Randomness and chaos in qubit models

Pak Hang Chris Lau (NCTS) with Chen-Te Ma (SCNU), Jeff Murugan (Cape Town U.) and Masaki Tezuka (Kyoto U)

Phys. Lett. B795 (2019) 230-235

Classical chaos

• Non-linear system • Double rod pendulum

• Weather

Definition

• Sensitivity to initial condition 𝛿𝛿𝛿𝛿 𝑡𝑡𝛿𝛿𝛿𝛿 0 = 𝑒𝑒𝜆𝜆𝜆𝜆

• Topologically transitive

𝑓𝑓:𝑋𝑋 → 𝑋𝑋 𝑓𝑓𝑛𝑛 𝐴𝐴 ∩ 𝐵𝐵 ≠ ∅ 𝐴𝐴,𝐵𝐵 ⊂ 𝑋𝑋

• Dense periodic orbits

Quantum chaos

• Quantum mechanics

• Schrodinger equation • Linear

• Classical definition of chaos ↔ Quantum ?

Quantum chaos

• Quantum ?→ Classical chaos

• Early time • Out of time-ordered correlator (OTOC)

• Late time • Spectral form factor

Diagnostic 1

• Out of time ordered correlator (OTOC)

• Quantum mechanics • Operators 𝛿𝛿�, �̂�𝑝

• 𝛿𝛿� 𝑡𝑡 , �̂�𝑝(0) = 𝑖𝑖 𝛿𝛿𝛿𝛿 𝜆𝜆𝛿𝛿𝛿𝛿 0

𝐶𝐶𝑇𝑇 = − 𝑊𝑊 𝑡𝑡 ,𝑈𝑈 0 2𝛽𝛽~𝑒𝑒2𝜆𝜆𝜆𝜆

OTOC

• Holography • Maximum bound

• Jackiw–Teitelboim gravity • 1+1d dilaton gravity

• Sachedev-Ye-Kitaev model • 0+1 Fermionic quantum mechanics with all to all couplings

𝜆𝜆 ≤2𝜋𝜋𝑘𝑘𝐵𝐵𝛽𝛽𝛽

J. Maldacena, S.H. Shenker, D. Stanford JHEP (2016) 08 106

Diagnostic 2

• Quantum → Probabilistic • Statistic properties of the system

• Random matrix theory (RMT) • Spectra of heavy nuclei

• Wigner 1955

Classical chaos & RMT

• Sinai billiard • Chaotic

• Quantum Sinai billiard • Spectrum

• RMT

Quantum spectrum

• Level spacing distribution 𝑃𝑃 𝑠𝑠 • Correlation of neighbour states

• Uncorrelated • Poisson distribution 𝑃𝑃 𝑠𝑠 = 𝑒𝑒−𝑠𝑠

• Correlated (RMT)

• Wigner-Surmise 𝑃𝑃 𝑠𝑠 ~𝑠𝑠𝛽𝛽𝑒𝑒−𝑘𝑘𝑠𝑠2

Diagnostic 2 - RMT

• Spectral form factor (SFF) • Correlation between eigenvalues

𝑔𝑔 𝑡𝑡 =

Z 𝛽𝛽, 𝑡𝑡 2

Z 𝛽𝛽, 0 2

Z 𝛽𝛽, 𝑡𝑡 = Tr 𝑒𝑒 −𝛽𝛽+𝑖𝑖𝜆𝜆 𝐻𝐻

J.S. Cotler et.al JHEP (2017) 1705: 118.

Ensemble averaged

• Annealed spectral form factor

• Quenched spectral form factor

𝑔𝑔𝑎𝑎𝑛𝑛𝑛𝑛 𝑡𝑡 =Z 𝛽𝛽, 𝑡𝑡 2

Z 𝛽𝛽, 0 2

𝑔𝑔𝑞𝑞𝑞𝑞𝑞𝑞 𝑡𝑡 =Z 𝛽𝛽, 𝑡𝑡 2

Z 𝛽𝛽, 0 2

Sachedev-Ye-Kitaev model

• Hamiltonian

• Majorana fermions 𝜓𝜓𝑖𝑖 ,𝜓𝜓𝑗𝑗 = 𝛿𝛿𝑖𝑖𝑗𝑗

• Random coupling

• Gaussian variance 𝑞𝑞 − 1 ! 𝐽𝐽2

𝑁𝑁𝑞𝑞−1

• Most studied case • 𝑞𝑞 = 4 large N (chaotic)

Model

• 𝑞𝑞 = 2 Sachdev-Ye-Kitaev model (SYK) • Majorona fermions

• Random coupling • Gaussian

• Integrable

Numerical calculation

• Jordan-Wigner transformation • Fermions → Spin (Pauli matrices)

• Hamiltonian

• 2𝑁𝑁2 × 2

𝑁𝑁2 matrix

• 2𝑁𝑁2 eigenvalues

• Order 1010 ensembles

SYK2 - Spectral form factor

J.S. Cotler et.al JHEP (2017) 2017: 118.

Spectral form factor

Small N SFF

Small N SFF

• Only 𝑁𝑁 = 2 • No dip-ramp-plateau

• 𝑁𝑁 = 2 solvable analytically

• Eigenvectors • Independent of coupling

• Origin of dip-ramp-plateau?

Transverse Ising model

• Spin system with 𝐿𝐿 sites

• Random coupling 𝐾𝐾𝑖𝑖

• Magnetic field 𝑀𝑀

• Analytically integrable

R B Stinchcombe (1973) Journal of Physics C 6(15) 2459

Transverse Ising model

• 𝐿𝐿 = 2

Transverse Ising model

• 𝐿𝐿 = 4

Comparison

• Same behaviour as the SYK4 model (chaotic) for 𝑁𝑁 > 2 • Transverse Ising model consistent

• General integrable spin system?

• Dip-ramp-plateau • Generate from large set of eigenstates

• 𝑁𝑁 = 2 eigenvectors independent of couplings

• Not sufficient condition to diagnose chaos

Connection to thermalisation

• Closed random quantum system • Pure state → Thermal state

• Decoherence

Set-up

• A general quantum state

• Random couplings

• Expectation value of operators • Ensemble averaged

Decoherence

• A particular weight

One qubit model

• Most general Hamiltonian

• Random couplings • Gaussian

• Operators

One qubit model

Summary

• Randomness can generate dip-ramp-plateau behaviour • Chaos like?

• Randomness leads to decoherence • Thermalisation of expectation values

• Relation to Chaos → Thermalisation?