Fock space localization ina perturbed Sachdev-Ye-Kitaev model
22 March 2022
The 17th Japan-Slovenia Seminaron Nonlinear Science (online)
Masaki TEZUKA (Kyoto University)

Plan
෡=
𝐻

• Sachdev-Ye-Kitaev model෍𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁• Maximally chaotic quantum mechanical model• SYK4+2• Departure from chaotic behavior

𝑁

෡=
𝐻

෍

𝑁

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑• Quantitative analysis of Fock-space localization• Many-body transition point• Spectral statistics, inverse participation ratio• Ergodicity of the eigenstates

1≤𝑎<𝑏

SYK model-related publications and collaborators• Sachdev-Ye-Kitaev model• Proposal for experiment: PTEP 2017, 083I01 and arXiv:1709.07189• with Ippei Danshita and Masanori Hanada• Black Holes and Random Matrices: JHEP 1705(2017)118• with J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, and A. Streicher• Scrambling time: JHEP 1807(2018)124 with Hrant Gharibyan, M. Hanada, and S. H. Shenker• SYK4+2• Chaotic-integrable transition: PRL 120, 241603 (2018)

Also see our reply [PRL 126, 109102 (2021)] to thecomment by J. Kim and X. Cao [PRL 126, 109101 (2021)]• with Antonio M. García-García, Bruno Loureiro, and Aurelio Romero-Bermúdez• Characterization of quantum chaos: JHEP 1904(2019)082 and Phys. Rev. E 102, 022213 (2020)• with Hrant Gharibyan, M. Hanada, and Brian Swingle• Related setups:• [short-range interactions] Phys. Rev. B 99, 054202 (2019) with A. M. García-García• Phys. Lett. B 795, 230 (2019) and J. Phys. A 54, 095401 (2021) with Pak Hang Chris Lau, Chen-Te Ma, and Jeff Murugan• Quantitative analysis of Fock-space localization in SYK4+2• Many-body transition point and inverse participation ratio• Phys. Rev. Research 3, 013023 (2021) with Felipe Monteiro, Tobias Micklitz, and Alexander Altland• Entanglement entropy• Phys. Rev. Lett. 127, 030601 (2021) with F. Monteiro, A. Altland, David A. Huse, and T. Micklitz

Recent works not directly related to SYK• 2112.12467 Universal properties of dissipative Tomonaga-Luttinger
liquids: A case study of a non-Hermitian XXZ spin chain(with Kazuki Yamamoto, Masaya Nakagawa, Masahito Ueda, and Norio Kawakami)• 2112.08763 Topological Lifshitz Transitions, Orbital Currents, andInteractions in Low-dimensional Fermi Gases in Synthetic Gauge Fields(with Chen-How Huang and Miguel A. Cazalilla) New J. Phys. (2022) in press• 2111.03857 Ferromagnetism in tilted fermionic Mott insulators(with Kazuaki Takasan)• 2110.03008 Numerical evidence for a Haagerup conformal field theory(with Tzu-Chen Huang, Ying-Hsuan Lin, Kantaro Ohmori, and Yuji Tachikawa)• 2012.14609 Local Operator Entanglement in Spin Chains(with Eric Mascot, Masahiro Nozaki)• Fuyuki Matsuda, Masaki Tezuka, and Norio Kawakami,

Two-Dimensional Thouless Pumping of Ultracold Fermions in Obliquely
Introduced Optical Superlattice, J. Phys. Soc. Jpn. 89, 114708 (2020)👆Sachdev-Ye-Kitaev model෡=𝐻෍[Kitaev, talks at KITP (2015)][Sachdev, PRX (2015)][Sachdev and Ye, PRL (1993)]

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑

𝑁4
𝑁≅ terms24
4

1≤𝑎<𝑏<𝑐<𝑑≤𝑁𝜒Ƹ 𝑎=1,2,…,𝑁 : 𝑁 Majorana fermions ( 𝜒Ƹ 𝑎 , 𝜒Ƹ 𝑏 = 2𝛿𝑎𝑏 )𝐽𝑎𝑏𝑐𝑑 : independent Gaussian random couplings (𝐽𝑎𝑏𝑐𝑑 2 = 𝐽2 , 𝐽𝑎𝑏𝑐𝑑 = 0)
𝑁 Majorana fermions• Solvable in the large-𝑁 limit[Maldacena, Shenker, and Stanford, JHEP08(2016)106]
low 𝑇• Maximally chaotic (𝜆Lyapunov → 2𝜋𝑘B 𝑇/ℏ: chaos bound)• Correspondence to 1+1d gravity, random matrix
Reviews e.g. [D. Chowdhury, A. Georges, O. Parcollet, and S. Sachdev, 2109.05037];[M. Tezuka, Kotai Butsuri (in Japanese) March 2022]

One term of the 10-Majorana fermion SYKq=4

𝜒𝑎 𝜒𝑐 𝜒𝑒 𝜒𝑔
𝑎

𝑐

𝑏

0

32-state Fock space: 5-dimensional hypercube
𝑒

𝑔

𝑖

𝑑

𝑓

ℎ

𝑗

1

2

3

4

5 qubits

𝐽𝑎𝑐𝑒𝑔 𝜒𝑎 𝜒𝑐 𝜒𝑒 𝜒𝑔4

104= 210 terms

2|11111⟩103|00000⟩|11110⟩Lyapunov exponent and out-of-time-ordercorrelators (OTOC)
𝑊 𝑡 = e𝑖𝐻𝑡 𝑊e−𝑖𝐻𝑡

Classical chaos:

Quantum dynamics:

Infinitesimally different initial coords

𝐶𝑇 𝑡 = 𝑥ො 𝑡 , 𝑝Ƹ 0

𝛿𝑥 𝑡 = 0𝛿𝑥 𝑡~𝑒 𝜆L𝑡 𝛿𝑥 𝑡 = 0

2

For operators 𝑉 and 𝑊, consider
𝐶 𝑡 = | 𝑊 𝑡 , 𝑉 𝑡 = 0 |2 = 𝑊 † 𝑡 𝑉 † 0 𝑊 𝑡 𝑉 0 + ⋯[Wiener 1938][Larkin & Ovchinnikov 1969]
OTOC ~ 𝑒 2𝜆L 𝑡 at long times, 𝜆L > 0: chaotic“Black holes are fastest quantum scramblers”
t=0

Real time t[P. Hayden and J. Preskill 2007] [Y. Sekino and L. Susskind 2008][Shenker and Stanford 2014]

𝜆L : Lyapunov exponent
𝜕𝑥 𝑡
𝜕𝑥 0

2

= 𝑥 𝑡 ,𝑝 0

PB2

→ 𝑒 2𝜆L𝑡

𝜆L ≤ 2π𝑘B 𝑇Τℏ (chaos bound)[J. Maldacena, S. H. Shenker, and D. Stanford, JHEP08(2016)106]

Out-of-time-ordered correlators (OTOCs)
𝜒Ƹ 𝑖 𝑡1 𝜒Ƹ 𝑖 𝑡2 𝜒Ƹ 𝑗 𝑡3 𝜒Ƹ 𝑗 𝑡4

Regularized OTOC can becalculated for large-N SYK model,
satisfies the chaos bound𝜆L = 2π𝑘B 𝑇Τℏ at low T limit

Γ 𝑡1 , 𝑡2 , 𝑡3 , 𝑡4 = Γ0 𝑡1 , 𝑡2 , 𝑡3 , 𝑡4 + න 𝑑𝑡𝑎 𝑑𝑡𝑏 Γ 𝑡1 , 𝑡2 , 𝑡𝑎 , 𝑡𝑏 𝐾 𝑡𝑎 , 𝑡𝑏 , 𝑡3 , 𝑡4

[Kitaev’s talks][J. Polchinski and V. Rosenhaus, JHEP 1604 (2016) 001]
[J. Maldacena and D. Stanford, Phys. Rev. D 94, 106002 (2016)]

Maximally chaotic systems
S. Sachdev, Phys. Rev. Lett. 105, 151602 (2010),
Phys. Rev. X 5, 041025 (2015);
J. Maldacena and D. Stanford,
Phys. Rev. D 94, 106002 (2016); …

1+1d
JT gravity

0+1d SY &SYK models

J. S. Cotler, G. Gur-Ari, M. Hanada, J.
Polchinski, P. Saad, S. H. Shenker, D.
Stanford, A. Streicher, and MT, JHEP
1705(2017)118; T. Nosaka and T.
Numasawa, JHEP 2008(2020)81; Y. Jia
and J. J. M. Verbaarschot, JHEP
2007(2020)193; ...

A. Almheiri and J. Polchinski, JHEP 1511(2015)014;
P. Saad, S. H. Shenker, and D. Stanford, arXiv:1903.11115;
D. Stanford and E. Witten, arXiv:1907.03363; …

RandommatrixGaussian random matrices
Gaussian distribution
𝐾

Density ∝ 𝑒

𝑎𝑖𝑗 𝑖,𝑗=1

𝛽: Dyson index
Real (𝛽 = 1): Gaussian Orthogonal Ensemble (GOE)
Complex (𝛽 = 2): G. Unitary E. (GUE)
Quaternion (𝛽 = 4): G. Symplectic E. (GSE)

𝛽𝐾
Tr𝐻 2
4

−

𝑎𝑖𝑗 = 𝑎𝑗𝑖∗

𝛽𝐾

= exp − 4 σ𝐾𝑖,𝑗 𝑎𝑖𝑗

Joint distribution function for
eigenvalues 𝑒𝑗

0Distribution of normalized
𝑒𝑗+1 −𝑒𝑗level separation 𝑠𝑗 =ҧ

𝑝 𝑒1 , 𝑒2 , … , 𝑒𝐾 ∝ෑ1≤𝑖<𝑗≤𝐾∆ 𝑒

GOE/GUE/GSE: 𝑃 𝑠 ∝ 𝑠 𝛽
2
at small 𝑠, has 𝑒 −𝑠 tailNeighboring gap ratio𝑟=𝑟

Uncorrelated: 𝑃 𝑠 = 𝑒 −𝑠
(Poisson distribution)2

min 𝑒𝑖+1 − 𝑒𝑖 , 𝑒𝑖+2 − 𝑒𝑖+1
max 𝑒𝑖+1 − 𝑒𝑖 , 𝑒𝑖+2 − 𝑒𝑖+1

Uncorrelated GOE

GUE

2log 2 – 1 =0.38629…

0.5996(1) 0.6744(1)0.5307(1)GSE[Y. Y. Atas et al. PRL 2013]

Level repulsion
𝐾𝑒𝑖 − 𝑒𝑗𝛽2ෑ 𝑒 −𝛽𝐾𝑒𝑖 Τ4
𝑖=1➔ SYK model: level correlation(𝑃 𝑠 , 𝑃 𝑟 , ⟨𝑟⟩, etc.)
indistinguishable fromcorresponding Gaussian ensemble
Majorana SYK4 with𝑁 ≡ 0 (mod 8): GOE
𝑁 ≡ 2, 6 (mod 8): GUE
𝑁 ≡ 4 (mod 8): GSE[Fidkowski and Kitaev PRB 2010, 2011][You, Ludwig, and Xu PRB 2017]

Eigenvalue spectrum

J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H.
Shenker, D. Stanford, A. Streicher, and MT, JHEP 1705(2017)118

cf. DoS for large N [A. M. García-García and J. J. M. Verbaarschot: PRD 94, 126010 (2016)]

Plan
෡=
𝐻

• Sachdev-Ye-Kitaev model෍𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁• Maximally chaotic quantum mechanical model• SYK4+2• Departure from chaotic behavior

𝑁

෡=
𝐻

෍

𝑁

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑• Quantitative analysis of Fock-space localization• Many-body transition point• Spectral statistics, inverse participation ratio• Ergodicity of the eigenstates1≤𝑎<𝑏A. M. García-García, A. Romero-Bermúdez, B. Loureiro,
and MT, Phys. Rev. Lett. 120, 241603 (2018)

SYK4+2

Also see our reply [PRL 126, 109102 (2021)] to thecomment by J. Kim and X. Cao [PRL 126, 109101 (2021)]

Q.: Minimum requirements for chaotic behavior? (→ gravity interpretation?)
Study a simple model with analytical + numerical methods

෡=
𝐻

𝑁

SYK4

෍

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑Gaussian random couplings

𝑁

SYK2

1≤𝑎<𝑏
6𝐽
𝐽𝑎𝑏𝑐𝑑 : average 0, standard deviation 3Τ2𝑁
𝐾𝑎𝑏 : average 0, standard deviation

𝐽 = 1: unit of energy

𝐾
𝑁

SYK4 as unperturbed Hamiltonian,
𝐾 controls the strength of SYK2 (one-body random term, solvable)
Both terms respect charge parity in complex fermion description➔ Full numerical exact diagonalization (ED) of 2N/2-1-dimensional matrix, 𝑁 ≲ 34 possible

Here we take (GUE)𝑁 ≡ 2,6 (mod 8)

Large-N calculation for Out-of-Time Order Correlator (OTOC)
𝑁

෡=
𝐻

෍

SYK4

𝑁

SYK2

𝐾𝑎𝑏 : standard deviation

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑Chaos bound [Maldacena,
Shenker, and Stanford 2016]

𝐾
𝑁

1≤𝑎<𝑏

SYK4 limit →SYKq + κ SYK2
Large-q limit(𝐾 = 0)non-chaotic

normalized
Lyapunov exponent

𝐾 = 0.2

chaotic

0.5
12

Antonio M. GarciaGarcia, Bruno Loureiro,
Aurelio RomeroBermudez, and MT,
PRL 120, 241603(2018)

𝛽 = 1Τ 𝑘B 𝑇 inverse temperature

Deviation from the chaos bound as SYK2 component is introduced

PRL 120, 241603 (2018)

RMT-like behavior lost as SYK2 term is introduced
𝑃 𝑠 : level spacing distribution
Ratio of consecutive level spacing 𝐸𝑖+1 − 𝐸𝑖to the local mean level spacing Δ(requires unfolding of the spectrum)

SYK4 limit (small K):
Obeys random matrix theory (RMT)(GUE (Gaussian Unitary Ensemble) if 𝑁 ≡ 2,6 (mod 8))

SYK2 (large K): Poisson (𝑒 −𝑆 )N=30, Central 10 % of eigenvalues
Also see: T. Nosaka, D. Rosa, and J. Yoon, JHEP 1809, 041 (2018) for other symmetry cases
cf. A. V. Lunkin, K. S. Tikhonov, and M. V. Feigel’man, PRL 121, 236601 (2018); Y. Yu-Xiang, F. Sun, J. Ye, and W. M. Liu, 1809.07577, …

SYK 𝑞≥4 + SYK 2 : breakdown of chaos
𝑁

෡=
𝐻

෍

SYK4

𝑁

SYK2

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑1≤𝑎<𝑏

PRL 120, 241603 (2018)

𝐾𝑎𝑏 : standard deviation = 𝜅ൗ 𝑁

GUE
(Gaussian Unitary Ensemble)
Averagedratio between
neighboringenergy levelseparations

Poisson(uncorrelated)

We consider 𝑁
Majorana fermions
with normalization𝜒Ƹ 𝑎 , 𝜒Ƹ 𝑏 = 𝛿𝑎𝑏 here

Deviation from Gaussian random matrix as SYK2 component is introduced

Plan
෡=
𝐻

• Sachdev-Ye-Kitaev model෍𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1≤𝑎<𝑏<𝑐<𝑑≤𝑁• Maximally chaotic quantum mechanical model• SYK4+2• Departure from chaotic behavior

𝑁

෡=
𝐻

෍

𝑁

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏

1≤𝑎<𝑏<𝑐<𝑑• Quantitative analysis of Fock-space localization• Many-body transition point• Spectral statistics, inverse participation ratio• Ergodicity of the eigenstates1≤𝑎<𝑏Many-body localization

ETH: “(almost) all eigenstates are thermal(expectation values of operators = microcanonical average)”• Anderson localization: concept in non-interacting systems• Localization of wavefunctions due to scatterings at impurities• Many experiments in cold atom gases, optical fibers, etc.

• MBL: does localization occur in interacting systems?
[Gornyi, Mirlin, Polyakov 2005, Basko, Aleiner, Altshuler 2006, Oganesyan and Huse 2007, … many others]• Memory of initial conditions remains accessible at long times• Reduced density matrix on a subsystem does not approach a thermal one• Energy eigenstates do not obey Eigenstate Thermalization Hypothesis (ETH)• Area law, rather than volume law, of entanglement entropy• “Standard model”: spin-1/2 Heisenberg model + random field in z direction• Much debate on the location of the localization transition• See e.g. Avalanches [Crowley and Chandran PRR 2020][A. Morningstar et al., 2107.05642]
No localization? [Sels and Polkovnikov PRE 2021]

𝑁

𝑁

෢𝑧
෡ = ෍ 𝑆෡𝑖 ∙ 𝑆෢𝐻
𝑖+1 + ෍ ℎ𝑖 𝑆𝑖𝑖𝑖ℎ𝑖 ∈ [−ℎ, ℎ] uniform distribution

F. Monteiro, T. Micklitz, MT, and A. Altland, Phys. Rev. Research 3, 013023 (2021)

Our model and choice of basis
SYK 4 + 𝛿 SYK 2
𝑁=2𝑁D

෡=−
𝐻

𝑁

𝐽′𝑎𝑏𝑐𝑑 𝜓෠𝑎 𝜓෠ 𝑏 𝜓෠𝑐 𝜓෠ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜓෠𝑎 𝜓෠ 𝑏

෍1≤𝑎<𝑏<𝑐<𝑑1≤𝑎<𝑏Block-diagonalize the SYK 2 part(the skew-symmetric matrix 𝐾𝑎𝑏 has eigenvalues ±𝑣𝑗 )
2𝑁D

෡=−
𝐻

෍

𝑁D

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝑣𝑗 𝜒Ƹ 2𝑗−1 𝜒Ƹ 2𝑗

1≤𝑎<𝑏<𝑐<𝑑Normalization of 𝐽𝑎𝑏𝑐𝑑 , 𝑣𝑗 :SYK4 bandwidth = 1,
Width of 𝑣𝑗 distribution = 𝛿

𝑗=1
We choose 𝜓෠𝑎 , 𝜓෠ 𝑏 = 𝜒Ƹ 𝑎 , 𝜒Ƹ 𝑏 = 2𝛿𝑎𝑏 as thenormalization for the 𝑁 = 2𝑁D Majorana fermions.
1
For 𝑐𝑗Ƹ = 2 𝜒Ƹ 2𝑗−1 + i𝜒Ƹ 2j we have 𝑐𝑖Ƹ , 𝑐𝑗Ƹ † = 𝛿𝑖𝑗 .

F. Monteiro, T. Micklitz, MT, and A. Altland, Phys. Rev. Research 3, 013023 (2021)

Our model and choice of basis
𝑁 = 2𝑁D = 14: 27 = 128 states Basis diagonalizing the complex fermion number operators𝑛ො𝑗 = 𝑐𝑗Ƹ † 𝑐𝑗Ƹ → Sites: the 2𝑁D vertices of an 𝑁D -dim. hypercube.
1
𝑐𝑗Ƹ = 𝜒Ƹ 2𝑗−1 + i𝜒Ƹ 2j
2
2𝑁D

෡=−
𝐻

𝑁D

෍

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝑣𝑗 𝜒Ƹ 2𝑗−1 𝜒Ƹ 2𝑗

1≤𝑎<𝑏<𝑐<𝑑
2𝑁D

=−

෍
1≤𝑎<𝑏<𝑐<𝑑

1≤𝑗≤𝑁
𝑁D

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + ෍ 𝑣𝑗 2𝑛ො𝑗 − 11≤𝑗≤𝑁Each term of SYK 4 connects vertices with distance = 0, 2, 4.
|0001100⟩

For 𝑁 = 14, each vertex is directly connected with
1 (distance=0, itself) + 21 (distance=2) + 35 (distance=4)vertices out of the possible 2𝑁 = 128 (64 per parity).

F. Monteiro, T. Micklitz, MT, and A. Altland, Phys. Rev. Research 3, 013023 (2021)

Our model and choice of basis
2𝑁D Fock states

Basis diagonalizing the complex fermion number operators𝑛ො𝑗 = 𝑐𝑗Ƹ † 𝑐𝑗Ƹ → Sites: the 2𝑁D vertices of an 𝑁D -dim. hypercube.
bandwidth = 1
SYK 4 + 𝛿 SYK 2 SYK hopping
4

2𝑁

෡=−
𝐻
4

𝒪(𝑁 ) neighbors෍1≤𝑎<𝑏<𝑐<𝑑width of 𝑣𝑗 dist. = 𝛿site energy
𝑁

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + ෍ 𝑣𝑗 2𝑛ො𝑗 − 11≤𝑗≤𝑁Each term of SYK 4 connects vertices with distance = 0, 2, 4.
For 𝑁 = 34, each vertex is directly connected with
1 (distance=0, itself) + 136 (distance=2) + 2380 (distance=4)vertices out of the possible 2𝑁/2 = 131072 (65536 per parity).

See T. Micklitz, F. Monteiro, and A. Altland, Phys. Rev. Lett. 123, 125701 (2019).

Picture for Fock space localization
𝐽𝑖𝑗𝑘𝑙

2

=

6𝐽2
, hopping amplitude 𝑡 =2𝑁 3

𝐽𝑖𝑗𝑘𝑙 ∼ 𝐽𝑁

3
2−

𝑛 = ෍ 2𝑗−1 𝑛𝑗

1≤𝑗≤𝑁
𝑣𝑛 : operator diagonal in the occupation number basis
Here (on this slide only): drawn from a box distribution of width Δ ∼ 𝑁 𝛼
1
2

1

SYK4 bandwidth ∼ 𝐽𝑁 set to unity, 𝐽 ∼ 𝑁−2
3Characteristic hopping amplitude by SYK4 Hamiltonian: 𝑡 ∼ 𝐽𝑁

−2

∼ 𝑁 −2𝜅Hybridization of two nearest neighbors occur for Δ of 𝒪 𝑁 4 neighbors
Number of neighbors
in resonance
By Fermi’s golden rule, 𝜅 ∼4𝑁 𝜅𝑡 2 Δ /𝜅.

Broadened energy

Typical level broadening: 𝜅1∴ 𝜅 ∼ Δ ∼ 𝑁 −𝛼 by self-consistency;𝜅−2 ∼ 𝑁 −2𝛼 .
∼
Δ
Δ

𝑁4
Typical # of resonant neighboring levels: Δ2 ∼ 𝑁 4−2𝛼 . If 𝛼 > 2, localization (fragmentation of the Fock space) is possible

What happens in the case that 𝑣𝑛 are given by SYK2 eigenvalues?

Four regimes of disorder strengths• Typical energy difference between
arbitrary pair of sites ≲ 1I

෡=𝐻෡4 + 𝐻෡2
𝐻width of 𝑣𝑗 dist. = 𝛿

Site energy of site #m:𝑁ΤII

• Typical energy difference > 1, but difference෡4 ≲ 1
between sites directly connected by 𝐻𝜖(𝑚=σ𝑗−1 𝑛 ) =𝑗
1≤𝑗≤𝑁 2III

෡4 > 1• Difference between sites connected by 𝐻

Width of 𝜖𝑚 dist. =෡4 = −
𝐻

෍

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1≤𝑎<𝑏<𝑐<𝑑

SYK4 bandwidth = 1

𝛿c

𝛿

𝑁D 𝛿

2𝑁

𝐸

IV

෍ −1 𝑛𝑗 −1 𝑣𝑗
1≤𝑗≤𝑁

𝐸

1

෡2 = ෍ 𝑣𝑗 2𝑛ො𝑗 − 1
𝐻
1≤𝑗≤𝑁

𝐸

𝑁D−1 2𝑁• Fock space localization (𝛿c ∼ 𝑁D2 ln 𝑁D for Bethe lattice)[Altshuler, Gefen, Kamenev, and Levitov, PRL 78, 2803 (1997)]
logarithmic correction from hybridization beyond nearest neighbors

Diagnostic quantities: Moments of wave functions
and spectral two-point correlation function• Moments of eigenstate wave functions
𝐼𝑞 = 𝜈 −1 ෍

𝜓 𝑛 2𝑞 𝛿 𝐸𝜓• Spectral two-point correlation function
𝐾 𝜔

𝐽

𝑛,𝜓

𝜈 = 𝜈 𝐸 ≃ 0 , 𝜈 𝐸 = ෍ 𝛿 𝐸 − 𝐸𝜓𝐽𝜓➔Parametrizes localization, allows
comparison with numerics
𝜓 𝑛 4 𝛿 𝐸𝜓𝐽

𝜈

𝐴𝐵 𝑐 = 𝐴𝐵 𝐽 − 𝐴 𝐽 𝐵 𝐽

𝜔
−
2

c
𝜔
−
2

0

𝜔
2

𝐸

➔ Reflects level repulsion if the spectrum israndom matrix-like:1
inverse participation ratio (IPR), ≤ 𝐼2 ≤ 1
𝐷

Equal weights
𝐷: dimension of |𝑛⟩ = 2𝑁−1𝜈𝜔2

c: connected partwith average density of states at band center

𝐼2 = 𝜈 −1 σ𝑛,𝜓= 𝜈 −2Single nonzero element

We calculate these quantities for large N
and compare against numerical results

Analytical results

PRR 3, 013023 (2021)

Method: Exact matrix integral representation;
mapping to a supersymmetric sigma model;
saddle point equations; effective medium approximation(𝑁D =• I: Average density of states (ADoS) at band center 𝜈 = 𝑐𝐷

I

Fully delocalized• 𝐼𝑞 = 𝑞! 𝐷1−𝑞

Eigenenergy spectral
statistics (for odd 𝑁 case𝛿Τ𝑁D−1 2II• II: ADoS 𝜈 =

𝑐𝐷
𝑁D 𝛿

, spread of wave functions 𝐷res ≃1−𝑞• 𝐼𝑞 = 𝑞! 𝐷resfor simplicity)𝐷
𝑁D 𝛿

sin2 𝑠𝑠෩𝐾 𝑠 =1− 2 +𝛿,𝑠𝜋
𝑠 = 𝜋𝜔𝜈 in I, II, III :agrees with Gaussian
Unitary Ensemble (GUE)

Restricted
𝛿

1

III
𝛿c

IV
𝛿

• III: ADoS 𝜈 =
•

𝑐𝐷
𝑁D 𝛿

, spread of wave functions 𝐷res ≃4 𝑁𝛿 21−𝑞
𝐼𝑞 = 𝑞! 𝐷res = 𝑞 2𝑞 − 3 ‼ 𝜋𝐷

𝑞−1

𝑁
, 𝑐 = O 1 , 𝐷 = 2𝑁D −1 )2𝐷𝑁D 𝛿 2Strongly restricted

2
𝑁D
𝛿c =
log 𝑁D for large 𝑁
4 3• IV: All eigenstates localized to 𝒪(1) sites

𝛿

IV: Poisson statistics

PRR 3, 013023 (2021)

Inverse participation ratio vs prediction for III
IPR 𝐼2 = average of σ𝑛 𝜓 𝑛 4 for normalized 𝜓,1≤ 𝐼2 ≤ 1
𝐷

Equal weights

𝑁D = 11

𝑞 2𝑞−3 ‼
𝜋𝐷
𝐼𝑞 = 2 1−𝑞𝛿
4 𝑁D

𝑁D = 131−𝑞= 𝑞 2𝑞 − 3 ‼

4 𝑁D 𝛿 2
2𝑁−1 𝜋Single non-zero element

𝑁D = 15

𝑞−1

in III

Central 1/7 of the energy spectrum

Higher moments ofeigenvectors
𝐼𝑞 = 𝜈 −1 ෍

𝜓 𝑛 2𝑞 𝛿 𝐸𝜓

Analytical prediction:𝑞 2𝑞−3 ‼
𝜋𝐷
𝐼𝑞 = 𝛿2 1−𝑞
4 𝑁D

PRR 3, 013023 (2021)

1−𝑞= 𝑞 2𝑞 − 3 ‼

4 𝑁D 𝛿 2
2𝑁−1 𝜋𝐽𝑛,𝜓Good agreement up to large 𝑞 for 𝛿 ∼ 1

𝑁D = 15

Central 1/7 of the energy spectrum

𝑞−1

in IIIPRR 3, 013023 (2021)

Spectral statistics: gap ratio distribution
Measure difference by Kullback-Leibler (KL)𝑃 𝑥
divergence: 𝐷KL (𝑃| 𝑄 = σ𝑥 𝑃 𝑥 log
.
𝑄 𝑥

𝜹

𝐷KL (𝑃(𝛿 , 𝑟)||𝑃Poisson 𝑟 )𝐷KL (𝑃(𝛿 , 𝑟)||𝑃GUE 𝑟 )30.36085 × 10−6140.12340.146340

0.00960.5705

𝑁D = 15𝑟=(Analytical prediction: 𝛿c =

𝑍
𝑊 2𝑍
2𝜌

𝜋 = 38.47)

min 𝐸𝑖+1 − 𝐸𝑖 , 𝐸𝑖+2 − 𝐸𝑖+1
max 𝐸𝑖+1 − 𝐸𝑖 , 𝐸𝑖+2 − 𝐸𝑖+1

PRR 3, 013023 (2021)

Departure from random matrix 𝑃 𝑟 occurs
after 𝐼2 has grown significantly

I
𝑁D = 15

II

III

IV

Summary so far…

Felipe Monteiro, Tobias Micklitz, Masaki
Tezuka, and Alexander Altland, Phys. Rev.
Research 3, 013023 (2021) arXiv:2005.12809

Fock space localization in
many-body quantum systems

Sachdev-Ye-Kitaev model
as tractable system

Analytical estimate of inverse
participation ratio, spectral
statistics

Numerical calculation of inverse
participation ratio, energy
spectrum correlation

Four regimes (I: ergodic, II: localization starts,
III: localization rapidly progresses, IV: MBL)
found in SYK4 + δ SYK2 system(in SYK2-diagonal basis);
I, II, III are chaotic while IV is not

Prediction for momenta of eigenstate
wavefunctions 𝐼𝑞 is verified by parameter freecomparison, and energy spectrum statistics is
consistent with GUE/Poisson transition wellafter entering regime III➔ Are eigenstates ergodic? Behavior of the entanglement entropy?

Physics just outside MBL (regions II & III)?
• Thermal phase smoothly connected to extended states (as those intranslationally invariant models)?
• Non-ergodic extended (NEE) states discussed for several models(Bethe lattice, random regular graphs, disordered Josephson junction chains, …)“golf course” potential energy landscape“Non-ergodic extended phase of the Quantum Random Energy Model”[L. Faoro, M. V. Feigel’man, L. Ioffe, Ann. Phys. 409, 167916 (2019)]

PRL 127, 030601 (2021)

Evaluation of entanglement entropyAB

𝑁𝐴 bits

𝑁𝐵 = 𝑁 − 𝑁𝐴 bits

Zero-energy eigenstate 𝜓 , density matrix 𝜌 = |𝜓⟩⟨𝜓|Reduced density matrix 𝜌𝐴 = tr𝐵 𝜌

Fock space ℱ = ℱ𝐴 ⊗ ℱ𝐵
𝑛 = (𝑙, 𝑚)

Entanglement entropy 𝑆𝐴 = −tr𝐴 (𝜌𝐴 ln𝜌𝐴 )Evaluate disorder averaged moments 𝑀𝑟 = ⟨tr𝐴 𝜌𝐴𝑟 ⟩, 𝑆𝐴 = −𝜕𝑟 𝑀𝑟 |𝑟=1 .
𝒩 = 𝑛1 , 𝑛2 , … , 𝑛𝑟 , 𝒩𝐴 = 𝑙1 , 𝑙 2 , … , 𝑙 𝑟 , 𝒩𝐵 = 𝑚1 , 𝑚2 , … , 𝑚𝑟
𝑙
𝑚

𝑙
𝑚

𝜌𝐴𝑟 =

෍
𝑙 1 ,…,𝑙𝑟
𝑚1 ,…,𝑚𝑟

1 1
𝜓 (𝑙 ,𝑚 ) 𝜓(𝑙 2 ,𝑚1 )2 2
𝜓 (𝑙 ,𝑚 ) 𝜓(𝑙 3 ,𝑚2 )𝑟 𝑟⋯ 𝜓 (𝑙 ,𝑚 ) 𝜓(𝑙1 ,𝑚𝑟 )PRL 127, 030601 (2021)

Evaluation of power of reduced density matrix
1

𝑛1
𝜌𝐴𝑟 =

෍

𝜓

(𝑙 1 ,𝑚1 )

𝑛2

𝑛
𝜓

(𝑙 2 ,𝑚1 )𝜓𝑛(𝑙2 ,𝑚2 )𝜓

2(𝑙 3 ,𝑚2 )⋯𝜓(𝑙𝑟 ,𝑚𝑟 )𝜓(𝑙1 ,𝑚𝑟 )𝑙 1 ,…,𝑙𝑟
𝑚1 ,…,𝑚𝑟

For this sum to survive disorder averaging,
1 2𝑟
𝒩 = 𝑛1 , 𝑛2 , … , 𝑛𝑟 and 𝒩 = 𝑛 , 𝑛 , … , 𝑛 should be equal as sets,
𝒩𝑖 = 𝒩

1

2

𝜎 𝑖

3

4

𝑛1 = 𝑛 , 𝑛2 = 𝑛 , 𝑛3 = 𝑛 , 𝑛4 = 𝑛 , 𝑛5 = 𝑛

5

1

𝟒

𝑟

𝑀𝑟 = tr𝐴 𝜌𝐴𝑟= ෍ ෍ ෑ 𝜓𝑛 𝑖
𝜎

𝒩 𝑖=1

3

𝟐

𝑛1 = 𝑛 , 𝒏𝟐 = 𝒏 , 𝑛3 = 𝑛 , 𝒏𝟒 = 𝒏 , 𝑛5 = 𝑛
2𝛿𝒩𝐴 , 𝜎∘𝜏 𝒩𝐴 𝛿𝒩𝐵 ,𝜎𝒩𝐵

5

PRR 3, 013023 (2021)

Four regimes of SYK4 + 𝛿SYK2• I: Uniform distribution of wave functions, 𝜈𝑛 = 𝜈I𝐸Τ𝑁D−1 2II

Δ4 = 𝒪(1)III

𝐸
Nearest neighbors remain energetically close, 𝛿 ≪ Δ4 , and level broadening 𝜅 = Δ4

• II, III: Global DoS 𝐷𝜈 ≈Only 𝒪𝐷
2𝜋𝑁D 𝛿

Δ4 2
𝛿

, spectral measure 𝜌𝑛 ≃1 Δ
𝑒
𝐷A Δ 𝐵of nearest neighbors remain in resonance, broadening reduced to 𝜅~Δ24 /𝛿
𝐸

𝛿c

IV
𝛿

𝑣2
− 𝐴2
2Δ𝐵

2
𝑁D
𝛿c =
log 𝑁D for large 𝑁
4 3• IV: All eigenstates localized to 𝒪(1) sites

PRL 127, 030601 (2021)

Regime I: maximally random case

𝑀𝑟 = ⟨tr𝐴 𝜌𝐴𝑟 ⟩, 𝑆𝐴 = −𝜕𝑟 𝑀𝑟 |𝑟=1

Uniform distribution of wave functions, 𝜈𝑛 = 𝜈𝑀𝑟 ≈ 𝐷𝐴1−𝑟 +𝑟 2−𝑟 −1
𝐷 𝐷𝐵
2 𝐴

Up to single transpositions

Difference from the thermal value 𝑆th = ln 𝐷𝐴
𝐷𝐴
𝑆𝐴 − 𝑆th = −2𝐷𝐵Exponentially small if 𝑁𝐴 ≪ 𝑁𝐵 ;𝑆𝐴 very close to the thermal value

Leading termSingle transpositions: next leading term

uniform𝑟

𝑀𝑟 = tr𝐴 𝜌𝐴𝑟

𝐷𝐴(𝐵) = 2𝑁𝐴(𝐵) −1= ෍ ෍ ෑ 𝜓𝑛 𝑖
𝜎

𝒩 𝑖=1

2

𝛿𝒩𝐴 , 𝜎∘𝜏 𝒩𝐴 𝛿𝒩𝐵 ,𝜎𝒩𝐵PRL 127, 030601 (2021)

Regimes II and III: reduced effective dimension• Assume ergodicity and calculate 𝑆𝐴• Energy shell: extended cluster ofresonant sites (width 𝜅) embedded in the
Fock space• Neighboring sites of 𝑛: energy 𝑣𝑚 =𝑣𝑛 ± 𝒪 𝛿 , much more likely to be in thesame shell because 𝛿 ≪ Δ2 = 𝑁D 𝛿

Additional assumptions• Exponentially large number of sites →self averaging(sum over site energies = average overapprox. Gaussian distributed
contributions of subsystem energies tothe total energy)• Total energy 𝐸 ~ 𝐸𝐴 + 𝐸𝐵

➔ Up to single transpositions (justified in 1 ≪ 𝑁𝐴 ≪ 𝑁D & replica limit):
1
𝑁D
𝑁𝐴
𝑁D 𝐷𝐴 in Regimes II, III
𝑆𝐴 − 𝑆th = − ln+−1(≪ 𝛿 < 𝛿c ~𝑁D2 ln 𝑁D )2
𝑁𝐵
2𝑁𝐷
2𝑁𝐴 2𝐷𝐵𝑁
D

𝐷𝐴
𝑆𝐴 − 𝑆th = −2𝐷𝐵
in Regime I

PRL 127, 030601 (2021)

Offset from the thermal value𝑆th − 𝑆𝐴𝑁D = 14 (𝑁 = 28 Majorana fermions)

𝑆th
Plateau expected➔ Numerically checked
𝐷𝐴
2𝐷𝐵

I

1

𝑁D

II

𝒪(1)

III IV
𝛿c

𝛿

1
𝑁D
𝑁𝐴
𝑁D 𝐷𝐴
𝑆𝐴 − 𝑆th = − ln
+
−
(< 0)
2
𝑁𝐵
2𝑁𝐷
2𝑁𝐴 2𝐷𝐵in Regimes II, III (1
𝑁D≪ 𝛿 < 𝛿c ~𝑁D2 ln 𝑁D )Phys. Rev. Research 3, 013023 (2021) arXiv:2005.12809
with Felipe Monteiro, Tobias Micklitz, and Alexander Altland

Phys. Rev. Lett. 127, 030601 (2021) arXiv:2012.07884

Summary

with F. Monteiro, A. Altland, David A. Huse, and T. Micklitz• The Sachdev-Ye-Kitaev (SYK) model: quantum mechanical model realizing
chaos bound (~ random matrix, black holes)• Several experimental proposals, small systems realized• SYK4+2: analytically tractable model for many-body localization (MBL)• Fock space: (N/2)-dimensional hypercube• Analytical results on eigenfunction moments and MBL point➔ Agreement with numerical results without free paramters• Evaluation of entanglement entropy 𝑆𝐴 assuming ergodicity in energy shells➔ Agreement between the numerical and analytical results• Sparse SYK