Quasinormal modes and complexity in saddle-dominated SU(N) spin systems

TL;DR

This paper investigates SU(N) spin systems that emulate particles in de Sitter space, revealing quasinormal modes and complex spectral properties, with early-time chaos-like behavior transitioning to integrability at late times.

Contribution

It introduces a class of SU(N) spin models with hyperbolic fixed points, demonstrating emergent quasinormal modes and analyzing their spectral and dynamical properties, highlighting a transition from chaos to integrability.

Findings

Quasinormal modes manifest as peaks in the density of states.

Early-time dynamics mimic chaos, late-time behavior shows integrability.

Spectral properties depend on the classical phase space structure.

Abstract

We study SU($N$) spin systems that mimic the behavior of particles in $N$-dimensional de Sitter space for $N=2,3$. Their Hamiltonians describe a dynamical system with hyperbolic fixed points, leading to emergent quasinormal modes at the quantum level. These manifest as quasiparticle peaks in the density of states. For a particle in 2-dimensional de Sitter, we find both principal and complementary series densities of states from a PT-symmetric version of the Lipkin-Meshkov-Glick model, having two hyperbolic fixed points in the classical phase space. We then study different spectral and dynamical properties of this class of models, including level spacing statistics, two-point functions, squared commutators, spectral form factor, Krylov operator and state complexity. We find that, even though the early-time properties of these quantities are governed by the saddle points -- thereby in some cases mimicking corresponding properties of chaotic systems, a close look at the late-time behavior reveals the integrable nature of the system.