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arXiv:2109.11522 · Physical Review Research

Fractional quantum Hall states on CP2 space

Jie Wang, Semyon Klevtsov, Michael R. Douglas

Published Apr 20, 2023• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.5.023042

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We study four-dimensional fractional quantum Hall states on CP2{\mathbb{CP}}^{2}CP2 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 NpointsN\mathrm{points}Npoints 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.
The quantum Hall effect (QHE) is a cornerstone of topological physics, inspiring extensive explorations of its high-dimensional generalizations such as experimental realizations in synthetic systems including cold atoms, photonic lattices, and metamaterials. However, the many-body effect in the higher dimensional QHE system remains poorly understood. We explore this problem by formulating the microscopic wave functions on a four-dimensional sphere inspired by Laughlin's seminal work. Employing a generalized pseudopotential framework, we derive an exact microscopic Hamiltonian consisting of two-body projection operators that annihilate the microscopic wave functions. Diagonalizations on finite system sizes show that the quasihole states remain zero energy while the quasiparticle states exhibit a finite gap, in consistency with an incompressible state. Furthermore, the pair distribution is calculated to substantiate the liquidlike nature of the wave function. Our Letter provides a preliminary understanding to the fractional quantum Hall states in high dimensions.
Fractional quantum Hall systems are often described by model wave functions, which are the ground states of pure systems with short-range interaction. A primary example is the Laughlin wave function, which supports Abelian quasiparticles with fractionalized charge. In the presence of disorder, the wave function of the ground state is expected to deviate from the Laughlin form. We study the disorder-driven collapse of the quantum Hall state by analyzing the evolution of the ground state and the single-quasihole state. In particular, we demonstrate that the quasihole tunneling amplitude can signal the fractional quantum Hall phase to insulator transition.
We introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors $\ensuremath{\nu}=1/q$) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the Z2{Z}_{2}Z2​ properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflect gapless edge states as was discussed by Li and Haldane.
We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. For this purpose, we study in detail the simplest nontrivial system beyond the Laughlin states, namely, bosons at filling factor $\ensuremath{\nu}=2/3$, and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four-particle interactions produces this state as the exact ground state and speculate that this remains true even when interaction terms involving a greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of $6\ensuremath{\hbar}$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $\ensuremath{\lambda}$ state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two states have a high overlap, they have the same root partition, the quantum numbers for their neutral excitations are identical, and the quantum numbers for the quasiparticle excitations also match. On the quasihole side, we find that even though the quantum numbers of the lowest energy states agree with the prediction from the composite-fermion theory, these states are not separated from the others by a clearly identifiable gap. This prevents us from making a conclusive claim regarding the topological equivalence of the $\ensuremath{\lambda}$ state and the composite-fermion state. Our study illustrates how new candidate states can be identified from constraining selected many-particle configurations and it would be interesting to pursue their topological classification.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
We motivate a close look on the usefulness of the Gaffnian and Haffnian quasihole manifold (null spaces of the respective model Hamiltonians) for well-known gapped fractional quantum Hall phases. The conformal invariance of these subspaces is derived explicitly from microscopic many-body states. The resultant conformal field theory (CFT) description leads to an intriguing emergent primary field with h=2,c=0h=2,\phantom{\rule{0.28em}{0ex}}c=0h=2,c=0, and we argue the quasihole manifolds are quantum mechanically well defined and well behaved. Focusing on the incompressible phases at $\ensuremath{\nu}=\frac{1}{3}$ and 25\frac{2}{5}52​, we show the low-lying excitations of the Laughlin phase are quantum fluids of Gaffnian and Haffnian quasiholes, and give a microscopic argument showing that the Haffnian model Hamiltonian is gapless against Laughlin quasielectrons. We discuss the thermal Hall conductance and shot-noise measurements at $\ensuremath{\nu}=\frac{2}{5}$, and argue that the experimental observations can be understood from the dynamics within the Gaffnian quasihole manifold. A number of detailed predictions on these experimental measurements are proposed, and we discuss their relationships to the conventional CFT arguments and the composite fermion descriptions.