Abstract
We numerically study the effect of short-ranged potential disorder on massless noninteracting three-dimensional Dirac and Weyl fermions, with a focus on the question of the proposed (and extensively theoretically studied) quantum critical point separating semimetal and diffusive-metal phases. We determine the properties of the eigenstates of the disordered Dirac Hamiltonian () and exactly calculate the density of states (DOS) near zero energy, using a combination of Lanczos on and the kernel polynomial method on . We establish the existence of two distinct types of low-energy eigenstates contributing to the disordered density of states in the weak-disorder semimetal regime. These are (i) typical eigenstates that are well described by linearly dispersing perturbatively dressed Dirac states and (ii) nonperturbative rare eigenstates that are weakly dispersive and quasilocalized in the real-space regions with the largest (and rarest) local random potential. Using twisted boundary conditions, we are able to systematically find and study these two (essentially independent) types of eigenstates. We find that the Dirac states contribute low-energy peaks in the finite-size DOS that arise from the clean eigenstates which shift and broaden in the presence of disorder. On the other hand, we establish that the rare quasilocalized eigenstates contribute a nonzero background DOS which is only weakly energy dependent near zero energy and is exponentially small at weak disorder. We also find that the expected semimetal to diffusive-metal quantum critical point is converted to an avoided quantum criticality that is “rounded out” by nonperturbative effects, with no signs of any singular behavior in the DOS at the energy of the clean Dirac point. However, the crossover effects of the avoided (or hidden) criticality manifest themselves in a so-called quantum critical fan region away from the Dirac energy. We discuss the implications of our results for disordered Dirac and Weyl semimetals, and reconcile the large body of existing numerical work showing quantum criticality with the existence of these nonperturbative effects.
12 More- Received 15 February 2016
- Corrected 13 July 2016
DOI:https://doi.org/10.1103/PhysRevX.6.021042
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Published by the American Physical Society
Physics Subject Headings (PhySH)
Corrections
13 July 2016
Erratum
Publisher’s Note: Rare-Region-Induced Avoided Quantum Criticality in Disordered Three-Dimensional Dirac and Weyl Semimetals [Phys. Rev. X 6, 021042 (2016)]
J. H. Pixley, David A. Huse, and S. Das Sarma
Phys. Rev. X 6, 039901 (2016)
Popular Summary
The question of whether newly discovered Dirac and Weyl semimetals can host a stable semimetal phase in the presence of disorder is a fundamental question of interest. Based on extensive theoretical investigations, two schools of thought have emerged: (i) The semimetal phase is stable and the resulting quantum phase transition exists (perturbative arguments), and (ii) nonperturbative effects of rare eigenstates convert the semimetal into a diffusive metal for an infinitesimal disorder strength (i.e., there is no semimetal phase or quantum phase transition). Previous numerical calculations seem to support scenario (i) and did not find any signs of the rare eigenstates. Here, we definitely solve this open problem by examining the existence of a stable semimetal phase and the proposed quantum phase transition into a diffusive metal in the presence of disorder.
Using numerically exact methods applied to massless Dirac and Weyl fermions occupying a cubic lattice, we explicitly demonstrate how to study the low-energy eigenstates and determine their contribution to the density of states. We find the existence of two different types of eigenstates at weak disorder: Perturbatively dressed Dirac eigenstates and nonperturbative quasilocalized rare eigenstates. We establish that the rare eigenstates contribute an exponentially small but nonzero density of states at zero energy for infinitesimal disorder. As a result, we definitively show that there is no semimetal phase for any nonzero disorder strength. Excitingly, we establish that the quantum critical point separating the semimetal and the diffusive metal is an avoided quantum phase transition, which explains why previous numerical studies seemed to agree with (i).
We expect that our findings will pave the way for future studies of the topological nature of Dirac and Weyl semimetals.