46 research outputs found
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
Eleven strategies for making reproducible research and open science training the norm at research institutions
Reproducible research and open science practices have the potential to accelerate scientific progress by allowing others to reuse research outputs, and by promoting rigorous research that is more likely to yield trustworthy results. However, these practices are uncommon in many fields, so there is a clear need for training that helps and encourages researchers to integrate reproducible research and open science practices into their daily work. Here, we outline eleven strategies for making training in these practices the norm at research institutions. The strategies, which emerged from a virtual brainstorming event organized in collaboration with the German Reproducibility Network, are concentrated in three areas: (i) adapting research assessment criteria and program requirements; (ii) training; (iii) building communities. We provide a brief overview of each strategy, offer tips for implementation, and provide links to resources. We also highlight the importance of allocating resources and monitoring impact. Our goal is to encourage researchers - in their roles as scientists, supervisors, mentors, instructors, and members of curriculum, hiring or evaluation committees - to think creatively about the many ways they can promote reproducible research and open science practices in their institutions
Emergent fractal phase in energy stratified random models
We study the effects of partial correlations in kinetic hopping terms of
long-range disordered random matrix models on their localization properties. We
consider a set of models interpolating between fully-localized Richardson's
model and the celebrated Rosenzweig-Porter model (with implemented
translation-invariant symmetry). In order to do this, we propose the
energy-stratified spectral structure of the hopping term allowing one to
decrease the range of correlations gradually. We show both analytically and
numerically that any deviation from the completely correlated case leads to the
emergent non-ergodic delocalization in the system unlike the predictions of
localization of cooperative shielding. In order to describe the models with
correlated kinetic terms, we develop the generalization of the Dyson Brownian
motion and cavity approaches basing on stochastic matrix process with
independent rank-one matrix increments and examine its applicability to the
above set of models.Comment: 27 pages, 6 figure