Finite-Layer Thickness Stabilizes the Pfaffian State for the 5/2 Fractional Quantum Hall Effect: Wave Function Overlap and Topological Degeneracy

We find the finite width, i.e., the layer thickness, of experimental quasi-two-dimensional systems produces a physical environment sufficient to stabilize the Moore-Read Pfaffian state thought to describe the fractional quantum Hall effect at filling factor $\nu =5/2$. This conclusion is based on exact calculations performed in the spherical and torus geometries, studying wave function overlap and ground state degeneracy.

Figure 1

Wave function overlap between the exact ground state, for a quasi-2D system modeled by an infinite square-well potential, and the Pf as a function of finite width $d$ for $N_e=8$, 10, and 12 electrons, respectively. The experimental $d$ of Refs. 24 (solid circle), 25 (star), and 13 (open square) are also indicated. The inset shows the excitation gap $\Delta$ for $N_e=8$ as a function of $d$ both in units of $e^2/ϵl$ (solid) and $e^2/ϵ\sqrt{l^2+d^2}$ (dashed).

Figure 2

Low-lying eigenenergies as a function of the pseudomomentum $\sqrt{K_x^2+K_y^2}$ (in physical units) for an aspect ratio of $a/b=0.75$. The left panels refer to zero width while the right panels correspond to the SQ potential of width $d=4l$ for (a)–(b) $N_e=10$, (c)–(d) $N_e=12$, (e)–(f) $N_e=14$.

Figure 3

Low-lying eigenenergies of $N_e=12$ electrons as a function $\mathbf{K}$ using the SQ potential of width $d=4l$ and aspect ratio $a/b=0.99$. There are doublets at the wave vectors of the Pf ground states corresponding to the Pfaffian-anti-Pfaffian symmetric and antisymmetric combinations. Because of the almost square symmetry, there are quasidegeneracies between states at $\mathbf{K}=(6,0)$ and $\mathbf{K}=(0,6)$.