Spectral Decimation of Quantum Many-Body Hamiltonians

Abstract

We develop a systematic theory of spectral decimation for quantum many-body Hamiltonians and show that it provides a quantitative probe of emergent symmetries in statistically mixed spectra. Building on an analytical description of statistical mixtures, we derive an explicit expression for the size of a characteristic symmetry sector (CSS), defined as the largest subsequence of levels exhibiting non-Poissonian correlations. The CSS dimension is shown to be the size-biased average of the underlying symmetry sectors, establishing a direct link between spectral statistics and Hilbert-space structure. We apply this framework to two paradigmatic settings: Hilbert-space fragmentation and disorder-induced many-body localization (MBL). In fragmented systems, the CSS reproduces the mixture prediction and isolates correlated subsectors even when the full spectrum appears nearly Poissonian. In the disordered Heisenberg chain, spectral decimation reveals the gradual emergence of integrability through a shrinking CSS, whose statistics exhibit signatures consistent with local integrals of motion. We introduce a characteristic symmetry entropy (CSE) as a finite-size scaling observable and extract, within accessible system sizes, the crossover exponents. Our results establish spectral decimation as a controlled, unbiased and computationally inexpensive diagnostic of hidden structure in many-body spectra, capable of distinguishing between chaotic dynamics, statistical mixtures, and emergent integrability.

Figures (8)

• Figure 1: Outcome of the decimation in the case the parent Hamiltonian is a) integrable, b) statistical mixture of chaotic spectra, c) non-interacting. In a) the periodic spin-1/2 XXX model is diagonalized for $L=22$ sites in the $SU(2)$ irreducible representation. The remainder gaps are Poissonian and the decimation succeeds for any $d_{\text{halt}}$. b) The parent Hamiltonian is a mixture of 24 random GOE (Gaussian Orthogonal Ensemble) matrices of same size $d = 10^4$. Here the decimation ends at the size of one GOE block and the gap statistics clearly shows signs of level repulsion. The small-gap expansion found in Section \ref{['s_mixtures']} closely follows the decimated gap statistics for $s \lesssim 1$. c) Decimated spectrum of a $L=10$ free fermionic chain with random on-site energies. In particular, they are i.i.d. uniformly in [0,1]. The gap statistics of the decimated gaps presents an excess in the zero-gap probability compared to the pure Poisson case, due to i.i.d. nature of the frequencies.
• Figure 2: Application of spectral decimation to the pair-flip model of Eq. \ref{['pair_flip']}. (a) The decimation output, $d_{\text{out}}$, follows closely the theoretical prediction Eq. \ref{['e_expectation_dout']}. (b) Despite the global distribution of the gaps follows closely the Poisson statistics, the distribution of the remainder gaps, the one surviving the decimation, feature clear sign of level repulsion and therefore the CSS is non-integrable.
• Figure 3: Histograms of gaps and $r$-ratios of the one-dimensional disordered Heisenberg chain for $L$ = 16 at weak ($W = 0.5$) and strong ($W = 6.5$) disorders, in a energy window of $\Delta E = 90 \%$ of the full spectrum. At weak disorder, one observe excellent agreement of the distribution of gaps (a) and $r$-ratios (b) with the Wigner-Dyson distribution of GOE random matrices. The value of $d_{\text{out}}/d$ is approximately 1 in both cases, the one obtained from the $r$-ratios subject to smaller statistical fluctuations. At strong disorder, on the other hand, while the $r$-ratio statistics (d) seems to be compatible with a semi-Poissonian (which eventually will be restored as $W\to\infty$), the gap statistics (c) clearly deviates from the Poisson one and accumulates towards $s\to0$. Sharp peaks are characteristic of effectively free or weakly interacting spectra, where spectra are regular and energy differences expressible as differences of linear combinations of one-particle energies. This distinction between free and strongly-interacting integrable system statistics is not captured by the $r$-ratio distribution, which is the same. We also observe that the CSS size extracted from r-ratio decimation deviates quantitatively from that of gap decimation, as discussed in Section \ref{['s_decimation']}.
• Figure 4: (a) Plot of the CSE over a large window of the spectrum ($\Delta E = 0.9$), varying the disorder strength $W$, for different volumes $L$. For $W\lesssim 3$, the CSE remains close to 0, indicating that the system is close to GOE, while it increases for larger $W$'s. For the disorder strengths and volumes studied, the CSE does not approach unity, even though the CSS may be as low as $1\%$ of the Hilbert space dimension. As such, the CSE is a sensible probe of the fine spectral structure of disordered quantum Hamiltonians. (b) FSS collapse of the various $\Sigma(W,L)$ curves. Fitted parameters, as well as the fitting form, are reported in the box on the bottom right.
• Figure 5: Results of finite-size scaling (FSS) of the characteristic symmetry entropy (CSE). (a) the FSS of the CSE is reported for few energy windows $\Delta E$, showing that as $\Delta E$ increases, the steepness of the curves, dictated by the critical exponent $\nu$ diminishes. (b) The same critical exponent $\nu$ is plotted against $\Delta E$ for the energy windows studied. (c) Similar plot for $W_*$. Finally, the fit constants are varying monotonically with the energy window (d).