Gapped quantum systems: from higher-dimensional Lieb-Schultz-Mattis to the quantum Hall effect (Q6200716)

Summary: 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. For the entire collection see [Zbl 07816359].