Effective dimension, level statistics,
and integrability of
Sachdev-Ye-Kitaev-like models
One-day workshop for QFT and string theory
Dec. 14, 2018 @ Osaka city university

Keywords:
Chaos and SYK model
the role of disorder in SYK

Eiki Iyoda(University of Tokyo)

E. Iyoda, H. Katsura, and T. Sagawa, Phys. Rev. D 98, 086020 (2018)

Background 1/3: chaos and SYK model
Chaos: "High sensitivity to initial conditions"e.g. the butterfly effectquantum "analogues" → quantum chaos

SYK model (Sachdev-Ye-Kitaev model)
Model of N randomly interacting fermions
Attracted attentions inholography, condensed matter,quantum information, etc....

Background 2/3: Blackhole information paradox
Quantum mechanics: information is never lost
Unitary time evolution<latexit sha1_base64="pB8AlDpt5wZIc0uUPNaljDLgjEs=">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</latexit>

General relativity: information is "lost"
because the Hawking radiation is "thermal"

Hawking radiation

Black
hole

<latexit sha1_base64="4gZa2TCo3TcKBsqitXaPUZ+Ppn4=">AAACcXichVHLSgMxFD0d3/VVdaO4KS2KINY7IiiuRDcufbWK2paZMdXBeTGTFrT4A/6AgqsKIuJnuPEHXPgJ4rKCGxfeTgdERb0hycnJPTcnie5ZZiCJnmJKS2tbe0dnV7y7p7evPzEwmAvcsm+IrOFarr+ta4GwTEdkpSktse35QrN1S2zpR8uN/a2K8APTdTblsSfytnbgmCXT0CRTBVGoTu3pQmrJldPpnWIiTRkKI/kTqBFII4pVN3GDPezDhYEybAg4kIwtaAi47UIFwWMujypzPiMz3Bc4RZy1Zc4SnKExe8TjAa92I9bhdaNmEKoNPsXi7rMyiTF6pFuq0wPd0TO9/1qrGtZoeDnmWW9qhVfsPxveePtXZfMscfip+tOzRAnzoVeTvXsh07iF0dRXTs7rGwvrY9VxuqIX9l+jJ7rnGziVV+N6TaxfIs4foH5/7p8gN5NRKaOuzaYXl6Kv6MQoUpjg957DIlawiiyf6+MCNVzF6sqIklRSzVQlFmmG8CWUyQ+wLo7T</latexit>

How to bridge these two conflicting theories?

Background 3/3: "Chaotic bound and SYK"
Thought experiment by Hayden-Preskill (2007)→ "Blackhole reflects quantum information like a mirror."
To make the phenomena possible,
BH should be a very good scrambler of quantum information.
Besides, BH is the "fastest scrambler" in nature.
Hayden-Preskill, Sekino-Susskind

In the sense of "the chaotic bound",proven by Maldacena-Shenker-Stanford.
SYK model saturates the chaotic bound.

Background 3/3: "Chaotic bound and SYK"
Thought experiment by Hayden-Preskill (2007)→ "Blackhole reflects quantum information like a mirror."
To make the phenomena possible,
BH should be a very good scrambler of quantum information.
Besides, BH is the "fastest scrambler" in nature.
Hayden-Preskill, Sekino-Susskind

In the sense of "the chaotic bound",proven by Maldacena-Shenker-Stanford.
SYK model saturates the chaotic bound.
We will review.

What I will talk about....
"SYK model and chaos" attracts attention
not only in high-energy physics,
but also in condensed matter physics.

What I will talk about....
"SYK model and chaos" attracts attention
not only in high-energy physics,
but also in condensed matter physics.
I study statistical physics.
- thermalization- dynamics of Q-info...

What I will talk about....
"SYK model and chaos" attracts attention
not only in high-energy physics,
but also in condensed matter physics.
I study statistical physics.
- thermalization- dynamics of Q-info...
1. Review
1.1 Chaos and quantum chaos
1.2 Information paradox and chaotic bound
1.3 SYK model
2. Wishart SYK model"The role of disorder in SYK model"
E. Iyoda, H. Katsura, and T. Sagawa,PRD 98, 086020 (2018) or 1806.10405

ChaosHigh sensitivity to initial conditions
e.g. the butterfly effect
Small deviation is amplified by chaotic dynamics.
<latexit sha1_base64="4qiEzFSrngIb0hx2nn3+D/rc+G4=">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</latexit>: Lyapunov exponent<latexit sha1_base64="EXZ15h8kry6xo4sJxmlxFK8pcEg=">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</latexit>

trajectory 2<latexit sha1_base64="FhGw+n3sGciia8LUeaz1EE64YOY=">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</latexit><latexit sha1_base64="En263AEUmrv0t1sv+wzVzgW/aOc=">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</latexit>

trajectory 1TimeChaosExamples
1. billiard potential
2. time dependent external field: Kicked rotator model
3. interaction: double pendulum, coupled oscillator

How about chaos in quantum systems?

"Absense" of chaos in quantum systems"Quantum systems do not have high sensitivity toinitial conditions."
Ex. Numerical experiment of recurrence
1. (forward) time evolution
2. time reverse operation
3. backward time evolution<latexit sha1_base64="/QreTvkVNNv/nKd0eRTWrqfaxIM=">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</latexit><latexit sha1_base64="a9ymGR2DYX4QjZ303up33FesD5A=">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</latexit><latexit sha1_base64="2eE5bk+LqD1UBVnFTf/8Z5pgqRE=">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</latexit>→ The system should go back to the initial state at<latexit sha1_base64="ijjYYJE+yn0yXFjMzBjbuU8HdqE=">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</latexit>

Classical systems:
recurrence is violated due to the numerical errors.
"initial" deviations at

Quantum systems: recurrence is observed.

<latexit sha1_base64="a9ymGR2DYX4QjZ303up33FesD5A=">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</latexit>.

"Absense" of chaos in quantum systems
Difference between classical and quantum systems
Ex. Baker's map
Chaos is also characterized by folding and stretchingof phase space.

→ an infinitely small structure is madeHowever, in quantum systems, a smaller structure
than the Planck cell cannot be made.
It there no chaos in quantum systems...?

Quantum chaos
In quantum systems,
how do we see the remnant of classical chaos?
BGS conjecture (Bohigas-Giannoni-Schmit)
PRL 52, 1 (1984)"Energy spectra of quantum systems, whose classicalcounterpart is chaotic, show the same fluctuation
properties as predicted by random matrix theory (RMT)."(The original paper studied quantum billiard and GOE.)(No proof. Many numerical evidences.)

Level-spacing distribution obeys Wigner-Dyson distribution
in non-integrable quantum many-body systems.
(Berry-Tabor)(Wigner-Dyson: GOE, GUE, or GSE in RMT)

Quantum chaos: level statistics
Eigenenergies
Level spacing<latexit sha1_base64="dnOh6Di1kxvRarYT3VTVPuWgRIY=">AAACcXichVHLSsNAFD2Nr1pfVTeKm2JRFKHciKCIi6IUXGprVfBRkjjq2DQJSVrQoh/gDyi4UhARP8ONP+DCTxCXFdy48DYNiIp6h5k5c+aeO2dmdMeUnk/0FFGamlta26LtsY7Oru6eeG/fqmeXXUPkDdu03XVd84QpLZH3pW+KdccVWkk3xZpeXKjvr1WE60nbWvEPHbFV0vYsuSsNzWdqO1OQc5nCQeJkTM4djBfiSUpREImfQA1BEmEs2fEbbGIHNgyUUYKABZ+xCQ0etw2oIDjMbaHKnMtIBvsCx4ixtsxZgjM0Zos87vFqI2QtXtdreoHa4FNM7i4rExihR7qlGj3QHT3T+6+1qkGNupdDnvWGVjiFntOB3Nu/qhLPPvY/VX969rGLmcCrZO9OwNRvYTT0laOzWm42O1IdpSt6Yf+X9ET3fAOr8mpcL4vsBWL8Aer35/4JVidTKqXU5alkej78iiiGMIwxfu9ppLGIJeT5XBfnuMRVpKYMKglluJGqREJNP76EMvEBtGyOVw==</latexit><latexit sha1_base64="vlDEPNhLpeFBMshT5L2ydvwd7U8=">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</latexit>

Sinai's billiard

1d Quantum spin chain
integrable

non-integrable

O. Bohigas, M. J. Giannoni,
and C. Schmit, PRL (1984)

L. Santos, J. Phys. A (2004)

Energy fluctuation is universal, which is characterized
by random matrix: GOE, GUE, or GSE.
Gaussian Orthogonal/Unitary/Symplectic Ensemble

Each ensemble corresponds to the symmetry of Hamiltonian

Quantum chaotic properties in many-body systems
Indicators of quantum chaos
Level statistics
integrable

Loschmidt Echo

Eigenstate thermalization
hypothesis

non-integrable

integrable

non-integrable
integrable
non-integrable
L. Santos, J. Phys. A (2004)

T. Gorin et al., Phys. Rep. (2006)

W. Beugeling et al., PRE (2014)

Recently, another quantum analogue of chaos,"scrambling", attracts attention in the context of- black hole information paradox- "A chaotic bound"Scrambling
Recently, another quantum analogue of chaos has
been investigated → scrambling (delocalization)
Example of quantum spins
Initial state

Final state

singletLocally encoded entanglement (correlation) is
delocalized by unitary time evolution.

Scrambling
Recently, another quantum analogue of chaos has
been investigated → scrambling (delocalization)
Example of quantum spins
Initial state

Final state

singletLocally encoded entanglement (correlation) is
delocalized by unitary time evolution.
Indicators: Decay of out-of-time-ordered correlator (OTOC)
Negativity of tripartite mutual information (TMI)
Hosur, Qi, Roberts, Yoshida, JHEP 02, 004 (2016)

Scrambling
OTOC
a squared commutatorLet<latexit sha1_base64="pxTAcZ+BVyVlqKD+OKHxbvXsrxI=">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</latexit>

comes from<latexit sha1_base64="+6Q3vvNsOIi8pJi4xRqFmR3Mrwc=">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</latexit>(position and momentum)

Replace the commutators by the Poisson brackets inthe semiclassical limit→ scrambling is another quantum analogue ofquantum chaos (in the sense of the above equation).
<latexit sha1_base64="2ko2315NNEisLcJzjPEioh2RMb0=">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</latexit>

Note: The squared commutator is more essential than OTOC...?
see Hamazaki, Fujimoto and Ueda, 1807.02360 for example

Blackhole information paradox
Thought experiment by Hayden and Preskill (2007)
Alice throws<latexit sha1_base64="dsmRjeFA4m+omZvwL7h3yiJ9PRA=">AAACb3ichVHLSsNAFD2Nr1ofrbpQEKRYFN2UGxEUV0U3LrW1VbAiSRzrYJqEJC1o9Qf8AF248AEi4me48Qdc9BPElSi4ceFtGhAV9Q4zc+bMPXfOzOiOKT2fqB5RWlrb2juinbGu7p7eeKKvv+DZFdcQecM2bXdN1zxhSkvkfembYs1xhVbWTbGq7y409lerwvWkba34e47YKGslS25LQ/OZKh4UHU8WXc0qmWIzkaI0BZH8CdQQpBDGkp24RhFbsGGggjIELPiMTWjwuK1DBcFhbgM15lxGMtgXOESMtRXOEpyhMbvLY4lX6yFr8bpR0wvUBp9icndZmcQYPdANvdA93dIjvf9aqxbUaHjZ41lvaoWzGT8ayr39qyrz7GPnU/WnZx/bmA28SvbuBEzjFkZTX90/ecnNZcdq43RJT+z/gup0xzewqq/G1bLIniLGH6B+f+6foDCVVimtLk+nMvPhV0QxjFFM8HvPIINFLCHP5zo4xhnOI8/KoDKiJJupSiTUDOBLKJMfsDyO5w==</latexit>(m-qubits) into the BH

Bob holds a quantum memory maximally entangled with the BH.
(For example, a half of the BH has evaporated and Bob hascollected the Hawking radiations.)
Bob's goal:reconstructremaining
BH<latexit sha1_base64="dsmRjeFA4m+omZvwL7h3yiJ9PRA=">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</latexit>from the output.

Hawkingradiation

unitary U
inside BH
Alice<latexit sha1_base64="dsmRjeFA4m+omZvwL7h3yiJ9PRA=">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</latexit>

Bob

Ref. see also B. Yoshida's talk slide@ISSP (Aug. 2018)

Blackhole information paradox
Thought experiment by Hayden and Preskill (2007)
Alice throws(m-qubits) into the BH<latexit sha1_base64="dsmRjeFA4m+omZvwL7h3yiJ9PRA=">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</latexit>

Bob holds a quantum memory maximally entangled with the BH.
(For example, a half of the BH has evaporated and Bob hascollected the Hawking radiations.)
Bob's goal:reconstructremaining
BH

unitary U
inside BH

from the output.

<latexit sha1_base64="dsmRjeFA4m+omZvwL7h3yiJ9PRA=">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</latexit>

Hawkingradiation→ collecting
qubits ofthe Hawking radiation is enough<latexit sha1_base64="kmdQm2QGjm2kDXWt7G+a0UBHEg4=">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</latexit>

BH reflects quantum information

Alice<latexit sha1_base64="dsmRjeFA4m+omZvwL7h3yiJ9PRA=">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</latexit><latexit sha1_base64="dsmRjeFA4m+omZvwL7h3yiJ9PRA=">AAACb3ichVHLSsNAFD2Nr1ofrbpQEKRYFN2UGxEUV0U3LrW1VbAiSRzrYJqEJC1o9Qf8AF248AEi4me48Qdc9BPElSi4ceFtGhAV9Q4zc+bMPXfOzOiOKT2fqB5RWlrb2juinbGu7p7eeKKvv+DZFdcQecM2bXdN1zxhSkvkfembYs1xhVbWTbGq7y409lerwvWkba34e47YKGslS25LQ/OZKh4UHU8WXc0qmWIzkaI0BZH8CdQQpBDGkp24RhFbsGGggjIELPiMTWjwuK1DBcFhbgM15lxGMtgXOESMtRXOEpyhMbvLY4lX6yFr8bpR0wvUBp9icndZmcQYPdANvdA93dIjvf9aqxbUaHjZ41lvaoWzGT8ayr39qyrz7GPnU/WnZx/bmA28SvbuBEzjFkZTX90/ecnNZcdq43RJT+z/gup0xzewqq/G1bLIniLGH6B+f+6foDCVVimtLk+nMvPhV0QxjFFM8HvPIINFLCHP5zo4xhnOI8/KoDKiJJupSiTUDOBLKJMfsDyO5w==</latexit>

Bob

like a mirror.

Ref. see also B. Yoshida's talk slide@ISSP (Aug. 2018)

Blackhole information paradox
Scrambling makes the phenomena predicted by Hayden
and Preskill possible.
How much time should does Bob need?
→ scrambling time<latexit sha1_base64="JgaMdE46KPyUk8rDLqyi5jCzk54=">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</latexit><latexit sha1_base64="OHGRrldsgGnWZhzjzriGhGXcyfo=">AAACZHichVHLSsNAFD2Nr1qrrRZBEEQsiqtyI4LiqujGZR+2CrWUJI4amiYhSQu1+AO6VVy4UhARP8ONP+CiPyCISwU3LrxNA6Ki3mFmzpy5586ZGdU2dNcjaoeknt6+/oHwYGQoOjwSi4+OFV2r7miioFmG5WypiisM3RQFT/cMsWU7QqmphthUq2ud/c2GcFzdMje8pi3KNWXP1Hd1TfGYypqVeJJS5Mf0TyAHIIkgMlb8GtvYgQUNddQgYMJjbECBy60EGQSbuTJazDmMdH9f4BAR1tY5S3CGwmyVxz1elQLW5HWnpuurNT7F4O6wchqz9EA39EL3dEtP9P5rrZZfo+OlybPa1Qq7EjuayL/9q6rx7GH/U/WnZw+7WPa96uzd9pnOLbSuvnFw9pJfyc225uiSntn/BbXpjm9gNl61q6zInSPCHyB/f+6foLiQkiklZxeT6dXgK8KYxAzm+b2XkMY6MijwuQLHOMFp6FGKSglpvJsqhQJNAl9CmvoA4kyJ7g==</latexit>: total number ofdegree of freedom

1. Blackhole complementarity : to avoid quantum cloning<latexit sha1_base64="B+j3Ls9FQrnabwY00MiHqinGzE4=">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</latexit>

2. Quantum information theory : signal should reach the whole system<latexit sha1_base64="B+j3Ls9FQrnabwY00MiHqinGzE4=">AAACf3ichVG7SgNBFD1ZXzG+ojaiTTD4asJdEXxUoo2lr6hgJOyuY7K4L3cngRgEa3/AwkpBRSz0H2z8AQs/QSwVbCy82SyIinqHmTlz5p47Z2Z0zzIDSfQYUxoam5pb4q2JtvaOzq5kd89a4JZ8Q2QN13L9DV0LhGU6IitNaYkNzxearVtiXd+dr+2vl4UfmK6zKiue2LK1gmPumIYmmcon+2U+Z2uy6NvVwPAPcgWxl8pZbmHUGcsn05ShMFI/gRqBNKJYdJOXyGEbLgyUYEPAgWRsQUPAbRMqCB5zW6gy5zMyw32BAyRYW+IswRkas7s8Fni1GbEOr2s1g1Bt8CkWd5+VKQzRA13RC93TNT3R+6+1qmGNmpcKz3pdK7x811Hfytu/KptnieKn6k/PEjuYCr2a7N0LmdotjLq+vH/8sjKzPFQdpjN6Zv+n9Eh3fAOn/GqcL4nlEyT4A9Tvz/0TrI1nVMqoSxPp2bnoK+IYwCBG+b0nMYsFLCLL5x7iAje4VWLKiJJRqJ6qxCJNL76EMv0Bst2Tug==</latexit>

Fast scrambling conjecture (Hayden-Preskill, Sekino-Susskind)<latexit sha1_base64="Qg1aF4zqtyINfdBAn/W1qasg+Og=">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</latexit>→ proven by Maldacena-Shenker-Stanford

"Chaotic bound" or "MSS bound"
Decay of OTOC has a universal upper bound.
Maldacena, Shenker, and Stanford, JHEP 08, 106 (2016)
Tsuji, Shitara, and Ueda, arXiv:1706.09160

If OTOC shows an exponential growth,<latexit sha1_base64="pgE7v5LX9gj7pl/DSHZbwA4wNw0=">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</latexit><latexit sha1_base64="P3ihZ/ye/oR7DDkpxh+b9Us9K9U=">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</latexit><latexit sha1_base64="eiA5PDWf+wUzyxkelnSbe3X9Rbw=">AAACbnichVHLSsNAFD2N7/poVRBBRLEorsqNCIqrohuX1doHVJEkjjo0TUKSFrT4A+7FhaAoiIif4cYfcNFPEDdCBTcuvE0DoqLeMJkzZ+65c+aO7pjS84nqEaWtvaOzq7sn2tvXPxCLDw7lPLviGiJr2KbtFnTNE6a0RNaXvikKjiu0sm6KvF5aae7nq8L1pG1t+AeO2Cpre5bclYbmM1XcrGqucDxp2tZ2PEFJCmLyJ1BDkEAYaTt+g03swIaBCsoQsOAzNqHB468IFQSHuS3UmHMZyWBf4AhR1lY4S3CGxmyJ/3u8KoasxetmTS9QG3yKycNl5SSm6ZFuqUEPdEdP9P5rrVpQo+nlgGe9pRXOdux4NPP2r6rMs4/9T9Wfnn3sYjHwKtm7EzDNWxgtffXwtJFZWp+uzdAVPbP/S6rTPd/Aqr4a12ti/QxRfgD1e7t/gtxcUqWkujafSC2HT9GNMUxhlvu9gBRWkUY26NgJznEReVFGlHFlopWqRELNML6EMvsB0byOgw==</latexit>: certain small positive expansion parameter

Then, the decay rate of the OTOC is upper-bounded:chaotic bound (MSS bound)<latexit sha1_base64="O+5i2gE5gFTBQ23GMlVUyEBSI8g=">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</latexit>

Note on MSS bound 1
MSS made physical assumptions on time-ordered
Maldacena, Shenker, and Stanford,
correlations.
JHEP 08, 106 (2016)<latexit sha1_base64="O2PT+ExVBSwrETDbGDjjubBwRmo=">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</latexit>

Meanwhile, Tsuji et al. obtained similar results without
Tsuji, Shitara, and Ueda,
making the above assumption.
arXiv:1706.09160, PRE 97, 012101
Some differences on assumptions.
Both have good and bad points.

Note on MSS bound 2
Quantum many-body models saturating the MSS bound・2d-conformal field theory with large-c limit
Roberts and Stanford, PRL (2015)

・Sachdev-Ye-Kitaev model with large-N limit
Kitaev (2014,2015)"nearly holographic dual" of AdS2

Maldacena and Stanford, PRD (2016)"nearly conformal quantum mechanics"

Sachdev-Ye-Kitaev model
Kitaev, Talks at KITP (2015)
Sachdev, PRX (2015)
Maldacena and Stanford,
PRD (2016)

Majorana fermions- disordered interaction- no kinetic termis sampled from Gaussian with varianceNotes
1. Kitaev introduced SYK with Majorana fermions→ tractable or "solvable" in large-N limit(not integrable)

Sachdev-Ye-Kitaev model

Maldacena and Stanford,
PRD (2016)

We can calculate many quantities in large-N limita. Two-point Green function(and taking disorder average)

Non-Fermi liquid(similar to TLL)
Tomonaga-Luttinger liquidb. Four-point functions, including OTOC

→ saturates "the chaotic bound" in large-N andconformal (low energy/strong coupling) limit

Sachdev-Ye-Kitaev model
Notes
2. SYK with complex fermions is also tractable
and has essentially similar features.
Sachdev, PRX (2015)
3. Sachdev-Ye originally investigated generalized
Sherrington-Kirkpatrick model to study spin liquid
Sachdev and Ye,
PRL (1993)

for spin glass

N sites, SU(M) spin
This model becomes bosonic/fermionic model in some limits.
Kitaev simplified them with Majorana fermions.

Table of contents
1. Review
1.1 Chaos and quantum chaos
1.2 Information paradox and chaotic bound
1.3 SYK model
2. Wishart SYK model"The role of disorder in SYK model"
2-1. Dynamics of tripartite mutual information→ disordered SYK and clean SYK
2-2. Wishart SYK・Huge degeneracy and fluctuation of OTOC
・Level statistics (quantum chaos) and integrability
E. Iyoda, H. Katsura, and T. Sagawa, Phys. Rev. D 98, 086020 (2018)

Model
SYK model with complex fermions: complex Gaussian with variancesatisfying
Clean SYK model<latexit sha1_base64="uvR3ZWXJNluFwCry4iMoSU8OFBM=">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</latexit>

Calculate tripartite mutual information (TMI) of SYK model→ indicator of scrambling
What is the role of disorder in the SYK model?

Tripartite mutual information

Hosur, Qi, Roberts, Yoshida,
JHEP 02, 004 (2016)
Cerf and Adami,
Physics D, 120, 62(1998)

・von Neumann entropy: Reduced density operator in region X

・Bipartite mutual information (BMI)

A

B
A∩B

Correlation between A and B

B

・Tripartite mutual information (TMI)

A
Meaning of TMI → see examples in the next slideCNegativity of TMIExamples of three classical bits
1.
2.
3.

:random(three-body correlation):random, independent:random, independent
Neither nor is individually correlated with .
The composite
is correlated with .
→ Information about is delocalized to and(no correlation)

Negativity of TMIExamples of three classical bits
1.
2.
3.

:random(three-body correlation):random, independent(no correlation):random, independent
Neither nor is individually correlated with .
The composite
is correlated with .
→ Information about is delocalized to and

TMI is negative when→ scrambling: delocalization of quantum information
Cf. Ryu-Takayanagi formula
Hayden, Headrick, Maloney, PRD 87, 046003 (2013)

SetupSchematic of the total system:- qubit A- SYK model (BCD)

1. Prepare
2. Apply the CNOT gate (A:control qubit, B:target qubit)→ Information about A is encoded in B through entanglement

3. Only BCD evolves with a HamiltonianIn this setup,
we calculate dynamics of TMI

Numerical result: TMI
E. Iyoda, and T. Sagawa, Phys. Rev. A 97, 042103 (2018)｝16 samples

Disordered SYK model exhibits scrambling evenin a single disorder realization
Larger temporal fluctuation in the clean SYK model→ We will discuss this.

Difference between disordered and clean SYK models
What is the origin of
temporal fluctuationsof the clean SYK model?
Introduce and investigate a variant of SYK model→ Wishart SYK model for fermions/bosons
Includes the clean SYK as a special case

Difference between disordered and clean SYK models
What is the origin of
temporal fluctuationsof the clean SYK model?
Introduce and investigate a variant of SYK model→ Wishart SYK model for fermions/bosons
Includes the clean SYK as a special case
SYK model<latexit sha1_base64="tLZuAr083rCv8SpfPsbkX7O1z/U=">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</latexit>

Clean SYKDifference between disordered and clean SYK models
What is the origin of
temporal fluctuationsof the clean SYK model?
Introduce and investigate a variant of SYK model→ Wishart SYK model for fermions/bosons
Includes the clean SYK as a special case
SYK model<latexit sha1_base64="tLZuAr083rCv8SpfPsbkX7O1z/U=">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</latexit>

Wishart SYKClean SYKVariants of SYKMany variants of the SYK model have been investigated
not only in high-energy physics,
but also in condensed matter physics→ Non-fermi liquid, quantum critical, disorder in correlated systems

q-point interactions
SUSY extensions

Maldacena and Stanford, PRD (2016)
Fu et al., PRD (2017); Sannomiya et al., PRD (2017)
Peng et al., JHEP (2017); Li et al., JHEP (2017)
Kanazawa and Wetting, JHEP (2017)

disorder-free tensor models

Peng et al., JHEP (2017)
Witten, arXiv: 1610.09758 (2016)

and Yao, PRL (2017)
lattice structure Jian
Gu et al., JHEP (2017); Berkooz et al., JHEP (2017)

coupled or perturbed system

Song et al., PRL (2017); Chen et al., PRL (2017);
Bi et al., PRB (2017); Chen et al., JHEP (2017);
Garcia-Garcia et al., JHEP (2017);
Garcia-Garcia et al., PRL (2018);
Zhang and Zhai, PRB (2018)

Proposals for experiments

Cold atom, photo association

Quantum wire with Majorana

Graphene device

Topologicalsuperconductor

Definition of Wishart SYK model
named after the Wishart matrices in random matrix theory: complex Gaussian (mean=, variance=): annihilation operator of complex fermionsNote
1. The total fermion number is conserved.
2. The clean SYK when
3. Positive semidefinite → eigenenergies are non-negative
4. Wishart SYK for hard-core bosons is defined in the same manner.

replaceGround-state degeneracy
Wishart SYK model has huge ground-state degeneracy

Dimension of the sector with: the binomial coefficient<latexit sha1_base64="mDODoN0T9u9p8gkr7qEpOFA2d+0=">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</latexit>

Q changes the dimension of the sector→ This is negative when

: the floor function

If Q decreases the dimension of the sector,
there exists eigenstates ofwith zero eigenenergy

Ground-state degeneracy
The number of zero-energy eigenstates in the sector with

By summing up with respect to

,

Note
1.

increases exponentially in→ The residual entropy is extensive.

2. We numerically confirmed that these bounds are saturated.
(We prove the equalities hold for the fermionic Wishart SYK.)Note:=2 SUSY SYKFu et al., PRD (2017)
Sannomiya et al., PRD (2017)
Kanazawa and Wetting, JHEP (2017)=2 SUSY SYKsupercharge
Fermionic Wishart SYK

1. SUSY SYK has zero-energy eigenstates
2. The number of annihilation operators is different
The fermionic parity

SUSY
Fermionic Wishart

Numerical result: energy spectrum
Conventional SYK

1.5

Wishart SYK
Non-negative
Ground-state degeneracy→ equals

Eigenenergy

Similar between
fermionic SYK andbosonic SYK

1.5(a) fermionic SYK

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5
1

0

4000

8000

12000

(c) fermionic Wishart SYK

-1.5
1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

4000

8000

12000

0

(b) bosonic SYK

0

0

4000

8000

12000

4000

8000

12000

(d) bosonic Wishart SYKIndex of eigenstate

Fermion: many degenerate excited statesNumerical result: OTOC
0.1

0.05

fermionic SYK

0

0

-0.1

-0.05

-0.2

-0.1

(a)

-0.3
10-1

(b)

fermionic Wishart SYK
100

101

102

103

104

bosonic SYK

105

-0.15
10-1

bosonic Wishart SYK
100

101

102

103

104

105

Fermionic Wishart SYK has
larger temporal fluctuations at late times#sample=128
Initial state: eigenstate
SYK
Wishart SYKTheorem for relaxation and effective dimension

Reimann, PRL (2008); Short and Farrely, NJP (2012)

Initial state of many-body systemdoes not depend on<latexit sha1_base64="vUEw8bo2fxP/kkJGQDRWrlEA1zg=">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</latexit><latexit sha1_base64="5PyFIs3w5K9EQ9/jpUyZ01NHzgo=">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</latexit>

Effective dimension: "the number of eigenstates" in<latexit sha1_base64="o9u9J/OjmlvQ63gfh3SntnQzBd0=">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</latexit>

Reimann, PRL (2008)
Short and Farrely, NJP (2012)

Theorem for temporal fluctuation around long time average
Expectation value of observable
relaxes if the effective dimension is large: maximum degeneracy of energy gaps

We expect a similar bound holds for OTOC.

Numerical result: OTOC vs effective dimension
0.1

Each point represents acomputational basis state.

fermionic SYK
fermionic Wishart SYKbosonic SYKbosonic Wishart SYK

0.08

0.06

We expect

0.04

0.02

0

0

0.5

1

1.5

2

2.5

3

can be valid
for the case of OTOC

Huge degeneracy in the ground state
decreases the effective dimension smaller,
which make the temporal fluctuation larger

Table of contents
1. Review
1.1 Chaos and quantum chaos
1.2 Information paradox and chaotic bound
1.3 SYK model
2. Wishart SYK model"The role of disorder in SYK model"
2-1. Dynamics of tripartite mutual information→ disordered SYK and clean SYK
2-2. Wishart SYK・Huge degeneracy and fluctuation of OTOC
・Level statistics (quantum chaos) and integrability
E. Iyoda, H. Katsura, and T. Sagawa, Phys. Rev. D 98, 086020 (2018)

Level statistics (quantum chaos)
Level statistics of energy level spacings is described by- Poisson distribution (integrable)- Wigner-Dyson distribution (non-integrable)→ GOE, GUE, GSE (random matrix)
Previous studies for SYK and SUSY SYK→ The conventional SYK is quantum chaotic.
Level statistics is GUE.
You et al, PRB (2017)
Kanazawa and Wetting, JHEP (2017)

We numerically investigate
the level statistics of the Wishart SYK models.

Level statistics (quantum chaos)#samples=24

2
1.5

GSE
GUE

1

GOE
Poisson

0.5
00

0.2

0.4

0.6

0.8

fermionSYK
Wishart
SYK

boson
SYK
WishartSYK

1

SYK and Boson Wishart: GUE
Fermionic Wishart: Poisson
The Poisson distribution implies the following possibilities.
1. integrable or 2. we missed another symmetry

Integrability of fermionic Wishart SYKFor simplicity, we assume

is even andis real.

1. Because J is real skew-symmetric matrix,
we can block-diagonalize it.
..

2. Introduce new fermion operators:.

Integrability of fermionic Wishart SYK

3. Hamiltonian is written with the new operators→ This is a special case of the Richardson-Gaudin model,
which is known to be integrable with algebraic Bethe ansatz.
Richardson, J. Math. Phys. (1965)
Balantekin, PRC (2007)

We can also show the integrability by explicitly constructing
conserved quantities.

Summary 1/2
In the first part, we reviewed chaos and SYK model.
In the second part, we investigated the Wishart SYK model.
→ to consider the role of disorder in SYKE. Iyoda, H. Katsura, and T. Sagawa,Phys. Rev. D 98, 086020 (2018)

1. Huge ground-state degeneracy affects
dynamics of OTOC through effective dimensions→ OTOC of the fermionic Wishart SYK models exhibits
large temporal fluctuations at late times

2. Level statistics

SYK: GUE
Bosonic Wishart: GUE or GOE
Fermionic Wishart: Poisson

3. Fermionic Wishart SYK is integrable

Summary 2/2
Introduce and investigate a variant of SYK model→ Wishart SYK model for fermions/bosons
Includes the clean SYK as a special case
SYK model

Clean SYK

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Wishart SYK

Although fermionic Wishart SYK is integrable (not chaotic),
we believe it serves as a reference for evaluating the effectof disorder on the original SYK model.