Breakthroughs in Condensed Matter Physics ============================================ .. include:: /_generated/nav/condensed.rst .. epigraph:: "More is different." -- P. W. Anderson, *Science*, 1972 Condensed matter physics is the study of what happens when enormous numbers of quantum particles are put together: emergent order, broken symmetries, and -- since the 1980s -- emergent *topology*. This chronology traces the major conceptual breakthroughs behind :mod:`physicskit.condensed`, from Bloch's 1928 theorem to the topological superconductors of the 21st century. Every stop has a pointer to the corresponding implementation in this package, a structural diagram of the system it describes, and a short, runnable example reproducing the milestone's signature observable. .. contents:: Timeline :local: :depth: 1 1928 -- Bloch's Theorem and Band Theory ------------------------------------------ Felix Bloch showed that the eigenstates of a single electron in a perfectly periodic crystal potential :math:`V(\mathbf{r}) = V(\mathbf{r} + \mathbf{R})` take the form .. math:: \psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}\, u_{n\mathbf{k}}(\mathbf{r}), \qquad u_{n\mathbf{k}}(\mathbf{r} + \mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r}), turning the Schrodinger equation :math:`H\psi = E\psi` for an infinite solid into a finite eigenvalue problem :math:`H(\mathbf{k})u_{n\mathbf{k}} = E_n(\mathbf{k}) u_{n\mathbf{k}}` at each crystal momentum :math:`\mathbf{k}`, periodic on the Brillouin zone torus. This single idea -- that a crystal's electronic structure decomposes into bands :math:`E_n(\mathbf{k})` -- is the foundation of every model in this package. *Implementation:* :class:`physicskit.condensed.tight_binding.Hamiltonian` constructs :math:`H(\mathbf{k})` directly by Bloch-summing real-space hoppings; :meth:`~physicskit.condensed.tight_binding.Hamiltonian.bands` returns :math:`E_n(\mathbf{k})`. :func:`~physicskit.condensed.tight_binding.build_finite_cluster` truncates that same translational symmetry to a finite, open chain, so :func:`~physicskit.condensed.visualizers.plot_lattice_structure` can draw the real-space structure Bloch's theorem is summing over directly, alongside the reciprocal-space band. *References:* F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. 52, 555-600 (1929). .. minigallery:: ../../examples/condensed/tight_binding/plot_bloch_bands.py 1930 -- Landau Levels and Quantum Diamagnetism -------------------------------------------------- Lev Landau solved the quantum mechanics of a charged particle in a uniform magnetic field, finding that the continuous kinetic energy spectrum collapses into discrete, macroscopically degenerate levels .. math:: E_n = \hbar\omega_c\left(n + \tfrac{1}{2}\right), \qquad \omega_c = \frac{eB}{m}, \qquad n = 0, 1, 2, \dots Quantized cyclotron orbits are the microscopic origin of orbital (Landau) diamagnetism in metals, and, decoupled from any lattice, the direct ancestor of the quantum Hall effect fifty years later. Each level's macroscopic degeneracy per unit area, .. math:: n_B = \frac{1}{2\pi\ell_B^2} = \frac{eB}{2\pi\hbar}, \qquad \ell_B = \sqrt{\frac{\hbar}{eB}}, set by the magnetic length :math:`\ell_B`, is what makes a partially filled Landau level's filling factor :math:`\nu = n_e/n_B` the natural variable of the quantum Hall effect below. *Implementation:* :func:`physicskit.condensed.landau_levels.landau_level_energies`, :func:`~physicskit.condensed.landau_levels.landau_degeneracy`, and :func:`~physicskit.condensed.landau_levels.filling_factor` implement this continuum solution directly; the Peierls substitution in :func:`physicskit.condensed.tight_binding.apply_peierls_phase` is the lattice (tight-binding) route to the same physics -- turning a uniform flux into Hofstadter-like, near-degenerate bands on a finite lattice, and sweeping the flux continuously from 0 to 1 traces out the full Hofstadter butterfly itself. *References:* L. Landau, "Diamagnetismus der Metalle," Z. Phys. 64, 629-637 (1930). .. minigallery:: ../../examples/condensed/landau_levels/plot_landau_levels.py .. minigallery:: ../../examples/condensed/tight_binding/plot_peierls_landau_levels.py 1950 -- Ginzburg-Landau Theory ------------------------------------ Vitaly Ginzburg and Lev Landau proposed a phenomenological theory of superconductivity built entirely on symmetry, seven years before BCS supplied a microscopic mechanism: expand the free energy in a complex order parameter :math:`\psi(\mathbf{r})` (the superconducting condensate wavefunction) and its gradient, .. math:: f[\psi] = a|\psi|^2 + \frac{b}{2}|\psi|^4 + \frac{\hbar^2}{2m}|\nabla\psi|^2, and let symmetry and stability alone fix the physics. Minimizing :math:`f` gives a nonzero equilibrium :math:`|\psi_0|^2 = -a/b` below the transition (:math:`a<0`) and, from the competition between the gradient term and the electromagnetic coupling absorbed into it, two emergent length scales: the coherence length :math:`\xi` over which :math:`\psi` heals from a boundary back to :math:`\psi_0`, and the magnetic penetration depth :math:`\lambda`. Their ratio, the Ginzburg-Landau parameter :math:`\kappa=\lambda/\xi`, alone decides whether flux is excluded entirely (Type I, :math:`\kappa<1/\sqrt2`) or threads the material as a vortex lattice (Type II, :math:`\kappa>1/\sqrt2`). This order-parameter, broken-symmetry language -- rather than any microscopic pairing mechanism -- is exactly what reappears at the 2D XY model's Kosterlitz-Thouless transition below. *Implementation:* :func:`physicskit.condensed.ginzburg_landau.gl_equilibrium_order_parameter`, :func:`~physicskit.condensed.ginzburg_landau.gl_coherence_length`, :func:`~physicskit.condensed.ginzburg_landau.gl_penetration_depth`, and :func:`~physicskit.condensed.ginzburg_landau.ginzburg_landau_parameter` compute exactly these quantities; :func:`~physicskit.condensed.ginzburg_landau.gl_order_parameter_profile` is the exact healing profile at a boundary. *References:* V. L. Ginzburg and L. D. Landau, "On the theory of superconductivity," Zh. Eksp. Teor. Fiz. 20, 1064-1082 (1950); the Type-II vortex lattice that :func:`~physicskit.condensed.ginzburg_landau.ginzburg_landau_parameter` selects between was itself solved by A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442-1452 (1957) [Sov. Phys. JETP 5, 1174-1182 (1957)]. .. minigallery:: ../../examples/condensed/ginzburg_landau/plot_ginzburg_landau_healing.py 1954 -- The Slater-Koster Tight-Binding Framework ------------------------------------------------------ John Slater and George Koster showed how to build realistic band structures from a *minimal* empirical basis: a linear combination of atomic orbitals (LCAO), with hopping matrix elements between neighboring orbitals as fitting parameters rather than computed from first principles. This traded first-principles rigor for tractability and physical transparency, and remains the standard language for model-building in condensed matter theory to this day. *Implementation:* :class:`physicskit.condensed.tight_binding.Lattice` and :class:`physicskit.condensed.tight_binding.Hamiltonian` are a direct, general-purpose realization of the Slater-Koster LCAO philosophy -- specify orbitals, specify hoppings, Bloch-sum the result. *References:* J. C. Slater and G. F. Koster, "Simplified LCAO Method for the Periodic Potential Problem," Phys. Rev. 94, 1498-1524 (1954). .. minigallery:: ../../examples/condensed/tight_binding/plot_slater_koster_bands.py 1957 -- BCS Theory of Superconductivity -------------------------------------------- Bardeen, Cooper, and Schrieffer explained superconductivity as a condensate of electron pairs (Cooper pairs) bound by an effective phonon-mediated attraction, however weak. Pairing opens a gap :math:`\Delta` in the single-particle excitation spectrum, and the resulting condensate flows without dissipation. The Bogoliubov-de Gennes (BdG) formalism recasts this as a single-particle-like problem in an enlarged Nambu (particle-hole) space, diagonalized to give the quasiparticle spectrum :math:`E(\mathbf{k}) = \sqrt{\xi(\mathbf{k})^2 + |\Delta(\mathbf{k})|^2}`. *Implementation:* :func:`physicskit.condensed.correlated.bdg_bcs_hamiltonian` and :func:`physicskit.condensed.correlated.bdg_spectrum` build and diagonalize exactly this s-wave BdG Hamiltonian. *References:* J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Theory of Superconductivity," Phys. Rev. 108, 1175-1204 (1957); the Bogoliubov-de Gennes formalism: N. N. Bogoliubov, Zh. Eksp. Teor. Fiz. 34, 58-65 (1958); J. G. Valatin, Nuovo Cimento 7, 843-857 (1958). .. minigallery:: ../../examples/condensed/correlated/plot_bcs_gap.py 1958 -- Anderson Localization ------------------------------------ Philip Anderson showed that quenched, random disorder is not a small perturbative correction to a metal's conductivity but can halt transport outright: interference between all the scattering paths off a random potential exponentially localizes *every* electronic eigenstate, at any nonzero disorder strength, in one and two dimensions (in three dimensions only above a critical disorder, the mobility edge). A localized state's probability density decays as .. math:: |\psi(x)|^2 \sim e^{-2|x-x_0|/\xi}, with localization length :math:`\xi`; an extended state instead spreads over the whole system, distinguished numerically by the inverse participation ratio :math:`\text{IPR}=\sum_i|\psi_i|^4/\left(\sum_i|\psi_i|^2\right)^2`, which is :math:`O(1/N)` when extended and :math:`O(1)` when localized. This is the disorder physics that broadens the sharp Landau levels above into the finite-width plateaus actually measured in the quantum Hall effect below. *Implementation:* :func:`physicskit.condensed.anderson_localization.anderson_chain_hamiltonian` builds the disordered 1D tight-binding chain; :func:`~physicskit.condensed.anderson_localization.inverse_participation_ratio` and :func:`~physicskit.condensed.anderson_localization.localization_length` diagnose localization directly from its eigenstates. *References:* P. W. Anderson, "Absence of Diffusion in Certain Random Lattices," Phys. Rev. 109, 1492-1505 (1958). .. minigallery:: ../../examples/condensed/anderson_localization/plot_anderson_localization.py 1963 -- The Hubbard Model and Correlated Electrons -------------------------------------------------------- John Hubbard (with independent, essentially simultaneous work by Martin Gutzwiller and Junjiro Kanamori) proposed the minimal model of interacting electrons on a lattice: band electrons that hop between sites, penalized by a Coulomb repulsion :math:`U` whenever two electrons of opposite spin occupy the same site. Deceptively simple, it is the standard model for Mott insulators, high-:math:`T_c` cuprate superconductivity, and quantum magnetism -- and, apart from special limits and one dimension, it remains unsolved in general. .. math:: H = -t \sum_{\langle ij \rangle, \sigma} c_{i\sigma}^\dagger c_{j\sigma} + U \sum_i n_{i\uparrow} n_{i\downarrow} Where :math:`t` is the nearest-neighbor hopping amplitude, :math:`U` the onsite Coulomb repulsion, and :math:`n_{i\sigma} = c_{i\sigma}^\dagger c_{i\sigma}` the number operator for spin :math:`\sigma` on site :math:`i`. As :math:`U/t \to \infty` at half filling, double occupancy is suppressed entirely and the itinerant metal is driven into a Mott insulator of singly occupied, localized moments. *Implementation:* :func:`physicskit.condensed.correlated.hubbard_1d_exact_diagonalization` builds and diagonalizes the full Fock-space Hamiltonian in a fixed :math:`(n_\uparrow, n_\downarrow)` sector for small clusters. *References:* J. Hubbard, "Electron Correlations in Narrow Energy Bands," Proc. R. Soc. A 276, 238-257 (1963); M. C. Gutzwiller, Phys. Rev. Lett. 10, 159-162 (1963); J. Kanamori, Prog. Theor. Phys. 30, 275-289 (1963). .. minigallery:: ../../examples/condensed/correlated/plot_hubbard_mott_transition.py 1973 -- The Kosterlitz-Thouless Phase Transition ------------------------------------------------------- Kosterlitz and Thouless (with foundational input from Berezinskii) identified a phase transition in 2D systems with continuous symmetry that has no local order parameter at all: it is driven instead by the unbinding of topological vortex-antivortex pairs. This was one of the earliest demonstrations that *topology*, not symmetry breaking alone, can organize a phase transition -- a theme that would come to dominate condensed matter theory a decade later. .. math:: E_{\text{vortex}} = \pi J \ln(L/a), \qquad k_B T_{KT} \approx \frac{\pi}{2}J Where :math:`J` is the spin-stiffness (coupling) of the 2D XY model, :math:`a` a microscopic core cutoff, and :math:`L` the system size: a single vortex costs energy that diverges logarithmically with system size, but its *entropy* of placement diverges the same way, so a bound vortex-antivortex pair (net energy independent of :math:`L`) is thermodynamically favored below :math:`T_{KT}` and unbinds above it. *Not implemented as a fermionic band or pairing model* -- :mod:`physicskit.condensed` targets those rather than classical XY spin textures, so this transition's actual implementation lives in :mod:`physicskit.statphys` instead: :class:`physicskit.statphys.chapters.ising_lattice.XYModel2D` Metropolis- samples the classical 2D XY model directly (built on the numba-jitted sweep and vorticity kernels in :mod:`physicskit.statphys.core.monte_carlo`), and :meth:`~physicskit.statphys.chapters.ising_lattice.XYModel2D.vortex_count` tracks exactly the bound-pair-to-plasma unbinding described above. Conceptually, within this package, it is the direct ancestor of the topological (rather than symmetry-breaking) organizing principle that TKNN and Haldane apply to electronic bands below. *References:* J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181-1203 (1973), with the two-years-earlier precursor V. L. Berezinskii, Sov. Phys. JETP 32, 493-500 (1971). .. minigallery:: ../../examples/statphys/xy_model/plot_kt_vortex_unbinding.py 1979 -- Su-Schrieffer-Heeger (SSH) Model and Topological Solitons --------------------------------------------------------------------- Su, Schrieffer, and Heeger showed that a 1D dimerized chain -- alternating strong (intracell) and weak (intercell) bonds, modeling polyacetylene -- hosts domain-wall solitons pinned to the boundary between the two possible dimerization patterns. Cast on an *open* finite chain, the same physics produces a pair of protected, exponentially localized zero-energy states, one at each end. It is the earliest and simplest example of the bulk-boundary correspondence that would come to define topological band theory: a bulk topological invariant (the Zak phase) predicting the existence of boundary modes. .. math:: H(k) = \begin{pmatrix} 0 & v + w e^{-ik} \\ v + w e^{ik} & 0 \end{pmatrix} Where :math:`v` is the intracell hopping amplitude and :math:`w` the intercell hopping amplitude, each real. The bulk gap closes only at :math:`v = w`; for :math:`v < w` the chain is topological (Zak phase :math:`\pi`, protected zero-energy edge modes under open boundaries), and for :math:`v > w` it is trivial (Zak phase :math:`0`, no edge modes). *Implementation:* :func:`physicskit.condensed.models.ssh_hamiltonian` (Bloch form) and :func:`physicskit.condensed.models.ssh_lattice_hamiltonian` (real-space builder, for use with :func:`~physicskit.condensed.tight_binding.build_ribbon`); :func:`physicskit.condensed.topology.zak_phase` computes the bulk invariant. *References:* W. P. Su, J. R. Schrieffer, and A. J. Heeger, "Solitons in Polyacetylene," Phys. Rev. Lett. 42, 1698-1701 (1979). .. minigallery:: ../../examples/condensed/topology/plot_ssh_edge_states.py 1980 -- The Integer Quantum Hall Effect -------------------------------------------- Klaus von Klitzing, Gerhard Dorda, and Michael Pepper discovered that the Hall conductance of a two-dimensional electron gas in a strong magnetic field is quantized to extraordinary precision, .. math:: \sigma_{xy} = \nu\,\frac{e^2}{h}, \qquad \nu \in \mathbb{Z}, independent of sample details, geometry, or disorder. The quantization was so precise it now defines the SI standard of electrical resistance; von Klitzing alone was awarded the 1985 Nobel Prize in Physics for the discovery. The puzzle of *why* an integer emerges with such robustness would be resolved two years later. *Implementation:* the effect itself is a bulk transport measurement, but its topological explanation is directly reproducible on the lattice: :func:`physicskit.condensed.models.harper_hofstadter_hamiltonian` builds the Bloch Hamiltonian of a 2D electron gas on a lattice threaded by a rational flux :math:`p/q` per plaquette (the lattice route to a magnetic field, complementing the continuum :mod:`physicskit.condensed.landau_levels` picture used to motivate it above), and feeding its :math:`q` magnetic sub-bands one at a time into :func:`~physicskit.condensed.topology.compute_chern_number` (the same TKNN machinery applied to Haldane's model below) gives the exactly quantized Hall conductance at every gap directly, as a sum of individually integer Chern numbers. *References:* K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494-497 (1980). .. minigallery:: ../../examples/condensed/topology/plot_iqhe_hall_conductance.py 1982 -- The TKNN Invariant -------------------------------- Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) showed that the quantized Hall conductance is a topological invariant: the integral of the Berry curvature :math:`\Omega(\mathbf{k})` of the occupied Bloch bands over the Brillouin zone, .. math:: C = \frac{1}{2\pi}\int_{\mathrm{BZ}} \Omega(\mathbf{k})\, d^2k \in \mathbb{Z}, now called the (first) Chern number. Because :math:`C` can only change when a bulk energy gap closes, it is invariant under any smooth perturbation that preserves the gap -- explaining the extreme precision of von Klitzing's measurement as a topological, rather than a merely material, fact. This is the single idea that turned "band theory" into "topological band theory." *Implementation:* :func:`physicskit.condensed.topology.compute_chern_number` and :func:`physicskit.condensed.topology.compute_berry_curvature` implement the Fukui-Hatsugai-Suzuki lattice discretization of exactly this integral, returning exactly quantized integers. *References:* D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405-408 (1982). .. minigallery:: ../../examples/condensed/topology/plot_chern_number.py 1982-1983 -- The Fractional Quantum Hall Effect and Laughlin's Wavefunction ------------------------------------------------------------------------------- Daniel Tsui, Horst Stormer, and Arthur Gossard found that an even cleaner two-dimensional electron gas in a strong magnetic field develops Hall plateaus not only at integer filling but at simple fractions, :math:`\nu = 1/3` foremost among them -- a result the single-particle Landau-level picture behind the integer effect above cannot explain at all, since a partially filled Landau level is macroscopically degenerate and gapless without electron-electron interactions. Robert Laughlin supplied the missing many-body mechanism the following year: a trial wavefunction .. math:: \Psi_{1/m}(z_1,\dots,z_N) = \prod_{i