Prime Suspects in a Quantum Ladder

Giuseppe Mussardo, Andrea Trombettoni, Zhao Zhang

Published 2020-05-04Version 1

In this paper we set up a suggestive number theory interpretation of a quantum ladder system made of ${\mathcal N}$ coupled chains of spin 1/2. Using the hard-core boson representation, we associate to the spins $\sigma_a$ along the chains the prime numbers $p_a$ so that the chains become quantum registers for square-free integers. The Hamiltonian of the system consists of a hopping term and a magnetic field along the chains, together with a repulsion rung interaction and a permutation term between next neighborhood chains . The system has various phases, among which there is one whose ground state is a coherent superposition of the first ${\mathcal N}$ prime numbers. We also discuss the realization of such a model in terms of an open quantum system with a dissipative Lindblad dynamics.