Metallic states in two-dimensional quantum matter have a long history and pose extremely challenging problems. A generic metallic state is described by a gapless system with a finite density of particles, along with disorders and interactions. Such correlated many-body systems are usually difficult to study, both analytically and numerically. In this thesis, we are dedicated to certain simplified cases which enable us to study via analytical approaches. Firstly, we study the effects of quenched disorder and a dissipative Coulomb interaction in the Dirac composite fermion theory describing the quantum phase transition of integer quantum Hall plateau and magnetic-field tuned 2D supercondutor
The renormalization group study is presented, by considering the quantum effect of disorder and gauge fluctuation. Secondly, we present a study of integer quantum Hall plateau transition using a mean-field theory of composite fermions with a gyromagnetic ratio equal to two. We investigate the stability problem in terms of semi-classical approach and derive the corresponding nonlinear sigma model. Thirdly, we study a single 2D Dirac fermion at finite density, subjected to a quenched random magnetic field. The low-energy theory can be mapped onto an infinite collection of 1D chiral fermions coupled by a random vector potential matrix. The theory is exactly solvable, and the electrical response is computed non-perturbatively. Lastly, we shift our focus to a disorder-free system formed by a collection of 1D wires. We provide an example of an Ersatz Fermi liquid by deforming the chiral Wess-Zumino-Witten model with level k greater than unity
