We study four-dimensional fractional quantum Hall states on ${\mathbb{CP}}^{2}$ geometry from microscopic approaches. While in 2d the standard Laughlin wave function, given by a power of Vandermonde determinant, admits a product representation in terms of the Jastrow factor, this is no longer true in higher dimensions. In 4d, we can define two different types of Laughlin wave functions, the determinant-Laughlin and Jastrow-Laughlin states. We find that they are exactly annihilated by, respectively, two-particle and three-particle short-ranged interacting Hamiltonians. We then mainly focus on the ground state, low-energy excitations, and the quasihole degeneracy of determinant-Laughlin state. The quasihole degeneracy exhibits an anomalous counting, indicating the existence of multiple forms of quasihole wave functions. We argue that these are captured by the mathematical framework of ``commutative algebra of $N\mathrm{points}$ in the plane.'' The microscopic wave functions and Hamiltonians studied in this work pave the way for a systematic study of a high-dimensional topological phase of matter that is potentially realizable in cold atom and optical experiments.We study four-dimensional fractional quantum Hall states on ${\mathbb{CP}}^{2}$ geometry from microscopic approaches. While in 2d the standard Laughlin wave function, given by a power of Vandermonde determinant, admits a product representation in terms of the Jastrow factor, this is no longer true in higher dimensions. In 4d, we can define two different types of Laughlin wave functions, the determinant-Laughlin and Jastrow-Laughlin states. We find that they are exactly annihilated by, respectively, two-particle and three-particle short-ranged interacting Hamiltonians. We then mainly focus on the ground state, low-energy excitations, and the quasihole degeneracy of determinant-Laughlin state. The quasihole degeneracy exhibits an anomalous counting, indicating the existence of multiple forms of quasihole wave functions. We argue that these are captured by the mathematical framework of ``commutative algebra of $N\mathrm{points}$ in the plane.'' The microscopic wave functions and Hamiltonians studied in this work pave the way for a systematic study of a high-dimensional topological phase of matter that is potentially realizable in cold atom and optical experiments.

arXiv:2109.11522Physical Review ResearchApr 20, 20231 citation

Effective hypernetted-chain study of even-denominator-filling state of the fractional quantum Hall effect

Orion Ciftja

The microscopic approach for studying the half-filled state of the fractional quantum Hall effect is based on the idea of proposing a trial Fermi wave function of the Jastrow-Slater form, which is then fully projected onto the lowest Landau level. A simplified starting point is to drop the projection operator and to consider an unprojected wave function. A recent study claims that such a wave function approximated in a Jastrow form may still constitute a good starting point on the study of the half-filled state. In this paper we formalize the effective hypernetted-chain approximation and apply it to the unprojected Fermi wave function, which describes the even-denominator-filling states. We test the above approximation by using the Fermi hypernetted-chain theory, which constitutes the natural choice for the present case. Our results suggest that the approximation of the Slater determinant of plane waves as a Jastrow wave function may not be a very accurate approximation. We conclude that the lowest Landau-level projection operator cannot be neglected if one wants a better quantitative understanding of the phenomena.The microscopic approach for studying the half-filled state of the fractional quantum Hall effect is based on the idea of proposing a trial Fermi wave function of the Jastrow-Slater form, which is then fully projected onto the lowest Landau level. A simplified starting point is to drop the projection operator and to consider an unprojected wave function. A recent study claims that such a wave function approximated in a Jastrow form may still constitute a good starting point on the study of the half-filled state. In this paper we formalize the effective hypernetted-chain approximation and apply it to the unprojected Fermi wave function, which describes the even-denominator-filling states. We test the above approximation by using the Fermi hypernetted-chain theory, which constitutes the natural choice for the present case. Our results suggest that the approximation of the Slater determinant of plane waves as a Jastrow wave function may not be a very accurate approximation. We conclude that the lowest Landau-level projection operator cannot be neglected if one wants a better quantitative understanding of the phenomena.

Physical review. B, Condensed matterApr 15, 199916 citations

We study four-dimensional fractional quantum Hall states on

{\mathbb{CP}}^{2}

geometry from microscopic approaches. While in 2d the standard Laughlin wave function, given by a power of Vandermonde determinant, admits a product representation in terms of the Jastrow factor, this is no longer true in higher dimensions. In 4d, we can define two different types of Laughlin wave functions, the determinant-Laughlin and Jastrow-Laughlin states. We find that they are exactly annihilated by, respectively, two-particle and three-particle short-ranged interacting Hamiltonians. We then mainly focus on the ground state, low-energy excitations, and the quasihole degeneracy of determinant-Laughlin state. The quasihole degeneracy exhibits an anomalous counting, indicating the existence of multiple forms of quasihole wave functions. We argue that these are captured by the mathematical framework of ``commutative algebra of

N\mathrm{points}

in the plane.'' The microscopic wave functions and Hamiltonians studied in this work pave the way for a systematic study of a high-dimensional topological phase of matter that is potentially realizable in cold atom and optical experiments.