A quantum many-body system has an energy spectrum — a list of allowed energies. If the system is chaotic, the spectrum follows random matrix statistics: level spacings obey the Wigner-Dyson distribution, and consecutive levels repel each other. If the system is integrable, levels are independent and spacings follow a Poisson distribution. This dichotomy — chaotic or integrable — has been the main question asked of energy spectra for decades.
He, Hutsalyuk, Mussardo, and Stampiggi (arXiv:2602.20256) show that asking this question of the full spectrum misses structure hiding in subsequences.
Spectral decimation is the method: take every nth energy level and analyze the statistics of the subsequence. The key insight is that different symmetry sectors of the Hilbert space contribute energy levels at different rates. When you decimate — sample every second or third level — you preferentially extract levels from the sector that contributes most densely. They call this the Characteristic Symmetry Sector (CSS), and derive an exact expression: its dimension equals the size-biased average of all symmetry sector dimensions.
The applications reveal what the full spectrum cannot see. In Hilbert-space fragmented systems, the full spectrum looks nearly random — close to Wigner-Dyson statistics. But decimation isolates correlated subsectors, showing that the apparent chaos is actually a statistical mixture of structured fragments. The system isn't chaotic; it is several non-chaotic systems layered on top of each other, and only the superposition looks random.
In disordered Heisenberg chains, decimation tracks the emergence of integrability through a shrinking CSS. As disorder increases, the characteristic sector contracts — fewer and fewer levels participate in the dominant statistical pattern. The transition from chaos to integrability is visible as a geometric shrinkage in Hilbert space, not just a change in level spacing statistics.
The deeper point: a spectrum is not just a set of numbers. It is a set of numbers with internal structure that depends on which subset you examine. The full set can deceive. The subsequence reveals.
This is a general principle in data analysis — aggregation destroys structure that disaggregation recovers — but it acquires special force in quantum mechanics, where the structure being hidden is a symmetry. The universe put the symmetry there. The spectrum scrambles it. Decimation unscrambles it by the simple act of skipping.