Topological order from disorder and the quantized Hall thermal metal: Possible applications to the $\nu =5/2$ state

Although numerical studies modeling the quantum Hall effect at filling fraction $5/2$ predict either the Pfaffian (Pf) or its particle-hole conjugate, the anti-Pfaffian (aPf) state, recent experiments appear to favor a quantized thermal Hall conductivity with quantum number $K=5/2$, rather than the value $K=7/2$ or $3/2$ expected for the Pf or aPF state, respectively. While a particle-hole symmetric topological order (the PH-Pfaffian) would be consistent with the experiments, this state is believed to be energetically unfavorable in a homogenous system. Here we study the effects of disorder that are assumed to locally nucleate domains of Pf and aPf. When the disorder is relatively weak and the size of domains is relatively large, we find that when the electrical Hall conductance is on the quantized plateau with ${\sigma }_{xy}=(5/2)(e^2/h)$, the value of $K$ can be only 7/2 or 3/2, with a possible first-order-like transition between them as the magnetic field is varied. However, for sufficiently strong disorder, an intermediate state might appear, which we analyze within a network model of the domain walls. Predominantly, we find a thermal metal phase, where $K$ varies continuously and the longitudinal thermal conductivity is nonzero, while the electrical Hall conductivity remains quantized at $(5/2)e^2/h$. However, in a restricted parameter range we find a thermal insulator with $K=5/2$, a disorder stabilized phase which is adiabatically connected to the PH-Pfaffian. We discuss a possible scenario to rationalize these special values of parameters.

Figure 1

(Top) Possible phase diagram containing four phases: Pf, aPf, PH-Pf, and the quantized Hall thermal metal. The vertical axis $\delta$ is primarily related to disorder strength, but could also be affected by detailed energetics. At a given disorder strength, three scenarios are possible as one varies the filling fraction $\nu$ across the symmetric point ${\nu }_c$ (which is expected to be shifted slightly away from $5/2$ due to Landau level mixing): a direct first-order-like transition from Pf to aPf phase (dashed blue line), an intermediate thermal metal (dashed red line) and an intermediate PH-Pf phase separated from Pf/aPf by a thermal metal (solid red line). The values of $K$ as functions of $\nu$ in these scenarios are shown in the lower panel.

Figure 2

Illustrations of inhomogeneous Pfaffian/anti-Pfaffian mixtures. The upper panel describes when a single phase (for example anti-Pfaffian) percolates, and the system behaves on macroscopic scales as the percolated phase, with the corresponding edge states. The lower panel describes when neither of the two phases percolates, but the domain walls between them percolate. If the neutral modes propagate diffusively on the domain walls, the system becomes a thermal metal (but is still a quantized electric Hall insulator). If the neutral modes are localized instead, the system becomes equivalent to a PH-Pfaffian state on macroscopic scales, with the edge modes that give $K=5/2$.

Figure 3

(a) Scattering at each domain wall junction, described by the scattering matrix $M$. (b) Illustration of the network model. Associated with each link is an $O\left(4\right)$ flavor mixing matrix, and with each vertex a scattering matrix $\tilde{M}$.

Figure 4

Phase diagram of the network model in scenario A, with ${\eta }_{i,j}=\eta$ for any $(i,j)$ and ${\alpha }_i=\alpha$ for any $i$. The Pfaffian/anti-Pfaffian transition in the clean limit ($\eta \to 0$ and $\alpha =\pi /4$) is broadened to a thermal metal phase upon introducing a generic disorder. Here, as in Figs. 6 and 7, points represent data, while lines are guides for the eye.

Figure 5

A PH-Pfaffian state can be stabilized if we assume the two pairs of Majorana chiral modes on each domain wall are spatially separated.

Figure 6

The reduced localization length ${\mathrm{\Lambda }}_L={\xi }_L/L$ as a function of the transverse dimension $L$. We choose ${\eta }_{1,2}={\eta }_{3,4}=1, {\eta }_{i,j}=0.1$ for other $(i,j)$ pairs. (Top) $tan{\alpha }_{1,2}=1/6$ and $tan{\alpha }_{3,4}=6$, the trend of localizing to a PH-Pfaffian state is clear. (Bottom) $tan{\alpha }_{1,2}=2/3$ and $tan{\alpha }_{3,4}=3/2$, a delocalized (metallic) state.

Figure 7

Phase diagram of the network model in scenario B. We assume a weak $\text{SO}\left(4\right)$ flavor mixing with ${\eta }_{i,j}=0.1$, but a different mixing between flavor (1,3) and (2,4), given by ${\eta }_{\mathrm{intra}}$ (vertical axis). The horizontal axis ${\alpha }_{\mathrm{average}}$ describes the average particle-hole asymmetry. We assume $\mathrm{\Delta }\alpha =\pi /6, {\alpha }_1={\alpha }_3=\text{max}({\alpha }_{\mathrm{average}}-\mathrm{\Delta }\alpha /2,\pi /100)$, and ${\alpha }_2={\alpha }_4=\text{min}({\alpha }_{\mathrm{average}}+\mathrm{\Delta }\alpha /2,\pi /2-\pi /100)$. We have also put in random vortices by having a random sign with equal probabilities on each link, which caused the metallic region in the small $\eta$ regime to significantly broaden.

Figure 8

Schematic phase diagram of the network model with an approximate $\mathrm{U}\left(1\right)$ symmetry. The shaded area represents a first-order-like transition in a finite sample.

Figure 9

Schematic phase diagram of the network model in scenario C2. The shaded area represents a first-order-like transition in a finite sample.