Floquet Topological Insulators and Majorana Modes

Transcription

1 Floquet Topological Insulators and Majorana Modes Manisha Thakurathi Journal Club Centre for High Energy Physics IISc Bangalore January 17, 2013

2 References Floquet Topological Insulators by J. Cayssol et al., arxiv : Generating Many Majorana Modes via Periodic Driving by Q. Tong et al., arxiv : Floquet Topological Insulator in Semiconductor Quantum Well by N.H. Linder et al., Nature Physics

3 Outline 1 Introduction Symmetry Integer Quantum Hall Effect Topological Insulator Bloch theory Topological Invariant 2 Floquet topological insulator Floquet Theory Floquet Formalism Floquet Chern Number 3 HgTe/CdTe heterostructures Generating Many Floquet Majorana modes Conclusions

4 Introduction Symmetry Symmetry Symmetry: Invariance under certain operation Most quantum states of matter are categorized by the continuous symmetries they break and by a local order parameter. For example, the crystallization of water into ice breaks translational symmetry or the magnetic ordering of spins breaks rotational symmetry. There are some states where continuous symmetries are not broken.

5 Introduction Integer Quantum Hall Effect Integer Quantum Hall Effect In the quantum Hall state, an external magnetic field perpendicular to a two-dimensional electron gas causes the electrons to circulate in quantized orbits. The bulk of the electron gas is an insulator, but along its edge, electrons circulate.

6 Introduction Topological Insulator Topological Insulator It has been realized that same robust conducting edge states that are found in the quantum Hall state could be found on the boundary of band insulators with large spin-orbit effect, called topological insulators. A topological insulator (TI) is a band insulator which is characterized by a topological number and which has gapless excitations at its boundaries.

7 Introduction Bloch theory Bloch theory Bloch theorem formulates a condition on the solutions ψ k ( r) of a lattice having periodic potential V ( r) ψ k ( r) = u k ( r) e i k. r k = Allowed wave vector for electron that is obtained for a constant potential u k ( r) = Arbitrary function but always with the periodicity of lattice u k ( r + R) = u k ( r) We can rewrite Bloch s theorem as ψ k ( r + R) = ψ k ( r) e i k. R

8 Introduction Topological Invariant Topological Invariant What is the difference between a quantum Hall state and an ordinary insulator? Chern (or TKNN) topological invariant (Thouless et al, 1982) where n m = 1 2π X A m d 2 k A m = i u m ( k) u m ( k) Insulator : n = 0 IQHE : σ xy = ne 2 /h Analogy : Genus of a surface : g = No. of holes

9 Floquet topological insulator Floquet topological insulator (FTI) The dearth of experimentally accessible TIs has motivated many proposals. One of the most intriguing proposals involves shining light on conventional insulator to yield systems with a topological nature. These kind of TI s called as Floquet Topological Insulator

10 Floquet topological insulator Floquet Theory Floquet Theory Quantum systems with Hamiltonian periodic function in time, follow Floquet theorem The Schrodinger equation for the 1D quantum system is With (H(x, t) i ) ψ(x, t) = 0 (1) t H(x, t) = H 0 (x) + V (x, t); V (x, t) = V (x, t + T ) According to the Floquet theorem, there exist solutions to Eq.(1) that have the form ψ(x, t + T ) = e i αt / ψ(x, t); α = Floquet characteristic exponent or Quasienergy

11 Floquet topological insulator Floquet Formalism Effective Hamiltonian In the absence of irradiation the Bloch Hamiltonian of a two band insulator is H 0 ( k) = 0 ( k)i 2x2 + d( k) σ Time dependent perturbation V ( k, t) = V 0 ( k) σ cos(ωt) Chern number for non irradiated insulator is given by C I = 1 ( ) d 2 ˆd(k) k ˆd(k) ˆd(k) 4π k x k y

12 Floquet topological insulator Floquet Chern Number Floquet Chern Number Under irradiation ψ(t) = U(t, t 0 )ψ(t 0 ) The evolution operator is U(t, t 0 ) = T t exp ( i Floquet Hamiltonian H F is defind as It can be parametrized as t t 0 dt H( k, t )) U k (T + t 0, t 0 ) = e ih F ( k)t H F ( k) = Floquet Chern number is given by C F = 1 ( d 2 ˆn(k) k 4π k x F ( k)i 2x2 + n( k) σ ˆn(k) ) ˆn(k) k y

13 New topological phases by light

14 HgTe/CdTe heterostructures HgTe/CdTe heterostructures Irradiation of light is able to produce transition between trivial insulating state (non- inverted regime) and the non trivial state (inverted regime). Let s start from a non inverted HgTe/CdTe quantum well : ( H 4x4 ( H k) = 0 ( ) k) 0 0 H0 ( k) Where and the vector H 0 ( k) = 0 ( k)i 2x2 + d( k) σ d( k) = (Ak x, Ak y, M B k 2 )

15 HgTe/CdTe heterostructures HgTe/CdTe heterostructures When a linearly polarized perturbation is added to the trivial phase (M/B < 0), the bands reshuffle in such a way that the Hamiltonian is characterized by inverted effective bands. Figure: Quasi-energy spectrum with ω = 2.3M, V = A = B = 0.2 M Ref. : N. H. Linder et. al. Nat. Phys.7, 490 (2011)

16 HgTe/CdTe heterostructures HgTe/CdTe heterostructures Figure: Density of edge mode for k x = 0 and k x = 0.84 respectively Ref. : N. H. Linder et al. Nat. Phys.7, 490 (2011)

17 Generating Many Floquet Majorana modes Majorana Fermion Majorana fermion provide a very robust way of storing information Majorana fermion operator is Hermitian and anticommutes with all other fermionic operator. a 2 = 1 and {a, b} = 0 Two such operators can be combined to give a single Dirac fermion operator: c = 1 2 (a + i b) and c = 1 2 (a i b) Majorana fermions are bound states at zero energy and called Majorana bound states or Majorana zero modes.

18 Generating Many Floquet Majorana modes Majorana Bound State We now consider the Hamiltonian H = N 1 n=1 J x σ x nσ x n+1 h N n=1 σ z n Define Jordan-Wigner transformation from N spin-1/2 s to 2N Majorana operators a n = ( n 1 ) j=1 σz j σ x n, b n = ( n 1 ) j=1 σz j σ y n H = i N 1 n=1 J x b n a n+1 + i N h n a n b n n=1

19 Generating Many Floquet Majorana modes Majorana Bound State Majorana bound state for N = 200, J x = 1, h = 0.9 :

20 Generating Many Floquet Majorana modes Generating Many Floquet Majorana Modes Formation of many Majorana modes need : Protection of time-reversal symmetry Longer range interaction in the system More long-ranged the interaction is, the more Majorana modes one might obtain Question : How to synthesize a long-range interaction in a topologically nontrivial condensed-matter system while maintaining time-reversal symmetry Ref. : Wong et al. arxiv

21 Generating Many Floquet Majorana modes Generating Many Floquet Majorana Modes Answer to this question is Periodic Driving of system Periodic driving is becoming an highly controllable and versatile tool in generating different topological states of matter Here periodic driving protocol is - switching Hamiltonian from H 1 for first half period to H 2 for second half period The Floquet operator U is given by U(T ) = e i H 2 T 2 e i H 1 T 2 e i H eff T

22 Generating Many Floquet Majorana modes Generating Many Floquet Majorana Modes Using Baker-Campbell-Hausdorff (BCH) formula, H eff H eff = H H 2 2 it 8 [H 2, H 1 ] T [(H 2 H 1 ), [H 2, H 1 ]] + The engineered H eff may still have long-range hopping or pairing terms via the nested-commutators even if H 1 or H 2 are short-range Hamiltonians

23 Generating Many Floquet Majorana modes Static Model 1D spinless p-wave superconductor H = µ N l=1 N 2 N 1 c l c l (t 1 c l c l c l c l+1 + h.c.) l=1 (t 2 c l c l c l c l+2 + h.c.) l=1 where a = a e iφa The relative phase φ = φ 1 φ 2 determines the topological class of H. For φ = 0 and π, H has time-reversal and particle-hole symmetries but other values of φ, H has particle-hole symmetry only.(s. Ryu et. al. New J. Phys. 12, (2010)

24 Generating Many Floquet Majorana modes Driven Model Driving Protocol: Switch between the two Hamiltonians H 1 and H 2. In the first half period, H 1 = H(φ 1, φ 2 ) whereas in the second half period, swap φ 1 and φ 2 so that H 2 = H(φ 2, φ 1 ). φ = φ 1 φ 2 = ±π/2, so within each half period, the Hamiltonian breaks time-reversal symmetry In addition to a possible generation of long-range interactions for H eff, this driving protocol is designed to recover time-reversal symmetry.

25 Generating Many Floquet Majorana modes Driven Model

26 Generating Many Floquet Majorana modes Theoretical results U(T ) = e i H 2 T 2 e i H 1 T 2 e i H eff T H eff = H H 2 2 it 8 [H 2, H 1 ] U(T ) u = e iɛt u Define c k = l c le ikl / N and introduce the particle-hole representation C k = [c k, c k ]T, then H eff is given by : H eff = k BZ C k H eff(k)c k, with H eff (k) = E k n(k) σ n 1 (k) = 0 so n(k) is in yz plane

27 Generating Many Floquet Majorana modes Theoretical results Topological invariant Z can be obtained by the integer winding number W = π dθ k π 2π Z, where θ k = arctan[n 3 (k)/n 2 (k)] Geometrically, W means the number of times the vector n(k) rotates around the origin point as k varies from π to π The number of pairs of Majorana modes under open boundary condition is then given by W As some system parameters continuously change, gap closing and consequently topological phase transitions occur

28 Generating Many Floquet Majorana modes Theoretical results W = -3 n n3 1 Π k 0 Π n2 Π k 0 Π 1 0 W = 0 W = 1 n Π k 0 Π 1 0 (a) 1 n2 3 W = -2 n 1 1 Π k 0 Π n2 1 n2 t b t 1 Figure: (a) Winding of the unit vector n(k), in different parameter regimes under our driving protocol.t 1 = 1; t 2 = 5 for W = 3, t 2 = 3 for W = 2, t 2 = 0 for W = 0,and t 2 = 3 for W = 1. (b) Topological phase diagram of driven system plotted in the (t 1, t 2 ) plane. Other parameters are µ = 10, 2 = 2.5 and T = 0.2 Ref. : Q. Tong et. al. arxiv: v1

29 Generating Many Floquet Majorana modes Numerical results Floquet Majorana modes have two flavors: one at ɛ = 0 and the other at ɛ = ±π/t.the second flavor is certainly absent in an undriven system. energy( Ε T) Quasi- Π 0 a Π t 2 Eenergy 20 0 b Figure: Comparison between quasi-energy spectrum for the driven case (a) and energy spectrum for the static case(b); t 1 = 1, N=200 Ref. : Q. Tong et. al. arxiv: v1 t 2

30 Generating Many Floquet Majorana modes Generating Even More Majorana In efforts to generate even more Majorana modes, extend direct numerical studies to other parameter regimes. The BCH formula indicates that as T increases, the nested commutators will have heavier weights. An increasing T can then induce longer-range interactions in H eff. Numerically H eff expanded as a quadratic function of the operators (c 1,, c N, c 1,, c N )T

31 l l l l Floquet Topological Insulators and Majorana Modes Generating Many Floquet Majorana modes Generating Even More Majorana 1 (a) 1 1 (b) m (c) m (d) m m Figure: The expansion coefficients of H eff for (a) T = 0.2 and (b) T = 2.0 and of the static H for the real (c) and imaginary (d) parts Ref. : Q. Tong et. al. arxiv: v1

32 Generating Many Floquet Majorana modes Generating Even More Majorana For large values of T, winding number W is no longer well-defind and one is forced to count the formed pairs of Majorana modes. Table: Number of Majorana modes localized at each boundary for different values of T t T = T = T =

33 Conclusions Conclusions A trivial insulator can be driven into a topological phase of matter by applying a proper time-periodic perturbation A periodic driving has the capacity to restore time-reversal symmetry and to induce an effective long-range interaction With these two mechanisms working at once, the generation of many Majorana modes is achieved using a standard p-wave superconductor model under certain periodic modulation.

34 Conclusions References Floquet Topological Insulators by J. Cayssol et al., arxiv : Generating Many Majorana Modes via Periodic Driving by Q. Tong et al., arxiv : Floquet Topological Insulator in Semiconductor Quantum Well by N.H. Linder et al., Nature Physics

35 Conclusions Acknowledgement I gratefully acknowledge Diptiman Sen and Abhiram Soori for the discussions. Thank you