Projective construction of the ${\mathbb{Z}}_k$ Read-Rezayi fractional quantum Hall states and their excitations on the torus geometry

Multilayer fractional quantum Hall wave functions can be used to construct the non-Abelian states of the ${\mathbb{Z}}_k$ Read-Rezayi series upon symmetrization over the layer index. Unfortunately, this construction does not yield the complete set of ${\mathbb{Z}}_k$ ground states on the torus. We develop an alternative projective construction of ${\mathbb{Z}}_k$ Read-Rezayi states that complements the existing one. On the multilayer torus geometry, our construction consists of introducing twisted boundary conditions connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. Furthermore, we analyze the neutral excitation modes above the Moore-Read on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations to these modes. Finally, we extend our symmetrization scheme to the plane and sphere geometries.

Figure 1

Bilayer system with different types of topological defects. (a) Bilayer system with no defect. (b) Bilayer system with a twist defect along the $y$ direction. If the system has periodic boundary conditions, it is transformed into a monolayer torus with a doubled length in the $x$ direction. (c) Bilayer system with a defect along the $x$ direction. If the system has periodic boundary conditions, it is transformed into a monolayer torus with a doubled length in the $y$ direction.

Figure 2

Trilayer system with different types of topological defects. (a) Trilayer system with no defect. (b) Trilayer system with a defect along the $y$ direction connecting the two upper layers. With periodic boundary conditions, this system is equivalent to a bilayer torus, with the first layer having aspect ratio 2, and the second layer having aspect ratio 1. (c) Trilayer system with a defect along the $y$ direction connecting respectively layers 1 (blue) and 2 (red), 2 and 3 (green), 3 and 1. With periodic boundary conditions, this system is equivalent to a torus with aspect ratio 3.

Figure 3

Schematic representation of the symmetrization procedure to obtain the ground state on the $T$ torus of dimensions $L_x\times L_y$ pierced by $N_{\varphi }$ flux quanta. The orbitals colored in red have become equivalent after symmetrization. (a) The $T$ torus bilayer of lengths $L_x\times L_y$, pierced by $N_{\varphi }$ flux quanta. (b) The ${\left(2T\right)}_x$ torus of lengths $2L_x\times L_y$, pierced by $2N_{\varphi }$ flux quanta. (c) The ${\left(2T\right)}_y$ torus of lengths $L_x\times 2L_y$, pierced by $2N_{\varphi }$ flux quanta.

Figure 4

Dispersion of the neutral excitation mode above the Moore-Read ground states on the torus as a function of the center of mass and relative momenta of the many-body states in the FQH Brillouin zone. Interpolations of the dispersions are obtained from energy eigenvalues of finite-size systems for (a) $N=N_{\varphi }=16$ and (b) $N=N_{\varphi }=17$. In accordance with the momentum sectors in Table 1, these systems feature three ground states and one ground state, respectively (black dots). This plot shows that the dispersions are centered around the inversion-symmetric momenta while comprising a single connected excitation mode throughout the FQH Brillouin zone. We clearly observe two types of neutral excitations depending on the presence or absence of a zero-energy state at the center of each dispersive part: In the absence of a ground state, the neutral mode has a much more pronounced minimum (green, neutral fermion mode). Above the ground states, we find the magnetoroton mode (yellow) without minimum. The slight anisotropy of the dispersive part of each mode near these momenta is a finite-size effect. For energies larger than $E\sim 1$, a continuum of excited states appears (not shown).

Figure 5

Low-energy spectrum of the model three-body contact interaction (Moore-Read or ${\mathbb{Z}}_2$ Read-Rezayi state) on the torus for as a function of the one-dimensional momentum $\kappa$ defined in Eq. (30). In each graph, we show a different domain of the Brillouin zone, which is highlighted in gray in the inset. The black dots (respectively crosses) in the inset represent the Moore-Read ground states for $N$ even (respectively odd). In each domain, $\kappa$ is defined as the distance to the center of the domain [see also Eq. (30)]. (a) For both $N$ even (dots) and $N$ odd (crosses), there is a zero-energy eigenstate lying at ${\mathit{k}}_0=0$. All points fall onto the same curve, defining a neutral excitation (the magnetoroton mode). (b) A zero-energy state is found at each inversion-symmetric momentum in this domain for $N$ even only. The neutral excitation energies fall onto two distinct curves, with the repartition determined by the parity of $N$. (c) No zero-energy ground state is found at ${\mathit{k}}_3=(\pi ,\pi )$. All neutral excitation energies fall onto the same curve.

Figure 6

Low-energy spectrum of the FQH on the torus as a function of the modulus of the momentum $\kappa$, for the model three-body contact interaction. We compare the energy of the trial states obtained by symmetrization (hollow squares) to the result of exact diagonalization (filled squares and crosses). The variational states are obtained by symmetrizing a decoupled bilayer system [(a), (b), (c)] or a single-layer system defined on the ${\left(2T\right)}_x$ torus [(d), (e), (f)]. The energies of these last states differ by less than a percent from the energies of the states obtained by the symmetrization of a system defined on the ${\left(gT\right)}_y$ system. We thus only show the former. See Sec. 3b for details on the construction of the trials states. In each graph, we show a different domain of the Brillouin zone, which is highlighted in gray in the inset. The black dots (respectively crosses) in the inset represent the Moore-Read ground states for $N$ even (respectively odd). In each domain, $\kappa$ is defined as the distance to the center of the domain [see also Eq. (30)].

Figure 7

Geometrical interpretation of the alternative symmetrization prescription on rotationally symmetric manifolds, such as (a) the cylinder or (b) the disk. In either case, the double layer is joined into a singly connected surface via a topological twist defect.

Figure 8

Sketch of the two possible symmetrizations for the double-layer torus with twisted boundary conditions in both directions. We look at the simple case where $L_x=L_y$. (a) and (d) show the initial system where the boundaries have to be glued according to the plain, dotted, or dashed lines at the edge. (a) requires the equivalent single-layer system (b) to be defined by the spanning vectors $L_x{\mathit{e}}_x+L_y{\mathit{e}}_y$ of length $\sqrt{L_x^2+L_y^2}$ and $2L_y{\mathit{e}}_y$. Analogously, the single-layer system (e) for (d) is set by the spanning vectors $2L_x{\mathit{e}}_x$ and $L_x{\mathit{e}}_x+L_y{\mathit{e}}_y$ of length $\sqrt{L_x^2+L_y^2}$. From (b) [resp. (e)], we apply the symmetrization procedure ${\mathcal{S}}_{{\left(2T\right)}_y\to T}$ (resp. ${\mathcal{S}}_{{\left(2T\right)}_x\to T}$) to obtain a Moore-Read state (c) [resp. (f)]. In both cases, one ends up with a Moore-Read state on a twisted torus. Notice that the orientation of the orbitals (along ${\mathit{L}}_2$) depicted here is chosen accordingly to the gauge used when expending the wave functions on the many-body basis.

Figure 9

(a) Schematic representation of the impossibility to find a thin-torus configuration respecting the $(1,2)$ exclusion principle for a bilayer torus with $N=7$ bosons, and $N_{\varphi }=7$ flux quanta. (b) Inserting a topological defect that connects the two tori relieves the frustration. A configuration respecting the exclusion principle is then possible.