Quantum states that spread out—but remain uneven

The question: spread out, or trapped?

Disorder can make a quantum particle’s wave function uneven. In a localized state, its probability is concentrated in a limited region. In an extended state, it reaches across the system. But reaching many sites does not necessarily mean that the wave is spread evenly: rare, large amplitudes can have a strong influence on measures that emphasize high-probability sites.

Carlo Vanoni and colleagues study a particle hopping between neighboring sites on an infinite Cayley tree, a branching network. Hopping strengths vary randomly from link to link, with no random on-site energies. The tree is bipartite: its sites can be divided into two groups, with links only between groups. This structure gives the model chiral symmetry, which strongly affects states at the center of the energy spectrum—the focus of the paper.

From random links to wave-function statistics

The researchers derive power-law tails in the distribution of the local density of states (LDoS), a measure of how strongly states near a particular energy occupy a site. Their derivation uses the cavity method: equations relating the quantum response at a site to responses on the branches connected to it. At the band center, chiral symmetry simplifies these equations.

They compare the predictions with population dynamics, a numerical method that repeatedly updates a large sample of cavity fields to estimate their distribution. They also use exact diagonalization, or direct numerical solution, on finite random regular graphs. These graphs are not the same as an infinite Cayley tree. In a finite Cayley tree, an imbalance between the two groups of sites can produce zero-energy states associated with the boundary. The authors note that these states would obscure a direct test of bulk behavior, so their finite-graph comparison instead uses random regular graphs.

To connect LDoS statistics to wave functions, the authors relate moments of the LDoS distribution to moments of site probabilities in finite systems. Moments of higher order are particularly sensitive to large probabilities, and so reveal unevenness that a simple count of occupied sites could miss.

The main finding: semi-fractality

The calculations indicate a regime in which the wave function’s support—the set of sites needed to account for its norm—grows in proportion to system size, while higher moments behave as they would for a multifractal state. The authors call this combination semi-fractality. The state is extensive in its support, but its intensity statistics remain uneven: rare large amplitudes matter disproportionately to higher moments.

In the paper’s notation, the dimension describing the support is one, while dimensions for sufficiently high moments fall below one. The authors also report a hierarchy in rank-ordered site probabilities in finite-graph calculations. This offers a possible microscopic picture, but connecting that pattern to asymptotic scaling requires assumptions that the finite-size data do not establish.

A parameter in the hopping distribution controls how common very weak links are. As it increases, the analysis predicts a change from semi-fractal behavior to localization. At the boundary, the authors describe an extreme intermediate case as semi-localized: the support remains extensive, while higher-moment measures indicate localization. They also find that semi-fractality occurs for Gaussian hopping strengths; a singular abundance of weak links is not required.

What remains uncertain

The predicted transition and the numerical estimate do not coincide exactly. The analytical criterion gives a critical parameter of one third, while fits to finite-range numerical tails suggest about 0.25. The authors say the difference is unresolved by their results. Estimates of power-law exponents depend on which range of a finite distribution is fitted, particularly for some parameter values. Finite system sizes and energy broadening also complicate comparisons between the infinite-system calculation and individual finite-graph eigenstates.

The derivation assumes a stationary cavity-field distribution with power-law tails; it does not establish that such a distribution exists or is stable in all cases. Matching the two tails also does not prove an inversion relation for the entire LDoS distribution. The authors test that relation numerically for a particular case, rather than establishing it generally. Away from the band center, or when on-site disorder is added, the distribution crosses over from the chiral band-center result.

This is a theoretical and numerical study, not an experimental observation. The source is an arXiv preprint; the supplied material does not establish its peer-review status. The conclusions concern the model studied and its band-center analysis, not all systems with chiral symmetry.