A new framework presents quasi-locality bounds for quantum lattice systems

In the early 1970s, Elliott Lieb and Derek Robinson mathematically determined the upper bound on the speed at which information can travel in non-relativistic systems. This theoretical foundation has been essential to major breakthroughs and applications in many-body physics and quantum computing. A new comprehensive review presents the proofs behind such breakthroughs and their applications for analyzing quantum lattice systems.

Nachtergaele et al. present a comprehensive set of proofs for quasi-locality properties in quantum lattice systems. The paper mainly focuses on bosonic lattice systems, but the results can also be applied, with minor changes, to lattice fermion systems. Using both generalizations and examining specific cases, the paper expands the theoretical framework for calculating quasi-locality bounds in quantum lattice systems based on previous works.

As an application of Lieb-Robinson bounds, the authors discuss the existence of the thermodynamic limit of the dynamics and demonstrate how to approximate quasi-local observables with strictly local ones using conditional expectation maps. They provided detailed discussions of general quasi-local maps and the auxiliary dynamics, known as the spectral flow, which has applications in defining gapped ground state phases. Looking at generalized cases, the paper covers systems with infinite-dimensional Hilbert spaces and Hamiltonians with unbounded on-site contributions, wherein the operator norm topology is frequently replaced with the strong operator topology.

According to the authors, the new paper will be the first of a series — the next paper will apply the quasi-locality bounds and spectral flow techniques to practical applications, including the discussion of the stability of gapped ground state phases.

Source: “Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms,” by Bruno Nachtergaele, Robert Sims, and Amanda Young, Journal of Mathematical Physics (2019). The article can be accessed at https://doi.org/10.1063/1.5095769 .