**Introduction**. – In the last decade, great strides have been made in the study of topological materials. A great number of topological phases have been classified in terms of spatial and non-spatial symmetries [1] and various topological materials have been discovered [2]. With these results in hand, attention is now pivoting towards the microscopic characterization of topological materials, the investigation of topological hybrid structures, and the study of topological responses. The concept of topological band theory is now being applied also to synthetic materials, e.g., ultracold atoms, photonic crystals, and electric circuit networks [3]. These synthetic systems have the advantage that they can be controlled and manipulated with high precision. In addition, they can readily accommodate non-equilibrium perturbations, leading to non-Hermitian physics [4].

The aim of these undergraduate research projects is to (i) study and characterize topological materials with nodal planes, (ii) design synthetic materials with nontrivial topology, (iii) investigate light-induced topological states, i.e., Floquet topological phases, and (iv) study superconducting hybrid structures. While these projects mainly aim at the design and fundamental understanding of new topological systems, the insights gained may have important consequences for applications, for example, spintronic, thermoelectric, and superconducting devices, which can be utilized for future quantum information technologies or for energy-efficient post-silicon electronics.

**(i) Topological metals with nodal planes**. – Topological metals and semi-metals exhibit band crossings near the Fermi energy, which are protected by the non-trivial topological character of the wave functions. These topological band degeneracies are protected against perturbations and give rise to exotic surface states and unusual magneto-transport properties. In this project we will study materials with topological nodal planes. The band structure of these materials exhibit topological band degeneracies on entire planes, whose existence is enforced by a combination of time-reversal symmetry with screw rotation symmetries [5]. We will study and characterize the exotic phenomena associated with the nontrivial topology of these nodal planes, in particular, the Fermi arcs at the surface, the anomalous Hall currents in the bulk, as well as photo-galvanic and magneto-electric effects.

**(ii) Synthetic materials with nontrivial topology**. – Synthetic lattices, which can be realized, for example, in photonic crystals or electric circuits, can have topological properties which are highly tunable. Moreover they can be brought out-of-equilibrium in a controlled manner, e.g., by particle gain and loss. In this project we will use non-Hermitian Hamiltonians to describe the non-equilibrium properties of these synthetic lattices [3]. We will investigate their topological properties and determine how they are protected by lattice symmetries. Particularly interesting will be the topology of exceptional points, i.e., band degeneracies at which two or more eigenstates become identical, which we will characterize using the discriminant number [6]. Possible applications in photonic crystals will be discussed, including topological lasing and sensor devices with ultrahigh sensitivity.

**(iii) Floquet topological phases.** – Optical driving is an exquisite tool to engineer and control topological properties of Dirac materials and other semimetals [7]. In this project, we will use circularly polarized laser light to modulate the band topologies and transport characteristics of various types of semimetals. We will compute the Floquet band structures and laser-induced anomalous Hall currents. Since the periodic driving can be viewed as an extra dimension, it is possible to realize four-dimensional topological states using Floquet systems. We will compute the surface states of these four-dimensional topological systems and find new ways to probe the surface states in experiments.

**(iv) Superconducting hybrid structures.** – In order to create the desired topological state it is often necessary to combine several materials in a hybrid structure. For example, a p-wave topological superconductor can be engineered in a hybrid structure at the interface between a hight-Tc superconductor and a semiconductor [8]. In this project we will investigate the Josephson currents, the differential conductance, and the noise spectroscopy of superconducting hybrid structures. A particular focus will be on superconductors interfaced with persistent-spin-helix materials [8]. We will investigate how the topological bound states (i.e., the Majorana bound states) affect the transport and noise properties, and how these can be manipulated with an external magnetic field.

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

[2] P. Liu, J. R. Williams, and J. J. Cha, Nature Rev. Mat. **7**, 479 (2019).

[3] Y. Rui, Y. X. Zhao, and A. P. Schnyder, Nat. Sci. Rev. **7**, 1288 (2020).

[4] W. B. Rui, Y. X. Zhao, and A. P. Schnyder, Phys. Rev. B **99**, 241110(R) (2019).

[5] M. A. Wilde *et al.*, Nature **594**, 374 (2021).

[6] Z. Yang, A. P. Schnyder, J. Hu, and C.-K. Chiu, Phys. Rev. Lett. **126**, 086401 (2021).

[7] M. S. Rudner and N. H. Lindner, Nature Rev. Phys. **2**, 229 (2020).

[8] S. Ikegaya, J. Lee, A. P. Schnyder, Y. Asano, Phys. Rev. B **104**, L020502 (2021).