We prove that quantum local Hamiltonians with generic interactions are gapless. In fact, we prove that there is a continuous density of states arbitrary above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for herein may include translational invariance in a disorder (i. e. , probabilistic) sense with some assumptions on the local distributions. We calculate the scaling of the gap with the system's size in the case that the local terms are distributed according to gaussian β−orthogonal random matrix ensemble. As a corollary there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. In addition to the lack of an energy gap, we prove that the ground state is degenerate when the local eigenvalue distribution is discrete. Time permitting we will present another very new result on the eigenvalue distribution of sums of matrices from the knowledge of the summands. This theory and techniques utilize modern free probability theory and other ideas from random matrix theory. References:RM- "Generic Local Hamiltonians are Gapless", Phys. Rev. Lett (2017) (RM-, Alan Edelman) Phys. Rev. Lett. 107, 097205 (2011)
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