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arXiv:cond-mat/9811328

Exact Density of States in a System of Disordered Two Dimensional Dirac Fermions in the Unitarity Limit: The d-wave Superconductor

C. Pépin, Patrick A. Lee

Published Nov 23, 1998• 0 citations
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Abstract

In this paper we present a method to compute the exact density of states induced by $N$ non magnetic impurities in a system of two dimensional Dirac fermions in the unitarity limit. We review the case of the $π$-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 $δρ(ω) \simeq n_i/ |ω(\ln^2 |ω/Δ_0| + (π/2)^2)|$.

DOI
10.48550/arxiv.cond-mat/9811328

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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)∣.
We show that weak nonmagnetic impurities in a two-dimensional d-wave superconductor introduce a nontrivial exponent in the energy dependence of the single-particle density of states: \ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}\ensuremath{\Vert}\ensuremath{\omega}${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\alpha}}}$. The exponent \ensuremath{\alpha}>0 is exactly calculated for two types of disorder. We argue that the appearance of a finite \ensuremath{\rho}(0) would be equivalent to breaking of a continuous symmetry, prohibited in two dimensions. The method we use is based on the Abelian and non-Abelian bosonization of the effective model of 1D interacting fermions.
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.
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 quantum interference effect on the quasiparticle density of states (DOS) is studied with the diagrammatic technique in two-dimensional d-wave superconductors with dilute nonmagnetic impurities both near the Born and near the unitary limits. We derive in details the expressions of the Goldstone modes (cooperon and diffuson) for quasiparticle diffusion. The DOS for generic Fermi surfaces is shown to be subject to a quantum interference correction of logarithmic suppression, but with various renormalization factors for the Born and unitary limits. Upon approaching the combined limit of unitarity and nested Fermi surface, the DOS correction is found to become a δδδ-function of the energy, which can be used to account for the resonant peak found by the numerical studies.
We solve the problem of NNN nonmagnetic impurities in the staggered flux phase of the Heisenberg model which we assume to be a good mean-field approximation for the spin-gap phase of the cuprates. The density of states is evaluated exactly in the unitary scattering limit and is proportional to $1/\ensuremath{\omega}{ln}^{2}(|\ensuremath{\omega}|/D)$, in analogy with the 1D case of doped spin-Peierls and two-leg ladder compounds. We argue that the system exhibits a quasi-long-range order at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 with instantaneous spin-spin correlations decreasing as ni/ln2(niRij){n}_{i}/{ln}^{2}({n}_{i}{R}_{\mathrm{ij}})ni​/ln2(ni​Rij​) for large distances Rij{R}_{\mathrm{ij}}Rij​ and we predict enhanced low energy fluctuations in neutron scattering.