N33.00007

arXiv:2111.03857

Ferromagnetism in tilted
fermionic Mott insulators
APS March Meeting 2022
March 2022

Kazuaki Takasan (UC Berkeley) andMasaki Tezuka (Kyoto University)

Introduction: tilted potential in periodic systems
Long history of research:
Response of electrons in solids
to electric field

Renewed interest in thecontext of localization ofquantum many-body systeme.g. Bloch oscillation, Zener tunnelinge.g. Hilbert space fragmentation,
Hilbert space shuttering,
quantum many-body scars

Introduction: Magnetism of tilted Mott insulators

Static tilt enhances 𝐽eff
[K. Takasan and M. Sato,PRB 100, 060408 (2019)]

Localized spin system?
→ Wannier-Stark
localization occurs!

What happens forlarger 𝐸?

Main result: Effective interaction
becomes ferromagnetic
cf. Mentink, Balzer and Eckstein, Nat. Commun. 6, 6708 (2015)
Reversible control of exchange interaction by periodic modulation

SetupOne-dimensional Hubbard model𝐿−1

෡ = −𝑡ℎ ෍ ෍𝐻

with a linear potential
𝐿𝑐𝑗+1,𝜎Ƹ † 𝑐𝑗,𝜎
Ƹ + h. c.

𝑗=1 𝜎=↑,↓[length gauge]

𝐿+ 𝑈 ෍ 𝑛ො𝑗,↑ 𝑛ො𝑗,↓ + ෍ ෍ 𝑗𝐸 𝑛ො𝑗,𝜎𝑗=1

𝑗=1 𝜎=↑,↓

Time-dependent gauge transformation

Initial state: antiferromagnetic| ↑ ↓ ↑ ↓ ↑ ↓ ⋯ ↑ ↓⟩𝑗,𝜎𝐿−1𝐿

෩ 𝑡 = −𝑡ℎ ෍ ෍ 𝑒 𝑖𝐸𝑡 𝑐𝑗+1,𝜎𝐻Ƹ † 𝑐𝑗,𝜎Ƹ + h. c. + 𝑈 ෍ 𝑛ො𝑗,↑ 𝑛ො𝑗,↓no double occupancy𝑡h = 1: unit of energy, 𝐸 = 𝑒 𝑎ℰ/ℏ
𝐿 = 10 in numerical simulations

෩ 𝑡 = 𝑈 † 𝐻𝑈 − 𝑖𝑈 † 𝜕𝑡 𝑈
𝑈 𝑡 = exp −𝑖𝐸𝑡 ෍ 𝑗𝑛𝑗,𝜎 , 𝐻

𝑗=1 𝜎=↑,↓

𝑒: elementary charge
𝑎: lattice constantℰ: electric field

𝑗=1[velocity gauge]

Time-periodic system

Period:2𝜋𝐸

arXiv:2111.03857

For 𝐸 = 0,4𝑡ℎ2
෡eff = σ𝑗 𝐽0 𝑺𝑖 ⋅ 𝑺𝑖+1 , 𝐽0 = .
𝐻

Effective spin Hamiltonian෡eff = ෍ 𝐽eff 𝐸 𝑺𝑖 ⋅ 𝑺𝑖+1 , 𝐽eff 𝐸 =𝐻𝑗

𝑈𝐽0𝐸 21− 𝑈

2𝑡ℎ2
𝐽+ 𝐸 =𝑈+𝐸
𝐸2𝑡ℎ2
𝐽− 𝐸 =𝑈−𝐸𝐽eff 𝐸 = 𝐽+ 𝐸 + 𝐽− (𝐸)See also [K. Takasan and M. Sato, PRB 100, 060408 (2019)]

Time reversal by the strong electric field
1Imbalance 𝐼 = 𝐿 σ𝐿𝑗=1 −1 𝑗 ⟨𝜓(𝑡)|(𝑛ො𝑗↓ − 𝑛ො𝑗↑ )|𝜓 𝑡 ⟩| ↑ ↓ ↑ ↓ ↑ ↓ ⋯ ↑ ↓⟩arXiv:2111.03857

𝐽eff 𝐸 =𝐽0
𝐸 21− 𝑈
𝐿 = 10𝑈 = 50
𝑡on = 50

Fast-forward dynamics

Reversed dynamics

𝐽eff 𝐸 > 𝐽0 for 0 < 𝐸 < 𝑈

𝐽eff 𝐸 < 0 for 𝐸 > 𝑈

𝐽0𝑡rescaled = 𝑡0 +𝐽eff 𝐸𝑡 − 𝑡0

arXiv:2111.03857

𝑈 = 50

Double occupancy?
𝐸=𝑈𝒩double =𝑡1 1‫ 𝐿 𝑡=𝑡׬‬σ𝐿𝑗=1 Ψ 𝑡 𝑛ො𝑗↑ 𝑛ො𝑗↓ Ψ 𝑡
0𝑡1 − 𝑡0

𝐸 = 𝑈/2𝑈 𝑈
𝐸 = 𝑈, , , …: resonances
2 3

𝐸=𝑈arXiv:2111.03857

Dynamics after limited-time field application
𝐸= 𝑈 = 50,2𝐽eff =𝐽0𝐸1−𝑈

2 = −𝐽0Time reversal only during
the application of the field

Same Δ𝑡 = 𝑡off − 𝑡on→ Same dynamics after time reversal

Potential application

cf. chaos in classical dynamics

Out-of-time ordered correlations:characterize operator scrambling[Wiener 1938][Larkin and Ovchinnikov, JETP 1969]

Non-commutating𝛿𝑥 𝑡 = 0𝑉𝑊 𝑡 ≠ 𝑊 𝑡 𝑉scrambling

Commuting
𝑉𝑊 0 = 𝑊 0 𝑉

𝐶 𝑡 = | 𝑊 𝑡 , 𝑉 𝑡 = 0 |2 = 𝑊 † 𝑡 𝑉 † 0 𝑊 𝑡 𝑉 0 + ⋯𝛿𝑥 𝑡~𝑒 𝜆L 𝑡 𝛿𝑥 𝑡 = 02𝜕𝑥 𝑡
𝜕𝑥 02= 𝑥 𝑡 , 𝑝 0 PB ~ 𝑒 2𝜆L 𝑡
𝜆L : Lyapunov exponent

OTOC ~ 𝑒 2𝜆L 𝑡 at long times with 𝜆L > 0: chaotic𝜆L ≤ 2π𝑘B 𝑇Τℏ (chaos bound)[Maldacena, Shenker, and Stanford, JHEP08(2016)106]

𝑊 𝑡 = e𝑖𝐻𝑡 𝑊 e−𝑖𝐻𝑡
Time evolution by −𝐻 is key to the measurement

N33.00007

Ferromagnetism in tilted fermionic Mott insulators
Kazuaki Takasan and Masaki Tezuka, arXiv:2111.03857

Summary• One-dimensional Mott insulator (half-filled Hubbard model) + tilt
• Effective spin interaction: 𝐽eff 𝐸 = 𝐽0 ൗ(1 −𝐸<𝑈𝐸 2)
𝑈𝐸>𝑈𝑈 𝑈• Double occupancy suppressed for 𝑈 ≫ 𝑡h away from 𝐸 = 𝑈, , , …
2 3• Sign reversal may be useful for measuring out-of-time-order correlators