Complexity, information geometry, and Loschmidt echo near quantum criticality

We consider the Nielsen complexity (NC) C-N, the Loschmidt echo (LE) 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 L similar to e(-CN). By computing a novel time-dependent quantum information metric, we show that in this regime, C-N similar to 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 NC and the LE 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 NC 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.

Automated summary

In this study, the relationship between Nielsen complexity, Loschmidt echo, and Fubini-Study complexity in the transverse XY model after a sudden quantum quench was investigated. It was found that NC and LE are related at small times and show enhanced temporal oscillations in the thermodynamic limit, especially when quenching from a close neighborhood of the critical line.