Feature spectrum topology and bulk boundary correspondence in energy and projective operator spectra

Abstract Bulk-boundary correspondence, a key result of ground-state topology, predicts gapless boundary states at interfaces between distinct phases. Motivated by the limitations of conventional band topology in describing boundary states under weak symmetry-breaking perturbations, we propose feature spectrum topology, a generalized framework based on the spectrum of projected operators of the form $$F=P\widehat{O}P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:mi>P</mml:mi> <mml:mover> <mml:mrow> <mml:mi>O</mml:mi> </mml:mrow> <mml:mo>̂</mml:mo> </mml:mover> <mml:mi>P</mml:mi> </mml:math> , where P projects onto a manifold of interest, and $$\widehat{O}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mrow> <mml:mi>O</mml:mi> </mml:mrow> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> is an operator representing physical observables. We rigorously establish a bulk-boundary correspondence in this projected spectrum, showing that the topological invariant guarantees gapless boundary states appearing in either the energy spectrum or the projected operator spectrum. This feature-energy complementarity uncovers boundary phenomena that are overlooked by traditional energy-based approaches. Our framework provides a unifying perspective for characterizing topological features tied to observables, offering new insights into systems that extend beyond the conventional paradigm. We demonstrate the application and discuss possible experimental observations of our theory in topological materials.