-Ninomiya theo-rem by giving up the hermiticity. From the point of viewof quantum anomaly, this can be understood by takingthe one higher dimensional bulk as the environment ofthe edge, then the chiral fermion on the edge may bedescribed by a non-hermitian lattice model with samespacetime dimension. In this paper, we construct a non-hermitian 1d lattice model for a 1d left-moving chiralfermion. Though there are still left-moving and right-moving fermions in our model, we will show that in thecontinuum limit, the lifetime of the right-moving fermionvanishes, while the lifetime of the left-moving fermiongoes to infinity. We also check the chiral anomaly ofour non-hermitian model in various approaches and showthat the non-hermitian model does describe a 1+1D left-moving chiral fermion. Our result shows that a latticemodel may achieve a nonzero chirality by coupling tosome kinds of environments. This may provide an al-ternative possibility, besides the spontaneous symmetrybreaking, for the origin of some phenomena with nonzerochirality in the nature. The Model.
A 1+1D left(right) moving chiral fermionfield Ψ L ( R ) ( x ) can be achieved by projecting the Diracspinor with the projection operator P L,R = (1 ∓ γ ) / L ( x ) and +1 for Ψ R ( x ), aregiven by the eigenvalue of γ . The action of a free leftmoving fermion is given by S Weyl = Z d t d x {− i ˙Ψ † L ( x )Ψ L ( x )+ Ψ † L ( x )[ − i∂ x + A ( x )]Ψ L ( x ) } , (1)where we set the velocity of fermion v = 1 and choose thegauge A = 0, A = A ( t ). The corresponding Hamilto- a r X i v : . [ c ond - m a t . o t h e r] O c t nian reads H = Ψ † L ( x )[ i∂ x − A ]Ψ L ( x ) . (2)In general, a lattice realization of a field theory is con-structed by replacing the differential operator ∂ µ withsome kind of difference operator on the lattice. Differentchoices of the difference operators may lead to differentlattice models. In this paper, we consider a 1d discretespatial lattice with continuous time, a natural choice ofthe difference operator is ∂ x Ψ( x ) → a [ c n +1 − c n − ] , (3)where a is lattice constant, and c n is the fermion annihila-tion operator at lattice site n . However, the correspond-ing lattice model has the well-known fermion doublingproblem, i.e. besides of the left-moving fermion at k ∼ k around π/a as shown in fig. 1(a). This is consistent with the Nielsen-Ninomiya theorem as discussed in the introduction. Inthe following, we refer the above difference operator andthe corresponding model as symmetric lattice operatorand symmetric lattice model.We may escape from the Nielsen-Ninomiya theorem byconsidering a non-hermitian model. This can be under-stood through a proof of the Nielsen-Ninomiya theorembased on the Poincare-Hopf theorem . According to thePoincare-Hopf theorem, the sum of indices of all the iso-lated zero modes of a 1d lattice model of chiral fermion iszero. For a local, hermitian, translation invariant model,the index of an isolated zero mode is just its chirality χ = ±
1. Thus one must have equal numbers of left-movers and right-movers. However, in the non-hermitiancase, the index of an isolated zero mode is in general 0.Therefore, it may be possible to have only one left-moversin a non-hermitian lattice model.The non-hermitian model can be constructed by con-sidering a different difference operator, the oriented lat-tice operator, ∂ x Ψ( n ) → a [ c n +1 − c n ] , (4)which leads to a non-hermitian lattice model, H − = n − X j =1 ia e iaA ( c † j c j +1 + c † n c ) − n X j =1 ia c † j c j . (5)This model describes a free left moving fermion and willbe referred as the oriented lattice model in the following.A Hamiltonian describing a free right moving fermion canbe simply obtained by acting a parity transformation on(5): H + = n X j =2 ia e iaA ( c † j c j − + c † c n ) − n X j =1 ia c † j c j . (6) 𝐸𝑎 𝑘𝑎(𝑏)𝐸𝑎 (𝑎) 𝑘𝑎
FIG. 1. The energy dispersion of the symmetric lattice model( a ) and the oriented lattice model ( b ) with periodic boundaryconditions and A = 0. Blue line: the real part of the energy;red line: the imaginary part of the energy. The spectrum.
In general, a non-hermitian model maystill have a real spectrum. For example, a non-hermitianHamiltonian with PT symmetry has only real eigenval-ues. If we consider a difference operator in the form D λ = a [ λ ( c n +1 − c n ) + (1 − λ )( c n − c n − )], the resul-tant Hamiltonian is PT symmetric if and only if λ = 1 / E k = ia (cid:0) e iak − (cid:1) , (7)in the case with periodic boundary conditions in the ab-sence of gauge field. The propagator of the fermion withenergy E k suggest that the real part Re E k can be un-derstood as the ordinary energy of the fermion. Note theimaginary part Im E k is negative for every k , it can be re-garded as a loss rate or inverse of lifetime of the fermiondue to the coupling with some kinds of environments. Asshown in fig. 1(b), Re E k is similar to the one of the sym-metric lattice model shown in fig. 1 (a). Thus we stillhave a left-moving fermion at k ∼ k ∼ π/a . However, these two fermions havedifferent lifetimes. For the left-moving fermion, we have E k ≈ − k − iak /
2, while for the right-moving fermion, E k ≈ k − i/a with k = k − π/a . Thus the lifetime ofleft-movers are much larger than the one of right-moversprovided a is small enough. And in the continuum limit a →
0, the left-mover has infinite lifetime, while the life-time of right-mover vanishes. This indicates that thelattice Hamiltonian will reduce to the Hamiltonian (2) inthe continuum limit. And also there is an emergent PTsymmetry in the continuum limit, though PT symmetryis explicitly broken in the lattice model.
Chirality and chiral symmetry.
A good lattice realiza-tion of a chiral fermion should respect chiral symmetryand have correct chirality. This can be checked with the 𝐴 (𝑏) 𝐸 (𝑎) 𝐴 𝐸 FIG. 2. Illustration of spectral flow of the band driven byvarying A from − π/L to π/L adiabatically are shown in ( a )and ( b ). Solid line means state is right-handed and dash lineindicates state is left-handed. Strength of the line impliesthe observation probability of the state. Here, we set a = 1, n = 100. Atiyah–Singer index theorem, which in 2D continuousEuclidean spacetime is given byindex( ˜ D + ) = n + − n − = X n h ϕ n | γ | ϕ n i = − π Z M F. (8)Here ˜ D ± = ˜ D (1 ± γ ) / χ = ± D is the Dirac operator, and n ± represents number of zero modes with chirality χ = ± ϕ n is the eigenstate of ˜ D with ˜ D | ϕ n i = λ n | ϕ n i . F is the strength tensor of the U(1) gauge field.To generalize the theorem to a lattice, we introduce adiscrete Dirac operator D ( n | m ) = γ e iaA ( τ ) δ n +1 ,m − δ n,m a + γ ∂ τ , (9)where n and m are indices of lattice sites. It can be easilyproved that the action of the generalized Dirac operatoris invariant under an infinitesimal chiral transformationΨ( n ) → e iαγ Ψ( n ) , ¯Ψ( n ) → ¯Ψ( n ) e iαγ . (10)Therefore we can define the corresponding Weyl opera-tors with D ± ≡ D (1 ± γ ) /
2. Please note that D − is justthe difference operator in the action of the lattice model(5) (after a Wick rotation). Thus our lattice Hamiltonianalso has the chiral symmetry.Please notice that D is also a non-Hermitian opera-tor. Therefore its right eigenstates are not orthogonalto each other, and a left eigenstate of D is not hermitianconjugate to the corresponding right eigenstate anymore.Thus, we introduce the so-called bi-orthogonal basis, i.e.the right eigenstates ϕ m,R as the basis of kets | ϕ i andleft eigenstates ϕ n,L as the basis of bras h ϕ | . It canbe easily proved that h ϕ n,L | ϕ m,R i = δ nm . With thebi-orthogonal basis, the lattice-version index theorem is written asindex( D + ) = n + − n − = X n h ϕ n,L | γ | ϕ n,R i = − π Z M F, (11)which will be proved in the following.Mathematically, the first equal sign in equation (11)holds for any elliptic operator, hence it holds for ourgeneralized Dirac operator D , because it is still an anelliptic operator . For the second equal sign, because D | ϕ n,R i = λ n | ϕ n,R i , h ϕ n,L | D = λ n h ϕ n,L | and { D, γ } = 0, we have λ n h ϕ n,L | γ | ϕ n,R i = h ϕ n,L | γ D | ϕ n,R i = −h ϕ n,L | Dγ | ϕ n,R i = − λ n h ϕ n,L | γ | ϕ n,R i . Therefore only zero modes contribute to the summationin the right hand side. Since the D operator commuteswith the chiral symmetry generator, we can write thezero modes in the basis with definite chiral charge, whichleads to X n h ϕ n,L | γ | ϕ n,R i = n + − n − The third equal sign can be proved in a way simi-lar to ref. . With the bi-orthogonal basis of the Non-Hermitian operator D , the Jacobian of the infinitesimalchiral transformation (10) in path integral reads J { α } = e − i R d x α P n h ϕ n,L | γ | ϕ n,R i , (12)which has exactly the same form as the one in continuumtheory except that the eigenstates of ˜ D is replaced by thebi-orthogonal basis. By further introducing Fujikawa’sregularization, we can reestablish the third equal sign forthe discrete case.The integration over the strength of the gauge field, orthe topological index, in the right hand side of eqn. (8)and (11), should be same. Thus, according to the twoindex theorems, we must have index( D + ) = index( ˜ D + ),which means that the D ± operator has the same chiral-ity as the continuum Weyl operators. Notice that D − operator defined above is just the one we used in theHamitlonian (5), our Hamitlonian should have correctchirality χ = − A from − π/L at time t = 0 to π/L at time t = T , where L = N a , N is total number of lattice sites. T shouldbe large enough for an adiabatic process. The result-ing spectral flow for the symmetric and oriented latticemodels are depicted in fig.2 (a) and (b), respectively. Forthe symmetric lattice model, one can find that the chi-ral charge is changed by 2 which is consistent with thefermion doubling discussed above. On the other hand,for the oriented lattice model, an additional decay factor e T Im E appears because of the time evolution. For T is 𝑘𝑅𝑒(𝐸) (𝑎) 𝑘(𝑏)𝐼𝑚(𝐸) FIG. 3. Illustration of band dispersion for the low energyeffect Hamiltonian of chiral fermion on the oriented lattice.We set a = 0 .
02 and V = 1. large enough ( T (cid:29) a ) but not too large ( aT (cid:28) E k ∝ /a , as illustrated in fig. 2 (b).This suggests that the chiral charge is changed by just 1,which is consistent with single left-handed chiral fermionin the continuum theory (2). Quantum anomaly.
As discussed in the introduction,the fermion doubling problem is closely related to thequantum anomaly of the chiral fermion theory. However,with the help of non-hermiticity, we may realize the quan-tum anomaly with a non-hermitian lattice model with thesame spacetime dimension. In the 1+1D chiral fermioncase, the field theory has the so-called chiral anomaly and gravitational anomaly . According to the latticeversion index theorem (11) and the spectral flow shownin fig. 2, the chiral anomaly is correctly tracked by ourlattice model.An anomaly indicator of the gravitational anomaly of1D chiral fermion is the chiral central charge. For ourlattice model, if the lattice constant a is small enough, wemay consider only the states around k = 0 and k = π/a ,whose spectrums are given by E Lk ∼ − k − iak / E Rk ∼ k − i/a, (13)where L and R stands for the left-mover and right-mover,respectively. If we ignore the imaginary part of the en-ergy, the spectrum is the same as the free fermion con-formal field theory (CFT), where the E L corresponds tothe holomorphic component with c = 1 and E R corre-sponds to the anti-holomorphic component with ¯ c = 1.As we discussed before, the imaginary part of the en-ergy corresponds to the lifetime of the fermions. Thusin the continuum limit, only the holomorphic componentsurvives. Therefore it corresponds to a CFT with chiralcentral charge c − ¯ c = 1, which indicates that our latticemodel have the same gravitational anomaly as the 1+1Dleft-moving chiral fermion. Stability under local perturbations.
The topological na-ture of the lattice-version index theorem suggests that the zero mode of our non-hermitian model is robust againstlocal perturbations. To check this, we consider a low en-ergy effective Hamiltonian with coupling between the twokinds of fermions H = X k − ( k + i k a c † ,k c ,k + ( k − ia ) c † π,k c π,k + V c † ,k c π,k + V c † π,k c ,k , where c ,k and c π,k corresponds to the left movingfermion around k = 0 and the right moving fermionaround k = π/a , respectively. For small enough a with ak (cid:28) , aV (cid:28)
1, we have (cid:15) + ≈ − k − V + k ai, (cid:15) − ≈ k − ia . Thus, the perturbation does not open a gap. It merelychanges the imaginary part of the energy (see fig.3 foran example), or the inverse lifetime of the fermions.However, the qualitative behavior of the lifetime of thefermions, i.e. the left moving fermion has infinite life-time in the continuum limit, while the lifetime of theright moving fermion is zero in the continuum limit, areunchanged.We also check the disorder effect by adding P j α j c † j c j terms into the Hamiltonian (5) with randomly generated α i ∈ [0 , . Discussion and Conclusion.
Finally, we want to dis-cuss the possible experimental implementation of ournon-hermitian model. The nonhermitian lattice modelmay be realized with ultracold atoms in optical lattices.By taking the gauge field A = π/a in eqn. (5), thismodel matches exactly the eqn. F(4) in Ref. with κ = − J = 1 /a , which has been proposed to be realizablein a system consists of two parallel fine-tuning optical lat-tices. Alternatively, the model can also be simulated withelectric circuits with diodes that induce left-right asym-metry and electrical inductors for the imaginary chemi-cal potential . The zero modes can be detected viaprominent two-point impedance peaks.In summary, we have constructed a local non-hermitian1D lattice model with a complex spectrum. We have pre-sented several evidences that this model describes a left-moving chiral fermion in various approaches, and shownthat it is stable against local perturbations. Our resultssuggest that with the help of non-hermiticity, an anoma-lous field theory may be realized in a non-hermitian lat-tice model with same spacetime dimension. The non-hermiticity tracks the coupling of the lattice model withits nontrivial one higher dimensional bulk. Our theoryalso indicates that a chiral theory may be realized by cou-pling to some kinds of environments. This provides analternative possibility, besides the spontaneous symmetrybreaking, for the origin of at least some chiral phenomenain the nature. Acknowledgement
We are grateful to the help-ful discussions with Zheng-Cheng Gu and Ling-Yan(Janet) Hung. This work was supported by Na-tional Key Research and Development Program of China (Grant No. 2016YFA0300300), NSFC (GrantsNo. 11861161001), the Science, Technology and In-novation Commission of Shenzhen Municipality (No.ZDSYS20190902092905285), and Center for Computa-tional Science and Engineering at Southern Universityof Science and Technology. ∗ These two authors contributed equally to this work. † [email protected] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.Savrasov, Phys. Rev. B , 205101 (2011). S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian,C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang,A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, andM. Z. Hasan, Science , 613 (2015). T. D. C. Bevan, A. J. Manninen, J. B. Cook, J. R. Hook,H. E. Hall, T. Vachaspati, and G. E. Volovik, Nature ,689–692 (1997). M. A. Silaev and G. E. Volovik, Phys. Rev. B , 214511(2012). K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. , 494 (1980). R. B. Laughlin, Phys. Rev. Lett. , 1395 (1983). C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang,M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji,Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C.Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue,Science , 167 (2013). H. Nielsen and M. Ninomiya, Physics Letters B , 219(1981). C. Callan and J. Harvey, Nuclear Physics B , 427(1985). M. Stone, Annals of Physics , 38 (1991). Y. Ashida, Z. Gong, and M. Ueda, (2020),arXiv:2006.01837. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, arXiv e-prints , arXiv:1912.10048 (2019), arXiv:1912.10048 [cond-mat.mes-hall]. K. Nagata and Y.-S. Wu, Phys. Rev. D , 065002 (2008),A non-hermitian lattice action for a chiral Chern-Simonstheory is first constructed, which then leads to a doubleChern-Simons theory after adding hermitian conjugate. Y.-S. Wu and Y. Yu, (1996), A paper on mean-fieldand perturbation theory of a quasi-hermitian compositefield theory for the half-filled Landau level., arXiv:cond-mat/9608061. W. Xi, Z.-H. Zhang, Z.-C. Gu, and W.-Q. Chen, (2019),arXiv:1911.01590. L. Yamauchi, T. Hayata, M. Uwamichi, T. Ozawa, andK. Kawaguchi, (2020), arXiv:2008.10852. M. N. Chernodub, Journal of Physics A: Mathematical andTheoretical , 385001 (2017). M. DeMarco and X.-G. Wen, (2018), arXiv:1805.03663. M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. ,065703 (2009). Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao,and D. N. Christodoulides, Phys. Rev. Lett. , 213901(2011). K. Esaki, M. Sato, K. Hasebe, and M. Kohmoto, Phys. Rev. B , 205128 (2011). Y. C. Hu and T. L. Hughes, Phys. Rev. B , 153101(2011). J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer,S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, Phys.Rev. Lett. , 040402 (2015). T. E. Lee, Phys. Rev. Lett. , 133903 (2016). D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, andF. Nori, Phys. Rev. Lett. , 040401 (2017). Y. Xu, S.-T. Wang, and L.-M. Duan, Phys. Rev. Lett. , 045701 (2017). H. Shen, B. Zhen, and L. Fu, Phys. Rev. Lett. , 146402(2018). L. Jin and Z. Song, Phys. Rev. B , 081103 (2019). H. Zhou and J. Y. Lee, Phys. Rev. B , 235112 (2019). F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J.Bergholtz, Phys. Rev. Lett. , 026808 (2018). S. Yao and Z. Wang, Phys. Rev. Lett. , 086803 (2018). T. Hofmann, T. Helbig, C. H. Lee, M. Greiter, andR. Thomale, Phys. Rev. Lett. , 247702 (2019). J. Y. Lee, J. Ahn, H. Zhou, and A. Vishwanath, Phys.Rev. Lett. , 206404 (2019). K. Kawabata, T. Bessho, and M. Sato, Phys. Rev. Lett. , 066405 (2019). X.-X. Zhang and M. Franz, Phys. Rev. Lett. , 046401(2020). D. S. Borgnia, A. J. Kruchkov, and R.-J. Slager, Phys.Rev. Lett. , 056802 (2020). S. Longhi, Phys. Rev. Lett. , 066602 (2020). L. Li, C. H. Lee, and J. Gong, Phys. Rev. Lett. ,250402 (2020). K. Zhang, Z. Yang, and C. Fang, Phys. Rev. Lett. ,126402 (2020). T. Yoshida, T. Mizoguchi, and Y. Hatsugai, Phys. Rev.Research , 022062 (2020). P.-Y. Chang, J.-S. You, X. Wen, and S. Ryu, Phys. Rev.Research , 033069 (2020). T. E. Lee and C.-K. Chan, Phys. Rev. X , 041001 (2014). A. McDonald, T. Pereg-Barnea, and A. A. Clerk, Phys.Rev. X , 041031 (2018). Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi-gashikawa, and M. Ueda, Phys. Rev. X , 031079 (2018). K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys.Rev. X , 041015 (2019). A. Beygi, S. P. Klevansky, and C. M. Bender, Phys. Rev.A , 062117 (2019). Y. Sunkyu, P. Xianji, and P. Namkyoo, Current Opticsand Photonics , 275 (2019). L. Herviou, J. H. Bardarson, and N. Regnault, Phys. Rev.A , 052118 (2019). L. Herviou, N. Regnault, and J. H. Bardarson, SciPostPhys. , 69 (2019). K.-I. Imura and Y. Takane, Phys. Rev. B , 165430 (2019). C.-X. Guo, X.-R. Wang, C. Wang, and S.-P. Kou, Phys.Rev. B , 144439 (2020). L. Xiao, T. Deng, K. Wang, G. Zhu, Z. Wang, W. Yi, andP. Xue, Nature Physics , 761–766 (2020). L. Pan, X. Chen, Y. Chen, and H. Zhai, Nature Physics , 767–771 (2020). M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren,M. Segev, D. N. Christodoulides, and M. Khajavikhan,Science (2018), 10.1126/science.aar4005. G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman,Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, andM. Segev, Science (2018), 10.1126/science.aar4003. T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany,T. Kiessling, L. Molenkamp, C. H. Lee, A. Szameit,M. Greiter, and R. Thomale, Nature Physics (2020),10.1038/s41567-020-0922-9. H. Zhao, X. Qiao, T. Wu, B. Midya, S. Longhi, andL. Feng, Science , 1163 (2019). M.-A. Miri and A. Alù, Science (2019), 10.1126/sci-ence.aar7709. S. K. Özdemir, S. Rotter, F. Nori, and L. Yang, NatureMaterials , 783 (2019). H. Schomerus, Opt. Lett. , 1912 (2013). C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev.Lett. , 270401 (2002). K. Jones-Smith and H. Mathur, Phys. Rev. D , 125014(2014). L. Lootens, R. Vanhove, J. Haegeman, and F. Verstraete,Phys. Rev. Lett. , 120601 (2020). L. H. Karsten, Physics Letters B , 315 (1981). M. F. Atiyah and I. M. Singer, Annals of Mathematics ,139 (1971). K. Fujikawa, Phys. Rev. Lett. , 1195 (1979). S. L. Adler, Phys. Rev. , 2426 (1969). J. S. Bell and R. W. Jackiw, Nuovo Cimento , 47 (1969). L. Alvarez-Gaumé and E. Witten, Nuclear Physics B ,269 (1984). M. Ezawa, Phys. Rev. B , 201411 (2019). K. Luo, J. Feng, Y. X. Zhao, and R. Yu, arXiv e-prints , arXiv:1810.09231 (2018), arXiv:1810.09231 [cond-mat.mes-hall]. S. Liu, S. Ma, C. Yang, L. Zhang, W. Gao, Y. J. Xiang,T. J. Cui, and S. Zhang, Phys. Rev. Applied13