Beyond characteristic equations: A unified one-dimensional non-Bloch band theory via wavefunction data

Non-Hermitian systems play a central role in nonequilibrium physics, where determining the energy spectrum under open boundary conditions is a fundamental problem. Non-Bloch band theory, based on the characteristic equation <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mtext>det</a:mtext><a:mo>[</a:mo><a:mi>E</a:mi><a:mo>−</a:mo><a:mi>H</a:mi><a:mo>(</a:mo><a:mi>β</a:mi><a:mo>)</a:mo><a:mo>]</a:mo><a:mo>=</a:mo><a:mn>0</a:mn></a:mrow></a:math>, has emerged as a key tool for this task. However, we show that this framework becomes insufficient in systems with certain symmetries, where identical characteristic equations can yield different spectra. To resolve this, we develop a unified theory that incorporates additional wavefunction information beyond the characteristic equation. Our framework accurately captures spectral properties such as the energy spectrum and the end-to-end signal response in a broad class of systems, particularly those with high symmetry. It reveals the essential role of wavefunction information and symmetry in shaping non-Hermitian band theory.