ワイルフェルミオンの非断熱なポンプとニールセン・二宮の定理の破れ
University of Tokyo

Sho HIGASHIKAWA
Collaborator
RIKEN CEMS

Masaya NAKAGAWA

University of Tokyo, RIKEN CEMS

Masahito UEDA

熱場の量子論とその応用 30/08/2017

Message

Exotic topological band structure can be realized
in periodically driven (Floquet) system.
e.g. Weyl fermion
・Topological semimetal
・Single Weyl fermion: Impossible
to realize in equilibrium lattice system(Nielsen-Ninomiya theorem)

T. Kitagawa, et.al., Phys. Rev. B 82, 235114 (2010).
M. S. Rudner, et.al., Phys. Rev. X 3, 031005 (2013).
N. Nielsen and M. Ninomiya, Nucl. Phys. B 185, 20 (1981), 193, 173 (1981).

Floquet theory
・Periodically driven (Floquet) system𝑖 𝜕𝑡 𝜓𝐹 𝑡 = 𝐻 𝑡 𝜓𝐹 𝑡
𝐻 𝑡 : driving Hamiltonian (𝐻 𝑡 + 𝑇 = 𝐻 𝑡 , 𝑇: period)→ Difficult to analyze → Map to static system.
1. Floquet operator 𝑼𝑭 , 2. Effective Hamiltonian 𝑯𝐞𝐟𝐟

Floquet theory
・Periodically driven (Floquet) system: 𝑖𝜕𝑡 𝜓𝐹 𝑡 = 𝐻 𝑡 𝜓𝐹 𝑡
𝑻

1. Floquet operator 𝑼𝑭 ≔ 𝓣 𝐞𝐱𝐩 −𝒊 ‫𝒕𝒅 𝒕 𝑯 𝟎׬‬Time evolution in one period: 𝜓𝐹 𝑡 + 𝑇 = 𝑈𝐹 𝜓𝐹 𝑡→ stroboscopic description 𝝍𝑭 𝟎 , 𝝍𝑭 𝑻 , 𝝍𝑭 𝟐𝑻 , 𝝍𝑭 𝟑𝑻 ⋯2. Effective Hamiltonian 𝑯𝐞𝐟𝐟 (𝑈𝐹 =: exp(−𝑖𝐻eff 𝑇))
Floquet dynamics mimic dynamics of 𝑯𝐞𝐟𝐟 : 𝝍𝑭 𝒏𝑻 = 𝝍𝑯 𝒏𝑻 .
𝝍 𝑻𝝍 𝟐𝑻

𝝍 𝟒𝑻

Dynamics of 𝑯𝐞𝐟𝐟
𝝍𝑯 𝒕 : 𝒊 𝝏𝒕 𝝍𝑯 = 𝑯𝐞𝐟𝐟 𝝍𝑯

𝝍 𝟎𝑖𝜕𝑡 𝜓𝐹 𝑡 = 𝐻 𝑡 𝜓𝐹 𝑡
𝝍 𝟑𝑻

𝝍𝑭 𝒕 + 𝑻 = 𝑼𝑭 𝝍𝑭 𝒕

Band topology in Floquet system
・Periodically driven system + lattice → momentum 𝒌:→ 𝑈𝐹 = σ𝒌 𝑈 𝒌 , 𝐻eff 𝒌 = σ𝒌 ℎ 𝒌Time evolution: 𝑈 𝒌 = σ𝜶 𝒆−𝒊෤𝝐𝜶 𝒌 𝑻 |𝒌, 𝜶ۧ‫𝒌ۦ‬, 𝜶|
Effective Bloch Hamiltonian: ℎ 𝒌 = σ𝜶 𝝐෤ 𝜶 𝒌 |𝒌, 𝜶ۧ‫𝒌ۦ‬, 𝜶|→ Band topology𝝐෤ 𝜶 𝒌 :Quasi energy
𝟐𝝅−𝒊(෤𝝐𝜶 𝒌 + 𝑻 )𝑻෥𝜶 𝒌 𝑻−𝒊
𝝐→ Ambiguity of 𝟐𝝅/𝑻: e=e
Periodicity in quasi energy→ Unique topology

c.f. Equilibrium system

No periodicity−𝝅/𝑻 +𝝅/𝑻
𝝐 = −∞𝝐෤𝝐𝝐=∞

1 dimensional analogue of the story

1 dim.
spinlessfermion

Equilibrium(lattice) system

Periodicallydriven system

Single chiral fermion→ Impossible

Single chiral fermion→ Possible
Thouless pump

1 dim Nielsen-Ninomiya

T. Kitagawa, et.al., Phys. Rev. B 82, 235114 (2010).

1 dim chiral fermion & Nielsen-Ninomiya theorem
・Spinless fermion in 1 dimension→ 2 types of (chiral) fermions near fermi surface
Chirality+1 fermion
𝝐 𝒌 = +𝒗 𝒌 − 𝒌𝑭

Chirality-1 fermion
𝝐 𝒌 = −𝒗 𝒌 − 𝒌𝑭

𝒗>𝟎𝝐𝑭 →𝒌𝑭𝒌𝑭

1 dim chiral fermion & Nielsen-Ninomiya theorem
・1 dim lattice system→ Single chiral fermion: Impossible !
(1 dim. Nielsen-Ninomiya theorem)

𝝐 𝒌

𝝐𝑭 →Periodicity
𝝐 𝝅 = 𝝐 −𝝅-1−𝝅-1𝝅+1+1

Chirality+1 fermion

Chirality-1 fermion

1 dimensional analogue of the story

1 dim.
spinlessfermion

Equilibrium(lattice) system

Periodicallydriven system

Single chiral fermion→ Impossible

Single chiral fermion→ Possible
Thouless pump

1 dim Nielsen-Ninomiya

T. Kitagawa, et.al., Phys. Rev. B 82, 235114 (2010).
D. J. Thouless, Phys. Rev. B 27 6083 (1983).

Thouless pump as periodically driven system・Thouless pump・1 dim lattice + adiabatic parameter change 𝐻 𝑡
Periodic: 𝑯 𝒕 + 𝑻 = 𝑯 𝒕

・1 cycle → 1 site translation: 𝝍 𝑻 = 𝐞𝐱𝐩 −𝒊𝒌𝒂 𝝍 𝟎𝑽 𝒕 : Time dependent potential

𝐻 0
t=0Periodic: 𝑽 𝒕 + 𝑻 = 𝑽 𝒕𝝍 𝟎

1 cyclet=T𝐻 𝑇 =𝐻 0
𝝍 𝑻 ≠𝝍 𝟎

Thouless pump as periodically driven system
・1 cycle → 1 site translation: 𝝍 𝑻 = 𝐞𝐱𝐩 −𝒊𝒌 𝝍 𝟎
Floquet operator 𝑼𝑭 [𝝍 𝑻 = 𝑼𝑭 𝝍 𝟎 ]→ Floquet operator 𝑈𝐹 = exp −𝑖𝑘→ Effective Hamiltonian 𝐻eff = 𝑖 log 𝑈𝐹 = 𝑘Single chiral fermion at 𝒌 = 𝟎→ Chiral transport in Thouless pumpChange in Topology
𝑼𝑭 = 𝐞𝐱𝐩 −𝒊෤𝝐 𝒌= 𝐞𝐱𝐩 −𝒊 𝝐෤ 𝒌 + 𝟐𝝅𝝐෤ 𝒌𝝐෤※ 𝑻 = 𝟏, 𝒂 = 𝟏𝝐෤Chiralfermion 𝝅@𝒌 = 𝟎𝝐෤ 𝒌−𝝅𝝐෤ = ±𝝅𝝅𝒌−𝝅𝝅−𝝅𝒌1 dimensional analogue of the story
Equilibrium(lattice) system

Periodicallydriven system

1 dim.
spinlessfermion

Single chiral fermion→ Impossible
1 dim Nielsen-Ninomiya

Single chiral fermion→ Possible
Thouless pump

3 dim.
spinfulfermion

Single Weyl fermion→ Impossible

Single Weyl fermion→ Possible

3-dim Nielsen-Ninomiya

Weyl fermion(WF) & chiral magnetic effect・Spinful (𝜎𝑥 , 𝜎𝑦 , 𝜎𝑧 ) fermion, 3D (𝑘𝑥 , 𝑘𝑦 , 𝑘𝑧 )Semimetal: 𝒉 𝒌𝑾 = 𝟎 → Dispersion near 𝒌𝑾 : 𝜹𝒌 = 𝒌 − 𝒌𝑾 ≃ 𝟎 ?
Left handed WF
・𝒉 𝒌 ≃ +𝝈 ⋅ 𝜹𝒌
1. Spin momentum locking෢𝒌 𝝈 𝒌 = +𝜹𝒌
2. Hedgehog

Right handed WF
・𝒉 𝒌 ≃ −𝝈 ⋅ 𝜹𝒌
෢
1. 𝒌 𝝈 𝒌 = −𝜹𝒌
2. Antihedgehog𝝈 ⋅ 𝜹𝒌 = 𝝈𝒙 𝜹𝒌𝒙 + 𝝈𝒚 𝜹𝒌𝒚 + 𝝈𝒛 𝜹𝒌𝒛෢
𝝈 = 𝜹𝒌෢𝝈 = −𝜹𝒌𝒌𝑾

𝒌𝑾

Weyl fermion & chiral magnetic effect・Single Weyl fermion + magnetic field 𝑩→ Current 𝒋 flows parallel to 𝑩
→ Left handed 𝒋𝑳 = +𝐶𝑩, Right handed 𝒋𝑹 = −𝐶𝑩・3D lattice system→ Equal number of left & right handed fermions(Nielsen-Ninomiya theorem)→ 𝒋𝐭𝐨𝐭 = 𝒋𝑳 + 𝒋𝑹 = 𝒋𝑳 + −𝒋𝑳 = 𝟎: No chiral current 𝒋𝐭𝐨𝐭 ∥ 𝑩  

K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D 78, 074033 (2008).
A. A. Zyuzin and A. A. Burkov, Phys. Rev. B 85, 165110 (2012).
A. A. Zyuzin, S. Wu, and A. A. Burkov, Phys. Rev. B 86, 115113 (2012).

Helical pump・”Spinful” Thouless pump: 1dim lattice + spin d.o.f+1 site translation for 𝝈𝒙 = +𝟏& -1 site translation for 𝝈𝒙 = −𝟏・Possible to realize in ultracold atomic gas

t=01 cyclet=1

𝜎𝑥 = +1

𝜎𝑥 = −1※ 𝑻 = 𝟏, 𝒂 = 𝟏

J. Budich, Y. Hu, and P. Zoller, Phys. Rev. Lett 118, 105302 (2017).

Helical pumpt=0

𝜎𝑥 = +1

1 cycle𝜎𝑥 = −1

t=1・Time evolution: 𝑈 𝑘𝑥 = 𝑒 −𝑖𝑘𝑥 |𝑘𝑥 , +ۧ‫ 𝑥𝑘ۦ‬, +| +𝑒 𝑖𝑘𝑥 |𝑘𝑥 , −ۧ‫ 𝑥𝑘ۦ‬, −|𝑈 𝑘𝑥 = exp(−𝑖𝜎𝑥 𝑘𝑥 ) → 𝐻eff 𝑘𝑥 = 𝑖 log 𝑈 𝑘𝑥 = 𝜎𝑥 𝑘𝑥
Helical fermion
𝐻 = 𝜎𝑥 𝑘𝑥
Helical pump
𝑈1 𝑘 = exp(−𝑖𝜎𝑥 𝑘𝑥 )Supplemental Materials

Helical pump in ultracold atomic gas
・𝑈 𝑘𝑥 = exp −𝑖𝑘𝑥 𝜎𝑥 = exp −𝑖𝐻2 𝑇/2 exp −𝑖𝐻1 𝑇/2= successive application of
𝐻1 = − 𝜋/𝑇 𝜎𝑥 cos 𝑘𝑥 − 𝜎𝑦 sin 𝑘𝑥 : spin-flip hopping
𝐻2 = 𝜋/𝑇 𝜎𝑥 : on-site spin flip

J. Budich, Y. Hu, and P. Zoller, Phys. Rev. Lett 118, 105302 (2017).

Relation to high-frequency expansion
ഥ 𝜔𝑡 (𝜔: frequency, 𝑡0 = 𝜔−1 )𝐻 𝑡 =𝐻
High-frequency limit (𝑡0 → 0): 𝐻 𝑡 → 𝐻0 (static Hamiltonian)
Adiabatic pump
Nonadiabatic pump

High-frequency
expansion𝒕𝟎 = ∞

𝒕𝟎

𝒕𝟎 = 𝟎𝝎=𝟎𝝎𝝎=∞Nielsen-Ninomiya
theorem