熱場の量子論とその応用 @ KEK, 2023/8/29

Quantum many-body scars

量子多体傷跡状態と関連する話題
桂 法称 (東京大学)

Acknowledgments
 Onsager’s scars• N. Shibata, N. Yoshioka, HK, PRL 124, 180604 (2020)

 Fermionic models• H. Yoshida & HK, PRB 105, 024520 (2022)• K. Tamura & HK, PRB 106, 144306 (2022)

 Integrable boundary states, scalar chirality, …
• K. Sanada, Y. Miao & HK, arXiv:2304.13624.

 Dzyaloshinskii-Moriya + Zeeman model• M. Kunimi, T. Tomita, HK & Y. Kato, arXiv:2306.05591

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Outline1. Introduction and Motivation• Eigenstate thermalization hypothesis (ETH)• Violation of ETH• Rydberg-atom array & PXP model• Quantum many-body scars (QMBS)

2. Onsager scars

3. Other scarred models
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.
But why?
Fundamental problem since von Neumann’s work (1929)• Typicality
A great majority of states with the sameenergy are indistinguishable bymacroscopic observables!
“thermal equilibrium”= common properties shared by
the majority of states
 Microcanonical (MC) ensemble works!

• Thermalization

The approach to these typical states

H. Tasaki, J. Stat. Phys.
163 (2016) and his book

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Experimental verification
S. Trotzky et al., Nat. Phys. 8 (2012)

1d Bose-Hubbard, 87Rb

Comparison with t-DMRG result Numerical verification
M. Rigol et al., Nature 452 (2008)[From Hamazaki-san’s slides]

Eigenstate thermalization hypothesis (ETH) 6/31
• 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
Biroli, Kollath & Lauchli, PRL 105 (2010);
Iyoda, Kaneko & Sagawa, PRL 119 (2017)

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Exceptions of strong ETH
1. Integrable systems
Many conserved charges
Strong ETH, Weak ETHEx.) S=1/2 Heisenberg chain

2. Many-body localized (MBL) systems
Emergent local integrals of motion
Strong ETH, Weak ETH
3. Hilbert-space fragmentation
Hilbert space splits into exp. many sectors
Strong ETH, Weak ETH&4.

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

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
Recommendation:
15-puzzle and Nagaoka ferromagnetism
Quanta magazine, January 2019. 「パズドラの数理と物理」

 One-body scars

東大理学部ニュース(2019年7月号)

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
vdW-typeinteraction

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

Fendley, Sengupta & Sachdev,
PRB 69 (2004); Lesanovsky &Katsura, PRB 86 (2012)

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

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Revivals of fidelity

2. Long-time oscillations are observed
3. Energy (E) v.s. entanglement
entropy (S)  Anomalously low S at high E• Exact QMBS
Lin & Motrunich, PRL 122, 173401 (2019).

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

Exact QMBS
• Embedding method

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Frustration-free system

Shiraishi & Mori, PRL 119 (2017)• AKLT models
Moudgalya, Regnault & Bernevig, PRB 98 (2018)
Mark, Lin & Motrunich, PRB 101 (2020)• 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)
Mizuta, Takasan & Kawakami, PRR 2 (2020)• Recent reviews
Serbyn, Abanin & Papic, Nat. Phys. 17 (2021)
Moudgalya, Bernevig & Regnault, Rep. Prog. Phys. (2022)
Chandran, Iadecola, Khemani & Moessner, ARCMP 14 (2023)

Scars in lattice gauge theories?
• U(1) quantum link model

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U.-J. Wiese, Ann. Phys. 525, 777 (2013)

Constraint: ice rule
Hamiltonian

Scar states
D. Banerjee & Arnab Sen. PRL 126, 220601 (2021)• Z2 gauge model (Fradkin, Kogut, Susskind, …)
O. Fukushima & R. Hamazaki, arXiv:2305.04984 (2023)• Regularized lattice Yang-Mills
T. Hayata & Y. Hidaka, arXiv:2305.05950 (2023)

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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• Constructing models with exact QMBS

 Using Onsager algebra

2d Ising model:
Phys. Rev. 65 (1944)

 Using integrable boundary states Using (restricted) spectrum generating algebra

…

Outline
1. Introduction and Motivation
2. Onsager scars•

Strategy•

Perturbed S=1/2 XY chain•

Higher-spin models

3. Other scarred models
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 Heisenberg Hamiltonian

Spin op. on j-th site:
Eigenstates take theBethe-ansatz form (1931)

17/31Strategy
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 actingwith the subalgebra on the reference state:
 QMBS in non-integrable H
They have the same energy as ψ05. Add perturbations that break the integrability ofbut do not hurt the tower of states18/31

Example: S=1/2 XY chain
 Model12

L

L: even
Periodic chain• Hamiltonian

Can be mapped to free fermions via Jordan-Wigner
Lieb-Schultz-Mattis (1961), Katsura (1962)• Conserved charges
Total Sz:“bi-magnon” operator:
An element of Onsager’s algebra! Infinitely many such.
• Reference eigenstate
All down state:19/31

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20/31

Desired perturbations• Tower of exact eigenstates (with fixed total Sz)• “Coherent state”
MPS with bond dim. 2• Possible perturbations

in any three consecutive sites
 Identify Hermitian operatorsthat annihilate

never appear
c’s can berandom!

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Properties of the perturbed model Level-spacing statistics

Pal, Huse, PRB 82 (2010)• Perturbed Hamiltonian• System size: L=16
• Only diagonal perturbation• Zero magnetization sector

 Entanglement diagnosis• Entanglement entropy (EE)
Volume law  Thermal
Sub-volume law  non-thermal
Mori et al., JPB 51 (2018)• QMBS states
Rigorous result:
EE of QMBS ≦ O (ln L)

H is non-integrable!
Half-chain EE
L=14

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

Revival at Numerical results• Fidelity• Entanglement23/31

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.)24/31

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

Matrix product state (MPS) with bonddimension 3. Desired perturbations
can be identified from this MPS.

• Half-chain entanglement• Fidelity

Outline
1. Introduction and Motivation
2. Onsager scars
3. Other scarred models•

Boundary scars & scalar chirality•

Dzyaloshinskii-Moriya int.+Zeeman

4. Summary

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Integrable boundary states• Integrable Hamiltonian:(1d nearest neighbor int.)• Boost operator:• Conserved charges:

is even / odd under parity: Example: S =1/2 Heisenberg chain

Scalar chirality• Integrable boundary states: Piroli, Pozsgay & Vernier, NPB 925 (2017)

Lattice version of boundary states in integrable QFT:
Ghoshal & Zamolodchikov, IJMP A9, 3841 (1994)

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Boundary scars
 Ifis an eigenstate of a non-integrable Hamiltonian

then it is an eigenstate of Example of a scarred model• H0 : Majumdar-Ghosh model [JMP 10 (1969)]

Dimer g.s. are annihilated by CSC!
• Hamiltonian Non-integrable (Wigner-Dyson)
 Energy v.s. EE plot Dimer g.s. is a scar!

L = 18, t = 8
Sz=0,Toward realization of spin models

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 Experimental setup• 1d array of Rb atoms• Effective spin states• Effective Hamiltonian S=1/2 XXZ chain in a rotating magnetic field Hamiltonian in spin-rotating frame

DH model• Tuning q, δ, etc.  Model with only Dzyaloshinskii-Moriya int.

and field in the z-direction [Kodama, Kato & Tanaka, PRB 107 (2023)]

Toward realization of spin models

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 Experimental setup• 1d array of Rb atoms• Effective spin states• Effective Hamiltonian S=1/2 XXZ chain in a rotating magnetic field Hamiltonian in spin-rotating frame

DH model• Tuning q, δ, etc.  Model with only Dzyaloshinskii-Moriya int.

and field in the z-direction [Kodama, Kato & Tanaka, PRB 107 (2023)]

QMBS states in DH model• Hamiltonian• Raising operator

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PBC orspecial OBC
Similar toin Schecter &Iadecola, PRL 123 (2019).

• They satisfy a restricted spectrum generating algebra (SGA)
See e.g., Moudgalya et al.,
PRB 102, 085140 (2020).

• Exact eigenstates Non-integrable (Wigner-Dyson)
 Energy v.s. EE plot, fidelity
 They are scars!

OBC, L=18, H=0.1D, Soliton num. = 5

Summary

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 Constructing models with QMBS
• Using Onsager algebra

Perturbed S=1/2 XY chain, higher-spin models• Using integrable boundary states
Majumdar-Ghosh + scalar chirality• Using restricted SGA
Dzyaloshinskii-Moriya + Zeeman
Proposal for an experiment Other models• Correlated hopping model: Tamura & HK, PRB 106 (2022)• Generalization of eta-pairing: Yoshida & HK, PRB 105 (2022)• S=1 AKLT + SU(3) scalar chirality• Perturbed S=1 scalar chirality in 1d and 2d32/31

Backup slides

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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!

Rydberg atom system (backup 1)

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S=1 AKLT + SU(3) scalar chirality

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The g.s. ofis an
integrable boundary state• Level statistics• Entanglement entropy

Perturbed S=1 scalar chirality (1)• S=1 scalar chirality
Exponentially many E=0 states• Model 1

Exact eigenstates• Entanglement entropy & fidelity
L=10, h=1; Sz = 0, Spin-flip sym. sector;

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Perturbed S=1 scalar chirality (2)• Model 2

Exact eigenstates• Entanglement entropy & fidelity
L=10, h=1; Sz = 0 sector;• Can also construct 2D models

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Rydberg atom system (backup 2)

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Rydberg atom system (backup 3)

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Rydberg atom system (backup 4)• Level statistics (weak to moderate field)

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