Interaction Effects in Quantum Wires and One-Dimensional Incoherent Semimetals

Abstract

In this thesis, we address the effect of interactions in low dimensional systems. In the first part, using bosonization, we study a microscopic model of parallel quantum wires constructed from two dimensional Dirac fermions in the presence of periodic topological domain walls. The model accounts for the lateral spread of the wave-functions $\ell$ in the transverse direction to the wires. The gapless modes confined to each domain wall are shown to form Luttinger liquids, which realize a well known smectic non-Fermi liquid fixed point when inter-wire Coulomb interactions are taken into account. Perturbative studies on phenomenological models have shown that the smectic fixed point is unstable towards a variety of phases such as superconductivity, stripe, smectic and Fermi liquid phases. Here, we show that the considered microscopic model leads to a phase diagram with only smectic metal and Fermi liquid phases. The smectic metal phase is stable in the ideal quantum wire limit $\ell\to0$. For finite $\ell$, we find a critical Coulomb coupling $\alpha_{c}$ separating the strong coupling smectic metal from a weak coupling Fermi liquid phase. We conjecture that the absence of superconductivity should be a generic feature of similar microscopic models. We also discuss the physical realization of this model with moiré heterostructures. In particular, our model may be of relevance to recent experiments on twisted bilayer $\mathrm{tWTe_{2}.}$ In the second part we study a two band dispersive Sachdev-Ye-Kitaev model in 1+1 dimension. We suggest a model that describes a semimetal with quadratic dispersion at half-filling. We compute the Green's function at the saddle point using a combination of analytical and numerical methods. Employing a scaling symmetry of the Schwinger-Dyson equations that becomes transparent in the strongly dispersive limit, we show that the exact solution of the problem yields a distinct type of non-Fermi liquid with sub-linear $\rho\propto T^{2/5}$ temperature dependence of the resistivity. A scaling analysis indicates that this state corresponds to the fixed point of the dispersive SYK model for a quadratic band touching semimetal.