Solutions of a two-particle interacting quantum walk

TL;DR

This paper derives and analyzes two-particle solutions of a discrete-time interacting Fermionic quantum walk, revealing unique scattering behaviors, bound states, and momentum transfer phenomena that differ from traditional Hamiltonian models.

Contribution

It provides the first derivation of two-particle solutions for a space-time discrete Fermionic automaton with interactions, highlighting novel scattering and bound state properties.

Findings

Automaton exhibits nontrivial momentum transfer and Fermion-doubled particles.

Bound states exist for all total momenta, even with zero coupling.

Numerical simulations support analytical solutions and reveal unique scattering behaviors.

Abstract

We study the solutions of the interacting Fermionic cellular automaton introduced in Ref. [Phys Rev A 97, 032132 (2018)]. The automaton is the analogue of the Thirring model with both space and time discrete. We present a derivation of the two-particles solutions of the automaton, which exploits the symmetries of the evolution operator. In the two-particles 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 even for vanishing coupling constant. As a complement to the analytical derivations we show numerical simulations of the interacting evolution.