Quantum-coherent transport in conductors whose linear dimensions are smaller than the phase-coherence length L_φ but larger than the elastic mean free path ℓ — i.e. the regime where the wave nature of electrons matters macroscopically yet…
mesoscopic-physics
Landauer-Büttiker formula: G = (2e²/h)·Σ T_n
Landauer 1957 / 1970 and Büttiker 1985-88 recast electrical conduction in the mesoscopic regime as a scattering problem: an elastic,…
Aharonov-Bohm effect: gauge-invariant phase, period Φ_0 = h/e
Aharonov-Bohm 1959 (Phys. Rev. 115:485) — a charged particle encircling a region of magnetic flux Φ acquires a gauge-invariant phase φ =…
Wiedemann-Franz law: κ/(σT) = π²k_B²/(3e²)
Wiedemann-Franz 1853 empirical law — for pure metals the ratio of thermal to electrical conductivity is proportional to temperature, with a…
Landauer: N open modes give G = N·G_0 exactly
Closed-form consequence of the Landauer-Büttiker formula at full transmission. For a ballistic, disorder-free conductor supporting N…
AB period Φ_0 = h/e, phase-per-period = 2π (exact)
Closed-form evaluation of the Aharonov-Bohm phase identity. Gauge-invariant phase φ(Φ) = (2π/Φ_0)·Φ with Φ_0 = h/e. Period: the phase…
Lorenz number L_0·e²/k_B² = π²/3 (exact rational π²)
Closed-form rational pin of the Sommerfeld Lorenz number. L_0 = π²·k_B²/(3·e²); rescaled ratio L_0·e²/k_B² = π²/3 exactly — a…
Kondo one-loop RG: scaling equation dJ/d ln(Lambda) = -rho J^2; dynamical scale T_K = Lambda_0 exp(-1/(2 rho J_0))
Kondo one-loop renormalisation-group flow framework. Setup: a magnetic impurity with spin S in a metal of density of states rho couples…
UCF conductance variance identity: det(A A^T) = sum_{|S|=m} det(A_S)^2 (Cauchy-Binet)
Universal Conductance Fluctuation (UCF) Cauchy-Binet framework. Setup: in the Landauer-Buettiker formalism for mesoscopic conductance G =…
Aharonov-Bohm periodicity: current I(Phi) has period Phi_0 = h/e (U(1) covering space of annulus)
Aharonov-Bohm mesoscopic-ring framework (AB 1959 Phys Rev 115, 485). Setup: a metal ring encloses a magnetic flux Phi; electrons encircle…
Theorem: Kondo dynamical scale T_K = Lambda_0 * exp(-1/(2 rho J_0)) (RG integration)
Theorem (Kondo dynamical-scale canonical): starting from the poor-man's scaling equation dJ/d ln(Lambda) = -rho * J^2 with initial coupling…
Theorem: Cauchy-Binet residual det(A A^T) - sum det(A_S)^2 = 0 (2x3 UCF test matrix)
Theorem (UCF Cauchy-Binet identity canonical): for any real 2 x 3 matrix A with rows a = (a_11, a_12, a_13), a' = (a_21, a_22, a_23), the…
Theorem: AB periodicity I(Phi + Phi_0) - I(Phi) = 0 for I(Phi) = cos(2 pi Phi / Phi_0)
Theorem (Aharonov-Bohm periodicity canonical): for I(Phi) = cos(2 pi Phi / Phi_0), the difference I(Phi + Phi_0) - I(Phi) simplifies…
Aharonov-Bohm (1959)
Y Aharonov-D Bohm 1959 magnetic-flux affects phase via vector-potential despite zero-B; experimental confirmation Tonomura 1986.
Landauer-Buttiker (1985)
R Landauer 1957 + M Buttiker 1985 G = (e^2/h) sum T_nm conductance from S-matrix; basis of mesoscopic transport + quantum-point-contact.
Kondo effect (1964)
J Kondo 1964 log-T resistivity-minimum from magnetic-impurity; modern numerical-renormalization-group + heavy-fermion universal scaling.
Quantum Hall (Klitzing 1980)
K von Klitzing 1980 (Nobel 1985) integer-QH plateaus h/e^2 = 25812.8 ohms; modern resistance-standard; foundation of…
FQHE (Tsui-Stormer 1982)
Tsui-Stormer-Gossard 1982 (Nobel 1998); fractional plateaus 1/3 etc; Laughlin 1983 wavefunction; basis of anyons +…
UCF (Altshuler 1985)
B Altshuler-P A Lee 1985 universal-conductance-fluctuations e^2/h regardless of size; basis of mesoscopic-transport-statistics;…
Landauer (1957)
R Landauer 1957 G=2e²/h × Σ T_n quantized-conductance; modern foundational text in mesoscopic-transport + dissipation-bound.
Büttiker (1986)
M Büttiker 1986 multi-terminal Landauer-Büttiker; modern 4-probe magnetoresistance + Aharonov-Bohm phase + edge-state physics.
Aharonov-Bohm (1959)
Y Aharonov-D Bohm 1959 vector-potential phase Φ/Φ_0; modern foundational test of QM + topological-pumping + 2D Dirac fermions.
QPC (van Wees 1988)
B van Wees 1988 + Wharam 1988 GaAs/AlGaAs QPC G=N×2e²/h; modern modern 0.7-anomaly + cavity-QED + topological-QPC.
Coulomb blockade (Fulton-Dolan 1987)
T Fulton-G Dolan 1987 single-electron-transistor; modern modern quantum-dot-charge-sensor + CCD-style architectures qubit readout.
RMT (Mehta 1960)
M L Mehta 1960 random-matrix-theory + Wigner-Dyson; modern foundational text universal-conductance-fluctuation 2-RMT-applications.