A8 Finite-Size Criteria for Spectral Gaps in Quantum Lattice Systems
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 [H04; BMNS12] and it has far-reaching consequences for the low-energy physics of the system, specifically for the ground state correlation properties [NS06;H07; AAG21]. 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:
(i) What is the wider scope of finite-size criteria? The last few years have seen various extensions and improvements of finite-size criteria for spectral gaps in frustration-free quantum spin systems. So far, a main focus has been on their quantitative improvement. By contrast, finite-size criteria for longranged and unbounded interactions are lacking. The first goal of this project is to extend finite-size criteria to long-ranged and unbounded interactions which occur in applications, e.g., in bosonic lattice gases. Afterwards, we will turn to broader and more ambitious extensions of these criteria, specifically by investigating (a) finite-size criteria for alternative notions like mobility gap or ground state correlations and (b) criteria for frustrated Hamiltonians. To summarize, the long-term vision is to grow finite-size criteria into a methodological paradigm that can be used to tackle spectral gap problems in a wide variety of quantum
(ii) In which concrete models can we verify finite-size criteria? In recent years, finite-size criteria have been successfully used to obtain spectral gaps in various frustration-free quantum spin systems, especially higher-dimensional ones. While these first successes are encouraging, the spectral gap problem remains open for many frustration-free Hamiltonians of wide interest. This even includes onedimensional Hamiltonians like the recently introduced Motzkin spin chains of area weight t < 1 and spin S ⩾ 2. Relevant 2D examples where the spectral gap problem remains open include the AKLT Hamiltonians on the square lattice and on the Kagome lattice. This project will apply improved finite-size criteria, including those developed in Part (i) to concrete model Hamiltonians with the goal of deriving a spectral gap. In practice, a finite-size criterion is verified either through rigorous analytical estimates, culminating in a mathematical theorem, or numerically with computer assistance. A related novel direction is to study the spectral gap of random translation-invariant frustration-free Hamiltonians. Finite-size criteria have already proven useful in this context. However, several open questions remain; in particular, whether there are physically natural conditions on the probabilitydistribution of the local interaction that ensure a random translation-invariant Hamiltonian is gapped with probability one.