Particle-hole symmetry breaking and the $\nu =\frac{5}{2}$ fractional quantum Hall effect

We report on the study of the fractional quantum Hall effect at the filling factor 5/2 using exact diagonalization method with torus geometry. The particle-hole symmetry breaking effect is considered using an additional three-body interaction. Both Pfaffian and anti-Pfaffian states can be the ground state depending on the sign of the three-body interaction. The results of the low-energy spectrum, the wave function overlap, and the particle-hole parity evolution, have shown clear evidence of a direct sharp transition (possibly first order) from the Pfaffian to the anti-Pfaffian state at the Coulomb point. A quantum phase diagram is established, where one finds further transitions from the Pfaffian or anti-Pfaffian state to the nearby compressible phases induced by a change of the pseudopotential.

Figure 1

(Color online) Low-lying energies as a function of the pseudomomentum $\sqrt{K_1^2+K_2^2}$ (in physical units) for a $N_e=12$ pure Coulomb system with the rectangular unit cell at the aspect ratios of (a) $L_1/L_2=0.99$, and (b) $L_1/L_2=0.97$, where a pair of lowest-energy states at the pseudomomentum (0,6) are nearly degenerate. The plots of (c) and (d) are low-lying energies (relative to the globe GS) as a function of $V_{3b}$ for the $N_e=12$ model system with the additional 3b interaction $H_3$ and $H_3^{\ast }$, respectively. The unit cell is the same as the one in (b). The circle, square and triangle symbols stand for the GSs (solid symbols) and the first excited states (open symbols) at the pseudomomenta of (0,6), (6,0), and (6,6), respectively. The insets in (c) and (d) are the zoom-in spectra for $-0.1<V_{3b}<0.1$.

Figure 2

(Color online) The projection of the system GS on the Pf and APf states as a function of $V_{3b}$. The aspect ratio $L_1/L_2$ are 0.94, 0.97, and 0.70 for the system with $N_e=10$, 12 and 14 electrons, respectively. The curves with solid (open) symbols stand for the projection on the Pf (APf) state. States of $N_e=12$ system with the additional 3b interaction of (a) $H_3$ and (b) $H_3^{\ast }$ at pseudomomenta (0,6), (6,0), and (6,6) are represented by the curves with circle, square and triangle symbols, respectively. (c) States of $N_e=10$ system with $H_3$ at pseudomomentum (5,5), $N_e=12$ at (0,6) and $N_e=14$ at (7,7) are represented by the curves with diamond, square, and triangle symbols, respectively.

Figure 3

(Color online) Detectors of the QPT for the model system with $H_3$ at $-0.1<V_{3b}<0.1$ and different $N_e$. (a) GS fidelity function, (b) second derivative of the GS energy, and (c) GS PH parity as function of $V_{3b}$. The curves with diamond, star, and triangle symbols represent the system with $N_e=10$, 12, and 14 electrons, respectively.

Figure 4

(Color online) Phase diagram of $N_e=12$ system near the Coulomb point with the 3b interactions of (a) $H_3$ and (b) $H_3^{\ast }$. The letters of S, L, Pf, and APf mark the phase regions of $\left(\delta V_1,V_{3b}\right)$ with the stripelike, composite-fermion-liquid-like, Pf and APf phase, respectively.