1. TYPE II SUPERCONDUCTIVITY

Superconductivity brings an example of spontaneous symmetry breaking,
motivated by real physics. The project involves a study of the
Ginzburg-Landau theory [1] and its extention to type II
superconductivity  [2], acquaintance with the concept of the
topologic defect - Abrikosov vortex. A number of problems could form
the mini-projects associated with this Literature review.

2.  BAND TAIL AND OPTIMAL FLUCTUATIONS

Different relevant problems of quantum mechanics are considered together
with a toy problem about a drunkard in a city full of policemen.
 

3. FUNCTIONAL INTEGRAL, PHASE TRANSITIONS AND
     TYPE I SPONTANEOUS SYMMETRY BREAKING

This project reviews a number of concepts: phase transitions of type I
and type II, spontaneous symmetry breaking, functional integral, scaling,
catastrophe theory, etc.  It involves a study of a seminal paper [6], which
shows that, in a compressible solids, any type II phase transition is
transformed into a type I transition. A mini-project on material of this review
is available.

4.  RATE OF DEPHASING FOR ELECTRONS IN METALS

The aim of this project to study the paper [17] about electron in a metal,
 which interacts with the other electrons. Such interaction leads,  at finite
temperatures,  to relaxation of the phase of the wave function of electron.
 Finally, the rate of this relaxation as function of temperature and the
 propertices of the  sample is calculated.    Meanwhile,  the student should
 learn about  the way such a rate is measured (weak localisation), about
 electromagnetic noise, Feynman path integral and other things.

5.  MOESSBAUER EFFECT AND ORTHOGONALITY CATASTROPHE

Project involves a review of two phenomena associated with a reaction of a
coupled system to a sudden pertirbation [11]. It involves a study of a two
classical many-body problems about the recoil-less emition of gamma-rays and
the motion of an external heavy particle in electron gas and X-ray absorption

6.  PHASE-SLIP-CENTRES, THERMALLY ACTIVATED DESTRUCTION OF
SUPERCONDUCTIVITY AND RESISTANCE OF  SUPERCONDUCTING WIRES

Superconductivity means zero resistance. The resistance would vanish if
not  the thermal fluctuations, which lead to temporary and local
destruction of superconductivity. As the result, a superconducting wire
obtains a finite resistance, which decreases exponentially with
decreasing  the temperature. Theoretical consideration of this problem [19]
involves a study of several basic ingredient, characteristic for modern
theory: order parameter and its time evolution; nucleation of the defects,
activation energy, rate of nucleation. This is a pretty demanding but
hugely rewarding project.

7.   HOT ELECTRONS IN SEMICONDUCTORS 

The aim of this project is to introduce the student to several basic notions of
Physical Kinetics [20]: Boltzmann equation, its transformation into Fokker-Planck
equation, stationary solution. All this could be seen while discussing electrons in
semiconductors, interacting with each other and with phonons.

8.   ANOMALOUS SKIN-EFFECT AND CYCLOTRON RESONANCE 

This project is introducing the student into kinetics of electrons in clean metals
in presence of high frequency electromagnetic filed. Two cases are under
consideration: zero magnetic field (skin-effect) and non-zero constant magnetic
field parallel to the surface of the sample. A further development of the subject
is also possible: a selective transparency; waves, etc.

9.  BOLTZMAN EQUATION AND COLLECTIVE MODES IN PLASMAS

This review allows to the student to get acquainted with several concenpts: Boltzmann
Equation, collective motion and plasma oscillations, their dispersion and attenuation,
plasma instabilities, echo.

10.  JOSEPHSON EFFECT AND ITS APPLICATIONS

The aim of this projects is:
To study physics associated with Josephson Effect;
To explore the Josephson Electronics, i.e. applications of Josephson elements for
precise measurement of different physical quantities;
To explore recent application of Josephson effect.

11.  CLASSICAL AND QUANTUM MECHANICS OF BLOCH ELECTRONS

This project should provide a guide to the facinating world of electrons in solids.
Sophisticated dispersion laws of Bloch electrons make their motion in external
electric and magnetic field, even in classical mechanics, an unusual and highly
adsorbing journey. Quantum mechanics makes it even more interesting.

12.  ELECTROMAGNETIC FLUCTUATIONS IN THE MEDIA
AND MOLECULAR FORCES BETWEEN MACROSCOPICAL BODIES

Quantum fluctuations of electromagnetic field in contineous media contribute to
their internal energy. This results in long-range forces between macrscopical bodies.
The project suggests to study relevant theory [22, 23] and diversed physics of
molecular forces.

13.  HALL EFFECT AND ITS ANALOGUES

The aim of this project is to introduce the student to various analogies of the ordinary
Hall effect: the Anomalous Hall effect in conducting Ferromagnets, the Spin-Hall effect,
the Thermal Hall effect, the Beenakker-Senftleben effect in a Gas of rotating molecules,
etc [24]. A facinating Zoo exibits to the student its Beasts, whose qualities are not
explored completely so far.

14.  APPLICATION OF ALGEBRAIC TOPOLOGY TO PHYSICS OF DEFECTS

It is suggested to study an art of homotopic classification of defects in the ordered media:
crystals, magnets, super-fluids, liquid crystals and so on. Statics and dynamics of disloca-
tions, disclinations, quantum vortices and other strangers of this world comes along [25, 26].

15.  ANDREEV REFLECTION

Andreev Reflection (AR) is a specific quantum process which occurs when an electron in a
normal metal falls at its boundary with a superconductor: as the result of AR, an electron
with momentum p transforms in a hole with momentum - p.
The student gets a chanse to learn about different phenomena AR could lead to.

16.  QUANTUM TUNNELING OF MAGNETISATION

A molecular cluster containing atoms of transition elements often has a macroscopic magne-
tisation . If such a cluster is positioned in a crystalline matrix its magnetisation might have
several minima of the energy of magnetic anysotropy. At low temperatures, the relaxation of
magnetisation requires magetisation to tunnel under the energy barrier. The project is in-
volved with theoretical study of the tunneling for spin degrees of freedom as well as an explo-
ration of the rich experimental material associated with discovery and further studies of this
phenomenon.

17.  YOUR OWN RESEARCH REVIEW

In case the student has his own subject of interest - either in physics or in theoretical
technology - come to me and discuss this. I could suggest the subject which would
meet your interest and the aims of Research Review.

REFERENCES

1.  V.L. Ginzburg and L.D. Landau, in L.D. Landau Collected Papers.

 2.  A.A. Abrikosov, JETP  (1957)

 3. J.S. Langer and J.H. Zittartz Phys Rev (1966)

 4. B.I. Halperin and M. Lax,  Phys Rev  (1966)

 5. E. Brezin and G. Parisi,   J.Phys.C  (1982)

 6.  A.I. Larkin and S.A. Pikin,
       JETP  33,  (1969).

 7.  L. Gunther, D. Bergman and Y. Imry ,
       Phys Rev Lett  27,  558 (1971).

 8.  Y. Imry,
      Phys Rev Lett.   33, 1304 (1974).

 11.  H. Lipkin,  Quntim Mechanics

 12.  P.W. Anderson  Phys. Rev. (1965)

 13.  P. Nozieres and C. de Dominicis  Phys. Rev. (1970)

 14.  L. Cooper, Phys. Rev. (1956)

 15.  J. Bardeen, L. Cooper and J.R. Schrieffer,  Phys. Rev. (1957)

 16. A.A. Abrikosov,  L.P. Gorkov, I.E. Dzyaloshinskii
        Methods of Quantum Field Theory in Statistical Physics

 17.  L.D. Landau and E.M. Lifshits, Statistical Physics. Part I

  18.   B.L. Altshuler, A.G. Aronov and D.E. Khmelnitskii
          Effect of electron-electron collisions with small energy transfer
              on Quantum  Localisation, J.Phys. C15, p7367 (1982)

 19.    L.S.  Langer and V. Ambegaokar,
         Intrinsic Resistive  Transition in Narrow Superconducting
         Channels.
         Phys. Rev.  164,  498   (1967)

 20.      E.M. Lifsits and L.P. Pitaevskii,
            Physical Kinetics,
             Pergamon Press

 21.      Niels Berglund and Turgay Uzer,
           Foundation of Physics, vol 31, p.283 (2001)

 22.      E.M. Lifshits and L.P. Pitaevskii,
           Statistical Physics, part 2

 23.       I.E. Dzyaloshinskii, E.M. Lifshits and L.P. Pitaevskii,
           Advances in Physics, vol 10, p.165 (1961)

 24.       A.F. Barabanov et al,
            Uspekhi, vol 185, p.480 (2015)

 25.        V.P. Mineev,
            Topologically stable defects and solitons in ordered media
            CRC Press 1998            

 26.        N.D. Mermin,
              Review of Modern Physics, vol 51, p.591 (1979)