Quantum critical matter: Casimir pairing, emergent orders and topological defects

In this thesis we address a few problems on critical phenomena in two- and three dimensionalquantum materials. Specifically we consider high temperature superconductors, dimerized quantum magnets and frustrated magnets.First, motivated by the recent discovery of a magnetic quantum critical point in hole-doped cuprates, we study the influence of the criticality on the superconducting pairing. We consider a fermion- fermion interaction in a 2D system at a quantum critical point. We found a new physical mechanism of fermion pairing by quantum critical fluctuations, that is similar to a Casimir effect. The Casimir pairing mechanism has conceptual similarities with chiral bag models in quantum chromodynamics and is generic for a wide class of quantum phase transitions.Second, we consider an impurity screening problem in a 3D magnet close to a quantum critical point. We show that a local magnetic moment of the impurity becomes screened by the cloud of critical magnons and we calculate the distribution of a magnetization in the cloud. Our results show that adding a small concentration of impurities can significantly affect critical properties of the system.Next, we perform a phenomenological study of incommensurate charge density wave (CDW) in cuprates. Proceeding with a combined analysis of recent experimental data on the CDW, we find the amplitude and the spatial pattern of the CDW. From nuclear magnetic resonance data we extract the s, s' and d-wave amplitudes of the CDW and rule out the checkerboard pattern of the CDW. We show that data potentially rules out a wide class of theoretical models of the CDW.Finally, we consider topological defects (skyrmions and merons) in frustrated magnets which are in the vicinity of a phase transition from a collinear to a spiral magnetic state.We show that isolated metastable skyrmions can exist in the absence of an external magnetic field in frustrated magnets with an easy-axis/easy-plane anisotropy. We found exotic skyrmion states with a large topological charge: skyrmion rings and meron rings.In the systems with easy-plane spin anisotropy we demonstrated that at the critical point a skyrmion with a unit topological charge ``fractionalizes'' into a pair of merons