Poincar\'{e} crystal on the one-dimensional lattice

TL;DR

This paper develops a quantum theory for particles with discrete Poincaré symmetry on a one-dimensional lattice, introducing a Lorentz-invariant many-body framework that respects lattice translational symmetry and analyzing particle localization.

Contribution

It introduces the first quantum theory incorporating discrete Poincaré symmetry on a lattice, including conditions for representations and Lorentz-invariant many-body Hamiltonians.

Findings

Green's functions show lattice-structured nonzero points.

Particles remain localized during propagation to preserve symmetry.

The theory includes long-range hopping Hamiltonians.

Abstract

In this paper, we develop the quantum theory of particles that has discrete Poincar\'{e} symmetry on the one-dimensional Bravais lattice. We review the recently discovered discrete Lorentz symmetry, which is the unique Lorentz symmetry that coexists with the discrete space translational symmetry on a Bravais lattice. The discrete Lorentz transformations and spacetime translations form the discrete Poincar\'{e} group, which are represented by unitary operators in a quantum theory. We find the conditions for the existence of representation, which are expressed as the congruence relation between quasi-momentum and quasi-energy. We then build the Lorentz-invariant many-body theory of indistinguishable particles by expressing both the unitary operators and Floquet Hamiltonians in terms of the field operators. Some typical Hamiltonians include the long-range hopping which fluctuates as the distance between sites increases. We calculate the Green's functions of the lattice theory. The spacetime points where the Green's function is nonzero display a lattice structure. During the propagation, the particles stay localized on a single or a few sites to preserve the Lorentz symmetry.