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arXiv:2206.03366 · Journal of Statistical Mechanics Theory and Experiment

Evolution of circuit complexity in a harmonic chain under multiple quenches

Kuntal Pal, Kunal Pal, Ankit Gill, Tapobrata Sarkar

Published May 1, 2023• 6 citations
arXivHTMLPDF

Abstract

Abstract We study Nielsen’s circuit complexity in a periodic harmonic oscillator chain, under single and multiple quenches. This simple system is amenable to analytical computations and yet offers considerable insight into the dynamics of quantum information. For a single quench scenario, we highlight some important differences between our results that explicitly use the wavefunction, as compared to the recently proposed covariance matrix method and point out the consequences. In a multiple quench scenario, the complexity shows remarkably different behaviour compared to the other information theoretic measures, such as the entanglement entropy and the out of time ordered correlator. In particular, the latter is known to show signs of chaos in this integrable system, but our results indicate the contrary. We further show the presence of a ‘residual complexity,’ i.e. after two successive quenches, when the frequency returns to its initial value, the complexity has a non-zero lower limit. Further, applying a large number of successive quenches, the complexity of the time evolved state can be increased to a high value, which is not possible by applying a single quench. Finally, we show that this simplistic model exhibits the interesting phenomenon of ‘complexity crossover’ between two successive quenches performed at different times.

DOI
10.1088/1742-5468/acd4b3

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•Sep 27, 2021•0 citations
Abstract We study Nielsen’s circuit complexity in a periodic harmonic oscillator chain, under single and multiple quenches. This simple system is amenable to analytical computations and yet offers considerable insight into the dynamics of quantum information. For a single quench scenario, we highlight some important differences between our results that explicitly use the wavefunction, as compared to the recently proposed covariance matrix method and point out the consequences. In a multiple quench scenario, the complexity shows remarkably different behaviour compared to the other information theoretic measures, such as the entanglement entropy and the out of time ordered correlator. In particular, the latter is known to show signs of chaos in this integrable system, but our results indicate the contrary. We further show the presence of a ‘residual complexity,’ i.e. after two successive quenches, when the frequency returns to its initial value, the complexity has a non-zero lower limit. Further, applying a large number of successive quenches, the complexity of the time evolved state can be increased to a high value, which is not possible by applying a single quench. Finally, we show that this simplistic model exhibits the interesting phenomenon of ‘complexity crossover’ between two successive quenches performed at different times.
Abstract We study the temporal evolution of the circuit complexity for a subsystem in harmonic lattices after a global quantum quench of the mass parameter, choosing the initial reduced density matrix as the reference state. Upper and lower bounds are derived for the temporal evolution of the complexity for the entire system. The subsystem complexity is evaluated by employing the Fisher information geometry for the covariance matrices. We discuss numerical results for the temporal evolutions of the subsystem complexity for a block of consecutive sites in harmonic chains with either periodic or Dirichlet boundary conditions, comparing them with the temporal evolutions of the entanglement entropy. For infinite harmonic chains, the asymptotic value of the subsystem complexity is studied through the generalised Gibbs ensemble.
The quantum dynamics of isolated systems under quench condition exhibits a variety of interesting features depending on the integrable/chaotic nature of system. We study the exact dynamics of trivially integrable system of harmonic chains under a multiple quench protocol. Out of time ordered correlator of two Hermitian operators at large time displays scrambling in the thermodynamic limit. In this limit, the entanglement entropy and the central component of momentum distribution both saturate to a steady-state value. We also show that reduced density matrix assumes the diagonal form long after multiple quenches for large system size. These exact results involving infinite-dimensional Hilbert space are indicative of dynamical equilibration for a trivially integrable harmonic chain.
Abstract We study the temporal evolution of the circuit complexity after the local quench where two harmonic chains are suddenly joined, choosing the initial state as the reference state. We discuss numerical results for the complexity for the entire chain and the subsystem complexity for a block of consecutive sites, obtained by exploiting the Fisher information geometry of the covariance matrices. The qualitative behaviour of the temporal evolutions of the subsystem complexity depends on whether the joining point is inside the subsystem. The revivals and a logarithmic growth observed during these temporal evolutions are discussed. When the joining point is outside the subsystem, the temporal evolutions of the subsystem complexity and of the corresponding entanglement entropy are qualitatively similar.
The harmonic oscillator is the paragon of physical models; conceptually and computationally simple, yet rich enough to teach us about physics on scales that span classical mechanics to quantum field theory. This multifaceted nature extends also to its inverted counterpart, in which the oscillator frequency is analytically continued to pure imaginary values. In this article we probe the inverted harmonic oscillator (IHO) with recently developed quantum chaos diagnostics such as the out-of-time-order correlator (OTOC) and the circuit complexity. In particular, we study the OTOC for the displacement operator of the IHO with and without a non-Gaussian cubic perturbation to explore genuine and quasi scrambling respectively. In addition, we compute the full quantum Lyapunov spectrum for the inverted oscillator, finding a paired structure among the Lyapunov exponents. We also use the Heisenberg group to compute the complexity for the time evolved displacement operator, which displays chaotic behaviour. Finally, we extended our analysis to N-inverted harmonic oscillators to study the behaviour of complexity at the different timescales encoded in dissipation, scrambling and asymptotic regimes.
We examine the temporal evolution of the modular entropy and capacity (in particular, the fluctuation of the entanglement entropy) for systems of time-dependent oscillators coupled by a (time-dependent) parameter. Such models, through the discretization procedure, fit into field theory problems arising from quench phenomena or nonstatic spacetimes. First, we compare the dynamics of the modular and R\'enyi entropies and derive the form of the modular capacity for the single time-dependent oscillator as well as chains with bipartite decompositions. In the latter case we analyze distinguished periodicities during the evolution and the role of various boundary conditions. Next, we focus on the dynamics of the capacity (fluctuation) of entanglement. We compare the results obtained with the predictions of quasiparticles models; in particular, we obtain a theoretical value of the initial slope of the capacity for abrupt quenches. We study also continuous protocols with the frequency that vanishes at plus (and minus) infinity, including a model in which the frequency tends to the Dirac delta. All the above issues are discussed with the emphasis on the analytical methods.
Entanglement and circuit complexity are fundamental concepts in quantum information theory which turned out to have considerable applications in various contexts ranging from many-body systems and condensed matter theory to quantum gravity and black hole physics. In order to find connections between these aspects, an insightful approach consists in comparing the complexity with some entanglement measures of a given bipartition, as for instance the Von Neumann entanglement entropy. Since, for a system in a given quantum state, the bipartite entanglement can be determined once a spatial subsystem and its complement are chosen, this comparison has to be made considering the complexity of reduced density matrices. This motivates the development of techniques for determining the circuit complexity of mixed states. In this thesis, focussing on free bosonic systems, we study the circuit complexity of mixed Gaussian states by employing the Fisher information geometry for the covariance matrices. This approach allows computing the complexity between reduced density matrices (subsystem complexity), as well as the complexity between thermal states. It also leads to a precise prescription for computing the spectrum and the basis complexity and the temporal evolution of the subsystem complexity after global and local quantum quenches. The study of the subsystem complexity of a given bipartition and a complete characterisation of its entanglement requires the detailed knowledge of the corresponding reduced density matrix, or, equivalently, of the entanglement Hamiltonian. A part of this thesis is devoted to determining the entanglement Hamiltonians in free theories. At equilibrium, we consider gapless systems, with the aim of recovering the underlying CFT results, and gapped lattice models. The evolution of entanglement Hamiltonians in quantum chains after a global quench is also studied. In addition, other entanglement quantifiers, as the entanglement spectrum and the contour for the entanglement entropies, are discussed in some of these settings. Finally, motivated by theoretical and experimental advances, we also address the issue of how the entanglement splits into the different charge sectors of a system endowed with a U(1) symmetry. We focus in particular on massive theories, either continuous or on the lattice, and we try to understand whether the entanglement equipartition, which is known to characterise gapless systems, survives away from criticality.