Fractional quantum Hall states by Feynman's diagrammatic expansion

TL;DR

This paper demonstrates that Feynman's diagrammatic expansion, combined with Monte Carlo methods, can reliably describe fractional quantum Hall states and their spectral properties directly from electronic interactions.

Contribution

It introduces a controlled diagrammatic Monte Carlo approach to study FQH states from first principles, showing the emergence of incompressible states and spectral gaps.

Findings

Observation of the 1/3-filled incompressible state at low temperature

Spectral properties indicating an energy gap at 1/3-filling

Results consistent with pseudogapped behavior at 1/2-filling

Abstract

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.