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arXiv:1711.03920 · Physical Review A

Thirring quantum cellular automaton

Alessandro Bisio, Giacomo Mauro D’Ariano, Paolo Perinotti, Alessandro Tosini

Published Mar 29, 2018• 37 citations
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Abstract

We analytically diagonalize a discrete-time on-site interacting fermionic cellular automaton in the two-particle sector. Important features of the solutions sensibly differ from those of analogous Hamiltonian models. In particular, we find a wider variety of scattering processes, we have bound states for every value of the total momentum, and there exist bound states also in the free case, where the coupling constant is null.

DOI
10.1103/physreva.97.032132

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We analytically diagonalize a discrete-time on-site interacting fermionic cellular automaton in the two-particle sector. Important features of the solutions sensibly differ from those of analogous Hamiltonian models. In particular, we find a wider variety of scattering processes, we have bound states for every value of the total momentum, and there exist bound states also in the free case, where the coupling constant is null.
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We present a fermionic cellular automaton model which describes massless Dirac fermions in 1+11+11+1 dimension coupled with local, number preserving interaction. The diagonalization of the two particle sector shows that specific values of the total momentum and of the coupling constant allow for the formation of bound states. Furthermore, we present a classification of the local number preserving interactions that are invariant under the isotropy group of the cellular automaton which simulates the Weyl equation.
The Thirring Quantum Cellular Automaton (QCA) describes the discrete time dynamics of local fermionic modes that evolve according to one step of the Dirac cellular automaton, followed by the most general on-site number-preserving interaction, and serves as the QCA counterpart of the Thirring model in quantum field theory. In this work, we develop perturbative techniques for the QCA path sum approach, expanding both the number of interaction vertices and the mass parameter of the Thirring QCA. By classifying paths within the regimes of very light and very heavy particles, we computed the transition amplitudes in the two- and three-particle sectors to the first few orders. Our investigation into the properties of the Thirring QCA, addressing the combinatorial complexity of the problem, yielded some useful results applicable to the many-particle sector of any on-site number-preserving interactions in one spatial dimension.
We study the dynamical behavior of a quantum cellular automaton which reproduces the Dirac dynamics in the limit of small wave vectors and masses. We present analytical evaluations along with computer simulations, showing that the automaton exhibits typical Dirac dynamical features, such as the Zitterbewegung and, considering the scattering from potential, the so-called Klein paradox. The motivation is to show concretely how pure processing of quantum information can lead to particle mechanics as an emergent feature, an issue that has been the focus of solid-state, optical, and atomic-physics quantum simulators.
We show that the Dirac quantum cellular automaton [A. Bisio, G. M. D'Ariano, and A. Tosini, Ann. Phys. (N.Y.) 354, 244 (2015)] shares many properties in common with the discrete-time quantum walk. These similarities can be exploited to study the automaton as a unitary process that takes place at regular time steps on a one-dimensional lattice, in the spirit of general quantum cellular automata. In this way, it becomes an alternative to the quantum walk, with a dispersion relation that can be controlled by a parameter that plays a similar role to the coin angle in the quantum walk. The Dirac Hamiltonian is recovered under a suitable limit. We provide two independent analytical approximations to the long-term probability distribution. It is shown that, starting from localized conditions, the asymptotic value of the entropy of entanglement between internal and motional degrees of freedom overcomes the known limit that is approached by the quantum walk for the same initial conditions and is similar to the ones achieved by highly localized states of the Dirac equation.
A simple relation of the order of nnn abstract objects generates an n−2n-2n−2 dimensional basis of three dimensional vectors. A cellular automaton-like model of evolution of this system is postulated. During this evolution, some quantities stabilize with time and form a discrete spectrum of values. The presented model may have some general aspects in common with a cellular automaton representation of a quantum system.