Spread and Spectral Complexity in Quantum Spin Chains: from Integrability to Chaos

TL;DR

This paper investigates how spread and spectral complexity evolve in quantum spin chains transitioning from integrability to chaos, revealing universal bounds, characteristic peaks, and the role of the thermofield double state in chaos detection.

Contribution

It introduces a detailed analysis of spread and spectral complexity across integrable and chaotic regimes, identifying universal bounds and the significance of the thermofield double state.

Findings

Peak in spread complexity signals chaos.

Saturation values depend on spectral statistics and state.

Thermofield double state saturates universal bounds.

Abstract

We explore spread and spectral complexity in quantum systems that exhibit a transition from integrability to chaos, namely the mixed-field Ising model and the next-to-nearest-neighbor deformation of the Heisenberg XXZ spin chain. We corroborate the observation that the presence of a peak in spread complexity before its saturation, is a characteristic feature in chaotic systems. We find that, in general, the saturation value of spread complexity post-peak depends not only on the spectral statistics of the Hamiltonian, but also on the specific state. However, there appears to be a maximal universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double state (TFD) at infinite temperature. We also find that the time scales at which the spread complexity and spectral form factor change their behaviour agree with each other and are independent of the chaotic properties of the systems. In the case of spectral complexity, we identify that the key factor determining its saturation value and timescale in chaotic systems is given by minimum energy difference in the theory's spectrum. This explains observations made in the literature regarding its earlier saturation in chaotic systems compared to their integrable counterparts. We conclude by discussing the properties of the TFD which, we conjecture, make it suitable for probing signatures of chaos in quantum many-body systems.