Konstantin Y. Bliokh, Daria Smirnova, Franco Nori. Center for Emergent Matter Science, RIKEN, Japan. Science 348, 1448 (2015)

Transcription

1 Konstantin Y. Bliokh, Daria Smirnova, Franco Nori Center for Emergent Matter Science, RIKEN, Japan Science 348, 1448 (2015)

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3 QSHE and topological insulators The quantum spin Hall effect means the presence of topologically protected edge modes at the interface between two 2D insulators. Such modes are characterized by strong spin momentum locking: opposite spins propagate in opposite directions: Spin Filtered or helical edge states vacuum QSH Insulator Kane & Mele, PRL (2005); Bernevig, Hughes, Zhang, Science (2006)

4 QSHE and topological insulators The QSHE modes provide a unidirectional spin current, they are protected from backscattering (without spin flip), and have a non removable degeneracy in their spectrum, connecting the valence and conductance bands of the two insulators: Spin Filtered or helical edge states vacuum QSH Insulator Edge band structure Edge states form a unique 1D electronic conductor 0 /a k Kane & Mele, PRL (2005); Bernevig, Hughes, Zhang, Science (2006)

5 M. Z. Hasan and C. L. Kane: Colloquiu QSHE and topological insulators 3D generalization of such states is a topological insulator. Such insulator exhibits 2D surface modes, which are helical massless fermions with spin momentum locking (vortex spin texture): Hiseh et al., Nature (2009); Hasan & Kane, RMP (2010)

6 QSHE and topological insulators A textbook example of topological insulator is the Dirac electron with an interface between the positive mass and negative mass regions: M. Z. Hasan and C. L. Kane: Jackiw & Rebbi, PRD (1976)

7 QSHE and topological insulators The simplest examples of the QSHE states appear in time reversal ( T ) symmetric systems, where ˆ T 2 = ±1 If the S z spin component is conserved in the system, it can be characterized by the spin Chern number and the corresponding topological Z 2 number: C spin = σ C σ, C σ = 1 2π σ Fσ d 2 k ν = C spin 2 mod2 = ( 0,1)

8 QSHE and topological insulators The general topological classification of fermionic systems also involves the particle hole symmetry Xˆ, and results in 10 classes, 5 of which have topological surface states: Z 2 TRS PHS SLS d=1 d=2 d=3 Standard A unitary Z - Wigner-Dyson AI orthogonal AII symplectic Z 2 Z 2 Chiral AIII chiral unitary Z - Z sublattice BDI chiral orthogonal Z - - CII chiral symplectic Z - Z 2 BdG D Z 2 Z - C Z - DIII Z 2 Z 2 Z CI Z Schnyder et al., PRB (2008)

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10 Basic spin properties of light The bulk modes for free light are propagating plane waves: E σ e σ exp( ikz), e σ = x + iσ y 2 Here σ = ±1 x E is the helicity, and photons carry spin: H y S z S = σ k k Longitudinal helicity dependent spin

11 Basic spin properties of light The transversality condition e σ k = 0 results in the spin orbit coupling between the polarization and momentum of light: e σ k. ( ) This results in the monopole Berry curvature associated with massless spectrum of photons: ω F σ = σ k k 3

12 Basic spin properties of light Since the longitudinal spin component (helicity) is conserved, we can calculate the spin Chern number (integral over 2D sphere in k space). Due to the bosonic nature of photons, this results in C spin = 4, ν = 0 and T ˆ 2 = +1 This is a topologically trivial situation: two pairs of counter propagating surface modes with opposite spins. I.e., any spin can propagate in any direction. Lu, Joannopoulos, Soljacic, Nature Photon. (2014)

13 Surface modes and transverse spin However, in reality only one pair of surface modes survives in Maxwell equations. These are surface modes between media with different parameters ε and µ. x k x = iκ k z > k y ε < 1 z Such modes have only one polarization (TE or TM), with zero helicity, so that we do not expect any spin properties in these linearly polarized surface waves.

14 Surface modes and transverse spin Nonetheless, they have spin! Let us consider their evanescent wave tails: k = k z z + iκ x H y E σ e σ exp( ik z z κ x), x E S z e σ = S = x i κ z k z Rek Im k (Rek) 2 Transverse helicity independent spin! Bliokh & Nori, PRA (2012); Bliokh et al., Nature Commun. (2014)

15 Surface modes and transverse spin The nature of this transverse spin is similar to the circular motion of water in surface ocean waves: M. Stone, Science (2015)

16 QSHE of light This unusual transverse spin (independent of the polarization) survives in the TE or TM surface modes. Most importantly, opposite directions of propagation correspond to opposite transverse spins: S surf = 1 ε k surf n Bliokh, Smirnova, Nori, Science (2015)

17 QSHE of light Dispersion of surface modes also has non removable degeneracy due to the light cone dispersion of photons:

18 QSHE of light Well known example: surface plasmon polaritons. Taking into account transverse spin, usual metals exhibit surface modes with spin momentum locking as in 3D topological insulators:

19 QSHE of light: experiments In several groups independently reported experiments on spin dependent unidirectional excitation of surface or waveguided Maxwell waves: Transversely-incident light with usual spin Scatterer: coupling to surface modes Spin-dependent direction of surface modes

20 QSHE of light: experiments O Connor et al., Nature Commun. (2014)

21 QSHE of light: experiments Petersen et al., Science (2014); Mitsch et al., Nature Commun. (2014) le Feber et al. Nature Commun. (2015)

22 Suppression of backscattering Vanishing backscattering (antilocalization) of chiral fermions: Suppressed backscattering of surface plasmons: Massless fermions Massless bosons Evlyukhin & Bozhevolnyi, PRB (2005) Bliokh, PLA (2005); Bliokh & Kravtsov

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24 Distinctions and questions Surface modes of Maxwell equations have strong spin momentum locking and massless dispersion, exactly as in QSHE and topological insulators for electrons. Therefore, we call this the QSHE of light. However, there are also strong distinctions between the QSHE for electrons and photons. The QSHE of electrons relies on their fermionic nature and has topological origin protected by time reversal symmetry Tˆ 2 = 1. The QSHE of light has a different (topological?) origin.

25 Distinctions and questions Difference between the spin and spinor (polarization) for relativistic particles: Spin has a spinor independent contribution (transverse spin). The QSHE of light is the spin momentum locking with trivial spinor properties, while the QSHE of electrons is the spinor momentum locking with trivial vanishing spin (for Dirac electrons):

26 Distinctions and questions Bispinor: W ( p) 1 m E 1+ m E w Plane wave spin: σ p p w Spinor and the rest frame spin: w = w 1 w 2 S =ψ Σψ = m E s + 1 m E, p( p s) p 2 s = 1 2 w σw S w = Evanescent wave spin: S =ψ evan Σψ evan = S w + S m E s + 1 m E ( ) 1 m ( Rep) 2 E Rep Rep s S = m E ( Rep Imp) ( Rep Imp) s p 2 ( Rep) 2 Rep Im p p 2 e 2Im p r e 2Imp r

27 Distinctions and questions For surface Dirac (Jackiw Rebbi) modes, 1 i W surf, w surf = 1 ±1 2 i Surface wave spin vanishes: propagating: because the spinor dependent (usual) and spinor Independent (transverse) contributions cancel each other: 1 i, S surf =ψ surf Σψ surf 0 S = S w Bliokh, Smirnova, Nori, Science (2015) ±z s surf = 1 2 y

28 Distinctions and questions Possible topological origin for the QSHE of light: The existence of surface Maxwell modes is intimately related to the breaking of the dual symmetry D. A discrete version of this symmetry resembles the particle hole symmetry for fermions: 0 i i 0 E H = ω E H ˆD = ˆD 2 = 1

29 Distinctions and questions This resembles the topologically nontrivial fermionic class CI, one of Bogolubov de Gennes superconductor classes. Z 2 TRS PHS SLS d=1 d=2 d=3 Standard A unitary Z - Wigner-Dyson AI orthogonal AII symplectic Z 2 Z 2 Chiral AIII chiral unitary Z - Z sublattice BDI chiral orthogonal Z - - CII chiral symplectic Z - Z 2 BdG D Z 2 Z - C Z - DIII Z 2 Z 2 Z CI Z Schnyder et al., PRB 2008

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31 ü Light possesses intrinsic QSHE, i.e., strong spin momentum locking in surface Maxwell modes. ü The transverse polarization independent spin in evanescent waves (stemming from the transversality and SOI of light) is responsible for it. ü It differs in its origin from the QSHE of electrons (fermions). Spin momentum coupling rather than spinor momentum coupling. ü It seems that the dual symmetry between magnetic and electric properties plays an important role in the QSHE of light, but it is not fully clarified yet.

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