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Determining the Number of Symmetry Sectors in Composite Quantum Spectra

This paper proposes a Best Subset Technique to determine the number of symmetry sectors in composite quantum spectra from both Hermitian and non-Hermitian systems by comparing spectral fluctuations to random matrix ensembles, thereby eliminating the need for prior de-symmetrization or knowledge of symmetry dimensions.

Original authors: Nisarg Vyas, S. Harshini Tekur, M. S. Santhanam

Published 2026-10-05

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CC BY 4.0

Original authors: Nisarg Vyas, S. Harshini Tekur, M. S. Santhanam

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of quantum physics, scientists often look for patterns in the chaotic jumble of energy levels that atoms and molecules possess. When a system is simple and predictable, these energy levels appear in a regular, uncorrelated fashion, much like raindrops hitting a roof at random intervals. However, when a system becomes complex and chaotic, the energy levels begin to interact and repel one another in a very specific way, creating a distinct statistical fingerprint. This fingerprint is so reliable that physicists use it to diagnose whether a system is behaving chaotically or not. But there is a catch: this diagnostic tool only works if the scientist is looking at a single, pure type of chaos. If the system has hidden rules, such as symmetries that split the energy levels into separate, independent groups, the resulting mix of levels looks like noise, hiding the true chaotic signal beneath a layer of confusion.

For decades, researchers have struggled with this problem when they cannot easily separate these hidden groups. If the groups are of equal size, there are ways to detect them, but real-world systems often have groups of vastly different sizes, making the task nearly impossible without knowing the system's internal structure beforehand. This is where a new approach, developed by researchers at the Indian Institute of Science Education and Research and MindBrug Research, offers a fresh perspective. They have devised a method to count these hidden groups directly from the messy, mixed data, without needing to first untangle the system or know its underlying rules.

The core of the challenge lies in how energy levels from different symmetry groups mix together. Imagine a choir where singers from three different sections are singing at once, but the sections have different numbers of people. If you listen to the whole group, the sound is a jumble. If the sections were equal in size, the pattern of the mix would be predictable, but when the sizes differ, the pattern becomes distorted and hard to read. Previous attempts to solve this relied on looking at the spacing between neighboring energy levels, but this method failed when the groups were unequal, often leading to the wrong count of how many groups were present. The researchers realized that simply looking at immediate neighbors was not enough; they needed to look further ahead in the sequence of levels to find a hidden rhythm.

The team introduced a technique they call the Best Subset Technique. Instead of trying to analyze the entire collection of energy levels at once, their method involves testing different slices of the data. They systematically select smaller and smaller portions of the spectrum, focusing on the regions where the different groups overlap most significantly. For each slice, they calculate the spacing between levels that are not just next to each other, but separated by several steps. By comparing the statistical distribution of these wider gaps against what is expected from pure chaos, they can identify which slice of the data reveals the true underlying structure. The method essentially searches for the specific fraction of the data and the specific spacing distance that best matches the theoretical signature of a single chaotic system.

When the researchers applied this technique to computer simulations of random matrices, which serve as the mathematical models for these quantum systems, the results were striking. In cases where traditional methods incorrectly identified the number of groups, the new technique successfully pinpointed the correct count. For instance, in a simulation involving four distinct groups of energy levels with different sizes, standard analysis suggested there were only three. The new method, by isolating the most informative subset of the data, correctly identified that there were four. This success held true across different types of random matrices, including those that model complex, non-reversible systems where energy can be gained or lost, not just the standard systems where energy is conserved.

The power of this approach was further demonstrated on actual physical models, such as a disordered magnetic system known as the transverse-field Ising model. In this system, the researchers constructed a composite spectrum from three different system sizes. Without the new technique, the analysis falsely concluded that there were four symmetry groups. However, by applying the Best Subset Technique and focusing on the optimal fraction of the data, the analysis correctly revealed the presence of three groups. Similarly, when testing a model of particles moving in a non-reciprocal way, where the rules of physics are slightly broken to allow for one-way transport, the method again confirmed the correct number of underlying symmetry sectors.

A key finding of this work is that the success of the technique depends heavily on the relative sizes of the hidden groups. When two groups are nearly the same size, the method requires almost the entire dataset to work. However, as the size difference between the groups grows, the method becomes more efficient, needing only a small, specific portion of the data to reveal the truth. Remarkably, this behavior depends on the fundamental symmetry class of the system rather than the specific details of the model, suggesting a universal principle at play. The researchers found that for systems with two groups, the optimal amount of data to analyze is determined by the type of symmetry and the ratio of the group sizes, not by the microscopic details of the particles involved.

This work provides a robust tool for physicists studying complex quantum systems, particularly those where the internal symmetries are unknown or too difficult to calculate. By allowing researchers to count the number of hidden symmetry sectors directly from the spectral data, the Best Subset Technique removes the need for complex, often impossible, mathematical transformations to separate the levels. It turns a previously ambiguous signal into a clear diagnostic, enabling a more accurate understanding of chaos in both theoretical models and potential future experiments with real quantum materials. The method does not require knowing the system's Hamiltonian, or its governing equations, beforehand, making it a versatile instrument for exploring the hidden architecture of the quantum world.

Technical Summary: Determining the Number of Symmetry Sectors in Composite Quantum Spectra

Problem Statement
Spectral fluctuations in complex quantum systems are typically modeled using Random Matrix Theory (RMT). Quantitative agreement with RMT ensembles (such as Wigner-Dyson or Ginibre) holds only when the analyzed spectrum constitutes a "pure sequence" from a single irreducible symmetry sector. However, physical systems often possess discrete symmetries (e.g., parity, reflection) that decompose the Hilbert space into $m$ invariant subspaces. The resulting spectrum is a composite superposition of $m$ statistically independent level sequences.

Analyzing such composite spectra without "desymmetrization" (separating the blocks) leads to ambiguous results, often masking the underlying RMT correlations and mimicking Poisson-like statistics. While previous methods exist to detect symmetries, they suffer from significant limitations:

They often require prior knowledge of the symmetry structure or the Hamiltonian.
They rely on "unfolding," necessitating prior knowledge of the level density.
Existing approaches using higher-order spacing ratios (HOSRs) are restricted to cases where symmetry blocks have equal dimensions and apply primarily to Gaussian Orthogonal Ensembles (GOE).
Distinguishing between different numbers of blocks ( $m$ ) becomes nearly impossible for modest values (e.g., $m=3, 4$ ) when block dimensions are unequal, as the spectral statistics of the composite spectrum do not uniquely identify the number of sectors.

The central challenge addressed is determining the number of symmetry sectors ( $m$ ) and their true spectral statistics from a composite spectrum of arbitrary block dimensions, without prior knowledge of the symmetry structure or the need for unfolding.

Methodology: The Best Subset Technique (BST)
The authors propose a numerical algorithm called the Best Subset Technique (BST) to infer $m$ using Higher-Order Spacing Ratios (HOSRs).

HOSR Definition: For an ordered sequence of eigenvalues (or singular values) $E_i$ , the $k$ -th order spacing ratio is defined as $r^{(k)}_i = \frac{E_{i+2k} - E_{i+k}}{E_{i+k} - E_i}$ .
Theoretical Basis: In a composite spectrum of $m$ equal-sized blocks, the "loose interleaving" property suggests that every $(m+1)$ -th eigenvalue likely originates from the same block. Consequently, the distribution of $k$ -th order spacing ratios ( $k$ -SRD) follows a specific scaling law where the effective Dyson index $\beta'$ relates to the number of blocks $m$ . Specifically, for equal blocks, $P^k(r, \beta, m) = P(r, \beta')$ , where $\beta' = k = m$ (for orthogonal classes) or $\beta' = k+1 = m+1$ (for unitary classes).
The Unequal Block Problem: When block dimensions ( $N_i$ ) are unequal, the eigenvalue support scales differently (e.g., $\Delta N \propto \sqrt{N}$ for Gaussian ensembles). This causes the "loose interleaving" property to break down across the entire spectrum, as levels from smaller blocks are sparser than those from larger blocks.
The BST Solution: The authors observe that a specific fraction ( $f$ $f$ ) of the composite spectrum (typically the region of maximal overlap, such as the center for Gaussian or the lower end for Wishart/Ginibre) retains sufficient information about all contributing blocks.
- The algorithm iterates over a range of subset fractions $f \in (0, 1)$ and HOSR orders $k$ .
- For each pair $(k, f)$ , it calculates the empirical $k$ -SRD on the subset.
- It computes the Kullback-Leibler (KL) divergence, $DKL(k, f)$ , between this empirical distribution and the theoretical RMT distribution $P(r, \beta')$ where $\beta'$ is determined by the symmetry class (e.g., $\beta' = k$ for GOE).
- The optimal parameters $k^*$ and $f^*$ are identified as the global minimum of the $DKL(k, f)$ surface.
- The number of symmetry sectors is then deduced self-consistently via the relation between $k^*$ and $m$ (e.g., $m = k^*$ for GOE).

Key Results
The BST was validated across several random matrix ensembles and physical models:

Random Matrix Ensembles:
- Gaussian (GOE/GUE): Applied to composite spectra of four GUE matrices with sizes $N \in \{5000, 10000, 15000, 20000\}$ . Without BST, standard HOSR analysis incorrectly identified $m=3$ (due to $k^*=3$ ). BST correctly identified $m=4$ by finding a global minimum at $k^*=4$ and a specific optimal fraction $f^* \approx 32\%$ .
- Wishart and Ginibre: The method successfully determined $m$ for composite spectra from Wishart-Laguerre and Ginibre ensembles (analyzing singular values for the latter), demonstrating applicability to non-Hermitian settings.
- Scaling of $f^*$ : For the case of two blocks ( $m=2$ ) with unequal dimensions $N_1$ and $N_2$ , the optimal fraction $f^*$ was found to depend primarily on the ratio $N_2/N_1$ and the Dyson index $\beta$ , rather than the specific ensemble details. As $N_2/N_1 \to 1$ , $f^* \to 1$ .
Physical Models:
- Transverse-Field Ising Model (TFIM): Applied to a disordered, all-to-all TFIM with chaotic dynamics. The composite spectrum was constructed from systems with $N \in \{14, 15, 16\}$ spins. Standard analysis suggested $m=4$ , while BST correctly deduced $m=3$ using only the best subset ( $f^* \approx 21\%$ ).
- Non-Hermitian Hardcore Boson Model: A one-dimensional non-reciprocal model at half-filling. The spectrum corresponds to the GinUE class. BST confirmed $m=3$ for composite spectra from systems of sizes $L \in \{16, 17, 18\}$ . In this specific case, the spectrum showed no size-dependent scaling, meaning the overlap was uniform, and BST confirmed the result found by standard analysis.

Significance and Claims
The paper claims that the BST provides a general, model-independent technique to determine the number of symmetry sectors in complex quantum systems. Its primary contributions are:

Handling Unequal Dimensions: It extends the utility of HOSRs to the general case where symmetry blocks have unequal dimensions, a scenario where previous methods fail.
No Desymmetrization Required: It allows for the inference of symmetry structure without constructing symmetry-adapted bases or performing spectral unfolding, which is crucial for experimental data or systems with hidden symmetries.
Universality: The technique applies equally to Hermitian (Gaussian, Wishart) and non-Hermitian (Ginibre) systems.
Robustness: The method relies on the "loose interleaving" property being recoverable within a specific spectral subset ( $f^*$ ), which is governed by the scaling of eigenvalue supports with matrix dimension.

The authors note that while the method is empirically robust, analytical support for the scaling of $f^*$ and the validity of the loose interleaving property in the general unequal-size case remains an open area for future investigation. They also suggest future work could extend these results to Wishart ensembles where $N \neq M$ , as both spectral edges scale non-trivially with system size in those cases.

Technical Summary: Determining the Number of Symmetry Sectors in Composite Quantum Spectra

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