Renormalization Group Approach to two Strongly Correlated Condensed Matter Models

Abstract

This thesis presents renormalization group (RG) analyses of two strongly correlated condensed matter systems.

In the first part, the phase diagram of the spin-$\frac{1}{2}$ Heisenberg antiferromagnetic model on a spatially anisotropic triangular lattice is discussed. This model, together with a Dzyaloshinskii-Moriya (DM) interaction, describes the magnetic properties of the layered Mott insulator Cs$_{2}$CuCl$_{4}$. Employing a real-space RG approach, it is found, in agreement with a previous similar study, that a fragile collinear antiferromagnetic (CAF) state can be stabilized at sufficiently strong anisotropies. The presented RG analysis only indicates the presence of the CAF and spiral states in the phase diagram, with no extended quantum-disordered state at strong anisotropies. Specifically, it reveals a fine-tuning of couplings that entails the persistence of ferromagnetic correlations between second-nearest chains over large length scales even in the CAF phase. This has important implications on how numerical studies on finite-size systems should be interpreted, and reconciles the presence of the CAF state with the observation of only ferromagnetic correlations in numerical studies. The effect of a weak DM interaction within this RG approach is examined. It is concluded that Cs$_{2}$CuCl$_{4}$ is well within the stability region of the spiral ordering.

In the second part, the fate of a neck-narrowing Lifshitz transition in two-dimensions and in the presence of weak interactions is studied. Such a transition is a topological quantum phase transition, with no change in symmetry. At the critical point of this transition, the density of states at the Fermi energy is logarithmically divergent and a van Hove singularity appears. It is found that, at the critical point, the Wilsonian effective action is intrinsically non-local. This non-locality is attributed to integrating out an emergent soft degree of freedom. Away from the critical point, a local perturbative RG description is presented, and it is shown that weak attractive interactions grow as $\log^2L$ ($L$ is the physical length). However, this local description is restricted to a finite momentum range that shrinks as the critical point is approached.