Complexity, Information Geometry, and Loschmidt Echo near Quantum Criticality

TL;DR

This paper investigates the relationships between different quantum complexity measures and the Loschmidt echo in the transverse XY model near criticality, revealing their dynamic behavior and potential violations of expected inequalities.

Contribution

It introduces a novel time-dependent quantum information metric and explores the connection between Nielsen complexity, Fubini-Study complexity, and Loschmidt echo near quantum critical points.

Findings

Nielsen complexity and Loschmidt echo are related by ${ m L} o e^{- ext{Nielsen complexity}}$ at small times.

Enhanced oscillations occur in complexity measures when quenching near criticality.

Possible violation of the triangle inequality for Nielsen complexity in certain parameter regions.

Abstract

We consider the Nielsen complexity ${\mathcal C}_N$, the Loschmidt echo ${\mathcal L}$, and the Fubini-Study complexity $\tau$ in the transverse XY model, following a sudden quantum quench, in the thermodynamic limit. At small times, the first two are related by ${\mathcal L} \sim e^{-{\mathcal C}_N}$. By computing a novel time-dependent quantum information metric, we show that in this regime, ${\mathcal C}_N \sim d\tau^2$, up to lowest order in perturbation. The former relation continues to hold in the same limit at large times, whereas the latter does not. Our results indicate that in the thermodynamic limit, the Nielsen complexity and the Loschmidt echo show enhanced temporal oscillations when one quenches from a close neighbourhood of the critical line, while such oscillations are notably absent when the quench is on such a line. We explain this behaviour by studying the nature of quasi-particle excitations in the vicinity of criticality. Finally, we argue that the triangle inequality for the Nielsen complexity might be violated in certain regions of the parameter space, and point out why one should be careful about the nature of the interaction Hamiltonian, while using this measure.