Complexity, information geometry, and Loschmidt echo near quantum criticality
Nitesh Jaiswal, Mamta Gautam, Tapobrata Sarkar
Abstract
Abstract We consider the Nielsen complexity (NC) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:math> , the Loschmidt echo (LE) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">L</mml:mi> </mml:math> , and the Fubini-study complexity τ in the transverse XY model, following a sudden quantum quench, in the thermodynamic limit. At small times, the first two are related by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">L</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msup> </mml:math> . By computing a novel time-dependent quantum information metric, we show that in this regime, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="script">C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∼</mml:mo> <mml:mi mathvariant="normal">d</mml:mi> <mml:msup> <mml:mrow> <mml:mi>τ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> , 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.