Departmental Doctoral Oral Examination (Thesis Title: “Topological quantum phase transitions and topological quantum criticality in superfluids and superconductors”)

Abstract:
Quantum phases of different topologies exist in superfluids and superconductors. These phases can be either fully gapped or gapless with nodal structures. Topological quantum phase transitions exist between these gapped phases or between gapped and nodal phases. The two phases on both sides of the phase transitions have the same local order but differ in topology. These topological quantum phase transitions cannot be described by the Landau paradigm of symmetry breaking. It is well-known that surface states change across these phase transitions as a result of change of topology in the bulk. However, a complete theory of topological quantum phase transitions has not been developed before. Here we construct an effective field theory to study the universality class of these topological quantum phase transitions. We find that quantum phase transitions between fully gapped phases with different topologies belong to the emergent relativistic Majorana field universality class. Quantum phase transitions between gapped and nodal phases belong to what we call quantum Lifshitz Majorana field universality classes. We also study the bulk signatures and energetics of these phase transitions. We find non-analyticities in certain thermodynamic quantities across these phase transitions, which can be viewed as a collective signature of Majorana fermions in the bulk. These topological quantum phase transitions only exist at zero temperature. At finite temperature, different states are connected by smooth crossovers. There exists a quantum critical region where physical properties are dictated by the topological quantum critical points (QCPs) at zero temperature. Thermodynamic quantities have universal scaling dependence on temperature that are also unique to each universality class. These temperature scalings can be used to probe and differentiate different topological QCPs.We also find that topological QCPs can exist on surfaces when time-reversal symmetry is broken. These surface topological QCPs belong to the emergent relativistic Majorana field universality class. These topological quantum phase transitions exist in various concrete models, such as chiral and time-reversal invariant $p$-wave superfluids, topological superconductors of emergent Dirac fermions, and topological superconducting model of Cu$_x$Bi$_2$Se$_3$.