Anomalous Reentrant $5/2$ Quantum Hall Phase at Moderate Landau-Level-Mixing Strength

A successful probing of the neutral Majorana mode in recent thermal Hall conductivity measurements opines in favor of the particle-hole symmetric Pfaffian (PH-Pf) topological order, contrasting the theoretical predictions of Pfaffian or anti-Pfaffian phases. Here we report a reentrant anomalous quantized phase that is found to be gapped in the thermodynamic limit, distinct from the conventional Pfaffian, anti-Pfaffian, or PH-Pf phases, at an intermediate strength of Landau level mixing. Our proposed wave function consistent with the PH-Pf shift in spherical geometry rightly captures the topological order of this phase, as its overlap with the exact ground state is very high and it reproduces low-lying entanglement spectra. A unique topological order, irrespective of the flux shifts, found for this phase, possibly corroborates the experimentally found topological order.

Figure 1

(a) Overlaps (shown as color map) of the exact ground states (found at $L=0$) of the Hamiltonian ${\widehat{H}}_{\mathrm{eff}}$ in Eq. (1) with LLM-corrected pseudopotentials [40] at Pf flux shift ($N=14$ and $N_{\varphi }=25$) for varying $\kappa$. Two topologically distinct phases are found as the overlaps of the ground states at different $\kappa$ values belonging to the same phase are closer to unity and two different phases are closer to zero. These two phases are intermediated by an unquantized regime (shown as crossed marks) of $\kappa$ where the ground state is not found at $L=0$. Overlaps have not been calculated for the gray zones, as one of the ground states corresponds to the unquantized regime. (b) Same as (a) but for the A-Pf flux shift ($N=12$ and $N_{\varphi }=25$). (c) Same as (a) but for PH-Pf flux shift ($N=14$ and $N_{\varphi }=27$).

Figure 2

Scaling of gap for a pair of quasiparticle-quasihole excitation at PH-Pf flux shift in the $\mathcal{A}$ phase for $\kappa =1.1$. and 1.2 with $1/N$.

Figure 3

(a) ES for PH-Pf flux shift with $N_{\varphi }=27$ at a $\kappa$ belonging to the $\mathcal{A}$ phase. $L_z^A$ represents the sum of the azimuthal components of angular momenta occupied by the particles in the $A$ part of the partition [41]. An equal number of electrons ($N_A=N_B=7$) in both partitions are considered for computing the corresponding ES. Here $\xi$ represents entanglement energy in an arbitrary unit. The entanglement gap is shown by a line with an arrow-headed top and bottom, and the counting of edge states is marked. Inset: Overlaps of the ground state of PH-Pf flux ($N_{\varphi }=27$) at $\kappa =1.2$ with the ground states at Pf flux ($N_{\varphi }=25$) and A-Pf flux ($N_{\varphi }=29$) for different values of $\kappa$ in the range of the $\mathcal{A}$ phase for $N=14$. The respective overlaps are ${\mathcal{O}}_{\mathrm{PH}-\mathrm{Pf}}^{\mathrm{Pf}}$ (circles) and ${\mathcal{O}}_{\mathrm{PH}-\mathrm{Pf}}^{\mathrm{A}-\mathrm{Pf}}$ (triangles). (b) The ES for Pf flux shift with $N_{\varphi }=25$, $N_A=N_B=7$, and $\kappa =1.2$. (c) Same as (b) but for A-Pf flux shift with $N_{\varphi }=25$, $N_A=N_B=6$.

Figure 4

Overlaps of ${\mathrm{\Psi }}_{\mathrm{MR}}$, ${\mathrm{\Psi }}_{\mathrm{MR}}^{\mathrm{phc}}$ (particle-hole conjugate of ${\mathrm{\Psi }}_{\mathrm{MR}}$), and ${\mathrm{\Psi }}_{\mathcal{A}}$ in Eq. (2) with the corresponding exact ground states for $(N=14,N_{\varphi }=25)$, $(N=12,N_{\varphi }=25)$, and $(N=14,N_{\varphi }=27)$, respectively, versus $\kappa$ at zero quantum well width. The unconnected zones refer to the unquantized regimes between two distinct topological quantized phases for each case.

Figure 5

ES for $N=10$ and $N_{\varphi }=19$ for the exact ground state (red dashed) and for ${\mathrm{\Psi }}_{\mathcal{A}}$ (blue diamonds) at $\kappa =0.8$ belonging to quantized $\mathcal{A}$ phase. Equal number of particles ($N_A=5$, $N_B=5$) have been considered for both the partitions for calculating the corresponding ES.