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Revision: | 2013 Dec 7 |

## Course Summary

Course of lectures is devoted to the theory of quantum phenomena in electronic nanosystems: the theory of random Hamiltonian matrices of Wigner-Dyson and thermodynamics of nanoclusters, Peierls transitions in quasi one-dimensional conductors, phase transition of Ising and Berezinskii-Kosterlitz-Thouless kind in two-dimensional systems, theory of spin fluctuations in one-dimensional Ising chain, Landauer theory of quantum conductance of quantum point contact. The aim of the course is to teach the basic concepts, laws and methods of quantum mechanics and physics of phase transitions in many-body low-dimensional systems. As a result of training the student should be able to analyze the physical picture of the distribution of quantum energy levels of electrons in metallic nanoparticles and solve specific tasks for calculation of temperature, field and frequency dependences of thermodynamic and kinetic characteristics of nanoparticles, which are of fundamental interest for physics of nanosystems and possess practical significance.

## Course Format

Hours of lecture | Hours of discussion | Hours of independent study | Hours total |
---|---|---|---|

34 | 34 | 52 | 120 |

**Please note** that students are expected to study outside of class for three hours for every hour in class.

## Course Content

The plan is to work through the following topics:

- Statistics of electronic spectra of metallic nanoclusters
- Quantum size effect
- Discreteness of eigenvalues of the Hamiltonian matrix of the electron in a metal cluster
- The concept of random Hamiltonian matrices of the electrons in the nanoclusters
- The theory of Gaussian ensembles of Hamiltonian matrices of Wigner-Dyson
- Geometric correlations of energy levels and Porter's
- Symmetry elements of the Hamiltonian matrix
- Effective repulsion between the random energy levels of electrons in the nanoclusters
- Transition from orthogonal ensemble of random Hamiltonian matrices to the unitary ensemble under a gradual increase in the magnetic field
- Exact solution by Mehta for a two-point correlation function in a magnetic field
- Ballistic and diffusive limits of the nature of the motion of electrons in the nanoclusters

- Thermodynamic properties of metal nanoclusters
- Partition function of nanocluster with equidistant energy levels
- Specific heat and magnetic susceptibility of nanocluster with equidistant energy levels
- Cases of even and odd number of electrons
- Basic principles of the theory of Gor'kov and Eliashberg
- Approximate methods of calculating the specific heat and magnetic susceptibility of nanocluster with a random distribution of energy levels in low-temperature limit.
- Averaging over the distributions of the Wigner-Dyson-Mehta and Poisson
- Experimental observation of crossover in the behavior of thermodynamic characteristics of nanoclusters

- Metal-insulator transition in quasi one-dimensional conductors
- The method of Bogolyubov -de Gennes equations
- The Frohlich’s Hamiltonian of electron-phonon interaction in one-dimensional conducting chain of atoms
- Doubling of the chain period and the Peierls phase transition in one-dimensional conductor in a state with a charge density wave (CDW)
- The theory of Keldysh and Kopayev of phase transition in a state with a charge density wave (CDW) in semimetals
- Metal-insulator transitions with the formation of spin density wave (SDW)
- Coupled SDW and CDW transitions in repulsive Hubbard model: analytical solution of the stripe phase in quasi 1D weak coupling limit close to half-filling of electron band
- The theory of Gor'kov and Lebed’ of lowering the dimensionality of electronic spectrum and the formation of Magnetic Field Induced SDW

- Magnetic phase transitions in low-dimensional systems
- Ising model on two-dimensional square lattice
- Onsager's exact solution for a ferromagnetic phase transition in two-dimensional Ising model
- Power-law behavior of the temperature dependence of thermodynamic quantities
- Critical indices of two-dimensional Ising model
- The phase transition of Berezinskii, Kosterlitz and Thouless (BKT) in two-dimensional xy-model
- Theorem of Mermin-Wagner-Hohenberg
- Dissociation of vortex-antivortex pairs in the theory of BKT
- A criterion for the transition to an ordered state in the theory of BKT
- Spin fluctuations in one-dimensional Ising chain
- Exact solution for the partition function and correlation length as a function of temperature
- Phase transition to long-range order at absolute zero

- Quantum transport: the theory of Landauer
- Basic principles of Landauer theory of quantum transport
- The resistance of an ideal quantum wire, the formula of Landauer-Buttiker.
- The summation of the transverse quantum modes of electric current through a ballistic quantum filament. Landauer formula.
- Quantum of conductance of quantum wire
- Scttering matrix
- Landauer theory for a quantum point contact
- The experimental realization
- Adiabatic point contact
- Step structure of energy-dependence of the transmission coefficient through the potential barrier

## Textbooks

Primary textbooks:

- Eduardo Fradkin.
Field Theories of Condensed Matter Physics. Cambridge University Press, Cambridge, 2 edition edition, April 2013. - Madan Lal Mehta. Random Matrices, Volume 142, Third Edition. Academic Press, Amsterdam; San Diego, CA, 3 edition edition, November 2004.

Additional textbooks:

- I.V. Lerner, B.L. Althsuler, V.I. Fal’ko, and T. Giamarchi. Strongly Correlated Fermions and Bosons in Low-Dimensional Disordered Systems. NATO Science Series II: Mathematics, Physics and Chemistry. Kluwer Academic Publishers, NY, 2002. URL:
http://www.springer.com/materials/book/978-1-4020-0748-4 .

Problems and solutions:

- Alexei L. Ivanov and Sergei G. Tikhodeev. Problems of Condensed Matter Physics: Quantum Coherence Phenomena in Electron-hole and Coupled Matter-light Systems. Oxford University Press, Oxford ; New York, February 2008.

## Homework Assignments

Weekly, 12 problem sets in total, due at the beginning of the lecture. You may also submit via e-mail before the due date/time. It is of outmost importance that you invest your own effort into solving problems. Should you consult any sources, please provide references. Homework assignments should be typed. Legible handwritten assignments are acceptable.

## Grading

Class participation | 10% |

Homework assignments | 20% |

Midterm exam | 20% |

Final exam | 50% |