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Physical review. B, Condensed matter

Numerical study of localization of Dirac fermions on a lattice in two dimensions

Yasuhiro Hatsugai, Patrick A. Lee

Published Aug 1, 1993• 36 citations
Publisher

Abstract

We studied the effects of impurities on Dirac fermions numerically. We used a tight-binding Hamiltonian of fermions with the magentic flux equal to \ensuremath{\pi} per plaquette for the model. A small amount of strong impurities create states near the zero energy and strong localization of these states is observed. Our results support a recent proposal of localization effects in a gapless d-wave superconductor.

DOI
10.1103/physrevb.48.4204

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We studied the effects of impurities on Dirac fermions numerically. We used a tight-binding Hamiltonian of fermions with the magentic flux equal to \ensuremath{\pi} per plaquette for the model. A small amount of strong impurities create states near the zero energy and strong localization of these states is observed. Our results support a recent proposal of localization effects in a gapless d-wave superconductor.
A lattice model for disordered d-wave superconductors with spin-rotational and time-reversal symmetries is reconsidered. Near the band center, the lattice model can be described by Dirac fermions with several species. Each fermion yields a Wess-Zumino-Witten (WZW) term for an effective action of the Goldstone mode, which is known to play a crucial role in critical phenomena in two dimensions. The WZW terms cancel out each other because of the fourfold symmetry of the model, which suggests that the quasiparticle states are localized. If the lattice model has, however, symmetry breaking terms which generate mass for any species of the Dirac fermions, remaining a WZW term which avoids the cancellation can derive the system to a delocalized strong-coupling fixed point.
Two-dimensional random Dirac fermions are studied numerically. They are realized on a square lattice by the $\ensuremath{\pi}$-flux model with random hopping. It preserves a symmetry denoted by ${H,\ensuremath{\gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ in an effective field theory. Although it belongs to the orthogonal ensemble, the zero-energy states do not localize but become critical. The density of states vanishes as $\ensuremath{\sim}{E}^{\ensuremath{\alpha}}$ and the exponent $\ensuremath{\alpha}$ changes with strength of the randomness (the critical line). Rapid enhancement of the Thouless number is observed near the zero energy. The level-spacing distribution is also investigated, which is consistent with the existence of the critical states at E=0E\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0E=0.
In this paper we present a method to compute the exact density of states induced by NNN non magnetic impurities in a system of two dimensional Dirac fermions in the unitarity limit. We review the case of the πππ-flux phase of the Heisenberg model and also treat the disordered d-wave superconductor. In both case we find additional states in the gap with δρ(ω)≃ni/∣ω(ln⁡2∣ω/Δ0∣+(π/2)2)∣δρ(ω) \simeq n_i/ |ω(\ln^2 |ω/Δ_0| + (π/2)^2)|δρ(ω)≃ni​/∣ω(ln2∣ω/Δ0​∣+(π/2)2)∣.
Effects of nonmagnetic impurities with unitarity scattering in two-dimensional d -wave superconductors are investigated by the numerical diagonalization of finite-size systems of up to 35×35 sites. It is shown that a localized state with almost zero excitation energy is formed around an impurity site if it is isolated. However, its localized state is extended in real space along the direction of the node of the d -wave gap, so that the localized states of adjacent impurities easily interfere with each other forming a kind of impurity band, i.e., resonance states around the zero energy. It is also pointed out that there exists a strong association between the resonance density of states due to impurity scattering in the unitarity limit and the so-called zero-bias anomaly of tunneling conductance.
Dirac Fermions on 2D lattice are studied in the presence of a single flux tube, a pair of flux tubes and an edge dislocation/anti-dislocation pair. Numerical studies have been carried out using exact diagonalization for each case, and the results indicate that the model supports fractionally charged particles with anyonic statistics. An analytical calculation is also provided, predicting a zero energy bound state in the continuum for Dirac Fermions in the presence of a single flux tube carrying one-half flux quantum, as well as a brief discussion of anyonic quasiparticles and their possible application to quantum computation.
We derive a solvable resonant-level-type model, to describe an impurity spin coupled to zero-energy bound states localized at the edge of a one-dimensional d-wave superconductor.This results in a two-channel Kondo effect with a quite unusual low-temperature thermodynamics.For instance, the local impurity susceptibility yields a finite maximum at zero temperature (but no ln-divergence) due to the splitting of the impurity in two Majorana fermions.Moreover, we make comparisons with the Kondo effect occurring in a two-dimensional d-wave superconductor.