Chaos and complexity of quantum motion

We consider a class of quantum lattice models in 1 + 1 dimensions represented as local quantum circuits that enjoy a particular 'dual-unitarity' property. In essence, this property ensures that both the evolution 'in time' and that 'in space' are given in terms of unitary transfer matrices. We show that for this class of circuits, generically non-integrable, one can compute explicitly all dynamical correlations of local observables. Our result is exact, non-pertubative, and holds for any dimension d of the local Hilbert space. In the minimal case of qubits (d = 2) we also present a classification of all dual-unitary circuits which allows us to single out a number of distinct classes for the behaviour of the dynamical correlations. We find 'non-interacting' classes, where all correlations are preserved, the ergodic and mixing one, where all correlations decay, and, interestingly, also classes that are are both interacting and non-ergodic.

show abstract

“…Let us consider the kicked Ising model [14,25,26], whose dynamics are determined by the following time-periodic Hamiltonian…”

Section: Quantum Circuit Formulation Of the Kicked Ising Modelmentioning

confidence: 99%

Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions

2019

Self Cite

View full text Add to dashboard Cite

show abstract

“…[50], both of them involve a repeated application of the function (1-qudit channel) F defined in Eq. (55), and hence depend essentially on the spectrum of F. Accordingly, the classification of unitary gates reported in Ref. [50] not only distinguishes the behavior of dynamical correlations at infinite temperature, but also applies to predict the qualitative features of two-point functions during the quantum dynamics arising from solvable initial MPSs .…”

Section: B the Two-point Correlation Functionsmentioning

confidence: 99%

Exact dynamics in dual-unitary quantum circuits

et al. 2020

Self Cite

View full text Add to dashboard Cite

We consider the class of dual-unitary quantum circuits in 1 + 1 dimensions and introduce a notion of "solvable" matrix product states (MPSs), defined by a specific condition which allows us to tackle their time evolution analytically. We provide a classification of the latter, showing that they include certain MPSs of arbitrary bond dimension, and study analytically different aspects of their dynamics. For these initial states, we show that while any subsystem of size reaches infinite temperature after a time t ∝ , irrespective of the presence of conserved quantities, the light-cone of two-point correlation functions displays qualitatively different features depending on the ergodicity of the quantum circuit, defined by the behavior of infinite-temperature dynamical correlation functions. Furthermore, we study the entanglement spreading from such solvable initial states, providing a closed formula for the time evolution of the entanglement entropy of a connected block. This generalizes recent results obtained in the context of the self-dual kicked Ising model. By comparison, we also consider a family of non-solvable initial mixed states depending on one real parameter β, which, as β is varied from zero to infinity, interpolate between the infinite temperature density matrix and arbitrary initial pure product states. We study analytically their dynamics for small values of β, and highlight the differences from the case of solvable MPSs.

show abstract

“…The main objective of this paper is to determine a minimal quantum chaotic model [56], with local interactions and finite local Hilbert space, allowing for an exact determination of the entanglement spreading. A candidate emerging naturally in our quest is the so called kicked Ising model [57][58][59], which describes a classical Ising model in the presence of a longitudinal magnetic field and periodically kicked with a transverse magnetic field. This model is quantum chaotic in the sense that its spectral statistics are described by the circular orthogonal ensemble of random matrices [60], but, as we recently proved [55], at some specific points of its parameter space it allows for exact calculations.…”

Section: Model Quench Protocol and Observables Of Interestmentioning

confidence: 99%

Entanglement Spreading in a Minimal Model of Maximal Many-Body Quantum Chaos

2019

Self Cite

View full text Add to dashboard Cite

The spreading of entanglement in out-of-equilibrium quantum systems is currently at the centre of intense interdisciplinary research efforts involving communities with interests ranging from holography to quantum information. Here we provide a constructive and mathematically rigorous method to compute the entanglement dynamics in a class of "maximally chaotic", periodically driven, quantum spin chains. Specifically, we consider the so called "self-dual" kicked Ising chains initialised in a class of separable states and devise a method to compute exactly the time evolution of the entanglement entropies of finite blocks of spins in the thermodynamic limit. Remarkably, these exact results are obtained despite the models considered are maximally chaotic: their spectral correlations are described by the circular orthogonal ensemble of random matrices on all scales. Our results saturate the so called "minimal cut" bound and are in agreement with those found in the contexts of random unitary circuits with infinite-dimensional local Hilbert space and conformal field theory. In particular, they agree with the expectations from both the quasiparticle picture, which accounts for the entanglement spreading in integrable models, and the minimal membrane picture, recently proposed to describe the entanglement growth in generic systems. Based on a novel "duality-based" numerical method, we argue that our results describe the entanglement spreading from any product state at the leading order in time when the model is non-integrable. CONTENTS

show abstract

Chaos and complexity of quantum motion

Cited by 71 publications

References 103 publications

Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions

Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions

Exact dynamics in dual-unitary quantum circuits

Entanglement Spreading in a Minimal Model of Maximal Many-Body Quantum Chaos

Contact Info

Product

Resources

About