Multifractal Analysis of Low-Energy Eigenstates

• Multifractal analysis of low-energy eigenstates is a method that quantifies scale-invariant, hierarchical distributions of quantum wavefunction amplitudes using participation ratios and generalized fractal dimensions.
• It employs techniques such as numerical diagonalization, real-space renormalization group, and power-law scaling to extract detailed spectral and scaling properties.
• The approach provides insights into quantum criticality, localization phenomena, and many-body transitions, bridging few-body observable bounds with complex quantum behavior.

Multifractal analysis of low-energy eigenstates investigates the scale-invariant, nontrivial distribution of quantum wavefunction amplitudes in a variety of quantum systems, particularly near or at the ground state. Unlike simple fractal or extended (ergodic) states, multifractal eigenstates display a spectrum of scaling exponents indicative of hierarchical structures in configuration or real space. This phenomenon is quantitatively characterized using a family of generalized fractal dimensions and a corresponding singularity spectrum, revealing fundamental aspects of quantum criticality, localization, and the structure of many-body Hilbert space.

1. Formalism and Key Quantities

The multifractal characterization of an eigenstate $\Psi = \sum_\sigma \Psi_\sigma |\sigma\rangle$ (with $|\sigma\rangle$ a basis of size $M$) is based on moments of the squared amplitudes, known as participation ratios: $P_q = \sum_\sigma |\Psi_\sigma|^{2q}$ In the thermodynamic limit, if the wavefunction is multifractal, $P_q$ displays nontrivial scaling: $$P_q \sim M^{-\tau(q)}, \qquad \tau(q) = -\lim_{M\to\infty} \frac{\ln P_q}{\ln M}$$ The Rényi entropy,

$S_R(q, M) = -\frac{1}{q-1} \ln P_q$

gives access to the generalized fractal dimension,

$$D_q = \lim_{M\to\infty} \frac{S_R(q, M)}{\ln M} = \frac{\tau(q)}{q-1}$$

The full scaling structure is captured by the singularity spectrum (via Legendre transform),

$$\alpha(q) = \frac{d\tau(q)}{dq}, \qquad f(\alpha) = q\alpha - \tau(q)$$

A nonlinear, non-constant $D_q$ and broad $f(\alpha)$ signal genuine multifractality, distinguishing these eigenstates from both localized and ergodic (random matrix-like) states (Atas et al., 2012).

2. Analytical Results for Prototype Models

Integrable Spin Chains

For the ground states of 1D quantum spin-$\frac12$ chains, such as the transverse-field Ising, XY, XXZ, and XYZ models, explicit closed-form or integral representations for $D_q$ are available in specific regimes:

• Transverse-Field Ising Model: Extreme fractal dimensions,

$$D_{\pm\infty}(\lambda) = \frac12 - \frac{1}{2\pi\ln2} \int_0^\pi \ln\left[1 \pm \frac{\lambda - \cos u}{\sqrt{1-2\lambda \cos u + \lambda^2}}\right] du$$

At the factorizing field and for special parameter values, the wavefunction becomes a product (binomial) measure, yielding

$$D_q = -\frac{\ln[\cos^{2q}\theta + \sin^{2q}\theta]}{(q-1)\ln 2}$$

• XXZ and XYZ Chains: At the “combinatorial point,” $D_q$ can be computed using special combinatorial amplitudes, resulting in nontrivial but explicit numerical values.

Numerical diagonalization confirms that across these and related models, the ground-state $D_q$ is a smooth, convex function of $q$ with analytic and finite-size agreement (Atas et al., 2012).

3. Methodologies: Numerical, Real-Space RG, and Power-Law Approaches

Numerical Diagonalization and Scaling

For non-integrable or non-exactly solvable systems, multifractal analysis proceeds via exact diagonalization up to maximal tractable sizes (e.g., $N\sim 13$ for spin chains), with participation ratios and entropies computed for all low-energy eigenstates. Extrapolation employs fits of $S_R(q, N)$ to linear forms in $N\ln2$ plus finite-size corrections. Results for $D_q$ stabilize rapidly and exhibit minimal parity or subleading effects (Atas et al., 2012).

Real-Space Renormalization Group (RG) in Quasiperiodic Chains

In aperiodic tight-binding chains (Fibonacci, Tribonacci, etc.), real-space RG combined with local resonator mode (LRM) analysis provides closed-form predictions for the scaling exponents in the low-energy (bottom band), e.g.,

$$D_q^{(\rm low)} = \frac{-q\ln s + \ln 2}{3(q-1)\ln\beta}$$

where $s$ and $\beta$ encode the RG reduction of amplitude and length, respectively. This framework confirms the critical, scale-invariant (i.e., multifractal) structure of the ground states in these models, with explicit agreement between RG and numerical values (Krebbekx et al., 2023).

Power-Law Decaying States

A generic principle emerges: eigenstates with power-law decay $|\psi(n)|^2 \sim n^{-2a}$ in $d$ dimensions satisfy

$$q_*=d/(2a),\quad D_q=1\text{ for } q\le q_*,\quad D_q=1-(1-D_\infty)\frac{q-q_*}{q-1}\text{ for } q>q_*$$

This allows a sharp distinction between simple fractal ($D_q = \text{const}$) and genuine multifractal ($D_q$ non-constant) phases. For example, in a 1D mean Hamiltonian mapped to a quantum harmonic oscillator, low-energy eigenstates display $a=1/4$, $d=1$, and thus multifractality for $q>2$ (Das et al., 28 Jan 2025).

4. Multifractality in Disordered and Correlated Systems

Strongly Correlated Disorder

Maximally correlated disorder in 1D tight-binding Hamiltonians leads to “anomalous” multifractal low-energy eigenstates. Unlike conventional critical multifractals with spiky, hierarchical amplitudes, these states exhibit uniform amplitudes over a contiguous support of length $\ell\sim L^\alpha$ ($0<\alpha<1$). The $D_q$ spectrum decreases smoothly from $1$ to $0.5$ with increasing disorder strength $W$, and dynamical behaviors include ballistic spreading up to a sub-extensive scale and saturation at $X^2\sim L^{2\beta}$, $\beta<1$. These properties contrast with both extended and Anderson-type critical states (Duthie et al., 2021).

Many-Body Localization and Spectral Multifractality

Across the many-body localization (MBL) transition, multifractal analysis using the spectral decomposition in a localized state basis reveals characteristic energy scales dividing ergodic and multifractal (localized) behavior. In the ergodic phase, generalized fractal dimensions $D_q$ approach $1$; in the MBL phase, $D_q$ decreases toward zero in the thermodynamic limit. The scaling functions $G_q$ governing this crossover are consistent with Kosterlitz-Thouless-type criticality: $$D_q(L,\delta g)=D_{q,c} + (1-D_{q,c}) G_q^{\rm erg}(L/\Lambda), \quad D_q(L,\delta g)=D_{q,c} G_q^{\rm MBL}((\ln L)/\xi)$$ Demonstrated numerically, this structure governs both mid-spectrum and low-energy eigenstates (Roy, 2024).

5. Bounding Fractal Dimensions via Observables

Participation entropies can be rigorously bounded above by the structure of few-body observables, particularly in systems obeying the eigenstate thermalization hypothesis (ETH). For fixed one- and two-body densities (e.g., local particle occupations and correlations), maximizing the entropy subject to these constraints yields sharp upper bounds $S_q^{\max}$ and therefore $D_q^{\max}$. In ergodic (chaotic) systems, these bounds saturate the actual $D_q$ as $L\to\infty$, while in localized phases the gap between the true $D_q$ and the bound (which grows as $\sim\ln Q$ with Hilbert space size) signals genuine multifractality beyond few-body expectation values. Moreover, the one-body bound connects directly to single-particle occupation-number entropies, thus providing a bridge between single-particle and Fock-space multifractal analyses (Vanhala et al., 29 Jan 2025).

6. Physical Interpretation and Comparison to Other Systems

Nontrivial $q$-dependence of $D_q$—i.e., multifractality of low-energy eigenstates—reflects neither localization (where amplitudes concentrate on a vanishing fraction of basis states) nor full ergodicity (where amplitudes are spread nearly uniformly). Instead, these states display a hierarchical, scale-invariant organization of amplitudes. This feature appears at quantum critical points, in aperiodic or quasiperiodic chains, and at the MBL transition, distinguished by the absence of external randomness in integrable cases and emergence purely from interaction-induced complexity of the Hilbert space. The tree-like structure of sparse many-body Hamiltonians with large effective branching numbers naturally supports this intermediate regime (Atas et al., 2012).

7. Applications, Generalizations, and Future Directions

Multifractal analysis of low-energy eigenstates serves as a fundamental diagnostic for quantum phase transitions, localization phenomena, and complexity growth in many-body quantum systems. Practical tools—such as observable-based entropy bounds, real-space RG prescriptions, and spectral decompositions—enable nuanced characterizations across integrable, disordered, and nonergodic systems. These methods establish direct links between few-body and many-body measures, as well as connections to single-particle multifractal localization. Continuing advances in diagonalization, RG methods, and experimental measurement of few-body observables are anticipated to further unify the understanding of multifractal structures across diverse physical settings (Atas et al., 2012, Vanhala et al., 29 Jan 2025, Das et al., 28 Jan 2025, Krebbekx et al., 2023, Duthie et al., 2021, Roy, 2024).