The Rashba effect as an electrically tunable spin-orbit interaction 1 is the base for a multitude of possible applications 2-4 such as spin filters 3 , spin transistors 5,6 , and quantum computing using Majorana states in nanowires 7,8 . Moreover, this interaction can determine the spin dephasing 9 and antilocalization phenomena in two dimensions. 10 However, the real space pattern of the Rashba parameter has never been probed, albeit it critically influences, e.g., the more robust spin transistors using the spin helix state 6,11,12 and the otherwise forbidden electron backscattering in topologically protected channels. 13,14 Here, we map this pattern down to nanometer length scales by measuring the spin splitting of the lowest Landau level using scanning tunnelling spectroscopy. We reveal strong fluctuations correlated with the local electrostatic potential for an InSb inversion layer with a large Rashba coefficient (~1 eVÅ).The novel type of Rashba field mapping enables a more comprehensive understanding of the critical fluctuations, which might be decisive towards robust semiconductor-based spintronic devices.

The conventional approach to circuit quantization is based on node fluxes and traces the motion of node charges on the islands of the circuit. However, for some devices, the relevant physics can be best described by the motion of polarization charges over the branches of the circuit that are in general related to the node charges in a highly nonlocal way. Here, we present a method, dual to the conventional approach, for quantizing planar circuits in terms of loop charges. In this way, the polarization charges are directly obtained as the differences of the two loop charges on the neighboring loops. The loop charges trace the motion of fluxes through the circuit loops. We show that loop charges yield a simple description of the flux transport across phase-slip junctions. We outline a concrete construction of circuits based on phase-slip junctions that are electromagnetically dual to arbitrary planar Josephson junction circuits. We argue that loop charges also yield a simple description of the flux transport in conventional Josephson junctions shunted by large impedances. We show that a mixed circuit description in terms of node fluxes and loop charges yields an insight into the flux decompactification of a Josephson junction shunted by an inductor. As an application, we show that the fluxonium qubit is well approximated as a phase-slip junction for the experimentally relevant parameters. Moreover, we argue that the 0-π qubit is effectively the dual of a Majorana Josephson junction.

We investigate the effect of quantum phase slips on a helical quantum wire coupled to a superconductor by proximity. The effective low-energy description of the wire is that of a Majorana chain minimally coupled to a dynamical Z 2 gauge field. Hence the wire emulates a matter-coupled gauge theory, with fermion parity playing the role of the gauged global symmetry. Quantum phase slips lift the ground-state degeneracy associated with unpaired Majorana edge modes at the ends of the chain, a change that can be understood as a transition between the confined and the Higgs-mechanism regimes of the gauge theory. We identify the quantization of thermal conductance at the transition as a robust experimental feature separating the two regimes. We explain this result by establishing a relation between thermal conductance and the Fredenhagen-Marcu string order parameter for confinement in gauge theories. Our work indicates that thermal transport could serve as a measure of nonlocal order parameters for emergent or simulated topological quantum order. Topological phases of matter cannot be characterized by any local order parameter and, hence, signatures of these phases are not accessible by a local experimental probe. For free fermions, the complete classification of topological phases has recently been established [1][2][3] and a connection between the (experimentally accessible) linear response properties of a system and the value of its topological invariant has been obtained. A prominent and illustrative example are one-dimensional (1D) topological superconductors [4][5][6][7], currently the subject of intense theoretical [8,9] and experimental investigation [10][11][12][13][14][15]. In this case, the topological phase is characterized by unpaired Majorana zero modes at the ends of the superconductor, whose presence allows nonlocal storage of one bit of quantum information encoded in the total fermion parity of the superconductor [4]. This topological phase can be recognized by striking transport properties [9]. Perfect Andreev reflection off a Majorana end mode leads to a quantized zero-bias conductance of [16][17][18][19]. The peak can only be removed if the system undergoes a phase transition into a phase without Majorana modes. Exactly at the transition, the two unpaired Majorana modes combine into a perfectly transmitting mode. As a consequence, the thermal conductance through the wire peaks at a value equal to its superconducting quantum 20]. The quantization of the peak is a way to identify the topological phase transition, even in a wire of finite size [20]. In the topologically trivial phase, both zero-bias Andreev and thermal conductance are zero.It is currently a challenge in condensed matter physics to extend the classification of topological phases to interacting fermionic systems (see ) and in particular, to provide a similar connection with experimental probes. Often, insight into interacting topological phases is offered by nonlocal order parameters [24,25]. However, such quantities lack an obvious therm...

The nonlocal nature of the fermionic mode spanned by a pair of Majorana bound states in a one-dimensional topological superconductor has inspired many proposals aiming at demonstrating this property in transport. In particular, transport through the mode from a lead attached to the left bound state to a lead attached to the right will result in current cross-correlations. For ideal zero modes on a grounded superconductor, the cross-correlations are however completely suppressed in favor of purely local Andreev reflection. In order to obtain a non-vanishing cross-correlation, previous studies have required the presence of an additional global charging energy. Adding nonlocal terms in the form of a global charging energy to the Hamiltonian when testing the intrinsic nonlocality of the Majorana modes seems to be conceptually troublesome. Here, we show that a floating superconductor allows to observe nonlocal current correlations in the absence of charging energy. We show that the non-interacting and the Coulomb-blockade regime have the same peak conductance e 2 /h but different shot-noise power; while the shot noise is sub-Poissonian in the Coulomb-blockade regime in the large bias limit, Poissonian shot noise is generically obtained in the non-interacting case.

For systems of lattice anyons like Majorana and parafermions, the unconventional quantum statistics determines a set of global symmetries (e.g., fermion parity for Majoranas) admitting no relevant perturbations. Any operator that breaks these symmetries explicitly would violate locality if added to the the Hamiltonian. As a consequence, the associated quasi-degeneracy of topologically nontrivial phases is protected, at least partially, by locality via the symmetries singled out by quantum statistics. We show that it is possible to bypass this type of protection by way of specifically engineered gauge fields, in order to modify the topological structure of the edge of the system without destroying the topological order completely. To illustrate our ideas in a concrete setting, we focus on the Z6 parafermion chain. Starting in the topological phase of the chain (sixfold ground degeneracy), we show that a gauge field with restricted dynamics acts as a relevant perturbation, driving a transition to a phase with threefold degeneracy and Z3 parafermion edge modes. The transition from the Z3 to the topologically trivial phase occurs on a critical line in the three-state Potts universality class. Hence, to all effects and purposes, the gauged Z6 chain realizes the Z3 parafermion chain. We also investigate numerically the emergence of Majorana edge modes when the Z6 chain is coupled to a differently restricted gauge field.