Topological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators

Transcription

1 Topological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators Satoshi Fujimoto Dept. Phys., Kyoto University Collaborator: Ken Shiozaki (Dept. Phys., Kyoto Univ.) ref. Phys. Rev. Lett.110, (2013)

2 Topological insulator/superconductor e.g. characterized by topological invariant in k-space (Z) Quantum Hall effect in 2D DEG (Thouless, Kohmoto et al.) first Chern number Z invariant (any integer) σ xy = e2 h n c QHE Electromagnetic properties Transport properties 3D time-reversal invariant topological insulator Z2 invariant or θ =0 θ = π L EM = θe2 2πh E J x = e2 2πh θe y M = e2 2πh θe (Kane, Zhang, Qi et al.) (only two values) (trivial insulator) (topological insulator) TI P = e2 2πh θ

3 For most classes of TI and TSC, relations between topological invariants characterizing bulk topological features and physically observable quantities have been well clarified. However, case of Z topological insulators/superconductors in odd spatial dimensions has not yet been well understood! e.g. 3D TRI topological SC 3 He ( phase) (class DIII) Cuxi2Se3? Li2Pt3? (NCS P+s-wave SC) (Fu-erg, Sasaki et al.) 1D or 3D topological insulators with chiral symmetry (class III) (sub-lattice symmetry) Kitaev s Majorana chain (1D spinless p-wave SC) (class DI) also, modified Su-Schrieffer-Heeger model In this case, the topological invariant takes any integer values N. However, it has not yet been well clarified in what physical quantities this topological invariant can be detected!

4 Z Topological insulator/superconductor in odd dimensions Periodic table of TI and TSC Symmetry spatial dimension Z Θ Ξ Π Z 0 III Z 0 Z I DI Z 0 0 D Z 2 Z 0 DIII Z 2 Z 2 Z II Z 2 Z 2 CII Z 0 Z 2 C Z 0 CI Z Θ Ξ Π = ΘΞ (Schnyder et al., Kitaev) Time-reversal sym. e.g. Kitaev model 3 He Li2Pt3 Cuxi2Se3 Charge conjugation (particle-hole sym.) [superconductivity] Chiral sym. (sublattice sym.) Z invariant in odd dimensions: winding number arises from chiral symmetry HΠ = ΠH Hψ = Eψ HΠψ = EΠψ 0 E3 -E3 E2 E1 -E1 -E2 k

5 n example of 3D class III topological insulator t 3 t 1 t 2 and insulator on cubic lattice with sublattice Electron hopping is only between - and -sublattices H = Energy levels at (E) and (E) sites are equal (put 0) 0 q(k) q (k) 0 q(k) =(k)+σ (k) (k) = 8t 2 cos k x cos k y cos k z 4t 3 [cos k x (cos 2k y + cos 2k z ) + cos k y (cos 2k z + cos 2k x ) + cos k z (cos 2k x + cos 2k y )] (k) =it 1 (sin k x, sin k y, sin k z ) (arise from SO int.+ magnetic order) chiral symmetry Π= HΠ = ΠH There is no diagonal element of H in sub-lattice space (sub-lattice symmetry)

6 3D TRI (class DIII) topological superconductor 3 He Li2Pt3 (NCS P+s-wave SC) Cuxi2Se3 There is no explicit sub-lattice structure chiral symmetry combination of time-reversal symmetry and particle-hole symmetry E E E TRS E PHS E k k k k k

7 Z invariant in odd dimensional TI / TSC: winding number N 3D case N = µνλ 24π 2 dktr[q kµ q q kν q q kλ q ] e.g. He 3, Li2Pt3 etc. for 3 He ( phase) q = ε k + id(k) σ ε 2 k + d(k) 2 d(k) =(d x,d y,d z ) : d-vector of SC 3D momentum space (rillouin zone) q 3D sphere n k / n k n k =(ε k,d x,d y,d z ) N is the winding of the mapping from 3D Z to 3D sphere homeomorphic to the Hilbert space of class III TI and class DIII TSC N is the total number of gapless surface Dirac (Majorana) fermions How does the winding number N appear in electromagnetic or thermal responses and transport phenomena?

8 Effective low energy theory for Z topological insulator in 3 D Ryu et al., Qi et al. 3D class III top. insulator L eff = θe2 2πh E θ = 0 or π (mod 2π) because of chiral symmetry e.g. Quantum anomalous Hall effect J x = Magnetoelectric effect M = e2 2πh θe e2 2πh θe y P = e2 2πh θ in analogy with xion electrodynamics of 3D TRI Z2 top. insulators However, only Z2 part ( θ = 0 or π ) is captured. The Z invariant N (winding number taking any integer) is actually the number of topologically protected gapless surface modes. However, it is hidden in the above electromagnetic responses. Is it possible to detect the Z invariant N in any electromagnetic responses?

9 Effective low energy theory for Z topological superconductor in 3D Ryu et al., Qi et al. 3D TRI (class DIII) top. superconductor basically, spin-triplet SC Since charge and spin are not conserved, topological characters (quantization of physical quantities) do not appear in electromagnetic responses surface gapless states are Majorana fermions However, instead, topological characters appear in thermal responses, because of energy conservation Thermal responses gravitational field theory E c 2 couples to gravity potential φ g (Luttinger (1964)) E g = φ g gravitoelectric field T T g gravitomagnetic field associated with circulating heat (energy) current

10 Effective low energy theory for Z topological superconductor in 3D Ryu et al., Qi et al.,nomura et al. gravitational field theory for low-energy effective theory of TRI TSC in 3D L = πk2 T 2 12h θ = 0 or π θ g E g Thermal responses! E g g gravitoelectric field (mod 2π) gravitomagnetic field T T e.g. Quantum anomalous thermal Hall effect J H x = πk2 T 12h θ T y Thermal-analogue of ME effect M H = πk2 T 2 12h θe g P H = πk2 T 2 12h θ g in analogy with xion electrodynamics of 3D TRI Z2 top. insulators However, only Z2 part ( θ = 0 or π ) is captured. Incomplete description for Z topological features! Is it possible to detect the Z invariant N in any thermal responses?

11 Electromagnetic and thermal responses which characterize Z topological non-triviality of class III TIs and class DIII TSCs : n idea using heterostructure systems

12 asic Key Idea: in 3D cases, we can make a connection between the winding number N and θ of the axion field theory by considering spatially (temporally) varying systems (e.g. heterostructure systems), which consist of chiralsymmetric topological insulators (or superconductors) and chiralsymmetry-broken trivial insulators (or superconductors). π 2 0 dφ 2π dθ(φ) dφ = N 2 chiral-s trivial insulator Π φ = π 2 H(k) H(φ) =H cos φ + Π sin φ Hamiltonian for heterostructure systems φ = φ(r,t) 0 φ π 2 Π = : Hamiltonian of class III topological insulator in 3D ε 0 0 ε HΠ = ΠH itself is the chiral-symmetry breaking field. H(k) φ =0 chiral-symmetric topological insulator (class III)

13 φ = π 2 φ =0 J total = Π = Quantum anomalous Hall effect π 2 0 chiral-s trivial insulator class III topological insulator ε 0 0 ε dφ dj(φ) dφ J Π = e2 2h N ˆn E E ˆn π 2 energy level difference between - sublattices (from axion EM) J = 0 e2 2πh dφ 2π Hall current carried by surface states surface mode gives quantized conductivity dz z θ E dθ(φ) dφ = N 2

14 Topological magnetoelectric effect M total = P total = class III topological insulator π 2 0 π 2 0 dφ dm(φ) dφ dφ M total (P total ) dp (φ) dφ = e2 2h NE = e2 2h N E () circulating current axion EM π 2 0 M = e2 2πh θe P = e2 2πh θ dφ 2π dθ(φ) dφ = N 2 trivial insulator with broken chiral symmetry

15 Case of class DIII (TRI) topological superconductor 3He (W phase) 1-band with spins spherical Fermi surface Cuxi2Se3 (p-wave SC) winding number N=1 1-band with spins winding number N=1? Li2Pt3 (noncentrosymmetric P+s-wave SC) 4 band systems with SO split pairs 137 D=0.4% S= D=0.5% S= D=12.1% S=0.70 winding number N>1 (or < -1) 140 D=52.4% S=0.75 (Shishidou and Oguchi) 141 D=21.2% S= D=9.1% S= D=2.5% S= D=1.9% S=0.77 8

16 Case of class DIII (TRI) topological superconductor What is chiral-symmetry-breaking perturbation? 3D p-wave SC (W-phase) ε H = k d k σiσ y iσ y d k σ ε k change basis H = 0 q(k) q (k) 0 d k k s-wave gap H = 0 s iσ y siσ y 0 H = i s s s = s chiral-symmetry breaking field is Time reversal symmetry breaking s-wave SC gap!

17 Quantum anomalous thermal Hall effect J H total = π2 k 2 T 2 12h trivial TR s-wave superconductor TRI topological superconductor = π2 k 2 T 12h N ˆn E g N ˆn T Im s =0 J H ˆn T N : winding number π 2 0 c.f. axion gravitational field theory J H x dφ 2π Heat Hall current carried by surface states dθ(φ) dφ = N 2 = πk2 T 12h θ T y p How to realize phase difference? c.f. Wang, Qi, Zhang, similar result based on argument of surface Majorana fermions. bias between s-wave SC and TSC dynamical effect, Im s (ω) = 0 for ω = 0due to inelastic scattering

18 Topological gravitomagnetoelectric effect (Thermal analogue of topological magnetoelectric effect) P H TRI topological superconductor P H total = π2 k 2 T 2 12h : heat polarization which induces temperature gradient P H total N g g N : winding number c.f. axion gravitational field theory P H = πk2 T 2 12h T θ g gravitomagnetic field (raised by mechanical rotation?) (Nomura et al.) trivial TR s-wave superconductor Im s =0 circulating heat current

19 Implications for dynamical axion K. Shiozaki, S.F., poster presentation on 13 June TRI topological insulator (Li,Wang,Qi,Zhang) magnetic fluctuations break quantization of θ dynamical axion TRI topological SC fluctuations of TRS s-wave OP dynamical axion Case of non-centrosymmetric SC admixture of p-wave gap p e iφ p and s-wave gap s e iφ s relative phase fluctuation a la Leggett mode is dynamical axion!! δφ p δφ s

20 ulk electromagnetic responses characterizing Z topological non-triviality : Chiral charge polarization

21 Chiral charge polarization: bulk quantity related to the winding number P 5 = e w nr Πˆr w nr n occ c.f. charge polarization: P = e w nr ˆr w nr n occ w nr : Wannier func. Π : operator for chiral symmetry For class III systems, is the difference of the charge polarization between -sublattice and -sublattice H = Π= q(k) q (k) 0 P 5 P 5 is gauge-invariant even for periodic.c.! polarization is gauge-dependent

22 Chiral charge polarization: bulk quantity related to the winding number 1D case (class III, class DI) chiral polarization is indeed equivalent to the winding number! P 5 = e N = 1 2πi n occ w nr Πˆx w nr dktr[q k q ] = e N 2 winding number in 1D P 5 corresponds to fractional charge (or # of Majorana fermions) emergent at open edges zero-energy edge state modified SSH model Kitaev s Majorana chain model

23 Chiral charge polarization: bulk quantity related to the winding number 3D class III P 5 is induced by an applied magnetic field, and, remarkably, related to the bulk winding number!! P 5 = e2 2h N N : winding number Topological magnetoelectric effect This relation is obtained by first-order perturbative calculation P 5 is gauge-invariant. (uniquely defined without heterostructure!) Topological ME effect from axion field theory (only Z2 part) P = e2 2πh θ θ = π (mod 2π) (gauge-dependent)

24 Summary (i) The Z topological invariant in 3D can be observed in the quantum anomalous (thermal) Hall effect and topological (gravito-) magnetoelectric effects in heterostructure systems which consist of the chiral-symmetric TI (TRI TSC) and chiral-symmetry-broken trivial insulators (superconductors). (ii) In 3D class III TI, the winding number appears in chiral charge polarization induced by an applied magnetic field. P 5 = e2 2h N