Disorder-Induced Topological Phases

• Disorder-induced topological phases are quantum states where randomness in system parameters creates nontrivial topology by renormalizing gap parameters and inverting band structure.
• They exhibit robust boundary, corner, and hinge states protected by chiral, time-reversal, or average crystalline symmetries, with quantized real-space invariants.
• Experimental and numerical studies using methods like SCBA and real-space diagnostics confirm these phases across engineered lattices, photonic systems, and solid-state materials.

Disorder-induced topological phases are quantum phases in which spatial disorder—randomness in parameters such as on-site potentials, hopping amplitudes, or lattice positions—does not merely degrade pre-existing band topology but actually creates new topological phases absent in the clean (disorder-free) limit. These phases, typically realized in systems with preserved protecting symmetries (e.g., chiral, time-reversal, or average crystalline symmetries), are characterized by quantized real-space invariants, robust boundary or corner states, and critical transport signatures. Of particular importance is the phenomenon whereby disorder renormalizes gap parameters or critical masses, enabling inversion of band topology or the emergence of higher-order and interaction-enabled phases beyond the clean symmetry-protected landscape.

1. Fundamental Mechanisms and Symmetry Protection

The archetypal disorder-induced topological phase is the Topological Anderson Insulator (TAI), first analyzed in 2D Dirac/Chern systems. The essential mechanism involves symmetry-preserving disorder generating a self-energy correction such that a trivial band mass $m$ is renormalized ($m \rightarrow m_\text{eff} = m + \Sigma$) to surpass a topological threshold. Analytically, this is captured by the self-consistent Born approximation (SCBA), which incorporates the disorder-averaged Green's function and yields phase boundaries by setting the renormalized mass to its critical value (e.g., $m_\text{eff}=0$ or equivalent criterion). This framework generalizes to systems with higher-order topology (e.g., quadrupole and octupole insulators), crystalline symmetries, Floquet drives, and interaction-enabled phases, provided that the relevant symmetry (chiral, time-reversal, inversion, average crystalline) remains unbroken or is only broken locally but restored in an averaged sense (Zhang et al., 20 Jan 2026, Lóio et al., 2023, Li et al., 2020, Chaou et al., 2024).

Higher-order topological Anderson insulators (HOTAI, TOTAI) emerge when disorder not only inverts bulk topological indices, but also triggers quantized higher multipole moments or crystalline-symmetry-derived invariants (quadrupole, octupole, etc.), resulting in protected boundary states of codimension $d>1$ (e.g., corner or hinge states). The critical insight is that chiral or crystalline symmetries—sometimes only statistically preserved—can enforce quantization of invariants and protection of boundary modes even under strong stochastic perturbations (Lóio et al., 2023, Yang et al., 2020, Peng et al., 2024, Chaou et al., 2024).

2. Classification, Topological Invariants, and Diagnostic Tools

Disorder-induced phases are classified according to the real-space analogs of clean invariants, as translational symmetry is generically lost under disorder. The most widely used diagnostics are:

• Winding numbers in real space (1D chiral classes): Constructed from projectors onto positive/negative energy states, commutators with the position operator, and traced per unit volume. Local markers and bulk-averaged winding numbers serve as the quantized invariant (Shi et al., 2021, Sircar, 2024).
• Noncommutative Chern or spin-Chern numbers (2D/3D): Evaluated from projectors and commutators with position, or via numerical Bott indices, converging to the Chern number in the thermodynamic limit for finite samples (Zhang et al., 20 Jan 2026, Chen et al., 2017).
• Multipole moments and boundary polarization: The quadrupole moment is computed as a many-body expectation value over projections of occupied states, with real-space formulas ensuring quantization in the presence of disorder if chiral symmetry is unbroken (Li et al., 2020, Yang et al., 2020, Peng et al., 2024). The octupole moment generalizes this to third-order TIs (Lóio et al., 2023).
• Symmetry-protected $\mathbb{Z}_2$ and other crystalline indicators (e.g., inversion, mirror, rotation): In systems with average symmetry, statistical (rather than strict) protection of topological indices and boundary signatures emerges, with group structures collapsing (e.g., $\mathbb{Z} \rightarrow \mathbb{Z}_2$ or $\mathbb{Z}_4$) under disorder (Chaou et al., 2024).
• Numerical signatures: Exact diagonalization, kernel polynomial method (KPM) for density of states (DOS), transfer-matrix evaluation for localization lengths, inverse participation ratios (IPR), and level-spacing statistics (GOE/Poisson) (Lóio et al., 2023, Yang et al., 2020).

3. Disorder-induced Phase Diagrams and Physical Regimes

The addition of disorder to a trivial insulator or semimetal may induce a rich sequence of topological and non-topological phases, as quantified by real-space invariants and transport observables. Canonical sequences observed in various models include:

• NI $\rightarrow$ TAI (or HOTAI/TOTAI) $\rightarrow$ diffusive metal $\rightarrow$ Anderson insulator: Increasing disorder strength first drives a trivial system into a topological phase (either first-order or higher-order depending on the model and symmetry), then to a metallic regime with extended states, and finally into a fully localized Anderson insulator (Lóio et al., 2023, Okugawa et al., 2020, Ding et al., 2024).
• Reentrant and multi-lobe topological regions: Particularly in systems with quasi-periodic modulations (e.g., mechanical quasicrystals, Aubry-André models), phase diagrams display multiple topological regions interrupted by trivial phases as disorder or modulation is varied (Sircar, 2024).
• Critical metallic and Griffiths regimes: In HOTAI systems, gapless topological regimes with localized bulk states but critical (multifractal) corner or edge states can emerge, as typified by fractal scaling of IPRs and spectral statistics (Yang et al., 2020, Silva et al., 2024).

Characteristic critical disorder strengths and corresponding phase boundaries are accurately predicted by SCBA for all non-chiral-symmetry-breaking disorder types (Lóio et al., 2023, Li et al., 2020, Okugawa et al., 2020, Chen et al., 2017, Chaou et al., 2024).

4. Higher-order, Statistical, and Amorphous Topology under Disorder

Disorder extends the notion of higher-order topology far beyond periodically ordered systems:

• Higher-order topological Anderson insulators: In 2D quadrupole insulators and 3D octupole models, disorder inverts band topology and yields gapped phases with quantized quadrupole (or octupole) moments and robust corner (3D: 8 corners) or hinge (2D: 4 corners) modes (Lóio et al., 2023, Li et al., 2020, Yang et al., 2020, Peng et al., 2024).
• Amorphous and quasicrystalline TAI/HOTAI/TOTAI: Structural (geometric) disorder in the lattice, such as random site displacements in quasicrystals, induces transitions from trivial amorphous insulators to quantum spin Hall (QSH) and higher-order topological phases, with quantized spin Bott indices, corner mode multiplicities uniquely determined by the system's global symmetry (e.g., 8 corner modes in Ammann-Beenker tilings) (Peng et al., 2024).
• Average (statistical) symmetry-protected topological phases: In systems where mirror, inversion, or rotation symmetry is preserved only on average, disorder collapses clean classification groups (e.g., $\mathbb{Z}$ winding to $\mathbb{Z}_2$ parity), yielding new statistical topological phases with boundary signatures (e.g., hinge or corner modes remain robust with critical, non-localized wavefunction statistics) (Chaou et al., 2024).

5. Interacting and Dynamical (Floquet) Regimes

Disorder-induced topological phases are not restricted to noninteracting systems:

• Interacting topological Anderson phases: Structural disorder in arrays of interacting spins or bosons (e.g., Rydberg-atom chains) can produce many-body symmetry-protected topological (SPT) phases, observable via ground-state degeneracy, edge-mode spectroscopy, and slow edge-magnetization decay in quench experiments. The quantized invariant (e.g., many-body polarization) remains pinned to $\{0, 0.5\}$ due to average inversion symmetry (Yue et al., 7 May 2025).
• Floquet topological Anderson (dynamical) phases: Periodically driven systems subject to disorder exhibit disorder-induced transitions between trivial and topological Floquet phases. This is analytically captured by closure of the disorder-averaged Dyson equation, with critical exponents for gap vanishing obtained from free-probability theory. Such transitions manifest in both 1D and 2D (e.g., driven Kitaev or Bernevig-Hughes-Zhang models) as robust edge modes or gap closings at distinct disorder thresholds (Shtanko et al., 2018, Ling et al., 2023).

6. Experimental Realizations and Material Relevance

Multiple experimental platforms have realized or are poised to realize disorder-induced topological phases:

• Artificial and engineered lattices: Superconducting qubit chains (measuring disorder-averaged chiral displacement and pumped charge), photonic lattices (quantized Bott index and transmission), acoustic metamaterials, and topolectrical circuits (impedance resonance at corners) demonstrate topological Anderson regimes and higher-order analogs (Zhang et al., 20 Jan 2026, Yang et al., 2020, Shi et al., 2021, Peng et al., 2024).
• Solid-state quantum materials: Disorder-induced transitions have been observed in 3D TIs such as Bi$_2$Se$_3$, MnBi$_4$Te$_7$, and narrow-gap semiconductors (HgCdTe, InAsSb), where random alloying, substitutional impurities, or nanoscopic grain size variation control effective disorder strength (Brahlek et al., 2016, Krishtopenko et al., 2022).
• Atomic and mechanical systems: Rydberg-atom chains under structural disorder realize interacting many-body SPT phases (Yue et al., 7 May 2025), while mechanical metamaterials (e.g., spring–mass chains, quasicrystals) host TAIs and re-entrant HOTAI transitions upon varying the disorder or quasi-periodic modulation (Shi et al., 2021, Sircar, 2024).

7. Generalizations, Universality, and Outstanding Problems

The universality of the disorder-induced topological Anderson mechanism is established across symmetry classes, lattice geometry, and disorder type:

• Universality of critical disorder and SCBA: Phase boundaries in disorder-induced topological transitions are quantitatively matched by SCBA, with universal scaling exponents at gap-closing transitions (e.g., $\nu z = 3/2$ in Floquet-TAIs, $\nu \sim 1.4$ in percolative Euler semimetals) (Shtanko et al., 2018, Jankowski et al., 2023).
• Stability against disorder type and correlations: Provided the disorder preserves (on average) the protecting symmetry, TAIs, HOTAIs, and TOTAI remain robust under correlated, structural, or quasiperiodic randomness (Chaou et al., 2024, Peng et al., 2024).
• Multifractality and statistical criticality: At disorder-driven phase boundaries and within Griffiths regimes, boundary or corner states may exhibit multifractality or critical wavefunction statistics fundamentally distinct from Anderson localization (Silva et al., 2024, Chaou et al., 2024, Yang et al., 2020).
• Open challenges: Interplay of disorder and strong correlations, precise classification of amorphous and statistical crystalline topologies, experimental stabilization of higher-order TAIs in real materials, dynamical and non-Hermitian extensions, and the impact of many-body localization on interacting TAIs remain at the forefront of ongoing research (Zhang et al., 20 Jan 2026).

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References

• "Recent progress on disorder-induced topological phases" (Zhang et al., 20 Jan 2026)
• "Third-order topological insulator induced by disorder" (Lóio et al., 2023)
• "Topological Phase Transitions in Disordered Electric Quadrupole Insulators" (Li et al., 2020)
• "Higher-order Topological Anderson Insulators" (Yang et al., 2020)
• "Disordered topological crystalline phases" (Chaou et al., 2024)
• "Structural disorder-induced topological phase transitions in quasicrystals" (Peng et al., 2024)
• "Disorder-Induced Topological Transitions in a Multilayer Topological Insulator" (Alisultanov et al., 2024)
• "Topological Phase Transitions Induced by Disorder in Magnetically Doped (Bi, Sb)$_2$Te$_3$ Thin Films" (Okugawa et al., 2020)
• "Observing structural disorder induced interacting topological phase in an atom array" (Yue et al., 7 May 2025)
• "Disorder driven topological phase transitions in 1D mechanical quasicrystals" (Sircar, 2024)
• "Disorder-induced phase transitions in higher-order nodal line semimetals" (Ding et al., 2024)
• "Disorder-induced instability of a Weyl nodal loop semimetal towards a diffusive topological metal with protected multifractal surface states" (Silva et al., 2024)
• "Disorder-induced topological quantum phase transitions in multi-gap Euler semimetals" (Jankowski et al., 2023)
• "Disorder induced topological phase transition in a driven Majorana chain" (Ling et al., 2023)
• "Stability of Periodically Driven Topological Phases against Disorder" (Shtanko et al., 2018)
• "Disorder-induced topological phase transition in HgCdTe crystals" (Krishtopenko et al., 2022)
• "Disorder-driven topological phase transition in Bi2Se3 films" (Brahlek et al., 2016)