Fractional quantum Hall states by Feynman's diagrammatic expansion

The fractional quantum Hall (FQH) effect arises from strong electron correlations in a quantising magnetic field, and features exotic emergent phenomena such as electron fractionalisation. Using the diagrammatic Monte Carlo approach with the combinatorial summation (CoS) algorithm, we obtain results with controlled accuracy for the microscopic model of interacting electrons in the lowest Landau level (LLL) in the thermodynamic limit. Starting from the macroscopically degenerate LLL at finite temperature, including interactions order by order, and applying a controlled resummation to the resulting series, we observe the emergence of the incompressible 1/3-filled state as the temperature is lowered. By analysing the long-time decay of the Green’s function, we find spectral properties consistent with an energy gap at 1/3-filling, whereas at 1/2-filling our results are consistent with the pseudogapped behaviour previously observed experimentally and suggested theoretically. Our work provides the first demonstration that fractionalised phases of matter can be reliably described with Feynman’s diagrammatic technique in terms of the fundamental electronic degrees of freedom, while also showing applicability of expansions in the bare Coulomb potential for precision calculations.

💡 Research Summary

In this paper the authors demonstrate that the fractional quantum Hall (FQH) effect can be captured directly from a diagrammatic expansion in the bare Coulomb interaction, without resorting to composite‑fermion constructions or effective field theories. They work with the microscopic Hamiltonian of spin‑polarised electrons confined to the lowest Landau level (LLL) in two dimensions, subjected to a perpendicular magnetic field B. The non‑interacting limit yields a macroscopically degenerate Landau level; the only energy scale is the interaction strength V. To obtain a controlled perturbative regime they introduce a finite temperature T, which provides an additional scale so that V/T can be made small. Starting from V/T≪1 they systematically add interaction corrections order by order, and then lower the temperature to explore the strong‑coupling regime where FQH states are expected.

The technical core consists of two innovations. First, the long‑range Coulomb potential is regularised by a Yukawa screening V(r)=e^{‑r/λ}/r with two representative screening lengths λ=ℓ_B/2 (short‑range) and λ=2ℓ_B (long‑range). By varying λ they can interpolate between a short‑range interaction that is numerically benign and the pure Coulomb limit λ→∞, where individual diagrams diverge. Second, they partially dress the propagator with a “bold Hartree‑Fock” self‑energy Σ_{bold HF}=V_{HF}ν, where ν is the exact filling fraction. The dressed propagator ˜G_0(iω)=(iω+μ−Σ_{bold HF})^{-1} is used as the building block of the diagrammatic series. Crucially, any diagram that contains an insertion already accounted for in Σ_{bold HF} is excluded, which eliminates double counting and dramatically enlarges the convergence radius.

All connected diagrams up to order n=8 are summed exactly using the deterministic combinatorial‑summation (CoS) algorithm, which enumerates O(n³3ⁿ) topologies for spinless fermions and exploits full vertex permutation symmetry to reduce Monte‑Carlo variance. The coefficients a_n of the Taylor series P_N(ξ)=∑_{n=0}^N a_n ξⁿ (with ξ the auxiliary coupling that is set to 1 at the end) are obtained with high statistical precision. To reconstruct physical observables at ξ=1 the authors employ Padé and Dlog‑Padé analytic continuation, generating a family of approximants. The spread among these approximants, together with the statistical errors of the coefficients, provides an estimate of the systematic uncertainty. As temperature is lowered the series becomes increasingly divergent, limiting the lowest accessible temperature to the point where the error remains below a few percent.

From the Green’s function G(τ) they extract the filling fraction ν(μ)=−G(β⁻) and the single‑particle spectral function ρ(ω). The equation of state ν(μ) shows a pronounced plateau at ν=1/3 when the temperature is sufficiently low, signalling an incompressible state with an energy gap. The gap size inferred from the width of the plateau is Δ≈0.01 e²/ℓ_B for λ=ℓ_B/2 and Δ≈0.07 e²/ℓ_B for λ=2ℓ_B, in good agreement with composite‑fermion theory estimates. At ν=1/2 no plateau appears; instead the compressibility remains finite and the spectral function exhibits a pseudogap— a suppression of low‑energy weight without a true gap—consistent with experimental tunnelling measurements and with the Halperin‑Lee‑Read composite‑fermion liquid picture.

A detailed analysis of the complex‑ξ plane reveals that the emergence of the ν=1/3 plateau is associated with a pair of complex‑conjugate singularities moving inside the unit circle |ξ|=1 as μ approaches the plateau region. The authors conjecture that higher‑order fractions (e.g., ν=2/5) would correspond to additional singularity pairs approaching the circle, but capturing them would require diagram orders beyond n=8 with comparable precision.

Overall, the work establishes that a bare‑Coulomb diagrammatic expansion, when combined with appropriate screening, partial dressing, and high‑order deterministic summation, can faithfully reproduce the hallmark features of fractional quantum Hall physics in the thermodynamic limit. This provides the first controlled, first‑principles demonstration that strongly correlated topological phases can be accessed via conventional Feynman diagram techniques, opening a new avenue for precision many‑body calculations in systems where traditional perturbation theory was previously thought to be inapplicable.