Prime Suspects in a Quantum Ladder

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