Extended Diagonal AAH Model

• Extended Diagonal AAH Model is a quasiperiodic lattice model with both on-site and hopping modulations, enabling studies of localization, mobility edges, and topological edge states.
• The analytic localization criterion and multifractal diagnostics clearly delineate phase transitions between delocalized and insulating regimes with quantized topological signatures.
• Experimental realizations in ultracold atoms, photonic arrays, and superconducting circuits provide versatile platforms for observing quantum control, topological pumping, and many-body localization.

The extended diagonal Aubry-André-Harper (AAH) model generalizes the seminal AAH (or Harper) model by incorporating both incommensurate on-site (diagonal) and hopping (off-diagonal) quasiperiodic modulations, often with independently tunable phases. This extension enables exploration of complex localization phenomena, richer topological structures, mobility edges, and multifaceted experimental realizations. The model's physical realizations span ultracold atoms in optical lattices, photonics, and superconducting circuits, with applications in topological pumping, many-body localization, and quantum control.

1. Model Definition and Hamiltonian Forms

The archetypal Hamiltonian for the extended diagonal AAH model in a 1D lattice with open or periodic boundaries is

$$H = \sum_{i=1}^{N} V_i\,c_i^\dagger c_i + \sum_{i=1}^{N-1} (J_i\,c_{i+1}^\dagger c_i + \text{h.c.}) \tag{1}$$

with quasiperiodic potentials: $$V_i = V\cos(2\pi\beta i + \phi), \quad J_i = J + \lambda\cos(2\pi\beta i + \phi) \tag{2}$$ where $c_i^\dagger$ is the fermionic creation operator at site $i$, $V$ and $\lambda$ are the amplitudes of on-site and hopping modulations, $J$ is the uniform hopping, $\beta$ is an irrational incommensurability parameter (commonly the inverse golden ratio), and $\phi$ is a tunable global phase (Zhao et al., 2017, 2206.13107, Liu et al., 2014).

Further generalizations include second-neighbor hopping terms, on-site pairing (notably $p$-wave), and sublattice structures, enabling the study of new phases such as mobility edges and Majorana modes (Yahyavi et al., 2019, Hetényi et al., 2024).

2. Localization Transition and Phase Diagrams

The primary diagnostic for localization in the extended AAH model is the inverse participation ratio (IPR): $$\mathrm{IPR}[\psi] = \sum_{i=1}^{N} |\psi_i|^4 \tag{3}$$ where $\mathrm{IPR}\sim 1/N$ for extended states (delocalized), and is finite for localized states.

A key result is an analytic criterion for the localization transition: $$\left(\frac{V}{2}\right)^2 + \lambda^2 = J^2 \tag{4}$$ This delineates the boundary between the "nonlocal" (extended) and localized phases. The resulting phase diagrams demonstrate critical lines that separate metallic (conductor-like) behavior from insulating phases, with the transition further corroborated by sharp changes in band fractality and occupation imbalance (Zhao et al., 2017, 2206.13107).

In extensions with higher-order hopping or $p$-wave pairing, the phase diagram acquires spikes—special fillings at which the localization transition shifts to lower potential strength—and mobility edges that separate localized and extended states even within the same spectrum (Hetényi et al., 2024, Yahyavi et al., 2019).

3. Topological Properties and Edge States

The extended diagonal model engenders band structures whose topology can be characterized by Chern numbers and winding numbers. For rational flux $\beta = p/q$, the 1D system maps onto a synthetic 2D Brillouin zone, and the Chern number for band $n$ is

$$C_n = \frac{1}{2\pi} \int_{k_x} \int_{\varphi} F_n(k_x, \varphi) \, dk_x \, d\varphi$$

where $F_n$ is the Berry curvature. These invariants underlie quantized charge or atom pumping and predict the existence and number of robust edge states as synthetic parameters are cycled (Liu et al., 2014, Li et al., 2024, Zhao et al., 2017).

The model supports topologically nontrivial bandstructures with multiple edge state pairs, transitions between topologically trivial and nontrivial phases, and adiabatic pumping phenomena governed by the phase shift between diagonal and off-diagonal potentials. These edge states remain robust even in the presence of quasiperiodic disorder and loss, provided chiral symmetry is maintained (Zhao et al., 2017, Yahyavi et al., 2019).

4. Dynamical and Many-Body Signatures

The extended AAH model reveals rich dynamical behavior:

• Adiabatic edge-state pumping: Slow ramping of phase parameters can transfer a boundary state from one edge to the other with high fidelity in the extended phase; localization blocks this transfer in the insulating regime (Zhao et al., 2017, Liu et al., 2014).
• Quantum Lyapunov control: Active quantum control protocols can prepare edge-localized states efficiently in the extended regime, but are suppressed beyond the localization threshold (Zhao et al., 2017).
• Transport and participation entropy: In experiments, single- or multi-excitation states show ballistic transport in the extended phase, pinning in the localized phase, and intermediate fractal dynamics at criticality. The $q$-th order participation entropy $S^{\mathrm{PE}}_q$ and fractal dimension $D_q$ quantify the many-body phase structure (2206.13107).
• Spectral characteristics: The spectrum exhibits self-similar band splitting (Hofstadter butterfly), fractality, and, with $p$-wave pairing, demonstrates Chern-number switching and multi-gap edge states (Yahyavi et al., 2019, Li et al., 2024).

5. Experimental Realizations

Realizations span multiple platforms:

• Ultracold atoms in optical lattices: Bichromatic or superlattice potentials realize incommensurate diagonal and off-diagonal modulations through the superposition of different optical wavelengths. The relevant tight-binding parameters can be engineered by projecting onto Wannier states and tuning lattice depths ($V_1 \gg V_2$) (Li et al., 2024, Zhao et al., 2017).
• Photonic waveguide arrays: Spatially modulated waveguide spacings and widths control on-site energies and couplings; losses and disorder can be introduced controllably; synthetic gauge fields and chiral pumping can be observed via fluorescence microscopy (Zhao et al., 2017, Zhao et al., 2017).
• Superconducting qubit chains: Arrays of fixed-frequency transmons coupled via tunable couplers provide access to full parameter control, permitting direct quantum simulation of extended AAH Hamiltonians and measurement of transport and participation entropy in many-body sectors (2206.13107).

Experimental work has confirmed predicted phase boundaries, topological pumping, and the nature of localized, extended, and critical phases across various platforms.

6. Extensions and Novel Phenomena

Further generalizations include the addition of $p$-wave pairing, resulting in richer phase diagrams with coexisting Majorana zero modes, critical extended phases with gradual IPR transitions, and Hofstadter butterfly evolution through topological transitions (Yahyavi et al., 2019). Second-neighbor hopping alters the localization transition, introducing filling-dependent "spikes" where localization occurs at anomalously low potential strength, related to special irrational fillings associated with the Fibonacci sequence (Hetényi et al., 2024).

Non-Hermitian extensions incorporating loss (e.g., on specific sublattices) expand the region supporting real-energy band closings and enable quantized displacement witnesses of topology under open system dynamics (Zhao et al., 2017).

7. Methods and Numerical Diagnostics

Localization, topology, and criticality are characterized analytically and numerically using:

• IPR and mean IPR for single and many-body states,
• Participation entropy $S^{\mathrm{PE}}_q$ and fractal dimension $D_q$,
• Geometric Binder cumulant from polarization theory,
• Multifractal analysis (scaling index $\gamma_{\min}$),
• Real-space renormalization and spectral analysis,
• Chern number computation via Berry curvature integration,
• Exact diagonalization and time-dependent Schrödinger evolution,
• Fidelity and transport observables (edge pumping, quantum control).

The analytical self-duality, phase boundary equations in terms of modulation amplitude and phase, and phase-resolved spectroscopic signatures enable a full theoretical and experimental mapping of the model's physics (Zhao et al., 2017, Liu et al., 2014, 2206.13107, Li et al., 2024).

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For comprehensive mathematical formulations, phase diagrams, and further applications, see (Zhao et al., 2017, 2206.13107, Liu et al., 2014, Hetényi et al., 2024, Zhao et al., 2017, Yahyavi et al., 2019), and (Li et al., 2024).