On a new exact relation for the connection matrices in case of a linear second-order ODE with non-analytic coefficients

We consider the phase-integral method applied to an arbitrary linear ordinary second-order differential equation with non-analytical coefficients. We propose a universal technique based on the Frobenius method which allows to obtain new exact relation between connection matrices associated with its general solution. The technique allows the reader to write an exact algebraic equation for the Stokes constants provided the differential equation has at most one regular singular point in a finite area of the complex plane. We also propose a way to write approximate relations between Stokes constants in case of multiple regular singular points located far away from each other. The well-known Budden problem is solved with help of this technique as an illustration of its usage.Comment: 7 pages, 2 figure

Generalized symmetry relations for connection matrices in the phase-integral method

We consider the phase-integral method applied to an arbitrary ordinary linear differential equation of the second order and study how its symmetries affect the connection matrices associated with its general solution. We reduce the obtained exact general relation for the matrices to its limiting case introducing a concept of the effective Stokes constant. We also propose a concept of an effective Stokes diagram which can be a useful tool of analysing of difficult equations. We show that effective Stokes domains which can be overlapped by a symmetry transformation are associated with the same effective Stokes constant and can be described by the same analytical function. Basing on the derived symmetry relations we propose a way to write functional equations for the effective Stokes constants. Finally, we provide a generalization of the derived symmetry relations for the case of an arbitrary order linear system of the ordinary linear differential equations. This work also contains an example of usage of the presented ideas in a case of a real physical problem.Comment: 22 pages, 4 figure

Geometry-dependent effects in Majorana nanowires

Starting from the Bogolubov-de Gennes theory describing the induced p-wave superconductivity in the Majorana wire of an arbitrary shape, we predict a number of intriguing phenomena such as the geometry-dependent phase battery (or a phi-Josephson junction with the spontaneous superconducting phase difference) and generation of additional quasiparticle modes at the Fermi level with the spatial position tuned by the external magnetic field direction. This tuning can be used to extend the capabilities of the braiding protocols in Majorana networks

Renormalization to localization without a small parameter

We study the wave function localization properties in a d-dimensional model of randomly spaced particles with isotropic hopping potential depending solely on Euclidean interparticle distances. Due to generality of this model usually called Euclidean random matrix model, it arises naturally in various physical contexts such as studies of vibrational modes, artificial atomic systems, liquids and glasses, ultracold gases and photon localization phenomena. We generalize the known Burin-Levitov renormalization group approach, formulate universal conditions sufficient for localization in such models and inspect a striking equivalence of the wave function spatial decay between Euclidean random matrices and translation-invariant long-range lattice models with a diagonal disorder

Geometry controlled superconducting diode and anomalous Josephson effect triggered by the topological phase transition in curved proximitized nanowires

We study the key features of the Josephson transport through a curved semiconducting nanowire. Based on numerical simulations and analytical estimates within the framework of the Bogoliubov-de Gennes equations we find the ground-state phase difference phi(0), between the superconducting leads tuned by the spin splitting field h driving the system from the topologically trivial to the nontrivial superconducting state. The phase phi(0) vanishes for rather small h, grows in a certain field range around the topological transition, and then saturates at large h in the Kitaev regime. Both the subgap and the continuum quasiparticle levels are responsible for the above behavior of the anomalous Josephson phase. It is demonstrated that the crossover region on phi(0)(h) dependencies reveals itself in the superconducting diode effect. The resulting tunable phase battery can be used as a probe of topological transitions in Majorana networks and can become a useful element of various quantum computation devices

Anatomy of the eigenstates distribution: a quest for a genuine multifractality

Motivated by a series of recent works, an interest in multifractal phases has risen as they are believed to be present in the Many-Body Localized (MBL) phase and are of high demand in quantum annealing and machine learning. Inspired by the success of the RosenzweigPorter (RP) model with Gaussian-distributed hopping elements, several RP-like ensembles with the fat-tailed distributed hopping terms have been proposed, with claims that they host the desired multifractal phase. In the present work, we develop a general (graphical) approach allowing a self-consistent analytical calculation of fractal dimensions for a generic RP model and investigate what features of the RP Hamiltonians can be responsible for the multifractal phase emergence. We conclude that the only feature contributing to a genuine multifractality is the on-site energies' distribution, meaning that no random matrix model with a statistically homogeneous distribution of diagonal disorder and uncorrelated off-diagonal terms can host a multifractal phase

Eleven strategies for making reproducible research and open science training the norm at research institutions

Across disciplines, researchers increasingly recognize that open science and reproducible research practices may accelerate scientific progress by allowing others to reuse research outputs and by promoting rigorous research that is more likely to yield trustworthy results. While initiatives, training programs, and funder policies encourage researchers to adopt reproducible research and open science practices, these practices are uncommon inmanyfields. Researchers need training to integrate these practicesinto their daily work. We organized a virtual brainstorming event, in collaboration with the German Reproducibility Network, to discuss strategies for making reproducible research and open science training the norm at research institutions. Here, weoutline eleven strategies, concentrated in three areas:(1)offering training, (2)adapting research assessment criteria and program requirements, and (3) building communities. We provide a brief overview of each strategy, offer tips for implementation,and provide links to resources. Our goal is toencourage members of the research community to think creatively about the many ways they can contribute and collaborate to build communities,and make reproducible research and open sciencetraining the norm. Researchers may act in their roles as scientists, supervisors, mentors, instructors, and members of curriculum, hiring or evaluation committees. Institutionalleadership and research administration andsupport staff can accelerate progress by implementing change across their institution

Eleven strategies for making reproducible research and open science training the norm at research institutions

Across disciplines, researchers increasingly recognize that open science and reproducible research practices may accelerate scientific progress by allowing others to reuse research outputs and by promoting rigorous research that is more likely to yield trustworthy results. While initiatives, training programs, and funder policies encourage researchers to adopt reproducible research and open science practices, these practices are uncommon inmanyfields. Researchers need training to integrate these practicesinto their daily work. We organized a virtual brainstorming event, in collaboration with the German Reproducibility Network, to discuss strategies for making reproducible research and open science training the norm at research institutions. Here, weoutline eleven strategies, concentrated in three areas:(1)offering training, (2)adapting research assessment criteria and program requirements, and (3) building communities. We provide a brief overview of each strategy, offer tips for implementation,and provide links to resources. Our goal is toencourage members of the research community to think creatively about the many ways they can contribute and collaborate to build communities,and make reproducible research and open sciencetraining the norm. Researchers may act in their roles as scientists, supervisors, mentors, instructors, and members of curriculum, hiring or evaluation committees. Institutionalleadership and research administration andsupport staff can accelerate progress by implementing change across their institution