In this project, we are going to extend our investigations of correlation-driven topological phases. The emphasis was hitherto on topological insulators in two and three spatial dimensions. We have discov- ered a quantum tricritical point in the phase diagram of the Bernevig-Hughes-Zhang-Hubbard model where the second-order topological phase transition ends in a first-order line. This quantum critical phenomenon, whose presence we have unveiled in 3D strong topological insulators too, represents in fact a way of connecting trivial and non-trivial phases with no single-particle analog. The topological transition no longer occurs through a semimetal, but rather via a jump of the gap which switches from positive to negative without closing, as a result of many-body correlations.
Strong correlations oppose thus nodal points. Yet, the zero-gap phases we have been considering so far are, even without Coulomb repulsion, the result of a fine tuning. They live indeed right at the transition between band and quantum spin Hall insulator. In the next funding period we want to see if more robust semimetals, not constrained to exist on a single line of the phase diagram, have a better fate. We are hence going to study nodal topological phases such as interacting Weyl semimetals and see whether or not first-order phase transitions appear. If so, the electron-electron interaction can activate alternative ways of moving Weyl nodes around in momentum space, beyond the continuous one that we know from single-particle theories. Novel annihilation mechanisms between sources and sinks of Berry flux might also emerge. Further, by considering semimetals with broken inversion- and time-reversal symmetry we can devise models in which the monopole and the antimonopole are differently influenced by many-body effects. The interest here is also connected to the fact that cor- related nodal semimetals can exhibit magnetic ordering. Within dynamical mean-field theory (DMFT) we can address their intermediate-to-strong coupling aspects and make contact with the currently hot topic of antiferromagnetic spintronics.
Another goal of this project regards higher-degeneracy Dirac particles. Some space groups host triply or quadruply degenerate eigenvalues at high-symmetry points in the Brillouin zone, whose existence is guaranteed by the symmetries of the underlying non-symmorphic space groups. Since the Coulomb interaction does not break such fundamental symmetry, the basic question is if electronic correlation is at all able to gap out these elementary excitations. During the first funding period we have explicitly focused on space-group Nr. 130 and analyzed how the Hubbard repulsion drives the double-Dirac semimetal towards a Mott antiferromagnetic phase. By studying more general model Hamiltonians, in the second funding period, we want to address the question of how the concept of degeneracy between bands at high-symmetry points enforced by the lattice symmetry can be extended to the many-body case, where single-particle eigenvalues are no longer meaningful.
The main part of C07 is therefore placed at the core activities of Area C. A third research line connects however to Area B of the SFB 1170. We plan to incorporate BCS triplet pairing at the level of the single-particle Hamiltonians and treat additional many-body interactions within DMFT. The re- sulting microscopic models can be related to the ones studied in the previous part of the project. Yet, we are now going to resum to all orders in the coupling constant not only normal but also anomalous self-energy local diagrams. Starting from a time-reversal symmetric helical superconductor, we are interested in the effect of a strong s-wave perturbation onto its Majorana modes, beyond the BCS level which has been demonstrated to be non-destructive. Another task of this approach is to see whether the p-wave topological superconductor can coexist with the ordered phases induced by a strong lo- cal repulsion. DMFT can indeed describe Mott antiferromagnets or charge-density wave phases in intermediate-coupling regimes inaccessible to perturbative treatments. How these influence topolog- ical superconductivity – even if at the BCS level – is an open question.