We study Nielsen complexity and Fubini-Study complexity for the transverse XY spin chain and establish their nonanalyticity at the critical lines. We obtain the scaling behavior of both complexities near quantum phase transitions in the thermodynamic limit, as a function of the system parameters. The scaling of the Nielsen complexity with the system size is also established, close to criticality. We also obtain analytic expressions for the Fubini-Study complexity in some special cases for this model, while a numerical analysis in more generic situations is carried out. Our study clearly demonstrates the differences in the two notions of complexity in the transverse XY model.
We study Nielsen complexity and Fubini-Study complexity for the transverse XY spin chain and establish their nonanalyticity at the critical lines. We obtain the scaling behavior of both complexities near quantum phase transitions in the thermodynamic limit, as a function of the system parameters. The scaling of the Nielsen complexity with the system size is also established, close to criticality. We also obtain analytic expressions for the Fubini-Study complexity in some special cases for this model, while a numerical analysis in more generic situations is carried out. Our study clearly demonstrates the differences in the two notions of complexity in the transverse XY model.
Abstract We consider the Nielsen complexity (NC) C N , the Loschmidt echo (LE) L , 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 L ∼ e − C N . By computing a novel time-dependent quantum information metric, we show that in this regime, C N ∼ d τ 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.
Abstract We study information theoretic quantities in models with three and four spin interactions. These models show distinctive characteristics compared to their nearest neighbour (NN) counterparts. Here, we quantify these in terms of the Nielsen complexity (NC) in static and quench scenarios, the Fubini–Study complexity (FSC), and the entanglement entropy (EE). The models that we study have a rich phase structure, and we show how the difference in the nature of phase transitions in these, compared to ones with NN interactions, result in different behaviour of information theoretic quantities, from ones known in the literature. For example, the derivative of the NC does not diverge but shows a discontinuity near continuous phase transitions, and the FSC may be regular and continuous across such transitions. We also study multiple quench scenarios in these models and contrast these with quenches in the transverse XY model. The EE shows a novel discontinuity both at first and second order quantum phase transitions.
We study information theoretic geometry in time dependent quantum mechanical systems. First, we discuss global properties of the parameter manifold for two level systems exemplified by i) Rabi oscillations and ii) quenching dynamics of the XY spin chain in a transverse magnetic field, when driven across anisotropic criticality. Next, we comment upon the nature of the geometric phase from classical holonomy analyses of such parameter manifolds. In the context of the transverse XY model in the thermodynamic limit, our results are in contradiction to those in the existing literature, and we argue why the issue deserves a more careful analysis. Finally, we speculate on a novel geometric phase in the model, when driven across a quantum critical line.
We provide a study of various quantum phase transitions occurring in the XY Heisenberg chain in a transverse magnetic field using the Meyer-Wallach (MW) measure of (global) entanglement. Such a measure, while being readily evaluated, is a multipartite measure of entanglement as opposed to more commonly used bipartite measures. Consequently, we obtain analytic expression of the measure for finite-size systems and show that it can be used to obtain critical exponents via finite-size scaling with great accuracy for the Ising universality class. We also calculate an analytic expression for the isotropic (XX) model and show that global entanglement can precisely identify the level-crossing points. The critical exponent for the isotropic transition is obtained exactly from an analytic expression for global entanglement in the thermodynamic limit. Next, the general behavior of the measure is calculated in the thermodynamic limit considering the important role of symmetries for this limit. The so-called oscillatory transition in the ferromagnetic regime can only be characterized by the thermodynamic limit where global entanglement is shown to be zero on the transition curve. Finally, the anisotropic transition is explored where it is shown that global entanglement exhibits an interesting behavior in the finite-size limit. In the thermodynamic limit, we show that global entanglement shows a cusp singularity across the Ising and anisotropic transition, while showing non-analytic behavior at the XX multicritical point. It is concluded that global entanglement, despite its relative simplicity, can be used to identify all the rich structure of the ground-state Heisenberg chain.
We study the time-dependent circuit complexity of the periodically driven transverse field Ising model using Nielsen's geometric approach. In the high-frequency driving limit the system is known to exhibit nonequilibrium phase transitions governed by the amplitude of the driving field. We analytically compute the complexity in this regime and show that it clearly distinguishes between the different phases, exhibiting a universal linear behavior at early times. We also evaluate the time-averaged complexity, provide evidence of nonanalytic behavior at the critical points, and discuss its origin. Finally, we comment on the freezing of quantum dynamics at specific configurations and on the use of complexity as a tool to understand quantum phase transitions in Floquet systems.
We study complexity in a spin system with infinite-range interaction, via the paradigmatic Lipkin-Meshkov-Glick (LMG) model, in the thermodynamic limit. Exact expressions for the Nielsen complexity (NC) and the Fubini-Study complexity (FSC) are derived, which helps us to establish several distinguishing features compared to complexity in other known spin models. In a time-independent LMG model, close to phase transition, the NC diverges logarithmically, much like the entanglement entropy. Remarkably, however, in a time-dependent scenario, this divergence is replaced by a finite discontinuity, as we show by using the Lewis-Riesenfeld theory of time-dependent invariant operators. The FSC of a variant of the LMG model shows different behavior compared to quasifree spin models. Namely, it diverges logarithmically when the target (or reference) state is near the separatrix. Numerical analysis indicates that this is due to the fact that geodesics starting with arbitrary boundary conditions are "attracted" toward the separatrix and that near this line, a finite change in the affine parameter of the geodesic produces an infinitesimal change of the geodesic length. The same divergence is shared by the NC of this model as well.