Random translation-invariant Hamiltonians and their spectral gaps

Abstract

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.

12 pages; 1 figure

References in corpus (7)

• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Valence Bond Solids for Quantum Computation
• Quantum computational capability of a 2D valence bond solid phase
• An area law for 2D frustration-free spin systems
• Gaplessness is not generic for translation-invariant spin chains
• Quantitatively improved finite-size criteria for spectral gaps
• Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems

Cited by in corpus (4)

• Typical Correlation Length of Sequentially Generated Tensor Network States
• Quantitatively improved finite-size criteria for spectral gaps
• On the critical finite-size gap scaling for frustration-free Hamiltonians
• Area law for steady states of detailed-balance local Lindbladians