Floquet Topological Insulator:

Floquet Topological Insulator: Understanding Floquet topological insulator in semiconductor quantum wells by Lindner et al. Condensed Matter Journal Club Caltech February 12 2014

Motivation Motivation Helical edge states require a band inversion (saw this in previous papers) *References: (RRL) 7, no. 1-2 (2013): 101-108

Motivation Motivation Helical edge states require a band inversion (saw this in previous papers) This requires careful band structure engineering *References: (RRL) 7, no. 1-2 (2013): 101-108

Motivation Motivation Helical edge states require a band inversion (saw this in previous papers) This requires careful band structure engineering The topological property of the material is permanent; cannot widen well or change strain (?) *References: (RRL) 7, no. 1-2 (2013): 101-108

Time periodic perturbations Motivation They can induce non-equilibrium protected edge states!

Time periodic perturbations Motivation They can induce non-equilibrium protected edge states! Controlling low-frequency electromagnetic modes is a mature technology.

Time periodic perturbations Motivation They can induce non-equilibrium protected edge states! Controlling low-frequency electromagnetic modes is a mature technology. Could allow fast switching of edge state transport and control the spectral properties (velocity) of the edge states.

Floquet Theory Back to Chern numbers Spatial Periodicity in Hamiltonians: A familiar case Consider a system that has some discrete translation symmetry by some lattice vectors R. Then, an eigenstate, ψ, of the Hamiltonian of that system, Ĥ, is of the following form: ψ k = e ik.ˆr u (1) where u = u (r) r dr is an arbitrary periodic wavefunction and k is an arbitrary wavevector called the Bloch wavevector. In addition, u is an eigenstate of the effective Bloch Hamiltonian

Effective Bloch Hamiltonian Floquet Theory Back to Chern numbers Let us substitute a Bloch wavefunction ψ k = e ik.ˆr u into the Schrodinger s equation He ik.ˆr = e ik.ˆr ( 1 ) 2 ) (ˆP k + V (ˆR ) = Ee ik.ˆr u (2) 2m and hence Ĥ bloch = 1 ) 2 (ˆR) (ˆP k + V 2m is the effective Bloch Hamiltonian for u (3)

Some Properties Floquet Theory Back to Chern numbers Let K be a reciprocal lattice vector. Hence, ψ k+k is the same eigenvector as ψ k and k is defined modulo K.

Some Properties Floquet Theory Back to Chern numbers Let K be a reciprocal lattice vector. Hence, ψ k+k is the same eigenvector as ψ k and k is defined modulo K. Mean velocity is given by where n is a band index (obtained by solving Ĥ bloch (k) u n = E n (k) u n ) v n (k) = 1 ke n (k) (4)

Floquet theorem Floquet Theory Back to Chern numbers Consider the differential equation i d dx ψ (x) = H (x) ψ (x) = ˆP ψ (5) then according to the Floquet theorem we have that the unitary operator is given by U (x) = B (x) e ih B x (6) where H B is the effective Bloch Hamiltonian and B (x) is a periodic function in R

Floquet Theory Back to Chern numbers From spatial periodicity to time periodicity Consider a time periodic Hamiltonian H (t). Its evolution is governed by: i t ψ = H (t) ψ (7) I will consider i t to be an operator, ˆω, that is the analogue as ˆP (and so t would be the analog of ˆR) Then ˆω ˆω ɛ (in analogy with ˆP ˆP k), where ɛ is called a quasi energy. Hence, Schrodinger s equation becomes (for the floquet states e iɛt φ ):(i t ɛ) φ = H (t) φ which can be rewritten as (H (t) i t ) φ = ɛ φ (8)

Some Properties Floquet Theory Back to Chern numbers H F (t) H (t) i t ; H F (t) φ = ɛ φ (10) In analogy to Bloch wavevectors being defined modulo the reciprocal lattice vectors K, the quasi-energies are defined modulo the frequency ω = 2π/T (where T is the periodicity of the Hamiltonian) The unitary evolution operator for ψ (t) = e iɛt is given by S k (t) = P b (t) exp [ ih F (k) t] ; P k (t) = P (t + T ) (11)

Definitions Floquet Theory Back to Chern numbers Berry curvature: ( ) F n k = unit cell where u n, k is a Bloch wave at band n. [ k u n, k ( r)] k u n, k ( r) d r (12)

Definitions Floquet Theory Back to Chern numbers Berry curvature: ( ) F n k = unit cell where u n, k is a Bloch wave at band n. Chern number: [ k u n, k ( r)] k u n, k ( r) d r (12) C n = 1 2π ( ) F n k d k (13) all BZ (a topological index is defined for each band) Useless for systems with time reversal symmetry (and other symmetries)

Properties of Chern number Floquet Theory Back to Chern numbers Integer valued Same as the number of chiral edge states one can have For a quantum hall system, it can be shown that the Hall conductivity is given by σ xy = Ce 2 / where C = occupied bands n C n. Why? F n = k A n where A n = i u n,k k u n,k is called the Berry connection and can be thought of as a gauge field that arises from the local symmetry: u n,k e iφn(k) u n,k. Hence, F n is like a B (has physical meaning); can think of them interchangeably. If.B 0 then we must have that q m = c n/2q e (main idea of derivation: singularities in A mean non-zero integrals).

2d quantum well Model institution-logo-filena Inspired from HgTe/CdTe systems with time reversal symmetry, we will work with 4-band time-reversal invariant insulators, which have Hamiltonians of the form Ĥ = k H (k) c k c k where ( ) H (k) H (k) = H ; H (k) = ɛ (k) I + d (k).σ (14) ( k)

2d quantum well Model Inspired from HgTe/CdTe systems with time reversal symmetry, we will work with 4-band time-reversal invariant insulators, which have Hamiltonians of the form Ĥ = k H (k) c k c k where ( ) H (k) H (k) = H ; H (k) = ɛ (k) I + d (k).σ (14) ( k) We can add a time dependent perturbation V (t) = V.σ cos (ωt) (15) We will work with the following tight binding model d (k) = (A sin k x, A sin k y, M 4B + 2B [cos k x + cos k y ]) (16) where A < 0, B > 0 and M depend on the thickness of the quantum well and on parameters of the materials. institution-logo-filena

Some properties of the free Hamiltonian H institution-logo-filena H (k) = ɛ (k) I + d (k).σ; (17) d (k) = (A sin k x, A sin k y, M 4B + 2B [cos k x + cos k y ]) (18) Its eigenvalues and eigenvectors are given by ɛ ± (k) = ɛ (k) ± d (k) ; u ± (k) = 1 N ± ( dz (k) ± d (k) d x (k) + id y (k) ) (19)

H s Chern number The Chern number for the the + (higher energy) and bands can be found by C ± = ± 1 [ ] d 2 kˆd (k). kx ˆd (k) ky ˆd (k) (20) 4π where ˆd (k) d (k) / d (k) (found by using that σ xy = Ce 2 / and then expressing σ xy with help of Kubo formula and Matsubara green function). Interpretation: Considering ˆd (k): T 2 S 2 as a mapping [ from the Brillouin] zone to the unit sphere, the integrand ˆd (k). kx ˆd (k) ky ˆd (k) is simply the Jacobian of this mapping. Thus the integration over it gives the total area of the image of the Brillouin zone on S 2, which is a topological winding number with quantized value 4πn, n Z / it counts the number of times the vector ˆd (k) wraps around the unit sphere as k wraps around the entire FBZ. institution-logo-filena

H s Chern Number institution-logo-filena H (k) = ɛ (k) I + d (k).σ; (21) d (k) = (A sin k x, A sin k y, M 4B + 2B [cos k x + cos k y ]) (22) The Chern Number is given by [ ( )] M C ± = ± 1 + sign /2 (23) B

Rotating Wave Approximation (RWA) The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization H (k) = ɛ (k) I + d (k).σ; (24) From projection [ operators ] onto eigenstates of H: P ± (k) = 1 2 I ± ˆd (k).σ, form following unitary operator: U (k, t) = P + (k) + P (k) e iωt then H I (t) = P + (k) ɛ + (k) + P (k) (ɛ (k) + ω) + U (t) V (t) U (t) (25)

Extra level of sophistication The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization Assuming that the perturbation is strongly resonant (i.e. detuning = (ɛ + ɛ ) ω (ɛ + ɛ ) + ω) V RWA U VU P + (k) V.σ 2 P (k) + P (k) V.σ 2 P + (k) (26)

Extra level of sophistication The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization Near γ, ˆn k has the opposite direction as ˆd (k) while away from it, perturbation is weak and ˆn k ˆd (k) so there is an extra winding of the sphere and we expect something like C F ± = C ± ± 1. (note we ignored the time dependence of ˆn k because it evolves unitarily/smoothly from the initial time; we expect smooth deformation to leave topological quantities invariant)

A realistic setup The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization To study edge states, we must have an edge. The simplest system with an edge is an infinitely long strip There is still translational symmetry along x but not along y. Hence, k x is still a good quantum number but k y is NOT.

Solving Floquet Schrodinger equation The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization (H (k, t) + V (k, t) i t ) φ (k, t) =ɛ (k) φ (k, t) (28) Move to Fourier space: φ (k, t) = e inωt φ (k, n) (29) n=

Solving Floquet Schrodinger equation The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization Can write this in matrix form as... V.σ 2 0 (... e iωt 1 e iωt... ) V.σ 2 H 0 ω V.σ 2 0 V.σ V.σ 0 2 H 0 2 0 V.σ 0 2 H 0 + ω V.σ 2 V.σ. 0.. 2 (... e iωt 1 e iωt... ) ɛ (k). φ (k, φ (k φ (k.. φ (k, φ (k φ (k.

Solving Floquet Schrodinger equation Diagonalize to obtain The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization (... e iωt 1 e iωt... ) U diag (..., λ 1, λ 0, λ 1,...) U then the U (... e iωt 1 e iωt... ) U ɛ (k) U. φ (k, 1) φ (k, 0). φ (k, 1) φ (k, 0) φ (k, 1).. φ (k, 1) φ (k, 0) φ (k, 1) are the eigenvectors from which you can. = institution-logo-filena

Spectrum The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

subtle points The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization The quasi-energies are periodic in ω so can have 2 crossings. The Floquet Hamiltonian, however, does not contain all information about the topological properties of our system, for example it cannot reveal the Z Z or Z 2 Z 2 topological invariants Edge states robust under perturbations that break T HT 1 = H ( t + T ) (31)

Magnetic field realization The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization A microwave-thz oscillating magnetic field polarized in the ẑ/growth direction. ( ) H (k) Our original Hamiltonian H is written in a basis ( k) with states with m J = ± (1/2, 3/2) (for Hg/CdTe) for lower and upper block respectively. g 20, µ B / 10 5 /10 15 10 10 s 1.T 1. 0.1 K is like a frequency of 10 10 Hz so need field strength on the order of 1/20 T or 50 mt.

Electric field realization The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization In-plane electric field: E = Re ( Ee iωt) i k (32) Works for circularly polarized light: E = E ( iˆx ŷ). In that case V (k) = A ( A 2 4BM ) E M 3 winds twice around the equator. [ ] 1 ( k 2 2 x ky 2 ) ˆx + kx k y ŷ For HgTe of thickness 58 angstroms, would need electric fields on the order of 10 V/m (accessible with powers < 1mW) (33)