# Exact results in Floquet coin toss for driven integrable models

## TL;DR

This paper derives exact analytical results for the residual energy in a driven integrable quantum model under random binary (coin-toss) driving, showing bounded heating and universal behavior in the asymptotic limit.

## Contribution

It introduces an exact formalism using a disorder operator to analytically compute the residual energy in a randomly driven integrable system, revealing universal asymptotic behavior.

## Key findings

- Residual energy saturates to a finite value with random driving.
- The formalism simplifies numerical calculations significantly.
- Asymptotic residual energy is universal, independent of protocol bias.

## Abstract

We study an integrable Hamiltonian reducible to free fermions which is subjected to an imperfect periodic driving with the amplitude of driving (or kicking) randomly chosen from a binary distribution like a coin-toss problem. The randomness present in the driving protocol destabilises the periodic steady state, reached in the limit of perfectly periodic driving, leading to a monotonic rise of the stroboscopic residual energy with the number of periods ($N$). We establish that a minimal deviation from the perfectly periodic driving would always result in a {\it bounded} heating up of the system with $N$ to an asymptotic finite value. Remarkably, exploiting the completely uncorrelated nature of the randomness and the knowledge of the stroboscopic Floquet operator in the perfectly periodic situation, we provide an exact analytical formalism to derive the disorder averaged expectation value of the residual energy through a {\it disorder operator}. This formalism not only leads to an immense numerical simplification, but also enables us to derive an exact analytical form for the residual energy in the asymptotic limit which is universal, i.e, independent of the bias of coin-toss and the protocol chosen. Furthermore, this formalism clearly establishes the nature of the monotonic growth of the residual energy at intermediate $N$ while clearly revealing the possible non-universal behaviour of the same.