Thickness-Driven Topological Phase Transition

• Thickness-driven topological phase transition is an abrupt change in a system's topological order induced by variations in geometric or effective thickness.
• It is characterized by the emergence, splitting, or annihilation of protected edge states along with gap closings and changes in invariants like Chern numbers or Z₂ indices.
• Its tunability through geometric design and external fields paves the way for advanced device applications in low-dissipation electronics and spintronics.

A thickness-driven topological phase transition is an abrupt, often quantized, change in the topological properties of a physical system induced by variation of its geometric thickness or an effective “thickness” parameter. This concept is relevant for condensed matter systems, magnetic films, nanolayers, thin films, and percolated lattices, where quantum confinement, interfacial hybridization, or topological connectivity can be directly tuned by film thickness or structural geometry. Such transitions manifest as changes in topological invariants (such as Chern numbers or $\mathbb{Z}_2$ indices), the emergence or disappearance of protected edge/surface states, or the splitting/merging of topological clusters, and are often accompanied by spectral gap closings or changes in boundary-localized observables. The mechanism, physical signatures, and mathematical description vary by context, yet all thickness-driven topological transitions are unified by their reliance on geometry as a nontrivial control parameter for topological order.

1. Physical Mechanisms of Thickness-Driven Topological Transitions

Thickness-driven topological phase transitions can arise from various physical sources:

• Quantum confinement and phase inversion: In ultrathin films (e.g., $\alpha$-Sn on CdTe(110)), quantum confinement modifies the energy band structure, leading to a crossover from a 2D topological insulator (TI) phase (thin limit) to a 3D TI phase (thicker limit) as film thickness increases. The critical thickness marks where coupling between quantum-well subbands (or hybridization between opposing surfaces) changes the topological invariant, generating or annihilating robust Dirac surface or edge states (Liu et al., 4 Sep 2025).
• Surface-state hybridization: For 3D TIs such as BiSbTeSe$_2$, when the film thickness approaches the surface-state penetration depth, hybridization of the top and bottom Dirac surface states opens a hybridization gap. The gap magnitude follows an exponential decay with thickness, typically $\Delta \sim \Delta_0 \exp(-d/\xi)$. At the critical thickness, this gap closes, and the system can oscillate between topologically trivial and nontrivial phases (Chong et al., 2020).
• Finite-size quantum effects: In Cr-doped Bi$_2$Se$_3$, topological quantum phase transitions are observed as oscillatory gap closings and reopenings with increasing thickness, a direct signature of quantum finite size effects in Dirac semimetals. This quantum confinement modulates the effective band inversion required for nontrivial topology (1711.01797).
• Magnetic coupling and hybridization: In magnetic topological insulators, the balance between quantum anomalous Hall (QAH) effect and insulating behavior is regulated by a competition of induced exchange fields and hybridization gaps, both of which are thickness-dependent. A transition from a chiral edge-conducting QAH phase to a trivial insulator is realized as thickness decreases and hybridization dominates, requiring external magnetic fields to recover quantization in ultra-thin films (Ji et al., 2021).
• Topological textures in magnetic films: In ferromagnetic films, the allowed 3D topological spin textures (hourglass vs. dome states) and their transitions are direct functions of the film’s thickness. Transitions occur via the formation or annihilation of point singularities (such as Bloch points), controlled by film geometry and micromagnetic energies (Mankenberg et al., 6 Oct 2024).
• Connectivity in percolated lattices: In generalized models, effective “thickness”—such as the connectivity of bonds in 1D or higher-dimensional topological models—regulates whether global topological order (as measured by invariants like polarization) is supported. Near the percolation threshold, a “fractured topological region” emerges, where macroscopic disconnected topological clusters exist but no global order is realized (Mondal et al., 2023).

2. Experimental Signatures and Observable Phenomena

Manifestations of thickness-driven topological transitions span spectroscopic, transport, and structural probes:

• Helicity-Dependent Photocurrent (HDPC): In $\alpha$-Sn/CdTe(110) thin films, the HDPC changes symmetry as thickness is increased across the 2D–3D TI transition. In 5 nm films, HDPC is odd in incident angle (indicating dominant 2D quantum well states); in 10/30 nm films, it becomes even and associated with robust topological surface states on both sides of the film. These symmetry changes are directly tied to a topological phase boundary (Liu et al., 4 Sep 2025).
• ARPES and Band Gaps: In Cr-doped Bi$_2$Se$_3$, angle-resolved photoemission spectroscopy directly reveals oscillatory closing and reopening of the Dirac point gap with changing thickness. Critical thicknesses correspond to gapless points indicative of Dirac semimetal phases (1711.01797).
• Transport Plateaus and Quantum Spin Hall Effect: In ultrathin TIs (e.g., BSTS), longitudinal conductance plateaus at $G \sim 2e^2/h$ when the Fermi level resides in the hybridization gap, diagnosing a quantum spin Hall state. The conductance vanishes in the trivial insulating regime (subcritical thickness) (Chong et al., 2020).
• Phase Diagrams and Symmetry Analyses: HR-TEM and symmetry analysis in $\alpha$-Sn/CdTe samples reveals the substrate-induced strain and reduced crystal symmetry driving the changes in topological response as a function of thickness (Liu et al., 4 Sep 2025).
• Magnetic Spin Textures: In magnetic films, real-space imaging and theoretical phase diagrams reveal regions of metastability and transitions between hourglass and dome topological states as film thickness and boundary skyrmion radius are tuned (Mankenberg et al., 6 Oct 2024).

3. Mathematical Framework and Topological Invariants

The mathematical and topological characterization of these transitions involves:

• Band topology and invariants: The calculation of $\mathbb{Z}_2$ invariants (using Wannier charge center evolution) identifies topological phase boundaries in nanolayers subject to electric fields, with phase diagrams in thickness–field strength space demarcating trivial, topological, and metallic phases (Liu et al., 2023).
• Effective Model Hamiltonians: For thin films and 1D models, the mean-field, tight-binding, or $k\cdot p$ Hamiltonians (e.g.,

$$H_k = [v + wp \cos(k)] \sigma_x - (wp \sin(k)) \sigma_y$$

) and their extensions (e.g., involving polylogarithms for long-range hopping) serve to predict when the system crosses a topological phase boundary as the effective “thickness” parameter changes (Mondal et al., 2023).

• Percolation thresholds and geometric criteria: In percolated SU–Schrieffer–Heeger (SSH) models, the emergence of a “fractured topological region” is identified by comparing percolation (connectivity) thresholds to topological transition points predicted by mean-field energetic criteria, clarifying when local zero modes abound without global order (Mondal et al., 2023).
• Hydrodynamic and micromagnetic theories: In evaporating films, phase transitions between regions of different thickness are governed by Darcy’s law,

$\mathbf{v} = -\frac{h^2}{3\nu} \nabla P$

where $h$ is the film thickness, and the evolution of the interface is described by an infinite set of exponentially-decaying harmonic moments. For magnetic films, the continuum micromagnetic Hamiltonian

$$\mathcal{H} = \int dx\,dy\,dz\, \left(-\frac{J}{2}[(\partial_x\mathbf{m})^2 + (\partial_y\mathbf{m})^2] - \frac{J_z}{2}(\partial_z\mathbf{m})^2 - \lambda m_z^2\right)$$

captures the energetic competition underlying hourglass–dome transitions (Mankenberg et al., 6 Oct 2024, Klein et al., 2012).

• Asymptotic topology and Betti numbers: Transitions in configuration/energy space manifolds in models such as the 2D $\phi^4$ are characterized by topological “neck” formation, causing the zeroth Betti number $b_0$ to jump asymptotically from 1 to 2 in the thermodynamic limit, with the critical “thickness” of the neck vanishing at transition (Gori et al., 2017).

4. Classes of Thickness-Driven Transitions: Comparative Table

System Type	Physical Effect	Observed Transition
TI Thin Films (α-Sn, BiSbTeSe₂, Bi₂Se₃)	Surface hybridization, quantum confinement	2D–3D TI, QSH state emergence, Dirac gap oscillation
Magnetic Topological Insulators (Cr-doped)	Exchange field vs. hybridization gap	QAH to trivial insulator, Chern # change
Magnetic Films	Hourglass-dome topological textures	State switching, Bloch point transitions
Percolated SSH Chain	Connectivity (bond percolation)	Topological–trivial, “fractured region”
Nanolayers + Electric Field	Band inversion and gap reopening	Trivial–topological (Z₂)–metallic
Evaporating Thin Films	Viscous scaling with thickness, harmonic decay	First-order topological–geometric transition

Each system realizes the thickness-driven transition via different competing energy scales or structural parameters—hybridization gap, exchange field, quantum confinement, geometric connectivity, or boundary-induced instabilities.

5. Theoretical and Practical Implications

The tunability of topological order using film thickness or geometrical control has wide practical and conceptual implications:

• Device functionality: The ability to induce and control QAH, QSH, and other topological states via thickness or field modulation opens routes to dissipationless edge conduction in logic, memory, and spintronic devices (Ji et al., 2021, Mankenberg et al., 6 Oct 2024, Chong et al., 2020).
• Sensitivity and diagnostics: Techniques such as helicity-dependent photocurrent (HDPC) detect subtle symmetry and topological changes at interfaces, allowing noninvasive characterization of topological phase boundaries and the emergence of TSS or edge states (Liu et al., 4 Sep 2025).
• Structural engineering: Modulating surface coupling, strain, and magnetic doping through growth techniques or applied fields provides direct access to phase diagrams with variable functional phases, including metastable coexistence regions and externally switchable states (Liu et al., 2023, Liu et al., 4 Sep 2025, Mankenberg et al., 6 Oct 2024).
• Understanding disorder and robustness: Studies of percolation-induced transitions clarify the fundamental limitations of global bulk–boundary correspondence in geometrically disordered systems, showing that local signatures of topological order (e.g., zero modes) can persist without macroscopic polarization or quantized response (Mondal et al., 2023).

6. Future Directions and Open Problems

Outstanding problems and future research avenues include:

• Extending phase diagrams: Further mapping of thickness- and field-tunable phase spaces in both conventional and magnetic TIs is needed, with precision determinations of topological invariants across critical points (1711.01797, Ji et al., 2021).
• Dynamical control and ultrafast switching: Exploring the dynamics of transitions between topological magnetic textures (e.g., hourglass↔dome), including current- and field-induced manipulation and the potential for information storage at high speed and low energy (Mankenberg et al., 6 Oct 2024).
• Topology under structural/functional disorder: Investigations of topological responses under controlled disorder, amorphous architectures, and fractal systems to understand the robustness and collapse of bulk–boundary correspondence (Mondal et al., 2023).
• Integration with quantum technologies: Application of electric-field-driven topological transitions in nanolayers as platforms for Majorana states and topological qubits, leveraging seamless integration with semiconductor and superconducting devices (Liu et al., 2023).
• Visualization and imaging: Experimental confirmation and real-space imaging of 3D topological solitons, edge states, and domain boundaries using next-generation characterization techniques (Mankenberg et al., 6 Oct 2024).

In summary, thickness-driven topological phase transitions unify diverse physical systems where geometric parameters modulate energetics, symmetry, and connectivity to effect quantized changes in topological order. The phenomenon provides both a fundamental testing ground for the interplay between topology, geometry, and disorder, and a flexible platform for future electronic, spintronic, and quantum applications.