Abstract. We present a method which enables one to construct isospectral objects, such as quantum graphs and drums. One aspect of the method is based on representation theory arguments which are shown and proved. The complementary part concerns techniques of assembly which are both stated generally and demonstrated. For that purpose, quantum graphs are grist to the mill. We develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects. We discuss the theoretical implications which include Sunada's theorem of isospectrality [2] arising as a particular case of this method. A gallery of new isospectral examples is presented and some known examples are shown to result from our theory.

Conventional s-wave superconductivity is understood to arise from singlet pairing of electrons with opposite Fermi momenta, forming Cooper pairs whose net momentum is zero [1]. Several recent studies have focused on structures where such conventional s-wave superconductors are coupled to systems with an unusual configuration of electronic spin and momentum at the Fermi surface. Under these conditions, the nature of the paired state can be modified and the system may even undergo a topological phase transition [2,3]. Here we present measurements and theoretical calculations of several HgTe quantum wells coupled to either aluminum or niobium superconductors and subject to a magnetic field in the plane of the quantum well. By studying the oscillatory response of Josephson interference to the magnitude of the in-plane magnetic field, we find that the induced pairing within the quantum well is spatially varying. Cooper pairs acquire a tunable momentum that grows with magnetic field strength, directly reflecting the response of the spin-dependent Fermi surfaces to the in-plane magnetic field. In addition, in the regime of high electron density, nodes in the induced superconductivity evolve with the electron density in agreement with our model based on the Hamiltonian of Bernevig, Hughes, and Zhang [4]. This agreement allows us to quantitatively extract the value ofg/vF , whereg is the effective g-factor and vF is the Fermi velocity. However, at low density our measurements do not agree with our model in detail. Our new understanding of the interplay between spin physics and superconductivity introduces a way to spatially engineer the order parameter, as well as a general framework within which to investigate electronic spin texture at the Fermi surface of materials.1 arXiv:1509.02940v1 [cond-mat.mes-hall] Sep 2015Below a critical temperature and magnetic field, certain materials undergo a phase transition to the superconducting state. Macroscopically identified through effects such as zero resistivity and the Meissner effect [5], superconductors may further be understood microscopically as arising due to pairing of electrons occupying opposite points on the Fermi surface and having opposite spin. Within a conventional setting this interaction results in Cooper pairs with zero net momentum. However, in certain materials the presence of both magnetic order and superconductivity can lead to intrinsically nonzero pairing momentum as the system enters the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [6,7]. Studies of both CeCoIn 5 and κ-(BEDT-TTF) 2 Cu(NCS) 2 under large external magnetic fields found evidence for coupled magnetic order and superconductivity, although in each material the field strength needed was in excess of 10 T [8,9].Exotic superconductivity has recently come under additional investigation through the goal of combining s−wave superconductors with materials whose properties are rarely found among the conventional superconductors. For example, inducing the singlet pairing of an s-wave supercon...

The electron number parity of the ground state of a semiconductor nanowire proximity coupled to a bulk superconductor can alternate between the quantized values ±1 if parameters such as the wire length L, the chemical potential μ, or the magnetic field B are varied inside the topological superconductor phase. The parity jumps, which may be interpreted as changes in the occupancy of the fermion state formed from the pair of Majorana modes at opposite ends of the wire, are accompanied by jumps δN in the charge of the nanowire, whose values decrease exponentially with the wire length. We study theoretically the dependence of δN on system parameters, and compare the locations in the μ-B plane of parity jumps when the nanowire is or is not proximity coupled to a bulk superconductor. We show that, despite the fact that the wave functions of the Majorana modes are localized near the two ends of the wire, the charge-density jumps have spatial distributions that are essentially uniform along the wire length, being proportional to the product of the two Majorana wave functions. We explain how charge measurements, say by an external single-electron transistor, could reveal these effects. Whereas existing experimental methods require direct contact to the wire for tunneling measurements, charge sensing avoids this issue and provides an orthogonal measurement to confirm recent experimental developments. Furthermore, by comparing density of states measurements which show Majorana features at the wire ends with the uniformly distributed charge measurements, one can rule out alternative explanations for earlier results. We shed light on a parameter regime for these wire-superconductor hybrid systems, and propose a related experiment to measure spin density.

Observation of non-Abelian statistics for the e/4 quasiparticles in the ν = 5 2 fractional quantum Hall state remains an outstanding experimental problem. The non-Abelian statistics are linked to the presence of additional low energy states in a system with localised quasiparticles, and hence an additional low-temperature entropy. Recent experiments, which detect changes in the number of quasiparticles trapped in a local potential well as a function of an applied gate voltage, VG, provide a possibility for measuring this entropy, if carried out over a suitable range of temperatures, T . We present a microscopic model for quasiparticles in a potential well and study the effects of non-Abelian statistics on the charge stability diagram in the VG − T plane, including broadening at finite temperature. We predict a measurable slope for the first quasiparticle charging line, and an even-odd effect in the diagram, which is a signature of non-Abelian statistics.