Local symmetries in one-dimensional quantum scattering

TL;DR

This paper introduces the concept of local parity symmetry in one-dimensional quantum scattering, revealing how it influences transport properties, resonances, and perfect transmission in aperiodic potentials.

Contribution

It defines local parity symmetry in quantum systems, links it to invariant currents, and proposes a design scheme for resonant transparent potentials, advancing understanding of quantum transport.

Findings

Invariant non-local currents exist in locally symmetric potential domains.

Local parity symmetry is necessary and sufficient for perfect transmission with asymmetric states.

Multiple resonances are linked to the presence of local parity symmetries at different scales.

Abstract

We introduce the concept of parity symmetry in restricted spatial domains -- local parity -- and explore its impact on the stationary transport properties of generic, one-dimensional aperiodic potentials of compact support. It is shown that, in each domain of local parity symmetry of the potential, there exists an invariant quantity in the form of a non-local current, in addition to the globally invariant probability current. For symmetrically incoming states, both invariant currents vanish if weak commutation of the total local parity operator with the Hamiltonian is established, leading to local parity eigenstates. For asymmetrically incoming states which resonate within locally symmetric potential units, the complete local parity symmetry of the probability density is shown to be necessary and sufficient for the occurrence of perfect transmission. We connect the presence of local parity symmetries on different spatial scales to the occurrence of multiple perfectly transmitting resonances and propose a construction scheme for the design of resonant transparent aperiodic potentials. Our findings are illustrated through application to the analytically tractable case of piecewise constant potentials.