2021/11/02 by Matthew B. Hastings, Hastings, Matthew B.
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Quantum and electron transport phenomena #Quantum many-body systems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2111.01854
openalex publication_date 2021/11/02 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
We consider many-body quantum systems on a finite lattice, where the Hilbert space is the tensor product of finite-dimensional Hilbert spaces associated with each site, and where the Hamiltonian of the system is a sum of local terms. We are interested in proving uniform bounds on various properties as the size of the lattice tends to infinity. An important case is when there is a spectral gap between the lowest state(s) and the rest of the spectrum which persists in this limit, corresponding to what physicists call a ``phase of matter". Here, the combination of elementary Fourier analysis with the technique of Lieb-Robinson bounds (bounds on the velocity of propagation) is surprisingly powerful. We use this to prove exponential decay of connected correlation functions, a higher-dimensional Lieb-Schultz-Mattis theorem, and a Hall conductance quantization theorem for interacting electrons with disorder.