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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Bounds on eigenvalues of related Hamiltonian operators

H. H. Aly, Roger C. Hill

Published Jan 1, 1976• 0 citations
Publisher
DOI
10.1007/bf02730323

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We consider uncertainty relations that give lower bounds to the sum of variances. Finding such lower bounds is typically complicated, and efficient procedures are known only for a handful of cases. In this paper, we present procedures based on finding the ground state of appropriate Hamiltonian operators, which can make use of the many known techniques developed to this aim. To demonstrate the simplicity of the method, we analyze multiple instances, that involve two or more observables, both bounded and unbounded.
Appendix V‑A develops a modular-theoretic extension of R-layer Mode Theory (RLMT) by introducing relative modular Hamiltonians and relative layered entropy associated with the tension operator E and its eigenvalue hierarchy {λn}. Building on the spectral foundations of Appendix V, we formulate relative modular theory in a stratified Hilbert space where the spectral projections Π(n) define discrete layers. We construct the relative Tomita operator, the relative modular operator, and the relative modular Hamiltonian Kψ,ϕ, and define the relative layered entropy H(ψ,ϕ). For nested layer subsets L1⊂L2⊂L3, we establish monotonicity bounds HL1(ψ,ϕ)≤HL2(ψ,ϕ)≤HL3(ψ,ϕ), providing a layered analog of entropy inequalities in algebraic QFT. We further analyze coherent layered states in the CCR framework and derive explicit expressions for their relative modular Hamiltonians and entropies. Numerical models with Weyl-type eigenvalue growth illustrate the monotonicity and convergence of layered entropy bounds. This appendix provides the modular-theoretic backbone of RLMT, linking spectral hierarchy, modular flow, and information-theoretic structure across all RLMT volumes.
We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H0, we prove that there exists a constant threshold ϵ>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H0+ϵV has well-defined spectral bands originating from low-lying eigenvalues of H0. These bands are separated from the rest of the spectra and from each other by a constant gap. The band originating from the smallest eigenvalue of H0 has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb–Robinson bound.