The aim of this Forum session is to give a relatively basic and comprehensible tour of the fun-packed world of spin physics. This refers to the kind of theory used to describe many insulators, where motion of charge is suppressed, and the only important remaining interactions are spin-flips on the sites of a crystal lattice.
(If we were to insist on buzz words, we could slyly mention "spintronics" and "parent compounds of the high-Tc superconductors" - but hey, you're not a funding council, right?)
Depending on the size of the spin, the geometry and dimension of the lattice, and whether the effective interaction between spins is ferro- or antiferromagnetic, a whole host of different behaviours can occur. This can all get a bit confusing, so I'll emphasise trying to give you some feeling for how to think about these systems, rather than just listing a massive quantity of examples.
We'll start with the case of two spins, and gradually work up to chains, planes, and 3D lattices, perhaps taking in spin ladders on the way. Then, if there's time, we can think a bit about how fiddling with the geometry changes the conventional picture that emerges from all that.
And, of course, it would be impossible to write an abstract
like this without mentioning Bethe, Heisenberg, and Anderson.
Bethe.
Heisenberg.
Anderson.
There - that's got that one off. See you on Friday!
BOOKS
-----
J. H. van Vleck, "Theory of Electric and Magnetic Susceptibilities"
(Clarendon, Oxford, 1932).
For those with an historical bent, have a look at what they
were doing when quantum mechanics was just getting started!
J. J. Sakurai, "Modern Quantum Mechanics" (Addison-Wesley, 1994).
For the basic properties of spin operators, and some nice
discussions of projection operators etc.
A. Auerbach, "Interacting Electrons and Quantum Magnetism"
(Springer-Verlag (New York), 1994).
The first half of this book provides a good overview of
techniques used in treating magnetic systems: almost
everything I mentioned (at least in part one of my two
Fora) is discussed here.
C. Kittel, "Quantum Theory of Solids" (Wiley and Sons, 1963).
Includes a good discussion of the RKKY interaction
between local moments in metals.
PAPERS
------
P. W. Anderson, Phys. Rev. 86, 694 (1952).
An absolutely beautiful discussion of the nature of
spontaneous symmetry breaking and spin-wave theory for
the antiferromagnet - a must read!
T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).
The original spin-waves-as-bosons paper.
C. Marshall, Proc. Roy. Soc. 232, 48 (1955).
Indranil - this is for you! A good discussion of the
extent to which the fictitious states (with more than
2S bosons) spoil the Holstein-Primakoff correspondence
in the ferro- and antiferromagnetic cases.
F. J. Dyson, Phys. Rev. 102, 1217 (1956).
F. J. Dyson, Phys. Rev. 102, 1230 (1956).
Spin wave theory for the ferromagnet.
S. D. Reger and A. P. Young, Phys. Rev. B 37, 5978 (1988).
The first paper, to my knowledge, where the sublattice
magnetisation for the spin-1/2 Heisenberg AFM on the
square lattice was calculated. The result is
amazingly close to the linearised spin-wave value!
H. Bethe, Z. Phys. 71, 205 (1931).
This is a beautiful article, though note that it is in
German! It is Bethe's exact solution of the spin-1/2
AFM Heisenberg chain, a mere five years or so after
quantum mechanics got off the ground. No long range
order (this is impossible in 1D, even at zero temperature),
but it is critical. See the second part of my Forum
couplet for more details
BOOKS ----- A. Auerbach, "Interacting Electrons and Quantum Magnetism" (Springer-Verlag (New York), 1994). Quite a lot about the one-dimensional spin chain, including a discussion of the even/odd effect as one varies the spin, S. I think this is the best book to get a physical feel for what's going on. A. Tsvelik, "Quantum Field Theory in Condensed Matter Physics" (Cambridge University Press, 1995). A more technical discussion, but consistently done within the path integral approach. See in particular chapters 15, 18, and 27. A. Gogolin, A. Nersesyan, and A. Tsvelik, "Bosonization and Strongly Correlated Systems" (Cambridge University Press, 1998). And if you thought the last one was technical, this is the real stuff! This is where you go if you want the full derivations behind formulas in Tsvelik (1995). I recommend it only if you intend to work in the field, read very quickly, or have problems taking things on trust. PAPERS ------ H. Bethe, Z. Phys. 71, 205 (1931). The original "Bethe Ansatz" paper: an exact solution for the spin-1/2 Heisenberg AFM chain. (In German, but still downright magnificent.) des Cloiseaux and Pearson, Phys. Rev. 128, 2131 (1962). Excitations of the spin-1/2 Heisenberg AFM chain. (A mere thirty-one years after Bethe's paper - just goes to show that an exact solution is usually not simple!) P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928). How to write a spin-1/2 in terms of fermions, in one dimension. S. Coleman, Phys. Rev. D 11, 2088 (1975). How to write a fermion as a boson, in one dimension. This is an absolutely beautiful paper, and in my opinion should be read by anyone who is faintly interested in the subject of quantum statistics. I. Affleck and F. D. M. Haldane, Phys. Rev. B 36, 5291 (1987). More technical information on the even/odd distinction, showing that the half-integer-spin chains are gapless by various cunning bits of conformal field theory. Only attempt this if you know a bit of CFT already. (If you don't, Tsvelik (1995) - see above - is quite a good brief introduction.) A classic work, but requires a bit of effort.