Solutions of a Two-Particle Interacting Quantum Walk

Abstract

Abstract:We study the solutions of an interacting Fermionic cellular automaton which is the analogue of the Thirring model with both space and time discrete. We present a derivation of the two-particle solutions of the automaton recently in the literature, which exploits the symmetries of the evolution operator. In the two-particle sector, the evolution operator is given by the sequence of two steps, the first one corresponding to a unitary interaction activated by two-particle excitation at the same site, and the second one to two independent one-dimensional Dirac quantum walks. The interaction step can be regarded as the discrete-time version of the interacting term of some Hamiltonian integrable system, such as the Hubbard or the Thirring model. The present automaton exhibits scattering solutions with nontrivial momentum transfer, jumping between different regions of the Brillouin zone that can be interpreted as Fermion-doubled particles, in stark contrast with the customary momentum-exchange of the one-dimensional Hamiltonian systems. A further difference compared to the Hamiltonian model is that there exist bound states for every value of the total momentum and of the coupling constant. Even in the special case of vanishing coupling, the walk manifests bound states, for finitely many isolated values of the total momentum. As a complement to the analytical derivations we show numerical simulations of the interacting evolution.

A review of Quantum Cellular Automata

“…Then the overall dynamics is given by U = V W , and this QCA is referred to as a Thirring QCA [157]. This is essentially a second-quantized version of the interacting quantum walks considered in [158,159]. For other studies of interacting two-particle quantum walks, see [160,161].…”

Lattice–Gas–Automaton Modeling of Income Distribution

“…As a type of cellular automata, the LGA is almost the simplest among the aforementioned effective models [ 27 , 28 , 29 ]. The LGA method was pioneered by the Hardy–Pomeau–de Pazzis model in 1973 [ 28 ].…”

Quantized Alternate Current on Curved Graphene

“…Quantum walks were first introduced by Aharonov and collaborators in 1993 [12], just a few months before the appearance of the first quantum lattice Boltzmann scheme [13], which was only recently recognized to be a quantum walk too [14]. Quantum walks [15,16] are currently utilized to investigate exotic states of quantum matter [17] and to design new materials and technologies for quantum engineering applications [18].…”