Workshop on Non-Hermitian Quantum Mechanics@ Univ. of Tokyo, 2022/07/04

Exact Bethe-ansatz solutions
for open quantum many-body systems
Masaya Nakagawa(Univ. of Tokyo)
in collaboration withMasahito Ueda (Tokyo), Kazuki Yamamoto, Masaki Tezuka, and Norio Kawakami (Kyoto)

MN, N. Kawakami, and M. Ueda, PRL 126, 110404 (2021)
K. Yamamoto, MN, M. Tezuka, M. Ueda, and N. Kawakami, PRB 105, 205125 (2022)

Quantum integrability■ Integrable model: interacting, many-body, but still exactly solvable!
e.g. 1D Heisenberg modelFirst solved by Hans Bethe: Bethe ansatz[H. Bethe, Z. Phys. 71, 205 (1931)]■ Triumphs in equilibrium systems & dynamics in isolated systems■ Open quantum system: coupled to environment, non-unitary dynamics
ρ(t): density matrix of the system: Liouvillian superoperator (non-Hermitian!)This talk:Exact solutions for open quantum many-body systems
Non-Hermitian Bethe ansatz

Outline
1. Introduction- Quantum master equation & Liouvillian superoperator

2. Exact solution of a 1D dissipative Hubbard model[MN, N. Kawakami, and M. Ueda, PRL 126, 110404 (2021)]

3. Exact solution of a 1D non-Hermitian XXZ model[K. Yamamoto, MN, M. Tezuka, M. Ueda, and N. Kawakami,PRB 105, 205125 (2022)]

4. Summary

Outline
1. Introduction- Quantum master equation & Liouvillian superoperator

2. Exact solution of a 1D dissipative Hubbard model[MN, N. Kawakami, and M. Ueda, PRL 126, 110404 (2021)]

3. Exact solution of a 1D non-Hermitian XXZ model[K. Yamamoto, MN, M. Tezuka, M. Ueda, and N. Kawakami,PRB 105, 205125 (2022)]

4. Summary

Quantum master equation■ Time-evolution eq. for Markovian open quantum systems[Gorini, Kossakowski, and Sudarshan, J. Math. Phys. 17, 821 (1976)][Lindblad, Commun. Math. Phys. 48, 119 (1976)]

Quantum master equation in the GKSL form
(ρ: density matrix of the system)

Hamiltonian part

Dissipation (Lα: Lindblad operator)

Question:
Exact solution of the quantum master equation?

Liouvillian spectrum
Liouvillian
superoperator(non-Hermitian)■ Diagonalization of Liouvillian: Liouvillian spectrum■ General properties⚫

(non-Hermiticity),⚫

: steady state⚫

: Liouvillian gap→ inverse of the relaxation time[cf. nontrivial cases: Haga, MN et al., PRL 127, 070402; Mori and Shirai, PRL 125, 230604]

Outline
1. Introduction- Quantum master equation & Liouvillian superoperator

2. Exact solution of a 1D dissipative Hubbard model[MN, N. Kawakami, and M. Ueda, PRL 126, 110404 (2021)]

3. Exact solution of a 1D non-Hermitian XXZ model[K. Yamamoto, MN, M. Tezuka, M. Ueda, and N. Kawakami,PRB 105, 205125 (2022)]

4. SummarySetup■ Model: 1D dissipative Fermi-Hubbard model
quantum master eq.

HubbardHamiltonian
interaction

hoppingdissipation: on-site 2-body loss
γthopping

Uinteraction

dissipation

Experimental advance: dissipative Hubbard model■ Dissipative Fermi-Hubbard model: experiment[Sponselee et al., Quant. Sci. Tech. 4, 014002 (2018)]

Ultracold atoms in an optical lattice → Fermi-Hubbard model
Atoms in an excited state → inelastic collisions (2-body loss)

Non-equilibrium steady state:
Nonzero number of particles remain
even in the presence of particle loss!

Dissipative Hubbard model:
Strongly interacting dissipative system!

Diagonalization of the Liouvillian■ Step 1: Decomposition of the Liouvillian“Quantum jump” term
→ decreases particle #Non-Hermitian Hubbard model■ Step 2: Diagonalize the non-Hermitian Hamiltonian(N: particle number → conserved quantity of Heff)

Diagonalization of the Liouvillian■ Step 3: Consider matrix elements in the particle-number basis
2-body loss
N→N-2

Triangular structure!
[Torres, PRA 89, 052133 (2014)]

Liouvillian eigenvalue
Liouvillian eigenoperator
Integrable non-Hermitian Hamil. + Triangular structure → Solvable Liouvillian[MN, N. Kawakami, and M. Ueda, PRL 126, 110404 (2021)]

Non-Hermitian Bethe ansatz■ Step 4: Solve the non-Hermitian Hubbard model

complex-valued Hubbard interaction strength!

■ 1D Hubbard model → solvable by Bethe ansatz [Lieb and Wu, PRL 20, 1445 (1968)]→ Generalized solution of the non-Hermitian case
[MN, N. Kawakami, and M. Ueda, PRL 126, 110404 (2021)]

Bethe eqs.

N : # of particles, M : # of down spins,
kj : quasi-momentum, λα : spin rapidity)Ferromagnetic steady state■ Steady state:■ Vacuum

ifis a trivial steady state:■ Other steady states?
Spin-polarized eigenstate→ No loss: ferromagnetic steady state!
Spin SU(2) symmetry → Ferromagnetic steady states are degenerate

Non-vacuum steady state

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

Dissipative spin-exchange interaction■ Physical origin of the ferromagnetic steady state:Spin-exchange interaction in the presence of inelastic collisions[MN, N. Tsuji, N. Kawakami, and M. Ueda, PRL 124, 147203 (2020)]◇ Antiparallel spin configuration → lower the energy,
but have a finite lifetime

inelasticcollision◇ Parallel spin configuration → higher energy, but free from loss!

Ferromagnetism is stabilized by dissipation!

Liouvillian gap■ Exact solution of Liouvillian eigenmodes near the steady state→ ferromagnetic spin-wave-like excitations
Dispersion relationfrom Bethe ansatz(ΔP ≃ 0)(Q0 = πN/L)

gapless excitation!
Liouvillian gap

Relaxation time τR diverges
as τR ～ L2 in L → ∞ limit(Dot: numerics, L = 240, N = 80. Black line: numerics in L → ∞. Green line: analytic, applicable to ΔP ≃ 0)

Correlation length■ Another eigenmode (w/ lifetime): Mott insulator (ground state if γ = 0)
■ Correlation length ξ defined from charge stiffness D[Stafford and Millis, PRB 48, 1409 (1993)]

Change of the eigenenergy E0
due to a twisted boundary condition
with angle Φ

Localized state cannot “feel” the change ofthe boundary condition
Exact result from Bethe ansatz

Dissipation-induced divergence of correlation length■ Correlation length as a function of dissipation strength

U/4t = 1.0

U/4t = 0.7

U/4t = 0.6✓ Localization due to strong dissipation: quantum Zeno effect✓ Dissipation-induced divergence of the correlation length!
✓ Negative correlation length: breakdown of analytic continuation

“Phase diagram” of a dissipative Mott insulator
Hubbard gap vanishes:crossover line
localization by dissipation

Exceptional point(Liouvillian is not diagonalizable)(Breakdown ofanalytic continuation)
localization by repulsion

Many-body physics from interplay between interaction and dissipation

Outline
1. Introduction- Quantum master equation & Liouvillian superoperator

2. Exact solution of a 1D dissipative Hubbard model[MN, N. Kawakami, and M. Ueda, PRL 126, 110404 (2021)]

3. Exact solution of a 1D non-Hermitian XXZ model[K. Yamamoto, MN, M. Tezuka, M. Ueda, and N. Kawakami,PRB 105, 205125 (2022)]

4. SummarySetup■ 1D two-component Bose-Hubbard model + 2-body loss
Non-Hermitian Bose-Hubbard model■ Unit-filling sector, strong interaction limit (Uσσ’ >> t)→ non-Hermitian XXZ spin chain: exactly solvable!

complex-valued Ising interaction parameter!

Non-Hermitian Tomonaga-Luttinger liquid■ Longest-lived state in the unit-filling sector→ Non-Hermitian Tomonaga-Luttinger liquid
effective field theory: complex-valued Tomonaga-Luttinger parameter■ Non-unitary quantum critical phenomena[Ashida, Furukawa, and Ueda, PRA 94, 053615 (2016)]

HermitianSingle criticalexponent

NonHermitianTwo criticalexponents!

New universality class: complex extension of conformal field theory

Bethe ansatz solution■ Exact Bethe-ansatz solution of the 1D non-Hermitian XXZ model→ Exact result on the complex-valued Tomonaga-Luttinger parameter(extracted from a finite-size correction to the ground-state energy)[K. Yamamoto, MN, M. Tezuka, M. Ueda, and N. Kawakami, PRB 105, 205125 (2022)]

Deviation fromHermitian universality
class

Comparison with DMRG calculation

Non-unitary universality class of non-Hermitian many-body systems!

Outline
1. Introduction- Quantum master equation & Liouvillian superoperator

2. Exact solution of a 1D dissipative Hubbard model[MN, N. Kawakami, and M. Ueda, PRL 126, 110404 (2021)]

3. Exact solution of a 1D non-Hermitian XXZ model[K. Yamamoto, MN, M. Tezuka, M. Ueda, and N. Kawakami,PRB 105, 205125 (2022)]

4. Summary

Summary■ Exact solution for open quantum many-body systems- Triangular structure of the Liouvillian- Non-Hermitian Bethe ansatz■ Exact solution of the 1D dissipative Hubbard model- Dissipation-induced ferromagnetism- Dissipation-induced divergence of the correlation length■ Exact solution of the 1D non-Hermitian XXZ model- Non-Hermitian Tomonaga-Luttinger liquid & its criticality

MN, N. Kawakami, and M. Ueda, PRL 126, 110404 (2021)

K. Yamamoto, MN, M. Tezuka, M. Ueda, and N. Kawakami, PRB 105, 205125 (2022)