Fathoming interplay between symmetry and topology of many-electron wave-functions has deepened understanding of quantum many body systems, especially after the discovery of topological insulators. Topology of electron wave-functions enforces and protects emergent gapless excitations, and symmetry is intrinsically tied to the topological protection in a certain class. Namely, unless the symmetry is broken, the topological nature is intact. We show novel interplay phenomena between symmetry and topology in topological phase transitions associated with line-nodal superconductors. The interplay may induce an exotic universality class in sharp contrast to that of the phenomenological Landau-Ginzburg theory. Hyper-scaling violation and emergent relativistic scaling are main characteristics, and the interplay even induces unusually large quantum critical region. We propose characteristic experimental signatures around the phase transitions in three spatial dimensions, for example, a linear phase boundary in a temperature-tuning parameter phase-diagram.
We theoretically study topological phase transitions in four generalized versions of the Kane-Mele-Hubbard model with up to $2\ifmmode\times\else\texttimes\fi{}{18}^{2}$ sites. All models are free of the fermion-sign problem allowing numerically exact quantum Monte Carlo (QMC) calculations to be performed to extremely low temperatures. We numerically compute the
Z2 invariant and spin Chern number ${C}_{\ensuremath{\sigma}}$ directly from the zero-frequency single-particle Green's functions, and study the topological phase transitions driven by the tight-binding parameters at different on-site interaction strengths. The
Z2 invariant and spin Chern number, which are complementary to each another, characterize the topological phases and identify the critical points of topological phase transitions. Although the numerically determined phase boundaries are nearly identical for different system sizes, we find strong system-size dependence of the spin Chern number, where quantized values are only expected upon approaching the thermodynamic limit. For the Hubbard models we considered, the QMC results show that correlation effects lead to shifts in the phase boundaries relative to those in the noninteracting limit, without any spontaneously symmetry breaking. The interaction-induced shift is nonperturbative in the interactions and cannot be captured within a ``simple'' self-consistent calculation either, such as Hartree-Fock. Furthermore, our QMC calculations suggest that quantum fluctuations from interactions stabilize topological phases in systems where the one-body terms preserve the
D3 symmetry of the lattice, and destabilize topological phases when the one-body terms break the
D3 symmetry.
In the present study, the interplay among interaction, topology, quasiperiodicity, and non-Hermiticity is studied. The hard-core bosons model on a one-dimensional lattice with asymmetry hoppings and quasiperiodic onsite potentials is selected. This model, which preserves time-reversal symmetry (TRS), will exhibit three types of phase transition: Real-complex transition of eigenenergies, topological phase transition, and many-body localization (MBL) phase transition. For the real-complex transition, it is found that the imaginary parts of the eigenenergies are always suppressed by the MBL. Moreover, by calculating the winding number, a topological phase transition can be revealed with the increase of potential amplitude, and we find that the behavior is quite different from the single-particle systems. Based on our numerical results, we conjecture that these three types of phase transition occur at the same point in the thermodynamic limit, and the MBL transition of quasiperiodic system and disordered system should belong to different universality classes. Finally, we demonstrate that these phase transitions can profoundly affect the dynamics of the non-Hermitian many-body system.