Solid
State III
Magnetism
This course will begin with the
well-established subjects: formation of local moments, classical theories of
ordered magnetic states in metals and insulators and proceed to modern subject of
exotic phases in frustrated quantum magnets and classical spin glasses.
Time: Monday, Wednesday, Friday (as indicated)
3:20-4:40.
Serin 401.
Synopsis.
Past and coming lectures.
1.
Sept 7. Plan
of the course. Introduction.
2.
Sept 12.
Formation of magnetic moments in atomic gases. Thermodynamics of paramagnetic
gas. Relaxation times (estimate).
3.
Sept 14.
Formation of magnetic moments in atoms in solids. Crystal field effect:
simplistic treatment (for group representation theory see [5], part 1). Hund’s
interaction versus crystal field splitting. Typical magnetic transition metal
oxides. [4]
4.
Sept 19.
Concept of effective Hamiltonian. Its
application to insulating magnets. Typical energy scales of moment formation
and their interaction. Direct exchange interaction. Effective Hamiltonian of
magnetic moments: anisotropy of single ion term, ferromagnetic and
antiferromagnetic spin-spin interaction. [1]
5.
Sept 21.
Mott insulators versus band insulators. Hubbard model at the half filling as a
simplest model of antiferromagnet. Classical (Neel) ground state of
antiferromagnets and its stability. [6]
6.
Sept 26.
Hubbard model away from half-filling: Nagaoka theorem and formation of
ferromagnets. Exotic states of one-dimensional chain. [6] and original work Y.
Nagaoka, Phys. Rev. 147, 392 (1966).
7.
Sept 28.
Spin excitations in doped Hubbard model. Competition between antiferromagnetic
and ferromagnetic states. Phase separation.
8.
Oct. 3.
Insulating ferromagnets: ground state and excitations. Quasiparticles: spin
waves and their expression through quasiparticle creation-annihilation
operators. Infinite density of low energy spin waves in low dimensional
magnets. [2]
9.
Oct. 5.
Classical nature of fluctuations in low dimensional ferromagnets. Power law
correlations in low temperature phase. Berezinskii-Kosterlitz-Thouless
transition.[7]
10. Oct. 10. Divergence of fluctuations in low
dimensional Heisenberg model. Renormalization group. [7,8]
11. Oct. 12. Quantum fluctuations in
antiferromagnets: canonical theory. [2]
12. Oct. 17. Single spin equation of motion and
their representation in terms of the effective action. [6]
13. Oct. 24. Effective action of quantum
antiferromagnets. [6]
14. Oct. 26. Topological terms in the effective
action and their role. Lieb-Schultz-Mattis theorem. [9].
15. Oct. 31. Topological order parameter in
quantum disordered antiferromagnets. Kitaev model. [10]
16. Nov. 2. Theories of quantum spin liquid
phases.
17. Nov. 7. Magnetic impurities in metals:
Kondo problem and RKKY interaction. [1,2]
18. Nov. 9. Ferromagnetism of metals: Stoner
instability. [1,2]
19. Nov. 14. Ferromagnet - paramagnet
transition in metals at low temperature. Millis-Hertz theory and its problems.
[12]
20. Nov. 16. Single spin decoherence. Bloch
equations. [1]
21. Nov. 28. Spin glasses – phenomenology.
Results of numerical simulations, upper critical dimensions. [13,14]
22. Nov. 30. High temperature and virial
expansion for RKKY glasses. Importance of non-linear susceptibility for glass
transition. [13]
23. Dec. 5.
Simplest theoretical model (Sherrington-Kirkpatrick) and its physical
properties. [13]
24. Dec. 7.
Instability of the low temperature phase of SK model. Ergodicity
violation at low temperatures. [13]
25. Dec. 12. Method replic. Replica symmetric
solution. Replica symmetry breaking in SK model. [13]
26. Dec. 14. Parisi solution and its physical
meaning. [13]
Plan for future lectures: Disordered
magnets.
1. Dynamical approach to spin glasses.
References.
General theory:
1. R. M. White “Quantum theory of magnetism”
2. D. C. Mattis, “The theory of magnetism”
volume 1
3. L. D. Landau and E. M. Lifshitz,
“Theoretical Physics”, vol. VIII chap. 5, vol IX chap. VII.
Special
topics:
4. P. A. Cox “Transition metal oxides” (Very
good review of moment formation and basic chemistry of these materials).
5. J.-P. Serre “Representation lineaires des
groups finis” (Excellent concise textbook on group representation theory)
6. E. Fradkin “Field theories of condensed
matter”(see Chapter 2 for the introduction to Hubbard model, Chapters 3-5 for
the modern theory of quantum fluctuations in magnets).
7. A. M. Tsvelik “Quantum field theory in
condensed matter”
8. A. M. Polyakov “Gauge Fields and strings”
section 2.2.
9. D. C. Mattis, “The many body problem”.
10. A. Y. Kitaev, Annals of physics, 303, 2,
(2003).
11. A. Hewson “The Kondo Problem to Heavy
Fermions”
12. A. J. Millis, Phys. Rev. B 48, 7183 (1993);
D. V. Efremov, J.J. Betouras and A. Chubukov, Phys. Rev. B77, 220401 (2008).
13. K. H. Fischer and J. A. Hertz “Spin
Glasses” (Cambridge Studies in Magnetism).
14. J. A. Mydosh “Spin Glasses: An Experimental
Introduction”