arXiv subjects

Karsten Flensberg

Publications and source records attributed to Karsten Flensberg.

At least 73 records · Page 4Linked to original sources

arXiv2016

Signatures of Majorana Kramers pairs in superconductor-Luttinger liquid and superconductor-quantum dot-normal lead junctions

Time-reversal invariant topological superconductors are characterized by the presence of Majorana Kramers pairs localized at defects. One of the transport signatures of Majorana Kramers pairs is the quantized differential conductance of $4e^2/h$ when such a one-dimensional superconductor is coupled to a normal-metal lead. The resonant Andreev reflection, responsible for this phenomenon, can be understood as the boundary condition change for lead electrons at low energies. In this paper, we study the stability of the Andreev reflection fixed point with respect to electron-electron interactions in the Luttinger liquid. We first calculate the phase diagram for the Luttinger liquid-Majorana Kramers pair junction and show that its low-energy properties are determined by Andreev reflection scattering processes in the spin-triplet channel, i.e. the corresponding Andreev boundary conditions are similar to that in a spin-triplet superconductor - normal lead junction. We also study here a quantum dot coupled to a normal lead and a Majorana Kramers pair and investigate the effect of local repulsive interactions leading to an interplay between Kondo and Majorana correlations. Using a combination of renormalization group analysis and slave-boson mean-field theory, we show that the system flows to a new fixed point which is controlled by the Majorana interaction rather than the Kondo coupling. This Majorana fixed point is characterized by correlations between the localized spin and the fermion parity of each spin sector of the topological superconductor. We investigate the stability of the Majorana phase with respect to Gaussian fluctuations.

cond-mat.mes-hall↗

arXiv2016

Quantum charge fluctuations of a proximitized nanowire

Motivated by recent experiment, we consider charging of a nanowire which is proximitized by a superconductor and connected to a normal-state lead by a single-channel junction. The charge $Q$ of the nanowire is controlled by gate voltage $e{\cal N}_g/C$. A finite conductance of the contact allows for quantum charge fluctuations, making the function $Q(\mathcal{N}_g)$ continuous. It depends on the relation between the superconducting gap $Δ$ and the effective charging energy $E^*_C$. The latter is determined by the junction conductance, in addition to the geometrical capacitance of the proximitized nanowire. We investigate $Q(\mathcal{N}_g)$ at zero magnetic field $B$, and at fields exceeding the critical value $B_c$ corresponding to the topological phase transition. Unlike the case of $Δ= 0$, the function $Q(\mathcal{N}_g)$ is analytic even in the limit of negligible level spacing in the nanowire. At $B=0$ and $Δ>E^*_C$, the maxima of $dQ/d\mathcal{N}_g$ are smeared by $2e$-fluctuations described by a single-channel "charge Kondo" physics, while the $B=0$, $Δ<E^*_C$ case is described by a crossover between the Kondo and mixed-valence regimes of the Anderson impurity model. In the topological phase, $Q(\mathcal{N}_g)$ is analytic function of the gate voltage with $e$-periodic steps. In the weak tunneling limit, $dQ/d\mathcal{N}_g$ has peaks corresponding to Breit-Wigner resonances, whereas in the strong tunneling limit (i.e., small reflection amplitude $r$ ) these resonances are broadened, and $dQ/d\mathcal{N}_g-e \propto r\cos(2π\mathcal{N}_g)$.

cond-mat.mes-hall↗

arXiv2016

Time scales for Majorana manipulation using Coulomb blockade in gate-controlled superconducting nanowires

We numerically compute the low-energy spectrum of a gate-controlled superconducting topological nanowire segmented into two islands, each Josephson-coupled to a bulk superconductor. This device may host two pairs of Majorana bound states and could provide a platform for testing Majorana fusion rules. We analyze the crossover between (i) a charge-dominated regime utilizable for initialization and readout of Majorana bound states, (ii) a single-island regime for dominating inter-island Majorana coupling, (iii) a Josephson-plasmon regime for large coupling to the bulk superconductors, and (iv) a regime of four Majorana bound states allowing for topologically protected Majorana manipulations. From the energy spectrum, we derive conservative estimates for the time scales of a fusion-rule testing protocol proposed recently [arXiv:1511.05153]. We also analyze the steps needed for basic Majorana braiding operations in branched nanowire structures.

cond-mat.mes-hall↗

arXiv2016

Yu-Shiba-Rusinov states in phase-biased S-QD-S junctions

We study the effects of a phase difference on Yu-Shiba-Rusinov (YSR) states in a spinful Coulomb-blockaded quantum dot contacted by a superconducting loop. In the limit where charging energy is larger than the superconducting gap, we determine the subgap excitation spectrum, the corresponding supercurrent, and the differential conductance as measured by a normal-metal tunnel probe. In absence of a phase difference only one linear combination of the superconductor lead electrons couples to the spin, which gives a single YSR state. With finite phase difference, however, it is effectively a two-channel scattering problem and therefore an additional state emerges from the gap edge. The energy of the phase-dependent YSR states depend on the gate voltage and one state can cross zero energy twice inside the valley with odd occupancy. These crossings are shifted by the phase difference towards the charge degeneracy points, corresponding to larger exchange couplings. Moreover, the zero-energy crossings give rise to resonant peaks in the differential conductance with magnitude $4e^2/h$. Finally, we demonstrate that the quantum fluctuations of the dot spin do not alter qualitatively any of the results.

cond-mat.mes-hall↗

arXiv2016

Milestones toward Majorana-based quantum computing

We introduce a scheme for preparation, manipulation, and readout of Majorana zero modes in semiconducting wires with mesoscopic superconducting islands. Our approach synthesizes recent advances in materials growth with tools commonly used in quantum-dot experiments, including gate-control of tunnel barriers and Coulomb effects, charge sensing, and charge pumping. We outline a sequence of milestones interpolating between zero-mode detection and quantum computing that includes (1) detection of fusion rules for non-Abelian anyons using either proximal charge sensors or pumped current; (2) validation of a prototype topological qubit; and (3) demonstration of non-Abelian statistics by braiding in a branched geometry. The first two milestones require only a single wire with two islands, and additionally enable sensitive measurements of the system's excitation gap, quasiparticle poisoning rates, residual Majorana zero-mode splittings, and topological-qubit coherence times. These pre-braiding experiments can be adapted to other manipulation and readout schemes as well.

cond-mat.mes-hall↗

arXiv2016

Phase-tunable Majorana bound states in a topological N-SNS junction

We theoretically study the differential conductance of a one-dimensional normal-superconductor-normal-superconductor (N-SNS) junction with a phase bias applied between the two superconductors. We consider specifically a junction formed by a spin-orbit coupled semiconducting nanowire with regions of the nanowire having superconducting pairing induced by a bulk $s$-wave superconductor. When the nanowire is tuned into a topologically non-trivial phase by a Zeeman field, it hosts zero-energy Majorana modes at its ends as well as at the interface between the two superconductors. The phase-dependent splitting of the Majorana modes gives rise to features in the differential conductance that offer a clear distinction between the topologically trivial and non-trivial phases. We calculate the transport properties of the junction numerically and also present a simple analytical model that captures the main properties of the predicted tunneling spectroscopy.

cond-mat.mes-hall↗

arXiv2016

Effects of spin-orbit coupling and spatial symmetries on the Josephson current in SNS junctions

We present an analysis of the symmetries of the interference pattern of critical currents through a two-dimensional superconductor-semiconductor-superconductor junction, taking into account Rashba and Dresselhaus spin-orbit interaction, an arbitrarily oriented magnetic field, disorder, and structural asymmetries. We relate the symmetries of the pattern to the absence or presence of symmetries in the Hamiltonian, which provides a qualitative connection between easily measurable quantities and the spin-orbit coupling and other symmetries of the junction. We support our analysis with numerical calculations of the Josephson current based on a perturbative expansion up to eighth order in tunnel coupling between the normal region and the superconductors.

cond-mat.mes-hall↗

arXiv2015

Environmental Coulomb blockade of topological superconductor-normal metal junctions

We study charge transport of a topological superconductor connected to different electromagnetic environments using a low-energy description where only the Majorana bound states in the superconductor are included. Extending earlier findings who found a crossover between perfect Andreev reflection with conductance $2e^2/h$ to a regime with blocked transport when the resistance of the environment is larger than $2e^2/h$, we consider Majorana bound states coupled to metallic dots. in particular, we study two topological superconducting leads connected by a metallic quantum dot in both the weak tunneling and strong tunneling regimes. For weak tunneling, we project onto the most relevant charge states. For strong tunneling, we start from the Andreev fixed point and integrate out charge fluctuations which gives an effective low-energy model for the non-perturbative gate-voltage modulated cotunneling current. In both regimes and in contrast to cotunneling with normal leads, the conductance is temperature independent because of the resonant Andreev reflections, which are included to all orders.

cond-mat.mes-hall↗

arXiv2015

Nonlocal Damping of Helimagnets in One-Dimensional Interacting Electron Systems

We investigate the magnetization relaxation of a one-dimensional helimagnetic system coupled to interacting itinerant electrons. The relaxation is assumed to result from the emission of plasmons, the elementary excitations of the one-dimensional interacting electron system, caused by slow changes of the magnetization profile. This dissipation mechanism leads to a highly nonlocal form of magnetization damping that is strongly dependent on the electron-electron interaction. Forward scattering processes lead to a spatially constant damping kernel, while backscattering processes produce a spatially oscillating contribution. Due to the nonlocal damping, the thermal fluctuations become spatially correlated over the entire system. We estimate the characteristic magnetization relaxation times for magnetic quantum wires and nuclear helimagnets.

cond-mat.mes-hall↗

arXiv2015

Braiding properties of Majorana Kramers Pairs

We consider the braiding of Kramers pairs of Majorana bound states. We derive the most general transformation on the many-body ground state that is applied as the result of such a braiding process. The result is derived in the context of a simple toy model, but we will show that it has the most general form that is compatible with local and global conservation of electron parity. In accordance with earlier work the resulting transformation turns out to be path dependent, which shows that Kramers pairs of Majorana bound states cannot be used for topological quantum computation. We also discuss under which conditions the result is path independent and corresponds to two independent exchanges of pairs of Majorana bound states.

cond-mat.mes-hall↗

arXiv2015

Local Adiabatic Mixing of Kramers Pairs of Majorana Bound States

We consider Kramers pairs of Majorana bound states under adiabatic time evolution. This is important for the prospects of using such bound states as parity qubits. We show that local adiabatic perturbations can cause a rotation in the space spanned by the Kramers pair. Hence the quantum information is unprotected against local perturbations, in contrast to the case of single localized Majorana bound states in systems with broken time reversal symmetry. We give an analytical and a numerical example for such a rotation, and specify sufficient conditions under which a rotation is avoided. We give a general scheme for determining when these conditions are satisfied, and exemplify it with a general model of a quasi 1D time reversal symmetric topological superconductor.

cond-mat.mes-hall↗

arXiv2015

Interaction effects on proximity-induced superconductivity in semiconducting nanowires

We investigate the effect of electron-electron interactions on proximity-induced $s$-wave superconductivity in one-dimensional nanowires. We treat the interactions on a self-consistent mean-field level, and find an analytic expression for the effective pairing potential in the presence of interactions, valid for a weakly tunnel coupled wire. We show that for a set of two nanowires placed in parallel on a superconducting substrate, the interaction-induced reduction of the pairing energy could result in the effective interwire pairing potential exceeding the intrawire potential, which is one of the requirements for creating a time-reversal symmetric topological superconducting state in such a two-wire system.

cond-mat.mes-hall↗

arXiv2015

Spin-lattice order in one-dimensional conductors: beyond the RKKY effect

We investigate magnetic order in a lattice of classical spins coupled to an isotropic gas of one-dimensional (1d) conduction electrons via local exchange interactions. The frequently discussed Ruderman-Kittel-Kasuya-Yosida (RKKY) effective exchange model for this system predicts that spiral order is always preferred. Here we consider the problem nonperturbatively, and find that such order vanishes above a critical value of the exchange coupling that depends strongly on the lattice spacing. The critical coupling tends to zero as the lattice spacing becomes commensurate with the Fermi wave vector, signalling the breakdown of the perturbative RKKY picture, and spiral order, even at weak coupling. We provide the exact phase diagram for arbitrary exchange coupling and lattice spacing, and discuss its stability. Our results shed new light on the problem of utilizing a spiral spin-lattice state to drive a one-dimensional superconductor into a topological phase.

cond-mat.mes-hall↗

arXiv2014

Tunnel spectroscopy of Majorana bound states in topological superconductor-quantum dot Josephson junctions

We theoretically investigate electronic transport through a junction where a quantum dot (QD) is tunnel coupled on both sides to semiconductor nanowires with strong spin-orbit interaction and proximity-induced superconductivity. The results are presented as stability diagrams, i.e., the differential conductance as a function of the bias voltage applied across the junction and the gate voltage used to control the electrostatic potential on the QD. A small applied magnetic field splits and modifies the resonances due to the Zeeman splitting of the QD level. Above a critical field strength, Majorana bound states (MBS) appear at the interfaces between the two superconducting nanowires and the QD, resulting in a qualitative change of the entire stability diagram, suggesting this setup as a promising platform to identify MBS. Our calculations are based on a nonequilibrium Green's function description and is exact when Coulomb interactions on the QD can be neglected. In addition, we develop a simple pictorial view of the involved transport processes, which is equivalent to a description in terms of multiple Andreev reflections, but provides an alternative way to understand the role of the QD level in enhancing transport for certain gate and bias voltages. We believe that this description will be useful in future studies of interacting QDs coupled to superconducting leads (with or without MBS), where it can be used to develop a perturbation expansion in the tunnel coupling.

cond-mat.supr-con↗

arXiv2014

Designing $π$-stacked molecular structures to control heat transport through molecular junctions

We propose and analyze a new way of using $π$ stacking to design molecular junctions that either enhance or suppress a phononic heat current, but at the same time remain conductors for an electric current. Such functionality is highly desirable in thermoelectric energy converters, as well as in other electronic components where heat dissipation should be minimized or maximized. We suggest a molecular design consisting of two masses coupled to each other with one mass coupled to each lead. By having a small coupling (spring constant) between the masses, it is possible to either reduce, or perhaps more surprisingly enhance the phonon conductance. We investigate a simple model system to identify optimal parameter regimes and then use first principle calculations to extract model parameters for a number of specific molecular realizations, confirming that our proposal can indeed be realized using standard molecular building blocks.

cond-mat.mes-hall↗

arXiv2014

Majorana fermions in superconducting nanowires without spin-orbit coupling

We show that confined Majorana fermions can exist in nanowires with proximity induced s-wave superconducting pairing if the direction of an external magnetic field rotates along the wire. The system is equivalent to nanowires with Rashba-type spin-orbit coupling, with strength proportional to the derivative of the field angle. For realistic parameters, we demonstrate that a set of permanent magnets can bring a nearby nanowire into the topologically non-trivial phase with localized Majorana modes at its ends. Without the requirement of spin-orbit coupling this opens up for a new route for demonstration and design of Majorana fermion systems.

cond-mat.mes-hall↗

arXiv2014

Majorana bound states in two-channel time-reversal-symmetric nanowire systems

We consider time-reversal-symmetric two-channel semiconducting quantum wires proximity coupled to an s-wave superconductor. We analyze the requirements for a nontrivial topological phase and find that necessary conditions are 1) the determinant of the pairing matrix in channel space must be negative, 2) inversion symmetry must be broken, and 3) the two channels must have different spin-orbit couplings. The first condition can be implemented in semiconducting nanowire systems where interactions suppress intra-channel pairing, while the inversion symmetry can be broken by tuning the chemical potentials of the channels. For the case of collinear spin-orbit directions, we find a general expression for the topological invariant by block diagonalization into two blocks with chiral symmetry only. By projection to the low-energy sector, we solve for the zero modes explicitly and study the details of the gap closing, which in the general case happens at finite momenta.

cond-mat.mes-hall↗

arXiv2013

Coupling spin qubits via superconductors

We show how superconductors can be used to couple, initialize, and read out spatially separated spin qubits. When two single-electron quantum dots are tunnel coupled to the same superconductor, the singlet component of the two-electron state partially leaks into the superconductor via crossed Andreev reflection. This induces a gate-controlled singlet-triplet splitting which, with an appropriate superconductor geometry, remains large for dot separations within the superconducting coherence length. Furthermore, we show that when two double-dot singlet-triplet qubits are tunnel coupled to a superconductor with finite charging energy, crossed Andreev reflection enables a strong two-qubit coupling over distances much larger than the coherence length.

cond-mat.mes-hall↗