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arXiv:2301.04070 · SciPost Physics

A simple theory for quantum quenches in the ANNNI model

Jacob H. Robertson, Riccardo Senese, Fabian H. L. Eßler

Published Jul 27, 2023• 12 citations
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Abstract

In a recent numerical study by Haldar et al. [Phys. Rev. X 11, 031062] it was shown that signatures of proximate quantum critical points can be observed at early and intermediate times after certain quantum quenches. Said work focused mainly on the case of the axial next-nearest neighbour Ising (ANNNI) model. Here we construct a simple time-dependent mean-field theory that allows us to obtain a quantitatively accurate description of these quenches at short times, which for reasons we explain remains a fair approximation at late times (with some caveats). Our approach provides a simple framework for understanding the reported numerical results as well as fundamental limitations on detecting quantum critical points through quench dynamics. We moreover explain the origin of the peculiar oscillatory behaviour seen in various observables as arising from the formation of a long-lived bound state.

DOI
10.21468/scipostphys.15.1.032

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In a recent numerical study by Haldar et al. [Phys. Rev. X 11, 031062] it was shown that signatures of proximate quantum critical points can be observed at early and intermediate times after certain quantum quenches. Said work focused mainly on the case of the axial next-nearest neighbour Ising (ANNNI) model. Here we construct a simple time-dependent mean-field theory that allows us to obtain a quantitatively accurate description of these quenches at short times, which for reasons we explain remains a fair approximation at late times (with some caveats). Our approach provides a simple framework for understanding the reported numerical results as well as fundamental limitations on detecting quantum critical points through quench dynamics. We moreover explain the origin of the peculiar oscillatory behaviour seen in various observables as arising from the formation of a long-lived bound state.
We investigate both the local and global persistence behaviour in the ANNNI (axial next-nearest-neighbour Ising) model. We find that when the ratio κ of the second neighbour interaction to the first neighbour interaction is less than 1, P ( t ), the probability of a spin to remain in its original state up to time t shows a stretched exponential decay. For κ > 1, P ( t ) has an algebraic decay but the exponent is different from that of the nearest-neighbour Ising model. The global persistence behaviour shows similar features. We also conduct some deeper investigations in the dynamics of the ANNNI model and conclude that it has a different dynamical behaviour compared to the nearest-neighbour Ising model.
We examine the validity of a potential extension of the adiabatic theorem to quantum quenches, i.e., nonadiabatic changes. In particular, the transverse field Ising model (TFIM) and the axial next nearest neighbor Ising (ANNNI) model are studied. The proposed extension of the adiabatic theorem is stated as follows: Consider the overlap between the initial ground state and the postquench Hamiltonian eigenstates for quenches within the same phase. This overlap is largest for the postquench ground state. In the case of the TFIM, this conjecture is confirmed for both the paramagnetic and ferromagnetic phases numerically and analytically. In the ANNNI model, the conjecture could be analytically proven for a special case. Numerical methods are employed to investigate the conjecture's validity beyond this special case.
In recent times it has been observed that signatures of equilibrium quantum criticality surprisingly show up in many-body systems, which are manifestly far from equilibrium. We explore such scenarios in interacting spin systems subject to a quench and develop a robust method to systematically probe ground-state critical physics through nonequilibrium post-quench dynamics. Analyzing late-time behavior of finite string-like observables, we find emerging sharp signatures of equilibrium criticality. Specifically, these observables accurately detect equilibrium critical points and universal scaling exponents after long times following a quench. This happens despite the fact that the analyzed systems are strongly chaotic/ergodic and is interestingly due to a strong memory of the initial conditions retained by these observables after quench. We find that our results can also be used to explain critical signatures in post-quench domain formation, seen in a recent experiment with trapped ion quantum simulators.
We study the quantum dynamics of a number of model systems as their coupling constants are changed rapidly across a quantum critical point. The primary motivation is provided by the recent experiments of Greiner et al. [Nature (London) 415, 39 (2002)] who studied the response of a Mott insulator of ultracold atoms in an optical lattice to a strong potential gradient. In a previous work, it had been argued that the resonant response observed at a critical potential gradient could be understood by proximity to an Ising quantum critical point describing the onset of density wave order. Here we obtain numerical results on the evolution of the density wave order as the potential gradient is scanned across the quantum critical point. This is supplemented by studies of the integrable quantum Ising spin chain in a transverse field, where we obtain exact results for the evolution of the Ising order correlations under a time-dependent transverse field. We also study the evolution of transverse superfluid order in the three-dimensional case. In all cases, the order parameter is best enhanced in the vicinity of the quantum critical point.
The theory of quantum quenches in translationally invariant isolated systems near quantum critical points is formulated from first principles. The theory generally explains the role of interactions among the quasi-particle excitations, the quantum mechanism at the origin of light cone spreading of correlations, and the presence of different time scales in the dynamics. In one dimension the theory clarifies the role of integrability and predicts that persistent oscillations of the order parameter arise depending on the coupling subjected to the quench. The oscillation frequencies correspond to the masses of the quasi-particle excitations. The predictions of the theory are confirmed by independent numerical studies. References: [1] G. Delfino, J. Phys. A: Math. Theor. 47 (2014) 402001. [2] G. Delfino and J. Viti, J. Phys. A: Math. Theor. 50 (2017) 084004. [3] G. Delfino, Phys. Rev. E 97, 062138 (2018).