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Quantum Spin Hall Effect and Topological Insulators for Light

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Quantum Spin Hall Effect and Topological Insulators for Light Quantum spin Hall effect and topological insulators for light Konstantin Y. Bliokh1,2 and Franco Nori1,3 1Center for Emergent Matter Science, RIKEN, Wako-shi, Saitama 351-0198, Japan 2Nonlinear Physics Centre, RSPhysE, The Australian National University, ACT 0200, Australia 3Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA We show that free-space light has intrinsic quantum spin-Hall effect (QSHE) properties. These are characterized by a non-zero topological spin Chern number, and manifest themselves as evanescent modes of Maxwell equations. The recently discovered transverse spin of evanescent modes demonstrates spin-momentum locking stemming from the intrinsic spin-orbit coupling in Maxwell equations. As a result, any interface between free space and a medium supporting surface modes exhibits QSHE of light with opposite transverse spins propagating in opposite directions. In particular, we find that usual isotropic metals with surface plasmon- polariton modes represent natural 3D topological insulators for light. Several recent experiments have demonstrated transverse spin-momentum locking and spin- controlled unidirectional propagation of light at various interfaces with evanescent waves. Our results show that all these experiments can be interpreted as observations of the QSHE of light. 1. Introduction Solid-state physics exhibits a family of Hall effects with remarkable physical properties. The usual Hall effect (HE) and quantum Hall effect (QHE) appear in the presence of an external magnetic field, which breaks the time-reversal ( T ) symmetry of the system. The HE represents charge-dependent deflection of electrons orthogonal to the magnetic field, whereas the QHE [1] offers distinct topological electron states, with unidirectional edge modes (charge-momentum locking), characterized by the topological Chern number [2]. The intrinsic spin Hall effect (SHE) can occur in T -symmetric electron systems with spin- orbit interactions. The SHE manifests itself as spin-dependent transport of electrons orthogonal to the external potential gradient (electric field) [3–5]. By analogy with the QHE, there is also the quantum spin Hall effect (QSHE) [6–8], which is characterized by topological edge states, where opposite directions of propagation are strongly coupled to two spin states of the electron. Such topological states with spin-momentum locking gave rise to a new class of materials: topological insulators [9,10]. Both the SHE and QSHE originate from the spin-orbit interactions and accompanying Berry-phase phenomena. The difference is that the SHE is a ‘weak’ spin-momentum coupling effect described by flexible geometric Berry curvature, while the QSHE and topological insulators are characterized by ‘strong’ spin-momentum locking and are described by quantized topological numbers (e.g., integrals of the Berry curvature). Alongside the extensive condensed-matter studies of electron Hall effects, their photonic counterparts were found in various optical systems. In particular, both the HE [11] and QHE with unidirectional edge propagation [12,13] have been reported in magneto-optical systems with broken T -symmetry. Furthermore, since photons are relativistic particles with spin 1, they naturally offer intrinsic spin-orbit interaction effects, including Berry phase [14,15] and the SHE [16–21] stemming from fundamental spin properties of free-space Maxwell equations [22,23]. Note that optical systems have some significant advantages compared to condensed-matter electronic systems because of the direct access to the local wave-function (electromagnetic field) 1 measurements and absence of many side effects (impurity scattering, temperature dependence, etc.). For instance, the first direct observation of the spin-dependent deflection of the particle trajectory (SHE) due to the Berry curvature was realized in optics [21]. The only missing optical part in the above family of Hall effects is the QSHE or topological insulators for photons. Recently, it was suggested that photonic topological insulators can be created in metamaterials, i.e., complex artificial electromagnetic analogues of natural crystals [24–26]. However, here we show that pure free-space light already possesses all the properties needed for the QSHE, and simple natural materials (such as metals supporting surface plasmon- polariton modes) represent perfect 3D photonic topological insulators. We show that the Berry curvature of free-space photons naturally provides a non-zero spin Chern number responsible for the QSHE. Remarkably, the QSHE edge modes with strong spin-momentum locking are well- known evanescent waves, which appear at any interface supporting surface waves. We show that recently discovered transverse spin in evanescent waves [27,28] and several very recent experimental demonstrations of the strong transverse spin-directional coupling at interfaces with evanescent surface modes [29–34] demonstrate inherent QSHE and topological-insulator properties of light. These properties are independent on the details of the interfaces and are determined by fundamental spin-orbit interaction features of free-space Maxwell equations. Thus, our theory solves an important puzzle and reveals new profound features in Maxwell’s theory of light, by combining several previously disconnected pieces into a unified and comprehensive picture. 2. Theory Propagating (bulk) free-space modes of Maxwell equations are polarized plane waves. Introducing the complex amplitude E r of the harmonic electric field E r,t = Re ⎡E r e−iωt ⎤ , ( ) ( ) ⎣ ( ) ⎦ the plane-wave solution with the wave vector k = kz can be written as x + my E ∝ eexp ikz , e = . (1) ( ) 2 1+ m Here e is the complex unit polarization vector, m is the complex polarization parameter [28,35], whereas x , y , and z denote the unit vectors of the corresponding axes. The spin states 2 of propagating light are determined by the helicity σ = 2Im m / 1+ m , so that the m = ±i ( ) modes correspond to the right-hand and left-hand circular polarizations with helicities σ = ±1. According to the relativistic massless nature of photons, the spin angular momentum is directed along the wave vector as S = σ k / k (we consider the spin density per photon in = 1 units). Generalizing Eq. (1) to plane waves with arbitrary direction of propagation, the polarization vector becomes momentum-dependent: e(k) . In fact, this vector is tangent to the k -space sphere due to the transversality condition E⋅k = 0 . This condition and the spherical k - space geometry underlies the Berry phase and spin-orbit interaction for photons [16–23]. In particular, introducing the unit polarization vectors for circularly-polarized states, eσ (k) σσ ′ σ σ ′ (helicity basis [22,23]), one can calculate the Berry connection A = −i e ⋅(∇k )e and σσ ′ σσ ′ curvature F = ∇k × A for photons. In agreement with the relativistic light-cone spectrum of photons with a double (helicity-degenerate) Dirac point at k = 0 , the Berry curvature is diagonal in the helicity basis, Fσσ ′ = δ σσ ′Fσ , and it forms two monopoles (σ = ±1) at the origin of the momentum space [16–18,21–23]: 2 k Fσ = σ . (2) k 3 This curvature is responsible for the spin-redirection Berry phase in optics and the SHE of light, i.e., ‘weak’ (geometric) spin-orbit interaction phenomena [16–23]. To characterize the ‘strong’ (topological) spin-orbit interaction effects, we now define the 1 topological Chern numbers for the two helicity states as Cσ = ∫ Fσ d 2k , where the integral is 2π taken over the k -space sphere [9,10]. The monopole curvature (2) immediately yields Cσ = 2σ . Note that the physical meaning of the Chern number in electron systems is the number of edge modes with fixed direction of propagation. To properly characterize photonic QHE and QSHE properties, we calculate the total Chern number C = ∑ Cσ and the spin Chern number σ =±1 σ [9,10,36]. (These quantities can be used because the helicity, i.e., spin Cspin = ∑ σC σ =±1 component normal to the k -space sphere, is conserved in free space.) This yields C = 0 , Cspin = 4 . (3) The vanishing total Chern number corresponds to the T -symmetry of free-space Maxwell equations and the absence of the photonic QHE states in free space. At the same time, the non- zero spin Chern number implies that free-space light naturally has QSHE modes, i.e., edge counter-propagating modes with strong locking between the spin and direction of propagation. The value Cspin = 4 implies that there should be two pairs of such modes. Fig. 1. Transverse spin in evanescent waves. Evanescent wave (4) propagates along the z - axis and decays exponentially in the x > 0 semi-space. The inset shows the instantaneous distributions of the electric and magnetic wave field for the case of the linear TM polarization m = 0 . The cycloidal (x,z)-plane rotation of the electric field generates the transverse spin S⊥ , Eq. (5) [27,28]. The sign of the transverse spin depends on the direction of propagation of the evanescent wave. 3 Indeed, the QSHE states of light exist, and they are well known. The photonic edge states of a bounded segment of free-space are evanescent waves [37]. For instance, assuming the x = 0 boundary, with the free space being at the x > 0 semi-space, the generic evanescent-wave solution of Maxwell equations can be written as [28] 1 ⎛ k κ ⎞ E ∝ e exp ik z −κ x , e = x + m y − i z . (4) evan evan ( z ) evan 2 ⎜ ⎟ 1+ m ⎝ kz kz ⎠ This wave propagates along the z -axis with wave number kz > ω / c and decays exponentially 2 2 away from the boundary with the decrement κ = kz − k . (One can consider the evanescent wave (4) as a plane wave with the complex wave vector k = kz z + iκ x .) Fig. 2. Quantum spin Hall effect edge modes at the boundary between free space and any medium supporting surface waves (e.g., surface plasmon-polaritons). Independently of the medium, the surface modes have evanescent tails (4) in free space. The direction of the transverse spin in free space is strongly coupled (locked) to the direction of propagation of the evanescent wave.
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