Atomic Collapse States in Condensed Matter

• Atomic Collapse States (ACSs) are quantum states emerging when potentials become supercritical, causing the breakdown of conventional bound state formation.
• ACS phenomena manifest in graphene and engineered nanostructures where singular potentials produce observable LDOS peaks and discrete energy spectra.
• ACS research bridges quantum mechanics and condensed matter physics, offering insights for electron confinement and innovative quantum device applications.

Atomic Collapse States (ACSs) are quantum states characterized by nonperturbative “collapse” phenomena arising from singular or supercritical potentials in otherwise well-controlled systems. Originally predicted in relativistic quantum mechanics for superheavy nuclei, ACSs have since been realized in various condensed matter platforms—most notably graphene and engineered nanostructures—where the thresholds for collapse become experimentally accessible. The concept has evolved to encompass not only relativistic collapse of Dirac electrons under Coulomb potentials but also collapse phenomena induced by higher-order singular potentials, geometric constraints, strong correlations, and collective molecular effects.

1. Fundamental Physical Principles of Atomic Collapse

ACSs fundamentally emerge when a quantum system’s governing potential becomes supercritical or sufficiently singular to overthrow conventional bound-state formation. In the classic Dirac–Coulomb scenario, an electron subjected to a nucleus with charge $Z > Z_c$ produces bound states whose energies dive into the negative-energy continuum, leading to particle–hole symmetry breaking and vacuum restructuring. In graphene, the effective fine structure constant $\alpha$ is large and the “supercriticality” threshold ($Z_c$) drops to order unity, as the Fermi velocity $v_F \ll c$.

Mathematically, for massless Dirac fermions: $$\left[v_F \sigma \cdot p - \frac{Ze^2}{\kappa r}\right] \Psi(r) = E \Psi(r),$$ with ACS emergence associated with the supercritical regime $\beta = Ze^2 / (\kappa \hbar v_F) > m$ (where $m$ is the total angular momentum quantum number, typically $m = 1/2$). The breakdown of continuous scale invariance and the appearance of towers of resonant states are particularly profound when the system is regularized by a short-distance cutoff, breaking the conformal symmetry (Gorsky et al., 2013).

The phenomenon generalizes: in singular potentials $V(r) \sim -\beta / r^\gamma$ with $\gamma > 1$, collapse can occur for arbitrarily small $\beta$ (Zhuang et al., 4 Sep 2025), and the energy spectra deviate from geometric (exponential) scaling to power-law sequences due to loss of discrete scale invariance.

2. Collapse States in Graphene and Engineered Materials

Graphene and similar Dirac materials are archetypes for ACS physics due to their emergent relativistic band structures. The critical ACS behavior has been observed experimentally in systems with charged impurities (Ca-dimer clusters, adatoms, vacancies), graphene quantum dots, and nanoribbons.

Key features in graphene:

• Critical charge for collapse: $Z_c \sim 1$–$2$—enabling scalable experimental studies (Wang et al., 2015).
• Resonance formation: As impurity charge or external potential is increased past $Z_c$, a bound state “dives” into the continuum, producing observable LDOS (local density of states) peaks at the Dirac point and below.
• Scaling and discrete scale invariance: In unscreened Coulomb potentials, collapse state energies obey exponential (Efimov-type) spacing, $E_n \propto \exp(-n\pi / \mu)$, reflecting cyclic RG flow and analogies with the conformal Calogero model (Gorsky et al., 2013, Shao et al., 2022).
• Molecular collapse: When multiple impurities, either subcritical or supercritical, are spatially proximate, collective quasi-bound states emerge (“molecular collapse”), supporting bonding, anti-bonding, and non-bonding orbitals with tunable spatial and energy characteristics (Pottelberge et al., 2019, Zheng et al., 2022).
• Higher-order potentials: For potentials $V(r) \sim -\beta / r^\gamma$ ($\gamma > 1$), ACSs may form with infinitesimal charge, and exist above as well as below the bulk Dirac point, a feature not present in Coulomb ($\gamma = 1$) collapse (Zhuang et al., 4 Sep 2025).
• Disorder and band gap effects: Lattice defects, puddle disorder, or finite gaps (substrate-induced) alter the critical threshold and the spatial structure of ACSs; disorder can increase $Z_c$ by up to $34\%$ (Polat et al., 2020), while gaps induce atomic-orbital-like splitting, including valley-dependent pseudospin polarization (Wang et al., 25 Jan 2025).

3. Mathematical Formalism and Energy Arrangement

Collapse state physics is governed by differential equations possessing singular solutions. The energy arrangement differs by potential type:

• Coulomb ($\gamma = 1$):

$$|E_n| \approx (\beta/r_0) \exp\left[ -\frac{n\pi}{\sqrt{\beta^2 - m^2}} \right ],$$

yielding geometric series and Efimov-like towers (Gorsky et al., 2013).

• High-order singular ($\gamma > 1$):

$$|E_n/\beta|^{(\gamma-1)/\gamma} \propto n, \quad \text{(power sequence law)}$$

e.g., for $\gamma = 2$, $|E_n| \sim (1/(4\beta)) (\beta/r_0 - n\pi )^2$ (Zhuang et al., 4 Sep 2025).

• Finite gap and valley: Supercritical charge for collapse grows sublinearly with gap $\Delta$:

$$Z_c = 0.5 \times (1 + 29.8\, \Delta[\mathrm{eV}])^{0.38}$$

(Wang et al., 25 Jan 2025).

• Topology and scale invariance: In topological materials (e.g., HfTe$_5$) with massless Dirac fermions, resonance energies obey log-periodic discrete scaling: $\epsilon_{n+1} = \lambda \epsilon_n$ for scale factor $\lambda = \exp(\pi/s_0)$ (Shao et al., 2022).

4. Collapse Phenomena Beyond Graphene

• Dice lattice: ACSs coexist with flat-band bound states, yielding ring-like charge localization spatially distinct from the collapse state centered at the impurity. Flat band energies scale linearly with impurity strength; hybridization leads to anti-crossings between flat-band and collapse states (Wang et al., 2021).
• Rhombohedral graphene: Adatomic collapse arises from virtual bound states formed within a Coulomb gap. The van Hove singularity in ABC multilayer graphene with flat bands leads to highly divergent DOS, facilitating collapse for effective critical atomic numbers as low as $\mathcal{Z}_c \sim 0.96$, thus making experimental realization feasible without large nuclear charge (Silva et al., 2023).
• Geometry-induced collapse: Quantum particles in non-Euclidean (e.g., conic) spaces experience geometric potentials of inverse-square form. For zero angular momentum, these potentials mimic the attraction leading to collapse, producing infinite LDOS oscillations at zero energy even in the absence of any relativistic or external field mechanism (Ye et al., 21 Feb 2024). Experimental nanostructures designed with Riemannian geometry may thus host geometry-induced wavefunction collapse.

5. Quantum Measurement, Control, and Collapse

Atomic collapse phenomena are not restricted to potential scattering. In ultracold gases within optical cavities (Mekhov et al., 2011), the quantum state of atoms evolves under a Hamiltonian with a phase term quadratic in the atom number. Upon measurement (photodetection), the atom number distribution narrows—first as $1/\sqrt{\tau}$, then exponentially with time—resulting in “collapse” to a sharply defined atom number state, a process relevant for quantum metrology and non-demolition measurement protocols.

6. Experimental Signatures and Applications

ACSs manifest as:

• Sharp LDOS/spectroscopy peaks near impurity centers, observed via STM/STS (Wang et al., 2015, Shao et al., 2022).
• Distinct spatial charge distributions in nanoribbons, quantum dots, and molecular collapse states detected as bonding/anti-bonding/non-bonding orbital features.
• Gate-tunable, magnetic-field-modified, or geometry-induced splitting and relocalization of collapse resonances (Moldovan et al., 2017, Wang et al., 2021, Zheng et al., 2021).
• Enhanced energy sensitivity enabling quantum device applications; ACSs may serve as “artificial atoms” for electron confinement, resonant tunneling, or valleytronic encoding.
• In strongly correlated or perturbed condensed matter (e.g., acoustic shock waves), anomalous electron wells with MeV-scale depth and nanometric width may be accessible, potentially enabling collapsed matter with dramatically enhanced density (Ivlev, 2016).

7. Outlook and Future Directions

The expanded landscape of ACSs—from single-particle Dirac physics to collective, molecular, geometric, and correlated environments—opens new paradigms for testing high-energy and quantum field theory concepts in condensed matter. Contemporary directions include:

• Systematic exploration of collapse regimes in high-order singular potentials, flat-band systems, and topological quantum materials.
• Engineering of controlled collapse phenomena with tunable geometry, impurity, or band gap parameters.
• Application to ultra-dense matter phases, quantum confinement, novel quantum device architectures, and low-dimensional valleytronic and topological systems.
• Extension to quantum measurement-induced control of collapse and squeezing in ultracold matter and quantum optical platforms.

Atomic Collapse States thus serve as a bridge between fundamental quantum mechanics, emergent condensed matter physics, and applied nanoscience, continually motivating new experimental and theoretical development.