Measuring anyon dispersion with tunneling probes

Abstract: Anyons are usually characterized by their topological data and their fractional quantum numbers under global symmetries. In lattice systems such as fractional Chern insulators (FCI), they are also mobile quasiparticles. Their motion controls the possible ground states of the dilute anyon gas obtained by doping an FCI, including possible superconducting states. We show how tunneling probes can measure this motion. In scanning tunneling spectroscopy, weak disorder produces spatially oscillating quasiparticle-interference patterns whose branches reveal the dispersion of fractionalized constituents. In quantum twisting microscopy, planar momentum-conserving tunneling selects the total momentum of the injected electron, so the continuum thresholds of fractionalized electron spectra encode the dispersion of the constituent anyons. The resulting spectra distinguish compact electron-like excitations, bound anyon molecules, and unbound anyon continuum.

• The paper introduces a dual tunneling probe methodology (STS-QPI and QTM) for directly measuring anyon dispersion in fractional Chern insulators.
• It details analytic derivations that quantify power-law activation exponents, distinguishing compact, molecular, and continuum excitations.
• The study outlines experimental strategies to extract kinetic parameters and binding hierarchies, paving the way for diagnosing anyon-driven superconductivity.

Measuring Anyon Dispersion via Tunneling Probes: An Expert Overview

Motivation and Theoretical Context

The paper "Measuring anyon dispersion with tunneling probes" (2605.29017) addresses the direct measurement of anyon kinetics—dispersion relations and mobility—in fractional Chern insulators (FCIs) and related lattice fractional quantum anomalous Hall (FQAH) systems. Traditionally, experimental characterization of anyons has focused on their topological features—fractional charge and braiding statistics—using techniques such as shot noise, interferometry, and thermodynamic gap measurements. However, the lattice context of FCIs permits anyons to acquire well-defined crystal momentum, rendering their dispersion an essential microscopic property governing the low-energy physics, particularly in doped regimes where dilute anyon gases may manifest superconducting or reentrant anomalous Hall phases.

The authors emphasize that anyon masses and bandwidths define the density of states for the dilute gas, influencing whether the carriers remain isolated, form bound molecular states, or condense. This kinetic data is pivotal for theories of anyon-driven superconductivity and the understanding of doped FCIs' phase behavior.

Tunneling Spectroscopy Frameworks

Scanning Tunneling Spectroscopy (STS): Quasiparticle Interference

The manuscript synthesizes an advanced STS protocol, leveraging Fourier-transform techniques to resolve energy-dependent local density of states (LDOS) modulations induced by weak disorder. In conventional systems, quasiparticle interference (QPI) can infer the scattering vectors and thus the band structure. The authors generalize this to fractionalized electron-addition states in FCIs. Specifically, weak impurities scatter one constituent anyon, yielding branches in the spatial LDOS oscillations whose energy-momentum profiles encode the dispersion of fractionalized constituents.

Critical to the analysis is the decomposition of the electron-addition spectrum into three sectors: compact electron-like excitations ($e$), bound anyon molecules ($ba$), and unbound anyon continuum ($aaa$). The clean activation threshold for tunneling distinguishes these sectors through the emergence of distinct power-law singularities, determined by contact exponents reflecting both phase-space and statistical contributions. For Laughlin-type $\nu=1/3$ states, these exponents are quantitatively provided ($\gamma_{ba}=2/3$, $\gamma_{aaa}=1$), yielding precise predictions for spectral onset behavior.

The QPI branch points are dictated by on-shell scattering of a single constituent, with analytic expressions for the energy-momentum relation ($\omega_s(q)=\Delta_s+q^2/8M_s$ for parabolic dispersion). Singularities in the spectral derivative concentrate near these branches, with exponents ($1/2$ for $ba$, $11/6$ for $ba$0 at $ba$1) clarified in the paper.

Quantum Twisting Microscopy (QTM): Momentum-Resolved Addition Spectroscopy

Quantum Twisting Microscopy (QTM) provides an orthogonal approach, utilizing planar momentum-conserving tunneling from a twisted monolayer graphene probe to the FCI target. The twist angle selects the injected electron's momentum, and the continuum thresholds in the fractionalized spectral function encode the dispersion of constituent anyons.

QTM measures the spectral function directly, at fixed total momentum, allowing the extraction of kinetic parameters free from the impurity-assisted Born kernel convolution intrinsic to STS-QPI. For compact and molecular sectors, QTM thresholds follow $ba$2 and $ba$3 respectively, with three-anyon continuum at $ba$4. The threshold exponents ($ba$5 for $ba$6, $ba$7 for $ba$8) are analytically computed and match the physical expectations for phase-space and statistical contributions in the fixed-momentum regime.

Comparative Analysis and Binding Diagnostics

The dual probe approach—STS-QPI and QTM—allows for cross-validation of extracted constituent masses and binding hierarchies. QPI branches result from equal-energy scattering of one anyon, while QTM edges arise from minimization under fixed total momentum. Concordance between extracted parameters strengthens the identification of spectral sectors and their binding properties.

Importantly, the power-law exponents at clean activation and QPI branches distinguish between compact, bound, and unbound excitations, providing a robust spectroscopic signature of the underlying anyon binding hierarchy. This capability extends to identifying whether the dominant low-energy excitation is a compact electron, molecular anyon, or fractionalized continuum.

Implications for Superconductivity and Phase Transitions

A significant practical implication is the potential diagnosis of anyon-driven superconductivity (ADSC) versus conventional BCS pairing. In ADSC scenarios, there is no inherent particle-hole symmetry around zero bias, unlike weak-coupling BCS. The presence of activation exponents and QPI/QTM dispersion tied to anyon bands—rather than electron-like quasiparticles—serves as a strong diagnostic in the proximate normal state above $ba$9. However, the authors note that particle-hole asymmetry is not exclusive to anyon physics, necessitating a combination of asymmetry and activation exponent analysis for definitive identification.

Limitations and Prospects

The authors are careful to delineate matrix-element constraints, emphasizing that tunneling selection rules and angular momentum channels can suppress or modify peak intensities and apparent exponents. The practical measurement requires conductance derivatives to resolve weak singularities, as raw $aaa$0 may not directly exhibit sharp peaks in certain sectors (e.g., $aaa$1 at $aaa$2). The analytic formulas provided are valid near isolated quadratic minima; non-parabolic band features might alter exponents, and further theoretical work is needed to generalize to arbitrary Bloch band structures.

The methodology establishes a promising framework for future microscopic evaluation of anyon kinetics in moiré FCIs, providing critical input to theories of emergent superconductivity, phase competition, and excitonic binding. Continued experimental development of STS-QPI and QTM platforms is expected to facilitate the systematic exploration of anyon dispersions, binding hierarchies, and their role in correlated lattice systems.

Conclusion

The paper presents a rigorous formalism for measuring anyon dispersion in FCIs using STS-QPI and QTM tunneling probes. Through detailed analytic derivations and sector-resolved spectral analysis, it offers a practical route to extracting constituent anyon kinetic parameters and diagnosing binding hierarchies. These measurements are foundational for advancing understanding of dilute anyon gases, emergent superconductivity, and complex phase behavior in strongly correlated lattice systems. The dual-probe approach provides robust theoretical and spectroscopic tools, with implications extending to the diagnosis of unconventional superconductivity and exploration of emergent fractionalized quasiparticle dynamics.