1.1 Newtonian model of classical mechanics with its inertial frames and action at a distance and the Standard Model of particle physics with its hierarchy and unnaturalness. We discuss famous models of condensed matter physics, an obvious subject for our center.
1.2 Condensed matter theory is filled with many hundreds of models because of the vast scope of the subject.
2. We provide and discuss the top-20 list of the most famous condensed matter physics models of the 20th century in this thread. Let us know what we missed. A top-20 list is necessarily subjective, particularly for a subject as broad as condensed matter physics.
2.1 First, a semantic note: To qualify for this list, the candidate theory must be referred to as a ‘model’ in the literature by everybody. Yes, it must be a model for it to make it to our list of top-20 models! The list is chronological and stops in 1990.
3. Drude model (1900) is still used extensively to understand the electrical conductivity of electronic materials, an amazing fact because the model is originally pre-quantum and assumes electrons to be essentially like little billiard balls moving around.
3.1 They collide with impurities and defects, but later quantum theories validate the basic kinetic approach of the model based on the existence of Fermi surface in metals.
4. Einstein’s (1906) model of quantized thermal excitations (‘phonons’ in analogy with ‘photons’, the quanta of radiation introduced by Planck in 1900) in solids, and its slightly modified version by Debye (1912), is the first theoretical model in quantum physics.
4.1 Originally introduced to understand the failure of the Dulong-Petit law of specific heat. These phonon models are still used extensively!
5. Ising Model (1920-25) is perhaps the most well-known and the most paradigmatic condensed matter model ever, originally developed in the context of magnetism, involving simple scalar (spin up or down) localized spin-spin coupling in lattices of arbitrary dimensions.
5.1 Both classical and quantum versions are studied extensively in critical phenomena. The 2D Ising Model solution by Onsager is one of the most important theoretical papers in physics. The 3D version is still unsolved. Curiously, the model was introduced by Lenz, not Ising!
6 Free Electron Jellium Model (originally introduced by Sommerfeld in 1927 and still used extensively) is the simplest model of electrons in solids, where background lattice effects are ignored, assuming the positive charge to be distributed as a jelly to keep the system neutral.
6.1 If band structure effects are included in the leading order scattering theory, it becomes the ‘nearly free electron model’. If interaction effects among the electrons are included in the theory, the model becomes the ‘electron gas’ or the ‘electron liquid’ model.
7. Heisenberg Model (1928) is a quantum vector version of the Ising Model where the spins can point in different spatial directions and are represented by Pauli matrices.
7.1 It is an extensively used spin model for studying magnetic properties and statistical mechanics. It is a paradigmatic model in the study of quantum critical phenomena.
8. Stoner (1938) applied the molecular field theory of Weiss (1907) to the problem of ferromagnetism in an itinerant electron gas with exchange interaction, deriving a simple formula (“Stoner criterion”) for how strong the exchange coupling must be for ferromagnetism to occur.
8.1 This is a classic mean field model of a phase transition, which is still used in some forms in numerical band theories of ferromagnetism for many practical applications.
9. Frohlich model (1950) of electron-phonon interaction is the first modern many body Hamiltonian studying interaction between almost-free electrons and thermal excitations leading eventually to the BCS model explaining superconductivity.
9.1 It remains a central model in condensed matter physics extensively applied to study electrical conductivity, lattice distortions, thermal conductivity, and superconductivity.
10. Tight binding model (1954), based on the linear combination of atomic orbitals (LCAO) theory of chemical bonding in molecules, is complementary to the free electron band model.
10.1 It describes electron bands as arising from almost localized electrons near lattice sites with electrons weakly tunneling (“hopping”) between lattice sites to produce band motion. It is used extensively in materials showing strong effects of electron correlations.
11. Fermi liquid model (1955-1965), a central model in condensed matter, assumes that strongly interacting electron systems are all adiabatically connected to the noninteracting Fermi gas in spite of strong interactions between electrons.
11.1 The properties of the Fermi liquid can be understood as arising from weakly interacting long-lived quasiparticles. The existence of bands in solids is consistent with the Fermi liquid model.
12. Anderson localization model (1958) is a paradigmatic model for studying disorder effects where electrons hop between lattice sites in the tight binding manner, with disorder in the lattice producing strong scattering of the electron waves.
12.1 Leading to localization if the scattering is strong. It is the standard model for studying disorder induced metal-insulator transition.
13. Holstein model (1959) is an electron-phonon interaction model, complementary to the Frohlich model, where the tight binding description is used for the electronic hopping, so the lattice effects are strong.
13.1 It is still used extensively to study effects of electron-phonon interactions in narrow band materials.
14. Anderson impurity model (1961) deals with interaction between tight binding electrons and local magnetic moments in the lattice due to magnetic impurities. A standard model for studying magnetic moments and magnetism and led to the subsequent Hubbard model and Kondo model.
15. Lieb-Mattis model (1962) is a paradigmatic model for antiferromagnetism and quantum phase transition, explicitly showing the emergence of spontaneous symmetry breaking in a bipartite lattice of interacting quantum spins.
16. Hubbard model (1963) is the extensively used paradigm for ‘strongly correlated materials’ where electrons hopping on a tight binding lattice interacts strongly only when they occupy the same lattice site (and hence of opposite spins by Pauli principle).
16.1 The model was introduced to explain ferromagnetism and is now ironically a model for antiferromagnetism in narrow band systems.
17. Kondo model (1964) describes the interaction between one localized magnetic moment and a free electron gas scattering from this moment. It was introduced to understand the logarithmically increasing resistivity in certain metals with lowering of temperature.
17.1 It was eventually solved by Wilson in his famous renormalization group theory in 1970.
18. LDA model (1965) is the cornerstone of band theory combining many particle Schrodinger equation in solids with Fermi liquid model providing a specific computational algorithm for calculating energy bands.
18.1 One of the most-used models in physics, LDA appears in some form in all of the most-cited physics publications.
19. Nonlinear sigma model, originally introduced in field theory in 1960, started seeing its extensive use in condensed matter physics starting around 1979 in the context of localization, spin physics, and critical phenomena, and has now become a standard condensed matter mode.
20. Haldane Model (1988) realized in a time reversal invariance broken honeycomb lattice demonstrates the existence of a quantum Hall effect without any external magnetic field.
20.1 The model has different topological phases with different Chern numbers and turns out to be a precursor to the active subject of quantum spin Hall effect and topological insulator during the 21st century.
21. t-J model (1988) is a specific simplification of the Hubbard model in the context of holes in doped antiferromagnets near half-filling, a situation considered important in understanding high-T_c cuprate superconductors.
21.1 The model dates back to the late 1970s in some form but became significant only after it was applied to cuprates to study their strongly correlated behavior.
22. RVB, the resonating valence bond, model (1988) is another model introduced in condensed matter physics in the context of cuprate superconductors. Its origin (1938) is in molecular chemistry in molecules like benzene where the chemical bonds are resonant.
22.1 However it took on a whole new life as a paradigm for strongly correlated non-Fermi liquid ground states such as spin liquids.
Postscript 1: We stop at 1990 since the 20 model limit is reached already, and the 1990-2010 period is recently covered in a separate tweet by CMTC’s own @MBarkeshli
Postscript 2: It is curious that certain topics recur over and over in different models, for example, Anderson impurity model, Hubbard model, Kondo model, and t-J model. Another example is Free Electron Jellium model, Fermi liquid model, and LDA model.
Postscript 3: The reason condensed matter theory must focus on models, in spite of the fundamental Hamiltonian controlling the system being well-known, is the extreme complexity of a large collection of interacting quantum particles.
Postscript 3.1: Ultimately, the quantum many body problem is an intractable problem in an exponentially large Hilbert space, and the only way to make progress is by making drastic approximations through models. See, “Are There Ultimate Laws of Physics?” physics.umd.edu/cmtc/blog.html
