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.

arXiv comment12 pages; 1 figure

References in corpus (43)

• Anyons in an exactly solved model and beyond
• String-net condensation: A physical mechanism for topological phases
• An Area Law for One Dimensional Quantum Systems
• Lieb-Schultz-Mattis in Higher Dimensions
• Spectral Gap and Exponential Decay of Correlations
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Topological quantum order: stability under local perturbations
• Valence Bond Solids for Quantum Computation
• Undecidability of the Spectral Gap (short version)
• Stability of Frustration-Free Hamiltonians
• Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource
• Power law violation of the area law in quantum spin chains
• The spectral gap for some spin chains with discrete symmetry breaking
• Quantum computational capability of a 2D valence bond solid phase
• Quasi-Locality Bounds for Quantum Lattice Systems. Part I. Lieb-Robinson Bounds, Quasi-Local Maps, and Spectral Flow Automorphisms
• Rigorous RG algorithms and area laws for low energy eigenstates in 1D
• Quantum phase transition from bounded to extensive entanglement entropy in a frustration-free spin chain
• Local gap threshold for frustration-free spin systems
• Persistence of exponential decay and spectral gaps for interacting fermions
• An area law for 2D frustration-free spin systems
• Spectral gaps of frustration-free spin systems with boundary
• Demonstrating the AKLT spectral gap on 2D degree-3 lattices
• Unfrustrated Qudit Chains and their Ground States
• Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice
• Undecidability of the Spectral Gap in One Dimension
• Hybrid valence-bond states for universal quantum computation
• When a local Hamiltonian must be frustration-free
• A class of two-dimensional AKLT models with a gap
• Generic Local Hamiltonians are Gapless
• Two-dimensional Valence Bond Solid (AKLT) states from $t_{2g}$ electrons
• AKLT models on decorated square lattices are gapped
• Improved local spectral gap thresholds for lattices of finite dimension
• Product Vacua and Boundary State Models in d Dimensions
• Spectral Gaps and Incompressibility in a $\nu =1/3$ Fractional Quantum Hall System
• Product vacua with boundary states and the classification of gapped phases
• Spectral Gap and Edge Excitations of $d$-dimensional PVBS models on half-spaces
• Gaplessness is not generic for translation-invariant spin chains
• Quantitatively improved finite-size criteria for spectral gaps
• Lie-Schwinger block-diagonalization and gapped quantum chains with unbounded interactions
• Gapped PVBS models for all species numbers and dimensions
• The AKLT model on a hexagonal chain is gapped
• Sharp implications of AGSPs for degenerate ground spaces
• Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems

Cited by in corpus (5)

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