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arXiv:1406.1021 · Physics Letters A

Path-integral solution of the one-dimensional Dirac quantum cellular automaton

Giacomo Mauro D’Ariano, Nicola Mosco, Paolo Perinotti, Alessandro Tosini

Published Sep 1, 2014• 18 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2014.09.020

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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 construct and analyze an explicit one-dimensional quantum cellular automaton (QCA) that provides a controlled lattice regularization of the free Dirac equation in (1+1) dimensions, and we extend it to a locally U(1) gauge-covariant version. The microscopic model is strictly local, causal in the Lieb–Robinson sense, and unitarily evolves a chain of finite-dimensional quantum systems by a translation invariant, discrete-time rule. At the representation-theoretic level, it realizes a chiral spin-1/2 degree of freedom per site. We prove that, in the joint limit of small lattice spacing and small time step with fixed physical scales, the QCA dynamics converges to the continuum Dirac evolution, with an explicit bound on the error valid for smooth, band-limited initial states. We then couple the QCA to a compact U(1) gauge field living on links, in a way that is exactly gauge-covariant at the microscopic level. A local Gauss constraint defines a gauge-invariant code subspace, and we show that the QCA update preserves this subspace and implements a gauge-covariant discrete Dirac dynamics. This provides a concrete example of a gauge-coded QCA with a controlled continuum limit, which can serve as a microscopic building block for more elaborate SU(3) × SU(2) × U(1) constructions. All results are stated with modest but explicit hypotheses and proofs, so that the framework is mathematically well-defined and amenable to further generalization.