Understanding Magnetism: From Quantum Mechanics to Many-Body Physics, Study notes of Applied Chemistry

Download Understanding Magnetism: From Quantum Mechanics to Many-Body Physics and more Study notes Applied Chemistry in PDF only on Docsity! Physics 216: Special topics in many-body physics, Spring 2003: http://socrates.berkeley.edu/˜ jemoore/phys216.html Lecture XV This lecture begins our study of magnetism. Although humans have used magnetism for naviga- tion and other purposes since antiquity, understanding how it emerges from the underlying physics of a solid or insulator is quite complex. In fact, only in the past few decades can we really claim to have understood magnetism. Magnetism is also a reminder that some of the seemingly abstract phenomena we discuss in this course can have important real applications. For instance, for a few hundred dollars you can purchase a 100 GB hard disk that is rewritable many times, stable for years with no power, and contains more information than an entire ancient library. Future read/write heads for such hard disks may be based on CMR (colossal magnetoresistance) and GMR (giant magnetoresistance) effects, which were discovered quite recently. In your undergraduate statistical mechanics course, you should have learned about the Ising model with energy E = − ∑ 〈ij〉 Jsisj (1) where the classical spin variables take values ±1 and the sum is over nearest neighbors. We will say a bit more about this model in a moment, but it is first important to understand how such a model can emerge from the underlying quantum mechanics of the underlying mobile, interacting electrons. A standard model for understanding the origin of magnetism is the Hubbard model Hamiltonian H = −t ∑ 〈ij〉,σ (c†iσcjσ + h.c.) + ∑ i (− µ)(ni↑ + ni↓) + ∑ i Uni↑ni↓. (2) (Actually several different variants are called the Hubbard model.) This model is relatively simple if U is a small perturbation, as then we have nearly free electrons from the hopping part t with a weak attractive or repulsive interaction. We will be more interested in the case where a repulsive U  t, as we will show that in this case the system is effectively antiferromagnetic at half filling (one electron per site on average). If t = 0 the model is trivially solvable (the ground state is any assignment of one electron per site). Similar to the above Hubbard model is the Anderson model of a single impurity coupled to noninteracting conduction electrons: H = H0 + H1, H0 = ∑ k,σ k(c † kσckσ + h.c.) + ( − µ)(nd↑ + nd↓) + Und↑nd↓, H1 = ∑ kσ Vkd(c † kσcdσ + h.c.) (3) Even this model that only has interactions at one point can show surprising correlation effects like the Kondo effect when the values of µ and U are chosen so that First, note that the above quantum Hubbard model has an SU(2) spin symmetry and hence should give rise, in the localized limit, to a spin model with Heisenberg rather than Ising symmetry. Suppose we start with the above degenerate ground states at t = 0 and ask how they are split once t is turned on: we are essentially obtaining an effective Hamiltonian for the basis of states with one 1 electron per site, valid on energy scales much less than the t = 0 gap (either µ −  or U +  − µ). Let us assume the symmetric repulsive case with  = −U/2 and µ = 0. Consider a single nearest-neighbor pair of sites. First-order perturbation theory in the degen- erate basis doesn’t do anything, since t moves the system to have two electrons on one site and one empty site, which takes the system out of the degenerate ground-state manifold. In second- order perturbation theory, there is no coupling induced if the spins on both neighboring sites are both up or both down. If site 1 is up and site 2 is down, there is both a spin-flip coupling and a spin-preserving coupling: H12 = 2t2 U ((Sz1S z 2 − 1 4 ) + 2S+2 S − 1 ). (4) Here the Si are spin-half operators. Adding the case with site 1 down and site 2 up, and ignoring an overall constant energy, we finally obtain the antiferromagnetic Hamiltonian H = J ∑ 〈ij〉 Si · Sj . (5) Here J = 4t 2 U and the S operators are spin-half quantum operators. It is possible to obtain ferromag- netic interactions instead from slightly different microscopic models, and some real ferromagnets are itinerant (the electrons whose spins produce the ferromagnetism are mobile), corresponding to the Stoner instability we discussed in the context of Fermi liquids. Note that, as usual, obtaining a significant magnetic interaction depends on underlying elec- tric rather than magnetic interactions. Aside from the symmetry difference between the above antiferromagnet and an Ising model, another important difference is that specifying the (static) energy in a classical model does not specify the dynamics, while specifying the Hamiltonian of a quantum problem determines both the energy levels on the dynamics. We will see some practical consequences of this below when discussing spin waves. The mean-field theory of the classical Ising Hamiltonian can be derived heuristically by imagin- ing that each individual spin moves in the field of its z nearest neighbors and possibly an external field H. This gives the self-consistency equation 〈s〉 = e β(Jz〈s〉+H〈s〉) − e−β(Jz〈s〉+H〈s〉) eβ(Jz〈s〉+H〈s〉) + e−β(Jz〈s〉+H〈s〉) = tanh(β(Jz + H)〈s〉) (6) This has a phase transition at H = 0 between ordered and disordered states at the temperature kT = β−1 = Jz. At zero temperature the system is perfectly ordered for both the ferromagnet and antiferromagnet. The Ising model can be understood as an anisotropic and classical limit of the Heisenberg model which we have derived from the underlying electrons, for cases when the electrons move in an anisotropic crystal, say. Now we can state another interesting difference between the quantum and classical models. On a bipartite lattice, a lattice that can be divided into two symmetric sublattices A and B so that every bond of the model joins one site from sublattice A to one from sublattice B (like the square lattice, for example), there is no thermodynamic difference in zero magnetic field between the classical ferromagnet and antiferromagnet. The idea is that flipping the sign of the coupling constant is the same as flipping all the spins on one sublattice, so that any quantity which sums over all spin configurations should be independent between the ferromagnet and antiferromagnet. Hence, for example, the critical temperature in the above mean-field theory is the same for the ferromagnet and antiferromagnet. Under the symmetry transformation, a magnetic field would go into a “staggered field” that alternates from one sublattice to another. 2