Complement of the Hamiltonian

Stefan K. Kehrein

Patrick Panella

The static selector for graph-local Hamiltonians vanishes exactly at decomposition classes in which the Hamiltonian is a sum of one-site terms and graph-supported two-site terms. We give a sufficient operator-algebraic criterion for uniqueness of such an admissible-support realization, and hence for exact identification of the underlying Hilbert-space decomposition on the physical quotient. For an exact graph-local realization and an omitted site, we study the unital *-algebra generated by all one-site and graph-supported two-site terms that avoid that site. A seed-and-propagation theorem shows that if full one-site matrix control is available at a seed vertex in each connected component of the omitted-site complement and operator-Schmidt coefficients propagate that control along complement edges, then this complement algebra is the full matrix algebra on the omitted-site complement, so the omitted one-site algebra is recovered exactly as a commutant. A pairwise rigidity theorem then shows that two complement-generated exact realizations with the same recovered site-family differ only by local basis changes and subsystem permutation. Consequently, any exactly graph-local Hamiltonian with frame-stable site reconstruction has a unique admissible-support realization and exact zero-set identifiability. The hypotheses are sufficient and are stated entirely in terms of the exact graph-local realization.

D. E. Ismoilova

In this paper, we examine a 3 × 3 operator matrix 𝒜 within the context of fermionic Fock space. This operator is associated with the energy operator (Hamiltonian). This energy operator is derived from a non-preserved system containing three interacting quantum particles. We portray the location of the σess(𝒜) and construct the first and second Schur complements associated with the investigated operator matrix. We analyze the primary spectral characteristics of these two Schur complements and provide a realization of the well-known Birman-Schwinger principle for the first Schur complement.

‎Louis Marchildon

M. Fannes, Reinhard F. Werner

For classical lattice systems, the Dobrushin-Lanford-Ruelle theory of boundary conditions states that the restriction of a global equilibrium state to a subsystem can be obtained as an integral over equilibrium states of the subsystem alone. The Hamiltonians for the subsystem are obtained by fixing a configuration for the variables in the complement of the subsystem, or more generally, by evaluating the full interaction Hamiltonian with respect to a state for the complement. We provide examples showing that the quantum mechanical version of this statement is false. It fails even if the subsystem is classical, but embedded into a quantum environment. We suggest an alternative characterization of the local restrictions of global equilibrium states by inequalities involving only local data.

A. Horzela