A scattering matrix formulation of the topological index of interacting fermions in one-dimensional superconductors

We construct a scattering matrix formulation for the topological classification of one-dimensional superconductors with effective time reversal symmetry in the presence of interactions. For a closed geometry, Fidkowski and Kitaev have shown that such systems have a $\mathbb{Z}_8$ topological classification. We show that in the weak coupling limit, these systems retain a unitary scattering matrix at zero temperature, with a topological index given by the trace of the Andreev reflection matrix, \mbox{tr}\, r_{\rm he}. With interactions, \mbox{tr}\, r_{\rm he} generically takes on the finite set of values $0$, $\pm 1$, $\pm 2$, $\pm 3$, and $\pm 4$. We show that the two topologically equivalent phases with \mbox{tr}\, r_{\rm he} = \pm 4 support emergent many-body end states, which we identify to be a topologically protected Kondo-like resonance. The path in phase space that connects these equivalent phases crosses a non-fermi liquid fixed point where a multiple channel Kondo effect develops. Our results connect the topological index to transport properties, thereby highlighting the experimental signatures of interacting topological phases in one dimension.Comment: 4 pages, 1 fi

Multiple Particle Scattering in Quantum Point Contacts

Recent experiments performed on weakly pinched quantum point contacts, have shown a resistance that tend to decrease at low source drain voltage. We show that enhanced Coulomb interactions, prompt by the presence of the point contact, may lead to anomalously large multiple-particle scattering at finite bias voltage. These processes tend to decrease at low voltage, and thus may account for the observed reduction of the resistance. We concentrate on the case of a normal point contact, and model it by a spinfull interacting Tomonaga-Luttinger liquid, with a single impurity, connected to non interacting leads. We find that sufficiently strong Coulomb interactions enhance two-electron scattering, so as these dominate the conductance. Our calculation shows that the effective charge, probed by the shot noise of such a system, approaches a value proportional to e* = 2e at sufficiently large backscattering current. This distinctive hallmark may be tested experimentally. We discuss possible applications of this model to experiments conducted on Hall bars.Comment: 5 pages, 2 figure

Low-energy subgap states in multichannel p-wave superconducting wires

One-dimensional p-wave superconductors are known to harbor Majorana bound states at their ends. Superconducting wires with a finite width W may have fermionic subgap states in addition to possible Majorana end states. While they do not necessarily inhibit the use of Majorana end states for topological computation, these subgap states can obscure the identification of a topological phase through a density-of-states measurement. We present two simple models to describe low-energy fermionic subgap states. If the wire's width W is much smaller than the superconductor coherence length ξ, the relevant subgap states are localized near the ends of the wire and cluster near zero energy, whereas the lowest-energy subgap states are delocalized if W≳ξ. Notably, the energy of the lowest-lying fermionic subgap state (if present at all) has a maximum for W∼ξ

Superconductor insulator transition in thin films driven by an orbital parallel magnetic field effect

We study theoretically orbital effects of a parallel magnetic field applied to a disordered superconducting film. We find that the field reduces the phase stiffness and leads to strong quantum phase fluctuations driving the system into an insulating behavior. This microscopic model shows that the critical field decreases with the sheet resistance, in agreement with recent experimental results. The predictions of this model can be used to discriminate spin and orbital effects. We find that experiments conducted by A. Johansson \textit{et al.} are more consistent with the orbital mechanism.Comment: 4 pages, 2 figure

A topological classification of interaction-driven spin pumps

When adiabatically varied in time, certain one-dimensional band insulators allow for the quantized noiseless pumping of spin even in the presence of strong spin orbit scattering. These spin pumps are closely related to the quantum spin Hall system, and their properties are protected by a time-reversal restriction on the pumping cycle. In this paper we study pumps formed of one-dimensional insulators with a time-reversal restriction on the pumping cycle and a bulk energy gap which arises due to interactions. We find that the correlated gapped phase can lead to novel pumping properties. In particular, systems with $d$ different ground states can give rise to $d+1$ different classes of spin pumps, including a trivial class which does not pump quantized spin and $d$ non-trivial classes allowing for the pumping of quantized spin $\hbar/n$ on average per cycle, where $1\leq n\leq d$. We discuss an example of a spin pump that transfers on average spin $\hbar/2$ without transferring charge.Comment: 5 pages, 2 figure

Transport properties and signatures of an emergent two-dimensional weak topological phase

We study a one-dimensional chain of 2N Majorana bound states, which interact through a local quartic interaction. This model describes for example the edge physics of a quasi-one-dimensional (1D) stack of 2N Kitaev chains with modified time-reversal symmetry TγiT−1=γi, which precludes the presence of quadratic coupling. The ground state of our 1D Majorana chain displays a fourfold periodicity in N, corresponding to the four distinct topological classes of the stacked Kitaev chains. We analyze the transport properties of the 1D Majorana chain, when probed by local conductors located at its ends. We find that for finite but large N, the scattering matrix partially reflects the fourfold periodicity, and the chain exhibits strikingly different transport properties for different chain lengths. In the thermodynamic limit, the 1D Majorana chain hosts a robust many-body zero mode, which indicates that the corresponding stacked two-dimensional bulk system realizes a weak topological phase

Relation between scattering matrix topological invariants and conductance in Floquet Majorana systems

We analyze the conductance of a one-dimensional topological superconductor periodically driven to host Floquet Majorana zero-modes for different configurations of coupling to external leads. We compare the conductance of constantly coupled leads, as in standard transport experiments, with the stroboscopic conductance of pulsed coupling to leads used to identify a scattering matrix topological index for periodically driven systems. We find that the sum of DC conductance at voltages multiples of the driving frequency is quantitatively close to the stroboscopic conductance at all voltage biases. This is consistent with previous work which indicated that the summed conductance at zero/pi resonance is quantized. We quantify the difference between the two in terms of the width of their respective resonances and analyze that difference for two different stroboscopic driving protocols of the Kitaev chain. While the quantitative differences are protocol-dependent, we find that generically the discrepancy is larger when the zero mode weight at the end of the chain depends strongly on the offset time between the driving cycle and the pulsed coupling period

Sharp Superconductor-Insulator Transition in Short Wires

Recent experiments on short MoGe nanowires show a sharp superconductor-insulator transition tuned by the normal state resistance of the wire, with a critical resistance of $R_c\approx R_Q= h/(4e^2)$. These results are at odds with a broad range of theoretical work on Josephson-like systems that predicts a smooth transition, tuned by the value of the resistance that shunts the junction. We develop a self-consistent renormalization group treatment of interacting phase-slips and their dual counterparts, correlated cooper pair tunneling, beyond the dilute approximation. This analysis leads to a very sharp transition with a critical resistance of $R_Q$. The addition of the quasi-particles' resistance at finite temperature leads to a quantitative agreement with the experimental results. This self-consistent renormalization group method should also be applicable to other physical systems that can be mapped onto similar sine-Gordon models, in the previously inaccessible intermediate-coupling regime.Comment: 11 pages, 5 figures. Contribution to the proceedings of "Fluctuations and phase transitions in superconductors", Nazareth Ilit, Israel, 2007. To be published in Physica C, vol. 46

Topologically protected heat pumping from braiding Majorana zero modes

Majorana zero modes are non-Abelian quasiparticles that emerge on the edges of topological phases of superconductors. Evidence of their presence has been reported in transport measurements on engineered superconducting-based nanostructures. In this manuscript we identify signatures of a topologically protected dynamical manipulation of Majorana zero modes via continuous transport measurement during the manipulation. Specifically, we show that, in a two-terminal geometry, the heat pumped across the terminals at low temperature and voltage bias is characterized by a universal value. We show that this feature is inherent to the presence of Majorana zero modes and discuss its robustness against temperature, voltage bias and the detailed coupling to the contacts