Random translation-invariant Hamiltonians and their spectral gaps

Ian Jauslin1 and Marius Lemm2

1Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
2Department of Mathematics, University of Tübingen, 72076 Tübingen, Germany

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We consider random translation-invariant frustration-free quantum spin Hamiltonians on $\mathbb Z^D$ in which the nearest-neighbor interaction in every direction is randomly sampled and then distributed across the lattice. Our main result is that, under a small rank constraint, the Hamiltonians are automatically frustration-free and they are gapped with a positive probability. This extends previous results on 1D spin chains to all dimensions. The argument additionally controls the local gap. As an application, we obtain a 2D area law for a cut-dependent ground state via recent AGSP methods of Anshu-Arad-Gosset.

When studying low-energy many-body physics, a key quantity that determines basic features of any ground state is the spectral gap of the Hamiltonian operator above the ground state sector. In particular, a positive lower bound on the spectral gap that is independent of the system size implies rapid (exponentially fast) decrease of correlations in the ground state and in some cases it controls the amount of entanglement in the ground state. In this case, we say the system is "gapped". A variety of frustration-free translation-invariant Hamiltonians, mostly in one spatial dimension, have been proven to be gapped.

Here we study "typical" frustration-free translation-invariant Hamiltonians on any graph and their spectral gaps. We randomly generate translation-invariant Hamiltonians on any graph and prove that they are automatically frustration-free under a small rank constraint. Then we prove that these Hamiltonians are gapped with a positive probability.

The argument automatically controls the size of all gaps on intermediate system sizes as well (these are called "local gaps"). This allows us to verify a recent criterion for an area law for the entanglement of the ground state in 2D by Anshu-Arad-Gosset, again with positive probability.

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[4] Radu Andrei, Marius Lemm, and Ramis Movassagh, "The spin-one Motzkin chain is gapped for any area weight $t<1$", arXiv:2204.04517, (2022).

[5] Marius Lemm and David Xiang, "Quantitatively improved finite-size criteria for spectral gaps", Journal of Physics A Mathematical General 55 29, 295203 (2022).

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