Neural quantum states put fractional Chern insulators and Wigner crystals on one phase diagram, resolving them at 36 electrons under strong band mixing
Synopsis
On the Aharonov-Casher continuum model, neural quantum state variational Monte Carlo accommodates both fractional Chern insulators and generalized Wigner crystals and yields phase diagrams at fractional filling for up to 36 electrons with strong band mixing. Two wave functions, a flux-attachment form and a Slater-Jastrow form, describe topological order and crystalline order respectively, and their energies match multiband exact diagonalization on small systems. As the quantum geometry becomes more nonuniform, the Laughlin-like state at 1/3 filling keeps nearly constant energy while the Jain-like state at 2/5 filling drops, so the two respond differently to discrete translation symmetry.
Interpretation
The phase diagram at fractional filling is no longer inferred only from calculations projected onto one or two bands; topological liquids and charge-ordered states are compared by energy directly in a continuum model that includes band mixing. Earlier studies of the Aharonov-Casher model relied on one- or two-band projections and were constrained by system size, since a lowest-band projection requires the interaction strength to stay below the band gap. For 24 electrons on 8 unit cells at 1/3 filling, the neural quantum state variational Monte Carlo ground-state energy per particle agrees with exact diagonalization converged at the three-band level, and the approach extends to larger systems.
The two kinds of ground state are carried by two physics-informed neural-network wave functions, one suited to topological order and one to crystalline order, and their energy crossing marks a phase transition. Accurately representing topological order in the bulk limit had remained a challenge for neural quantum state approaches, and learning fractional quantum Hall correlations became harder as system size grew. For 24 electrons on 8 unit cells at 1/3 filling, the flux-attachment wave function has lower energy than the Slater-Jastrow wave function over a range of parameters around the Landau level limit; as the magnetic field modulation strengthens, the latter falls roughly linearly and eventually drops below.
Quantum geometry nonuniformity and band mixing together decide whether fractional states give way to crystals, and which crystal structure and Chern number appear. Phase competition at intermediate band mixing and quantum geometry nonuniformity had not been studied, particularly at 2/5 filling. For 32 electrons on 16 unit cells at 2/5 filling, a Jain-like fractional Chern insulator and three crystalline phases are identified; strong band mixing lets electronic orbitals localize further and lowers the Coulomb energy, so the electron generalized Wigner crystal wins over the hole generalized Wigner crystal.
Perspective
The scope is the Aharonov-Casher continuum model with Coulomb interactions, a triangularly periodic magnetic field, and finite systems on a torus with periodic boundary conditions, at 1/3 and 2/5 filling and up to 36 electrons; the conclusions apply to this model and these fillings, not directly to a particular experimental sample.
At 1/3 filling the freezing of the Laughlin state occurs at a parameter value for the 24-electron system that lies outside the plotted phase diagram, and a recent 24-electron study reported a transition value that differs, which the authors attribute to finite-size effects and a slightly different variational wave function. At 2/5 filling the critical value for freezing of the Jain state has not been reported before. In the intermediate negative region at 2/5 filling, band-projected exact diagonalization with a Haldane pseudopotential interaction had suggested a possible superconductor, whereas with Coulomb interactions no evidence of superconductivity appears within the parameter range studied.