# Quantum spin chains with multiple dynamics

## TL;DR

This paper investigates a quantum spin-1 bilinear-biquadratic chain revealing multiple emergent time scales and gapless modes with varying dynamical exponents, using large-scale simulations and continuum models to understand complex quantum dynamics.

## Contribution

It introduces a new understanding of multiple dynamical exponents and gapless modes in a solvable quantum spin chain through advanced numerical and analytical methods.

## Key findings

- Lowest excitations have a dynamical exponent z from 2 to 3.2.
- An additional gapless mode with a varying exponent 2 to <2.7.
- Large dynamical exponent due to subdiffusive spin motion.

## Abstract

Many-body systems with multiple emergent time scales arise in various contexts, including classical critical systems, correlated quantum materials, and ultra-cold atoms. We investigate such non-trivial quantum dynamics in a new setting: a spin-1 bilinear-biquadratic chain. It has a solvable entangled groundstate, but a gapless excitation spectrum that is poorly understood. By using large-scale DMRG simulations, we find that the lowest excitations have a dynamical exponent $z$ that varies from 2 to 3.2 as we vary a coupling in the Hamiltonian. We find an additional gapless mode with a continuously varying exponent $2\leq z <2.7$, which establishes the presence of multiple dynamics. In order to explain these striking properties, we construct a continuum wavefunction for the groundstate, which correctly describes the correlations and entanglement properties. We also give a continuum parent Hamiltonian, but show that additional ingredients are needed to capture the excitations of the chain. By using an exact mapping to the non-equilibrium dynamics of a classical spin chain, we find that the large dynamical exponent is due to subdiffusive spin motion. Finally, we discuss the connections to other spin chains and to a family of quantum critical models in 2d.