Hidden Localization Transition

• Hidden localization transition is a phenomenon where quantum states abruptly change their confinement due to intrinsic mechanisms such as landscape bifurcation and symmetry breaking.
• It is characterized by emergent mechanisms including hidden percolation, non-Hermitian modulations, and analog gravity effects, unifying diverse quantum and many-body systems.
• Recent studies leverage inhomogeneous landscape theory, resource-theoretic diagnostics, and advanced experimental protocols to identify and characterize these elusive transitions.

Hidden localization transition refers to abrupt or emergent changes in the localization properties of quantum states or quasiparticles, mediated by subtle mechanisms or parameters that are not directly linked to conventional disorder-driven Anderson transitions. Such transitions manifest as critical changes in transport, spectral, or structural observables—often associated with hidden percolation, symmetry breaking, landscape bifurcations, topological invariants, or emergent geometry. They unify phenomena ranging from electronic and bosonic systems with correlated or self-generated disorder, quasiperiodic and non-Hermitian models, to analogs of gravitation and quantum phase transitions. Recent advances leverage landscape theory, generalized Hamiltonian constructions, exceptional points, resource-theoretic diagnostics, and experimental protocols to reveal transitions that were previously obscured in conventional analyses.

1. Landscape Approach: Unified Description of Hidden Localization Mechanisms

The landscape theory (Filoche et al., 2011, Lyra et al., 2014) introduces an inhomogeneous Dirichlet problem for elliptic operators, yielding a “landscape function” $u(x)$ or $u_i$ (in discrete lattices) defined via

$(L u)(x) = 1, \quad u|_{\partial\Omega}=0.$

The valleys of $u(x)$ partition the domain into subregions $\Omega_\alpha$ separated by high effective barriers $W(x) = 1/u(x)$. Any normalized eigenfunction at energy $\lambda$ is exponentially suppressed in regions where $u(x) < 1/\lambda$, confining localization to pockets defined by the landscape.

In discrete models, duality emerges: low-energy states are confined by $u_i$ while high-energy states localize in regions determined by a dual landscape $v_i$ constructed via Hamiltonian symmetries (e.g., particle-hole exchange at $k = \pi/a$) (Lyra et al., 2014). The transition between these regimes is not apparent from disorder strength alone but is encoded in hidden bifurcations of the effective confining landscape, yielding pseudo-mobility edges even in one-dimensional noninteracting systems.

2. Hidden Localization in Correlated and Self-Generated Disorder

In certain models, localization transitions are not governed by external random potentials but by emergent or conserved quantities. The disorder-free Ising–Kondo lattice (Yang et al., 2019) is an archetype: local conservation of Ising spins creates a static, binary potential that, at high temperature, mimics quenched disorder for conduction electrons. The phase diagram unveiled by density of states (DOS), inverse participation ratio (IPR), and resistivity reveals regions of Fermi liquid (FL), Anderson localized (AL), and Mott insulator (MI) phases, with AL intervening between FL and MI. This hidden AL regime is diagnosed by saturation of IPR and scaling of entanglement entropy, demonstrating localization transitions in fully translation-invariant systems.

3. Phase-Induced and Non-Hermitian Hidden Transitions

Subtle control parameters such as global hopping phase or non-Hermitian modulations can drive unexpected localization transitions. In long-range quasiperiodic models (Liu et al., 2024), the global phase $\phi$ in hopping terms, usually considered superficial, cannot be gauged away due to broken translation symmetry by the potential. Tuning $\phi$ from 0 to $\pi$ transitions the system from a critical spectrum to clean mobility edge physics, analytically characterized by a Lyapunov exponent and supported numerically by IPR and multifractal analysis.

Non-Hermitian Aubry–André–Harper models (Tang et al., 2021) exhibit localization boundaries parameterized by nonreciprocal hopping $g$ and complex potential phase $\phi$, with transitions accompanied by topological winding numbers of complex eigenenergies. A hidden real–complex spectral transition, invisible in standard transport diagnostics but sharp in eigenvalue analysis, can precede or coincide with many-body localization in interacting cases, connecting spectral and localization properties in non-Hermitian matrices.

4. Hidden Localization in Generalized AA Models and Analog Gravity

Generalized Aubry–André models (Marra, 28 Jan 2026) constructed by replacing position and momentum operators with arbitrary canonically conjugate pairs $(\hat O, \hat K)$ display transitions where wavefunctions localize with respect to one operator but remain extended in the conjugate basis. The self-dual point $|J/K| = 1$ or, equivalently, vanishing normalized participation ratio (NPR), marks the hidden transition. Unexpectedly, at this point, the lattice model coincides with the Hamiltonian of a massless Dirac fermion in a quasiperiodically curved spacetime—indicating a structural bridge between many-body localization and analog gravity. Critical spectra display Cantor-set fractality, and transport exhibits anomalous diffusion corresponding to the underlying metric structure.

5. Percolative and Two-Fluid Hidden Transitions in Correlated Oxides

In nickelates such as oxygen-deficient NdNiO$_3$ films (Guo et al., 2020), the hidden localization transition emerges as metallic regions (“puddles”) percolate through an otherwise localized matrix, revealed only when hysteresis appears near the percolation threshold $p_c$. Conductivity in the coexistence regime requires a two-fluid model: localized (activated) and metallic (power-law) channels summed in parallel. Full oxygenation yields linear-in-$T$ resistivity and a scattering rate saturating to the Planckian bound $1/\tau = \alpha k_BT/\hbar$. Theoretical interpretation in terms of random interacting Hubbard models supports the universality of this mechanism.

6. Quantum Coherence and Hidden Transitions: Resource-Theoretic Diagnostics

Recent advances recast localization transitions in terms of resource-theoretic quantum coherence (Styliaris et al., 2019). The escape probability, closely related to the time-averaged return probability, is naturally a coherence monotone measuring the ergodicity of Hamiltonian eigenstates in the configuration basis. Coherence-generating power and differential coherence (encoded in a Riemannian metric on parameter space) track the uniformity of the basis transformation and, at infinite temperature, reproduce the dynamical conductivity—a quantifier of ergodic versus many-body localized regimes. These diagnostics unify spectral, entropic, and transport order parameters, proving sensitive to hidden localization transitions in both single-particle and interacting systems.

7. Experimental Signatures and Extensions

Hidden localization transitions can be probed via advanced experimental protocols:

• Thermal hysteresis in oxide films reveals percolative onset of conduction (Guo et al., 2020).
• Time-of-flight and in-situ imaging enable detection of extended versus localized wavepackets in cold atom setups with cavity backaction (Rojan et al., 2016).
• Photon-number growth and quench dynamics in driven Dicke models expose exceptional-point–driven transitions even in globally Hermitian Hamiltonians (He et al., 2024).
• Measurements of entanglement entropy and level statistics in interacting and non-Hermitian models distinguish many-body localized phases from ergodic regimes (Tang et al., 2021, Lisiecki et al., 2024).
• Resource-based coherence metrics coalesce localization diagnostics with quantum information-theoretic principles (Styliaris et al., 2019).

8. Context, Controversies, and Outlook

The discovery of hidden localization transitions extends classic Anderson theory into domains governed by emergent disorder, nontrivial topological invariants, phase-induced symmetry breaking, and analog gravitational metrics. These transitions reshape the phase diagrams of correlated, quasiperiodic, and disordered-free many-body systems, challenging conventional views on the role of randomness, the universality of the mobility edge, and the nature of the metal–insulator transition. The interplay between local conservation, landscape bifurcation, topological jumps, real–complex spectral breaks, and analog spacetime transformation offers novel perspectives for both fundamental theory and quantum simulation.

Continued research explores the generality of these transitions, their scaling laws, and their manifestation in higher-dimensional, strongly correlated, and open quantum systems, with resource-theoretic and landscape-based frameworks providing powerful tools for rigorous diagnosis and characterization.