 Thank you very much. I want to start by thanking the organizers for inviting me to this wonderful conference celebrating Boris's 60th birthday and his wonderful achievements in physics Like many speakers before me. I was fortunate to experience Boris's Inspirational influence and mentorship in my case. I was lucky to have been his PhD students But it was probably unique the only one may be here to have taken a course by Boris and I Want to say Boris that it was the best course I've taken my life and I said it before I thought it used the opportunity to do it So I want to wish you not only many more years of productive research, but also Inspiring and teaching new generations of physicists And so now I'll turn to the subject of my talk So I will present the work that was done together with Igor Aleynar in Valery Vinokur and I am very happy that it actually builds on the work by Boris Altshuler Arkady Aronov and Boris Pivak Which predicted the existence of Aronov bomb oscillations in disordered metallic cylinders So in this work we show that Aronov bomb oscillations Exist not only in conductors with holes such as cylinders or rings, but also in singly connected conductors So the phenomena I will be discussing are rooted in quantum coherence of electron motion Which plays a very important role in disordered conductors probably the most Profound phenomenon that follows from quantum coherence is Anderson localization Which occurs in strongly disordered samples where the mean free path is shorter than the Fermi wavelength of electrons and in the opposite weakly disordered metallic regime Electron motion at the Fermi level is semi-classical and so basically most of the transport phenomena can be understood Classically whereas quantum coherence results only in small corrections, which are small as Small in one or KFL and for that reason even long after Anderson's discovery of localization It was believed that one can basically neglect quantum corrections in metals But that changed with the advent of mesoscopic physics of which Boris is a funny father So in the late 70s and 80s people realized that although these quantum corrections are small They're also very sensitive to Extraordinary magnetic fields and temperature and as a result they actually dominate temperature and magnetic field Dependences of physical properties of disordered metals and so there are many spectacular phenomena that arise from quantum coherence Here is here a few examples negative magnetor resistance at small fields our own of bomb oscillations of Metallic cylinders and rings and the universal conductance fluctuations all of them bear Boris's name And in the talkable in the workable present today We are building on the second topic on Boris's work with Arkady Aronov and Boris Spivak so Aronov bomb oscillations mean that properties of Conductors depend non-monotonic really as a function of the external flux through the system. So if you apply flux phi the dependence of physical properties is monotonic non-monotonic and oscillatory and Probably the first discussion of Aronov bomb oscillations in metals was made by Bogacic Who considered clean metallic cylinders? And he showed that if you apply a magnetic field along along the axis of the cylinder then Its properties will oscillate as a function of flux with the oscillation period given by hc over e and in this case the oscillations arise from the Semiclassical electron trajectories in the whispering gallery mode. So that's a picture full from Wikipedia showing the acoustic pressure profile in the whispering gallery mode at St. Paul's Cathedral and so because of This reliance on Friction integrability these oscillations are actually very sensitive to this disorder and they are destroyed upon by disorder averaging But they occur not only in conductors with holes, but also in singly connected conductors And then Boris in this work with Arkady Aronov and Boris Spivak showed that even in Dissorted conductors. There are Aronov bomb oscillations which are not not washed out by disorder They have a different physical origin. They arise from quantum interference corrections They have the different oscillation period given by double the electron charge And they survive disorder averaging But in this work it was what was considered are Essentially very thin cylinders where one could neglect the flux of the magnetic field through the sample and only consider the flux through the hole And so what We showed in this work is that Aronov bomb oscillations Also exist in solid conductors singly without without holes The oscillation period is also given by the Cooper pair charge twice electron charge. They survive disorder averaging and What I will try to show is that these oscillations arise for also from in quantum interference corrections but due to a special class of Dissorted Diffusive electron trajectories Which are near the sample boundary So the main probably physical point of conclusion of this work is that Magnetic field affects quantum interference corrections in the bulk and near the surface in a different way and this different Difference in the way the quantum interference corrections are suppressed allows one to separate the bulk contribution to quantum interference corrections from the surface contribution and the Aronov bomb oscillations arise from this surface contribution, so Let me just qualitatively discuss the origin of the effect The sensitivity of quantum interference corrections to a magnetic field is described by the cooperon that's kind of the main object of the theory and the cooperon describes Interference of Feynman amplitudes of time-reversed pads Connecting points are one and are two and in terms of semi classical trajectories and can it can be written as a sum over The classical trajectories here w sub l is the probability of realization of a given path and the influence of the magnetic field is described by this phase factor, so theta is the Aronov bomb phase accumulated by the electron as it moves along the classical trajectory And so at zero magnetic field the phase factor goes to unity and this corresponds to constructive interference between time-reversed pads But at a finite magnetic field this phase factor starts oscillating and this leads to suppression of quantum interference corrections in a magnetic field and so the One can actually Express this Aronov bomb phase in terms of the oriented area Which is confined between the electron trajectory shown here and the straight line connecting the end points of the cooperon and so And so these oriented areas fluctuates and So the statistics of these oriented areas measured in units of the magnetic length Describe the suppression of the quantum interference corrections And it turns out that the statistics of these oriented areas are different In the bulk and near the boundary The difference arises because the in the bulk when both points are far from the boundary the Summation over pads is unrestricted and so we take the functional integral over all diffusive pads and compute statistics And this results in the Gaussian dependence of the cooperon on the distance between the two points But near the boundary there is a hard wall geometrical constraint on the allowed trajectories and as a result If you want to find how the cooperon falls off with the distance at large distances You can take this functional integral by saddle point and just like in analysis The saddle point can lie in the middle of the integration region or near the boundary in this case It lies sort of near the boundary in the functional space and this results first of all in the Exponential decay of the cooperon with the distance between the end points not Gaussian and the second important Distinction is that the non-random part of the phase Is measured along the boundary because the boundary as the optimal Trajectory hugs the the boundary and so this the phase should be measured along the boundary of the sample and So therefore if we take a cooperon at Distances larger than the magnetic length What we see is that the surface contribution to the cooperon becomes much larger than the bulk contribution because instead of a Gaussian decay we get an exponential decay and This is similar to the enhancement surface enhancement of electron tunneling in strong magnetic fields Discussed by Hayet-Skin-Schklovsky but this similarity is Formal because we are not dealing with the tunneling regime. We are dealing with semi-classical corrections and Because the cooperon obeys the same Schrodinger Schrodinger equation as a particle tunneling electron in magnetic field we get the same formal formal similarity so So as a result if you take For example two points of the cooperon somewhere near the boundary the This cooperon is dominated by not by going through the bulk diffusion through the bulk, but rather by Diffusive trajectories which Grays the sample boundary and so what one can get from between the two points going by would say counterclockwise or clockwise and the two interferon sample the two amplitudes interfere and this results in the oscillatory Dependence of the cooperon on the flux total flux through the system So for example a cooperon at coinciding points Will contain some non-oscillatory contribution and also an oscillatory contribution Which will oscillate with the flux and the period of the oscillations corresponds to the superconducting flux quantum and the amplitude of these oscillations is Exponentially suppressed in the length of the perimeter of the boundary P But exponentially not in the Gaussian way. Okay, so how can we Understand this in a quantitative treatment the origin of the oscillations We can write the cooperon Formally in terms of the Eigen values of the Schrodinger equation in a magnetic field and Eigen functions in this form And the boundary conditions that should be imposed on the cooperon corresponds to the vanishing of the normal component of the current So it differs from the boundary conditions for electrons and So below I will consider the simplest example of a two-dimensional disc although the conclusions are Very general and independent of the shape of the sample so in the case of a disc the Eigen values of the cooper problem can be labeled by two quantum numbers The Landau level index and the angular momentum. So we can just formally write the cooperon as a sum over The Landau level index which is denoted by n and the angular momentum m and so the wave functions are known and It's convenient to introduce dimension less Eigen values, which are measured the units of the Landau level spacing and so So each lambda so if the angular momentum of M is small so that the Landau orbital lies entirely in the interior of the sample then the Landau level spectrum is flat Plotted here so if So what's plotted here is This dimensionless Eigen values of the cooper problem as a function of the angular momentum. So there's the Small angular momentum correspond to the left part of the graph but as the angular momentum increases and the Landau orbital starts approaching the boundary of the sample the degeneracy of the Landau level is broken and importantly the Level so the spectrum doesn't just turn up, but the experience is a dip And so this dip occurs for angular momentum Approximately equal to the dimensionless flux through the sample And so the the dip in the spectrum is What is responsible for the appearance of superconductivity in magnetic field first near the boundary? So this is the lowest lowest angular value of the cooper problem and then it spreads into the bulk at higher fields and In our case, so the oscillations arise for the following reason that so the as you change the flux in the system this Minimum of the spectrum correspond to m star shifts, let's say right to left and so these quantized Quantized Eigen del is traverse the minimum and as a result of this traversal this all properties of the system oscillate with the flux in the units of the superconducting flux quantum and So using this we can just analyze the cooper on near the boundary at large separations angular separations and At large separations the fall of the cooper on is dominated by the minimum of the spectrum here and One can get this Expected exponential decay of the cooper with the distance Okay, and so what does it what are the implications of these? findings for physical observable, so let us consider as an example Fluxation correction to the free energy of a superconductor in a magnetic field And again, I want to stick to the example of a solid disk into dimensions So the fluxation corrections are well known one can find this formula for example in the review by Boris and Arkady Aronov And so what's important for me is that this correction depends only on the spectrum of the cooper on and not the shape of the eigenfunctions, so for these two-dimensional discs We just here express the correction in terms of the dimensionless eigenvalues of the cooper on and so this in this expression K is the Matsubara frequency index and N and M are the lambda level index and the angular momentum and so now if you know the spectrum you you get the fluxation correction so if you Approximate this the sum over the quantum eigenvalues as In a kind of a leading approximation as the area of the sample times the density of states for the Landau levels You don't get the oscillations that you get the bulk contribution to this fluxation correction but if you Actually treat the sum exactly it's the difference between the sum and this Integral approximation gives you the surface contribution and in particular the Aronov bomb oscillations So how do they so you you can? simplify This general expression near the apocritical field hc3 and so using the Poisson summation formula We can get the the simpler expression for the oscillatory part of the free energy So you can see that the amplitude of the different free harmonic the decay exponentially with the perimeter of the boundary P but the characteristic length of this exponential decay is given by the coherence length for superconducting flutations Near the boundary and this length diverges at the apocritical field h surface critical field hc3 So they need these fluctuations need not be small and this hyperbolic contingent describes the crossover between fluctuation regime dominated by classical fluctuations zero months above frequency or quant fluctuations and so these oscillations can in principle be measured in experiment by By measuring persistent currents But of course, they should exist not only in thermodynamic quantities, but also in transport and for example Aslamoza-Flarkin corrections in singly connected superconductors above the apocritical field should also give rise to Aronov bomb oscillations and can be measured for example in the fourth terminal geometry following this paper And finally, I'd like to just mention that these oscillatory flux dependence of transport properties of conductors of singly connected conductors has been observed in several experiments so notably there was a experiment published in 2005 from the group of Danny Shahar And also a later work from Nina Markovich's group at Johns Hopkins so which saw Solitary flux dependence of resistance of Superconducting wires in the normal state there was a theoretical work by David Packer, Giller-Frayer and Paul Goldbart, which attributed these oscillations to formation of vortices inside these wires And in terms of repulsion of these discrete vortices and so they called it Weber blockade And so in this work what we showed is that the oscillatory dependence on the flux through the system of various properties is Much more general phenomenon. It does not require formation of vortices it exists even when there is no mean field order parameter in the system and In this case oscillations arise from quantum interference corrections and so I Think that's it. It's an conclusion We've shown that the oscillations exist in singly connected conductors They arise from a quantum interference corrections due to a special class of trajectories that graze the sample boundary They are especially pronounced in Superconducting samples in the normal state and diverge near the surface critical field hd3 and if you consider general three-dimensional samples with or without holes then each Maximal or extremal cross section of the sample normal to the field contributes an oscillation period Thank you very much for your attention