In the paper [1]  the resistance  of strongly interacting two-dimensional electron gas is calculated as a function of electron spin polarization caused by external  magnetic field . To avoid  possible influence of orbital effects, it was assumed that the magnetic  field is parallel to the plane of the two-dimensional electrons.
       In the approximation in which the problem was solved in [1], one could easily solve  it much earlier , at least twenty years before. In this sense, the problem can be classified as forgotten
or missed task. This task became very interesting  after it was experimentally found  that  in 2D gas of silicon high mobility field effect structures (Si-MOSFETs )  one  achieves complete spin polarization of electron system in parallel to interface magnetic field [2,3].
       The idea of the solution is to take into account the variation in the screening of the Coulomb scattering center caused by the spin polarization increase . The effect becomes significant only at r_s~ 1 , where the parameter r_s  characterizes  the interaction strength between electrons . (In the simplest case, r_s is the ratio of the characteristic potential energy to the typical kinetic energy.) On the other hand, approximation used in [1] (RPA with Hubbard corrections) is valid also only in the region where r_s is slightly higher than unity. Thus, the formal calculation is valid  in a narrow range of electron densities  corresponding to the values of r_s close to unity.  Another  drawback
is taking into account   Hartree part of the scattering potential only .
        Despite these shortcomings, the paper [1] gained great popularity among  the experimentalists: the paper is cited about a dozen times each  year. It turned out that the results adequately describe the experiment  (both qualitatively and quantitatively), even in the limit of r_s >> 1, and in addition, the better description of the changes in the transport properties of two-dimensional electron systems caused by  spin polarization is still not avaliable.

In one year after publication of [1]  its results  were reproduced in [4].

1. V. T. Dolgopolov, A. Gold JETP Letters 71, 27 (2000).
2. T. Okamoto, K. Hosoya, S. Kawaji, A. Yagi Phys.Rev. Lett., 82, 3875 (1999)
3. T. Okamoto, K. Hosoya, S. Kawaji, et al, Cond-mat/ 9906425 (1999)
4. Igor F. Herbut Phys. Rev. B, 63, 113102 (2001)
 

The discovery of superfluid phases of liquid $^3$He  in 1972 put forward the problem of mathematical description of vortices in these superfluids. Vortices  in a fluid are characterized by circulation of velocity ${\bf v}$ given by integral $\oint_\gamma {\bf v}d{\bf l}$ over  closed contour $\gamma$. There was known that unlike to ordinary fluids where the circulation of velocity   can take an arbitrary value the circulation of superfluid velocity in superfluid $^4$He is quantized being equal to $N\frac{h}{m}$.   Here, $h$ is the Planck constant and $m$ is mass of $^4$He atom.   Hence, the  vortices differ each other by the integer number of circulation quanta N. This in particular means that vortex lines are either closed or terminated at the walls or on the free surface of the helium.
    The situation in superfluid $^3$He has proved to be different. There was shown that  the superfluid $^3$He-A admits the existence of the vortex lines with free ends [1]. The superfluid velocity field around such vortices coincides with the field of the vector potential  of the Dirac monopole [2]. This astonishing theoretical discovery pointed out that  one need to search  some general mathematical approach  to description of singular and nonsingular order parameter distributions in superfluid  phases of $^3$He. This was done in the paper by Volovik and Mineev [3].
 The idea is simple: the mathematical description of order parameter distributions is given in terms of mappings between real space filled by an ordered media for example by the superfluid $^3$He and the space of order parameter variations called {\it the space of degeneracy} which leave the energy invariant but not the  state of the superfluid. The  stable singularities  and rules of their coalescence were classified  in correspondence with elements of homotopy group of the particular  space of degeneracy and the rules of their multiplications. Unlike to similar approach developed at the same time by french scientists G.Toulouse and M. Kleman [4] in the paper [3] there was stressed that the topological stability determined by the energy of relevant interactions.  The latter are  different at different space scales.As result the types of topologically stable defects is also scale dependent.
Most of the exotic singular and nonsingular orders in superfluid phases of $^3$He described theoretically in the 1976 have been experimentally discovered in the 1980s and 1990s  by means of Nuclear Magnetic Resonance on liquid helium under rotation [5, 6]. Among the more recent experimental achievements based on the predictions done in the paper [3]  it is necessary to mention the discoveries of half-quantum vortices: in  mesoscopic samples of spin-triplet  superconductor Sr$_2$RuO$_4$ [7], in exciton-polariton condensate [8], in antiferromagnetic spinor Bose-Einstein condensate [9], in the polar phase of superfluid $^3$He [10].
Passed about four decades since the development of the topological approach to the classification of defects in ordered media. The use of topology for the treatment of unusually complex ordering in superfluid phases of $^3$He was innovative and offered new areas of applications. As it was with other mathematical tools, topological methods have been proved very effective for the description of many phenomena in different branches of physics. Chern classes, skyrmions and instantons  are encountered in  theory of quantum Hall effect and in quantum field theory. Monopole-like objects have been observed in  liquid crystals and in spin ice media and were discovered recently in the Bose-Einstein condensate of cold gas of $^{87}$Rb atoms [11]. The braid groups are applied in  theory of quantum computers.  Another new and vast area of topological applications is opened with discovery of so called topological insulators and theoretical studies of topological superfluids and superconductors [12].

[1] G. E. Volovik and V. P. Mineev, Pis'ma Zh.Exp. Teor. Fiz.  23, 647 (1976)[JETP Lett.  23, 593 (1976)].

[2] P.A.M.Dirac, Proc. R.Soc. A 133, 60 (1931).

[3] G. E. Volovik and V. P. Mineev, Pis'ma Zh.Exp. Teor. Fiz.24, 605 (1976)[JETP Lett.  24, 562 (1976)].

[4] G. Toulouse and M. Kleman, J. de Phys. Lettr.  37, L-149 (1076).

[5] M. M. Salomaa, G. E.Volovik, Rev. Mod. Phys. 59, 533 (1987).

[6] O. V. Lounasmaa, E. Thuneberg, Proc. Natl. Acad. Sci.  96, 7760 (1999).

[7] J.Jang, D. G. Ferguson, V. Vakaryuk  et al., Science 331, 186 (2011).

[8] K. G. Lagourdakis, T.Ostatnicky, A.V. Kavokin et al., Science 326 974 (2009).

[9] Sang Wong Seo, Seji Kang, Woo Jin Kwon, and Yong-il Shin, Phys. Rev.Lett.  115, 015301 (2015).

[10] S.Autti, V. V. Dmitriev,V.B. Eltsov et al, arXiv:1508.02197[cond-mat.]

[11]  M.W.Ray, E.Ruokokoski, S.Kandel et al, Nature 505}, 657 (2014).

[12] T Mizushima, Y.Tsutsumi, T.Kawakami et al, arXiv:1508.00787 [cond-mat.]


 

The conductivity of the metal is determined by the scattering processes without the conservation of the momentum of the electron system. At low temperatures,  the residual resistance is caused by  scattering on randomly placed impurities. In the frames  of the standard quasi-classical approach, the only characteristic of the disorder, a term of the Drude formula for conductivity is transport scattering time. How fair is such  approach?
To give  answer on this question it is necessary to consider the system of finite size. The conductivity is  not convenient for description of electron transport in the systems. More appropriate  is the description   in terms of the conductance $ G $: the reverse resistance of the sample (with a certain way of connecting external circuit). In the quasi-classical approximation, the average conductance of $ d $ -dimensional cube with a side of $ L $, measured between opposite faces, is $ G_0 (L) = \sigma_0L ^ {d-2} $, where $ \sigma_0 $ is Drude conductivity.
The conductance of the sample with randomly distributed impurities is also a random variable. Different samples of the same form with the same nominal average defect density will be showing slightly different conductances. The magnitude of the fluctuations is  characterized by a standard deviation of $ \langle \delta G ^ 2 \rangle = \langle (G-G_0)^ 2 \rangle $. In the framework of classical physics it is natural to expect the self-averaging $ G(L) $ with an increase of  the size of the system. Indeed, with the growth of $ L $ the number of scattering centers $ N_{\text {imp}} $ grows as $ L ^ d $, and the relative fluctuation of $ N _ {\text {imp}} $ decreases as $ [\delta N_ { \text {imp}} / \langle N _ {\text {imp}}\rangle] ^ 2 \propto L ^ {- d} $. Thus, fluctuations in the conductance of $ \delta G ^ 2 (L) \propto L ^ {d-4} $ should decrease with increasing $ L $ (in spaces of dimension $ d <4 $), confirming the hypothesis of self-averaging.
In the quantum world, however, situation is different. A mechanism, leading to much greater fluctuations  of the conductance was predicted theoretically in the pioneering work of B.L. Altshuler [1] (and independently published a few months later by P.A. Lee and A.D. Stone [2]). The mechanism is based on the phenomenon of quantum interference in the scattering of electrons on impurities.
Interference manifests itself most clearly in mesoscopic samples  where the inelastic processes do not break the coherence of the electron, i.e., under the condition $ L_ \phi \gg L $ (where $ L_ \phi $ is dephasing length). In this limit, $ \langle \delta G ^ 2 \rangle \sim (e ^ 2 / h) ^ 2 $ a factor of order unity, depending only on the shape of the sample. The fact that this ratio is independent of the degree of disorder and the size of the sample, allowed the authors \cite {LeeStone} to call such fluctuations universal conductance fluctuations.
Thus, the conductance of mesoscopic sample is not a self-averaging quantity: regardless of size $ L $, at sufficiently low temperatures $ G (L) $ fluctuates on the order of the quantum conductance of $ e ^ 2 / h $. For good metal relative magnitude of fluctuations is small, but near the transition to an insulator when $ G_0 \sim e ^ 2 / h $, fluctuations become strong. In this case, to describe the electron transport one should take into account  the distribution function of the conductance  $ P (G) $.
Universal conductance fluctuations are caused  by diffusion modes (diffuson and cooperons) moving  in  mesoscopic system. This  long-range interaction  is  responsible for the violation of the classical scaling $ \delta G ^ 2 (L) \propto L ^ {d-4} $. Classical scaling restores in the limit of $ L_ \phi \ll L $. Under this condition  the sample actually splits into incoherent subsystems having the  size $ L_ \phi $.
Another consequence of the universal conductance fluctuations  determined by long-range diffusion modes is their sensitivity to the symmetry of the problem. Spin symmetry or  time reversal symmetry  breaking reduces the number of long-range diffuson and cooperons which leads to a change in the expression for prefactor $ \langle \delta G ^ 2 \rangle $. Accurate analysis of the dependence of the conductance fluctuation on the symmetry of the system and the temperature can be found in [3].
The theory of universal conductance fluctuations has been  confirmed in many experiments. Normally, investigations are not carried out in a variety of different samples.The properties of a single sample are changed by  applying a weak magnetic field or changing the chemical potential of the electrons due to  the field-effect..
The universal fluctuations manifest themselves in experiment  as reproducible conductance irregularities when changing the control parameter.
The idea  suggested in [1], proved to be extremely fruitful. It has changed the language for  electrical transport description by switching  attention from size independent conductivity to  the conductance of the finite-size sample and led eventually to the emergence of a new field of research - mesoscopic physics, where quantum properties significantly affect the electron transport on a large scale.

[1] Altshuler  B.L.  JETP LETTERS  41, 648    (1985)
        [2] P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).
        [3] Altshuler, Shklovskii B.I., JETP 91, 220 (1986).
 

The papers [1] and [2] appeared in the time  of emergence of mesoscopic physics, when  the basis  of the theory of quantum transport in disordered metals was created.The initial impulse was given by the work of Gor'kov, Larkin and Khmelnytsky (1979) [3], where  weak-localization correction  to the conductivity of the metal was found. The correction occurs due to the electron interference on different paths. In this work by examining the "fan" diagrams, it was shown that in  the first approximation  quantum correction to the conductivity is determined by the probability of electron coming back to the starting point  during quantum diffusion in a random potential.Further research on this topic in the early 1980-ies can be called a confirmation and development of results[3].

The influence  of magnetic field on weak localization was investigated in the paper[4].  Fairly weak magnetic fields were considered, in which the curvature of the trajectories under the action of the Lorentz force can be neglected, and the main effect is the additional phase shift, different for different trajectories. Additional phase destroys the interference of long trajectories, reducing localization correction. As a result electron system demonstrates a positive magnetoresistance in low magnetic field: resistance growth at magnetic field increase. Such behaviour  Is  unexpected in the classical theory of metals. The characteristic value of the magnetic field is given by the condition  $L_H\sim L_\phi$, where $L_H=\sqrt{\hbar c/2eB}$ is magnetic length, and  $L_\phi$ is phase coherence length restricted by the inelastic scattering.

 In the paper \cite{AAS81}  ALTSHULER, ARONOV, and SPIVAK    continued study of the influence of magnetic field on  weak-localization correction to the conductivity. In this paper  authors  consider the case of thin-walled metal cylinder in a magnetic field parallel to cylinder axis. In the limit of small thickness of the wall  the magnetic field has no effect on the classical motion of an electron on the surface of the cylinder, however, through the vector potential it  is included in the quantum equation of motion.In this geometry, the vector potential cannot be eliminated by gauge transformation (mathematically this is due to the non-triviality of the fundamental group of the circle $\pi_1(S^1)=\mathbb{Z}$).

As a result, all single-electron properties are periodic functions of the magnetic flux  through the cylinder with period $hc/e$ (Aharonov-Bohm effect).However, according to [3], the quantum correction to the  onductivity is determined by cooperon describing the diffusive motion of the two electrons. In this case, the equation for cooperon formally coincides with the Schrodinger equation for a particle of mass  $\hbar^2/2D$ ($D$ is the diffusion coefficient) and a doubled charge $2e$.

Therefore, as it was predicted in the work [1], the quantum correction to the conductivity of the thin-walled cylinder must experience oscillations of resistance as a function of magnetic flux with twice shorter period $hc/2e$. Later, this prediction was experimentally confirmed by Sharvin and Sharvin [5].

In the paper  [2] Altshuler and Aronov studied quantum correction to the conductivity of thin films and wires in a magnetic field parallel to the film plane or axis of the wire. In this case, the magnetic field also suppresses weak-localization correction to the conductivity, leading to a positive magnetoresistance. However, the effect is less pronounced than in the perpendicular field; the characteristic value of the magnetic field is determined by the thickness of  film or wire $a$: $L_H\sim\sqrt{aL_\phi}$.

Quantum correction to the conductivity is small compared with semiclassical Drude conductivity, so it is problematic to distinguish contribution of quantum correction by conventional conductivity measurements.  Significantly different dependence  of this correction and Drude conductivity on the magnetic field comes to the aid. Currently  the study of the magnetoresistance  is the standard method for the experimental observation of weak localization. It is based on  the theory  developed in the papers [1, 2, 4].

Seminal works of the early 1980s led to the realization that quantum effects in electronic transport in disordered metals should be described in terms of interacting diffusion modes but not in the language of single-particle excitations  [6]. Subsequently, this knowledge has led to the formulation of the nonlinear sigma model [7] - one of the most powerful methods of modern theoretical condensed matter physics.



[1] ALTSHULER B.L., ARONOV  A.G.,SPIVAK  B.Z. JETP LETTERS 33   94  (1981)
[2] ALTSHULER B.L., ARONOV  A.G., JETP LETTERS 33, 499 ( 1981)
[3] GORKOV L.P.. LARKIN A.I., KHMELNITSKII D.E.  JETP LETTERS 30, 228 (1979).
[4] B. L. Altshuler, D. Khmel'nitzkii, A. I. Larkin, and P. A. Lee,Phys. Rev. B 22, 5142 (1980).
[5] D. Yu. Sharvin and Yu. V. Sharvin JETP LETTERS  34, 272 (1981).
[6] K.B. Efetov, A.I. Larkin, D.E. KhemlÒnitskii, Sov. Phys. JETP 52, 568 (1980).
[7] K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, New York, 1997).
 

The paper [1] was aimed at explaining the quantized conductance of a point contact formed in a two-dimensional electron gas (2DEG). The quantization was observed (see Refs. [2] and [3] in the paper) in experiments with contacts formed in GaAs heterostructures by means of gate depletion of 2DEG.  The two-terminal conductance was varying with the gate voltage controlling the width of the contact in steps of fundamental height, 2e²/h. The quantization effect was puzzling: first, it was observed in the absence of a magnetic field, so it had nothing to do with the quantum Hall effect; second, the conductance quantization ran contrary to the conventional ideas of electron diffraction in scattering off obstacles.

In the paper [1]   the point contact is considered as a constriction of 2DEG with boundaries smooth on the scale of Fermi wavelength. That allowed us to use a version of the Born-Oppenheimer approximation to solve the quantum transport problem for electrons moving through the constriction. In our case, the “slow” and “fast” degrees of freedom were, respectively, the electron motion in the directions along and across the constriction. The corresponding adiabatic separation of variables was controlled by a small parameter d/R, i.e., by the ratio of the width of the constriction d to the curvature radius of its boundaries R. The very same parameter ensured the dichotomy between the electron modes reflected from and propagating through the constriction, and, as a consequence, the sharpness of steps between the conductance plateaus. The relative width of the steps in the considered model is of the order of (d/2p⁴R)^1/2. Note the fortunate presence of a small numerical factor in the estimate, which makes the steps sharp even for R~d. (The conjecture of smooth boundaries got support from the developed later [2] microscopic understanding of the electrostatic depletion of 2DEG used to form the constrictions.)

The paper  [1] introduced the concept of adiabatic electron transport in mesoscopic semiconductor devices. Among the numerous applications of this concept, probably the most interesting ones are those treating the electron interaction in constrictions and quantum wires, see, e.g., [3,4] for early works in that direction. Current interest to this paper is associated, in part, with the investigation of mesoscopic thermopower and Coulomb drag effect [5], electron kinetics in one dimension [6], and with attempts [7] to explain a mysterious “0.7 anomaly” in the conductance of quantum point contacts.

[1] Glazman L. I. , Lesovik G. B., Khmel'nitskii D. E., Shekhter R. I. JETP Letters 48, 238 (1988)

[2] L.I. Glazman and I.A. Larkin, Semiconductor Science and Technology 6, 32 (1991)

[3 K.A. Matveev, Phys. Rev. B 51, 1743 (1995)

[4] D.L. Maslov and M. Stone, Phys. Rev. B 52, R5539 (1995)

[5] A. Levchenko and A. Kamenev, Phys. Rev. Lett. 101, 216806 (2008)

[6] A. Imambekov, T.L. Schmidt, L.I. Glazman, Rev. Mod. Phys, 84, 1253 (2012)

[7] A.M. Burke, O. Klochan, I. Farrer, et al., Nano Letters, 12, 4495 (2012)

D.E.Khmelnitskii
Cavendish Laboratory, University of Cambridge J.J.Thomson ave, Cambridge, CB3 0HE, UK

Sent to publication in January of 1980, this short paper finalised the stormish development in 1979 of what was later called the weak localisation. The year begun with publication of the famous "gang of four" paper [1], in which its authors proposed the scaling theory of localisation and claimed that all states in a 2D conductor with an arbitrary disorder are localised. A microscopic derivation of the scaling equation and the arguments in support of the conjecture of renormalisation group (RG) were given in the paper [2]. One month later, F. Wegner [3] proposed - phenomenologically - a model of the Field Theory, which possessed the RG and, in the limit of a weak disorder, gave the scaling equation conjectured by the authors of Ref [1].
The further development of the microscopic theory indicated the strong dependence of the quantum correction to conductivity on magnetic field [4], which led to suppression of the negative corrrection to conductivity and contributed to a negative mag-netirestance. This finding shifted attention to dependence of the resistance on magnetic field, which proved to be an easier way to conduct an experimental study of the whole phenomenon. It also brought up the necessity to discuss the influence of various factors on magneto-resistance. One month later, Hikami, Larkin and Nagaoka investigated effect of spin-orbit interaction on magnetoresistance [5] and found out that the latter changes its sign. Now, in this paper, Larkin attracted attention to the contribution to magneto-resistance of the Maki-Thompson correction to conductivity [6], discovered ten years earlier. The Maki-Thompson correction is connected to the scattering of the electrons on the Cooper pairs, which arose due to fluctuations. Originally, this correction was studied at the temperature close to the critical temperature of superconducting transition because it results in a smoothing out the sharp resistance drop. Still, the Cooper pairs could arise due to fluctuations far from superconducting transition and even for the repulsive sign of electron-electron interaction.
Larkin pointed out that the sensitivity of the wave function of the Cooper pair to external magnetic field results in the contribution to magneto-resistance. Dependence of this contribution upon the magnetic field turned out to be identical to that due to weak localisation and its magnitude being dependent on the magnitude of the strength g of electron-electron interaction in the "Cooper channel". As the result, the magnitude of mag-netoresistance due to weak localisation must be corrected by the account of new contribution.  The factor a (a = 1 for potential scattering and a = -0.5 for the case of strong spin-orbit interaction) at the front of the standard expression for magneto-resistance must be replaced by the combination $a - \beta $.
After exposition of the argument in favor of the new effect and the analytical derivation of the formulae, Larkin presents -in the forme of a tableaux - the results of numerical calculation of the factor $\beta $ for different values of the coupling constant g. For all the values of the coupling constant, the factor $\beta $ remains positive.
The results of this paper were quickly adsorbed by all practicing researchers and remain now a part of the uniformorly used formula for magneto-resistance.
References
[1] E. Abrahams, P.W. Anderson, D.C. Licciardello & T.V. Ra-makrishnan, Phys Rev Lett, 42, 673, (1979)
[2] L.P. Gorkov, A.I. Larkin & D.E. Khmelnitskii, JETP Letters, 30, 248, (1979)
[3] F. Wegner, Z.f Phys., B 35, 207 (1979)
[4] B.L. Altshuler, D.E. Khmelnitskii, A.I. Larkin, P.A.Lee, Phys Rev , B22, 5142 (1980)
[5] S. Hikami, A.I. Larkin & Y. Nagaoka, Progr. Theor. Phys., 63 , 707 (1980)
[6] R. Thompson, Phys. Rev., B1, 327 (1970)
 

   Usually propagation of an electromagnetic wave (light) is represented schematically as sequence of wave-front surfaces equidistant shifted on one wavelength and running with the light velocity. However, in this ideal structure of a monochromatic wave an existence of defects is possible similar to dislocations which appear in a crystal lattice [ 1 ]. Moreover with the presence of multiple defects light propagates in a manner of a turbulent flow. At the time of the article preparation the dislocations in complex speckle-fields of a laser radiation scattered in a non-uniform medium were just known. There was also known that laser cavities are able to emerge the output in a shape of a “doughnut” mode i.e. with axial zero of intensity and a wave-front in a form of a helicoid. These “optical vortices ” were considered as exotic and rather useless objects without practical applications.

  In the article [ 1 ]  it was shown that the generation of beams with optical vortices is possible in laboratory conditions by low-angle scattering of a beam in an optical fiber. More substantial was the ability shown for the transformation of an ordinary He-Ne laser beam to an optical vortex beam, that is a beam with screw dislocation, with the aid of a simple diffraction grating printed on a paper and photocopied with reduction to a film. The peculiarity of this diffraction grating was central “fork” i. e. splitting of a stripe into two (or more) stripes. The idea was a synthesis of an elementary hologram being an interference pattern of a plane wave and a beam with azimuthal dependence of the phase (along full turn around beam axis the phase varies on 2mp, where integer m is called topological charge). Laser beam diffraction on such a grating restores the structure of a beam which was used in the hologram synthesis, i. e. carrying optical vortex of corresponding charge. According to the laws of diffraction, in the first and minus-first diffraction orders the charges of vortices are opposite. Apart of the proposal of a simple technique of beams with optical vortices creation, first time the beams were generated with topological charges higher than unity.

    One year after the article publication similar paper was independently published by Australian researchers. Then the number of reports began grow quickly. A wavefront tear corresponding to the phase jump (singularity) was accepted for a common appellation “singular beams”. As a result 10 years after a new direction in a physical optics appeared, called “singular optics”

[1] BAZHENOV V.Y.; VASNETSOV M.V.; SOSKIN M.S.  JETP Letters, 52 , 490 (1990)

The first magnetic superconductors with a regular sublattice of magnetic atoms have been discovered in 1976. This generated a lot of interest to the problem of the coexistence between superconductivity and magnetism. In singlet superconductor the Cooper pairs comprise the electrons with the opposite spins and a very strong internal magnetic field in the ferromagnet (so called exchange field) acts on the electrons on the way to align their spins. This leads to the destruction of the Cooper pairs and explains while practically all know magnetic superconductors are antiferromagnets, where the averaged (on the scale of the Cooper pair) internal magnetic field is zero. Only recently (after 2000) the first three ferromagnetic superconductors where discovered and without any doubts they are triplet superconductor, where the Cooper pairs are formed from the electrons with parallel spin orientation.

            As the coexistence between the singlet superconductivity (S) and ferromagnetism (F) occurred to be impossible in the bulk compounds, it was natural to address a question what will be the proximity effect near the S/F interface?  Namely this problem was addressed in the referred article [1]. The authors where the first to discover the damping oscillatory behavior of the superconducting wave function in the ferromagnet and predicted the realization of the “p” state in S/F/S Josephson junctions. In such a “p” junction in the ground state the sign of the superconducting order parameter is opposite on the banks of the junction (the phase difference is “p”, which explain the name of such a junction).

The analysis performed in the referred paper corresponded to the clean ferromagnet, where the electron transport is ballistic. The subsequent studies demonstrated that the oscillatory behavior of the superconducting wave function is a very general phenomenon, which is robust toward the scattering on the impurities and then it is present in all type of the ferromagnetic junctions – ballistic or diffusive. In fact these oscillations of the order parameter are in some sort the manifestation to the non-homogeneous superconducting Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) state [2,3].   LOFF state was predicted to occur in the clean superconducting ferromagnet. Due to the incompatibility of ferromagnetism and singlet superconductivity it is not easy to verify this prediction on experiment. However in the S/F systems the Copper pairs penetrating inside the ferromagnet occur to be in the situation similar to that considered by Larkin, Ovchinnikov and Fulde, Ferrell. Moreover the damping oscillatory behavior of the superconducting wave function in the ferromagnet must be present in both clean and dirty limits.

            The S/F/S “p” junctions have been subsequently realized on experiment [4]. The “p” junctions incorporated into the superconducting circuits serve a source of the phase shift and generate the non-dissipative current. The recent experimental studies demonstrated that the “p” junctions permit to realize a “quiet” superconducting qubit – a building block of future quantum computer [5].

The current interest to the S/F/S systems is also stimulated by the perspectives they open for the emergence of the superconducting spintronics.  

[1] Буздин А.И., Булаевский Л.Н., Панюков С.В. Письма ЖЭТФ,  35, 147 (1982), JETP LETTERS  35, 178   (1982) 

[2] Larkin A. I. and Y. N. Ovchinnikov,  Zh. Eksp. Teor. Fiz. 47, 1136 (1964) Sov. Phys. JETP 4, (1965)

[3]  Fulde P. and R. A. Ferrell ,  Phys. Rev. 135,1550  (1964)

[4]  Ryazanov V. V., V. A. Oboznov, A. Yu. Rusanov, A. V. Veretennikov, A. A. Golubov, and J. Aarts,  Phys. Rev. Lett. 86, 2427 (2001).

[5] A. K. Feofanov, V. A. Oboznov, V. V. Bol’ginov, J. Lisenfeld, S. Poletto, V. V. Ryazanov, A. N. Rossolenko, M. Khabipov, D. Balashov, A. B. Zorin, P. N. Dmitriev, V. P. Koshelets, and A. V. Ustinov, Nat. Phys. 6, 593 (2010).

L.L.Frankfurt,

Department of particle physics,Tel Aviv University,Israel

In the paper [1] derived formula of impulse approximation for the collisions of ultra relativistic composite systems by exploring non relativistic quark model  of hadrons suggested by Zweig (1965,unpublished ). Application of this formula to the collisions of ultra relativistic hadrons as the systems consisting of few constituent quarks allowed to derive  relations between cross sections of hadron-hadron collisions. These relations agreed with numerous data. Agreement with data of the prediction: σ(pp) /σ (πp) = 3/2 i.e. to the ratio of the number of constituents in the wave functions of proton(anti proton) and pion  becomes one of
fundamental  confirmations of quark hypothesis.
Some concepts introduced in the paper were incorporated into current theoretical approaches .  The assumption that the radius of a quark is significantly smaller than radius of a hadron found explanation in terms  of asymptotic freedom in quantum chromodynamics. The assumed in the paper dominance of hadron-hadron collisions at central impact parameters and neglect  by scattering off meson field of a nucleon  contradicted to the basic ideas of S matrix of that time  but they are in line with  current data and theoretical approaches . The Lorentz transformation of the rest frame wave function of a hadron into frame where this hadron is energetic uses light cone variables  like fraction of energetic hadron momentum carried by interacting quark. This was incorporated later into Feynman parton model. 


[1] E.M. Levin and L.L. Frankfurt   Pis'ma Zh.Eksp. Teor. Fiz. 2, 65 (1965)