京大素論セミナー（2022/1/26）

量子多体傷跡状態の代数的構成法
桂 法称 (東京大学・物理学専攻)
Collaborators:
柴田直幸 (東京大学)
吉岡信行 (東京大学)
 N. Shibata, H.K., and N. Yoshioka,
Phys. Rev. Lett. 124, 180604 (2020)
Selected as Editors’ Suggestion

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Outline
1. Introduction and Motivation• Eigenstate thermalization hypothesis (ETH)• Violation of ETH• Experiment on Rydberg-atom array• Quantum many-body scars (QMBS)

2. Models with exact QMBS
3. Results and generalizations

4. Summary

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Foundation of equilibrium stat-mech

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An isolated macro classical/quantum system
relaxes towards a steady state at late times.

• Typicality
A great majority of states with the sameenergy are indistinguishable bymacroscopic observables!

H. Tasaki, J. Stat. Phys.
163 (2016) and his book“thermal equilibrium”= common properties shared by
the majority of states
 Microcanonical (MC) ensemble works!

• Thermalization
The approach to these typical states• Experiments and numerics
S. Trotzky et al., Nat. Phys. 8 (2012)
M. Rigol et al., Nature 452 (2008), …

1d Bose-Hubbard, 87Rb

Eigenstate thermalization hypothesis (ETH) 4/27
• Setup: Hamiltonian,: (normalized) energy eigenstate,: macroscopic observable,: MC ensemble,
Energy shell:• Thermal states
A stateis said to be thermal if• Strong ETH: Allin the energy shell are thermal.

Believed to be true for a large class of non-integrable systems
Concept: von Neumann, Deutsch, Srednicki, Tasaki, …
Numerical evidence: D’Alessio et al., Adv. Phys. 65 (2016).

• Weak ETH: Almost allin the energy shell are thermal.

Proved under certain conditions: translational sym., local interaction
Biroli, Kollath, Lauchli, PRL 105 (2010), Iyoda, Kaneko, Sagawa, PRL 119 (2017)

Violation of ETH
 Exceptions of strong ETH
1. Integrable systems
Many conserved charges
Strong ETH, Weak ETH
2. Many-body localized (MBL) systems
Emergent local integrals of motion
Strong ETH, Weak ETHEx.) S=1/2 Heisenberg chain

3. Hilbert-space fragmentation
Hilbert space splits into exponentially many sectors
Strong ETH, Weak ETH&
4.

Quantum many-body scarred (QMBS) systems
Strong ETH, Weak ETH
Non-integrable but have scarred states which
do not thermalize for an anomalously long time!

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What are scars?

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 A very nice blog article“Quantum Machine Appears to Defy
Universe’s Push for Disorder”,
Marcus Woo, Quanta magazine, March 2019.

 One-body scars
1-particle wave function in a Bunimovich stadium
E. Heller, PRL 53 (1984)(From Shibata’s PhD thesis)

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Experiment on Rydberg atom arrays
Bernien et al., Nature 551 (2017)

87Rb: el. in 5s  70s• Rydberg atoms

Atoms in which one of the electrons is in an excitedstate with a very high principal quantum number.

• Rydberg blockade+ --Van der
Waals-type
interaction

Never have adjacent excited states• A surprising finding!
Special initial states

Exhibit robust oscillations. Other initialstates thermalize much more rapidly.

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PXP model (1)• Hamiltonian

Turner et al., Nat. Phys. 14, 745 (2018)• Example: 4-site with PBC
Dimension of Hilbert space: F3+F5 = 7
State graph

1 2 3 4

Hamiltonian

PXP model (2)• Properties
1. Level statistics
 Wigner-Dyson, non-integrable
2. Long-time oscillations are observed
3. Energy (E) v.s. entanglement entropy (S) Anomalously low S at high E• Exact QMBS
Lin and Motrunich, PRL 122, 173401 (2019).

Exact eigenstates of HPXP in the formof matrix product states (MPS) Low entanglement states at high energy

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Revivals of fidelity

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Exact QMBS
• Embedding method
Shiraishi, Mori, PRL 119 (2017)• AKLT models
Moudgalya, Regnault, Bernevig, PRB 98 (2018)
Mark, Lin, Motrunich, PRB 101 (2020)

S=1 AKLT• Ising and XY-like models
Iadecola, Schecter, PRB 101 (2020)
Chattopadhyay, Pichler, Lukin, Ho, PRB 101 (2020)• Floquet scars
Driven PXP: Sugiura, Kuwahara, Saito, PRR 3 (2021)
PXP, PYP, …: Mizuta, Takasan, Kawakami, PRR 2 (2020)• Recent review
Serbyn, Abanin, Papic, Nat. Phys. (2021) [arXiv:2011.09486]
Moudgalya, Bernevig, Regnault, [arXiv:2109.00548]

Today’s subject• Quantum many-body scars (QMBS)- Non-thermal eigenstates of non-integrable Hamiltonians- Finite-energy density- Entanglement entropy does not obey a volume law• A new class of exact QMBS via Onsager algebra• Spin-S and the interaction range can be arbitrary• Models allow for spatially varying couplings (disorder)

Dolan-Grady
relation

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Outline
1. Introduction and Motivation
2. Models with exact QMBS
• Exactly solvable models• Algebraic approach• Example: perturbed S=1/2 XY chain

3. Results and generalizations
4. Summary

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Exactly solvable models (Crude) Classification• Integrable systems
Free fermions/bosons, Bethe ansatz
Many conserved charges

Not exclusive!

• Frustration-free systems
Ground state (g.s.) minimizes each local Hamiltonian
Explicit g.s., but hard to obtain excited states(A few exact excited states in AKLT chains)

 Heisenberg Hamiltonian

Spin op. on j-th site:

Eigenstates take theBethe-ansatz form (1931)

14/27(Generalized) Shiraishi-Mori
 Frustration-free system
Definition. A Hamiltonian
if there exists a statesuch thatis said to be frustration-freefor all j.

• Universal form

: minimum
eigenvalue of hj• Zero-energy g.s.

Can we cook up a model withexact/explicit excited states? YES!

 Embedding method• New Hamiltonian
The g.s. ofunless

may not be the g.s. of• Shiraishi-Mori PRL 119, 030601 (2017)
Particular case where15/27

Algebraic approach Strategy
1. Starting point:
Integrable model with conserved charges
They commute with the Hamiltonian

2. Take a subalgebra
3. Find a reference eigenstateψ0: simple state, e.g., product state or MPS

4. Find a tower of eigenstates generated by acting withthe subalgebra on the reference state:
 QMBS in non-integrable H

They have the same energy with ψ0
5. Add perturbations that break the integrability ofbut leave the tower of states unaffected16/27Example: S=1/2 XY chain
 Model12

L

S=1/2 at each site
L: even
Periodic chain• Hamiltonian

Can be mapped to free fermions via Jordan-Wigner
Lieb-Schultz-Mattis (1961), Katsura (1962)• Conserved charges
Total Sz:

Other charges are not very
obvious in the spin basis...

“bi-magnon” operator:• Reference eigenstate
All down state:
E=0 is in the middle of the spectrum

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Magnon eigenstates ``Motion” of flipped spin
12N

not an eigenstate of
Flipped spin hops tothe neighboring sites.

 Bloch stateis an exact eigenstate of Bi-magnon state with momentum

is an exact zero-energy stateTower of exact eigenstates• Eigenstates with fixed total Sz• “Coherent state”• Matrix-product operator (MPO)• Coherent state = Matrix-product state with bond dimension 2

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 Telescoping trick

19/27• Proves.
isn’t so important (?)• A similar idea was used in Baxter’s solution of XYZ chain• Also in stochastic integrable models (“hat relation”)

 Possible perturbations• We never havein any three consecutive sites• Identify Hermitian operatorsthat annihilate
Couplings can be random!

Outline
1. Introduction and Motivation
2. Models with exact QMBS
3. Results and generalizations• Level statistics  Non-integrability• Entanglement and dynamics• Longer-range extensions• Higher-spin generalizations

4. Summary

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Is the perturbed model non-integrable?

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 Level-spacing statistics• Perturbed Hamiltonian• Energy levels• Level spacing• H is integrable
 Poisson distribution, PDF:
• H is non-integrable (GOE) Wigner-Dyson distribution,

 Numerical result• System size: L=16
• Only diagonal perturbation• Zero magnetization sector
Clearly Wigner-Dyson!
H is non-integrable!

Casati et al, PRL 54 (1985),Pal, Huse, PRB 82 (2010)

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Entanglement diagnosis

A Half-chain entanglement• Reduced density matrix

B

L/2
L

• Entanglement entropy (EE)• Thermodynamic entropy ~ EEMori et al., J. Phys. B 51, 112001 (2018)

Volume law
 Thermal
Sub-volume law  non-thermal (including area law) Results• Coherent state
MPS, area-law EE• Energy eigenstates?

QMBS:Rigorous result: EE of QMBS ≦ O (ln L)

L=14

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Dynamics
 Initial state = coherent state• Hamiltonian• Coherent state• Time evolution

Revival at Numerical results• Fidelity• Entanglement24/27

Onsager algebra• Hamiltonian

Unitarily equivalent to• Commuting operators(Quantum) Ising!
Phys. Rev. 65 (1944)

Any polynomial incommutes with H2• Dolan-Grady relation• Defining relations of algebraAllcommute with H2Allows for scarred models withlonger-range interactions!

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What about S >1/2 ?
 Self-dual U(1)-invariant clock model
Vernier, O’Brien, Fendley, J. Stat. Mech. (2019)• Matrices• Hamiltonian

Truly interacting for n>2!

H2 boils down to (twisted) XY, H3  S=1 Fateev-Zamolodchikov• U(1) symmetry• Self-duality (in the• Onsager algebra!

rep.)26/27

S=1 (n=3) model• Integrable Hamiltonian• Coherent state

Again a matrix product state (MPS). The bond dimension is 3.
Desired perturbations can be identified from this MPS.
• Half-chain entanglement• Fidelity

Summary

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 A new construction of QMBS
• Perturbed S=1/2 XY chain• Exact Onsager scars• Level-spacing statistics  non-integrability• Entanglement, fidelity, dynamics, …

 Generalizations• Use other elements of Onsager algebra• Higher-spin models

 Future directions• More realistic models?
J1-J2 Heisenberg + DM int.: Mark, Motrunich, PRB 102 (2020)• Continuous models?
Chiral fermions in 1+1 D: Schindler, Regnault, Bernevig, arXiv: 2110.15365.