Electronic Transport in Topological and Strongly Correlated Systems

Abstract

In this dissertation, we attack the problem of strongly correlated and topological systems via a creative variety of approaches in the hopes of extracting and elucidating meaningful electronic transport phenomena. In the first part, we study the long-standing puzzle of the anomalously large and superuniversal correlation length exponent $\nu$ in the fractional quantum Hall effect. To tackle this, we utilize the newly conjectured Chern-Simons dualities as a powerful non-perturbative tool. By exploring the new descriptions of FQHE transitions afforded by the dualities, we find that large flavor expansions compare unfavorably to the experimental $\nu$. However, the non-Abelian nature of these dualities motivated us to try large color expansions. Utilizing duality techniques in combination with modular transformations, we were able to use large color expansions to demonstrate superuniversality across FQHE transitions. This was the first theoretical demonstration of superuniversality, as well as one of the first uses of the non-Abelian Chern-Simons dualities. In the second part, we critically examine transport features of the strongly-correlated electron hydrodynamic regime. Electron hydrodynamics has been claimed to be observed in a number of experiments, generating much excitement. However, clear demonstration of this regime is tricky since direct measurement of the electron-electron scattering length is difficult. Measurements of non-local transport behavior have been argued to be a signature of viscous flow and therefore provide indirect evidence of a short electron-electron scattering length. We begin by showing, on the contrary, that non-local transport behavior can occur even for disordered non-interacting fermionic systems which sits far from the hydrodynamic regime. Therefore, non-local transport is not unique to hydrodynamics. Furthermore, the linearized Navier-Stokes equation is structurally equivalent to common momentum-dependent Ohm's law; disentangling the hydrodynamic contribution requires precise understanding of the phenomenological parameters. By contrast, the fully nonlinear Navier-Stokes equation is distinct from the linear Ohm's law and can give rise to distinctive signatures. We therefore proposed three experiments to manifest unique nonlinear phenomena well-known in the classical fluids literature - the Bernoulli effect, Eckart streaming, and Rayleigh streaming. Analysis of experimental parameters suggests that these proposals are feasible and therefore provide strong signatures of a hydrodynamic regime. Moreover, as one of the first works to comprehensively study nonlinear effects, we hope that it would motivate further exploration of nonlinear electron fluid dynamics. In the third part, we look for optical signatures of the chiral anomaly in Weyl semimetals. Direct detection of the chiral anomaly via a negative longitudinal magnetoresistance has been difficult as this signature can arise from other mechanisms. Other works have proposed anomalous IR reflectance signatures as a smoking gun for the chiral anomaly in non-mirror-symmetric Weyl semimetals. However, they neglected that the presence of a magnetic field, necessary for the chiral anomaly, will generically break mirror symmetries. We go on to argue that the background magnetic field can break mirror symmetry strongly enough in physical systems to yield observable IR signatures of the chiral anomaly, even for mirror-symmetric crystals. In the fourth part, we study transport along topological domain wall networks in moir\'e systems. While most excitement around moir\'e physics have focused around the moir\'e miniband, recent experiments have suggest that moir\'e systems can also feature sharp domain walls and provide a natural setting to study networks of 1D topological modes. Previous works focused either on non-interacting models or utilized interacting models to find gapped correlated phases by imposing a single-particle gap. However, away from commensurate fillings we expect intervalley scattering to be suppressed so that a single-particle gap cannot open. Therefore, we study a triangular network of valley-helical Luttinger wires where we enforce no intervalley scattering. We find that transport in this network is inherently non-local, distinct from the local diffusive behavior of a resistor network. In particular, at strong repulsive interactions we predict a novel orbital antiferromagnetic-ordering phase.