Many-body localization in the SYK model: quantitative study as a Fock space localization

8pL2-5SYK模型における多体局在：
フォック空間での局在としての定量的解析
arXiv:2005.12809

Table of contents
1. Introduction:a.
b.

Many-body localization and Fock-spacelocalization
Sachdev-Ye-Kitaev model and deformation

2. Analytical predictions
3. Numerical results
4. Summary

Masaki TEZUKA (Kyoto University)
in collaboration with Felipe Monteiro, Tobias Micklitz, and Alexander Altland1

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Localization phenomena in quantum systems:
what is Fock space localization?
Anderson localization of (uncorrelated)single particle wave functions:
Analytical techniques, goodagreements with numerics

Fock space localization

Many-body localization (MBL) in 1D &higher dimensions with correlation:
Few analytical results, numerics
exponentially hard, many open issues
Ψ → 𝜙1 𝜑2 |𝜁3 ⟩ ⋯ |𝜉𝑁 ⟩• Spatially confined quantum systems (e.g. quantum dots)• Many-body eigenstates localizing in the Fock space[Altshuler-Gefen-Kamenev-Levitov PRL 1997, Silvestrov PRL 1997, PRB 1998, …]

ergodic,
uniform states

Nonergodic, extended(NEE) states

nonergodic,localized statesparameter2

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The Sachdev-Ye-Kitaev model
N Majorana- or Dirac- fermions with all-to-all random couplings[Majorana version]
෡=𝐻3!
𝑁 3/2

෍

[Dirac version]
෡=𝐻

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1
† †෍
𝐽𝑐
Ƹ
𝑐𝑗Ƹ 𝑐𝑘Ƹ 𝑐𝑙Ƹ
𝑖𝑗;𝑘𝑙𝑖
2𝑁 3/2𝑖𝑗;𝑘𝑙

1≤𝑎<𝑏<𝑐<𝑑≤𝑁[Kitaev’s talks][S. Sachdev: PRX 5, 041025 (2015)]“Two-body random ensemble”
since 1970s

𝐽𝑎𝑏𝑐𝑑 independently chosen fromGaussian random ensemble[A. Kitaev: talks at KITP (2015)]

cf. S. Sachdev and J. Ye, PRL 70, 3339 (1993)

Solvable in the large-N limit(after sample average ⋯𝐽 )➔ Maximally chaotic (𝜆L = 2π𝑘B 𝑇Τℏ)

See e.g. [J. Polchinski and V. Rosenhaus, JHEP 1604 (2016) 001]
[J. Maldacena and D. Stanford, PRD 94, 106002 (2016)][段下一平, 手塚真樹, 花田政範: 日本物理学会誌 73(8), 569 (2018)]

Candidate of quantum system holographically corresponding to black holes (experimental study of quantum gravity)

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SYK 𝑞≥4 + SYK 2 : breakdown of chaos𝑁෡=𝐻෍

SYK4

𝑁

SYK2𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏1≤𝑎<𝑏<𝑐<𝑑1≤𝑎<𝑏𝐾𝑎𝑏 : standard deviation = 𝜅ൗ 𝑁GUE(Gaussian Unitary Ensemble)
Averaged ratio
betweenneighboringenergy levelseparations

Poisson(uncorrelated)

A. M. Garcia-Garcia, B. Loureiro,
A. Romero-Bermudez, and MT,
PRL 120, 241603 (2018)(cf. T. Nosaka, D. Rosa, and J. Yoon,JHEP09(2018)041 for GOE, GSE;
Yu Yi-Xiang, Fadi Sun, Jinwu Ye, W.M. Liu,
arXiv:1809.07577 for odd N cases)
We consider 𝑁
Majorana fermions
with normalization𝜒Ƹ 𝑎 , 𝜒Ƹ 𝑏 = 𝛿𝑎𝑏 here

Deviation from Gaussian random matrix as SYK2 component is introduced

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arXiv:2005.12809

Our model and choice of basis
SYK 4 + 𝛿 SYK 22𝑁

෡=−𝐻෍
1≤𝑎<𝑏<𝑐<𝑑2𝑁𝐽′𝑎𝑏𝑐𝑑 𝜓෠𝑎 𝜓෠ 𝑏 𝜓෠𝑐 𝜓෠ 𝑑 + 𝑖 ෍ 𝐾𝑎𝑏 𝜓෠𝑎 𝜓෠ 𝑏
1≤𝑎<𝑏

Block-diagonalize the SYK 2 part(the skew-symmetric matrix 𝐾𝑎𝑏 has eigenvalues ±𝑣𝑗 )
2𝑁

෡=−𝐻

෍
1≤𝑎<𝑏<𝑐<𝑑𝑁𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + 𝑖 ෍ 𝑣𝑗 𝜒Ƹ 2𝑗−1 𝜒Ƹ 2𝑗
1≤𝑗≤𝑁

Normalization of 𝐽𝑎𝑏𝑐𝑑 , 𝑣𝑗 : We choose 𝜓෠𝑎 , 𝜓෠ 𝑏 = 𝜒Ƹ𝑎 , 𝜒Ƹ𝑏 = 2𝛿𝑎𝑏 as thenormalization for the 2𝑁 Majorana fermions.
SYK4 bandwidth = 1,1†Width of 𝑣𝑗 distribution = 𝛿 For 𝑐𝑗Ƹ = 2 𝜒Ƹ2𝑗−1 + ⅈ𝜒Ƹ2j we have 𝑐𝑖Ƹ , 𝑐𝑗Ƹ = 𝛿𝑖𝑗 .

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arXiv:2005.12809

Our model and choice of basis
𝑁 = 7: 27 = 128 states

Basis diagonalizing the complex fermion number operators𝑛ො𝑗 = 𝑐𝑗Ƹ † 𝑐𝑗Ƹ → Sites: the 2𝑁 vertices of an 𝑁-dim. hypercube.
1𝑐𝑗Ƹ = 𝜒Ƹ 2𝑗−1 + ⅈ𝜒Ƹ 2j22𝑁

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

1≤𝑎<𝑏<𝑐<𝑑2𝑁=−෍1≤𝑎<𝑏<𝑐<𝑑1≤𝑗≤𝑁
𝑁

𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + ෍ 𝑣𝑗 2𝑛ො𝑗 − 11≤𝑗≤𝑁

Each term of SYK 4 connects vertices with distance = 0, 2, 4.
|0001100⟩

For 𝑁 = 7, 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).

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Our model and choice of basis
2𝑁 Fock states

Basis diagonalizing the complex fermion number operators𝑛ො𝑗 = 𝑐𝑗Ƹ † 𝑐𝑗Ƹ → Sites: the 2𝑁 vertices of an 𝑁-dim. hypercube.

SYK 4 + 𝛿 SYK 22𝑁

෡=−𝐻𝒪(𝑁 4 ) neighbors෍1≤𝑎<𝑏<𝑐<𝑑𝑁𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑 + ෍ 𝑣𝑗 2𝑛ො𝑗 − 11≤𝑗≤𝑁

Each term of SYK 4 connects vertices with distance = 0, 2, 4.
For 𝑁 = 17, each vertex is directly connected with
1 (distance=0, itself) + 136 (distance=2) + 2380 (distance=4)vertices out of the possible 2𝑁 = 131072 (65536 per parity).

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

෡=𝐻
෡4 + 𝐻෡2
𝐻IIwidth of 𝑣𝑗 dist. = 𝛿

Site energy of site #m:𝑁• Typical energy difference > 1, but difference෡4 ≲ 1
between sites connected by 𝐻III

෡4 > 1• Difference between sites connected by 𝐻෍ −1 𝑛𝑗 −1 𝑣𝑗𝜖(𝑚=σ𝑁𝑗−1 𝑛 ) =𝑗1≤𝑗≤𝑁 21≤𝑗≤𝑁Width of 𝜖𝑚 dⅈst. = 𝑁𝛿𝐸1

෡2 = ෍ 𝑣𝑗 2𝑛ො𝑗 − 1𝐻1≤𝑗≤𝑁𝐸𝑁 −1Τ2𝑁2𝑁

෡4 = −𝐻෍𝐽𝑎𝑏𝑐𝑑 𝜒Ƹ 𝑎 𝜒Ƹ 𝑏 𝜒Ƹ 𝑐 𝜒Ƹ 𝑑1≤𝑎<𝑏<𝑐<𝑑

SYK4 bandwidth = 1𝐸𝛿cIV𝛿• Fock space localization (𝛿c ∼ 𝑁 2 log 𝑁 for Bethe lattice)[Altshuler, Gefen, Kamenev, and Levitov, PRL 78, 2803 (1997)]

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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 𝜔𝐽𝑛,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
−2c
𝜔
−
20𝜔
2𝐸

➔ Reflects level repulsion if the spectrum israndom matrix-like:1
inverse participation ratio (IPR), ≤ 𝐼2 ≤ 1
𝐷Equal weights
𝐷: dimension of |𝑛⟩ = 2𝑁−1𝜈𝜔2c: connected partwith average density of states at band center

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

We calculate these quantities for large Nand compare against numerical results
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Analytical results

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

Iergodic• 𝐼𝑞 = 𝑞! 𝐷1−𝑞II• II: ADoS 𝜈 =𝑐𝐷, spread of wave functions 𝐷res ≃𝑁𝛿

1−𝑞• 𝐼𝑞 = 𝑞! 𝐷resfor simplicity)𝐷𝑁𝛿

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

weakly non-ergodic
𝛿1III• III: ADoS 𝜈 =•𝑐𝐷
, spread of wave functions 𝐷res ≃𝑁𝛿

4 𝑁𝛿 21−𝑞
𝐼𝑞 = 𝑞! 𝐷res = 𝑞 2𝑞 − 3 ‼ 𝜋𝐷𝑞−1(𝑐 = O 1 , 𝐷 = 2𝑁−1 )Eigenenergy spectral
statistics (for odd 𝑁 case𝛿𝑁 −1Τ2arXiv:2005.12809

𝐷𝑁𝛿 2strongly non-ergodic
𝛿𝛿cIV𝛿𝑁2𝛿c =log 𝑁 for large 𝑁4 3

IV: Poisson statistics• IV: All eigenstates localized to 𝒪(1) sites
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Inverse participation ratio vs prediction for III
IPR 𝐼2 = average of σ𝑛 𝜓 𝑛 4 for normalized 𝜓,1≤ 𝐼2 ≤ 1
𝐷

Equal weights

𝑁 = 11

𝑁 = 13𝑞 2𝑞−3 ‼ 𝜋𝐷 1−𝑞
4 𝑁𝛿 2𝐼𝑞 = 2 1−𝑞= 𝑞 2𝑞 − 3 ‼ 𝑁−1
𝛿4 𝑁2𝜋Single non-zero element

𝑁 = 15𝑞−1

in III

Central 1/7 of the energy spectrum

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Analytical prediction:Moments of eigenvectors

𝑞 2𝑞−3 ‼ 𝜋𝐷 1−𝑞4 𝑁𝛿 2𝐼𝑞 = 𝛿2 1−𝑞 4 𝑁= 𝑞 2𝑞 − 3 ‼ 2𝑁−1 𝜋

Good agreement up to large 𝑞 for 𝛿 ∼ 1𝑞−1

Central 1/7 of the energy spectrumin III10

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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 )30.3608

5 × 10−6

140.1234

0.1463400.0096

0.5705𝑁 = 15𝑟=(𝛿c =𝑍2𝜌𝑊 2𝑍 𝜋 = 38.47)

mⅈn 𝐸𝑖+1 − 𝐸𝑖 , 𝐸𝑖+2 − 𝐸𝑖+1max 𝐸𝑖+1 − 𝐸𝑖 , 𝐸𝑖+2 − 𝐸𝑖+111

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

IIIIIIIV

𝑁 = 1512

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Summary

Felipe Monteiro, Tobias Micklitz,
Masaki Tezuka, and Alexander Altland,
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 inverseparticipation 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