# Slow scrambling and hidden integrability in a random rotor model

## TL;DR

This paper investigates the dynamics of quantum rotors with random interactions, revealing slow scrambling without exponential growth and uncovering integrability and connections to random-matrix theory.

## Contribution

It demonstrates that in a large N, M limit, the model is integrable and exhibits slow scrambling, with a small exponential growth due to irrelevant terms.

## Key findings

- Squared commutators do not grow exponentially in the large N, M limit.
- The model is integrable and connected to random-matrix theory.
- Irrelevant terms cause a small exponential growth in commutator dynamics.

## Abstract

We analyze the out-of-time-order correlation functions of a solvable model of a large number, $N$, of $M$-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. We focus on the growth of commutators of operators at a temperature $T$ above the zero temperature quantum critical point separating the spin-glass and paramagnetic phases. In the large $N,~M$ limit, the squared commutators of the rotor fields do not display any exponential growth of commutators, in spite of the absence of any sharp quasiparticle-like excitations in the disorder-averaged theory. We show that in this limit, the problem is integrable and point out interesting connections to random-matrix theory. At leading order in $1/M$, there are no modifications to the critical behavior but an irrelevant term in the fixed-point action leads to a small exponential growth of the squared commutator. We also introduce and comment on a generalized model involving $p$-pair rotor interactions.