** 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.

random potential. The lowest states belong to, so called, band

tail, which is formed by the states, bound to the rare fluctuations

of the potential. The student gets familiar with the concept of optimal

fluctuation (instanton). Techncally, the problem involves the steepest

descend technique of calculation of the functional integrals and

is very useful for many problems of quantum field theory and statistical

mechanics [3-5] .

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)

*Contact D.E. Khmelnitskii*
*Mott 521*
*tel: 37 289*
*e-mail: DEK12@cam.ac.uk*