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

Quantum heterodyne Hall effect from

• scillating 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?

Bluetooth Wifi, TV, radio

2.40xGHz

superheterodyne device

kHz

What is a heterodyne? fLO fS1-fLO

fS1-fLO

fS

from wikipedia “heterodyne” Local oscillator = periodically driven system

2.401GHz 2.402GHz

Linear response theory ja(t) =

• dt′σab(t, t′)Eb(ω)e−iωt′

w: signal frequency

Linear response theory ja(t) =

• dt′σab(t, t′)Eb(ω)e−iωt′

w: signal frequency If static, σ(t − t′)

= σab(ω)Eb(ω)e−iωt′

ac-conductivity

Linear response theory for periodically driven system

ja(t) =

• dt′σab(t, t′)Eb(ω)e−iωt′

Ω

Periodic driving input signal

Heterodyne (Floquet state)

• utput signal

ω ω + l Ω

=

• n

σn

ab(ω)e−i(ω+nΩ)tEb(ω)

w: signal frequency W: drive frequency

TO, Bucciantini, PRB’16

n

w + n W

Heterodyne Hall effect

TO, Bucciantini, PRB’16

Application: Dissipationless frequency conversion Ultra-low power consuming Bluetooth!?

σn

xy(ω)

Heterodyne Hall effect

TO, Bucciantini, PRB’16

In the following, I will focus on the resonant case

w=W

σn

xy(ω)

B(t) = B cos Ωt

time dependent magnetic field

Example 1: Classical particle in an oscillating B field m d dt + η

• v(t) = e
• E(t) + 1

c v × B(t)

• Newton’s equation

z

TO, Bucciantini, PRB’16

wc/W=3.0 6.0 5.0

B(t) = B cos Ωt

wc=qB/mec

cyclotron frequency

TO, Bucciantini, PRB’16

no E-field

Example 1: Classical particle in an oscillating B field

wc/W=3 6 5

(t=0.05, Ey=1) dc-conductivity with static Ey-field

ja(t) =

• dt′σab(t, t′)Eb(ω)e−iωt′

=

• n

σn

ab(ω)e−i(ω+nΩ)tEb(ω)

Periodic orbits

wc/W=2.41 8.66 5.52

(t=0.0, Ey=0)

(i) (iii) (ii) winding per half cycle a=1 a=2 a=3

Heterodyning Hall current

wc/W =3 6 5

TO, Bucciantini, PRB’16

exact results

σn,m

ab

= σn−m

ab

(mΩ)

Summary: Example I

TO, Bucciantini, PRB’16 ja(t) =

• n

σn

ab(ω)e−i(ω+nΩ)tEb(ω)

Classical particle in an oscillating B-field @ magic frequencies ((i), (ii), (iii),.. zeros of J0(B/W))

σ0

xx(0) = 0

σ−1

xy (Ω) ∼ 1

B

heterodyne Hall ``insulator”

B(t) = B cos Ωt

time dependent magnetic field

Example 2: Quantum particle in oscillating B field

m d dt + η

• v(t) = e
• E(t) + 1

c v × B(t)

• Newton’s equation

z

TO, Bucciantini, PRB’16

Schrodinger equation

H = 1 2m

• ˆ

px

2 + (py − B0 cos(Ωt)x)2

Ay

=

Solvable by Husimi transformation

Quantization: static B

ϕn(x) = e−x2/2l2

BHn(x/lB)

lB = B−1/2

H = 1 2m

• ˆ

px

2 + (py − Bx)2 x

momentum-position locking

X = py B

Ay

=

wc=qB/mec

level spacing

y

k

E

n=0 n=2 n=1 n

cyclotron frequency

• A. THz metamaterial
• Y. Mukai, K. Tanaka, et al. (Kyoto grp.) New J. Phys.’16

1 Tesla, 1 THz!!

enhancement factor of B

cf) E-field enhancement (Liu, Nelson, Averitt, et al. Nature 12)

How to realize ?

Ay = B cos(Ωt)x

① incoming laser ② resonating current

B(t)

③ resonating electromagnet

H = 1 2m

• ˆ

px

2 + (py − B0 cos(Ωt)x)2

Ay

=

Bz = B cos(Ωt) Ey = BΩ sin(Ωt)x

EM-fields

Quantization: time-oscillating B

H = 1 2m

• ˆ

px

2 + (py − B0 cos(Ωt)x)2

B0=0.2, W=1, py=5

|Φα(x, t)|2

[H − i∂t] |Φα⟩ = εα|Φα⟩

Floquet state

x

Ay

= dotted line wave function

Floquet theory see A. Eckardt RMP’16 Solvable by Husimi transformation

Quantization: time-oscillating B

H = 1 2m

• ˆ

px

2 + (py − B0 cos(Ωt)x)2 [H − i∂t] |Φα⟩ = εα|Φα⟩

B0=1.89, W=1, py=5

Floquet state

|Φα(x, t)|2

x

Ay

=

periodic micromotion

dotted line

Floquet theory see A. Eckardt RMP’16

εn(py) = En + p2

y

2m∗

e

Floquet quasi-energy Spectrum

Floquet quasi-energy B

εn Ω

ωeff(n + 1/2)

En

temporal mixture n-th Landau levels

εn(py) = En + p2

y

2m∗

e

Floquet quasi-energy Spectrum

new Landau quantization

inverse effective mass

me/m*

e

c

ω /Ω

r=r1

q

r=r2

q

r=r3

q

r=1.89 5.07 8.22

flat band y

k

E

n=0 n=2 n=1 n

m∗ → ∞

py e

me/m*

e

c

ω /Ω

r=r1

q

r=r2

q

r=r3

q

quantum r=1.89 5.07 8.22 flat band classical r=wc/W =1.89 r=5.07 r=8.22 r=wc/W =2.40 r=8.66 r=5.52 x

B(t) = B cos Ωt B E jdc

Dissipationless Heterodyne Hall current

jdc

y

k

E

n=0 n=2 n=1

E =0

x 1

n

σ0,1

yx = e2

h Qν, where ν = Ne/NΦ

LL filling

Integer heterodyne Hall effect pre-factor Q is non-universal

jdc

y = σ0,1 yx E1 x E1

x cos Ωt

B(t) = B cos Ωt

time dependent magnetic field

Example3: 2D Dirac electron in oscillating B field

z

TO, Kitamura, Nag, Saha, Bucciantini, in prep

Dirac equation

Ay

= graphene, surface of 3D TI, …

HDirac = σxˆ px + σy(py − B cos Ωtx)

honeycomb, zigzag edge W=0.6, B/a=0.000, Ex=0.0

Spectrum ky

A(k, ω) = −1 π ImTrˆ Pstatic 1 ω − ˆ Hk + iδ

honeycomb, zigzag edge W=0.6, B/a=0.0010, Ex=0.0

Spectrum ky

A(k, ω) = −1 π ImTrˆ Pstatic 1 ω − ˆ Hk + iδ

honeycomb, zigzag edge W=0.6, B/a=0.0020, Ex=0.0

Spectrum ky

A(k, ω) = −1 π ImTrˆ Pstatic 1 ω − ˆ Hk + iδ

honeycomb, zigzag edge W=0.6, B/a=0.0030, Ex=0.0

Spectrum ky

A(k, ω) = −1 π ImTrˆ Pstatic 1 ω − ˆ Hk + iδ

zigzag W=0.6, B/a=0.0030, Ex=0.0

Spectrum W/2

• W/2

W

• p-Flat bands at e=±W/2
• A series of bands around them
• electron-hole resonant state

preserved

A(k, ω) = −1 π ImTrˆ Pstatic 1 ω − ˆ Hk + iδ

“p-Landau levels”

Dirac node

W/2

• W/2

W Effective Hamiltonian “p-Landau levels”

HDirac = σxˆ px + σy(py − B cos Ωtx)

rotating frame transformation Landau levels of 2D Dirac system n=0 n=1 n=-1

“p-flat band”

n=0 n=1 n=-1

“p-flat band”

x

two n=0 states two n=1 states |y|2

x x x

εn =

• Ω2 − p2

z

√ Bn ± Ω/2 The flat band is protected by time-glide symmetry (Morimoto-Po-Vishwanath’17)

localized @ center

B(t) = B cos Ωt

time dependent magnetic field

z

Ex = E1

x cos Ωt

Heterodyne Hall effect additional ac-electric field

Heterodyne Hall effect (add B and E)

honeycomb, zigzag edge W=0.6, B/a=0.0020, Ex=0.20

ky

honeycomb, zigzag edge W=0.6, B/a=0.0020, Ex=0.20

ky W/2

• W/2

W

p-Flat bands at e=±W/2 tilts = p-chiral “center” mode → current in y-direction axial chiral magnetic effect-like band → current in (-y)-direction

“p-Landau levels”

Dirac node

CME: 3D Weyl in E,B fields

Fukushima-Kharzeev-Warringa’08

Heterodyne Hall effect (add B and E)

cf) p-edge state: Rudner-Lindner-Berg-Levin ‘13

Heterodyne Hall effect

honeycomb, zigzag edge W=0.6, B/a=0.0020, Ex=0.20

ky

chiral center state axial CME state

Summary

TO, Bucciantini, PRB’16 TO, Kitamura, Nag, Saha, Bucciantini in prep

σn

xy(ω)

y

k

E

n=0 n=2 n=1

E =0

x 1

n

• Heterodyne Hall effect in three examples was studied
• They are characterized by the heterodyne response functions

classical particle quantum 2DEG quantum 2DDirac

• ngoing: Relation with topology, interaction (fractional state)

Heterodyne Kubo formula

Jn

i (ω) = σn ij(ω)Ej(ω)

σn

ij(ω) = 1

iω

• k,α,β,m

fβ [εkα − εkβ − mΩ][(εkα − εkβ) + (n − m)Ω] (εkα − εkβ) + (n − m)Ω − ω − iδ Am

βiαA(n−m) αjβ

−[εkβ − εkα − mΩ][(εkβ − εkα) + (n − m)Ω] (εkβ − εkα) + (n − m)Ω − ω − iδ Am

αiβA(n−m) βjα

• ⟨φkβ(t)|∂kiφkα(t)⟩ =
• m

eimΩtAm

βiα

time dependent problem

Floquet theory (non-perturbative in driving)

review: A. Eckardt, RMP’16

eigenvalue problem

ψ(t) = e−iεtφ(t)

φ(t + T) = φ(t)

Floquet state

Hφα = εαφα

H = H(t) − i∂t

~ absorption of m “photons”

e: Floquet quasi-energy

Floquet Hamiltonian

Fourier transformation

φ(t) =

• m

φme−imΩt

general theory 1/3

How to construct the effective Hamiltonian?

Heff = i ln U(T)/T

• Mathematically ill defined in many-body systems
• Many expansion schemes (non-convergent)

Pershan, van der Ziel, Malmstrom Phys. Rev. 1966

(i) 2nd order perturbation (ii) 1/W expansions (van Vleck, Floquet-Magnus, Brillouin-Wigner)

Heff = H0 +

• m>0

[H−m, Hm] mΩ + . . .

(iii) 1/(DEab-nW), 1/(U-nW) expansions (Brillouin-Wigner, …) general theory 2/3 relations between schemes: Mikami, et al. PRB ’16

Brillouin-Wigner expansion

higher order monopoles in the Floquet Weyl semimetal

general theory 3/3

HW,±

eff

=

• |k| − Ω + A2 |k|2 + k2

3 ± Ωk3

|k|(4|k|2 − Ω2)

• σ3 − A2(|k| + k3)±1

2|k|Ω(2|k| − Ω)(k2∓1

+

σ+ + h.c.)

Bucciantini, Roy, Kitamura, Oka, arXiv’16 (appendix)

Projection I Projection II

1/W expansions

Heff = H0 +

• m>0

[H−m, Hm] mΩ + . . .

Mikami et al. PRB’16

P P=

different from Magnus expansion

Q=1-P

Husimi transformation

= driven Harmonic oscillator with an oscillating potential

(ii) Classical driven oscillator (i) Quantum oscillator without driving Husimi (Taniuti) PTP ’53 solution pseudo-energy

+

L = 1 2me ˙ X2 − 1 2meω(t)2X2 + XF(t)

Ψn(x, t) = e− i

Enteikyyϕn(x − X(t), t) exp

i {me ˙ X(t)(x − X(t)) + t dt′L(t′) − L0t}

• En = εn + 2k2

y

2m∗

En = εn + 2k2

y

2me − 1 T T L(t′)dt′