We study the mixed state in an extreme type-II lattice ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductor in the experimentally most relevant regime of intermediate magnetic fields
Hc1⪡H⪡Hc2. We analyze the low energy spectrum of the problem, dominated by nodal Dirac-like quasiparticles with momenta near ${\mathbf{k}}_{F}=(\ifmmode\pm\else\textpm\fi{}{k}_{D},\ifmmode\pm\else\textpm\fi{}{k}_{D})$, and find that the spectrum exhibits characteristic oscillatory behavior with respect to the product of
kD and magnetic length
l. The Simon-Lee scaling, predicted in this regime, is satisfied only on average, with the magnitude of the oscillatory part of the spectrum displaying the same ${l}^{\ensuremath{-}1}$ dependence as its monotonous ``envelope'' part. In general, the spectrum obeys a scaling law ${E}_{n\mathbf{k}}=(\ensuremath{\hbar}{v}_{F}∕l){\mathcal{E}}_{n}(\mathbf{k}l,t∕\ensuremath{\Delta},{k}_{D}l)$, where
E is a dimensionless universal $2\ensuremath{\pi}$-periodic function of
kDl. The oscillatory behavior of the spectrum is due to the internodal interference enhanced by the singular nature of the low-energy eigenfunctions near vortices. Our results constitute an example of a finite size scaling of the Dirac-type quantum criticality. We also study a separate problem of a single vortex piercing an isolated superconducting grain of size $L\ifmmode\times\else\texttimes\fi{}L$. Here we find that the periodicity of the quasiparticle energy oscillations with respect to
kDL is doubled relative to the case where the field is zero and the vortex is absent, both such oscillatory behaviors being present at the leading order in ${L}^{\ensuremath{-}1}$. Finally, we review the overall features of the tunneling conductance experiments in $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ (YBCO) and ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ (BSCCO), and suggest an interpretation of the peaks at
5−−20meV observed in the tunneling local density of states in these materials. We find that in the case of a pure
d-wave superconducting order parameter with featureless vortex cores, the zero bias conductance peak (ZBCP) appears only on the sites that are the immediate nearest neighbors of vortex locations, while all the other sites in the close vicinity of vortices exhibit no such ZBCP and instead display pronounced peaks at subgap energies, typically at about a half or less of the coherence peak energy. Furthermore, we find that the on-site ZBCP can be strongly suppressed by enhanced local pairing near a vortex.
We examine the quantum theory of the spontaneous breaking of lattice rotation symmetry in
d-wave superconductors on the square lattice. This is described by a field theory of an Ising nematic order parameter coupled to the gapless fermionic quasiparticles. We determine the structure of the renormalization group to all orders in a
1/Nf expansion, where
Nf is the number of fermion spin components. Asymptotically exact results are obtained for the quantum critical theory in which, as in the large
Nf theory, the nematic order has a large anomalous dimension, and the fermion spectral functions are highly anisotropic.
Topological phases are quantum phases of matter that appear close to absolute zero temperature. In contrast to commonly known condensed phases of matter, such as a magnet or a superconductor, distinct topological phases may have the same symmetry. Instead, these phases are distinguished by the topology of the electronic wavefunction. Already the single-particle excitations described by band theory in an insulator or a metal or by Bogoliubov-de Gennes theory in a superconductor may have novel anomalous properties associated to their topology. This topological band theory is the subject of this thesis. We aim towards identifying the anomalous boundary and defect states associated to the topology of the band structure in crystalline insulators and superconductors. Furthermore, we determine criteria to easily identify the topological phases from the band structure of a given insulator or superconductor. In Chapter 2 we show that crystalline topological insulators and superconductors may host anomalous gapless or in-gap excitations on corners of a two-dimensional crystal or hinges of a three-dimensional crystal. These topological phases are called "second-order" according to the codimension "2" of their anomalous boundary states. We discuss the precise conditions for their appearance and when they can be linked to the bulk topology. Our discussion includes mirror, twofold-rotation and inversion as crystalline symmetries. In Chapter 3 we discuss the appearance of anomalous gapless or ingap states at defects in a crystalline insulator or superconductor. In particular, we link the existence of anomalous states at disclinations -- lattice defects that violate a rotation symmetry only locally -- to second-order topological crystalline phases. As a side product, we identify the possible contributions of other topological phases to the disclination anomaly. The results of this chapter allow to determine precisely how the disclination anomaly is linked to the bulk topology. In Chapter 4 we present how to construct "symmetry-based indicators" for topological superconductors that admit a description within the Bogoliubov-de Gennes framework. Symmetry-based indicators are necessary criteria for a given Bogoliubov-de Gennes Hamiltonian to realize a topological crystalline phase with anomalous boundary exitations. These criteria are formulated in terms of data from a small set of high-symmetry momenta only. Our approach guarantees to extract the maximal information on the anomalous boundary exitations that can be extracted from a point-wise evaluation of the Bogoliubov-de Gennes Hamiltonian. In the limit of weak superconducting pairing, our criteria can be formulated in terms of data from the normal-state band structure alone.
Disorder in electronic systems suppresses diffusion and can lead to localization. The origin of this behavior is enhanced backscattering due to quantum interference. In addition, disorder leads to a broadening of the density of states (DoS). However, the localization properties of a disordered system are usually not discernible from a study of the DoS. It is necessary to calculate two-particle properties or consider the influence of boundary conditions on energy eigenvalues as in the study of Thouless numbers to obtain information about localization. Another feature of disordered systems is the independence of their localization properties from details of the disorder in the system. For example, the range of the disorder affects non-universal quantities like the mean-free path but does not influence whether or not the system shows localization at all. Recently, new symmetry classes came into focus that do not conform to the standard expectations about disordered systems outlined above. As we will show in this contribution, details of disorder do matter for these systems and localization properties leave their mark on the DoS. In particular, we look at a system that presents an idealization of a disordered d-wave superconductor. In our study we neglect effects of self-consistency and concentrate on the d-wave symmetry and influence of different kinds of disorder. We consider the lattice quasi-particle Hamiltonian
We study the effect of gapless quasiparticles in a d-wave superconductor on the
T=0 end point of the Kosterlitz-Thouless transition line in underdoped high-temperature superconductors. Starting from a lattice model that has gapless fermions coupled to three-dimensional
XY phase fluctuations of the superconducting order parameter, we propose a continuum field theory to describe the quantum phase transition between the d-wave superconductor and the spin-density-wave insulator. Without fermions, the theory reduces to the standard Higgs scalar electrodynamics (HSE), which is known to have the critical point in the inverted
XY universality class. Extending the renormalization group calculation for the HSE to include the coupling to fermions, we find that the qualitative effect of fermions is to increase the portion of the space of coupling constants where the transition is discontinuous. The critical exponents at the stable fixed point vary continuously with the number of fermion fields N, and we estimate the correlation length exponent $(\ensuremath{\nu}=0.65)$ and the vortex field anomalous dimension $({\ensuremath{\eta}}_{\ensuremath{\Phi}}=\ensuremath{-}0.48)$ at the quantum critical point for the physical case
N=2. The stable critical point in the theory disappears for the number of Dirac fermions
N>Nc, with ${N}_{c}\ensuremath{\approx}3.4$ in our approximation. We discuss the relationship between the superconducting and the chiral (SDW) transitions, and point to some interesting parallels between our theory and the Thirring model.
Within numerically exact solutions of the Bogoliubov-de Gennes equations, we demonstrate that discrepancies between predicted low-energy quasiparticle properties in disordered 2D d-wave superconductors occur because of the unanticipated importance of disorder model details and normal state particle-hole symmetry. For the realistic case, which is best described by a binary alloy model without particle-hole symmetry, we predict density-of-state suppression below an energy scale which appears to be correlated with the corresponding single impurity resonance.