Temperature Enhancement of Thermal Hall Conductance Quantization

The quest for non-Abelian quasiparticles has inspired decades of experimental and theoretical efforts, where the scarcity of direct probes poses a key challenge. Among their clearest signatures is a thermal Hall conductance with quantized half-integer value in units of ${\kappa }_0={\pi }^2{k}_B^{2}T/3h$ ($T$ is temperature, $h$ the Planck constant, $k_B$ the Boltzmann constant). Such values were recently observed in a quantum-Hall system and a magnetic insulator. We show that nontopological “thermal metal” phases that form due to quenched disorder may disguise as non-Abelian phases by well approximating the trademark quantized thermal Hall response. Remarkably, the quantization here improves with temperature, in contrast to fully gapped systems. We provide numerical evidence for this effect and discuss its possible implications for the aforementioned experiments.

Figure 1

Left: The unit cell of the model (shaded area) contains two sites, indicated by solid and open circles, each hosting a single Majorana mode. The signs of ($i$ times) the nearest-neighbor hopping $v_1$ and next-nearest-neighbor hoppings $v_2$ are indicated by the direction of arrows on the corresponding bonds. Right: Band structure of the model in a ribbon geometry, infinite in the horizontal direction and consisting of 30 unit cells in the vertical direction, using $v_1=1$ and $v_2=-0.2$. Bulk states are shown in green; states on the top and bottom edges are shown in red and blue, respectively.

Figure 2

The average transmission of a $80\times 80$ unit cells system, computed using 1000 disorder realizations. (a) The horizontal axis is $v_2$ and the vertical axis is disorder strength $V$. The topological transition between phases with Chern numbers $\mathcal{C}=\pm 1$ evolves into a thermal metal phase with increasing disorder strength. The dashed blue line indicates $V=1.3$. (b) For fixed $V=1.3$, the average transmission is plotted as a function of $v_2$ and energy $E$. Away from the particle-hole symmetric line, $E=0$, the thermal metal phase disappears; it is replaced by topologically trivial insulators ($\mathcal{C}=0$).

Figure 3

The dimensionless longitudinal and transverse thermal conductances (blue, orange) are computed in a six-terminal geometry (inset) at $T=0$ and $T=0.05$ (solid, dashed). We use a rectangular system composed of 80 zigzag chains in the vertical direction and 160 hexagonal plaquettes in the horizontal direction, and average over 1000 disorder realizations ($V=1.3$), with line thickness indicating the error bars. The thermal metal phase present in the $v_2=0$ region of the plot is converted into an insulating phase at nonzero temperature, leading to a quantized plateau in both longitudinal and transverse conductance. Notice the smaller error bars in the finite-temperature curves, which occur because the energy integral Eq. (4) provides additional disorder averaging.

Figure 4

(a) The numerically computed thermal conductance shows order-unity variations with $v_2$ at low temperatures. At higher temperatures it approaches the time-reversal symmetric value ${\kappa }_{xy}=0$ with very weak dependence on $v_2\in [-0.07,0.07]$. (b) Proposed finite-temperature phase diagram of $\nu \approx 5/2$ quantum Hall states. The thermal metal extends to finite temperatures due to residual interactions between neutral quasiparticles. At higher temperatures, but still below the charge gap, the mechanism discussed here sets in. Near ${\nu }^{*}$ it results in an large region where ${\kappa }_{xy}$ approaches the particle-hole symmetric value $5/2$. Hatched regions denote quantization of ${\kappa }_{xy}$ that is better than a threshold value, say 1%. The approximate plateaus are separated by relatively sharp crossovers as shown in Fig. 3.