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Spectral-topological downfolding of quantum Hamiltonians: from matrix mechanics to effective spin models
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DOI：10.1007/s40509-026-00393-y.png)
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Abstract
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Heisenberg's matrix formulation of quantum mechanics provides a natural operator-theoretic framework for quantum systems with internal degrees of freedom (spin, band and multi-level structure), in which physical properties arise from the spectral structure of self-adjoint Hamiltonians. In practical applications across atomic, condensed-matter and quantum-information physics, however, finite-dimensional effective models-such as spin-only Hamiltonians and few-level truncations-are often applied far beyond their natural domain of validity. This article develops a rigorous spectral-topological downfolding scheme interpolating between a microscopic self-adjoint Hamiltonian and low-dimensional effective spin models. Starting from an analytic family of self-adjoint Hamiltonians H(lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\lambda )$$\end{document}, with lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} ranging in a smooth parameter manifold M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document}, we construct a finite-rank spectral bundle Erel -> U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\textrm{rel}} \rightarrow U$$\end{document} over a suitable open set U subset of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U \subset M$$\end{document}, and introduce a spectral-topological transform T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}$$\end{document} which assigns to each H(lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\lambda )$$\end{document} a meromorphic field Phi(& centerdot;;lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (\cdot ;\lambda )$$\end{document} on an auxiliary parameter space. Topological invariants of Phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} (winding numbers, Chern-type indices, Euler characteristics) detect changes in the spectral bundle and provide a robust criterion for the validity of spin-only matrix models. Under spectral gap and topological triviality assumptions, we prove a downfolding theorem: there exists a smoothly varying family of effective spin Hamiltonians Heff(lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\textrm{eff}}(\lambda )$$\end{document} acting on a fixed finite-dimensional Hilbert space such that low-energy observables of H(lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\lambda )$$\end{document} are reproduced by Heff(lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\textrm{eff}}(\lambda )$$\end{document} with quantitatively controlled error, in the spirit of adiabatic and space-adiabatic perturbation theory. Within a topologically stable phase, the parameters of Heff(lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\textrm{eff}}(\lambda )$$\end{document} (effective g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document}-factors, splittings and couplings) can be expressed as smooth functionals of spectral-topological invariants extracted from Phi(& centerdot;;lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (\cdot ;\lambda )$$\end{document}. As a concrete illustration, we construct a simple spectral-topological transform based on resolvents and compute explicitly its action on a 2 & times;2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} model Hamiltonian, obtaining a meromorphic function Phi(z;lambda)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (z;\lambda )$$\end{document} whose poles and residues encode the low-energy spectrum. Conceptually, the formalism clarifies in which precise spectral-topological sense a finite-dimensional spin Hamiltonian can be regarded as a legitimate reduction of an underlying many-body Hamiltonian, and where it must necessarily fail once spectral gaps close or eigenbundles undergo topological transitions, as exemplified throughout standard treatments of few-level systems and effective models in quantum mechanics textbooks.
Keywords:
Spectral-topological downfolding
Quantum Hamiltonians
Effective spin models
Spectral bundles
Berry curvature
Adiabatic perturbation theory
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