The Art and Science of Model-Building

1. The Art and Science of Model-Building Model is a word with too many meanings. Here I want to discuss two particular types of “model” useful in physics: 1] The realistic or quasirealistic Low Energy Effective Theory 2] The “Model” as an aid to understanding a general type of behavior or a perceived anomaly in experiment. I am not particularly interested in 3] Models for the sake of models: answers in search of a question; but this is a very active specialty.

2. low-energy effective theories This concept has a long and honorable history in CM theory. The oldest example, I think, is projective theories, starting with Van Vleck’s effective Hamiltonians for dilute magnetic ions in insulating solids in the ‘30’s-40’s leading to “ligand field theory” (Orgel etc) where the ion’s wave functions mix with those of surrounding atoms, but the form of the Hamiltonian is still a simple pronuct of HionxHsubstrate, and one can project out the substrate. This was the scheme I followed in creating the “Hubbard model” in ‘59 for interacting magnetic ions, introducing Mott-Hubbard “U” to split off wrong valencies. I saw it as a realistic description of the low-energy subspace, and the papers go to great trouble to make it so--not a plaything. Hubbard generalized to a theory of metals, (see slide) and Rice and others introduced the projective canonical transformation to the “t-J model” (slide). NOTE on projective theories: they work surprisingly well because matrix elements which mix the spaces tend to make the energy gap between them larger, not smaller.

3. Strong correlation in CuO2 plane Cu2+ Large U charge-transfer gap Dpd ~ 2 eV best evidence for large U metal? Mott insulator antiferromagnet J~1400 K doping Hubbard t = 0.3 eV, U = 2 eV, J = 4t2/U = 0.12 eV

4. THE t-J model This comes by a hypothetically exact perturbative transformation of the Hubbard model. It is H= For what J does see next slide; t is the PROJECTED kinetic energy

5. continuous RNG methods Familiar to high-energy theorists: the “running coupling constants” of QED, QCD from the true renormalization group of relativistic field theories. But this is NOT the same as the RNG of condensed matter theory--we introduced concepts like relevance and universality: the Kadanoff-Wilson-Fisher theory of critical phenomena (I hope this has been discussed) Still a third way: the “poor-man’s” methods of PWA and Wilson for the Kondo problem, borrowed by Shankar for the Landau problem of the Fermi surface (see slide) (Recently I’ve added another wrinkle, irrelevant here but cute--see slide)

6. shankar renormalization cutoffW k-space Fermi surface U

7. final stage k space U Fermi Surface only interactions left: forward and backward scattering

8. But if U>W??? It doesn’t work! project, break connection,then renormalize! k space Philip Anderson: break connection W U pper Hubbard Fermi surface the broken connection means a new k-space: “hat” operators.

9. unrealistic models used for understanding Examples: localization, spin glass. Experiment defies the accepted canon; what to do? (Lehrer: “plagiarize, plagiarize, plagiarize!”) Answer: build a model where the accepted canon “obviously” works, show that it doesn’t and why! Localization: Experiments unequivocally defied canon (show) There are several “AM”s but actually the “model” aspect was to neglect interaction =non-linearity. In a linear system without disorder transport obviously occurs; I showed that when disordered enough, it doesn’t. Second example: spin glass. It took a little imagination to see crude experiments defied canon: the key is spectacular non-linearity (turned out later to indeed be a deeply-involved measure of non-ergodicity). Random alloy of Cu spins in Au--no exact model useful. Instead created the Edwards-Anderson model; regular lattice with random exchange interactions:

10. schematic of P donors in Si with 5%Si29 (S=1/2)

11. EPR spectra as function of donor concentration

12. Dig a “hole” at one field and study EPR as function of a low(NMR) frequency: each Si29 retains its own frequency.

14. Turns out to be soluble in mean field (Edwards-Anderson) but not quite as simply as we expected (Sherrington-Kirkpatrick) Key concepts: “frustration” (PWA, Toulouse); “Non-ergodicity” (Palmer) ramifications in computer science, neural nets, glass?, etc example of a bad model: the Mattis model Ji,j=f(I)f(j) No frustration, no physics.

15. A recent canon violation: the vortex liquid nonlinear diamagnetism in “normal” metals (cuprates above Tc) The unrealistic model: Kosterlitz-Thouless, 2D superfluid: show divergent diamagnetic susceptibility above Tc conclusion: normal Bose fluids are not “normal”.

16. for discussion are Kondo-Anderson and Hubbard models class 1 or class 2? Other examples? The sandpile as class 2? fluid dynamics as class 1? Are economic models ALL bad ?