Effective dimension, level statistics, and integrability of - - PowerPoint PPT Presentation

Effective dimension, level statistics, 
 and integrability of 
 Sachdev-Ye-Kitaev-like models

Eiki Iyoda

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

One-day workshop for QFT and string theory

• Dec. 14, 2018 @ Osaka city university

(University of Tokyo) Keywords: Chaos and SYK model the role of disorder in SYK

quantum "analogues" → quantum chaos

Background 1/3: chaos and SYK model

Chaos: "High sensitivity to initial conditions" SYK model (Sachdev-Ye-Kitaev model)

e.g. the butterfly effect Model of N randomly interacting fermions Attracted attentions in

holography, condensed matter, quantum information, etc....

Quantum mechanics: information is never lost General relativity: information is "lost" because the Hawking radiation is "thermal"

Background 2/3: Blackhole information paradox

Hawking radiation

Black hole

Unitary time evolution

How to bridge these two conflicting theories?

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.

Background 3/3: "Chaotic bound and SYK"

Besides, BH is the "fastest scrambler" in nature. In the sense of "the chaotic bound", proven by Maldacena-Shenker-Stanford. SYK model saturates the chaotic bound.

Hayden-Preskill, Sekino-Susskind

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.

Background 3/3: "Chaotic bound and SYK"

SYK model saturates the chaotic bound. We will review. Besides, BH is the "fastest scrambler" in nature. In the sense of "the chaotic bound", proven by Maldacena-Shenker-Stanford.

Hayden-Preskill, Sekino-Susskind

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....

• 1. Review

1.1 Chaos and quantum chaos 1.2 Information paradox and chaotic bound 1.3 SYK model "SYK model and chaos" attracts attention not only in high-energy physics, but also in condensed matter physics. I study statistical physics.

• E. Iyoda, H. Katsura, and T. Sagawa,

PRD 98, 086020 (2018) or 1806.10405

• 2. Wishart SYK model

"The role of disorder in SYK model"

• thermalization
• dynamics of Q-info...

Chaos

High sensitivity to initial conditions e.g. the butterfly effect : Lyapunov exponent Small deviation is amplified by chaotic dynamics.

Time trajectory 1 trajectory 2

Chaos

Examples

• 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 to initial conditions."

• Ex. Numerical experiment of recurrence
• 1. (forward) time evolution
• 2. time reverse operation
• 3. backward time evolution

→ The system should go back to the initial state at .

Classical systems: recurrence is violated due to the numerical errors. Quantum systems: recurrence is observed.

"initial" deviations at

"Absense" of chaos in quantum systems

Difference between classical and quantum systems

• Ex. Baker's map

Chaos is also characterized by folding and stretching

• f phase space.

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

Quantum chaos

In quantum systems, how do we see the remnant of classical chaos? "Energy spectra of quantum systems, whose classical counterpart is chaotic, show the same fluctuation properties as predicted by random matrix theory (RMT)."

(The original paper studied quantum billiard and GOE.)

Level-spacing distribution obeys Wigner-Dyson distribution in non-integrable quantum many-body systems.

(No proof. Many numerical evidences.) (Berry-Tabor)

BGS conjecture (Bohigas-Giannoni-Schmit)

PRL 52, 1 (1984)

(Wigner-Dyson: GOE, GUE, or GSE in RMT)

Quantum chaos: level statistics

Level spacing

Eigenenergies Sinai's billiard 1d Quantum spin chain Energy fluctuation is universal, which is characterized by random matrix: GOE, GUE, or GSE.

Each ensemble corresponds to the symmetry of Hamiltonian

Gaussian Orthogonal/Unitary/Symplectic Ensemble

• L. Santos, J. Phys. A (2004)

integrable non-integrable

• O. Bohigas, M. J. Giannoni,

and C. Schmit, PRL (1984)

Level statistics

Indicators of quantum chaos

Quantum chaotic properties in many-body systems

Loschmidt Echo

Eigenstate thermalization hypothesis

• L. Santos, J. Phys. A (2004)
• T. Gorin et al., Phys. Rep. (2006)
• W. Beugeling et al., PRE (2014)

integrable non-integrable non-integrable integrable non-integrable integrable

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)

Initial state Example of quantum spins Final state Locally encoded entanglement (correlation) is delocalized by unitary time evolution. singlet

Scrambling

Recently, another quantum analogue of chaos has been investigated → scrambling (delocalization)

Initial state Example of quantum spins Final state Locally encoded entanglement (correlation) is delocalized by unitary time evolution. singlet Indicators: Decay of out-of-time-ordered correlator (OTOC)

Hosur, Qi, Roberts, Yoshida, JHEP 02, 004 (2016)

Negativity of tripartite mutual information (TMI)

Scrambling

OTOC Let → scrambling is another quantum analogue of quantum chaos (in the sense of the above equation). comes from a squared commutator

(position and momentum)

Replace the commutators by the Poisson brackets in the semiclassical limit

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) unitary U inside BH

Bob holds a quantum memory maximally entangled with the BH.

Alice throws (m-qubits) into the BH

(For example, a half of the BH has evaporated and Bob has collected the Hawking radiations.) Alice Bob

remaining BH Hawking radiation

Bob's goal: reconstruct from the output.

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

Blackhole information paradox

Thought experiment by Hayden and Preskill (2007) unitary U inside BH

Bob holds a quantum memory maximally entangled with the BH.

Alice throws (m-qubits) into the BH

(For example, a half of the BH has evaporated and Bob has collected the Hawking radiations.) Alice Bob

remaining BH Hawking radiation

Bob's goal: reconstruct from the output.

→ collecting qubits of the Hawking radiation is enough

BH reflects quantum information 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?

• 1. Blackhole complementarity : to avoid quantum cloning
• 2. Quantum information theory : signal should reach the whole system

→ scrambling time Fast scrambling conjecture (Hayden-Preskill, Sekino-Susskind) → proven by Maldacena-Shenker-Stanford

: total number of 
 degree of freedom

"Chaotic bound" or "MSS bound"

Decay of OTOC has a universal upper bound. If OTOC shows an exponential growth, Then, the decay rate of the OTOC is upper-bounded:

Maldacena, Shenker, and Stanford, JHEP 08, 106 (2016)

: certain small positive expansion parameter

Tsuji, Shitara, and Ueda, arXiv:1706.09160

chaotic bound (MSS bound)

Note on MSS bound 1

MSS made physical assumptions on time-ordered correlations. Meanwhile, Tsuji et al. obtained similar results without making the above assumption.

Maldacena, Shenker, and Stanford, JHEP 08, 106 (2016) Tsuji, Shitara, and Ueda, 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 ・Sachdev-Ye-Kitaev model with large-N limit ・2d-conformal field theory with large-c limit

"nearly holographic dual" of AdS2

Maldacena and Stanford, PRD (2016) Roberts and Stanford, PRL (2015)

"nearly conformal quantum mechanics"

Kitaev (2014,2015)

Sachdev-Ye-Kitaev model

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

Notes

• Majorana fermions
• disordered interaction
• no kinetic term

is sampled from Gaussian with variance

• 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 limit

• a. Two-point Green function
• b. Four-point functions, including OTOC

Non-Fermi liquid → saturates "the chaotic bound" in large-N and conformal (low energy/strong coupling) limit (similar to TLL)

Tomonaga-Luttinger liquid

(and taking disorder average)

Sachdev-Ye-Kitaev model

Notes SYK with complex fermions is also tractable and has essentially similar features.

Sachdev, PRX (2015)

Sachdev-Ye originally investigated generalized Sherrington-Kirkpatrick model to study spin liquid N sites, SU(M) spin

Sachdev and Ye, PRL (1993)

2. 3. This model becomes bosonic/fermionic model in some limits. Kitaev simplified them with Majorana fermions.

for spin glass

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

: complex Gaussian with variance satisfying SYK model with complex fermions Clean SYK model Calculate tripartite mutual information (TMI) of SYK model → indicator of scrambling What is the role of disorder in the SYK model?

Tripartite mutual information

・von Neumann entropy

: Reduced density operator in region X

A B C A B A∩B

・Bipartite mutual information (BMI) ・Tripartite mutual information (TMI)

Meaning of TMI → see examples in the next slide Correlation between A and B

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

Negativity of TMI

Examples of three classical bits

:random 1.

(three-body correlation)

:random, independent 2.

(no correlation)

:random, independent 3. → Information about is delocalized to and Neither nor is individually correlated with . The composite is correlated with .

Negativity of TMI

Examples of three classical bits

:random 1.

(three-body correlation)

:random, independent 2.

(no correlation)

• Cf. Ryu-Takayanagi formula

Hayden, Headrick, Maloney, PRD 87, 046003 (2013)

TMI is negative when

→ scrambling: delocalization of quantum information

:random, independent 3. → Information about is delocalized to and Neither nor is individually correlated with . The composite is correlated with .

Setup

Schematic of the total system:

• qubit A
• SYK model (BCD)
• 1. Prepare
• 3. Only BCD evolves with a Hamiltonian

In this setup, we calculate dynamics of TMI

• 2. Apply the CNOT gate (A:control qubit, B:target qubit)

→ Information about A is encoded in B through entanglement

Numerical result: TMI

Larger temporal fluctuation in the clean SYK model → We will discuss this. Disordered SYK model exhibits scrambling even in a single disorder realization 16 samples ｝

• E. Iyoda, and T. Sagawa, Phys. Rev. A 97, 042103 (2018)

Difference between disordered and clean SYK models

Introduce and investigate a variant of SYK model → Wishart SYK model for fermions/bosons Includes the clean SYK as a special case What is the origin of 
 temporal fluctuations 


• f the clean SYK model?

Difference between disordered and clean SYK models

Introduce and investigate a variant of SYK model → Wishart SYK model for fermions/bosons Includes the clean SYK as a special case What is the origin of 
 temporal fluctuations 


• f the clean SYK model?

SYK model

Clean SYK

Difference between disordered and clean SYK models

Introduce and investigate a variant of SYK model → Wishart SYK model for fermions/bosons Includes the clean SYK as a special case What is the origin of 
 temporal fluctuations 


• f the clean SYK model?

SYK model

Clean SYK Wishart SYK

Variants of SYK

→ Non-fermi liquid, quantum critical, disorder in correlated systems

Many variants of the SYK model have been investigated
 not only in high-energy physics, but also in condensed matter physics q-point interactions lattice structure

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 SUSY extensions

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

coupled or perturbed system

Jian and Yao, PRL (2017) Gu et al., JHEP (2017); Berkooz et al., JHEP (2017) 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 Graphene device Quantum wire with Majorana Topological superconductor

Definition of Wishart SYK model

• 1. The total fermion number is conserved.

: complex Gaussian (mean= , variance= ) Note

• 2. The clean SYK when

: annihilation operator of complex fermions

• 4. Wishart SYK for hard-core bosons is defined in the same manner.

replace

named after the Wishart matrices in random matrix theory

• 3. Positive semidefinite → eigenenergies are non-negative

Ground-state degeneracy

Q changes the dimension of the sector If Q decreases the dimension of the sector, there exists eigenstates of with zero eigenenergy Wishart SYK model has huge ground-state degeneracy Dimension of the sector with

: the binomial coefficient

→ This is negative when

: the floor function

Ground-state degeneracy

The number of zero-energy eigenstates in the sector with By summing up with respect to , Note

• 2. We numerically confirmed that these bounds are saturated.
• 1. increases exponentially in

→ The residual entropy is extensive.

(We prove the equalities hold for the fermionic Wishart SYK.)

Note: =2 SUSY SYK

• 1. SUSY SYK has zero-energy eigenstates

=2 SUSY SYK Fermionic Wishart SYK

• 2. The number of annihilation operators is different

The fermionic parity SUSY Fermionic Wishart supercharge

Fu et al., PRD (2017) Sannomiya et al., PRD (2017) Kanazawa and Wetting, JHEP (2017)

Numerical result: energy spectrum

• 1.5
• 1
• 0.5

0.5 1 1.5 4000 8000 12000

• 1.5
• 1
• 0.5

0.5 1 1.5 4000 8000 12000 0.2 0.4 0.6 0.8 1 4000 8000 12000 0.2 0.4 0.6 0.8 1 4000 8000 12000

(a) fermionic SYK (b) bosonic SYK (c) fermionic Wishart SYK (d) bosonic Wishart SYK

Eigenenergy

Index of eigenstate

Conventional SYK Wishart SYK Ground-state degeneracy Similar between fermionic SYK and bosonic SYK Non-negative → equals Fermion: many degenerate excited states

Fermionic Wishart SYK has larger temporal fluctuations at late times

• 0.3
• 0.2
• 0.1

0.1 10-1 100 101 102 103 104 105 -0.15

• 0.1
• 0.05

0.05 10-1 100 101 102 103 104 105

fermionic Wishart SYK fermionic SYK bosonic SYK bosonic Wishart SYK

(a) (b)

#sample=128 Initial state: eigenstate SYK Wishart SYK

Numerical result: OTOC

Initial state of many-body system

Effective dimension: "the number of eigenstates" in

: maximum degeneracy of energy gaps

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

Theorem for relaxation and effective dimension

We expect a similar bound holds for OTOC.

Theorem for temporal fluctuation around long time average Expectation value of observable relaxes if the effective dimension is large

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

does not depend on

Numerical result: OTOC vs effective dimension

We expect

0.02 0.04 0.06 0.08 0.1 0.5 1 1.5 2 2.5 3

fermionic SYK fermionic Wishart SYK bosonic SYK bosonic Wishart SYK

can be valid for the case of OTOC Each point represents a computational basis state. 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)

You et al, PRB (2017) Kanazawa and Wetting, JHEP (2017)

Previous studies for SYK and SUSY SYK → The conventional SYK is quantum chaotic. Level statistics is GUE. Level statistics of energy level spacings is described by

• Poisson distribution (integrable)
• Wigner-Dyson distribution (non-integrable)

→ GOE, GUE, GSE (random matrix) We numerically investigate the level statistics of the Wishart SYK models.

SYK and Boson Wishart: GUE Fermionic Wishart: Poisson The Poisson distribution implies the following possibilities.

• 1. integrable or 2. we missed another symmetry

0.5 1 1.5 2 0.2 0.4 0.6 0.8 1

GOE GUE GSE Poisson

SYK

Wishart SYK

SYK fermion boson

Wishart SYK

#samples=24

Level statistics (quantum chaos)

Integrability of fermionic Wishart SYK

For simplicity, we assume is even and is 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. We can also show the integrability by explicitly constructing conserved quantities.

Richardson, J. Math. Phys. (1965) Balantekin, PRC (2007)

Summary 1/2

• 1. Huge ground-state degeneracy affects 


dynamics of OTOC through effective dimensions

• 2. Level statistics
• 3. Fermionic Wishart SYK is integrable

SYK: GUE Bosonic Wishart: GUE or GOE Fermionic Wishart: Poisson → OTOC of the fermionic Wishart SYK models exhibits large temporal fluctuations at late times

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

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 SYK

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 Wishart SYK Although fermionic Wishart SYK is integrable (not chaotic), we believe it serves as a reference for evaluating the effect

• f disorder on the original SYK model.

Summary 2/2