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.
Highlights
Quantum walks (QWs) describe the evolution of one-particle quantum states on a lattice, or more generally, on a graph
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
Summary
Quantum walks (QWs) describe the evolution of one-particle quantum states on a lattice, or more generally, on a graph. New research perspectives are unfolding in the scenario of multi-particle interacting quantum walks where two or more walking particles are coupled via non-linear (in the field) unitary operators. The properties of these systems are still largely unexplored. First we see that the set of possible scattering solutions is larger in the discrete-time case: for a fixed value total momentum, a non trivial transfer of relative momentum can occur besides the simple exchange of momentum between the two particles, differently from the Hamiltonian case. We show that in the set of solutions for the interacting walk there are perfectly localized states (namely states which lie on a finite number of lattice sites) and, differently from the Hamiltonian systems, bound states exist for vanishing coupling constant. In addition to the exact analytical solution of the dynamics we show the simulation of some significant initial state
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