One-dimensional quantum systems have the extraordinary property that their low-energy degrees of freedom can be described (almost) exactly, even in the presence of interactions, using the framework of bosonization and Luttinger liquid theory. This theory allows the calculation of thermodynamic properties and correlation functions, and its predictions have been verified in numerous experiments.

The only approximation within Luttinger liquid theory is the assumption that the single-particle spectrum is strictly linear. In contrast, non-linearities of the spectrum lead to corrections which become increasingly important at higher energies, and which make an exact solution impossible.

As important examples, such non-linearities lead to notable modifications of the dynamic correlation functions and also introduce qualitatively new effects, like relaxation and thermalization, which are entirely absent in Luttinger liquid theory. We study those effects which require extending Luttinger liquid theory in various directions.

Recently, we have focused in particular on one-dimensional systems with Rashba spin-orbit coupling, which is present for instance in indium arsenide or indium antimonide quantum wires. When coupled to superconductors, these wires have become an important resource for engineering topological states, such as Majorana bound states or parafermionic states which could have important applications in quantum computing.

**References:**

- Thomas L. Schmidt and Giacomo Dolcetto and Christopher J. Pedder and Karyn Le Hur and Peter P. Orth,
*Mechanical resonances of mobile impurities in a one-dimensional quantum fluid*, Phys. Rev. Lett.**123**, 075302 (2019) - Christopher J. Pedder and Tobias Meng and Rakesh Tiwari and Thomas L. Schmidt,
*Missing Shapiro steps and the $8π$-periodic Josephson effect in interacting helical electron systems*, Phys. Rev. B**96**, 165429 (2017) - Pedder, Christopher J. and Meng, Tobias and Tiwari, Rakesh P. and Schmidt, Thomas L.,
*Dynamic response functions and helical gaps in interacting Rashba nanowires with and without magnetic fields*, Phys. Rev. B**94**, 245414 (2016)

Majorana fermions have been theoretically predicted more than 70 years ago, but whether they exist as fundamental particles remains an open question to this day. The prediction and subsequent discovery of Majorana fermions as quasiparticles in certain solid-state systems has therefore sparked a flurry of research activity. In such systems, Majorana bound states (MBS) emerge due to the interplay of superconductivity and spin-orbit interaction.

An exciting prospect is the use of MBS as building blocks for topological qubits, on which certain operations can be performed in a decoherence-free way using “braiding”, i.e., spatially moving particles around each other. However, for MBS these protected operations are insufficient to allow for universal quantum computing, and other methods to manipulate MBS need to be developed. For this purpose, we investigated the coupling between MBS and microwave photons and showed that MBS can be efficiently manipulated in microwave cavities.

Parafermionic bound states are generalizations of MBS which occur in strongly correlated systems, such as fractional quantum Hall states. They are more suitable for topological quantum computation than MBS because their braiding allows additional protected operations. We explore ways to experimentally realize such parafermionic states and how to use them for quantum computation.

**References:**

- Thomas L. Schmidt,
*Bosonization for fermions and parafermions*, arXiv:1907.11413 [cond-mat.mes-hall] (2019) - Groenendijk, Solofo and Calzona, Alessio and Tschirhart, Hugo and Idrisov, Edvin G. and Schmidt, Thomas L.,
*Parafermion braiding in fractional quantum Hall edge states with a finite chemical potential*, Phys. Rev. B**100**, 205424 (2019)

Topological insulators are systems which behave like band insulators in the bulk, but have gapless, metallic states on their surfaces. It was theoretically predicted and experimentally confirmed that electron backscattering in these surface states is strongly suppressed, making these systems interesting for low-dissipation electronic devices. Moreover, they hold great promise in the field of spintronics. In topological insulators, the orbital motion of electrons is strongly correlated with their spin, as electrons carrying opposite spins generally propagate in opposite directions in these surface states.

Topological insulators could have important applications in nanoelectronics, and indeeed recent discoveries of new materials brings the aim closer to exploit their effects even at room temperatures. In this context, our main interests are (i) to study more in detail their properties as electronic conductors, and (ii) to learn how their nontrivial spin properties affect their response to electromagnetic radiation.

**References:**

- Moors, Kristof and Schüffelgen, Peter and Rosenbach, Daniel and Schmitt, Tobias and Schäpers, Thomas and Schmidt, Thomas L.,
*Magnetotransport signatures of three-dimensional topological insulator nanostructures*, Phys. Rev. B**97**, 245429 (2018)

Quantum dots are extremely useful tools for studying quantum effects at the mesoscopic scale. Due to their small sizes, electronic transport usually involves only a small number of electronic degrees of freedom. In particular, this makes it possible to use them experimentally to study fundamental quantum effects in a controlled environment. We are most interested in the effects arising from time-dependent perturbations of the system as well as from the presence of nonequilibrium environments.

In recent years, we have worked in particular on quantum dots with electron-phonon coupling, where the electronic level energy is modulated by mechanical vibrations of the quantum dot. Such a situation can be realized for instance in transport through single molecules. We found that in this case, the response to external stimuli is very strong due to an effect known as Franck-Condon blockade.

Beyond purely electronic transport, we also study transport in other thermodynamic quantities, such as heat, work, and entropy. Quantum thermodynamics is still a relatively new field whose aim it is to extend the concepts of nonequilibrium thermodynamics to open quantum systems. In particular, our aim is to understand the basic laws of thermodynamics in situations where the coupling between system and environment is not small, a condition which can be easily realized in quantum dots.

**References:**

- Haughian, Patrick and Esposito, Massimiliano and Schmidt, Thomas L.,
*Quantum thermodynamics of the resonant-level model with driven system-bath coupling*, Phys. Rev. B**97**, 085435 (2018)