Anyon-Trion: Exciton-Quasihole Composite

• Anyon-trion is a composite bound state where an interlayer exciton binds to a fractionally charged quasihole in a quantum Hall liquid, demonstrating fractionalized charge behavior.
• The binding energy, which scales linearly with the fractional charge, is measured through optical blueshift/redshift in TMD–graphene heterostructures.
• Device architectures using static IX emitters in high magnetic fields enable precise optical detection and control of anyon-trions for studying anyonic many-body correlations.

Searching arXiv for the specified papers to ground the article in current literature. An anyon-trion is a composite bound state in which a long-lived interlayer exciton binds to a fractionally charged quasihole in a fractional quantum Hall liquid. In the formulation introduced for atomically thin semiconductor heterostructures, the relevant platform is a transition-metal dichalcogenide heterobilayer hosting interlayer excitons placed near graphene hosting a Laughlin state; the bound state arises because the interlayer exciton interacts repulsively with the graphene electrons, so a local depletion of electron density at a quasihole reduces the excitonic blueshift and stabilizes a bound configuration (Mostaan et al., 11 Jul 2025). The concept extends the familiar notion of a trion from an exciton bound to a charge carrier of magnitude $|e|$ to an exciton bound to a quasihole of fractional charge $e^{\ast}=e/m$ at filling $\nu=1/m$, thereby importing fractionalized charge and anyonic many-body correlations into an excitonic bound state.

1. Definition and physical picture

The anyon-trion proposed in MoSe$_2$/WSe$_2$–graphene heterostructures is formed when an interlayer exciton (IX) in the transition-metal dichalcogenide bilayer binds to a positively charged quasihole in a nearby fractional quantum Hall liquid. The IX interacts repulsively with electrons in graphene, and its optical resonance is blueshifted by the local electron density. At the position of a quasihole, where the electron density is depleted, the IX experiences reduced repulsion and hence a reduced blueshift, which is a net redshift relative to the uniform background. When the IX overlaps spatially with the quasihole, the composite bound state is the anyon-trion (Mostaan et al., 11 Jul 2025).

This construction is explicitly related to, but distinct from, two more familiar objects. First, conventional trions in 2D semiconductors consist of an exciton bound to an additional charge carrier of magnitude $|e|$. Second, the same optical mechanism yields a “magnetic trion” in an integer quantum Hall state, where the IX binds to a Landau-quantized hole of full charge $e$. In the fractional case the corresponding bound object is instead tied to a quasihole with $q_h=+e/m$, and the fractional charge modifies the binding strength. The paper identifies this modification through the relation

$$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$

so that the binding-energy ratio itself becomes a probe of fractional charge.

A common terminological ambiguity concerns whether any composite involving anyons and three constituents should be called an anyon-trion. In the heterostructure setting of the optical proposal, the term denotes an exciton bound to a single quasihole rather than a three-anyon composite. That distinction becomes important in multilayer topological systems, where “trion” can instead mean a three-body object built from anyons distributed across layers.

2. Device architecture and microscopic Hamiltonians

The proposed device consists of a MoSe$_2$/WSe$e^{\ast}=e/m$0 heterobilayer placed at distance $e^{\ast}=e/m$1 above a monolayer graphene sheet, all embedded in hBN with dielectric constant $e^{\ast}=e/m$2, in a uniform perpendicular magnetic field $e^{\ast}=e/m$3. The interlayer spacing of the TMD bilayer is $e^{\ast}=e/m$4, setting the IX dipole moment $e^{\ast}=e/m$5. The magnetic length and Coulomb scale are

$e^{\ast}=e/m$6

For the representative field $e^{\ast}=e/m$7, the magnetic length is $e^{\ast}=e/m$8, the graphene Landau-level spacing is $e^{\ast}=e/m$9, and the IX binding energy is quoted as $\nu=1/m$0, so Landau-level and IX Rydberg admixtures are suppressed by $\nu=1/m$1 (Mostaan et al., 11 Jul 2025).

In the integer-quantum-Hall few-body problem, the magnetic trion Hamiltonian includes the IX electron, IX hole, and the graphene hole,

$\nu=1/m$2

with $\nu=1/m$3 for the IX constituents and $\nu=1/m$4 for the graphene hole in the symmetric gauge $\nu=1/m$5. After a Gor’kov–Dzyaloshinskii-type gauge transformation, the authors obtain a tractable form that separates center-of-mass and internal coordinates. The magnetic trion binding energy is defined as

$\nu=1/m$6

where $\nu=1/m$7 is the interacting ground-state energy and $\nu=1/m$8 is the ground-state energy with $\nu=1/m$9.

In the fractional case, the IX and $_2$0 electrons are placed on a sphere of radius $_2$1 threaded by a monopole of strength $_2$2. The full Hamiltonian is

$_2$3

with

$_2$4

In spherical geometry this is written in terms of Haldane pseudopotentials $_2$5 and LLL-projected density operators $_2$6. For a Laughlin $_2$7 state, one takes $_2$8, and a single quasihole is introduced by increasing the flux by $_2$9, $_2$0. In this subspace the quasihole configurations form an $_2$1 multiplet with $_2$2 degenerate ground states when the IX is absent.

3. Rotational-frame reduction and exact diagonalization

The mobile-IX problem greatly enlarges the Hilbert space. To control that growth, the analysis introduces a spherical Lee–Low–Pines (LLP) transformation,

$_2$3

which exploits rotational symmetry and effectively localizes the IX at the north pole in a co-moving frame while leaving the quantum Hall degrees of freedom to rotate (Mostaan et al., 11 Jul 2025). The transformed kinetic term obeys

$_2$4

so the IX motion couples directly to the total angular momentum of the quantum Hall fluid. This recasts the problem into an impurity-in-a-quantum-Hall-fluid framework amenable to exact diagonalization.

Exact diagonalization is performed on the sphere with $_2$5–$_2$6 electrons at $_2$7 using $_2$8. Energies are reported in units of $_2$9. In the mobile case, the transformed Hamiltonian is diagonalized in sectors of total angular momentum $|e|$0. In the static case, relevant to a localized IX at the tip of a quantum twisting microscope (QTM), the limit $|e|$1 is taken and the problem is diagonalized in sectors of $|e|$2.

The extraction of binding in the fractional case is based on the IX blueshift induced by the repulsive IX–electron interaction. Defining

$|e|$3

the robust binding energy used to combine integer and fractional Hall data is

$|e|$4

This definition compensates for the reduced electron density at $|e|$5 and minimizes contamination by excitations. Applying

$|e|$6

to the tabulated values yields $|e|$7, within $|e|$8 of $|e|$9.

4. Spectra, correlations, and quantitative binding

The low-energy spectrum of the mobile anyon-trion at $e$0 shows a ground state at $e$1, where $e$2. This state corresponds to an IX with zero center-of-mass angular momentum bound to the quasihole. Around $e$3, the center-of-mass dispersion is quadratic, indicating a finite anyon-trion mass. Two gaps, $e$4 and $e$5, separate the ground state, the first excited state, and the onset of the continuum, and both remain finite as $e$6 (Mostaan et al., 11 Jul 2025).

The internal structure of the bound state is characterized through the IX–electron two-particle correlation function

$e$7

On the sphere, $e$8 in the ground state at $e$9 and $q_h=+e/m$0 closely matches that of a free quasihole localized at $q_h=+e/m$1. This is presented as evidence that the mobile IX binds the quasihole and inherits its depleted electron environment. The first excited state shows slightly enhanced electron density near the IX, consistent with an internal edge-like excitation, whereas at the continuum onset the electron density near the IX is strongly enhanced.

The numerical energy scales are meV-scale and differ sharply between mobile and pinned excitons. The abstract reports a mobile anyon-trion binding energy of approximately $q_h=+e/m$2 and a static anyon-trion binding energy of about $q_h=+e/m$3. In the exact-diagonalization analysis, the kinetic-energy offset is approximately $q_h=+e/m$4 at both $q_h=+e/m$5 and $q_h=+e/m$6. The linear charge dependence is central: because the IX binds via its dipolar potential to a positively charged hole or quasihole, the binding energy scales linearly with the charge of the bound object, $q_h=+e/m$7.

Quantity	Mobile IX	Static IX
Binding energy	$q_h=+e/m$8	$q_h=+e/m$9
$$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$0 single-quasihole blueshift $$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$1	$$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$2–$$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$3	$$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$4–$$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$5
Kinetic-energy offset	$$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$6	eliminated

The same data show that a naive use of the mobile-IX value for charge extraction produces a $$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$7 underestimation, giving $$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$8–$$e^{\ast}/e = E_{B,\mathrm{AT}}(\nu)/E_{B,\mathrm{AT}}(1),$$9. This is the basis for the claim that static IX spectroscopy is preferable for quantitative metrology of fractional charge.

5. Static anyon-trions and optical detection in QTM geometry

For a localized IX, realized as a quantum emitter in a QTM tip, the static limit $_2$0 changes both the symmetry classification and the phenomenology. Exact diagonalization at $_2$1 and $_2$2 shows that introducing the static IX lifts the quasihole ground-state degeneracy: the configuration with the quasihole at the north pole, $_2$3, becomes a unique non-degenerate ground state, while other quasihole configurations form a gapped band separated from the continuum (Mostaan et al., 11 Jul 2025). In this geometry, the bound state is therefore not only spectroscopically distinct but also spatially anchored to the emitter location.

The experimental proposal is a quantum optical version of the quantum twisting microscope, with a localized IX emitter scanned over graphene in high magnetic field. In uniform regions of the incompressible quantum Hall liquid, the IX exhibits a homogeneous blueshift $_2$4 due to repulsive IX–electron interactions. When the tip overlaps with a localized quasihole, the local electron density is reduced, the blueshift is reduced, and the photoluminescence line undergoes a redshift relative to the background. In the tabulated values, the relevant scales are:

• $_2$5–$_2$6,
• $_2$7–$_2$8,
• $_2$9–$e^{\ast}=e/m$00,
• $e^{\ast}=e/m$01–$e^{\ast}=e/m$02 for the QTM emitter.

Scanning the tip and recording photoluminescence spectra therefore images quasiholes as localized spots of reduced blueshift. The same measurement protocol yields the anyon-trion binding energy through differences between Laughlin and quasihole regions, and then the fractional charge through the ratio $e^{\ast}=e/m$03. The proposal further states that linewidths in state-of-the-art TMD quantum emitters are narrow enough that meV-scale shifts should be resolvable, and that the long-lived exciton permits scanning and spectroscopy without perturbing the incompressible quantum Hall state.

A broader implication is that the static anyon-trion localizes the quasihole at the IX position. This suggests a route to controlled manipulation, and potentially braiding, by transporting the bound state along prescribed tip trajectories, provided the incompressibility gap is maintained and disorder is not so strong as to prevent tip-induced repositioning.

6. Relation to other bound states, limitations, and the trilayer “trion” distinction

The anyon-trion of the TMD–graphene proposal belongs to a wider family of excitonic and topological bound-state ideas, but it is not interchangeable with them. The paper explicitly distinguishes it from conventional trions, from magnetoexcitons, and from exciton-polarons. Conventional trions are exciton-plus-charge-$e^{\ast}=e/m$04 complexes in zero or weak magnetic field. Magnetic trions in the graphene problem arise purely due to Landau quantization, whereas in graphene at $e^{\ast}=e/m$05 the linear dispersion precludes binding. Exciton-polarons involve an exciton dressing by many-body excitations of a Fermi sea; by contrast, the present object dresses with correlated quasihole excitations of a quantum Hall liquid and is described as lying within a broader quantum Hall polaron perspective. The work also contrasts this optical bulk probe with earlier anyon detection strategies such as edge interferometry and shot noise, emphasizing complementary access to individual bulk anyons (Mostaan et al., 11 Jul 2025).

The stated limitations are standard but consequential: lowest-Landau-level projection, neglect of Landau-level mixing and finite layer-thickness corrections, a simplified IX–electron potential, neglect of the graphene valley degree of freedom, spherical-geometry finite-size effects with $e^{\ast}=e/m$06, and the assumption of clean low-temperature conditions. Disorder is described as double-edged: it can localize quasiholes and thereby assist detection, but it may also broaden spectral lines. The distinction between mobile and static IXs is also presented as a limitation of metrology, since mobile-IX binding remains robust but is quantitatively shifted by kinetic-energy correlations.

A separate but relevant use of the word “trion” appears in the trilayer quantum Hall study of an anyon–exciton condensate at $e^{\ast}=e/m$07 (Wang et al., 31 Jul 2025). There, the condensed object is not a trion but the neutral bi-exciton

$e^{\ast}=e/m$08

whose total charge vanishes and whose self-statistics is $e^{\ast}=e/m$09, so it is bosonic. The paper states that a genuine three-body bound state built from one fundamental anyon from each layer is generically a fermion for $e^{\ast}=e/m$10, with self-statistics $e^{\ast}=e/m$11, and is confined in the anyon–exciton condensate because its mutual statistics with $e^{\ast}=e/m$12 is generically non-integer. Accordingly, that work rules out trion condensation in the intermediate phase and identifies the low-energy state as an anyon–exciton condensate with a $e^{\ast}=e/m$13 Laughlin topological sector plus a neutral Goldstone mode. This terminological contrast clarifies that the optically active anyon-trion in atomically thin heterostructures is a distinct concept: an exciton–quasihole bound state used for detection and control of individual anyons, rather than a three-anyon condensate.