Green's function based Renormalisation Group Methods for disordered systems

Abstract

The goal of the thesis is to study systems with random and quasiperiodic disorders using Renormalisation Group (RG) methods that preserve the Green’s function of the system. An exact real-space RG for decimating fermionic degrees of freedom from a lattice system in a tight-binding model was proposed by Aoki. In the first part, these RG rules are applied to the case of the Anderson Model in 2D and the Aubry-Andre Model in 1D, where we decimate most of the sites except a couple of sites. The statistics of the renormalised parameters of the surviving sites are used to characterise the phase of the system. The on-site energies of the surviving sites remain finite at all system sizes in all the phases. In the localised phase, the typical value of the hopping parameter exponentially decays with system size. In contrast, it does not scale with system size in the delocalised phase. Strong Disorder Renormalisation Group, where the sites and bonds with maximum energy are iteratively decimated, is applied to the Aubry-Andre Model. By analysing the distribution of on-site energies and hopping parameters as the RG flows and by tracking the progress of RG in terms of bond and site decimation, we characterise the phase of the system. In the second part, a generalisation of this RG is derived within Keldysh Non-equilibrium Field Theory to deal with systems connected to baths. We obtain an RG rule for the inverse Green’s function of the system for decimating a degree of freedom from the system. Using this, we study the effect of connecting two semi-infinite leads to the 1D Aubry-Andre Model. We find that the phases of the system are not affected in any significant way due to the coupling of baths. The coupling of the bath causes the dissipation of particles into the bath, which we characterise by the imaginary part of the effective on-site energy. We find that the sites deep in the system do not have significant dissipation in the localised phase. In the delocalised and critical phase, there is finite dissipation even for sites far away from the connection to leads.