Chiral symmetry breaking in high-temperature superconductors and birefringent cold atoms, helicity modulus in layered bosons and phase diagram of superconductor-insulator transition

This work is a compilation of several research projects with them main theme being high-temperature superconductivity. We construct the field theory of underdoped cuprates beginning with a well-defined d-wave superconductor and adding the vortex degree of freedom using a singular gauge transformation. The symmetries of the theory both in the presence and absence of a quasi-particle mass are studied. Nodal quasi-particles are known to obey a relativistic Lorentz symmetry while their massless nature represent another symmetry which we will identify as a chiral SU(2) symmetry. It is shown that 2+1 quantum electrodynamics is the effective theory that describes underdoped cuprates in the zero- temperature pseudogap regime. We focus on the mechanism of dynamical mass generation in three dimensional quantum elec- trodynamics and theories with four-fermion interactions. This is a field that has been subject of extensive research in last two decades. However, our momentum-shell renormalization group ap- proach is new to the field and through that we are able to estimate the conditions for the mass generation mechanism and also work out the phase diagram of the theory for charge and inter- action strength. We discuss the applications of momentum-shell renormalization group to other four-fermionic theories in the absence of a gauge field. The justification for this is the fact that in the superconducting regime the system can be described by a massive gauge field theory coupled to relativistic quasi-particles which effectively represent a four-fermionic theory. Inspired by the field theory constructed for underdoped cuprates we discuss the superfluid re- sponse of the underdoped materials using an anisotropic bosonic model and compare it to experiment. The idea is to see how c-axis superfluid density measurements can help one set the parameters in our field theory for underdoped cuprates. The behaviours of the superfluid responses in both out- of-plane and in-plane measurements has been detailed as a function of temperature and density (doping) and it is shown that there is disagreement with the measured c-axis response using the iii conjectured bosonic Hamiltonian.