Dirac-Bogoliubov-deGennes quasiparticles in a vortex lattice

Abstract

In the mixed state of an extreme type-II $d$-wave superconductor and within a broad regime of weak magnetic fields $({H}_{c1}⪡H⪡{H}_{c2})$, the low-energy Bogoliubov-deGennes quasiparticles can be effectively described as Dirac fermions moving in the field of singular scalar and vector potentials. Although the effective linearized Hamiltonian operator does not formally depend on the structure of vortex cores, a singular nature of the perturbation requires choosing a self-adjoint extension of the Hamiltonian by imposing additional boundary conditions at vortex locations. Each vortex is described by a single parameter $\ensuremath{\theta}$ that effectively represents all effects arising from the physics beyond linearization. With the value of $\ensuremath{\theta}$ properly fixed, the resulting density of states of Dirac Hamiltonian exhibits full invariance under arbitrary singular gauge transformations applied at vortex positions. We identify the self-adjoint extensions of the solutions found earlier, within the framework of the linearized Hamiltonian diagonalized by expansion in the plane wave basis, and analyze the relation between fully self-consistent formulation of the problem and the linearized model. In particular, we construct the low-field scaling form of the nodal quasiparticle spectra, which incorporates the self-adjoint extension parameter $\ensuremath{\theta}$ explicitly and generalizes the conventional Simon-Lee scaling. In a companion paper, we also present a detailed numerical study of the lattice $d$-wave superconductor model and examine its low-energy, low-magnetic-field behavior with an eye on determining the proper self-adjoint extension(s) of the linearized continuum limit. In general, we find that the density of quasiparticle states always vanishes at the chemical potential, either linearly or by virtue of a finite gap. The low-energy continuum limit is thus faithfully represented by Dirac-like fermions, which are either truly massless, massless at the linearized level (mass $\ensuremath{\sim}H$), or massive (mass $\ensuremath{\sim}{H}^{1∕2}$), depending on the mutual commensuration of magnetic length and lattice spacing.