Zero-D Topological States

• Zero-dimensional topological states are discrete, energetically pinned modes that arise from topological invariants and symmetry principles.
• They manifest in systems like quantum dots, molecular rings, and synthetic lattices, characterized using Berry phases and generalized bulk–boundary correspondences.
• Their robust protection against perturbations paves the way for applications in quantum computing, photonics, and nanoscale device engineering.

Zero-dimensional topological states are discrete, energetically pinned modes whose existence and protection arise from topological invariants, symmetry principles, or synthetic constructions—often independent of the spatial dimensionality of the host system. Originally discussed in the context of boundary or defect modes in higher-dimensional topological phases, the concept now encompasses robust eigenstates manifesting as flat bands, defect-bound states, edge/corner modes in higher-order topological insulators, topological modes in zero-dimensional quantum systems (such as molecules and quantum dots), and even synthetic or phase-space platforms, provided appropriate symmetry or bulk–boundary (or generalized) correspondences are met. The following sections review foundational results, classifications, characterization methods, symmetry mechanisms, implementations, and their implications.

1. Topological Invariants and Classification in Zero-Dimensional Systems

Zero-dimensional systems, despite lacking continuous spatial periodicity, can host nontrivial topology governed by their underlying symmetry groups and discrete structure. In molecular rings, clusters, or quantum dots with cyclic (rotational) symmetry, eigenstates form “discrete bands” parametrized by momentum-like quantum numbers $k = 2\pi l/m$, with $l=0,\ldots,m-1$ (Yin et al., 1 May 2025). Such systems admit well-defined Berry (Zak) phases along “Wilson polygons”—ordered sequences of overlap matrices over the discrete symmetry group’s elements: $$\gamma_n = -\mathrm{Im}\,\ln\!\Biggl(\prod_{l=1}^{m} \langle k_l,n \mid k_{l+1},n \rangle\Biggr).$$ Summing these phased overlaps over occupied “bands” and normalizing modulo $2\pi$, a $\mathbb{Z}_2$ topological index characterizes the phase: $$\mathcal{Z}_{2,N} = \left(\sum_{n=1}^{N} \frac{\gamma_n}{\pi}\right)\!\!\pmod{2}.$$ Transitions in $\mathcal{Z}_2$ are triggered by energy-level crossings (gap closings) as system parameters (e.g., interunit hopping amplitudes in nanohoops) are tuned.

Beyond finite real-space systems, analogous invariants appear for synthetic Fock-state lattices of single-atom quantum systems (Lee et al., 30 Jun 2025). There, topological properties are encoded in a phase-space winding number $W$ associated with parameter-dependent coherent state trajectories: $$W = \frac{1}{2\pi i} \int_{0}^{2\pi} d\phi \, \frac{d}{d\phi} \ln \alpha(\phi).$$ The corresponding geometric phase difference (“Zak phase”) between spin sublattices is quantized, reflecting the system’s topological character.

2. Symmetry Mechanisms and Protection

Zero-dimensional topological states rely on symmetries (chiral, inversion, particle–hole, subsystem exchange, or generalized equivalents) that quantize topological invariants and impede hybridization with trivial states.

• Generalized chiral symmetry governs robust corner states in higher-order topological insulators (HOTIs). For instance, in the dimerized Kagome lattice, the generalized chiral operator $$\Gamma_3=\operatorname{Diag}(1, e^{2\pi i/3}, e^{-2\pi i/3})$$ ensures that, under suitable parameter configurations ($t_a < t_b$), zero-energy modes localize on corners and remain immune to symmetry-preserving perturbations (Kempkes et al., 2019).
• Inversion symmetry undergirds pinned edge or interface states in dimerized models such as the Hofstadter-Aubry-André-Harper system (Lau et al., 2015). Here, the integer-valued invariant

$\mathcal{N} = |n_1 - n_2|$

(with $n_{1,2}$ counting negative parity eigenvalues at inversion-symmetric points) predicts the emergence or disappearance of degenerate localized modes upon band inversion.

• Particle–hole and time-reversal symmetry (class D) enables $\mathbb{Z}_2$ protected Majorana-like zero modes in superconducting quantum dots under Zeeman splitting (Marra et al., 2018). The corresponding invariant is the fermion parity $P_\varphi = \operatorname{sgn}(E_\varphi^2 - B^2)$, which changes at topological phase transitions and is observable via discontinuities in the current-phase relation.
• Subsystem exchange symmetry can partition larger quasi-one-dimensional lattices into independent blocks, some of which fall into the BDI (chiral symmetry) or AI (absence of chiral symmetry) classes. The protection of zero-energy modes depends sensitively on the (blockwise) preservation of inversion and, where applicable, chiral symmetry (Jangjan et al., 2022).

3. Bulk–Boundary Correspondence and Generalized Extensions

The conventional bulk–boundary correspondence, which connects topological invariants of the bulk to (d–n)-dimensional boundary states in a d-dimensional system, is extended in multiple ways:

• Gapless topological phases (such as Fermi surface or nodal-line semimetals and superconductors) are described as momentum-space defects in $d_\text{BZ}$ dimensions (Matsuura et al., 2012). A hypersphere of codimension $d_k = d_\text{BZ}-q-1$ encloses the Fermi surface of dimension $q$, and integer winding ($\nu_1$) or $\mathbb{Z}_2$ invariants (Pfaffian-based) determined there guarantee the existence of zero-dimensional boundary modes—either as non-dispersing (flat) surface bands or localized point modes.
• Higher-order topology results in zero-dimensional states localized at corners or defect terminations (HEND states) in 2D and 3D systems, including insulators with crystalline defects or $\pi$-flux insertion (Schindler et al., 2022). Explicit conditions such as $B\cdot M_\nu^F \bmod 2\pi = \pi$ (with $B$ the Burgers vector and $M_\nu^F$ a weak topological invariant) predict robust defect-bound zero modes, supported by analytical and numerical evidence in PbTe and SnTe lattices as well as Kramers pairs tied to flux tubes in HOTIs.
• Generalized correspondence at domain walls: Zero-energy states can also arise at domain walls between trivial bulk domains, even in classes forbidden by standard tenfold tables. The key theoretical underpinning is a quantized Berry phase difference between domains—typically $\Delta\gamma = \pi$—verified by tight-binding, low-energy, and topological field theory (Han et al., 2023).
• Symbolic and zero-dimensional extensions: In topological dynamics, any topological flow admits a principal zero-dimensional extension—that is, a symbolic model (often a subshift) faithfully representing entropy and orbit structure (Burguet et al., 2021).

4. Physical Manifestations and Experimental Realizations

Zero-dimensional topological states manifest in diverse settings, spanning condensed matter, photonics, acoustics, and synthetic or quantum simulators.

Table: Selected Manifestations of Zero-Dimensional Topological States

System Type	Example Phenomenon	Reference
2D Kagome HOTI, artificial lattices	Robust 0D corner states via STM manipulation	(Kempkes et al., 2019)
Dimerized Hofstadter/Aubry-André-Harper	Edge-state doublet at inversion-symmetric $k$	(Lau et al., 2015)
Superconducting quantum dot	Zeeman-driven 0D $\mathbb{Z}_2$ phase, CPR jumps	(Marra et al., 2018)
Square lattice+staggered flux (SMF)	ZCNTI with quantized polarization; edge modes	(Chen et al., 6 Mar 2025)
Composite molecular nanohoops, [m]ITN/CPP	Topological boundary states via Wilson polygons	(Yin et al., 1 May 2025)
Germanene nanoribbons	Electric field-controlled topological end modes	(Eek et al., 19 Jun 2025)
Photonic cavity with synthetic dimensions	Quadrupole 0D corner states; entanglement protection	(Zhang et al., 2019)
Acoustic Kekulé lattice	Majorana-like bound state, Thouless pumping	(Gao et al., 2020)
Dipole sphere (astrophysical context)	Zero-flux, flat-band, chiral 0D modes	(Murugan, 6 Jul 2025)

Experimental detection techniques include STM spectroscopy (for spatially resolved corner and end modes), conductance and current-phase relation measurements in superconducting circuits, photonic waveguide arrays mapping wavefunction localization, and in-phase field extension in photonic SSH lattices via double-zero-index media (Dong et al., 5 Aug 2025).

5. Synthetic and Phase-Space Extensions

Zero-dimensional topological states are not confined to real-space platforms; synthetic dimensions extend these concepts:

• Synthetic dimensions in photonics: Quadrupole topological states are implemented in nominally 0D cavities by mapping internal degrees of freedom (frequency, orbital angular momentum, polarization) onto axes of an effective higher-dimensional lattice (Zhang et al., 2019). Topological corner modes in such synthetic lattices protect multi-photon entanglement.
• Phase-space topology in quantum optics: In a single-atom quantum Rabi system, the synthetic Fock-state lattice enables phase-space winding numbers and corresponds to quantized Zak phase differences, leading to robust zero-energy spin-polarized defect modes (Lee et al., 30 Jun 2025).

6. Robustness and Limitations

Symmetry or topology-protected zero-dimensional states are generically robust to disorder, perturbations, or finite-size effects provided the protecting invariants and symmetries are preserved. However, care must be taken to distinguish genuine topological zero modes from fine-tuned, trivial proximity-induced states in hybrid or open systems (Califrer et al., 2022). Diagnostics based solely on energy or spatial localization may be insufficient—detailed symmetry, spectral pairing, and parameter dependence analyses are required for unambiguous identification.

Moreover, the geometric interpretation of topological states can be fundamentally altered by engineered surroundings. Double-zero-index media allow the spatial extension of corner or interface-localized modes without loss of their topological protection, effectively breaking conventional bulk–edge correspondence and enabling scalable, less geometrically constrained topological devices (Dong et al., 5 Aug 2025).

7. Implications and Future Prospects

Zero-dimensional topological states are central to fields ranging from robust information and energy transfer, quantum computation (protected qubits via fractionalized end/corner charges or Majorana modes), to the simulation of unexpected quantum phases in engineered platforms. Their realization in systems with synthetic dimensions, photonics, acoustics, and artificial molecules opens new routes for device integration, ultra-compact memory, and neuromorphic architectures. Control schemes—such as electric fields modulating symmetry-breaking potentials in nanoribbons—demonstrate practical, reversible switching of topological character near ambient experimental conditions (Eek et al., 19 Jun 2025).

In astrophysical contexts, the realization of flat-band zero-dimensional modes in magnetic dipole spheres suggests the existence of robust topological quantum matter in extreme environments, with implications for both laboratory and cosmic settings (Murugan, 6 Jul 2025).

The theoretical underpinning continues to deepen with advances in K-theory classifications, symmetry-based indicators, Wilson loop and polygon prescriptions, and generalized bulk–boundary or domain wall correspondences, ensuring the continued relevance and expansion of the concept of zero-dimensional topological states across physics and engineering.