Interaction induced bi-skin effect in an exciton-polariton system
Xingran Xu, Huawen Xu, S. Mandal, R. Banerjee, Sanjib Ghosh, T.C.H. Liew
IInteraction induced bi-skin effect in an exciton-polariton system
Xingran Xu, ∗ Huawen Xu, S. Mandal, R. Banerjee, Sanjib Ghosh, and T.C.H. Liew
1, 2, † Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore 637371, Singapore MajuLab, International Joint Research Unit UMI 3654,CNRS, Universit´e Cˆote d’Azur, Sorbonne Universit´e,National University of Singapore, Nanyang Technological University, Singapore (Dated: March 1, 2021)The non-Hermitian skin effect can be realized through asymmetric hopping between forward andbackward directions, where all the modes of the system are localized at one edge of a finite 1Dlattice. However, achieving such an asymmetric hopping in optical systems is far from trivial. Herewe show theoretically that in a finite chain of 1D exciton-polariton micropillars with symmetrichopping, the inherent non-linearity of the system can exhibit a bi-skin effect, where the modes ofthe system are localized at the two edges of the system. To show the topological origin of suchmodes, we calculate the winding number.
Introduction.—
Exciton-polaritons are bosonic quasi-particles arising from the strong coupling between ex-citons and photons in microcavities [1–4]. The combina-tion of photons and excitons allows polaritons to possessthe characteristics of both its root parts, for example, loweffective mass from the photonic part and strong nonlin-earity from the excitonic part. Moreover, due to the pho-tons escaping from the microcavity, exciton-polaritonsare noticeably an open quantum (non-Hermitian) sys-tem, which requires additional pumping to maintain asteady state in the system [5]. This character makes theexciton-polariton system an ideal platform to study non-Hermitian transitions with gain and loss [6, 7]. Althoughvarious research has been done in the Hermitian regimeto realize polariton topological phases with the interplaybetween Zeeman shift resulting from the application ofmagnetic field and the transverse electric-transverse mag-netic (TE-TM) splitting of the photonic modes [8–12],accounting for nonlinearity [13, 14], and using the po-larization splitting of elliptical micropillars [15], severalworks have been reported in recent years to explore thenon-Hermitian physics in the exciton-polariton system.Exceptional points (EPs) in the exciton-polariton systemhave been realized in Refs. [16–18]. Recently, the mea-surement of non-Hermitian topological invariants [19],topological end mode lasing [20], and non-reciprocal po-laritons [21] have been reported.Non-Hermitian physics has attracted a tremendous at-tention in recent years due to the discovery of the non-Hermitian skin effect, where all the modes are localizedto one end of a lattice [22–45]. This is different tothe case of Hermitian systems, where modes within atopological bandgap are localized at the edges of a fi-nite sample due to bulk-boundary correspondence. Dueto non-Hermiticity, the Bloch theory does not even holdapproximately for finite sized non-Hermitian systems andone must turn to the generalized Brillouin zone (GBZ)theory based on a complex momentum to explain thenon-Hermitian skin effect [35, 46–50]. To realize the skin effect, the simplest model is the so-called Hatano-Nelson(HN) model [51, 52] without disorder where thehopping in a lattice is different in different directions.Along with the same idea of the HN model, the Su-Schrieffer-Heeger(SSH) model with asymmetric hoppinghas been proposed or realized in many systems likewaveguides [53, 54], photonic lattices [39, 55], circuits [56]etc. The non-Hermitian topological bulk-boundary cor-respondence can be obtained by using the GBZ theoryand by solving the boundary equations with complexmomentum [35–37, 49, 57–60]. Due to the chiral symme-try, the system will have an energy pair of ( E , − E ) andthey will collapse at zero energy modes. The windingnumber can be well defined by the calculation with theperiodic boundary energies with the GBZ. Meanwhile,more and more work is going to explain the topologicalorigin of the skin modes and the GBZ is not necessaryfor line gap non-Hermitian Hamiltonians where the ba-sis can be enlarged to construct a Hermitian Hamilto-nian [29, 35, 37, 48, 61, 62]. Although the winding in thewhole Bloch region is still zero for zero energy pairs, thesystem can still have a topological non-trivial phase if en-ergy pairs can be made to collapse at non-zero (complex)energy.In this letter, we use the particle-hole symmetry of thefluctuation theory of exciton-polariton systems to real-ize “bi-skin” modes where all the wavefunctions are lo-calized at both sides of a one-dimensional lattice. Be-ing compatible with recent experiments, we considera non-equilibrium polariton condensate correspondingto a plane-wave with non-zero momentum. By theBogoliubov-de-Gennes(BdG) transformation [63, 64], thefluctuations of the condensate will have two modes withparticle-hole symmetry. By considering a suitable BdGlattice by tight-binding theory, the bi-skin effect will ap-pear with symmetric hopping between sites. To classifythe symmetry and the topologies of the system, we calcu-late the winding number of the system without assumingthe GBZ theory. Because of the particle-hole symme- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b 𝐽 𝐽𝐽−𝐽 −𝐽𝐽 𝜓 ! 𝜓 !" 𝜓 !$ 𝐹 ! " 𝐹 ! 𝐹 ! $ 𝑢 ! 𝑢 !" 𝑢 !$ −𝑖𝛾 % −𝑖𝛾 % −𝑖𝛾 % −𝜓 (",$%&)∗) 𝜓 (",$%&)) −𝜓 (",$)∗’ 𝜓 (",$)’ −𝜓 (",$())∗’ 𝜓 (",$())’ (a)(b) 𝑣 ! 𝑣 !" 𝑣 !$ Figure 1. Scheme of polariton lattice (a) and the relatedBougoliubov lattice (b) with u n and v n representing the dif-ferent Bogoliubov fluctuation amplitudes on each site. try [65], the energy spectrum will not collapse at zeromodes but some purely imaginary energies. The differenttopological phases can be well defined by the winding ofthe EPs and the localization of the wavefunction. Finally,we analyze the stability of the system and calculate theenergy dispersion, which can be observed via photolumi-nescence [66, 67] or four-wave mixing spectroscopy [68]. Model.—
We consider a simple chain with symmet-ric hopping J between nearest neighbours, formed byexciton-polariton micropillars, as shown in Fig. 1(a).Such chains have been realized experimentally [69]. Typ-ically the behavior of a polariton condensate is studied,which itself has no topological features in such a system.The system can be described by the driven-dissipativenonlinear Gross-Pitaevskii (GP) equation i (cid:126) ∂ψ n ∂t = (∆ − iγ C ) ψ n + J ( ψ n − + ψ n +1 )+ g | ψ n | ψ n + F n , (1)Here, ∆ = E − (cid:126) ω p is the onsite detuning between thebare polariton mode having energy E and a coherentdrive F n with frequency ω p . g represents the nonlin-ear interaction strength and γ C is the effective dissipa-tion. In order to find the optimum parameter range wemove to the dimensionless units by making the follow-ing substitutions: t → t (cid:126) /γ C , J → J/γ C , ∆ → ∆ /γ C , ψ n → ψ n (cid:112) γ C /g , and F n → F n (cid:112) g/γ C . One of the mainingredients of our scheme is a plane-wave like station-ary state ψ ,n = Ae ik p n (as shown in Fig. 2(a) ) realizedby spatially modulating a coherent field [70–73] and theappropriate choice of F n is obtained by setting the timederivative in Eq. (1) to zero: F n = (cid:104) − ∆ + i − J cos( k p ) − | A | (cid:105) Ae ik p n , (2) where A is the square root of the density of the conden-sate and k p is the pumping momentum and n (cid:54) = 1 or N .Because of the boundary, we should set the first and thelast site pumping by replacing 2 J cos( k p ) in Eq. (2) with Je ik p and Je − ik p , respectively. Figure 2. The time evolution of the absolute square of thewavefunction | ψ n | (a) described by Eq. (1) and the spatialprofiles of Bogoliubov modes u n and v n are plotted in (b) and(c), which are obtained by Eqs.(3)- (4). The parameters areused ∆ (cid:48) /γ C =-0.2, J =0.4, A =1, and k p = π/ To show the bi-skin effect we consider the fluctuations,by introducing the fluctuation amplitudes u n and v n , andwriting ψ n = ψ (0 ,n ) + u n e iEt/ (cid:126) + v ∗ n e − iE ∗ t/ (cid:126) . The BdGmodes as shown in Fig. 1(b) are derived by substituting ψ n into the mean-field GP equation, which yields thetime-independent equations: Eu n = (∆ (cid:48) − i ) u n + J ( u n +1 + u n − ) + ψ ,n ) v n , (3) Ev n = − ( i + ∆ (cid:48) ) v n − J ( v n +1 + v n − ) − ψ ∗ ,n ) u n . (4)Here, ∆ (cid:48) = ∆ + 2 (cid:12)(cid:12) ψ (0 ,n ) (cid:12)(cid:12) is an effective detuning. TheBdG transformation ensures that u n and v n modes havethe particle-hole symmetry, so the hopping energies for u n and v n are ± J and the inter hopping energies inone unit cell are ± ψ ( ∗ )2(0 ,n ) . We need to mention thatthe hopping strength between different pairs of neigh-boring sites have the same magnitude but differ in theirphases because of the symmetry. The Bogoliubov ma-trix H n with N sites obtained by the Eqs. (3)-(4) canbe diagonalized by V − ˆ H n V = diag (cid:0) E N × N , − E ∗ N × N (cid:1) with V the matrix formed by eigenvectors, meanwhile, S x V − ˆ H n V S x = diag (cid:0) E N × N , − E ∗ N × N (cid:1) where S x V canalso be the eigenvectors of the system [63, 64], here, S x,y,z = ˆ I N,N ⊗ σ x,y,z and σ x,y,z are the Pauli matrices.On the other hand, the system has pseudo-Hermiticity[47, 74] defined as S z ˆ H n S z = ˆ H n † . The particle-holesymmetry and pseudo-Hermiticity guarantee that thesystem can have a 1D topological transition [35, 74] andthe energy spectrum in real space is highly symmetrical. The bi-skin modes.—
The stationary states of the bulkpolaritons can have non-zero momentum induced by thespecific F n pumping laser and the nonlinearity makes thefluctuations modes u n and v n have a momentum shift.The momentum shift will just influence the phases of theenergy spectrum under the periodic boundary condition(PBC), however, the energy spectrum under the openboundary condition (OBC) will completely change forthe fluctuations at the edge can not go further and thenbecome localized.The model is equivalent to say that the hopping en-ergies between the u n and v n sites are different, but westress that this is an effect induced by nonlinear interac-tions, where the fluctuations are affected by the non-zeromomentum of the considered polariton condensate (sta-tionary state). The underlying physical system is stillnone other than a regular polariton lattice with symmet-ric Hermitian hopping between lattice sites. As is shownin Figs. 2(b) and (c), all eigenstates for u n and v n sitesare localized at both sides. The energy spectrum of u n sites are ( E, E ∗ ) − i pairs, while the energy spectrum of v n sites are ( − E ∗ , − E ) − i pairs. All eigenvectors of E − i and − E ∗ − i will be localized at one side and wavefunc-tions of the E ∗ − i and − E − i will be localized at theopposite side. Remarkably, the localization here does notmean the polariton’s density localization but the fluctu-ations on the top of the polaritons. Figure 3. Profiles of the real (a) and imaginary (b) eigenstatesas a function of u n sites and eigenenergies. (c) Numericalresults of the energy spectrums solved by Eqs. (3)-(4) withcontinuous pumping and the dashed lines are the fitting curvesof the energy dispersion. Parameters are the same as in Fig.2. Stability.—
Our model is based on the fluctuations ofexciton-polaritons with a non-zero momentum station-ary state. Because of the symmetry of the system, theimaginary parts of the eigenenergies can be beyond zero.However, with sufficient dissipation, there will be a totalenergy shift iγ C that keeps all imaginary energy compo-nents below zero.Figs 3(a) and (b) show the distribution of intensityof the real and imaginary parts of the fluctuation eigen-states in real space. All fluctuation eigenstates are local- Figure 4. The real (a) and the imaginary (b) energy spec-trum obtained by Eqs.(3)-(4). Other parameters are the sameas in Fig. 2. ized at the edges of the system, while we find that thosewith least negative imaginary part are localized on theleft side.Because the imaginary parts of the energy spectrumare larger than the real parts, a fundamental problem iswhether the spectrum can be observed experimentally.To prove this, we numerically calculate the BdG lattice’senergy dispersion in Fig. 3(c). We take u n and v n asperturbations with small random number compared with ψ (0 ,n ) and solve the BdG equations. The dashed fittingcurves have the same energy dispersion with diagonaliza-tion results but doubly shrinks the period for two innersites here. The imaginary parts of the energy dispersionare different for the linewidth of the energy dispersionare different as a function of momentum. Remarkably,we solve the Eqs. (3)-(4) without coherent pumping here,which reveals our model is universal and can be extendedto other systems. Non-Hermitian topological invariants–
The effectiveHamiltonian can be written into the momentumspace with use of the Fourier transformation u n → √ N (cid:80) k ˆ a k e ikn and v n → √ N (cid:80) k ˆ b k e ikn with ˆ a k and ˆ b k the annihilation operators for the momentum k , giving H k = (cid:18) ∆ (cid:48) + 2 J cos ( k ) A − A − ∆ (cid:48) − J cos ( k − k p ) (cid:19) − i ˆ I, (5)where ˆ I is the 2 × a k , ˆ b k − k p ). The imaginary energy shift i can be ig-nored for the theoretical study, however, it can determinethe stability of two localization and is necessary to beobserved by the energy dispersion. The non-Hermitiantopology of H k with respect to the reference point E R is equivalent to the Hermitian topology of the followingdoubled Hamiltonian [43, 48, 75] with the basis vector˜ v = (ˆ a k , ˆ b k , ˆ a k +2 k p , ˆ b k − k p ) (cid:62) can be obtained in the sup-plementary materials where the open boundary conditionis considered.˜ H k = (cid:18) [∆ (cid:48) + 2 J cos( k )] ˆ σ z igA ˆ σ y igA ˆ σ y ∆ (cid:48) ˆ σ z + h ( k, k p ) (cid:19) , (6)where h ( k, k p ) = 2 J (cid:104) cos( k ) cos(2 k p )ˆ σ z − sin( k ) sin(2 k p ) ˆ I (cid:105) and ˆ I is the 2 × u n to v n will move to the − k p direction (ˆ a k ˆ b † k − k p term), meanwhile the polariton fluctuations hopping from v n to u n will move to the + k p direction (ˆ b k ˆ a † k +2 k p term).Therefore, the absolute squre of the eigenstates of theBdG lattice are localized at both sides of the lattice asshown in Figs. 2(b)-(c).The construction of the Hermitian Hamiltonian in theenlarged Hilbert space not only allows interpretationwith the Bloch theory [37, 43, 48, 75], but also highlightsthe differences of the four basis operators under the openboundary condition. The boundary will let the momen-tum be cut off at the different values and cause the fluc-tuations to move to the edge and then be localized. Forexample, the range of k is ( − π, π ), however, the range of k ± k p is ( − π ± k p , π ± k p ). Some related work hasbeen done in the SSH model with asymmetric hoppingand interpreted with the GBZ theory [32, 74, 76].There are two EPs in the energy spectrum, so we needto calculate each winding separately. By unitary transi-tion, the enlarged Hamiltonian can be written into a twoblocks off-diagonal matrix and for each one we can definethe winding around the reference energy E R [37] W E = 12 πi (cid:73) π − π d log det [ H ( k ) − E R I × ] dk dk. (7)The total winding number is zero for the vanishing of E and E ∗ because the winding direction is completely op-posite. However, we can get the integer winding number ± E and − E ∗ . The energies as a function of Hop-ping energies in the real space with OBC are shown inFigs. 4 (a) and (c). The critical points are labeled for thecallapse of the imaginary energies.The different topological phases can be distinguishedby the eigenenergy spectrum of Eq.(5) and how manyOBC energies in real space are encircled by the energyband in momentum space with PBC (see supplementarymaterial): • Nontrivial phase < | J | < ( A +∆ (cid:48) ) sec( k p )2 : Alleigenstates are localized at both sides and all OBCenergies are circled by the PBC energies. • Intermediate phase ( A +∆ (cid:48) ) sec( k p )2 (cid:54) | J | < ( A − ∆ (cid:48) ) sec( k p )2 : Parts of eigenstates are localizedand parts of OBC energies including EPs are encir-cled by the PBC energies. • Trivial phase J = 0 or J (cid:62) ( A − ∆ (cid:48) ) sec( k p )2 : Parts Figure 5. The real space energy profile with OBC (‘circles’)and the momentum space energy profile with PBC (‘dots’). J =0.4 and other parameters are the same as in Fig. 2. or no eigenstates are localized and no EPs are en-circled by the periodic boundary spectrum.Here, the critical points are calculated by the phases ofthe eigenvalues of Eq. (5) and this result is correspondingwith the winding number obtained by Eq. (7) (see thesupplementary materials). If the PBC energy dispersionhas different linewidth (imaginary parts of the energies)as a function of momentum, the skin modes will appear.Although the skin modes can appear in the above threephases, the topological invariants are only ± γ C = 0 . µ eV, which can be tuned by adjusting the overlapbetween the micropillars [77]. The non-linearity inducedblueshift becomes 0.1 meV which is a routine observationin GaAs based samples [78]. Given the room tempera-ture polariton micropillar chain has been demonstratedexperimentally [79, 80], our scheme is also compatiblewith perovskite and organic samples. To make it moreexperimental friendly, the fluctuations can be made tohave different polarization from that of the steady stateby proper choice of the resonant driving and exciton-photon detuning [14, 81]. Conclusion—
We consider the behaviour of fluctua-tions on a polariton mean-field stationary state in a one-dimensional lattice of coupled micropillars. By takingthe stationary state as a plane wave, which can be res-onantly injected with a suitably patterned optical field,we find that interactions in the system allow the pres-ence of a bi-skin effect where fluctuations are localizedat both edges of the lattice. Even though the underlyinghopping of the system is Hermitian, the interactions al-low an effective non-reciprocal coupling between particle-hole fluctuations. This results in a non-trivial topology,confirmed by calculations of the winding number.
Acknowledgements—
This work was supported by theSingaporean Ministry of Education, via the Tier 2 Aca-demic Research Fund project MOE2018-T2-2-068. ∗ [email protected] † [email protected][1] Iacopo Carusotto and Cristiano Ciuti, “Quantum fluidsof light,” Rev. Mod. Phys. , 299–366 (2013).[2] Hui Deng, Hartmut Haug, and Yoshihisa Yamamoto,“Exciton-polariton bose-einstein condensation,” Rev.Mod. Phys. , 1489–1537 (2010).[3] Jonathan Keeling and Natalia G. Berloff, “Exciton-polariton condensation,” Contemp. Phys. , 131–151(2011).[4] T. Byrnes, N. Y. Kim, and Y. Yamamoto, “Exciton-polariton condensates,” Nat. Phys. , 803–813 (2014).[5] P. G. Savvidis, C. Ciuti, J. J. Baumberg, D. M. Whit-taker, M. S. Skolnick, and J. S. Roberts, “Off-branchpolaritons and multiple scattering in semiconductor mi-crocavities,” Phys. Rev. B , 075311– (2001).[6] A. V. Nalitov, H. Sigurdsson, S. Morina, Y. S.Krivosenko, I. V. Iorsh, Y. G. Rubo, A. V. Kavokin, andI. A. Shelykh, “Optically trapped polariton condensatesas semiclassical time crystals,” Phys. Rev. A , 033830(2019).[7] Xingran Xu, Zhidong Zhang, and Zhaoxin Liang,“Nonequilibrium landau-zener tunneling in exciton-polariton condensates,” Phys. Rev. A , 033317(2020).[8] Charles-Edouard Bardyn, Torsten Karzig, Gil Refael,and Timothy C. H. Liew, “Topological polaritons andexcitons in garden-variety systems,” Phys. Rev. B ,161413 (2015).[9] A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, “Po-lariton Z topological insulator,” Phys. Rev. Lett. ,116401 (2015).[10] Chunyan Li, Fangwei Ye, Xianfeng Chen, Yaroslav V.Kartashov, Albert Ferrando, Lluis Torner, andDmitry V. Skryabin, “Lieb polariton topological insula-tors,” Phys. Rev. B , 081103 (2018).[11] D. R. Gulevich, D. Yudin, I. V. Iorsh, and I. A. Shelykh,“Kagome lattice from an exciton-polariton perspective,”Phys. Rev. B , 115437 (2016).[12] S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler,R. Ge, M. A. Bandres, M. Emmerling, L. Worschech,T. C. H. Liew, M. Segev, C. Schneider, and S. H¨ofling,“Exciton-polariton topological insulator,” Nature ,552–556 (2018).[13] Charles-Edouard Bardyn, Torsten Karzig, Gil Refael,and Timothy C. H. Liew, “Chiral bogoliubov excitations in nonlinear bosonic systems,” Phys. Rev. B , 020502–(2016).[14] H. Sigurdsson, G. Li, and T. C. H. Liew, “Spontaneousand superfluid chiral edge states in exciton-polariton con-densates,” Phys. Rev. B , 115453– (2017).[15] R. Banerjee, T. C. H. Liew, and O. Kyriienko, “Realiza-tion of hofstadter’s butterfly and a one-way edge mode ina polaritonic system,” Phys. Rev. B , 075412– (2018).[16] T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D.Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. H¨ofling,Y. Yamamoto, F. Nori, Y. S. Kivshar, A. G. Truscott,R. G. Dall, and E. A. Ostrovskaya, “Observation of non-hermitian degeneracies in a chaotic exciton-polariton bil-liard,” Nature , 554–558 (2015).[17] T. Gao, G. Li, E. Estrecho, T. C. H. Liew, D. Comber-Todd, A. Nalitov, M. Steger, K. West, L. Pfeiffer, D. W.Snoke, A. V. Kavokin, A. G. Truscott, and E. A. Ostro-vskaya, “Chiral modes at exceptional points in exciton-polariton quantum fluids,” Phys. Rev. Lett. , 065301–(2018).[18] Weilu Gao, Xinwei Li, Motoaki Bamba, and JunichiroKono, “Continuous transition between weak and ultra-strong coupling through exceptional points in carbonnanotube microcavity exciton–polaritons,” Nat. Photon-ics , 362–367 (2018).[19] Rui Su, Eliezer Estrecho, D. Biega´nska, Yuqing Huang,Matthias Wurdack, Maciej Pieczarka, Andrew G. Tr-uscott, Timothy C. H. Liew, Elena A. Ostrovskaya, andQihua Xiong, “Direct measurement of a non-hermitiantopological invariant in a hybrid light-matter system,”(2020), arXiv:2012.06133.[20] P. Comaron, V. Shahnazaryan, W. Brzezicki, T. Hyart,and M. Matuszewski, “Non-hermitian topological end-mode lasing in polariton systems,” Phys. Rev. Research , 022051 (2020).[21] S. Mandal, R. Banerjee, Elena A. Ostrovskaya, andT. C. H. Liew, “Nonreciprocal transport of exciton po-laritons in a non-hermitian chain,” Phys. Rev. Lett. ,123902 (2020).[22] Miguel A. Bandres, Steffen Wittek, Gal Harari, MidyaParto, Jinhan Ren, Mordechai Segev, Demetrios N.Christodoulides, and Mercedeh Khajavikhan, “Topolog-ical insulator laser: Experiments,” Science , eaar4005(2018).[23] Huitao Shen, Bo Zhen, and Liang Fu, “Topological bandtheory for non-hermitian hamiltonians,” Phys. Rev. Lett. , 146402 (2018).[24] Kohei Kawabata, Takumi Bessho, and Masatoshi Sato,“Classification of exceptional points and non-hermitiantopological semimetals,” Phys. Rev. Lett. , 066405(2019).[25] Stefano Longhi, “Probing non-hermitian skin effect andnon-bloch phase transitions,” Phys. Rev. Research ,023013 (2019).[26] Ananya Ghatak and Tanmoy Das, “New topological in-variants in non-hermitian systems,” Journal of Physics:Condensed Matter , 263001 (2019).[27] Junpeng Hou, Ya-Jie Wu, and Chuanwei Zhang, “Two-dimensional non-hermitian topological phases induced byasymmetric hopping in a one-dimensional superlattice,”(2020), arXiv:1906.03988 [cond-mat.mes-hall].[28] Sebastian Weidemann, Mark Kremer, Tobias Helbig, To-bias Hofmann, Alexander Stegmaier, Martin Greiter,Ronny Thomale, and Alexander Szameit, “Topological funneling of light,” Science , 311–314 (2020).[29] Ching Hua Lee and Ronny Thomale, “Anatomy of skinmodes and topology in non-hermitian systems,” Phys.Rev. B , 201103 (2019).[30] Tobias Hofmann, Tobias Helbig, Frank Schindler,Nora Salgo, Marta Brzezi´nska, Martin Greiter, TobiasKiessling, David Wolf, Achim Vollhardt, Anton Kabaˇsi,Ching Hua Lee, Ante Biluˇsi´c, Ronny Thomale, and Ti-tus Neupert, “Reciprocal skin effect and its realization ina topolectrical circuit,” Phys. Rev. Research , 023265(2020).[31] Hengyun Zhou and Jong Yeon Lee, “Periodic tablefor topological bands with non-hermitian symmetries,”Phys. Rev. B , 235112 (2019).[32] Chun-Hui Liu, Kai Zhang, Zhesen Yang, and Shu Chen,“Helical damping and dynamical critical skin effect inopen quantum systems,” Phys. Rev. Research , 043167(2020).[33] Xi-Wang Luo and Chuanwei Zhang, “Higher-order topo-logical corner states induced by gain and loss,” Phys.Rev. Lett. , 073601 (2019).[34] Tao Liu, James Jun He, Tsuneya Yoshida, Ze-Liang Xi-ang, and Franco Nori, “Non-hermitian topological mottinsulators in one-dimensional fermionic superlattices,”Phys. Rev. B , 235151 (2020).[35] Nobuyuki Okuma, Kohei Kawabata, Ken Shiozaki, andMasatoshi Sato, “Topological origin of non-hermitianskin effects,” Phys. Rev. Lett. , 086801 (2020).[36] Xiao-Ran Wang, Cui-Xian Guo, and Su-Peng Kou,“Defective edge states and number-anomalous bulk-boundary correspondence in non-hermitian topologicalsystems,” Phys. Rev. B , 121116 (2020).[37] Kai Zhang, Zhesen Yang, and Chen Fang, “Correspon-dence between winding numbers and skin modes in non-hermitian systems,” Phys. Rev. Lett. , 126402 (2020).[38] Linhu Li, Ching Hua Lee, Sen Mu, and Jiangbin Gong,“Critical non-hermitian skin effect,” Nat. Comm. (2020), 10.1038/s41467-020-18917-4.[39] Lei Xiao, Tianshu Deng, Kunkun Wang, Gaoyan Zhu,Zhong Wang, Wei Yi, and Peng Xue, “Non-hermitianbulk–boundary correspondence in quantum dynamics,”Nat. Phys. , 761–766 (2020).[40] Fei Song, Shunyu Yao, and Zhong Wang, “Non-hermitian skin effect and chiral damping in open quan-tum systems,” Phys. Rev. Lett. , 170401 (2019).[41] Fei Song, Shunyu Yao, and Zhong Wang, “Non-hermitian topological invariants in real space,” Phys.Rev. Lett. , 246801 (2019).[42] Yiling Song, Weiwei Liu, Lingzhi Zheng, Yicong Zhang,Bing Wang, and Peixiang Lu, “Two-dimensional non-hermitian skin effect in a synthetic photonic lattice,”Phys. Rev. Applied , 064076 (2020).[43] Ching Hua Lee, Linhu Li, and Jiangbin Gong, “Hybridhigher-order skin-topological modes in nonreciprocal sys-tems,” Phys. Rev. Lett. , 016805 (2019).[44] Yongxu Fu and Shaolong Wan, “Non-hermitiansecond-order skin and topological modes,” (2020),arXiv:2008.09033 [cond-mat.mes-hall].[45] Ryo Okugawa, Ryo Takahashi, and Kazuki Yokomizo,“Second-order topological non-hermitian skin effects,”Phys. Rev. B , 241202 (2020).[46] Flore K. Kunst, Elisabet Edvardsson, Jan Carl Budich,and Emil J. Bergholtz, “Biorthogonal bulk-boundary cor-respondence in non-hermitian systems,” Phys. Rev. Lett. , 026808 (2018).[47] Kohei Kawabata, Ken Shiozaki, Masahito Ueda, andMasatoshi Sato, “Symmetry and topology in non-hermitian physics,” Phys. Rev. X , 041015 (2019).[48] Zongping Gong, Yuto Ashida, Kohei Kawabata, KazuakiTakasan, Sho Higashikawa, and Masahito Ueda, “Topo-logical phases of non-hermitian systems,” Phys. Rev. X , 031079 (2018).[49] Shunyu Yao and Zhong Wang, “Edge states and topo-logical invariants of non-hermitian systems,” Phys. Rev.Lett. , 086803 (2018).[50] Simon Lieu, “Topological phases in the non-hermitian su-schrieffer-heeger model,” Phys. Rev. B , 045106 (2018).[51] Naomichi Hatano and David R. Nelson, “Localizationtransitions in non-hermitian quantum mechanics,” Phys.Rev. Lett. , 570–573 (1996).[52] Naomichi Hatano and David R. Nelson, “Vortex pinningand non-hermitian quantum mechanics,” Phys. Rev. B , 8651–8673 (1997).[53] Julia M. Zeuner, Mikael C. Rechtsman, Yonatan Plotnik,Yaakov Lumer, Stefan Nolte, Mark S. Rudner, MordechaiSegev, and Alexander Szameit, “Observation of a topo-logical transition in the bulk of a non-hermitian system,”Phys. Rev. Lett. , 040402 (2015).[54] Kazuki Yokomizo and Shuichi Murakami, “Non-blochband theory and bulk–edge correspondence in non-hermitian systems,” Prog. Theor. Exp. Phys. (2020), 10.1093/ptep/ptaa140.[55] Ramy El-Ganainy, Mercedeh Khajavikhan, Demetrios N.Christodoulides, and Sahin K. Ozdemir, “The dawnof non-hermitian optics,” Commun. Phys. (2019),10.1038/s42005-019-0130-z.[56] T. Helbig, T. Hofmann, S. Imhof, M. Abdelghany,T. Kiessling, L. W. Molenkamp, C. H. Lee, A. Sza-meit, M. Greiter, and R. Thomale, “Generalizedbulk–boundary correspondence in non-hermitian topolec-trical circuits,” Nat. Phys. , 747–750 (2020).[57] Tao Liu, Yu-Ran Zhang, Qing Ai, Zongping Gong,Kohei Kawabata, Masahito Ueda, and Franco Nori,“Second-order topological phases in non-hermitian sys-tems,” Phys. Rev. Lett. , 076801 (2019).[58] Shunyu Yao, Fei Song, and Zhong Wang, “Non-hermitian chern bands,” Phys. Rev. Lett. , 136802(2018).[59] Zhesen Yang, Kai Zhang, Chen Fang, and JiangpingHu, “Non-hermitian bulk-boundary correspondence andauxiliary generalized brillouin zone theory,” Phys. Rev.Lett. , 226402 (2020).[60] Ken-Ichiro Imura and Yositake Takane, “Generalizedbulk-edge correspondence for non-hermitian topologicalsystems,” Phys. Rev. B , 165430 (2019).[61] J S Liu, Y Z Han, and C S Liu, “A new way to constructtopological invariants of non-hermitian systems with thenon-hermitian skin effect,” Chin. Phys. B , 010302(2020).[62] Jong Yeon Lee, Junyeong Ahn, Hengyun Zhou, andAshvin Vishwanath, “Topological correspondence be-tween hermitian and non-hermitian systems: Anomalousdynamics,” Phys. Rev. Lett. , 206404 (2019).[63] Xingran Xu, Ying Hu, Zhidong Zhang, and ZhaoxinLiang, “Spinor polariton condensates under nonresonantpumping: Steady states and elementary excitations,”Phys. Rev. B , 144511 (2017).[64] R. Banerjee, S. Mandal, and T. C. H. Liew, “Coupling between exciton-polariton corner modes through edgestates,” Phys. Rev. Lett. , 063901 (2020).[65] Nobuyuki Okuma and Masatoshi Sato, “Topologicalphase transition driven by infinitesimal instability: Majo-rana fermions in non-hermitian spintronics,” Phys. Rev.Lett. , 097701 (2019).[66] S. Utsunomiya, L. Tian, G. Roumpos, C. W. Lai, N. Ku-mada, T. Fujisawa, M. Kuwata-Gonokami, A. L¨offler,S. H¨ofling, A. Forchel, and Y. Yamamoto, “Observa-tion of bogoliubov excitations in exciton-polariton con-densates,” Nat. Phys. , 700–705 (2008).[67] Maciej Pieczarka, Marcin Syperek, (cid:32)Lukasz Dusanowski,Jan Misiewicz, Fabian Langer, Alfred Forchel, MartinKamp, Christian Schneider, Sven H¨ofling, Alexey Ka-vokin, and Grzegorz S., “Ghost branch photolumines-cence from a polariton fluid under nonresonant excita-tion,” Phys. Rev. Lett. , 186401 (2015).[68] V. Kohnle, Y. L´eger, M. Wouters, M. Richard, M. T.Portella-Oberli, and B. Deveaud-Pl´edran, “From singleparticle to superfluid excitations in a dissipative polari-ton gas,” Phys. Rev. Lett. , 255302 (2011).[69] K. Winkler, O. A. Egorov, I. G. Savenko, X. Ma, E. Es-trecho, T. Gao, S. M¨uller, M. Kamp, T. C. H. Liew, E. A.Ostrovskaya, S. H¨ofling, and C. Schneider, “Collectivestate transitions of exciton-polaritons loaded into a peri-odic potential,” Phys. Rev. B , 121303 (2016).[70] H. Ohadi, Y. del Valle-Inclan Redondo, A. J. Ramsay,Z. Hatzopoulos, T. C. H. Liew, P. R. Eastham, P. G. Sav-vidis, and J. J. Baumberg, “Synchronization crossoverof polariton condensates in weakly disordered lattices,”Phys. Rev. B , 195109 (2018).[71] S. Alyatkin, J. D. T¨opfer, A. Askitopoulos, H. Sigurds-son, and P. G. Lagoudakis, “Optical control of couplingsin polariton condensate lattices,” Phys. Rev. Lett. ,207402 (2020).[72] L. Pickup, H. Sigurdsson, J. Ruostekoski, and P. G.Lagoudakis, “Synthetic band-structure engineering in po-lariton crystals with non-hermitian topological phases,”Nat. Comm. (2020), 10.1038/s41467-020-18213-1.[73] Iacopo Carusotto and Cristiano Ciuti, “Probing micro-cavity polariton superfluidity through resonant rayleighscattering,” Phys. Rev. Lett. , 166401 (2004).[74] Kazuki Yokomizo and Shuichi Murakami, “Non-blochband theory in bosonic bogoliubov-de gennes systems,”(2020), arXiv:2012.00439 [cond-mat.mes-hall].[75] Ryo Okugawa and Takehito Yokoyama, “Topological ex-ceptional surfaces in non-hermitian systems with parity-time and parity-particle-hole symmetries,” Phys. Rev. B , 041202 (2019).[76] Zhesen Yang, “Non-perturbative breakdown ofbloch’s theorem and hermitian skin effects,” (2020),arXiv:2012.03333 [cond-mat.mes-hall].[77] S. Michaelis de Vasconcellos, A. Calvar, A. Dousse,J. Suffczy´nski, N. Dupuis, A. Lemaˆıtre, I. Sagnes,J. Bloch, P. Voisin, and P. Senellart, “Spatial, spec-tral, and polarization properties of coupled micropillarcavities,” Applied Physics Letters , 101103 (2011).[78] E. Estrecho, T. Gao, N. Bobrovska, D. Comber-Todd,M. D. Fraser, M. Steger, K. West, L. N. Pfeiffer, J. Levin-sen, M. M. Parish, T. C. H. Liew, M. Matuszewski, D. W.Snoke, A. G. Truscott, and E. A. Ostrovskaya, “Directmeasurement of polariton-polariton interaction strengthin the thomas-fermi regime of exciton-polariton conden-sation,” Phys. Rev. B , 035306 (2019). [79] Rui Su, Sanjib Ghosh, Jun Wang, Sheng Liu, CaroleDiederichs, Timothy C. H. Liew, and Qihua Xiong,“Observation of exciton polariton condensation in a per-ovskite lattice at room temperature,” Nature Physics ,301–306 (2020).[80] M. Dusel, S. Betzold, O. A. Egorov, S. Klembt, J. Ohmer,U. Fischer, S. H¨ofling, and C. Schneider, “Room temper-ature organic exciton–polariton condensate in a lattice,”Nature Communications (2020), 10.1038/s41467-020-16656-0.[81] S. Mandal, R. Ge, and T. C. H. Liew, “Antichiral edgestates in an exciton polariton strip,” Phys. Rev. B ,115423 (2019). STATIONARY AND EXCITED STATES OF THE EXCITON POLARITONS UNDER RESONANT PUMP
In this section, we consider the behaviour of the Gross-Pitaevskii equation directly in the presence of fluctuations,which are modelled numerically. This will allow us to study the stability of the polariton mean-field solutions andaccess the fluctuation directly. We begin with the Gross-Pitaevskii equation for a driven-dissipative one dimensionalmicropillar chain, identical to Eq. (1) of the main text but accounting for an additional noise term [69] i (cid:126) ∂ψ n ∂t = (∆ − iγ C ) ψ n + (cid:88)
In this section, we will calculate the eigenvalues and the winding number in detail. The OBC energies have twoexceptional points (EPs) as is shown in Fig. 8(b). We need to calculate each winding of them separately. The
Figure 7. Energy dispersions for the mean-field (the first column), the mean-field with fluctuations (the second column), andonly the fluctuations (the third column) obtained from Eq.(8) under the coherent pumping. Parameters are used: ∆=-0.22meV, J =40 µ eV, k p = π/
6, and without the nonlinearity the dissipation in the first row and gA =0.1meV, γ C =0.1meV in the secondrow. Hamiltonian in the real space is H n = (∆ (cid:48) − iγ C ) (cid:88) n u ∗ n u n − (∆ (cid:48) + iγ C ) (cid:88) n v ∗ n v n + J (cid:88) n (cid:0) u ∗ n +1 u n − v ∗ n +1 v n + h.c. (cid:1) + gA (cid:88) n (cid:0) e ik p u ∗ n v n − e − ik p v ∗ n u n (cid:1) , (9)where u n and v n are different Bogoliubov modes. If we just consider the nearest neighbouring hopping and use theFourier transformation u n → √ N (cid:80) k ˆ a k e ikn and v n → √ N (cid:80) k ˆ b k e ikn with ˆ a k and ˆ b k the annihilation operators forthe momentum k , we can obtain the Hamiltonian with H = ∆ (cid:48) (cid:88) k (cid:16) ˆ a † k ˆ a k − ˆ b † k ˆ b k (cid:17) + gA (cid:88) k (cid:104) ˆ a † k ˆ b k − k p − ˆ b † k ˆ a k +2 k p (cid:105) + 2 J (cid:88) k (cid:16) ˆ a † k ˆ a k − ˆ b † k ˆ b k (cid:17) cos k − iγ C . (10)When the system is large enough and the periodic boundary condition is taken, we can get the 2 × a k , ˆ b k − k p ) as illustrated in the main text. To prove this, we can build a unit cell that contains l = 4 π/k p inner sites H ( k ) = (cid:32) H diag (cid:0) A , · · · , A e ik p ( l − (cid:1) l × l diag (cid:0) − A , · · · , − A e − ik p ( l − (cid:1) l × l − H ∗ (cid:33) , (11)with H = (∆ (cid:48) − iγ C ) ˆ I l × l + diag ( J, − l × l + diag ( J, l × l . The matrix diag ( J, ±
1) are the upper and lower shiftdiagonal matrix with the matrix element J and the Bloch theory gives H ,l = Je ik and H l, = Je − ik . The energyprofile is corresponding with the 2 × k changes continuously, the effective Hamiltonianin the momentum space with vectors v = (ˆ a k , ˆ b k , ˆ a k +2 k p , ˆ b k − k p ) (cid:62) ignoring the energy shift γ C can be written as˜ H ( k, k p ) = ∆ (cid:48) + 2 J cos( k ) 0 0 gA − ∆ (cid:48) − J cos( k ) − gA gA ∆ (cid:48) + 2 J cos( k + 2 k p ) 0 − gA − ∆ (cid:48) − J cos( k − k p ) . (12)The Hamiltonian in the momentum space has the symmetry of ˜ H ( − k, − k p ) = ˜ H ( k, k p ), S z ˜ H ( k, k p ) S z = ˜ H ( k, k p ) † ,and S x ˜ H ( k, k p ) S x = ˜ H ( k, k p ) with S x,y,z = I × (cid:78) σ x,y,z with σ i is the Pauli matrix. The profile of the energy0dispersion is highly symmetric and having two EPs if the Hamiltonian is non-Hermitian with A is a purely realnumber.By diagonalization, the four eigenenergies can be obtained E , ( k ) = ± (cid:112) f ( k ) − J sin( k p ) sin( k − k p ) , (13)with f ( k ) = − g A + ∆ (cid:48) + J cos(2 k p ) cos(2 k − k p ) + J + 2∆ (cid:48) J cos( k ) + 2 J cos( k − k p )(∆ (cid:48) + J cos( k )) , (14)and E , ( k )= E , ( k + 2 k p ). Meanwhile, the related wavefunctions are | ψ (cid:105) = − ∆ (cid:48) − J cos k p cos( k − k p ) ± √ f ( k ) √ A / √ , | ψ (cid:105) = − ∆ (cid:48) − J cos k p cos( k + k p ) ± √ f ( k +2 k p ) √ A / √ , (15)where, the function f ( k ) determine all the topological properties of the system. The real energy will collapse at Figure 8. (a) Phase transition diagram along with the change of the hopping energy J and (b)-(f) the energy profiles obtainedby OBC ‘dots’ and PBC ‘circles’ . Parameters are used: ∆ (cid:48) =-20 µ eV, γ C =0.1meV, gA =0.1meV, k p = π/
6, and J =40 µ eV,46 µ eV, 60 µ eV, 69 µ eV, 80 µ eV for (b)-(f). k = ± π/ E , ( k ) = E , ( k ), but the imaginary energy will have a jump across the EPs. To define each windingof the EPs, Eq.(12) can be written into two blocks off-diagonal form˜ H ( k ) = U H ( k ) U − = − ∆ (cid:48) − J cos( k ) − gA gA ∆ (cid:48) + 2 J cos( k + 2 k p ) − ∆ (cid:48) − J cos( k − k p ) − gA gA ∆ (cid:48) + 2 J cos( k ) 0 0 , (16)1with the unitary transition U = , and the winding number around two EPs can be calculated by W = − (cid:73) dk πi ∂ k (cid:26) log (cid:20) Det (cid:18) − ∆ (cid:48) − J cos( k ) − gA gA ∆ (cid:48) + 2 J cos( k + 2 k p ) (cid:19) − E R ˆ I (cid:21)(cid:27) , (17) W = − (cid:73) dk πi ∂ k (cid:26) log (cid:20) Det (cid:18) − ∆ (cid:48) − J cos( k − k p ) − gA gA ∆ (cid:48) + 2 J cos( k ) (cid:19) − E ∗ R ˆ I (cid:21)(cid:27) , (18)where E R and E ∗ R are two reference energies with open boundary condition(OBC) in real space. If we take the loop from the whole Bloch zone, the total winding number is zero, but for each winding, we have W , = ± E and − E ∗ will give a positive winding number while the E ∗ and − E give the negative one. As is marked in Figs. 8(b)-(f) withred circles, all EPs are occurring at the pure imaginary energies. The eigenenergies that go around the left EPs cangive the winding 1 and − E and E ∗ . CRITICAL POINTS OF THE TOPOLOGICAL TRANSITION
In this section, we will calculate the critical points of the topological phase transition. As is defined by Eqs. (17)-(18), two EPs have opposite windings. Based on the winding number and the localization of the wavefunction, wecan define three phases: topological, intermediate, and trivial phase. (a) (b) (c)(d) (e) (f)
Figure 9. The wavefuntions of u n (the first row) and v n (the second row) sites obtained by OBC. The parameters are used∆ (cid:48) =-20 µ eV, gA =0.1meV, γ C =0.1meV, k p = π/
6, and J =60 µ eV (the first column), 80 µ eV(the second column), 0.2meV(thethird column). The HN model [51, 52] can be easily theoretically realized by H = J e ikx + J e − ikx with J and J are asymmetrichoppings between two nearest sites and the energy spectrum can be obtained by: H = ( J + J ) cos ( kx ) − i ( J − J ) sin ( kx ) . (19)The skin modes will vanish for J = J and appear for J (cid:54) = J . Actually, if the system has the same energy dispersionlike Eq. (19) where different directions of the momenta have gain and loss at the same time, the skin modes willappear even with the same hopping by inducing other terms into the Hamiltonian.2To realize the skin effect, the complex dispersion will be like E k = C sin( k + C ) , (20)with C i must be complex numbers. The complex number of C i ensures the condensates have gain in one directionand loss in another direction forming skin modes.The derivative of Eq. (14) is f (cid:48) ( k ) = − J cos( k p ) sin( k − k p ) [∆ (cid:48) + 2 J cos( k p ) cos( k − k p )] , (21)with the zero points k = k p for the maximum and k = arcos( − ∆ (cid:48) J cos k p ) + k p for the minimum. If Eq. (14) is apurely imaginary number in the whole Bloch zone, all eigenstates are localized. The critical points for the topologicaltransition are J = (cid:0) gA + ∆ (cid:48) (cid:1) sec( k p ) / J = ( gA − ∆ (cid:48) ) sec( k p ) / J ≈ .
46 in Fig. 8 (a) and (c) and J ≈ .