UCLA

\abstract {

In chapter 2 we offer a pedagogic introduction to quantum phase transitions and quantum critical behavior in scalar $\phi^4$ theory. We focus on characterizing the quantum critical fan, and we explicitly show how one can identify the borders of said fan using the familiar arguments introduced by S. Chakravarty, B. I. Halperin, and D. R. Nelson\cite{chn} in 1989. By calculating the renormalization group flow equations to one loop we are able to approximately calculate the correlation lengths in the theory, and from the behavior of the obtained correlation length we are then able to identify the phases of our theory as ordered, quantum critical, or quantum disordered.

\begin{abstract}

In chapter 3 we investigate the effect that density wave states have on the Hofstadter Butterfly. We first review the problem of the $d$-density wave on a square lattice and then numerically solve the $d$-density wave problem when an external magnetic field is introduced. As the $d$-density wave condensation strength is tuned the spectrum evolves through three topologically distinct butterflies, and a relativistic quantum Hall effect is observed. The chiral $p+ip$-density wave state demonstrates drastically different Hofstadter physics--inducing a destruction of the gaps in the butterfly which causes electrons' cyclotron orbits to not obey any type of Landau quantization, and the creation of a large gap in the spectrum with Hall conductance $\sigma_{xy}$=0. To investigate the quantum phases in the system we perform a multifractal analysis of the single particle wavefunctions. We find that tuning the $d$-density wave strength at a generic value of magnetic flux controls a metal-metal transition at charge neutrality where the wavefunction multifractality occurs at energy level crossings. In the $p+ip$ case we observe another metal-metal transition occurring at an energy level crossing separated by a strongly multifractal quasi-insulating island state occurring at charge neutrality and strip dimerization of the lattice.\end{abstract}

\begin{abstract}

In chapter 4 we discuss recent anomalous transport measurements that have been observed through a wide doping range in the cuprates. We investigate the effects of a state that shares many features consistent with those of the pseudogap, the mixed triplet-singlet $d$-density wave state, and examine whether its presence could help explain these observations. For a sufficiently doped system Li

amp; Lee [arXiv:1905.04248v3] showed that that these density wave states produce a nonzero thermal Hall effect. Through the effect that density waves have on the localized spins of a square lattice in a magnetically ordered phase, we find that the mixed triplet-singlet $d$-density wave state induces \emph{stable} Dzyaloshinskii-Moriya (DM) interactions among the localized spins in the presence of an external magnetic field. As similar antisymmetric exchange couplings have yielded nonzero thermal Hall contributions, we examine this induced DM interaction by applying Holstein–Primakoff (HP) transformations to study the resulting magnon excitations of the spin models for both antiferromagnetic and ferromagnetic backgrounds--relevant to the near-half-filling and heavily overdoped regimes respectively. Furthermore, because the triplet-singlet $d$-density wave is experimentally challenging to detect directly, we discuss the magnetic signatures that this state can possibly induce away from the pseudogap regime. We calculate the magnon dispersion for La${}_{2-x}$Sr${}_x$CuO${}_4$ (LSCO) and find that the density wave induces a weak $d_{x^2-y^2}$ anisotropy; upon calculating the non-abelian Berry curvature for this magnon branch, we show explicitly that the magnon contribution to $\kappa_{xy}$ is zero. Finally, we calculate corrections to the magnetic ground state energy, spin canting angles, and the spin-wave dispersion due to the topological density wave for ferromagnetic backgrounds. We find that terms \emph{linear} in the HP bosons can affect the critical behavior, a point previously overlooked in the literature.

\end{abstract}