# Theory of Topological Quantum Matter

Topological quantum materials exhibit unusual properties, such as exotic surface states and anomalous transport phenomena, that are unaffected by deformations, e.g., stretching, compressing, or twisting of the material, or adding disorder [1]. The aim of this proposal is to develop a theoretical framework to describe these topological properties, and to find new ways how to use them in the laboratory and for device applications. We seek to classify topological materials in terms of symmetries and to discover new remarkable examples. The main focus will be on the properties of: (i) nodal-line materials, (ii) magnetic (semi-)metals, and (iii) bosonic topological phases. We will study these systems using both analytical and numerical techniques.

(i) Photo-induced currents in nodal-line materials.

Nodal-line materials, such as CaAgP and CaAgAs, are three-dimensional analogues of graphene [4]. They exhibit a Dirac band crossing along a nodal loop, which is protected by a π-Berry phase. By the bulkboundary correspondence, the π-Berry phase leads to the appearance of drumhead surface states. The aim of this project is to study and characterize the exotic phenomena associated with the nontrivial topology of the nodal line. One example, is the parity anomaly, which leads to anomalous transport characteristics [5]. We will investigate the topological nature of the nonlinear optical effects in nodal-line materials. Using the Folquet formalism and the Keldysh Green’s function technique we will compute the electronic response of nodal-line materials in the presence of circularly polarized incident light. This theoretical investigation will be accompanied by experimental works on CaAgP at UBC (David Jones group and Damascelli group) and MPI (Stefan Kaiser group), in particular, time-resolved ARPES and pump probe experiments, which will allow to measure the light-induced Floquet bands.

(ii) Magnetic (semi-)metals

Magnetic topological (semi-)metals combine long-range magnetic order with nontrival band topology.In contrast to nonmagnetic topological systems, these magnetic topological materials are highly tunable by external magnetic fields. In the magnetic antiperovskite Eu3PbO, for example, one can split the Dirac points into Weyl points by applying a magnetic field, see Fig. 1. Moreover, the Weyl points can then be pair-annihilated or moved in energy or momentum by adjusting the magnetization direction. This, in turn, modifies the Berry curvature of the bands, and thus the topological transport properties. We will study this interdependence between magnetism, topology, and anomalous transport properties in Eu3PbO and other magnetic topological (semi-)metals. Experimental investigations on these compounds will be performed at UTokyo (Zenji Hiroi group) and at MPI (Hide Takagi group). Furthermore, using the co-representation theory of magnetic space groups, we will devise and apply a strategy based on non-symmorphic symmetries to discover new magnetic topological (semi-)metals.

(iii) Topological band structures of bosonic degrees of freedom

Not only fermionic but also bosonic quasiparticles can exhibit topological band structures. Bosonic topological band structures occur intrinsically in a variety of systems, e.g., as phonon band structures in isostatic lattices or as magnon band structures in quantum magnets [2,3]. Bosonic quasiparticles with topological properties can also be created artificially in a number of synthetic systems, such as in photonic lattices or in microcavities. It is interesting to study these artificial band structures both in isolation, but also when they are coupled to the environment via particle gain and loss. In the latter case, the band structure is effectively described by a non-Hermitian bosonic Hamiltonain, which may exhibit a number of exotic phenomena. The aim of this proposal is to develop a comprehensive topological band theory for bosonic degrees of freedom described by Hermitian and non-Hermitian Hamiltonians. We will explore the advantages of such topological bosonic structures for applications, e.g., for photonic wave guides or magnon edge channels that are not affected by disorder and noise.

*FIG. 1. Surface states of Eu _{3}PbO for (a) the antiferromagnetic phase and (b) the ferromagnetic phase. The net magnetic moment of the ferromagnetic phase splits the Dirac points into Weyl points (green and blue dots). The red arrows display the spin polarization of the surface states.*

References

[1] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).

[2] D. G. Joshi, A. P. Schnyder, arXiv:1809.06387 (submitted).

[3] D. G. Joshi, A. P. Schnyder, Phys. Rev. B 96, 220405(R) (2017).

[4] Y.-H. Chan, C.-K. Chiu, M. Y. Chou, A. P. Schnyder, Phys. Rev. B 93, 205132 (2016).

[5] W. B. Rui, Y. X. Zhao, A. P. Schnyder, Phys. Rev. B 97, 161113(R) (2018).

[6] J. Zhang, et al., Phys. Rev. Materials 2, 074201 (2018).

Principal investigators

Andreas P. Schnyder (MPI), a.schnyder@fkf.mpg.de

Dirk Manske (MPI), d.manske@fkf.mpg.de

Marcel Franz (UBC), franz@phas.ubc.ca

Masaki Oshikawa (UTokyo), oshikawa@issp.u-tokyo.ac.jp