Griffiths phase in the thermal quantum Hall effect

Two-dimensional disordered superconductors with broken spin-rotation and time-reversal invariance, e.g., with $p_x+i{p}_y$ pairing, can exhibit plateaus in the thermal Hall coefficient (the thermal quantum Hall effect). Our numerical simulations show that the Hall insulating regions of the phase diagram can support a sub-phase where the quasiparticle density of states is divergent at zero energy, $\rho \left(E\right)\sim \mid E{\mid }^{1∕z-1}$, with a nonuniversal exponent $z>1$, due to the effects of rare configurations of disorder (Griffiths phase).

Figure 1

2D Ising models in a representation of fermions moving on a chiral network: (a) Clean system; spin sites of the Ising model in its original spin representation are also shown, (b) network carrying an isolated vortex (hatched plaquettes). Vortex pairs are realized by inserting two half integer flux quanta into a pair of plaquettes sharing a single node, and (c) a string of four vortex pairs.

Figure 2

Phase diagram of TQHE in RBIM-network model representation, disorder $p$ vs interplaquette coupling strength (temperature) $t_l$. Fixed points: High temperature (HT), low temperature (LT), clean Ising (I), multicritical point (N), and percolation $\left({\mathrm{P}}_{\mathrm{c}}\right)$. Phase boundary (solid, precise location calculated in Ref. 20) separates two Hall insulators (called ferro and para using RBIM terminology). The Nishimori line: a line which supports a local ${\mathrm{Z}}_2$ symmetry and where the internal energy of the model is analytic. Shaded region indicates Griffiths phase with diverging single particle density of states, $\rho \left(E\right)$; in plain white regions DOS has (pseudo)-gap. Line segments represent scans for $\rho \left(E\right)$ displayed in Fig. 3. In Fig. 4, scan is along phase boundary from $\mathrm{N}$ to ${\mathrm{P}}_{\mathrm{c}}$.

Figure 3

DOS at disorder value $p=0.08$ deep in the ferromagnetic phase. As coupling between plaquettes is progressively reduced, pseudo-gap gives way to power law divergency. Plots show averages for two system sizes, 128 (256) with 20,000 (10,000) realizations. Data comprise 8 lowest lying eigenstates; $t_l=0.029$ corresponds to location on Nishimori line.

Figure 4

Distribution of logarithm of lowest eigenvalue for string of defects of length $\ell =8–56$ inserted in disordered system in ferromagnetic phase ($p=0.08$, $t_l=0.029$, $L=128$). Inset: Typical lowest eigenvalue as a function of $\ell$.