Time-translation invariance symmetry breaking hidden by finite-scale singularities

TL;DR

This paper investigates how finite-scale singularities in a disordered quantum system's renormalization group flow can conceal a phase transition that breaks time-translation invariance, especially in the large N limit.

Contribution

It introduces a novel renormalization group approach based on the Wigner spectrum for disordered quantum systems, revealing hidden symmetry-breaking phase transitions.

Findings

Finite-scale singularities occur near the Gaussian region for strong disorder.

These singularities conceal a phase transition that breaks time-translation invariance.

The flow equations depend on scale, preventing fixed points and indicating complex critical behavior.

Abstract

In this paper, we consider a renormalization group perspective on the quantum dynamics of a particle moving in the Euclidean $\mathbb{R}^N$ space through the complex landscape provided by a disordered Hamiltonian of type $2+p$. We focus on the large $N$ limit, where the coarse-graining procedure is unconventional: it is based on the Wigner spectrum of the rank-2 disorder. The main consequence of this choice is that canonical dimensions depend on the scale, and the flow equations fail to become autonomous, preventing the existence of global fixed points. One of the main features of the underlying renormalization group flow is the existence of finite-scale singularities for initial conditions sufficiently close to the Gaussian region and for rank-$p$ disorder intensity large enough. Using the Luttinger-Ward formalism, we show that these finite-scale singularities hide (and should be resolved by) a phase transition that breaks time-translation invariance.