Generalized Aubry-André Formula

• Generalized Aubry–André formula is a framework of exact analytical conditions that determine localization-delocalization transitions in one-dimensional quasiperiodic systems.
• It extends the self-duality principle of the standard model to include complex modulations, energy-dependent phase transitions, and analytically accessible mobility edges.
• The formula provides a deterministic mechanism using duality transformations, enabling precise mapping of critical boundaries and design of synthetic quantum matter.

The generalized Aubry–André formula refers to the set of exact analytical conditions that determine the localization–delocalization (metal–insulator) phase boundaries in a broad class of one-dimensional deterministic aperiodic lattice models exhibiting quasiperiodic potentials or hopping modulations. Building on the self-duality of the original Aubry–André (AA) model, these generalizations extend the concept to models with more complex modulations, energy-dependent phase transitions, and settings where mobility edges—boundaries separating localized from extended eigenstates within the energy spectrum—are analytically accessible. Such results have enabled rigorous mapping of mobility edges in models ranging from the standard AA chain to the generalized Aubry–André–Harper (GAAH) model, almost Mathieu operators with various spectral regularities, and models incorporating long-range hopping or multi-harmonic potentials.

1. Generalized Aubry–André Models and Exact Mobility Edge Formulas

The canonical generalized Aubry–André model is defined by a one-dimensional nearest-neighbor tight-binding Hamiltonian with a site-dependent potential,

$$H = -J \sum_{n} \left( c_{n+1}^\dagger c_n + c_n^\dagger c_{n+1} \right) + \sum_{n} V_n c_n^\dagger c_n,$$

where the potential

$$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$

depends on the overall modulation strength $V$, incommensurate frequency $b$, phase $\phi$, and a deformation parameter $\alpha$ with $|\alpha| < 1$ (An et al., 2020, Cui et al., 2021, Alexanian, 25 Mar 2026, Lu et al., 10 Apr 2025). For $\alpha = 0$, this reduces to the standard AA model with a pure cosine potential.

The analytic mobility edge separating localized ($|E| > E_c$) from extended ($|E| < E_c$) single-particle states is given by the energy-dependent self-duality relation,

$$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$0

or, equivalently,

$$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$1

with the AA transition $$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$2 recovered as $$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$3 (An et al., 2020, Cui et al., 2021, Alexanian, 25 Mar 2026).

In models with generalized hopping (e.g., exponential or power-law decay), the critical energy for localization is determined from the self-duality condition,

$$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$4

where $$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$5 characterizes the hopping decay and $$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$6 the hopping scale (Biddle et al., 2010). The corresponding mobility edge,

$$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$7

captures the localization boundary for broad classes of deterministic models, including non-nearest-neighbor or long-range couplings.

2. Analytical Derivation and Physical Mechanisms

Duality transformations underpin the derivation of mobility-edge formulas. The standard AA model exhibits self-duality under Fourier transformation, effecting a map between spatial and momentum representations and exchanging the roles of potential and hopping. In the generalized model, the duality maps the real-space equation onto a form whose parameters reflect energy-dependent changes: $$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$8 Self-duality is achieved when $$V_n = \frac{V \cos(2\pi b n + \phi)}{1 - \alpha \cos(2\pi b n + \phi)},$$9, leading to an energy-dependent mobility edge (An et al., 2020, Cui et al., 2021).

In the presence of multiple modulations or complex hopping, generalized duality mappings—sometimes involving auxiliary parameters or basis rotations—yield closed-form mobility edge expressions. For example, in models with power-law onsite potentials ($V$0, $V$1), the mobility edge is given by the set of $V$2 for which

$V$3

with $V$4 (Wang et al., 17 Nov 2025).

3. Extensions: Long-Range Hopping, Topological Phases, Power-Law Potentials

Beyond nearest-neighbor models, the generalized Aubry–André formula applies to Hamiltonians with exponentially decaying, Gaussian, or power-law hopping: $V$5 with the duality condition $V$6 marking the mobility edge (Biddle et al., 2010). For sufficiently fast-decaying hopping, this formula approximates critical boundaries even in the presence of next-nearest or more general hopping terms.

The framework also encompasses topological considerations. In the presence of commensurate and incommensurate hopping modulations (Cestari et al., 2016), the phase diagram features boundaries for both bulk localization ($V$7) and survival of topological edge states ($V$8).

When onsite potentials are raised to powers ($V$9), the model generates mixed regimes with mobility edges for $b$0. The phase boundaries and IPR peaks are explicitly determined by the generalized duality conditions and the universal formula $b$1 (Wang et al., 17 Nov 2025).

4. Spectral Measures and Continuity: Almost Mathieu and Beyond

For the almost Mathieu operator (AMO), the generalized Aubry–André formula extends to describe the Lebesgue measure and its moments for the intersection spectrum,

$b$2

with the measure restricted to the intersection,

$b$3

The $b$4-th moment of the restricted measure is

$b$5

where $b$6 are explicit trigonometric polynomials in $b$7 (Gorodetski et al., 26 Apr 2026). Continuity and analyticity of the spectral measures in the parameters $b$8, except at the duality point $b$9, are rigorously established.

5. Physical Observable Criteria: Localization Indicators

Localization is typically characterized by quantities such as survival probability,

$\phi$0

and the inverse participation ratio (IPR),

$\phi$1

with both sharply distinguishing localized from delocalized eigenstates (Cui et al., 2021). Phase space diagnostics (e.g., Wigner entropy (Lu et al., 10 Apr 2025)) and participation ratios in rotated or dual bases (Marra, 28 Jan 2026) can also signal the precise localization threshold.

A unique result is that mobility edges coincide with energies where localization–delocalization transitions occur in twisted or rotated (canonical) bases, and, at criticality, the Hamiltonian can even map onto equivalent models of massless Dirac fermions in (quasi)periodic curved space, revealing connections to analog gravity (Marra, 28 Jan 2026).

6. Physical Implications and Novel Phenomena

The existence of analytically tractable, energy-dependent mobility edges in generalized Aubry–André models provides a deterministic mechanism for single-particle localization transitions not driven by randomness but by quasiperiodicity and spectral self-duality (An et al., 2020, Cui et al., 2021). Notably, in strongly localized regimes, localization can enhance rather than protect quantum state dissipation, defying common expectations about the robustness of localized states to bath coupling (Cui et al., 2021).

Furthermore, in models with multiple harmonics or nontrivial hopping, localized, extended, and mixed (coexistence) phases can occur, each with distinct topological and dynamical responses. Mixed regimes demonstrate critical scaling, Wigner entropy maxima, and anomalous transport characteristics—e.g., transitions from ballistic to subdiffusive dynamics (Wang et al., 17 Nov 2025, Lu et al., 10 Apr 2025).

7. Summary Table: Key Generalized Aubry–André Mobility-Edge Formulas

Model	Mobility Edge Formula / Transition Condition	Reference
GAA Model	$\phi$2	(An et al., 2020, Cui et al., 2021, Alexanian, 25 Mar 2026)
Exponential hopping	$\phi$3	(Biddle et al., 2010)
Off-diagonal commensurate/incommensurate	$\phi$4 (bulk); $\phi$5 (edge)	(Cestari et al., 2016)
Power-law onsite potential, $\phi$6	$\phi$7	(Wang et al., 17 Nov 2025)
AMO spectrum measure moments	$\phi$8 (see text)	(Gorodetski et al., 26 Apr 2026)

These exact generalized Aubry–André formulas systematically encode the boundaries of localized and extended quantum states for a large variety of deterministic quasiperiodic systems, thereby providing both theoretical and experimental foundations for the analysis and design of synthetic quantum matter with tunable localization landscapes.