Josephson junction between anisotropic superconductors

The sin-Gordon equation for Josephson junctions with arbitrary misaligned anisotropic banks is derived. As an application, the problem of Josephson vortices at twin planes of a YBCO-like material is considered. It is shown that for an arbitrary orientation of these vortices relative to the crystal axes of the banks, the junctions should experience a mechanical torque which is evaluated. This torque and its angular dependence may, in principle, be measured in small fields, since the flux penetration into twinned crystals begins with nucleation of Josephson vortices at twin planes.Comment: 6 page

High-field vortices in Josephson junctions with alternating critical current density

We study long Josephson junctions with the critical current density alternating along the junction. New equilibrium states, which we call the field synchronized or FS states, are shown to exist if the applied field is from narrow intervals centered around equidistant series of resonant fields, $H_m$. The values of $H_m$ are much higher than the flux penetration field, $H_s$. The flux per period of the alternating critical current density, $\phi_i$, is fixed for each of the FS states. In the $m$-th FS state the value of $\phi_i$ is equal to an integer amount of flux quanta, $\phi_i =m\phi_0$. Two types of single Josephson vortices carrying fluxes $\phi_0$ or/and $\phi_0/2$ can exist in the FS states. Specific stepwise resonances in the current-voltage characteristics are caused by periodic motion of these vortices between the edges of the junction.Comment: 4 pages, 5 figure

Thermal suppression of surface barrier in ultrasmall superconducting structures

In the recent experiment by Cren \textit{et al.} [Phys. Rev. Lett. \textbf{102}, 127005 (2009)], no hysteresis for vortex penetration and expulsion from the nano-island of Pb was observed. In the present paper, we argue that this effect can be associated with the thermoactivated surmounting of the surface barrier by a vortex. The typical entrance (exit) time is found analytically from the Fokker-Planck equation, written in the form suitable for the extreme vortex confinement. We show that this time is several orders of magnitude smaller than 1 second under the conditions of the experiment considered. Our results thus demonstrate a possibility for the thermal suppression of the surface barrier in nanosized low-$T_{c}$ superconductors. We also briefly discuss other recent experiments on vortices in related structures.Comment: 12 pages, 2 figure

Suppression of surface barrier in superconductors by columnar defects

We investigate the influence of columnar defects in layered superconductors on the thermally activated penetration of pancake vortices through the surface barrier. Columnar defects, located near the surface, facilitate penetration of vortices through the surface barrier, by creating ``weak spots'', through which pancakes can penetrate into the superconductor. Penetration of a pancake mediated by an isolated column, located near the surface, is a two-stage process involving hopping from the surface to the column and the detachment from the column into the bulk; each stage is controlled by its own activation barrier. The resulting effective energy is equal to the maximum of those two barriers. For a given external field there exists an optimum location of the column for which the barriers for the both processes are equal and the reduction of the effective penetration barrier is maximal. At high fields the effective penetration field is approximately two times smaller than in unirradiated samples. We also estimate the suppression of the effective penetration field by column clusters. This mechanism provides further reduction of the penetration field at low temperatures.Comment: 8 pages, 9 figures, submitted to Phys. Rev.

Shapiro steps in Josephson junctions with alternating critical current density

We treat theoretically Shapiro steps in tunnel Josephson junctions with spatially alternating critical current density. Explicit analytical formulas for the width of the first integer (normal) and half-integer (anomalous) Shapiro steps are derived for short junctions. We develop coarse-graining approach, which describes Shapiro steps in the voltage-current curves of the asymmetric grain boundaries in YBCO thin films and different superconductor-ferromagnet-superconductor Josephson-type heterostructures.Comment: 5 pages, 3 figures, accepted for publication in Phys. Rev.

REPLY TO COMMENTS FROM S.G. SKUBLOV, A.V. BEREZIN, AND L.I. SALIMGARAEVA ON THE ARTICLE AUTHORED BY M.V. MINTS AND K.A. DOKUKINA – THE BELOMORIAN ECLOGITE PROVINCE (EASTERN FENNOSCANDIAN SHIELD, RUSSIA): MESO-NEOARCHEAN OR LATE PALEOPROTEROZOIC?

In their research, the authors of the comments have focused on the Late Paleoproterozoic rims of zircons, but ignored many important details of their own data. Their comments are based on a misconception that eclogite zircons have unique geochemical (REE, Th/U) and isotopic (Lu-Hf, δ18O) characteristics that do not depend on rock types and pressure rates (that were high or ultrahigh) during metamorphism. This idea leads to false unambiguous dating of the eclogite facies metamorphism based on single samples of the rocks

МЕЗОНЕОПРОТЕРОЗОЙСКИЙ ГРЕНВИЛЛ-СВЕКОНОРВЕЖСКИЙ ВНУТРИКОНТИНЕНТАЛЬНЫЙ ОРОГЕН: ИСТОРИЯ, ТЕКТОНИКА, ГЕОДИНАМИКА

The objective of this paper is to represent the main features inherent to Grenville-Sveconorwegian Orogen (GSNO) and to propose a model of tectonic and geodynamic evolution of this orogen based on the results of research concerning similar Precambrian tectonic units in the East European Craton. The studies of the conditions and settings related to origin and evolution of GSNO are of special interest, because it is located geographicaly and in a certain sense ideologically in the center of Rodinia, a supposed Neoproterozoic supercontinent. GSNO originated in the MezoNeoproterozoic in the inner region of the Lauroscandia continent. At present, the synformal tectonic structure of GSNO is divided into two portions: Grenville sector along the southeastern margin of the Canadian Shield, and Sveconorwegian sector in the southwestern Scandinavia. The integrity of Lauroscandia was twice disturbed in the MezoNeoproterozoic when oceanic structures resembling the Atlantic Ocean were formed. Later on, the continuity of the continent was restored with the involvement of oceanic lithosphere subduction and accretion and obduction of the island-arc and oceanic terranes. We distinguish two stages in the GSNO history: (1) ‘preparatory’ stage (from ~1.90 to ~1.16 Ga), and (2) formation of GSNO proper (from ~1.19 to ~0.90 Ga). The manifestations of granulite-facies metamorphism were repeatedly recorded before the Grenville Orogeny at 1.67–1.66, 1.47–1.45, 1.37–1.35, and 1.20–1.18 Ga. The Ottawan stage of the Grenville metamorphism proper is dated between 1.16 and 1.05–1.03 Ga. Metamorphism at the base of Allochthonous Belt corresponds to high-pressure granulite facies and, in a number of places, to hightemperature eclogite facies (800–900 °C at pressure in the range between 14 and 20 kbar). The age of metamorphism of rocks within Paraautochthonous Belt is 1.05–0.95 Ga; metamorphic grade increases from the greenschist facies near the Grenville front to the high-pressure amphibolite facies near Allochthon Boundary Thrust showing an inverted metamorphic zoning. High-pressure granulite-facies metamorphism is characteristic of Sveconorwegian sector; and high-temperature eclogites are observed locally at the base of the allochthonous complexes and within the paraautochthonous complexes. A distinctive feature of GNSO is the abundant occurrence of specific intrusive magmatism. Massifs of anorthosite-mangerite-charnockite-granite (AMCG) and anorthosite-rapakivi granite (ARG) complexes formed 1.8–1.5 Ga ago frame the orogen as a wide arc. In the internal region of GSNO, these complexes were formed successively at 1.16–1.13, 1.09–1.05, 0.99–0.96, and 0.93–0.92 Ga. Later on, after the intrusion, the massifs unevenly underwent granulite-facies metamorphism. The high-temperature magmatism and metamorphism, numerous repeated thermal pulses and enormous crustal body that underwent high-temperature transformation point to a mantle plume as the most adequate source of thermal energy. The model of intracontinental development of GSNO comes into conflict with popular ideas, which assume origination of this orogen as a result of the collision and welding of the formerly distant continents (Laurentia, Baltica and Amazonia), which, as suggested, completed the assembly of Rodinia supercontinent. A conclusion is drawn that the concept of tectonic position and geodynamic evolution of GSNO, which is not a counterpart of the Tibet-Himalayan Orogen, should be revised.Цель статьи – представить главные особенности Гренвилл-Свеконорвежского орогена (ГСНО) и предложить модель тектонической и геодинамической эволюции этого орогена, опираясь на результаты исследований докембрийских тектонических структур аналогичного типа в пределах Восточно-Европейского кратона. Исследование условий и обстановок возникновения и эволюции ГСНО представляет специальный интерес в связи с тем, что он территориально и в определенном смысле «идейно» размещен в центре Родинии – предполагаемого неопротерозойского суперконтинента. ГСНО был сформирован в мезонеопротерозое во внутренней области континента Лавроскандия. В настоящее время синформная тектоническая структура ГСНО разделена на два сектора: Гренвиллский вдоль юго-восточной окраины Канадского щита и Свеконорвежский на юго-западе Скандинавии. Единство Лавроскандии в мезонеопротерозое дважды нарушалось в результате формирования океанических структур, подобных Атлантическому океану. Целостность континента через некоторое время восстанавливалась при участии процессов субдукции океанической литосферы, аккреции и обдукции островодужных и океанических террейнов. В истории формирования ГСНО выделяются два периода: первый («подготовительный») в интервале от ~1.90 до ~1.16 млрд лет и второй (собственно формирование ГСНО) – в интервале от ~1.19 до ~0.90 млрд лет. Проявления гранулитового метаморфизма неоднократно зафиксированы до Гренвиллского орогенеза: 1.67–1.66, 1.47–1.45, 1.37–1.35 и 1.20–1.18 млрд лет назад. Оттавский этап собственно Гренвиллского метаморфизма датирован 1.16 и 1.05–1.03 млрд лет. Метаморфизм в основании Аллохтонного пояса соответствует гранулитовой фации повышенных давлений и в ряде мест – высокотемпературной эклогитовой фации (800–900 °С при давлении в интервале между 14 и 20 кбар). Возраст метаморфизма в породах Параавтохтонного пояса равен 1.05–0.95 млрд лет, уровень метаморфизма возрастает от зеленосланцевой фации близ Гренвиллского фронта до амфиболитовой фации повышенных давлений близ Пограничного надвига аллохтона, демонстрируя инвертированную метаморфическую зональность. Для Свеконорвежского сектора характерен гранулитовый метаморфизм повышенных дав- лений и локально, в основании аллохтонных и в пределах параавтохтонного комплексов, отмечены высокотемпературные эклогиты. Отличительной особенностью ГСНО является широкое проявление специфического интрузивного магматизма. Массивы анортозит-мангерит-чарнокит-гранитоидного (АМЧГ) и анортозит-рапакиви-гранитного (АРГ) комплексов, сформированные 1.8–1.5 млрд лет назад, широкой дугой обрамляют ороген. Во внутренней области ГСНО массивы этих комплексов были последовательно сформированы 1.16– 1.13 и 1.09–1.05, 0.99–0.96 и 0.93–0.92 млрд лет назад; вслед за внедрением массивы неравномерно подверглись гранулитовому метаморфизму. Высокотемпературный характер магматизма и метаморфизма, многократное повторение термальных импульсов и грандиозный объем коры, подвергшейся высокотемпературным преобразованиям, указывают на мантийный плюм в качестве наиболее адекватного источника тепловой энергии. Модель внутриконтинентального развития ГСНО противоречит популярным представлениям о возникновении этого орогена в результате столкновения и объединения ранее удаленных континентов (Лаврентии, Балтики и Амазонии), которые, как предполагается, завершили сборку суперконтинента Родиния. Сделан вывод о необходимости пересмотра представлений о тектонической позиции и геодинамической эволюции ГСНО, который не является аналогом Тибет-Гималайского орогена

The Epsilon Calculus and Herbrand Complexity

Hilbert's epsilon-calculus is based on an extension of the language of predicate logic by a term-forming operator $\epsilon_{x}$. Two fundamental results about the epsilon-calculus, the first and second epsilon theorem, play a role similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.Comment: 23 p

The Lambek calculus with iteration: two variants

Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is $\Pi_1^0$-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations