Floquet Topological Insulator: UnderstandingFloquet topological - - PowerPoint PPT Presentation

Introduction Building up our toolbox The Model Analysis

Floquet Topological Insulator:

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

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Motivation

Motivation

Helical edge states require a band inversion (saw this in previous papers)

*References: (RRL) 7, no. 1-2 (2013): 101-108 Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis 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 Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis 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 Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis 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 (?)

... What about time periodic perturbations?

*References: (RRL) 7, no. 1-2 (2013): 101-108 Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Motivation

Time periodic perturbations

They can induce non-equilibrium protected edge states!

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Motivation

Time periodic perturbations

They can induce non-equilibrium protected edge states!

Controlling low-frequency electromagnetic modes is a mature technology.

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Motivation

Time periodic perturbations

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 TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Motivation

Time periodic perturbations

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.

We will treat these perturbations classically, and to solve for the dynamics, we will need Floquet theory.

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis 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, ˆ H, is of the following form: |ψk = e−ik.ˆ

R |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

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Effective Bloch Hamiltonian

Let us substitute a Bloch wavefunction |ψk = e−ik.ˆ

R |u into the

Schrodinger’s equation He−ik.ˆ

R

= e−ik.ˆ

R

1 2m

• ˆ

P − k 2 + V

• ˆ

R

• = Ee−ik.ˆ

R |u

(2) and hence ˆ Hbloch = 1 2m

• ˆ

P − k 2 + V

• ˆ

R

• (3)

is the effective Bloch Hamiltonian for |u

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Some Properties

Let K be a reciprocal lattice vector. Hence, |ψk+K is the same eigenvector as |ψk and k is defined modulo K.

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Some Properties

Let K be a reciprocal lattice vector. Hence, |ψk+K is the same eigenvector as |ψk and k is defined modulo K. Mean velocity is given by vn (k) = 1 ∇kEn (k) (4) where n is a band index (obtained by solving ˆ Hbloch (k) |un = En (k) |un)

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Some Properties

Let K be a reciprocal lattice vector. Hence, |ψk+K is the same eigenvector as |ψk and k is defined modulo K. Mean velocity is given by vn (k) = 1 ∇kEn (k) (4) where n is a band index (obtained by solving ˆ Hbloch (k) |un = En (k) |un) Weak periodic potentials break degeneracies.

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Some Properties

Let K be a reciprocal lattice vector. Hence, |ψk+K is the same eigenvector as |ψk and k is defined modulo K. Mean velocity is given by vn (k) = 1 ∇kEn (k) (4) where n is a band index (obtained by solving ˆ Hbloch (k) |un = En (k) |un) Weak periodic potentials break degeneracies. Plus one more very important property ...

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Floquet theorem

Consider the differential equation −i d dx ψ (x) = H (x) ψ (x) = ˆ P |ψ (5) then according to the Floquet theorem we have that the unitary operator is given by U (x) = B (x) e−iHBx (6) where HB is the effective Bloch Hamiltonian and B (x) is a periodic function in R

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis 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, ˆ ω, that is the analogue as ˆ P (and so t would be the analog of ˆ R) Then ˆ ω → ˆ ω − ǫ (in analogy with ˆ P → ˆ P − k), where ǫ is called a quasi energy. Hence, Schrodinger’s equation becomes (for the floquet states eiǫt |φ):(i∂t − ǫ) |φ = H (t) |φ which can be rewritten as (H (t) − i∂t) |φ = ǫ |φ (8)

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis 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, ˆ ω, that is the analogue as ˆ P (and so t would be the analog of ˆ R) Then ˆ ω → ˆ ω − ǫ (in analogy with ˆ P → ˆ P − k), where ǫ is called a quasi energy. Hence, Schrodinger’s equation becomes (for the floquet states eiǫt |φ):(i∂t − ǫ) |φ = H (t) |φ which can be rewritten as (H (t) − i∂t) |φ = ǫ |φ (8) In analogy to the effective Bloch Hamiltonian, we have the effective Floquet Hamiltonian/operator: HF (t) ≡ H (t) − i∂t (9)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Some Properties

HF (t) ≡ H (t) − i∂t; HF (t) |φ = ǫ |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)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Some Properties

HF (t) ≡ H (t) − i∂t; HF (t) |φ = ǫ |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−iǫt is given by Sk (t) = Pb (t) exp [−iHF (k) t] ; Pk (t) = P (t + T) (11)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Definitions

Berry curvature: Fn

• k
• =
• unit cell
• ∇

kun, k (

r) ∗ × ∇

kun, k (

r) d r (12) where un,

k is a Bloch wave at band n.

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Definitions

Berry curvature: Fn

• k
• =
• unit cell
• ∇

kun, k (

r) ∗ × ∇

kun, k (

r) d r (12) where un,

k is a Bloch wave at band n.

Chern number: Cn = 1 2π

• all BZ

Fn

• k
• d

k (13) (a topological index is defined for each band)

Useless for systems with time reversal symmetry (and other symmetries)

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Properties of Chern number

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 σxy = Ce2/ where C = occupied bands

n

Cn.

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis Floquet Theory Back to Chern numbers

Properties of Chern number

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 σxy = Ce2/ where C = occupied bands

n

Cn.

Why? Fn = ∇k × An where An = −i

• un,k|∇

k|un,k

• is called the

Berry connection and can be thought of as a gauge field that arises from the local symmetry: |un,k → eiφn(k) |un,k. Hence, Fn is like a ’B’ (has physical meaning); can think of them interchangeably. If ∇.B = 0 then we must have that qm = cn/2qe (main idea of derivation: singularities in A mean non-zero integrals).

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis

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 ˆ H =

k H (k) c† kck where

H (k) =

• H (k)

H∗ (−k)

• ; H (k) = ǫ (k) I + d (k) .σ

(14)

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis

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 ˆ H =

k H (k) c† kck where

H (k) =

• H (k)

H∗ (−k)

• ; H (k) = ǫ (k) I + d (k) .σ

(14) We can add a time dependent perturbation V (t) = V.σ cos (ωt) (15)

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis

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 ˆ H =

k H (k) c† kck where

H (k) =

• H (k)

H∗ (−k)

• ; H (k) = ǫ (k) I + d (k) .σ

(14) 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 kx, A sin ky, M − 4B + 2B [cos kx + cos ky]) (16) where A < 0, B > 0 and M depend on the thickness of the quantum well and on parameters of the materials.

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis

Some properties of the free Hamiltonian H

H (k) = ǫ (k) I + d (k) .σ; (17) d (k) = (A sin kx, A sin ky, M − 4B + 2B [cos kx + cos ky]) (18) Its eigenvalues and eigenvectors are given by ǫ± (k) = ǫ (k) ± |d (k)| ; u± (k) = 1 N± dz (k) ± |d (k)| dx (k) + idy (k)

• (19)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis

H’s Chern number

The Chern number for the the + (higher energy) and − bands can be found by C± = ± 1 4π

• d2kˆ

d (k) .

• ∂kx ˆ

d (k) × ∂ky ˆ d (k)

• (20)

where ˆ d (k) ≡ d (k) / |d (k)| (found by using that σxy = Ce2/ and then expressing σxy with help of Kubo formula and Matsubara green function). Interpretation: “Considering ˆ d (k): T 2 → S2 as a mapping from the Brillouin zone to the unit sphere, the integrand ˆ d (k) .

• ∂kx ˆ

d (k) × ∂ky ˆ d (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 S2 , which is a topological winding number with quantized value 4πn, n ∈ Z” /“it counts the number of times the vector ˆ d (k) wraps around the unit sphere as k wraps around the entire FBZ” .

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis

H’s Chern Number

H (k) = ǫ (k) I + d (k) .σ; (21) d (k) = (A sin kx, A sin ky, M − 4B + 2B [cos kx + cos ky]) (22) The Chern Number is given by C± = ±

• 1 + sign

M B

• /2

(23)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Rotating Wave Approximation (RWA)

H (k) = ǫ (k) I + d (k) .σ; (24) From projection operators onto eigenstates of H: P± (k) = 1

2

• I ± ˆ

d (k) .σ

• , form following unitary operator:

U (k, t) = P+ (k) + P− (k) eiωt then HI (t) = P+ (k) ǫ+ (k) + P− (k) (ǫ− (k) + ω) + U† (t) V (t) U (t) (25)

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Rotating Wave Approximation (RWA)

H (k) = ǫ (k) I + d (k) .σ; (24) From projection operators onto eigenstates of H: P± (k) = 1

2

• I ± ˆ

d (k) .σ

• , form following unitary operator:

U (k, t) = P+ (k) + P− (k) eiωt then HI (t) = P+ (k) ǫ+ (k) + P− (k) (ǫ− (k) + ω) + U† (t) V (t) U (t) (25) Diagonalize HI (t) to obtain

• ψ±

I (k, t)

• and the pseudospin vector

ˆ nk (t) = ψ− (k, t) |ˆ σ|ψ− (k, t) which is like expectation value of spin in the negative band

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Extra level of sophistication

Assuming that the perturbation is strongly resonant (i.e. detuning ∆ = |(ǫ+ − ǫ−) − ω| ≪ (ǫ+ − ǫ−) + ω) VRWA ≡ U†VU ≈ P+ (k) V.σ 2 P− (k) + P− (k) V.σ 2 P+ (k) (26)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Extra level of sophistication

Assuming that the perturbation is strongly resonant (i.e. detuning ∆ = |(ǫ+ − ǫ−) − ω| ≪ (ǫ+ − ǫ−) + ω) VRWA ≡ U†VU ≈ P+ (k) V.σ 2 P− (k) + P− (k) V.σ 2 P+ (k) (26) Decompose V into component ⊥ to d then VRWA = V⊥ (k) .σ (27)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Extra level of sophistication

Assuming that the perturbation is strongly resonant (i.e. detuning ∆ = |(ǫ+ − ǫ−) − ω| ≪ (ǫ+ − ǫ−) + ω) VRWA ≡ U†VU ≈ P+ (k) V.σ 2 P− (k) + P− (k) V.σ 2 P+ (k) (26) Decompose V into component ⊥ to d then VRWA = V⊥ (k) .σ (27) Then on γ, ˆ nk = −V⊥ (k) / |V⊥ (k)|

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Extra level of sophistication

Near γ, ˆ nk has the opposite direction as ˆ d (k) while away from it, perturbation is weak and ˆ nk ≈ ˆ d (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 ˆ nk because it evolves unitarily/smoothly from the initial time; we expect smooth deformation to leave topological quantities invariant)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

A realistic setup

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, kx is still a good quantum number but ky is NOT.

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Solving ’Floquet Schrodinger equation’

(H (k, t) + V (k, t) − i∂t) |φ (k, t) =ǫ (k) |φ (k, t) (28) Move to Fourier space: |φ (k, t) =

∞

• n=−∞

einωt |φ (k, n) (29)

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Solving ’Floquet Schrodinger equation’

(H (k, t) + V (k, t) − i∂t) |φ (k, t) =ǫ (k) |φ (k, t) (28) Move to Fourier space: |φ (k, t) =

∞

• n=−∞

einωt |φ (k, n) (29) Then

∞

• n=−∞

einωt

• (H0 + nω) |φ (k, n) + V.σ

2 |φ (k, n + 1) + V.σ 2 |φ (k, n − 1)

• (30)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Solving ’Floquet Schrodinger equation’

Can write this in matrix form as

• ...

e−iωt 1 eiωt ...

• 

       ...

V.σ 2 V.σ 2

H0 − ω

V.σ 2 V.σ 2

H0

V.σ 2 V.σ 2

H0 + ω

V.σ 2 V.σ 2

...                 . . . |φ (k, |φ (k, |φ (k, . . .

• ...

e−iωt 1 eiωt ...

• ǫ (k)

        . . . |φ (k, |φ (k, |φ (k, . . .

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Solving ’Floquet Schrodinger equation’

Diagonalize to obtain

• ...

e−iωt 1 eiωt ...

• U†diag (..., λ−1, λ0, λ1, ...) U

        . . . |φ (k, −1) |φ (k, 0) |φ (k, 1) . . .         =

• ...

e−iωt 1 eiωt ...

• U†ǫ (k) U

        . . . |φ (k, −1) |φ (k, 0) |φ (k, 1) . . .         then the U        . . . |φ (k, −1) |φ (k, 0) |φ (k, 1)        are the eigenvectors from which you can

Floquet TIs Journal club

institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Spectrum

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

subtle points

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 Z2 × Z2 topological invariants” Edge states robust under perturbations that break T HT −1 = H (−t + T) (31)

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Magnetic field realization

A microwave-THz oscillating magnetic field polarized in the ˆ z/growth direction. Our original Hamiltonian H (k) H∗ (−k)

• is written in a basis

with states with mJ = ± (1/2, 3/2) (for Hg/CdTe) for lower and upper block respectively. g ≈ 20, µB/ ≈ 10−5/10−15 ≈ 1010 s−1.T −1. 0.1 K is like a frequency of 1010 Hz so need field strength on the order of 1/20 T

• r 50 mT.

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institution-logo-filenam Introduction Building up our toolbox The Model Analysis The perturbation induces a topological phase (Heuristic argument) The perturbation induces a topological phase (Exact argument) Experimental Realization

Electric field realization

In-plane electric field: E = Re

• Eeiωt

i∇k (32) Works for circularly polarized light: E = E (−iˆ x − ˆ y). In that case V⊥ (k) = A

• A2 − 4BM
• E

M3 1 2

• k2

x − k2 y

• ˆ

x + kxky ˆ y

• (33)

winds twice around the equator. For HgTe of thickness 58 angstroms, would need electric fields on the order of 10 V/m (accessible with powers < 1mW)

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