Fractionally Quantized Anomalous Hall Effect

• Fractionally Quantized Anomalous Hall (FQAH) effect is a topological state defined by quantized Hall conductance at rational fractions without an external magnetic field.
• Experimental observations in twisted TMD bilayers and graphene/hBN moiré superlattices reveal robust conductance plateaus, minimized resistance, and tunable phase transitions.
• Theoretical frameworks such as composite fermion models, Hartree-Fock, and exact diagonalization illuminate the role of flat bands and electron correlations in stabilizing exotic fractional excitations.

The fractionally quantized anomalous Hall (FQAH) effect is a strongly correlated topological state that manifests as a quantized Hall conductance at rational, non-integer multiples of $e^2/h$ in the complete absence of an external magnetic field. FQAH phases arise in systems with flat, topological Chern bands and strong electron interactions, often stabilized by spontaneous time-reversal symmetry breaking and manifesting robust chiral edge states. The effect shares phenomenological similarities with the fractional quantum Hall effect in Landau levels, but occurs in periodic lattices, stabilized by band topology and many-body electron correlations rather than external quantizing fields.

1. Theoretical Origins and Mechanisms

The FQAH effect fundamentally requires: (i) a nearly flat, isolated band with nonzero Chern number, (ii) strong Coulomb interactions so that kinetic energy is quenched, and (iii) a mechanism for spontaneous time-reversal symmetry breaking.

In early proposals (Zhang et al., 2014), a weak hexagonal periodic potential is imposed on a 2D electron gas at even-denominator Landau filling, such that composite fermions (CFs) experience an effectively staggered (but zero net) magnetic field. The resulting effective Hamiltonian for the CFs is analogous to the Haldane model on a honeycomb lattice, with the phase parameter $\phi$ set by the spatial distribution of the staggered field. The topological CF band structure yields a modified Hall conductivity:

$$\sigma_{xy}^e = -\frac{e^2}{h} \frac{\mathcal{C}}{2p \mathcal{C} + 1}$$

where $\mathcal{C}$ is the CF Chern number and $p$ the number of attached flux pairs. For $\mathcal{C} = -1$, $\sigma_{xy}^e = -e^2/h$ (IQAH) for $p=1$ and $-e^2/3h$ (FQAH) for $p=2$.

In moiré superlattices (e.g. twisted TMD homobilayers (Reddy et al., 2023), rhombohedral graphene (Lu et al., 2023), and others), the interplay of narrow bandwidth, nontrivial Chern number, and enhanced interaction scales enables the emergence of FQAH ground states at commensurate fractional fillings (e.g. $\nu = 2/3$). The quantum geometry of the Bloch bands, Berry curvature uniformity, and Ising ferromagnetism (which suppresses valley/spin fluctuations) are crucial microscopic controls on phase stability (Reddy et al., 2023).

Alternative mechanisms have been proposed, e.g. "strip of stripes" models with helical magnetization and spin-selective inter-stripe tunneling, producing FQAH phases with fractional charge edge excitations in the absence of net magnetization via bosonization and resonance of interaction-induced multi-particle scattering terms (Klinovaja et al., 2014).

2. Experimental Observations: Moiré Materials and Graphene Platforms

Recent advances in sample fabrication and low-temperature transport have enabled the robust observation of FQAH plateaus at zero magnetic field in multiple systems:

• Twisted MoTe$_2$ Bilayers (Park et al., 2023, Cai et al., 2023, Xu et al., 2023):
  • Quantized Hall resistance plateaus observed at $R_{xy} = h/e^2$ for $\nu = -1$ and $R_{xy} = 3h/2e^2, 5h/3e^2$ for $\nu = -2/3, -3/5$, respectively.
  • Plateaus occur with vanishing $R_{xx}$ and are stable across wide filling and field ranges.
  • Magnetic circular dichroism reveals spontaneous ferromagnetism, which stabilizes the broken time-reversal symmetry phase.
  • Optical (Landau fan) and transport signatures follow the Streda formula $\partial n / \partial B = C / \Phi_0$ for both integer and fractional Chern number $C$.
  • Electric displacement field can tune the system between FQAH, trivial Wigner crystal/CDW, and metallic regimes.
• (Multilayer) Graphene/hBN Moiré Superlattices (Lu et al., 2023, Xie et al., 27 May 2024, Lu et al., 19 Aug 2024):
  • Pentalayer and hexalayer graphene/hBN devices (twist angles down to $\sim 0.2^\circ$) display FQAH states at numerous fractions ($\nu=2/3$, $3/5$, $4/7$, etc.), as well as an extended QAH (EQAH) state with $R_{xy}=h/e^2$ across broad filling ranges ($0.5\lesssim \nu \lesssim 1.3$).
  • FQAH plateaus are robust to field ($>99\%$ quantization), with clear $R_{xx}$ minima, and persist up to hundreds of mK.
  • Magnetic hysteresis, sign reversals in $\rho_{xy}$, and field-induced phase transitions (e.g. between composite quasi-electron and quasi-hole FQAH states) reveal intricate interaction-topology interplay.
  • The moiré potential is proven indispensable for engineering the flat Chern bands required for FQAH. However, in the limit of weak or vanishing moiré, interaction-driven topological Wigner crystals can host FQAH phases that are only pinned by residual periodicity (Zhou et al., 2023).

3. Theory, Numerical Methods, and Edge Physics

The theoretical understanding and modeling of FQAH ground states employs several frameworks:

• Parton/Composite Fermion Theories: Physical electrons or holes are decomposed into fractions (partons), each carrying part of the quantum numbers (e.g. charge, Chern number). The combined wavefunctions and associated Chern-Simons effective field theories reproduce the fractional quantization (e.g. $K$-matrix approaches for FQAH-crystal states coexistence with CDW order (Song et al., 2023), parton Green function convolution for the spectral function (Pichler et al., 9 Oct 2024)).
• Hartree-Fock + Exact Diagonalization: In rhombohedral multilayer graphene, a self-consistent HF calculation is used to account for interaction-driven isolation of flat $C=1$ bands; ED is then performed in the projected subspace to confirm FQAH state stability and entanglement spectra (Huang et al., 11 Jul 2024, Huang et al., 9 Aug 2024).
• Thermodynamics and Excitations: FQAH states support low-energy neutral collective modes (magneto-rotons at finite momentum), which set the functional onset temperature ($T^*$) for quantized transport well below the charge gap $T_{cg}$. These neutral modes induce experimental signatures: charge density oscillations near impurities, thermally activated behavior in $R_{xx}$, and entropic edge contributions (Lu et al., 2023).

4. Competing Phases, Phase Transitions, and Tunability

Competition with other ordered phases is a prominent feature:

• Charge Density Wave (CDW) and Wigner Crystal Orders: At higher twist angles or different filling, the FQAH state can give way to a charge-ordered CDW, trivial insulator, or metallic phase depending on the structure of Bloch wavefunctions, band dispersion, and geometric properties of the Chern band (Reddy et al., 2023, Song et al., 2023).
• Extended/Anomalous Hall Crystals (EQAH/AHC): In parts of the phase diagram, extended integer QAH states that spontaneously break translation symmetry dominate at low $T$ and low current. As temperature or bias increases, an entropy-driven transition to the FQAH phase is observed, stabilized by the multiplicity of gapless edge modes and their higher associated entropy (Shavit, 4 Sep 2024, Lu et al., 19 Aug 2024, Patri et al., 21 Aug 2024).
• Experimental Tunability: Both electrical displacement field $D$ and magnetic field (perpendicular and in-plane) can drive quantum phase transitions between IQAH, FQAH, EQAH, metallic, and various CDW/Wigner phases. These transitions are reflected in quantized or sign-reversed $\rho_{xy}$, field-dependent activation energies, and breakdown or recovery of quantized plateaus (Xie et al., 27 May 2024, Xu et al., 2023).

5. Exotic Excitations and Computational Implications

The elementary excitations in FQAH states exhibit fractionalization and anyonic statistics:

• Fractionalized Quasiparticles: Edge and bulk excitations carry fractional charge (e.g. $e/3$, $e/5$), characterized by Abelian or, in certain proposed scenarios, non-Abelian braiding statistics (Klinovaja et al., 2014, Cai et al., 2023, Lu et al., 2023). This is underpinned by bosonization and Chern-Simons field theory in both continuum and lattice implementations.
• Thermodynamic and Edge Entropy Effects: At higher temperature, the rich structure of chiral edge modes in the FQAH phase can stabilize it via entropic reduction of the free energy, with the phase boundary's temperature dependence being mesoscopic and device-size dependent (Shavit, 4 Sep 2024). Clausius–Clapeyron relations quantify the displacement field dependence of these entropy-driven transitions.
• Composite Fermi Liquid: Near half-filling, e.g. $\nu=1/2$, compressible composite Fermi liquid (CFL) states can emerge at zero magnetic field, with Hall resistance linear in $\nu$ and no clear $R_{xx}$ minimum, paralleling conventional half-filled Landau level physics (Park et al., 2023, Lu et al., 2023).

6. Open Directions, Disorder, and Device Engineering

Challenges and future perspectives include:

• Role of Disorder and Contact Resistance: Quantized FQAH plateaus are more sensitive to disorder than IQAH. Large background/contact resistance can obscure true quantization, making improved device engineering crucial for metrological applications (Sarma et al., 10 Jan 2024).
• Material Innovation and Tunability: Beyond TMDs and graphene, the FQAH effect can in principle be realized in other flat-band Chern insulator platforms (e.g. engineered cold atom lattices with singular flat bands (Yang et al., 3 May 2024), or via Coulomb-imprinted moiré superlattices (Zhou et al., 2023)).
• Spectral and Tunneling Probes: The energy- and momentum-resolved single-particle spectral function calculated via parton theory and exact diagonalization provides a direct route to distinguish fractionalized excitations and identify the topological order of FQAH states (Pichler et al., 9 Oct 2024). Experimental access is now possible in devices with spatially resolved spectroscopy.

7. Summary Table: Paradigms and Phenomenology

Platform	Key Mechanism	Observed Plateaus
Twisted TMD bilayers	Flat Chern bands + Ising FM	$\nu = -1$, $-2/3$, $-3/5$
Rhombohedral graphene	Interaction-induced topological WC	$1$, $2/3$, $3/5$, $4/7$
Singular flat band	Bipartite SFB with protected touching	$1/3$, $2/3$
"Strip of stripes"	Helical magnetism + spin-selective hop	$1$, $1/q$

FQAH systems, by realizing charge fractionalization, anyonic excitation, and tunable phase transitions at zero field, establish a paradigm for exploring strongly correlated topological phases and robust platforms for topological quantum computation. Their experimental realization in moiré and crystalline materials, in the absence of external quantizing fields, marks a major advance in controlling correlated flat band physics.