Microscopic geometric theory for gapped excitations in fractional topological fluids

Abstract

We propose a geometric description of all gapped excitations in fractional quantum Hall phases that reveals several fundamental understandings with experimental consequences. These include a duality between the Hilbert space of multiple gapped ``graviton-like" spin 2 excitations at $ν=1/3$ Laughlin phase, and that of non-Abelian quasiholes of an irrational Haffnian conformal field theory. This leads us to construct microscopic wave functions for multiple higher-spin gapped neutral modes in the Laughlin phase. Carrying spin $s \ge 2$, they emerge from higher-order geometric deformations of the topological ground state, and live within the Gaffnian conformal Hilbert space asymptotically. We show that the full many-body Hilbert space of an FQH phase can be generated from superpositions of such geometric deformations, supporting a concrete geometric interpretation of all gapped excitations. We analyse the scattering between these higher spin modes and conjecture that they can have a long lifetime, and propose methods for their experimental detection.

Figures (6)

• Figure 1: a. Schematic of ground-state deformations. Starting from the uniform FQH ground state, the lowest-order area-preserving deformation gives spin-$2$ graviton modes associated with an anisotropic metric $g_{\mu\nu}$. Higher-spin modes arise from higher-order deformations, naturally described by higher-rank tensor structures that cannot be reduced to products of local metrics without redundancy. b. Haffnian conformal Hilbert space (CHS) as a multiple Laughlin-graviton space.$V_1$-resolved spectrum in the Haffnian CHS, defined as the nullspace of a three-body model Hamiltonian at $N_o=3N_e-2$. Markers denote system size, and colors label the number $N_g$ of spin-2 excitations. The highest-$L$ multi-graviton states exhibit linear dispersion (dashed line), consistent with quantized excitations. Exact Haffnian quasihole counting matches that of bosonic spin-2 gravitons, showing that the CHS of non-Abelian anyons can equivalently be spanned by geometric multi-Laughlin-graviton states.
• Figure 2: Density distribution of $\hat{U}_{s}(\alpha) |0\rangle$. Here, we set the deformation strength $\alpha = 0.2$. Note that there is a $C_s$ symmetry in the wave packet acted by $\hat{U}_{s}$.
• Figure 3: (a) Finite-size scaling of HS mode energies. Their energies enter the continuum (above dashed line) and approach the GM energy. While the absolute values are non-universal and depend on system size, the thermodynamic extrapolation indicates convergence to the vicinity of the GM energy. (b) Finite-size scaling of HS modes' cumulative overlaps within Gaffnian nullspace. All the Laughlin HS modes asymptotically live within this space. (c) Spectral function of HS mode with $s=3$. Even after entering the continuum, the spectral peak is still sharp, indicating a long lifetime similar to the $s=2$ GM. (d) Spectrum from deformation basis. For $N_e=8$, including deformations up to third order, reproduces the full Hilbert space and thus recovers the complete spectrum of $\hat{V}_1$.
• Figure B.1: Dispersion of neutral gapped excitations with respect to the $V_1$ pseudopotential. The magnetoroton branch is clearly resolved, and its overlap with the exact diagonalization eigenstates is unity. The other neutral excitations, shown in lighter colors, are labeled according to their squeezing order: modes with lighter colors have root configurations that can be obtained by squeezing those shown in darker colors. These states lie at higher energies and are therefore deeply embedded in the continuum.
• Figure E.1: (a)-(f) Spectral functions of higher spin modes with $s = 2$ to $7$.

Theorems & Definitions (15)

• Definition 1
• Definition 2
• Theorem 1
• Definition 3: Fixed-particle-number Hilbert space
• Definition 4: Neutral excitations
• Lemma 2: Action of $\mathcal{C}_{a,b}$ on a Slater determinant
• proof
• Lemma 3: Constructive map between two occupation configurations
• proof
• Theorem 4: Generation of $\mathcal{H}_{N_e,N_o}$ by single-particle moves