University of St Andrews

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 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.

The picture above shows on the left the main topics of our current research and on the right the goal to bring all these topics to use for the general idea of quantum information processing. The lines and dots indicate the connections between the different topics on which we have been working recently. A few examples of how this looks in practice are described below.

• Coherent many-body dynamics between systems and environments

  The biggest enemy of quantum computing is decoherence, the uncontrolled degrading of a well defined quantum superposition. This occurs because a quantum system is always embedded in a wider environment with a macroscopic number of degrees of freedom. The latter lead to a destructive interference for any interaction with the system and this feeds back as the phenomenon of decoherence.

  Since this acts as a drain of quantum information large efforts are made to isolate the system from the environment, to suppress environments or to use special driving protocols to undo some decoherence processes. But it is also interesting to ask how exactly the decoherence builds up, if we can use this phase to learn something about the system or the environment and if there is a way to use it for quantum information processing.

  One example is a spin in a metal as shown in the figure on the right. In its pure quantum form this corresponds to the Kondo model, beyond the Kondo regime this is the basis for magnetic resonance techniques in the solid state. Most properties of such a system are thus known since long. However, mostly left aside was the regime of very short time scales in which the spin and a part of the metallic environment have a joint coherent evolution, in which in particular coherent excitations in the metal act back on the spin dynamics. The coherent many-body effect of a local excitation, such as from a spin flip, on the environment runs under the name of orthogonality catastrophe or Fermi edge singularity and we have worked on extending the techniques to access this physics for various situations over many years. Very recently we have also set up an approach allowing us to systematically investigate the backaction on the spin as well. We are now working towards using these approaches as tools for gaining information on correlated systems and towards understanding how quantum information can propagate coherently through an otherwise incoherent environment.

  Further reading

  • Spatiotemporal Spread of Fermi-edge Singularity as Time Delayed Interaction and Impact on Time-dependent RKKY Type Coupling
    C. Jackson, B. Braunecker
    Phys. Rev. Res. 4, 013119 (2022)   [arXiv:2106.09059]   [PDF]
    Fermi-edge singularity and Anderson’s orthogonality catastrophe are paradigmatic examples of non- equilibrium many-body physics in conductors, appearing after a quench is created by the sudden change of a localized potential. We investigate if the signal carried by the quench can be used to transmit a long ranged interaction, reminiscent of the RKKY interaction, but with the inclusion of the full many-body propagation over space and time. We calculate the response of a conductor to two quenches induced by localized states at different times and locations. We show that building up and maintaining coherence between the localized states is possible only with finely tuned interaction between the localized states and the conductor. This puts bounds to the use of time controlled RKKY type interactions and may limit the speed at which some quantum gates could operate.
  • Coherent backaction between spins and an electronic bath: Non-Markovian dynamics and low temperature quantum thermodynamic electron cooling
    S. Matern, D. Loss, J. Klinovaja, B. Braunecker
    Phys. Rev. B 100, 134308 (2019)   [arXiv:1905.11422]   [PDF]
    We provide a versatile analytical framework for calculating the dynamics of a spin system in contact with a fermionic bath beyond the Markov approximation. The approach is based on a second-order expansion of the Nakajima-Zwanzig master equation but systematically includes all quantum coherent memory effects leading to non-Markovian dynamics. Our results describe, for the free induction decay, the full time range from the non-Markovian dynamics at short times, to the well-known exponential thermal decay at long times. We provide full analytic results for the entire time range using a bath of itinerant electrons as an archetype for universal quantum fluctuations. Furthermore, we propose a quantum thermodynamic scheme to employ the temperature insensitivity of the non-Markovian decay to transport heat out of the electron system and thus, by repeated reinitialization of a cluster of spins, to efficiently cool the electrons at very low temperatures.
  • Response of a Fermi gas to time-dependent perturbations: Riemann-Hilbert approach at non-zero temperatures
    B. Braunecker
    Phys. Rev. B 73, 075122 (2006)   [arXiv:cond-mat/0510680]   [PDF]
    
    We provide an exact finite temperature extension to the recently developed Riemann-Hilbert approach for the calculation of response functions in nonadiabatically perturbed (multichannel) Fermi gases. We give a precise definition of the finite temperature Riemann-Hilbert problem and show that it is equivalent to a zero temperature problem. Using this equivalence, we discuss the solution of the nonequilibrium Fermi-edge singularity problem at finite temperatures.
  • Fermi edge singularity in a non-equilibrium system
    B. Muzykantskii, N. d'Ambrumenil, and B. Braunecker
    Phys. Rev. Lett. 91, 266602 (2003)   [arXiv:cond-mat/0304583]   [PDF]
    
    We report exact nonperturbative results for the Fermi-edge singularity in the absorption spectrum of an out-of-equilibrium tunnel junction. We consider two metals with chemical potential difference V separated by a tunneling barrier containing a defect, which exists in one of two states. When it is in its excited state, tunneling through the otherwise impermeable barrier is possible. Our nonperturbative solution of this nonequilibrium many-body problem shows that, as well as extending below the equilibrium threshold, the line shape depends on the difference in the phase of the reflection amplitudes on the two sides of the barrier. These results have a surprisingly simple interpretation in terms of known results for the equilibrium case but with (in general complex-valued) combinations of elements of the scattering matrix replacing the equilibrium phase shifts.
  • On solutions of the nonequilibrium x-ray edge problem
    B. Braunecker
    Phys. Rev. B 68, 153104 (2003)   [arXiv:cond-mat/0211511]   [PDF]
    
    We rediscuss a nonequilibrium x-ray edge problem which in recent publications led to discrepancies between the results of the perturbative and of an extended Nozières-De Dominicis approach. We show that this problem results from an uncritical separation of momenta of the scattering potential, and we propose a corrected Nozières-De Dominicis solution.

• Designed topological phases

  Topological phases and topological excitations have become a topic of massive research activity over the last years. Topology refers here to a classification of quantum states that is not directly measurable through a local observable but which has consequences whenever a system of one topological class has an interface to a system with a different topological class. The most prominent interface effect is the appearance of conducting interface states between two topological insulators or topological superconductors.

  

  Based on the maxim of designing rather than discovering we are investigating how the interaction between two quantum systems can be tailored to provide new, and especially topological effects. An important insight is that if we start from a sufficiently complex but conventional quantum state, the topological properties can be changed by projecting out a part of the wave function. Such a projection can occur through the interaction between two systems and in the best case the result is robust in the sense that no fine tuning is required. An example is shown on the right in which nuclear spins in a one-dimensional conductor order in a spiral pattern due to their interaction with the electrons, but in turn scattering on this pattern causes the opening of a gap in the electron bands as well and what is left as conducting states is known as a helical conductor. The robustness comes in because the combined state is self-stabilising through the feedback mechanism indicated in the figure.

  Helical conductors are an important component to obtain further topological states. For instance, when a helical conductor is brought in contact with a superconductor, the Cooper pairs from the superconductor are projected onto one half of their wave function component when they tunnel into the conductor by proximity effect. The resulting induced superconductivity has then a changed topology class, and the interesting topological interface properties arise in the form of Majorana bound states at the ends of the conductor (the interface to the vacuum), as illustrated in the figure next. We are currently exploring how the underlying physics can be used further to obtain topological textures directly in the superconducting substrates.

  Further reading

  • Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. I. Continuum description
    C. J. F. Carroll, B. Braunecker
    Phys. Rev. B 104, 245133 (2021)   [arXiv:1709.06093]   [PDF]
    We consider subgap bands induced in a two-dimensional superconductor by a densely packed chain of magnetic moments with ferromagnetic or spiral alignments. We show that by contrast with sparsely packed chains a consistent description requires that all wavelengths are taken into account for the scattering at the magnetic moments. The resulting subgap states are a composition of Yu-Shiba-Rusinov-type states and magnetic scattering states, whose mixture becomes especially important to understand the nature and dimensional renormalization of gap closures for spiral magnetic alignments under increasing scattering strength, particularly as the spiral becomes commensurate with the Fermi wavelength. The results are fully analytic in the form of Green’s functions and provide the tools for further analysis of the properties of the subgap states.
  • Subgap states at ferromagnetic and spiral-ordered magnetic chains in two-dimensional superconductors. II. Topological classification
    C. J. F. Carroll, B. Braunecker
    Phys. Rev. B 104, 245134 (2021)   [arXiv:2108.05768]   [PDF]
    We investigate the topological classification of the subgap bands induced in a two-dimensional superconductor by a densely packed chain of magnetic moments with ferromagnetic or spiral alignments. The wave functions for these bands are composites of Yu-Shiba-Rusinov-type states and magnetic scattering states and have a significant spatial extension away from the magnetic moments. We show that this spatial structure prohibits a straightforward extraction of a Hamiltonian useful for the topological classification. To address the latter correctly, we construct a family of spatially varying topological Hamiltonians for the subgap bands adapted for the broken translational symmetry caused by the chain. The spatial dependence in particular captures the transition to the topologically trivial bulk phase when moving away from the chain by showing how this, necessarily discontinuous, transition can be understood from an alignment of zeros with poles of Green’s functions. Through the latter, the topological Hamiltonians reflect a characteristic found otherwise primarily in strongly interacting systems.
  • Self-stabilising temperature-driven crossover between topological and non-topological ordered phases in one-dimensional conductors
    B. Braunecker and P. Simon
    Phys. Rev. B 92, 241410(R) (2015) [arXiv:1510.06339] [PDF]
    We present a self-consistent analysis of the topological superconductivity arising from the interaction between self-ordered localised magnetic moments and electrons in one-dimensional conductors in contact with a superconductor. We show that due to a gain in entropy there exists a magnetically ordered yet non-topological phase at finite temperatures that is relevant for systems of magnetic adatom chains on a superconductor. Spin-orbit interaction is taken into account, and we show that it causes a modification of the magnetic order yet without affecting the topological properties.
  • Spin-selective Peierls transition in interacting one-dimensional conductors with spin-orbit interaction
    B. Braunecker, G. I. Japaridze, J. Klinovaja, and D. Loss
    Phys. Rev. B 82, 045127 (2010) [arXiv:1004.0467] [PDF]
    
    Interacting one-dimensional conductors with Rashba spin-orbit coupling are shown to exhibit a spin-selective Peierls-type transition into a mixed spin-charge-density-wave state. The transition leads to a gap for one-half of the conducting modes, which is strongly enhanced by electron-electron interactions. The other half of the modes remains in a strongly renormalised gapless state and conducts opposite spins in opposite directions, thus providing a perfect spin filter. The transition is driven by magnetic field and by spin-orbit interactions. As an example we show for semiconducting quantum wires and carbon nanotubes that the gap induced by weak magnetic fields or intrinsic spin-orbit interactions can get renormalised by 1 order of magnitude up to 10 - 30 K.
  • Nuclear magnetism and electron order in interacting one-dimensional conductors
    B. Braunecker, P. Simon, and D. Loss
    Phys. Rev. B 80, 165119 (2009) [arXiv:0908.0904] [PDF]
    
    The interaction between localised magnetic moments and the electrons of a one-dimensional conductor can lead to an ordered phase in which the magnetic moments and the electrons are tightly bound to each other. We show here that this occurs when a lattice of nuclear spins is embedded in a Luttinger liquid. Experimentally available examples of such a system are single wall carbon nanotubes grown entirely from ¹³C 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.
  • Nuclear Magnetism and Electronic Order in ¹³C Nanotubes
    B. Braunecker, P. Simon, and D. Loss
    Phys. Rev. Lett. 102, 116403 (2009) [arXiv:0808.1685] [PDF]
    
    Single wall carbon nanotubes grown entirely from ¹³C 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 ¹³C 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.

• New physics with topological superconductors

  One of the prime applications of topological quantum states will be with quantum computation, and much research worldwide is focused on this aspect, in addition to the question of how to obtain and manipulate topological phases. Less explored, and the topic of this branch of our research, is to use topological states as a pathway to new physical phenomena.

  For instance, a most interesting situation arises in the system shown to the right. A topological superconductor (TSC) with two Majorana bound states as consequence of its topological class is left ungrounded that it acts as a capacitor. In such a situation it is possible to tune the energy levels of the Majorana states to form a low-energy two-level system similar to a magnetic moment. This resembles the Kondo model which describes an isolated magnetic moment that is screened by a complicated modification of the state of nearby conducting electrons. The Kondo model been used as the archetype of strong correlation physics over many decades.

  A similar model is realised here with the 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, which from a merger of Kondo and Majorana we have named Kondorana. This state is a novel many-body state that extends across metallic, Majorana and superconducting electrons and complements the family of Kondo related models in a most curious way.

  In a follow-up work we have investigated a similar set-up under driven situations and could identify a rich dynamics brough in by the Majorana states. Through these examples it has become clear that using topological excitations as an interface to novel phenomena has a large potential as the next step in topological physics research. In addition it is a very fruitful playground to explore how complicated many-body physics can be set up step by step by adding further interactions and interfaces.

  Further reading

  • Non-Equilibrium Charge Dynamics in Majorana-Josephson Devices
    I. J. van Beek, A. Levy Yeyati, B. Braunecker
    Phys. Rev. B 98, 224502 (2018)   [arXiv:1809.04701]   [PDF]
    We investigate the impact of introducing Majorana bound states, formed by a proximitized semiconducting nanowire in the topological regime, into a current biased capacitive Josephson junction, thereby adding delocalized states below the superconducting gap. We find that this qualitatively changes the charge dynamics of the system, diminishing the role of Bloch oscillations and causing single-particle tunneling effects to dominate. We fully characterize the resulting charge dynamics and the associated voltage and current signals. Our work reveals a rich landscape of behaviors in both the static and time-varying driving modes. This can be directly attributed to the presence of Majorana bound states, which serve as a pathway for charge transport and enable nonequilibrium excitations of the Majorana-Josephson device.
  • Non-Kondo many-body physics in a Majorana-based Kondo type system
    I. J. van Beek, B. Braunecker
    Phys. Rev. B 94, 115416 (2016) [arXiv:1606.08634] [PDF]
    We carry out a theoretical analysis of a prototypical Majorana system, which demonstrates the existence of a Majorana-mediated many-body state and an associated intermediate low-energy fixed point. Starting from two Majorana bound states, hosted by a Coulomb-blockaded topological superconductor and each coupled to a separate lead, we derive an effective low-energy Hamiltonian, which displays a Kondo-like character. However, in contrast to the Kondo model which tends to a strong- or weak-coupling limit under renormalization, we show that this effective Hamiltonian scales to an intermediate fixed point, whose existence is contingent upon teleportation via the Majorana modes. We conclude by determining experimental signatures of this fixed point, as well as the exotic many-body state associated with it.

• 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

  • Non-collinear spin-orbit magnetic fields in a carbon nanotube double quantum dot
    M. C. Hels, B. Braunecker, K. Grove-Rasmussen, J. Nygård
    Phys. Rev. Lett. 117, 276802 (2016) [arXiv:1606.01065] [PDF]
    We demonstrate experimentally that noncollinear intrinsic spin-orbit magnetic fields can be realized in a curved carbon nanotube two-segment device. Each segment, analyzed in the quantum dot regime, shows near fourfold degenerate shell structure allowing for identification of the spin-orbit coupling and the angle between the two segments. Furthermore, we determine the four unique spin directions of the quantum states for specific shells and magnetic fields. This class of quantum dot systems is particularly interesting when combined with induced superconducting correlations as it may facilitate unconventional superconductivity and detection of Cooper pair entanglement. Our device comprises the necessary elements.
  • Entanglement detection from conductance measurements in carbon nanotube Cooper pair splitters
    B. Braunecker, P. Burset, and A. Levy Yeyati
    Phys. Rev. Lett. 111, 136806 (2013) [arXiv:1303.6196] [PDF] [Supplement]
    Spin-orbit interaction provides a spin filtering effect in carbon nanotube based Cooper pair splitters that allows us to determine spin correlators directly from current measurements. The spin filtering axes are tunable by a global external magnetic field. By a bending of the nanotube, the filtering axes on both sides of the Cooper pair splitter become sufficiently different that a test of entanglement of the injected Cooper pairs through a Bell-like inequality can be implemented. This implementation does not require noise measurements, supports imperfect splitting efficiency and disorder, and does not demand a full knowledge of the spin-orbit strength. Using a microscopic calculation we demonstrate that entanglement detection by violation of the Bell-like inequality is within the reach of current experimental setups.