**Project leader: Teemu Ojanen **Personnel

##### Background

The notion of symmetry has traditionally served as a guiding principle in understanding ordered phases of matter. In the last two decades, topology has been recognized as complementary but equally fundamental concept. Topological properties of quantum many-body systems have remarkably far-reaching consequences on observable properties on macroscopic scale. For example, topological properties give rise to quantized responses, dissipationless currents, novel metallic states and exotic emergent quantum particles in condensed matter systems.

The topological matter research concentrates on predicting, fabricating and characterizing novel phases of matter and exploring their application potential in the future technology. The low-energy properties of topological quantum materials often manifest high-energy physics phenomena. Furthermore, condensed matter systems offer very promising platforms to explore quantum particles beyond the boson-fermion paradigm with non-abelian exchange statistics.

##### Research Plan

*Topological matter in random geometry*

Crystalline systems exhibit high-degree of spatial symmetry. Time-reversal invariant spatial symmetry protected topological states have been completely classified recently. Moreover, the general classification of noninteracting topological phases is approaching completion. In contrast, the topological properties of systems with amorphous or random geometry is largely unexplored. The geometric randomness is fundamentally different from the usual notion of disorder which is pervasive in condensed matter systems. Recently we have established the scaling theory of amorphous topological systems and proved that, indeed, amorphous systems support well-defined topological phases in the thermodynamic limit. In the next stages we will study their characteristic properties such as the critical transport and extract the critical exponents of various topological phase transitions. This work will require application of scaling ideas of the Anderson localization and percolation theory, elements of quantum transport theory and numerical evaluation of configurational averages of topological indices and physical observables. Another major goal in this area is to establish the topological properties of amorphous lattices in magnetic field. On regular lattices the magnetic spectrum reflects the fractal Chern number structure known as the Hofstadter butterfly. In the weak field limit the Hofstadter spectrum reduces to the celebrated Landau levels that each carry a unit Chern number. We will study the outstanding open problem of spectral topology of random lattices in magnetic field.

*Topological superconductivity in magnetic hybrid systems*

Hybrid systems combining materials with magnetization and superconductivity offer a promising route towards topological superconductivity. This pursuit illustrate the modern approach to condensed-matter physics: theorists can design novel phases of matter that will be subsequently fabricated by combining different elements and materials in the lab by experimentalist. We are particularly interested in 2d topological superconductors characterized by finite Chern number and chiral Majorana edge modes. The promising platform for this phase is provided by quasi-2d material NbSe_2 in contact with magnetic thin film material. Our work is motivated by the work of our experimental collaborators.

* **Curved-space physics in inhomogeneous Weyl semimetals*

Weyl semimetals are materials where the low-energy excitations are described by massless chiral fermions. In translationally invariant systems this gives rise to an emergent Lorentz symmetry. In our recent work we established that inhomogeneities due to smoothly varying material parameters give rise to effective curved-spacetime geometry. More precisely, the effective low-energy theory describes chiral fermions in torsional spacetimes with non-flat metric. The consequences of torsional geometry to observables will likely give rise to a class of novel phenomena. We will investigate this question in detail and propose experimental setups where the predicted phenomena could be measured.

** ****Collaboration**

** **The main body of work is carried out in the *Quantum Matter- *group at Tampere University. In the random topological systems we will collaborate with Jens Bardarson (KTH Stockholm) and Emil Bergholtz (Stockholm University). We are collaborating with the group of Timo Hyart (Warsaw) on machine learning methods to identify topological phases. Our work on topological superconductors are motivated by the experiments in Peter Liljeroth’s group at Aalto University. Also other collaborations with Aalto groups are in planning. On Weyl semimetals we will collaborate with Long Liang (Nordita).