Theory of Disorder-Induced Half-Integer Thermal Hall Conductance

The thermal Hall conductance in the half-filled first Landau level was recently measured to take the quantized noninteger value ${\kappa }_{xy}=5/2$ (in units of temperature times ${\pi }^2{k}_B^2/3h$), which indicates a non-Abelian phase of matter. Such exotic states have long been predicted to arise at this filling factor, but the measured value disagrees with numerical studies, which predict ${\kappa }_{xy}=3/2$ or $7/2$. We resolve this contradiction by invoking the disorder-induced formation of mesoscopic puddles with locally ${\kappa }_{xy}=3/2$ or $7/2$. Interactions between these puddles generate a coherent macroscopic state that exhibits a plateau with quantized ${\kappa }_{xy}=5/2$. The non-Abelian quasiparticles characterizing this phase are distinct from those of the microscopic puddles and, by the same mechanism, could even emerge from a system comprised of microscopic Abelian puddles.

Figure 1

Proposed phase diagram of the approximately half-filled first excited Landau level. Without disorder, the ground state is either Pfaffian or anti-Pfaffian, depending on whether the system is more particlelike $(\nu \lesssim {\nu }^{*})$ or holelike $(\nu \gtrsim {\nu }^{*})$ with an expected first-order transition at $\nu ={\nu }^{*}\approx \frac{5}{2}$. With disorder, this transition splits into four continuous phase transitions where the thermal Hall conductance changes by $\mathrm{\Delta }{\kappa }_{xy}=1/2$. For even stronger disorder, the system may enter a thermal metal, with nonquantized ${\kappa }_{xy}$. In contrast, the electrical Hall conductance ${\sigma }_{xy}=\frac{5}{2}(e^2/h)$ remains quantized across these transitions.

Figure 2

Possible edge modes of three kinds of Pfaffian phases. All three phases have identical charge modes, but differ in the number and chirality of neutral Majorana modes (indicated by the number and direction of arrows) and consequently in their thermal Hall conductance.

Figure 3

Microscopic puddles and schematic network model. (a) Puddles of anti-Pfaffian (of size $\sim \xi$) embedded in a Pfaffian phase. Each Pfaffian–anti-Pfaffian boundary hosts four copropagating Majorana fermions. (b) Adding a topologically trivial pair of counterpropagating Majorana modes next to the sample boundary, followed by suitable hybridization of counterpropagating modes results in the PH-Pfaffian edge on top of a Chalker-Coddington model containing four Majorana fermions.

Figure 4

Emergence of non-Abelian quasiparticles from puddles of Abelian phases. (a) In the Abelian “host” system, a fractional charge $e/4$ is associated with a pair of Majorana zero modes, which can hybridize with one another. (b) At the transition into the non-Abelian phase, only one of the two Majorana fermions (along with its zero mode) delocalizes throughout the system. (c) In the non-Abelian phase, the previously percolating Majorana fermion forms an edge state at the outer sample boundary, while still carrying the exact zero-energy mode.