NQS2022 @ Kyoto
25 Nov. 2022

Exact eigenstates of multicomponent Hubbard models:SU(N) magnetic η pairing and weak ergodicity breaking
Masaya Nakagawa(University of Tokyo)
Collaborators: Hosho Katsura (Tokyo), Masahito Ueda (Tokyo & RIKEN)
MN, H. Katsura, and M. Ueda, arXiv:2205.07235

Outline
1. Introduction- Weak ergodicity breaking- η pairing
2. Exact eigenstates of the N-component Hubbard model
3. Weak ergodicity breaking in the N-component Hubbard model

4. Partial integrability
5. Summary

Outline
1. Introduction- Weak ergodicity breaking- η pairing
2. Exact eigenstates of the N-component Hubbard model
3. Weak ergodicity breaking in the N-component Hubbard model

4. Partial integrability
5. Summary

Quantum thermalization■ Thermalization in isolated quantum many-body systems

Thermodynamics fromunitary time evolution?

■ (Strong) Eigenstate Thermalization Hypothesis (ETH)[Deutsch, PRA 43, 2046 (1991); Srednicki, PRE 50, 888 (1994); Rigol et al., Nature 452, 854 (2008)]

for all energy eigenstates
ENon-integrable systems → strong ETH → thermalization?

Weak ergodicity breaking■ Weak ergodicity breaking in non-integrable systems→ Special initial states do not thermalize! Violation of strong ETH
 Quantum many-body scar

[Turner et al., PRB 98, 155134 (2018)]

 Hilbert space fragmentation[Moudgalya et al., Rep. Prog. Phys. 85, 086501 (2022)]

Non-integrability does not guarantee ergodicity!

Hubbard model■ Fermi-Hubbard model

Hopping

Interaction■ Minimal model of interacting fermions→ quantum magnetism, (high-Tc) superconductivity, …
■ Quantum simulation with cold atomsAntiferromagnetic long-range order[Mazurenko et al., Nature 545, 462 (2017)]

η pairing■ η pairing: exact eigenstate of the Hubbard model(on hypercubic lattice, arbitrary dim.) [C. N. Yang, PRL 63, 2144 (1989)]

nonzero C.O.M. momentum

on-site Cooper pair(“doublon”)✓ Hidden (“dynamical”) η symmetry of the Hubbard model[C. N. Yang and S. C. Zhang, Mod. Phys. Lett. B 04, 759 (1990)]✓ Off-diagonal long-range order: “BEC” of doublons with momentum Q✓ Excited eigenstateη pairing■ η pairing: exact eigenstate of the Hubbard model(on hypercubic lattice, arbitrary dim.) [C. N. Yang, PRL 63, 2144 (1989)]

nonzero C.O.M. momentum

on-site Cooper pair(“doublon”)

This talk: Construction of exact eigenstates of the N-component Hubbard model Off-diagonal long-range order coexisting with SU(N) magnetism
 Weak ergodicity breaking in the N-component Hubbard model for N ≥ 3

Outline
1. Introduction- Weak ergodicity breaking- η pairing
2. Exact eigenstates of the N-component Hubbard model
3. Weak ergodicity breaking in the N-component Hubbard model

4. Partial integrability
5. Summary

Multicomponent Hubbard model■ N-component Fermi-Hubbard model (on d-dim. hypercubic lattice)

hopping: SU(N) sym.

Interaction: not necessarily SU(N)e.g., N = 3: 6Li atoms under a high magnetic field[Ottenstein et al., PRL 101, 203202 (2008)]■ SU(N) Hubbard model (Uα,β = U)
→ alkaline-earth-like atoms (173Yb: N = 6, 87Sr: N = 10)[Taie et al., Nat. Phys. 8, 825 (2012)]■ SU(M)×SU(N - M) Hubbard model→ 171Yb-173Yb mixture (SU(2)×SU(6)) [Taie et al., PRL 105, 190401 (2010)]

Generalized η pairing■ Generalized η-pairing stateη pair of “color” α & βAll η pairs must contain the same “color” β = 1■ Three-component case is special

Arbitrary η pairs are allowed!

1

2

3

Main result:
These states are exact eigenstates of the N-component Hubbard model

Generalized η-pairing states are exact eigenstates■ Proof Kinetic term
η pair = pair of fermions with momenta k & Q - kVanish!
→ eigenstate of kinetic term Interaction term

→ multiple η pairs cannot occupy the same site (Pauli exclusion)
N-component case:Pauli exclu. via “red” particles→ constant interaction energy
3-component case:Pauli exclusion between
arbitrary pairs!

SU(N) magnetic & off-diagonal long-range order■ Off-diagonal long-range order: (N - 1) types of pair correlations=±1
Site-indep. magnitude(Ns: # of sites, Nf: # of particles)■ Simultaneous condensation of multiple pairs→ fragmented condensate [Mueller et al., PRA 74, 033612 (2006)]■ Long-range magnetic correlation

SU(N) spin operator
Site-indep. correlation!
SU(N) magnetic order

Coexistence of SU(N) magnetism & off-diagonal long-range order!

Spectrum generating algebra■ Original spin-1/2 Hubbard model → hidden η symmetry■ No η symmetry in the N-component Hubbard model for N ≥ 3

Generalized η-pairing states
do not “feel” the residual term!

■ “Symmetry” in subspace: (restricted) spectrum generating algebra[Moudgalya et al., PRB 102, 085140 (2020); Mark et al., PRB 101, 195131 (2020)]

W: Hilbert subspace spanned byat each siteOutline
1. Introduction- Weak ergodicity breaking- η pairing
2. Exact eigenstates of the N-component Hubbard model
3. Weak ergodicity breaking in the N-component Hubbard model

4. Partial integrability
5. Summary

Sub-volume law entanglement entropy■ Entanglement entropy of a generalized η-pairing state[Two-component case: Vefek et al., SciPost Phys. 3, 043 (2017)]

AB
Sub-volume law!
(Ns,A: number of sites in subsystem A)■ Violation of the eigenstate thermalization hypothesis (ETH)
ETH:microcan. ensemble

volume law(extensive entropy)

Generalized η-pairing states violate the ETH!

Non-thermalizing dynamics■ Violation of the ETH (failure of thermalization)- Integrable system, many-body localization (many conserved quantities)- Quantum many-body scar (weak ergodicity breaking)→ no conserved quantity, but exceptional eigenstates violating ETH[Turner et al., Nat. Phys. 14, 745 (2018)]

Note: η-pairing state of the original Hubbard model is NOT a scar (∵ η symmetry)[See also Moudgalya et al., PRB 102, 085140 (2020); Mark et al., PRB 102, 075132 (2020)]■ Non-thermalizing dynamics of generalized η-pairing states
Raman coupling between hyperfine states[cf. 173Yb: Tusi et al., Nat. Phys. 18, 1201 (2022)]# of doublons
Persistent oscillation→ quantum many-body scar dynamics!

Construction of quantum many-body scars■ A systematic construction of quantum many-body scar states[Mark et al., PRB 101, 195131 (2020); O’Dea et al., PRR 2, 043305 (2020) etc.]

Symmetry-breaking term that satisfies
Symmetric Hamiltonian that has a “simple” eigenstate(e.g., ferro., η pairing, …)■ Generalized η pairing in the N-component Hubbard model
free fermion part→ generalized η symmetry(⊂ SO(2N) symmetry)
Interaction term breaks the η symmetry
if N ≥ 3 !

Generalized η pairing in N(≥ 3)-component systems
is a natural quantum many-body scar!

Outline
1. Introduction- Weak ergodicity breaking- η pairing
2. Exact eigenstates of the N-component Hubbard model
3. Weak ergodicity breaking in the N-component Hubbard model

4. Partial integrability
5. Summary

Partial integrability
H0 : SU(N) symmetric one-body term
Hint : (arbitrary) on-site two-body interactions■ Spin-polarized sector: equivalent to free fermions (integrable)
Pauli exclusion → no on-site interaction!

■ SU(N) ferromagnetic state (: SU(N) spin operator)

Spin-polarized eigenstateSpin flips
No double occupancy!

SU(N) sym. of H0 → exact eigenstate

N-component Fermi-Hubbard model has an integrable sector!
Partial integrability

Non-thermalization due to partial integrability■ Dynamics from an initial state in the integrable sector

Arbitrary spin-polarized state
For simplicity, assume translation invariancemomentum distribution

Conserved!

Non-thermalization in the integrable sector!

Weak ergodicity breaking due to partial integrability■ Integrable systems do not satisfy the strong ETH■ ETH → should be tested within each symmetry sector■ No SU(N) symmetry in the interaction term
→ The integrable sector cannot be distinguished from
non-integrable one by symmetry eigenvalues!
Integrable sector

Symmetry sector

Weak ergodicity breaking due to partial integrability!

Dissipation-induced non-thermalization■ Initial states in the integrable sector do not thermalize■ How to prepare such initial states? → Control of dissipation!
Lindblad eq.
On-site two-body loss → realized in cold atoms
SU(N) FM states are dark (steady) states! No double occ.

Experiment: SU(N) Hubbard + two-body loss

Ferromagnetic spin correlation[Sponselee et al., Quant. Sci. Tech. 4, 014002 (2018)][Honda, … , MN, and Takahashi, arXiv:2205.13162]

Outline
1. Introduction- Weak ergodicity breaking- η pairing
2. Exact eigenstates of the N-component Hubbard model
3. Weak ergodicity breaking in the N-component Hubbard model

4. Partial integrability
5. Summary

Summary■ Generalized η-pairing eigenstates of the N-component Hubbard model- Simultaneous condensation of multicomponent η pairs- Coexistence of SU(N) magnetism & off-diagonal long-range order■ Weak ergodicity breaking in the N-component Hubbard model for N ≥ 3- No η symmetry → quantum many-body scar
- Partial integrability- Dissipation-induced non-thermalization

Reference:MN, H. Katsura, and M. Ueda, arXiv:2205.07235