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From the Topological Spin-Hall Effect to the Non-Hermitian Skin Effect in An

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From the Topological Spin-Hall Effect to the Non-Hermitian Skin Effect in An From the topological spin-Hall effect to the non-Hermitian skin effect in an elliptical micropillar chain S. Mandal,∗ R. Banerjee,† and T.C.H. Liew‡ Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore The topological spin-Hall effect causes different spins to propagate in opposite directions based on Hermitian physics. The non-Hermitian skin effect causes the localization of all modes of a system at its edges. Although these effects have separately attracted tremendous interest, their completely different origins have made them hard to observe in a single system. Here we propose a system based on exciton-polariton elliptical micropillars hosting both the effects. The polarization splitting and different orientation of the elliptical micropillars naturally give rise to the topological spin-Hall effect in a one dimensional lattice. By making the effective decay rates of the different spin (σ±) modes unequal, the system makes a phase transition from the Hermitian regime to non-Hermitian regime showing the non-Hermitian skin effect. Introduction.— The field of topological insulators (TIs) started with the discovery of the quantum Hall effect [1] followed by the Haldane model [2], where electrons propagate robustly along the edges of a finite 2D sample. Later, Kane and Mele came up with the spin dependent version of the Haldane model, where they predicted that Unit cell electrons with different spins would propagate robustly in Hermitian case: Non-Hermitian case: opposite directions [3]. This is known as the topological spin-Hall effect (TSHE), which is expected to play an important role in the field of spintronics [4]. In the last decade, the concept of TIs has not only been Intensity Sites extended from fermions to different bosonic systems such Spin Hall effect Skin effect as photonics [5–9] and acoustics [10], but higher order TIs have also been demonstrated in those systems [11– FIG. 1: Schematic of an exciton-polariton elliptical micropil- 15]. In photonic system the role of spins can be played by lar chain. Proper orientations of the elliptical micropillars the photonic circular polarizations (σ±). However, due naturally give rise to the spin-Hall effect. If the polaritons to the presence of the splitting in energy between the having opposite circular polarizations (σ±) have unequal de- transverse-electric (TE) and transverse-magnetic (TM) cay rates, the non-Hermitian skin effect appears. modes (which effectively couples the σ± polarizations), realizing the TSHE for photons is far from trivial and often needs extremely careful fabrication in order to re- termined by the bulk-edge correspondence [29]. Due to duce the TE-TM splitting [16]. Instead of the photonic the exotic edge localization, any injected signal, inde- polarization, if the role of spin is played by an external pendent of its excitation position, always propagates in degree of freedom such as the angular momentum [17– one direction while the propagation in the opposite di- 19] or valley degree of freedom [20–22], then realizing the rection is strongly suppressed [25, 30]. Such an effect TSHE in photonic systems is more readily achieved. can be extremely useful in suppressing feedback in opti- cal circuits, which is a necessity [31]. However, achiev- Recently, non-Hermitian topological phases, especially ing the non-reciprocal coupling between two sites in an the non-Hermitian skin effect (NHSE), has been an in- optical system is difficult and often requires complicated tense area of research. Typically in a finite 1D lattice if experimental setups, which are hard to be integrated into arXiv:2103.05480v1 [cond-mat.mes-hall] 9 Mar 2021 the coupling between the adjacent sites is made different micro/nano-scale [30]. in the forward and backward directions (non-reciprocal coupling), then all the modes of the system get local- Exciton-polaritons are quasiparticles arising due to ized at the edges of the finite lattice leaving no modes the strong coupling of cavity photons and quantum well located in the bulk [23–28]. This is unlike the topolog- excitons [32]. Their fascinating properties, such as fi- ically non-trivial Hermitian systems, where the number nite lifetime, sensitivity to external fields, strong non- of edge modes residing within the bulk band gaps is de- linearity inherited from excitons, spin polarizations, etc, make them ideal for studying both the Hermitian and non-Hermitian topological phases. The polariton Chern insulator in presence of Zeeman splitting and TE-TM ∗Corresponding author: [email protected] splitting has been studied theoretically [33–41] and re- †Corresponding author: [email protected] alized experimentally [42]. Nonlinear topological polari- ‡Corresponding author: [email protected] tons have also been an intense area of research [43–46], 2 where in some cases the nonlinearity alone induces topo- (a) J (b) logical behavior [47–50]. The inherent non-Hermitian na- ture of the polaritons has lead to the realization of excep- J tional points [51, 52], non-Hermitian topological phases [52, 53], and other related theoretical predictions [54– y 56]. The spin-orbit coupling due to the TE-TM split- y Finite x ting inside the microcavity has played a vital role in ob- Periodic taining nontrivial spin related effects, such as the optical Finite x spin Hall effect [57–59], meron polarization textures [65], Periodic nontrivial bands with spin orbit interaction Hamiltonians (c) (d) [60–67]. 2 In this work, we propose a system based on an exciton- J 1 polariton elliptical micropillar chain, where both the TSHE and NHSE can be realized in a single set up (see 0 Fig. 1). By proper orientation of the otherwise identical Energy/ 1 elliptical micropillars along with the naturally present polarization splitting due to the shape anisotropy give 2 rise to a band structure, where the positive and nega- tive momentum states have opposite circular polariza- 0 0 tions (σ±). As a result σ± polaritons propagate in the opposite directions in the chain giving rise to the TSHE. If the effective decay rates of the σ± polaritons are made FIG. 2: (a) The unit cell of a square lattice with real couplings unequal by subjecting the chain to a circularly polarized along the x direction and complex couplings along the y di- spatially uniform incoherent pump, the non-Hermitian rection such that each plaquette has total accumulated phase physics takes over. The behavior of the system changes φ = 2π/3. (b)The same lattice as in (a), but having only two sites along the y direction. (c-d) The band structure of the drastically and the modes of the system get localized at system presented in (a-b), respectively. The bulk bands are one end of the chain, giving rise to the NHSE [23], where shown in grey, whereas red and green being the edge states σ± polaritons propagate in the same direction, while the located at the opposite edges. Parameters: ∆/J = 1. propagation in the other direction is strongly suppressed. The key advantage of our scheme is that here both the topological Hermitian and non-Hermitian phases can be Hofstadter butterfly if the eigenenergies of the system studied on the same sample without the need of spatial are plotted against φ [17, 68], where for φ = 2π/q, each patterning of the pump. Besides the robust polariton Bloch band splits into q sub-bands. In Fig. 2(a) the unit propagation here is in a one dimensional lattice, which is cell of the system, which is periodic along the x direc- more compact and ideal for potential information trans- tion, is shown for φ = 2π/3. The band structure of the port applications. system is shown in Fig. 2(c), where the bulk bands are Topological spin-Hall effect. — In order to show the shown in grey, while red and green represent the edge TSHE, we first consider a square lattice, which is pe- states located at the two edges. As expected there are riodic along the x direction and has Ny sites along the y three bulk bands, which are connected by the topologi- direction (see Fig. 2(a)). The coupling between the near- cally protected edge states. est sites along the x direction is taken as J, while the Next, we reduce the width of the lattice and keep only coupling between the nearest sites along the y direction two sites along the y direction (see Fig. 2(b)). The band is taken as complex ∆einxφ, where n is the site num- x structure of such a lattice is shown in Fig. 2(d). Remark- ber along the x direction. The total phase in each pla- ably the band structure in this case is exactly the same as quette becomes φ ( φ) while hopping counter-clockwise the one Fig. 2(c), however all the bulk modes disappear (clockwise). Any nonzero− phase φ introduces an effective leaving behind only the edge modes. Achieving com- magnetic field, which breaks the time reversal (TR) sym- plex hopping between sites for polaritons is extremely metry in the system. The Hamiltonian of the system can challenging, which makes the realization of the system be represented as shown in Fig. 2(a) difficult in practice. However, consid- ering the red (green) sites as σ+ (σ−) spin states of the Hˆ = Jaˆ† aˆ + ∆einxφaˆ† aˆ nx,ny nx+1,ny nx,ny nx,ny+1 polaritons, the system shown in Fig. 2(b) can be mapped h− i nXx,ny to a chain of elliptical micropillars (see Fig. 3(a)), where + h.c. (1) ∆ is the polarization splitting inside the micropillars due to shape anisotropy [69] and φ represents the orienta- Here ny is the site index along the y direction, tion angle.
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