Solvable Random Unitary Dynamics in a Disordered Tomonaga-Luttinger Liquid

TL;DR

This paper derives an analytical expression for the frame potential in a disordered Tomonaga-Luttinger liquid, revealing its decay and saturation behavior, with implications for quantum simulation algorithms.

Contribution

It provides the first analytical treatment of the frame potential in an interacting 1D disordered system, connecting quantum information diagnostics with many-body physics.

Findings

Frame potential decays as a power law at early times.

Saturates to a plateau controlled by a single parameter.

Randomness is maximized near the Heisenberg ferromagnetic point.

Abstract

Disordered one-dimensional interacting systems have long been characterized via conventional correlation functions. A complementary quantum-information perspective quantifies the randomness of the unitary ensemble dynamics generated by a quantum system through the frame potential, which serves as a practical diagnostic for quantum algorithmic performance. However, no analytical treatment has yet been achieved for experimentally accessible interacting one-dimensional systems. In this Letter, we derive a closed-form expression for the frame potential of a Tomonaga-Luttinger liquid with quenched Gaussian forward-scattering disorder. Exploiting the exactly quadratic structure of the disorder-averaged Keldysh action, we show that the frame potential decays as a power law at early times and saturates to a late-time plateau controlled by a single coupling parameter. Taking the random field XXZ spin chain as a specific microscopic realization, we show that the strongest randomness is achieved near the Heisenberg ferromagnetic point and can be exponentially enhanced through a multiple-quench protocol. We validate our results across the entire gapless phase, with direct implications for algorithm design in analog quantum simulation platforms.