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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)

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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

E∝ eexp ikz( ) , e = x +my

1+ m 2. (1)

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

of propagating light are determined by the helicity σ = 2Im m / 1+ m

2( ) , 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 =σk / 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( ) (helicity basis [22,23]), one can calculate the Berry connection Aσ ′σ = −ieσ ⋅ ∇k( )e ′σ and curvature Fσ ′σ = ∇k ×A

σ ′σ 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, Fσ ′σ = δ σ ′σ Fσ , and it forms two monopoles (σ = ±1) at the origin of the momentum space [16–18,21–23]:

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Fσ =σ kk 3

. (2)

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

topological Chern numbers for the two helicity states as Cσ = 1

2πFσ∫ d 2k , where the integral is

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

σ =±1∑ and the spin Chern number

Cspin = σCσ

σ =±1∑ [9,10,36]. (These quantities can be used because the helicity, i.e., spin

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.

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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]

Eevan ∝ eevan exp ikzz −κ x( ) , eevan =1

1+ m 2x +m k

kzy − i κ

kzz

⎛⎝⎜

⎞⎠⎟

. (4)

This wave propagates along the z -axis with wave number kz >ω / c and decays exponentially

away from the boundary with the decrement κ = kz2 − k2 . (One can consider the evanescent

wave (4) as a plane wave with the complex wave vector k = kzz + 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.

Importantly, the transversality (spin-orbit interaction) condition E ⋅k = 0 generates the

imaginary longitudinal z -component in the polarization vector (4) eevan , in contrast to the purely transverse polarization e in propagating waves (1). Recently it was shown [27,28] that this component produces x, z( ) -plane rotation of the wave electric field and thereby generates extraordinary transverse spin in evanescent waves (4), see Fig. 1. This transverse spin is

independent of the helicity σ and has the form S⊥ = κkzy (in the units used above) [27,28].

Generalizing this result for an arbitrary direction of propagation k , we represent the transverse spin as

S⊥ = κkz2 k × n , (5)

where n is the normal to the interface [ n = x for the wave (4)] and κ > 0 . Equation (5) demonstrates strong spin-momentum locking, precisely as in 3D topological insulators in solids

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[9,10,38] (see Fig. 3b below). In particular, for the z -propagating evanescent waves, the kz > 0 and kz < 0 modes will have opposite transverse spins Sy > 0 and Sy < 0 , respectively (see Figs. 2–4). Thus, an interface between free-space and any medium supporting surface-propagating modes with evanescent free-space tails (4) exhibits the QSHE state of light, with counter-propagating opposite-transverse-spin edge modes. This is the first key point of our work.

Note that the polarization parameter m still determines two degenerate helicity states σ = ±1 in the free-space evanescent modes (4), and the transverse spin is equally locked for both the helicity states. Thus, we have two degenerate pairs of counter-propagating QSHE modes in free space, in agreement with Eq. (3). However, depending on properties of a particular interface, only one pair of evanescent QSHE modes with fixed polarization m can survive (as in the surface plasmon-polariton case considered below). Such interfaces correspond to the spin Chern number Cspin = 2 , and the system acquires nontrivial topological insulator properties

described by the Z2 topological invariant ν =Cspin2mod2 [9,10,36]:

Cspin = 2 , ν = 1. (6)

3. Examples

We are now in a position to consider explicit examples of the photonic QSHE and topological insulators. First, we have to enclose a segment of free space with a boundary which supports surface modes with evanescent tails (4), Fig. 2. One of the examples of such boundary is the metal-vacuum interface supporting surface plasmon-polaritons (SPP) [39,40]. The free-space part of the SPP mode represents an evanescent wave (4) with the fixed polarization parameter m = 0 (TM polarization with zero helicity σ = 0 ) and fixed longitudinal wave number kz = kSPP , which is determined by the dispersion of the metal. Since σ = 0 , the spin of the free-space part of the SPP is purely transverse: S = S⊥ , Eq. (5) [27]. Figure 2 shows a quasi-2D example of a free-space area bounded by a metal. The counter-propagating SPP modes have opposite transverse spins (5), precisely as in electron QSHE systems [6–10]. Figure 3a shows the dispersion of the z -propagating SPPs on the x = 0 surface of a metal characterized by plasma frequency ω p and dielectric constant ε = 1−ω 2 /ω p

2 [39,40]. Originating from the light cone, the SPP spectrum represents two crossing branches with the opposite transverse spins – again, entirely analogous to the QSHE dispersion for electrons [6–10] and the Z2 topological properties (6). Importantly, the crossing (Dirac point) at the k = 0 origin is topologically protected and cannot be split by a frequency gap (ω ≠ 0 at k = 0 ). Indeed, surface modes with free-space evanescent tails (4) can exist only below the light cone, i.e., at kz >ω / c . It is the transversality (spin-orbit interaction) condition E ⋅k = 0 for Maxwell modes that links the surface-mode condition kz >ω / c to the transverse spin (5), which is in turn locked to the momentum.

Of course, a segment of free space (the whole in Fig. 2) cannot be a topological insulator, because it supports propagating (bulk) modes (1) at all frequencies. However, we can invert the problem and consider a metallic sample surrounded by free space (as shown in Fig. 4). Obviously, the spin-direction locking of the surface SPP modes remains the same, but the metal represents an insulator for light. Indeed, the propagating modes (1) exist only outside the gap −ω p,ω p( ) , i.e., at ω >ω p , as shown in Fig. 3a. At the same time, the SPPs represent surface

QSHE modes inside the optical gap of the metal. This consideration can be straightforwardly extended to the 2D surface of a 3D metal sample. The corresponding 2D dispersion of the SPP represents the Dirac-like cone shown in Fig. 3b. The transverse spin (5) is locked to the

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momentum, such that the spin direction exhibits the vortex-like texture on the SPP dispersion cone. This precisely mimics the properties of 3D topological insulators for electrons in solids [9,10,38]. Thus, we conclude that usual metals represent natural isotropic 3D topological insulators for light. This is the second key point of our work.

Fig. 3. Metals are natural topological insulators for light. (a) Dispersion of the bulk and surface modes in a metal characterized by the plasma frequency ω p . The metal is transparent for bulk (propagating) electromagnetic waves (1) for ω >ω p . In the gap ω <ω p the metal has surface plasmon-polariton modes [39,40] with fixed TM polarization (m = 0 ) but non-zero transverse spin (5) locked to the surface-plasmon momentum. The oppositely-propagating modes with opposite transverse spins (shown in red and blue) form a topologically-protected QSHE pair, cf. [6–10]. (b) Dispersion of the surface plasmon-polaritons on the 2D y, z( ) surface of a 3D metal. The topologically-protected Dirac point at the origin and the spin-momentum locking [a vortex-like texture of the transverse spin (5)] precisely correspond to 3D topological insulators [9,10,38].

We emphasize that only the polarization and the wave number of SPPs are determined by

the properties of the metal. At the same time, all the crucial QSHE and spin-momentum locking

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features are determined exclusively by free-space Maxwell equations and are independent of the details of the boundary supporting surface evanescent modes (4). Very recently, several experiments [29–34] in different optical systems reported similar results on the strong transverse spin-direction coupling and unidirectional spin-dependent propagation of light. All these experiments dealt with various interfaces supporting evanescent waves (4): metal surfaces, nanofibers, and photonic-crystal waveguides. Figure 4 shows a typical schematic of the experiments [29–34]. A transversely propagating light with different spin states was coupled into surface-evanescent modes via some scatterer (e.g., a nanoparticle or an atom). In doing so, the opposite transverse spin states excited the surface modes running in the corresponding opposite directions. Thus, these experiments clearly demonstrate the spin-momentum locking and QSHE of light, which are determined by the fundamental properties of the evanescent waves (4).

Fig. 4. Schematic of recent experiments [29–34], showing strong spin-to-direction coupling via evanescent modes. The incident y -propagating light (shown in green) is coupled to surface modes with evanescent tails (4) via scattering by a nanoparticle or another scatterer. Depending on the y -directed spin of the incident light with the helicity σ = ±1 (shown here by the circular-polarization arrows), the surface-evanescent waves with positive or negative kz are excited. This is a clear demonstration of the counter-propagating QSHE or topological-insulator surface modes with spin-momentum locking.

4. Conclusions

To summarize, we have shown that free-space light has intrinsic quantum spin-Hall effect (QSHE) properties. These are characterized by non-zero topological spin Chern number, and manifest themselves as evanescent edge modes of Maxwell equations with transverse spin-momentum locking. Therefore, any interface between free space and a medium supporting surface modes (e.g., a metal, nanofiber, or photonic crystal) exhibits the QSHE state of light. Moreover, we have shown that usual isotropic metals with surface plasmon-polariton modes represent natural 3D topological insulators for light. Remarkably, several very recent experiments have demonstrated strong transverse spin-momentum locking and spin-controlled unidirectional propagation of light at various interfaces with evanescent waves. Our work shows

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that all these experiments can be interpreted as observations of the QSHE of light. Independently of the details of the interfaces, the QSHE of light is determined by fundamental free-space properties of Maxwell equations. In this manner, the intrinsic spin-orbit coupling (transversality condition) of light determines remarkable spin properties of propagating (bulk) and evanescent (surface) modes.

Our theory solves an important puzzle and reveals intriguing new features in Maxwell’s theory of light. Strikingly, it seems that examples of optical topological insulators were known at least since the discovery of surface plasmon-polaritons in the 1960s. However, it required the discovery of topological insulators in condensed matter in 2005 [6–10], the transverse spin in evanescent waves in 2012 [27,28], and convincing spin-direction coupling experiments in 2013-2014 [29–34], before all these pieces were assembled together in a consistent and fundamental picture.

Acknowledgements

We acknowledge fruitful discussions with Akira Furusaki, Yuri Bliokh, Elena Ostrovskaya, and Alexander Khanikaev. This work was partially supported by the RIKEN iTHES Project, MURI Center for Dynamic Magneto-Optics, JSPS-RFBR contract no. 12-02-92100 and a Grant-in-Aid for Scientific Research (S).

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