A one-dimensional spin-orbit coupled nanowire with proximity-induced pairing from a nearby s-wave superconductor may be in a topological nontrivial state, in which it has a zero-energy Majorana bound state at each end. We find that the topological trivial phase may have fermionic end states with an exponentially small energy, if the confinement potential at the wire's ends is smooth. The possible existence of such near-zero-energy levels implies that the mere observation of a zero-bias peak in the tunneling conductance is not an exclusive signature of a topological superconducting phase, even in the ideal clean single channel limit.

Certain band insulators allow for the adiabatic pumping of quantized charge or spin for special time-dependences of the Hamiltonian. These "topological pumps" are closely related to two dimensional topological insulating phases of matter upon rolling the insulator up to a cylinder and threading it with a time dependent flux. In this article we extend the classification of topological pumps to the Wigner Dyson and chiral classes, coupled to multi-channel leads. The topological index distinguishing different topological classes is formulated in terms of the scattering matrix of the system. We argue that similar to topologically non-trivial insulators, topological pumps are characterized by the appearance of protected gapless end states during the course of a pumping cycle. We show that this property allows for the pumping of quantized charge or spin in the weak coupling limit. Our results may also be applied to two dimensional topological insulators, where they give a physically transparent interpretation of the topologically non-trivial phases in terms of scattering matrices. Topological insulating states of matter differ from regular band insulators by the fact that they support protected gapless surface states. Holding promise for numerous applications, this observation has considerably motivated the search for materials that realize such topological phases. The first experimental observation of a topologically nontrivial insulator dates back to the discovery of the quantum Hall effect 30 years ago [1,2]. The recent discovery of the quantum spin Hall effect [3][4][5], has lead to a full classification of insulators with topological order based on their underlying symmetries and spatial dimensions [6].The study of the quantum Hall effect has instigated numerous theoretical works dedicated to the understanding of its non-trivial topological nature. One particular appealing argument was introduced by Laughlin [7], who considered a pump formed by placing the two dimensional system on a cylinder and threading it with a time dependent magnetic flux. As the flux is varied periodically in time, an integer number of charges are transferred across the pump upon completing one cycle. This charge quantization is directly related to the quantized Hall conductance of the underlying Hall insulator [8][9][10][11][12][13][14].In this communication we extend Laughlin's considerations to pumps formed by two dimensional insulators belonging to the Wigner Dyson and the chiral classes. Based on their underlying symmetries, applying the above construction imposes a symmetry constraint on the pumping cycle. This allows for the classification of topological pumps in terms of invariants of their scattering matrix and gives rise to a physically transparent interpretation of the topologically non-trivial phases in terms of quantized pumping properties. Similarly to topologically non-trivial two-dimensional insulators, topologically nontrivial pumps are characterized by the appearance of gapless end states during the course of a pumping cy...

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. The search for Majorana fermions has attracted a great deal of interest in the last few years [1]. Notably their nonlocal properties and non-abelian braiding statistics make Majorana fermion systems attractive candidates for fault tolerant quantum computation [2][3][4]. The present wave of interest is driven by a number of proposals that suggest ways of realizing and manipulating Majorana states in solid state systems, most prominently interfaces of s-wave superconductors and topological insulators [5,6], half-metallic ferromagnets [7-9], or semiconductor films or wires [10][11][12], where the latter stand out because Majorana manipulation require a mere series of gate operations [13]. In all these proposals, the proximity coupling to the s-wave superconductor effectively turns the normal metal into a p-wave superconductor, which is well known to harbor Majorana fermions at its ends or edges [14][15][16][17].Majorana bound states at ends of what is effectively a pwave superconducting wire can be analyzed most straightforwardly if these wires are strictly one dimensional, with only a single propagating mode at the Fermi level in the absence of superconductivity [10,11]. Nevertheless, Majorana end states can also exist in a quasi one-dimensional geometry. The effect of multiple transverse channels, present in most realistic realizations, has been addressed in Refs. [12,[18][19][20][21]. Specifically one sees that a complex p-wave superconductor in a strip geometry undergoes a series of oscillatory quantum phase transitions between topologically trivial and topologically nontrivial phases (without and with Majorana end states, respectively) as the strip width W or chemical potential µ are varied. Both with and without Majorana end states, a range of subgap states is found [19], analogous to the sub-gap states in vortex cores of bulk superconductors [22]. Although the mere presence of sub-gap states does not prohibit the use of Majorana end states for topological quantum computation [23], the presence of low-lying sub-gap states clearly obstructs an unambiguous experimental verification of the Majorana states.The purpose of this paper is to systematically analyse the energi...

Recent experiments on short MoGe nanowires show a sharp superconducting-insulating transition at the universal resistance R Q h= 4e 2 , contrary to the expectation of a smooth temperature dependence of the resistance for such Josephson-like systems. We present a self-consistent renormalization-group treatment of interacting quantum phase slips in short superconducting wires, which reproduces this sharp universal transition. Our method should also apply to other systems in the sine-Gordon universality class, in the previously inaccessible intermediate-coupling regime. DOI: 10.1103/PhysRevLett.98.187001 PACS numbers: 74.78.Na, 73.21.Hb, 74.20.ÿz, 74.40.+k One of the most intriguing problems in low-dimensional superconductivity is the understanding of the mechanism that drives the superconductor-insulator transition (SIT). Experiments conducted on quasi-one-dimensional (1D) systems have shown that varying the resistivity and dimensions of thin metallic wires can suppress superconductivity [1,2], and in certain cases lead to an insulating-like behavior [3][4][5][6].Particularly interesting are recent experiments conducted on short MoGe nanowires [6] that explore the SIT tuned by the wire's normal state resistance with a critical resistance R c R Q . The universal critical resistance may suggest that at a temperature much lower than the meanfield transition temperature, T T c , the wire acts as a superconducting (SC) weak link resembling a Josephson junction (JJ) connecting two SC leads. Schmid [7] and Chakravarty [8] showed that a JJ undergoes a SIT as a function of the junction's shunt resistance, R s , with a critical resistance of R Q , and that the resistance across the junction obeys the power law R T / T 2 R Q =R s ÿ1 . The theory was later extended to JJ arrays and SC wires [9][10][11]. Within these theories, a similar power-law temperature dependence of the resistance at low temperature prevails.Contrary to this prediction, Bollinger et al. [6] observe that the resistance of quasi-1D MoGe nanowires exhibits a much stronger temperature dependence, even close to the SIT. In fact, the resistance could be fitted with a modified LAMH theory [12,13] of thermally activated phase slips down to very low temperatures. Nevertheless, for these narrow wires it appears that the LAMH theory is valid only in a narrow temperature window [14,15]. Moreover, the LAMH analysis does not explain the appearance of a critical resistance R c R Q .We propose an approach that captures both the critical resistance of R c R Q at the SIT and the sharp decay of the resistance as a function of temperature. As in previous works, we treat the SIT in nanowires as a transition governed by quantum phase-slip (QPS) proliferation. This picture alone, however, cannot account for the observed strong temperature dependence of the resistance. We note that in SC wires, collective excitations involving phase and amplitude fluctuations have considerably different propagation velocities. Hence, there is an intermediate regime for which QPSs that occur...

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