Finite-Size Criteria for Spectral Gaps in Quantum Lattice Systems

01.01.2023 bis 31.12.2026

A central question in the study of quantum lattice systems is whether the Hamiltonian operator exhibits a spectral gap above the ground state sector in the thermodynamic limit. Existence of a spectral gap is at the heart of the classification of quantum phases and it has far-reaching consequences for the low-energy physics of the system, specifically for the ground state correlation properties. Many open problems in mathematical physics concern the existence of a spectral gap for the Hamiltonian operator, but the mathematical techniques for rigorously deriving spectral gaps are limited. This research proposal focuses on the approach of deriving spectral gaps via finite-size criteria. Finite-size criteria allow to rigorously conclude the existence of a spectral gap at arbitrary system size from a single finite-size calculation. They have recently emerged as an effective tool for deriving spectral gaps in higher-dimensional frustration-free quantum spin systems. This proposal is guided by two related questions: (a) What is the wider scope of finite-size criteria? Here we explore long-range spin systems and bosonic lattice gases, for example. (b) In which concrete models can we verify finite-size criteria? Here we consider Motzkin Hamiltonians and certain families of random Hamiltonians, for example.