Volume-Independent Spectral Stability of Energy-Truncated Effective Hamiltonians in Quantum Spin Systems

TL;DR

This paper establishes a volume-independent spectral stability result for energy-truncated effective Hamiltonians in quantum spin systems, ensuring low-energy spectral properties are robust in the thermodynamic limit.

Contribution

It extends the effective-Hamiltonian framework by providing volume-uniform spectral-overlap bounds applicable to infinite systems, strengthening previous finite-volume results.

Findings

Proves volume-uniform spectral-overlap bounds for quantum spin systems.

Demonstrates stability of low-lying eigenvalues in infinite volume.

Provides exponential decay of cutoff-dependent remainders in spectral bounds.

Abstract

We prove a volume-uniform effective-Hamiltonian theorem for bounded finite-range quantum spin systems on possibly infinite lattices. For any finite target region, we construct an energy-truncated Hamiltonian and prove a volume-uniform spectral-overlap bound controlling the leakage of its low-energy spectral subspace into the high-energy spectral subspace of the original Hamiltonian. The bound may contain non-exponential spectral-window terms, but its cutoff-dependent remainder decays exponentially in the cutoff. In finite volume, this yields stability of low-lying eigenvalues, with eigenvalue errors controlled by the exponentially small cutoff-dependent remainder. In infinite volume, we prove the corresponding spectral-overlap estimate in the GNS representation of an infinite-volume ground state. Thus, for bounded finite-range interactions, we extend and strengthen the effective-Hamiltonian mechanism of Arad, Kuwahara, and Landau by replacing the finite-volume operator-norm formulation with a volume-uniform spectral-overlap formulation applicable in the thermodynamic limit.