# Recent research activities

Condensed matter theory is currently moving from its traditional role of modelling and interpreting existing matter to an active design and control of physical many-body quantum processes with desired physical properties. This is a strong requirement for the quantum technologies that will shape our future.

Luckily this is also accompanied by many new challenges of fundamental character that, since they are different from the traditional questions in physics, will lead to a leap forward in our basic understanding of quantum matter. Our research is focused on this aspect: it is driven only by curiosity, but gently directed towards the needs of future technology.

## New physics with topological superconductors and Majorana states

### Kondorana

Since the theoretical description of the Mott insulator in 1937, it has become clear that electron-electron interactions can lead to behaviour of a condensed matter system that is entirely different from the non-interacting case. A prominent example of interaction-dominated physics is the Kondo Effect, in which an isolated magnetic moment is screened by a complicated modification of the state of nearby conducting electrons. This has become something of a poster child for Philip Anderson's assertion that "more is different", i.e. that a system comprised of simple, well understood, elements can exhibit non-trivial collective behaviour that is of a fundamentally new nature.

In recent years, there has been growing interest in Majorana zero modes in condensed matter, in no small part due to their potential applications in the field of topologically protected quantum computing. Like the Kondo Effect, these Majorana modes are a many-body phenomenon that results from the collective behaviour of an electronic system: for the Kondo Effect a normal electronic conductor, and for the Majorana modes a topological superconductor. The fact that Majorana zero modes can become localised in a similar way to a magnetic moment leads to the question of whether or not a Majorana system can be tuned to exhibit Kondo type physics.

It turns out that a most interesting situation arises in the system shown to the right.
Two isolated Majorana states arise at the ends of a specially tuned quantum wire that becomes a
topological superconductor (TSC) through its contact with a regular superconductor (SC).
Since the system is ungrounded, it acts like a capacitor with a charging energy.
The metal gate behind the system (or a backgate) can then be tuned such that the Majorana states together
form a low-energy two-level system similar to a magnetic moment. Taking into account the influence of
high-energy interaction effects, we have demonstrated that one may realise an analogue to the Kondo
model using Majorana modes. However, in contrast to the Kondo system, in which the renormalisation
by the interactions leads either to a strongly or a very weakly correlated phase, we find an unusual
intermediate behaviour. The result indeed indicates the existence of a novel many-body state that
extends across metallic, Majorana and superconducting electrons, and we call this behaviour
**Kondorana** physics. In the reference given below, we provide details of the analysis and show
that this state should be testable by direct transport measurements.

#### Further reading

I. J. van Beek, B. Braunecker

Phys. Rev. B

**94**, 115416 (2016) [arXiv:1606.08634] [PDF]

## Self-ordered phases of electrons and magnetic moments

The interaction between localised magnetic moments and conduction electrons characterises a vast number of modern materials. It is on the basis of nuclear magnetism, magnetic semiconductors, and heavy fermion materials of the Kondo-lattice type. Traditionally, ordered phases in such systems can be separated into two classes. First the class, in which localised moments and the electrons form a joint correlated state, such as in the Kondo lattice systems at temperatures below the Kondo temperature. Second the class of the type of nuclear magnets in 3D metals, in which nuclear spins order due to the presence of electrons, yet the electrons themselves are unaffected by the nuclear spins.

Recently we have added an intermediate class, in which nuclear spins and electrons order individually but are tightly bound together through a self-consistent feedback mechanism. In this case, the nuclear spins order due to their effective interaction with the electrons, but do not form a coherent correlated state with them. Yet through their ordering they generate a magnetic potential that acts back on the electrons and changes the system properties of the latter as well. In turn, this causes a further stabilisation of the nuclear spin order. This mutual influence describes the feedback mechanism, and we have shown that electron-electron interactions play a decisive role in stabilising this joint ordered phase. The figure next to this text illustrates this process (with a few more specialised keywords) for one-dimensional conductors.

These results are the first examples of new self-organised phases that can emerge from the interaction between localised magnetic moments and electrons. Many further aspects and the consequences of stronger or weaker interaction strengths than considered in the examples above remain to be explored. The list of references below gives a hint of what can be done. The description of the Kondorana physics above provides another example of the new physics that can arise in similar systems.

#### Further reading

B. Braunecker, P. Simon, and D. Loss

Phys. Rev. B

**80**, 165119 (2009) [arXiv:0908.0904] [PDF]

^{13}C and GaAs-based quantum wires. In these systems the hyperfine interaction between the nuclear spin and the conduction electron spin is very weak, yet it triggers a strong feedback reaction that results in an ordered phase consisting of a nuclear helimagnet that is inseparably bound to an electronic density wave combining charge and spin degrees of freedom. This effect can be interpreted as a strong renormalisation of the nuclear Overhauser field and is a unique signature of Luttinger liquid physics. Through the feedback the order persists up into the millikelvin range. A particular signature is the reduction of the electric conductance by the universal factor 2.

^{13}C Nanotubes

B. Braunecker, P. Simon, and D. Loss

Phys. Rev. Lett.

**102**, 116403 (2009) [arXiv:0808.1685] [PDF]

^{13}C form an ideal system to study the effect of electron interaction on nuclear magnetism in one dimension. If the electrons are in the metallic, Luttinger liquid regime,we show that even a very weak hyperfine coupling to the

^{13}C nuclear spins has a striking effect: The system is driven into an ordered phase, which combines electron and nuclear degrees of freedom, and which persists up into the millikelvin range. In this phase the conductance is reduced by a universal factor of 2, allowing for detection by standard transport experiments.

B. Braunecker, G. I. Japaridze, J. Klinovaja, and D. Loss

Phys. Rev. B

**82**, 045127 (2010) [arXiv:1004.0467] [PDF]

B. Braunecker and P. Simon

Phys. Rev. B

**92**, 241410(R) (2015) [arXiv:1510.06339] [PDF]

## Entanglement generation and detection in nanostructures

The controlled generation and detection of entanglement is necessary if we want to use a quantum state for application. In a nanostructure, this task typically refers to controlling the entanglement of selected pairs of electrons. Such pairs must be generated by a source, for which a promising candidate is a Cooper pair splitter. The latter is a system as shown in the figure to the right, consisting of a superconductor that naturally contains pairs of spin-entangled electrons in the form of Cooper pairs. The superconductor is connected to two quantum dots, and the injection of a Cooper pair (hourglass shape in the figure) into the double quantum dot system is tuned such that preferably one electron of the pair is injected into the left dot, and the other electron into the right dot. Such split pairs supposedly maintain their entanglement, yet a direct proof by an experiment that this is indeed the case is very challenging and could not be realised so far.

To overcome the difficulties inhibiting the progress so far, we have
suggested a new setup of a Cooper pair splitter, in which spin filters are
placed directly at the superconductor (and we showed that this is
achievable actually with ordinary carbon nanotubes as sketched in the figure).
We demonstrated that a proof of entanglement can be given by standard
transport measurements of the conductances of the Cooper pair splitter
without the requirement of measuring noise correlators.
Necessary instead is some tunability of the spin filter settings, which for
the carbon nanotubes is an intrinsic property, accessible by applying a constant
magnetic field *B* and controlling resonant electron transport by
quantum dot side gates (see the figure).

#### Further reading

B. Braunecker, P. Burset, and A. Levy Yeyati

Phys. Rev. Lett.

**111**, 136806 (2013) [arXiv:1303.6196] [PDF] [Supplement]

M. C. Hels, B. Braunecker, K. Grove-Rasmussen, J. Nygård

Phys. Rev. Lett.

**117**, 276802 (2016) [arXiv:1606.01065] [PDF]