Quantum heterodyne Hall effect from oscillating magnetic fields - PowerPoint PPT Presentation

Quantum heterodyne Hall effect from oscillating magnetic fields Takashi Oka Max Planck institute PKS and CPfS (Dresden) “Nonequilibrium Quantum Matter” 1. Introduction 2. Classical heterodyne Hall effect 3. Quantum heterodyne Hall effect 4. Quantum Dirac heterodyne Hall effect TO, Bucciantini, PRB’16 TO, Kitamura, Nag, Saha, Bucciantini in prep

What is a heterodyne? Bluetooth Wifi, TV, radio

What is a heterodyne? 2.40xGHz Bluetooth Wifi, TV, radio superheterodyne device kHz

What is a heterodyne? 2.401GHz 2.402GHz f S f S1 -f LO f LO Local oscillator f S1 -f LO = periodically driven system from wikipedia “heterodyne”

Linear response theory w : signal frequency � dt ′ σ ab ( t, t ′ ) E b ( ω ) e − i ω t ′ j a ( t ) =

Linear response theory w : signal frequency � dt ′ σ ab ( t, t ′ ) E b ( ω ) e − i ω t ′ j a ( t ) = If static, σ ( t − t ′ ) = σ ab ( ω ) E b ( ω ) e − i ω t ′ ac-conductivity

Linear response theory for periodically driven system � dt ′ σ ab ( t, t ′ ) E b ( ω ) e − i ω t ′ j a ( t ) = � ab ( ω ) e − i ( ω + n Ω ) t E b ( ω ) σ n = n w : signal frequency W : drive frequency Periodic driving Ω input signal output signal n ω + l Ω w + n W ω Heterodyne (Floquet state) TO, Bucciantini, PRB’16

Heterodyne Hall effect TO, Bucciantini, PRB’16 σ n xy ( ω ) Application: Dissipationless frequency conversion Ultra-low power consuming Bluetooth!?

Heterodyne Hall effect TO, Bucciantini, PRB’16 σ n xy ( ω ) In the following, I will focus on the resonant case w = W

Example 1: Classical particle in an oscillating B field TO, Bucciantini, PRB’16 time dependent magnetic field B ( t ) = B cos Ω t z Newton’s equation � d � � E ( t ) + 1 � m dt + η v ( t ) = e c v × B ( t )

Example 1: Classical particle in an oscillating B field TO, Bucciantini, PRB’16 B ( t ) = B cos Ω t no E-field w c / W =3.0 5.0 6.0 w c = qB/m e c cyclotron frequency

� dt ′ σ ab ( t, t ′ ) E b ( ω ) e − i ω t ′ j a ( t ) = � σ n ab ( ω ) e − i ( ω + n Ω ) t E b ( ω ) = n dc-conductivity with static E y -field ( t =0.05, E y =1 ) w c / W =3 5 6

Periodic orbits ( t =0.0, E y =0 ) w c / W =2.41 5.52 8.66 (i) (ii) (iii) a =1 a =2 a =3 winding per half cycle

Heterodyning Hall current TO, Bucciantini, PRB’16 w c / W =3 5 6 exact results σ n,m = σ n − m ( m Ω ) ab ab

Summary: Example I TO, Bucciantini, PRB’16 Classical particle in an oscillating B-field heterodyne Hall ``insulator” xy ( Ω ) ∼ 1 σ 0 σ − 1 xx (0) = 0 B @ magic frequencies ((i), (ii), (iii),.. zeros of J 0 ( B / W ) ) � ab ( ω ) e − i ( ω + n Ω ) t E b ( ω ) σ n j a ( t ) = n

Example 2: Quantum particle in oscillating B field TO, Bucciantini, PRB’16 time dependent magnetic field B ( t ) = B cos Ω t 0 z Newton’s equation � d � � � E ( t ) + 1 m dt + η v ( t ) = e c v × B ( t ) Schrodinger equation 1 2 + ( p y − B 0 cos( Ω t ) x ) 2 � � H = p x ˆ 2 m = A y Solvable by Husimi transformation

Quantization: static B 1 2 + ( p y − Bx ) 2 � � H = p x ˆ 2 m = A y E n n =2 ϕ n ( x ) = e − x 2 / 2 l 2 l B = B − 1 / 2 B H n ( x/l B ) n =1 n =0 k y x X = p y level spacing w c = qB/m e c B momentum-position locking cyclotron frequency

How to realize ? A y = B cos( Ω t ) x ① incoming laser A. THz metamaterial enhancement factor of B ③ resonating electromagnet B(t) ② resonating current Y. Mukai, K. Tanaka, et al. (Kyoto grp.) New J. Phys.’16 1 Tesla, 1 THz!! cf) E -field enhancement (Liu, Nelson, Averitt, et al. Nature 12)

1 2 + ( p y − B 0 cos( Ω t ) x ) 2 � � H = p x ˆ 2 m = A y EM -fields B z = B cos( Ω t ) E y = B Ω sin( Ω t ) x

Quantization: time-oscillating B dotted line 1 2 + ( p y − B 0 cos( Ω t ) x ) 2 � � H = p x ˆ 2 m = Solvable by Husimi transformation A y B 0 =0.2, W =1, p y =5 | Φ α ( x, t ) | 2 wave function x Floquet theory Floquet state [ H − i ∂ t ] | Φ α ⟩ = ε α | Φ α ⟩ see A. Eckardt RMP’16

Quantization: time-oscillating B dotted line 1 2 + ( p y − B 0 cos( Ω t ) x ) 2 � � H = p x ˆ 2 m = A y B 0 =1.89, W =1, p y =5 | Φ α ( x, t ) | 2 periodic micromotion x Floquet theory Floquet state [ H − i ∂ t ] | Φ α ⟩ = ε α | Φ α ⟩ see A. Eckardt RMP’16

Floquet quasi-energy Spectrum ε n ( p y ) = E n + p 2 y 2 m ∗ e Floquet quasi-energy B ε n E n � ω e ff ( n + 1 / 2) Ω temporal mixture n -th Landau levels

Floquet quasi-energy Spectrum ε n ( p y ) = E n + p 2 y 2 m ∗ e inverse effective mass m ∗ → ∞ e E n flat band n =2 e m e / m * n =1 r =1.89 5.07 8.22 q q n =0 q r=r 1 r=r 3 r=r 2 k p y ω / Ω y c new Landau quantization

flat band e m e / m * r =1.89 5.07 8.22 q q q r=r 1 r=r 3 r=r 2 ω / Ω c r = w c / W r =5.07 r =8.22 =1.89 quantum x classical r= w c / W r= 8.66 r= 5.52 =2.40

Dissipationless Heterodyne Hall current B ( t ) = B cos Ω t B jdc jdc E E 1 x cos Ω t 1 E E =0 n x j dc y = σ 0 , 1 yx E 1 n =2 x LL filling yx = e 2 σ 0 , 1 where ν = N e /N Φ h Q ν , n =1 Integer heterodyne Hall effect n =0 pre-factor Q is non-universal k y

Example3: 2D Dirac electron in oscillating B field TO, Kitamura, Nag, Saha, Bucciantini, in prep time dependent magnetic field B ( t ) = B cos Ω t z graphene, surface of 3D TI, … Dirac equation H Dirac = σ x ˆ p x + σ y ( p y − B cos Ω tx ) = A y

Spectrum A ( k, ω ) = − 1 1 π ImTrˆ P static ω − ˆ H k + i δ k y W =0.6, B/a =0.000, E x =0.0 honeycomb, zigzag edge

Spectrum A ( k, ω ) = − 1 1 π ImTrˆ P static ω − ˆ H k + i δ k y W =0.6, B/a =0.0010, E x =0.0 honeycomb, zigzag edge

Spectrum A ( k, ω ) = − 1 1 π ImTrˆ P static ω − ˆ H k + i δ k y W =0.6, B/a =0.0020, E x =0.0 honeycomb, zigzag edge

Spectrum A ( k, ω ) = − 1 1 π ImTrˆ P static ω − ˆ H k + i δ k y W =0.6, B/a =0.0030, E x =0.0 honeycomb, zigzag edge

Spectrum A ( k, ω ) = − 1 1 π ImTrˆ P static ω − ˆ H k + i δ “ p -Landau levels” p -Flat bands at e = ± W /2 • W /2 • A series of bands around them • electron-hole resonant state W Dirac node -W /2 preserved W =0.6, B/a =0.0030, E x =0.0 zigzag

Effective Hamiltonian “ p -Landau levels” n=1 H Dirac = σ x ˆ p x + σ y ( p y − B cos Ω tx ) n=0 W /2 “ p -flat band” rotating frame transformation n=-1 W n=1 -W /2 n=0 “ p -flat band” n=-1 Landau levels of 2D Dirac system √ � Ω 2 − p 2 ε n = Bn ± Ω / 2 z two n =0 states two n =1 states | y | 2 localized @ center x x x x The flat band is protected by time-glide symmetry (Morimoto-Po-Vishwanath’17)

Heterodyne Hall effect time dependent magnetic field B ( t ) = B cos Ω t z E x = E 1 x cos Ω t additional ac-electric field

Heterodyne Hall effect (add B and E) k y W =0.6, B/a =0.0020, E x =0.20 honeycomb, zigzag edge

Heterodyne Hall effect (add B and E) “ p -Landau levels” p -Flat bands at e = ± W /2 tilts = p -chiral “center” mode W /2 → current in y-direction W cf) p -edge state: Rudner-Lindner-Berg-Levin ‘13 Dirac node -W /2 axial chiral magnetic effect-like band → current in (-y)-direction CME: 3D Weyl in E,B fields Fukushima-Kharzeev-Warringa’08 k y W =0.6, B/a =0.0020, E x =0.20 honeycomb, zigzag edge

Heterodyne Hall effect chiral center state axial CME state k y W =0.6, B/a =0.0020, E x =0.20 honeycomb, zigzag edge

Summary • Heterodyne Hall effect in three examples was studied • They are characterized by the heterodyne response functions σ n xy ( ω ) ongoing: Relation with topology, interaction (fractional state) classical particle quantum 2DEG quantum 2DDirac 1 E E =0 n x n =2 n =1 n =0 TO, Bucciantini, PRB’16 k y TO, Kitamura, Nag, Saha, Bucciantini in prep

Heterodyne Kubo formula J n i ( ω ) = σ n ij ( ω ) E j ( ω ) � [ ε k α − ε k β − m Ω ][( ε k α − ε k β ) + ( n − m ) Ω ] ij ( ω ) = 1 β i α A ( n − m ) � σ n A m f β α j β i ω ( ε k α − ε k β ) + ( n − m ) Ω − ω − i δ k, α , β ,m − [ ε k β − ε k α − m Ω ][( ε k β − ε k α ) + ( n − m ) Ω ] � α i β A ( n − m ) A m β j α ( ε k β − ε k α ) + ( n − m ) Ω − ω − i δ � e im Ω t A m ⟨ φ k β ( t ) | ∂ k i φ k α ( t ) ⟩ = β i α m

general theory 1/3 Floquet theory (non-perturbative in driving) review: A. Eckardt, RMP’16 time dependent problem eigenvalue problem H φ α = ε α φ α ψ ( t ) = e − i ε t φ ( t ) H = H ( t ) − i ∂ t e : Floquet quasi-energy φ ( t + T ) = φ ( t ) Floquet state Fourier transformation Floquet Hamiltonian � φ m e − im Ω t φ ( t ) = m ~ absorption of m “photons”