%0 Generic %1 hastings2021gapped %A Hastings, Matthew B. %D 2021 %K quantum spin system %T Gapped Quantum Systems: From Higher Dimensional Lieb-Schultz-Mattis to the Quantum Hall Effect %U http://arxiv.org/abs/2111.01854 %X 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.

@misc{hastings2021gapped, abstract = {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.}, added-at = {2021-11-04T14:43:44.000+0100}, author = {Hastings, Matthew B.}, biburl = {https://www.bibsonomy.org/bibtex/25aaeee34df2debec9e8f6eeae6165925/gzhou}, description = {Gapped Quantum Systems: From Higher Dimensional Lieb-Schultz-Mattis to the Quantum Hall Effect}, interhash = {ed02b360d9ae41dd4de4a769f9c4b453}, intrahash = {5aaeee34df2debec9e8f6eeae6165925}, keywords = {quantum spin system}, note = {cite arxiv:2111.01854Comment: Contribution to Proceedings of ICM 2022. 22 pages, 2 figures}, timestamp = {2021-11-04T14:43:44.000+0100}, title = {Gapped Quantum Systems: From Higher Dimensional Lieb-Schultz-Mattis to the Quantum Hall Effect}, url = {http://arxiv.org/abs/2111.01854}, year = 2021 }