Algebraic structure of Fock-state lattices

TL;DR

This paper explores the algebraic structure of Fock-state lattices (FSLs) using Lie algebras, revealing their geometric and symmetry properties and connecting them to phase-space concepts.

Contribution

It introduces a systematic algebraic framework for analyzing FSLs, linking their structure to Lie algebras and phase-space geometry, including curvature and symmetry insights.

Findings

FSLs can be constructed from Lie algebra generators, with sites and bonds defined by Cartan and root generators.

The algebraic approach reveals the dimensionality, connectivity, and symmetry constraints of FSLs.

Some Hamiltonians do not admit an underlying Lie algebra, especially nonlinear ones, and superalgebras may be needed for mixed systems.

Abstract

We analyze Fock-state lattices (FSLs) from an algebraic viewpoint. Starting from a Lie algebra, we associate a FSL constructed from the action of its generators: diagonal (Cartan) generators define the lattice sites, while off-diagonal (root) generators determine the lattice bonds. This construction reveals that identifying an underlying algebraic structure provides direct physical insight into FSLs, including their dimensionality, connectivity, symmetry constraints, and possible transport and revival phenomena. By examining several common Lie algebras, we identify not only their associated FSLs but also the corresponding Lie phase spaces, thereby establishing a systematic connection between FSL dynamics and phase-space geometry. In many cases, both the phase space and the FSL exhibit nontrivial curvature, opening possibilities for exploring quantum dynamics in curved synthetic spaces. We further address whether every integrable Hamiltonian admits an underlying Lie algebra that reproduces the same FSL structure. We show that this is not generally the case, particularly for Hamiltonians that are nonlinear in the generators, and that for systems combining different types of degrees of freedom the appropriate underlying structure may instead be a Lie superalgebra.