Many-body systems with random spatially local interactions

Abstract

We extend random matrix theory to consider randomly interacting spin systems with spatial locality. We develop several methods by which arbitrary correlators may be systematically evaluated in a limit where the local Hilbert space dimension $N$ is large. First, the correlators are given by sums over 'stacked' planar diagrams which are completely determined by the spectra of the individual interactions and a dependency graph encoding the locality in the system. We then introduce 'heap freeness' as a generalization of free independence, leading to a second practical method to evaluate the correlators. Finally, we generalize the cumulant expansion to a sum over 'dependency partitions', providing the third and most succinct of our methods. Our results provide tools to study dynamics and correlations within extended quantum many-body systems which conserve energy. We further apply the formalism to show that quantum satisfiability at large-$N$ is determined by the evaluation of the independence polynomial on a wide class of graphs.

arXiv comment33 pages, 6 figures

References in corpus (20)

• A bound on chaos
• Random Matrix Theories in Quantum Physics: Common Concepts
• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Black holes as mirrors: quantum information in random subsystems
• Fast Scramblers
• Operator Spreading in Random Unitary Circuits
• Localized shocks
• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws
• The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual
• Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws
• Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation
• Solution of a minimal model for many-body quantum chaos
• Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems
• Onset of Random Matrix Behavior in Scrambling Systems
• When a local Hamiltonian must be frustration-free
• A Diagrammatic Theory of Random Scattering Matrices for Normal-Superconducting Mesoscopic Junctions
• Lattices of matrices (revised)
• Epsilon-noncrossing partitions and cumulants in free probability
• Quantum Lovász Local Lemma: Shearer's Bound is Tight

Cited by in corpus (7)

• Eigenstate Thermalization Hypothesis and Free Probability
• Modelling equilibration of local many-body quantum systems by random graph ensembles
• Hamiltonian Dynamics of a Sum of Interacting Random Matrices
• The spectrum of local random Hamiltonians
• Matrix models for $\epsilon$-free independence
• Energy dynamics in a class of local random matrix Hamiltonians
• Quantum Lovász Local Lemma: Shearer's Bound is Tight