strongly correlated electronic systems

condensed matter special topics course #248

Spring 2006

Monday & Wednesday, 10:00-11:30am
room: RH 306 (Roland Hall)

Instructor: Dr. Sasha Chernyshev, Assistant Professor
Office: FRH 2158
Phone: (949)-824-6440
E-mail: sasha@uci.edu
Web page: http://www.physics.uci.edu/faculty/chernyshev.html
Office hours: open office policy

Introduction

This course is the Condensed Matter theory special topics course.
Within it I am going to discuss several theoretical models commonly used to describe physics
of the strongly correlated electronic systems. Various theoretical approaches to solving these models
will be described.
It is assumed that you are familiar with the second quantization, as it will be used routinely throughout this course.
This course can be thought as a continuation of the "Many-body theory" course, 212C.
Below is the preliminary list of topics that will be covered.

Topics to be covered

Hubbard model
- Origin of the models: "early" magnetism
- Ferromagnetic exchange
- Antiferromahnetism, superexchange, Heitler-London picture
- Hubbard model: Hilbert space, atomic limit, Hubbard operators
- Hubbard algebra, relation to spin algebra, fermionic/bosonic-like operators
- Hubbard model in Hubbard operators, diagonal, off-diagonal parts
- General idea of the projection to the low-energy subspace
- Canonical-transformation approach
- Hubbard model in the strong coupling limit
- t-J model + correlated hopping, "Big Picture".
- "Another" Hubbard model or weak-coupling approach
- RPA, correlators
- Instabilities in the spin channel, Stoner criteria
- Qualitative U-n phase diagram
- Square lattice Hubbard model, SDW instability close to 1/2-filling
- Spin-gap calculation, small-U, large-U
- Improvements on the phase diagram.
- some theorems, "Big Picture"
3-band Hubbard model
- d-orbitals, their role in the physics of transition metal compounds
- role of crystal symmetry, examples of FFA (A-type AF) in manganites
- example of titanium oxides. Zaanen-Sawatzky-Allen classification scheme.
- CuO₂ planes, electronic structure, orbital structure, role of hybridization.
- 3-band Hubbard model, motivation, high-T_c, etc.
- mapping, Zhang-Rice singlet idea, orthogonalization of the states
- role of the cluster-like structure of the CuO₂ plane, emergence of the small parameter
- rigorous mapping, superexchange, effective Hubbard gap, effective hopping of the singlet
- consistency check. "Big picture".
t-J model
- motivation
- effective charge carriers, 1-hole problem
- Ising limit, strings, confinement
- Brinkman-Rice picture, variational approach
- continnum approximation, Trugman loops
- relevance to the isotropic case, variational derivation of the E_k
- form of the band, minima positions, mass asymmetry
- spin-polarons, analogy with polarons
- "survival" of the quasiparticle, dressing of the hole, quasiparticle residue
- effective Migdal's theorem, SCBA, spectral function
- hole's wavefunction, dipolar distortion of the AF background
- "rigorous" approach: spinless fermion - Schwinger boson representation for the Hubbard operators
- derivation of the quasiparticle Hamiltonian
- pairing, relevance of the 2-hole problem to the many-hole problem
- Cooper argument, inapplicability to the higher partial waves
- short-range attraction, no distiction between the symmetries
- spin-wave exchange is the reason for the s-wave repulsion and d-wave attraction
- instability of the AF order, stripes. "Big picture".
Spin systems, etc.

singlet-triplet physics of Pr₃In (byVictor Fanelli)
spin waves (by Shiu Liu)
bosonic Hubbard model (by Jurij Smakov)
an alternative projection technique for the large-U Hubbard model (by Simeng Yan)
Anderson model (by Haihui Guo)

Recommended Books

Assa Auerbach, Interacting Electrons and Quantum Magnetism, Springer-Verlag, 1994.

G. D. Mahan, Many-particle physics, Kluwer 2000.

Homeworks, etc.

There will be a homework assigned regularly. It will be collected several times (approximately once every two or three weeks).
During the course, you will be asked to give a lecture presentation on one of the strongly correlated topics of your choice (1 hour) followed by the discussion.
Your grade will be based on the homework, your presentation, and your activity in the classroom. (Roughly, 45% - 45% - 10%).

Homework, weeks #1, #2, and #3: AA book: Chapter 2.4, problems #1 and #2 and Chapter 3.4, problems #1, #2, #3, #4.

Homework, weeks #4, #5, and #6: AA book: Chapter 4.3, problems #1 and #2. Mahan's book: Chapter 6: problems 11 and 22 (pp. 431-432).