Random Matrices
andHydrodynamics
Brian Swingle (Brandeis)
Extreme Universe Colloquium
October 2, 2023

[Midjourney-S]

What do these have in common?
Atoms/Nuclei[image: Wiki]

Billiards[image: E. Yeung]

Quantum spins[image: R. Aguiar]

Black holes[image: NASA]Random matrices
Atoms/Nuclei

Quantum spins

Billiards

Black holesHydrodynamics, construed broadly
Atoms/Nuclei

Quantum spins

Billiards

Black holes[image: MIT ME]

Are these two widespread phenomena related?

Are these two widespread phenomena related?

YES!

Are these two widespread phenomena related?

YES!

Spectral form factor (SFF)

Total return probability (TRP)

SFF(T) = SFFRMT(T) × TRP(T)Random matrix theory (RMT) in physics• Many complex quantum systems
have an “unstructured” energy
spectrum, especially far away fromthe ground state• Wigner’s idea: we can model thespectrum of such “unstructured”
systems using random* Hermitianmatrices [Wigner, Dyson, Mehta, …]Level spacings of different nuclei with the same spin/parity:[Bohigas-Haq-Pandey]*Many types of random matrices, but we’ll consider primarily Gaussian ensembles

Spectrum and statistics

n=8 spins, 256 energy levels

Ensemble of Hamiltonians with random fields:[chaos-RMT conjecture: Bohigas-Giannoni-Schmit]In a conventional liquid, we probe the particle
positions using scattering, e.g. of neutrons
POSITION / MOMENTUM
In our eigenvalue liquid, we can also probe theenergies by “scattering” (Fourier transform)
ENERGY / TIME

𝜌 𝐸𝑆𝐹𝐹(𝑇) = න 𝑑𝐸 න 𝑑𝜖𝑒−𝑖𝜖𝑇/ℏ𝜖𝜖
𝜌 𝐸+ 𝜌 𝐸−22

RMT and SFF

data: Dyson index and potentialf = filter function[review: Haake]For fun … liquid Argon vs liquid eigenvalues

data: Dyson index and potential[Yarnell-Katz-Wenzel-Koenig]

f = filter function

Quantum ruless’
The SFF can be written as a sum over return amplitudes,so let us recall the quantum rules for such amplitudes𝛾

A s → s’) = A(γ ⊂ 𝑠 exp(−𝑖𝐻𝑡/ℏ) 𝑠′
P(s → s’) =|A(s → s′)|2Probabilities are squares of amplitudess𝛾′𝛾

𝑆𝐹𝐹 = ෍ 𝐴 𝛾 𝐴 𝛾′∗𝛾,𝛾′sTerms in the SFF sum tend to destructively
interfere unless the paths 𝛾, 𝛾′ are the sames=𝑡=0sors’[𝐴 𝛾′ ]∗𝐴 𝛾𝑡=1sorThis residual freedom is responsible for the linear-in-T ramp

𝑡=2…
sToy example: coupled billiards
0SFF = 2 x RMT0

Short time:
SFF = 2 x RMTLong time:
SFF = RMT

Toy example: coupled billiards
Short time:
SFF = 2 x RMTLong time:
SFF = RMT

[Winer-S]

Toy example: coupled billiards
Short time:
SFF = 2 x RMTLong time:
SFF = RMTSFF RMT(T)TRP(T)[Winer-S]

Energy diffusion

diffusion + “noise” withappropriate correlations• Imagine breaking all other symmetries: all that remains is energy
diffusion → minimal slow dynamics in a local Hamiltonian system• At time T, there are an extensive number of almost conserved modes:• If each sector is random matrix like, then the SFF should correspondto a sum of many almost-independent ramps → sectors are labelled
by amplitudes of nearly-conserved energy fluctuations

Linear diffusion

exclude zero mode,quasi-continuous
wavevector regime

TRP(T)[Winer-S, large-q d=1 Floquet model Friedman et al. ‘19]

periodic box

• Consistent with numerical
data from [Friedman et al.],which derives the previous
formula (in the context ofU(1) conservation) in d=1
with large onsite dimension• We show that it arises
generally from linearized
diffusion; and we cancompute correctionsln(𝑆𝐹𝐹 Τ𝑡)Comparison with numerical data

[data from Friedman et al. 1906.07736]

Theorist’s cornerVS

SFF effective theory should be related to an effective
theory on a Schwinger-Keldysh contour → hydro!

Theorist’s corner

exactly reproduces prior calculation[Winer-S]

Are these two widespread phenomena related?

YES!

Spectral form factor (SFF)

Total return probability (TRP)

SFF(T) = SFFRMT(T) × TRP(T)But there is a twist …

But there is a twist …

???

SFF(T) = SFFRMT(T) × TRP(T)We first observed this suppression in SFFs
in a “block Rosenzweig-Porter” model[Barney-Winer-Baldwin-Galitski-S]∞Detour: Zeta zeros

1
𝜁 𝑠 =෍ 𝑠𝑛𝑛=11• Riemann zeta has zeros on the “critical” line 𝑠 = + 𝑖𝑡; amazingly,2

these critical line zeros are distributed like the eigenvalues of arandom Hermitian matrix! [Montgomery, Odlyzko…]• Hilbert, Polya, Berry, Keating, and others’ inspiring idea: what if thezeta zeros are secretly the energies of a quantum chaotic system?
• Still incomplete ... but one fruit is the Riemann-Siegel lookalike
formula – a resummation formula for the Gutzwiller trace formulainspired by the Riemann-Siegel formula for zeta, a relationshipbetween the contributions of short and long periodic orbits

SFF “sum rule”• Valid for any system with enough level repulsion → early timeenhancements must be “paid for” with a late time suppression• We conjecture a specific formula for GUE-type problems (derivationin special cases from the Riemann-Siegel lookalike [Berry-Keating])[Winer-S]

But there is a twist …

Convolution with“hydro” modes

SFF(T) = SFFRMT(T) × TRP(T)Summary• Random-matrix-like energies are common to many quantum systems• But real systems are not literally random matrices, and deviations
from RMT are controlled by hydro, construed broadly• We didn’t emphasize it, but there is even a sense in which this hydrobased “effective theory of spectral correlations” yields the ramp itself• So far, we’ve applied this theory to spin chains and simple blockmodels; what about nuclei and elsewhere?
• Quantum information has provided many new inspirations, e.g. thegrowth of complexity, scrambling and thermalization, and more …

Outlook – chaos and quantum information
Quantum physics ofclassically chaotic
systems

Random-matrix-likeenergy levels…

Information
scrambling“QUANTUM CHAOS”

Thermalization andHydrodynamics

Complexity growthEigenstate
thermalization

Today – linking hydro and RMTQuantum physics ofclassically chaoticsystems

Random-matrix-likeenergy levels…

Information
scrambling“QUANTUM CHAOS”

Thermalization andHydrodynamics

Complexity growth

Eigenstate
thermalization

Thanks and references• Mostly based on results with my student Mike Winer• Hydrodynamic theory of the connected spectral form factor, 2012.01436• Reappearance of Thermalization Dynamics in the Late-Time Spectral Form Factor, 2307.14415• Quantum chaos, e.g. Altshuler-Shklovskii ’86, D'Alessio-Kafri-Polkovnikov-Rigol, …
• Analytic results: Bertini-Kos-Prosen, Dubertrand-Muller, Chan-De Luca-Chalker,
Saad-Shenker-Stanford, Garcia-Garcia-Verbaarschot, Altland-Sonner, …
• RMT Onset: Schiulaz-Torres-Herrera-Santos, Gharibyan-Hanada-Shenker-Tezuka,
Friedman-Chan-De Luca-Chalker, Altland-Bagrets, …
• Fluctuating hydro: Dubovsky-Hui-Nicolis-Son, Grozdanov-Polonyi, HaehlLoganayagam-Rangamani, Crossley-Glorioso-Liu, Jensen-Pinkani-Fokeeva-Yarom,
Chen-Lin-Delacretaz-Hartnoll, …