Many-body physics of solids, liquids, and soft matter. Band theory, Bloch theorem, Fermi surfaces, phonons, superconductivity (BCS), superfluidity, magnetism, quantum Hall effect, topological phases.
Condensed Matter Physics
Bloch's theorem
Eigenstates of a periodic Hamiltonian are ψ_{nk}(r) = e^{ik·r} u_{nk}(r) with u periodic; labels = band index n, crystal momentum k.
Brillouin zone
Wigner–Seitz cell of the reciprocal lattice. The first BZ is the fundamental domain of crystal momentum under lattice translations.
Electronic band theory
Electrons in a crystal occupy Bloch states grouped into bands. Metal/insulator/semiconductor determined by whether the Fermi energy lies…
Fermi surface
Surface in k-space of states at the Fermi energy for a metal. Shape/topology determines transport, thermodynamic, and magnetic properties.
Phonon
Quantised lattice vibration; bosonic excitation with dispersion ω(q). Responsible for heat capacity (Debye), sound, phonon-mediated…
BCS superconductivity
Bardeen–Cooper–Schrieffer (1957): phonon-mediated Cooper pairing of electrons opens a gap Δ ~ 2ℏω_D e^{−1/NV}; causes zero resistivity and…
Meissner effect
Expulsion of magnetic field from a superconductor below T_c; not merely infinite conductivity but a thermodynamic phase transition.
Superfluidity
Frictionless flow of a Bose–Einstein condensed fluid (⁴He below 2.17 K, ³He Cooper-paired, cold-atom BECs). Landau critical velocity…
Ferromagnetism
Spontaneous ordering of electronic spins below the Curie temperature, driven by exchange interaction. Heisenberg Hamiltonian H = −J Σ…
Ising model
Lattice of classical ±1 spins with H = −J Σ s_i s_j − h Σ s_i. Exactly solved in d = 1 (no transition) and d = 2 (Onsager 1944). Paradigm…
Quantum Hall effect
Quantised transverse conductance σ_xy = ν e²/h in a 2D electron gas under strong B, at integer (IQHE) or fractional (FQHE) filling factors…
Topological insulator
Bulk-insulating phase with symmetry-protected conducting surface/edge states. Characterised by a Z₂ invariant (strong TI) arising from…
Density functional theory (DFT)
Hohenberg–Kohn theorems: ground-state observables are functionals of the density n(r); Kohn–Sham equations map the interacting system to a…
Electronic band structure
The set of single-particle energy eigenvalues E_n(k) of electrons in a crystal, parameterised by band index n and crystal momentum k in the…
Anyons & non-Abelian statistics
2D quasiparticles with exchange phase ≠ ±1 (Abelian) or non-commuting (non-Abelian). Fibonacci/Ising anyons; basis of topological quantum…
DFT in solids (KS + pseudopotentials)
Kohn-Sham in periodic BZ with planewaves + pseudopotentials (NCPP, USPP, PAW). LDA, GGA, hybrid (HSE). Workhorse for materials simulation.
Hubbard model
t-U competition on a lattice; Mott metal-insulator transition at half-filling, large U. Minimal model for correlated electrons; cuprate…
Kondo effect
Resistivity minimum at low T from magnetic-impurity/conduction-electron spin-flip scattering. RG flow to strong coupling; screening of…
Landau Fermi liquid theory
Interacting electrons adiabatically connected to free gas; quasiparticles with renormalized m*, Z; F_s, F_a Landau parameters; breaks down…
Luttinger liquid (1D)
Bosonization: charge/spin separation; no quasiparticles; power-law correlators controlled by Luttinger parameter K; observed in carbon…
Mott insulator and Hubbard U
Strong on-site repulsion U > W drives metal-insulator transition at half filling; super-exchange J = 4t²/U for AFM Heisenberg limit.
Fractional quantum Hall effect
σ_xy = (p/q) e²/h at rational fillings ν = 1/3, 2/5...; Laughlin wavefunction; anyonic quasi-particles with fractional charge/statistics.
Laughlin wavefunction
Ψ_m = Π(z_i-z_j)^m e^(-Σ|z_i|²/4); exact FQH ground state at ν=1/m for odd m; quasi-holes with charge e/m.
Z₂ topological insulator
Time-reversal-protected surface states with odd number of Dirac crossings; ν = 0 or 1 mod 2; HgTe/CdTe quantum wells, Bi₂Se₃ 3D.
Weyl and Dirac semimetals
Linearly dispersing band crossings protected by symmetry; Weyl nodes carry Berry-curvature monopoles ±1; Fermi arcs, chiral anomaly in TaAs.
Chern insulator (QAH)
Broken-TRS band with nonzero Chern number C = (1/2π)∫F; quantized anomalous Hall σ_xy = Ce²/h; Haldane model (1988), observed in…
BCS gap equation
1 = gN(0)∫tanh(√(ξ²+Δ²)/2T)/(2√(ξ²+Δ²)) dξ; T_c = 1.14 ω_D e^(-1/gN(0)); 2Δ(0)/T_c ≈ 3.53.
Ginzburg–Landau theory
F = α|ψ|² + (β/2)|ψ|⁴ + (1/2m*)|(−iℏ∇−e*A/c)ψ|² + B²/8π; coherence ξ, penetration depth λ; κ=λ/ξ distinguishes type I/II.
Abrikosov vortex lattice
Type-II superconductor in H_c1<H<H_c2 hosts triangular flux-line lattice; each vortex carries quantized flux Φ₀ = h/2e.
Josephson effect
DC: I = I_c sin φ; AC: V = (ℏ/2e) dφ/dt, radiates at 483.6 MHz/μV; basis of SQUIDs (10⁻¹⁵ T sensitivity) and voltage standard.
Kosterlitz–Thouless transition
2D XY-model vortex unbinding at T_KT; infinite-order topological transition; superfluid films, XY magnets; 2016 Nobel.
Anderson localization
Sufficient disorder localizes quantum states ψ ~ e^(-r/ξ); scaling theory: 1D/2D always localized, 3D has mobility edge; experiments in…
Kondo screening
Magnetic impurity in metal dynamically screened by conduction e⁻ below T_K = D e^(-1/2Jρ); resistance minimum; exactly solved by numerical…
Spin glass & replica symmetry breaking
Frustrated disordered magnets (e.g. CuMn); Edwards–Anderson order parameter; Parisi RSB solution of SK model (2021 Nobel).
High-T_c cuprate superconductors
T_c up to 133 K (HgBa₂Ca₂Cu₃O₈); d-wave pairing; pseudogap, strange metal normal state; likely unconventional electronic pairing mechanism.
Graphene Dirac cones & anomalous QHE
2D honeycomb; massless Dirac fermions at K, K'; σ_xy = (n+½)(4e²/h); Klein tunneling; ballistic transport over μm (2010 Nobel).
Heavy-fermion systems
4f/5f intermetallics (UPt₃, CeCu₂Si₂): m*/m ~ 100-1000 from Kondo-lattice; unconventional SC near quantum critical point.
Quantum critical points
T=0 continuous transitions driven by g; quantum fluctuations yield non-Fermi-liquid behaviour; marginal Fermi liquid, strange metals.
Quantum spin liquids
No magnetic order at T=0 despite strong interactions; fractionalized spinons/visons; candidates: herbertsmithite kagome, Kitaev α-RuCl₃.
Magnetic skyrmions
Topologically protected nanoscale spin textures in chiral magnets (MnSi, FeGe); topological charge Q integer; low-current depinning for…
Charge/spin density waves
Fermi-surface nesting at 2k_F drives periodic ρ(r) modulation with gap opening; Peierls transition in 1D conductors (TTF-TCNQ); CDW in…
Mott metal–insulator transition
V₂O₃, NiS₂₋ₓSeₓ tuned by U/W, P, doping; DMFT captures crossover; relevant to cuprate parent compounds.
Topological superconductors & Majoranas
p-wave pairing hosts Majorana zero modes at edges/vortices; non-Abelian braiding proposed for topological qubits (Kitaev chain, Sr₂RuO₄).
Tight-binding chain on a cycle graph (condensed-matter)
Condensed-matter application of L0 graph and spectral graph theory. Nearest-neighbour electron hopping with amplitude t on an N-site ring…
Su–Schrieffer–Heeger model (dimerised bipartite chain, condensed-matter)
Condensed-matter application of L0 bipartite graph and cycle graph. The Su–Schrieffer–Heeger 1979 model is a 1-D chain with alternating…