UCLA

Particle-hole condensates in the angular momentum $l=2$ channel, known as $d$-density wave orders, have been suggested to be realized in strongly correlated electron systems. In this dissertation, we study singlet and triplet $d$-density wave orders with a form factor of $d_{x^2-y^2}$ as well as a novel topological mixed singlet-triplet $d$-density wave with a form factor of $(i sigma d_{x^2-y^2}+d_{xy})$, and discuss the connections of these states to high-temperature superconductors and heavy-fermion materials.

In Chapter 2, we discuss the spin susceptibility of the singlet $d$-density wave, triplet $d$-density wave, and spin density wave orders with hopping anisotropies. From the numerical calculation, we find nearly vertical dispersion relations for spin excitations with anisotropic incommensurability at low energies in agreement with the inelastic neutron scattering experiments in the pseudogap state of the high-temperature superconductor YBa$_{2}$Cu$_{3}$O$_{6.6}$. The vertical dispersion is a distinct feature of all three density wave states in contrast to the superconducting state, which shows an hourglass shape dispersion experimentally.

In Chapter 3, we explore a mixed singlet-triplet $d$-density wave state in a two-dimensional square lattice, which is topologically nontrivial and exhibits quantum spin Hall effect. We also study the bulk-edge correspondence and Lifshitz transition in the system. In Chapter 4, we show that the skyrmions in the system carry charge $2e$ and can condense into a superconducting state. The phase transition between the density wave and superconductivity likely leads to deconfined quantum critical points. We suggest connections of this exotic state to the hidden-order state in the heavy-fermion compound URu$_{2}$Si$_{2}$.

The st-DDW model is generalized to three dimensions in Chapter 5, where we propose a novel pairing mechanism in URu$_{2}$Si$_{2}$. We assume the charge $2e$ skyrmionic spin texture in the mixed singlet-triplet $d$-density wave state fractionalizes into merons and antimerons at the deconfined quantum critical point. The interaction between these fractional particles results in a spin-singlet chiral $d$-wave superconducting state consistent with experiments. The unconventional superconductivity breaks time reversal symmetry, so we expect a polar Kerr effect at the onset of the superconductivity, but not in the time-reversal-invariant hidden-order state except perhaps for magnetic impurities.