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arXiv:cond-mat/0002002 · Physical review. B, Condensed matter

Localization-delocalization transition of disorderedd-wave superconductors with nonmagnetic impurities

Takahiro Fukui

Published Oct 1, 2000• 11 citations
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Abstract

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.

DOI
10.1103/physrevb.62.r9279

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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.
We analyse how the range of disorder affects the localization properties of quasiparticles in a two-dimensional d-wave superconductor within the standard non-linear σ-model approach to disordered systems. We show that for purely long range disorder, which only induces intra-node scattering processes, the approach is free from the ambiguities which often beset the disordered Dirac-fermion theories, and gives rise to a Wess–Zumino–Novikov–Witten action leading to vanishing density of states and finite conductivities. We also study the crossover induced by inter-node scattering due to a short range component of the disorder, thus providing a coherent non-linear σ-model description in agreement with all the various findings of different approaches.
The low-energy limits of models with disorder are frequently described by $\ensuremath{\sigma}$ models. In two dimensions, most $\ensuremath{\sigma}$ models admit either a Wess-Zumino-Witten term or a $\ensuremath{\theta}$ term. When such a term is present the model can have a stable critical point with gapless excitations. We describe how such a critical point appears, in particular in two-dimensional superconductors with disorder. The presence of such terms is required by the underlying (anomalous) symmetries of the original electron model. This indicates that the usual symmetry classes of disordered systems in two dimensions can be further refined. Conversely, our results also indicate that models previously thought to be in different universality classes are in fact the same once the appropriate extra terms are included.
A disordered fermion model defined on a two-dimensional square lattice with $\ensuremath{\pi}$ flux is studied. Time-reversal symmetry and one of four possible chiral symmetries are imposed. The continuum limit of the model near zero energy yields Dirac fermions with chiral symmetry. It is shown that one of them has $\mathrm{Sp}(n)\ifmmode\times\else\texttimes\fi{}\mathrm{Sp}(n)$ symmetry whereas others have U(2n)(2n)(2n) symmetry, both of which are broken to Sp(n),(n),(n), and the fluctuation around a saddle point is described, respectively, by the Sp(n)2{(n)}_{2}(n)2​ Wess-Zumino-Witten model and U(2n)(2n)(2n)/Sp(n)(n)(n) nonlinear sigma model.
We study the symmetry class for localization which arises from models of noninteracting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation invariance. Twodimensional systems in this category, which is known as class D, can display phases with three different types of quasiparticle dynamics: metallic, localized, or with a quantized (thermal) Hall conductance. Correspondingly, they can show a variety of delocalization transitions. We illustrate this behavior by investigating numerically the phase diagrams of network models with the appropriate symmetry and show the appearance of the metallic phase. We also study level statistics for this symmetry class and find that the nearest neighbor spacing distribution (NNSD) at the critical energy follows the Wigner’s surmise for Gaussian Unitary Ensembles (GUE), reflecting therefore only basic discrete symmetries of the system (time reversal violation) and ignoring particle-hole symmetries and other finer details (criticality). In the localized regime level repulsion is suppressed.
Anderson localization is studied for two flavors of massless Dirac fermions in two-dimensional space perturbed by static disorder that is invariant under a chiral symmetry (chS) and a time-reversal symmetry (TRS) operation which, when squared, is equal either to plus or minus the identity. The former TRS (symmetry class BDI) can, for example, be realized when the Dirac fermions emerge from spinless fermions hopping on a two-dimensional lattice with a linear energy dispersion such as the honeycomb lattice (graphene) or the square lattice with $\ensuremath{\pi}$ flux per plaquette. The latter TRS is realized by the surface states of three-dimensional Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological band insulators in symmetry class CII. In the phase diagram parametrized by the disorder strengths, there is an infrared stable line of critical points for both symmetry classes BDI and CII. Here we discuss a ``global phase diagram'' in which disordered Dirac fermion systems in all three chiral symmetry classes, AIII, CII, and BDI, occur in four quadrants, sharing one corner which represents the clean Dirac fermion limit. This phase diagram also includes symmetry classes AII [e.g., appearing at the surface of a disordered three-dimensional Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological band insulator in the spin-orbit (symplectic) symmetry class] and D (e.g., the random bond Ising model in two dimensions) as boundaries separating regions of the phase diagram belonging to the three chS classes AIII, BDI, and CII. Moreover, we argue that physics of Anderson localization in the CII phase can be presented in terms of a non-linear-$\ensuremath{\sigma}$ model (NL$\ensuremath{\sigma}$M) with a Z2{\mathbb{Z}}_{2}^{\phantom{\rule{4pt}{0ex}}}Z2​-topological term. We thereby complete the derivation of topological or Wess-Zumino-Novikov-Witten terms in the NL$\ensuremath{\sigma}$M description of disordered fermionic models in all ten symmetry classes relevant to Anderson localization in two spatial dimensions.