Some exotic gapless states in insulating and conducting many-body systems

This thesis investigates two distinct classes of gapless, strongly correlated quantum systems. In the first part of this thesis, we examine quantum spin-1 systems, which allow for non-geometric frustration between bilinear and biquadratic interactions. For these systems, frustration leads to a rich phase diagram hosting exotic spin nematic phases with highly entangled ground states near SU(3)-symmetric points. We use tensor network renormalisation group methods to study the phase diagram of the spin-1 bilinear biquadrati cHeisenberg model with focus on the phase transitions nearSU(3)-symmetric points. Tensor network renormalisation methods are helpful in the study of both non-critical and critical phases and give insight to new mechanisms of spin-nematic phases and exotic ground states. In the second part of this thesis, we study non-Fermi liquids, characterised by the absence of quasiparticles and unusual symmetries. Based on the recent conjecture that non-Fermi liquids have an emergent loop-U(1) group at the IR, we show the essential tools to quantise a loop-U(1)group. Using coadjoint orbits and their quantisation as the technology of choice, we propose a modest approach that might be helpful in the more profound problem of understanding the phenomenology of non-Fermi liquids.