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

• URL: http://arxiv.org/abs/2009.09441v1
• Date: Sun, 20 Sep 2020 14:44:42 GMT
• Title: Poincar\'{e} crystal on the one-dimensional lattice
• Authors: Pei Wang
• Abstract summary: We develop the quantum theory of particles with discrete Poincar'e symmetry on the one-dimensional Bravais lattice. We find the conditions for the existence of representation, which are expressed as the congruence relation between quasi-momentum and quasi-energy. During the propagation, the particles stay localized on a single or a few sites to preserve the Lorentz symmetry.
• Score: 8.25487382053784
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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