Finite size scaling
the time crystalline criticality:
Spectral Form Factor

Sagnik Ghosh •Uni Bonn •ML4Q Retreat •July 30, 2024

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sghosh@uni-bonn.de

Summer School: (Bonn, Sept 23-27, 2024)

Defintions:

[Typical systems]

For systems that scrambles, any expectation value of local operator at infinite time under finite size scaling is always zero.

Defintions:

[Time Fluids]

Time fluids are systems, for which there exists local operators, whose expectation value at the thermodynamic limit, under long time dynamics is finite and non-zero.

Eg: MBL

\lim_{t\to\infty} \lim_{L\to\infty} \braket{\mathcal{O}(t)} \neq 0

Defintions:

[Time Crystals and Time Glasses]

Time crystals are time glasses where the time translation symmetry is spontaneously broken.

If at large times, at the thermodynamic limit, the expectation value is a periodic function of t then the system is a time crystal.

If <O>(t)is a non-periodic function of t then it is a time glass.

In these cases typically you would look at two time correalator of local operators under the defined limit.

Bar Lev, Lazadires (2024) arXiv:2403.01912

The Circuit: Ippoliti PRXQuantum030346.2021

DTC in chip

Features of DTC: (Ippoliti PRXQuantum030346.2021)

Dynamical

Signatures:

Features of DTC:

Eigenstate Signatures:

Spectral Form Factors of an Unitary

The SFF is defined as K(t)=|Tr(U(t))|^2.
Using the spectral decomposition, one can reduce it to a more familiar form.

SFF is the fourier transform of the two point correlator of the level density.

Motivation:

Behaviour of SFF around Thermal Point

Diagonal elements are exactly 1

The ramp is a signature from Random Matrix Theory

Behaviour of SFF at PM-MBL ETH transition

Diagonal elements are exactly 1

Deep in MBL, the Poission statistics destroy the ramp

Behaviour of SFF at DTC transition

Diagonal elements are still exactly 1.

But now, there are also exactly -1 in the sum, which comes from the cat-states phase separated by pi.

Why?

But now, there are also exactly -1 in the sum, which comes from the cat-states phase separated by pi. In the even periods they add up, in odd periods the cancels.
Hence oscillations between 0,2n.

SFF Undersampling