Amorphous solids and glasses contain two-level systems (TLS): atoms tunneling between two nearly degenerate configurations. TLS dominate glassy anomalies at low temperature.
TLS (Anderson, Halperin, Varma 1972): Atoms in glasses tunnel between two local energy minima. Distribution of parameters P(Δ,Δ₀) = const.
Energy splitting: E = √(Δ²+Δ₀²)
Tunneling rate: Γ ~ Δ₀²/E · coth(E/2kT)
Low-T glass anomalies:
Heat capacity: C ~ T (linear) from TLS
vs C ~ T³ for Debye phonons
Thermal conductivity: κ ~ T² from phonon scattering by TLS.
Quantum coherence: TLS as natural qubits (charge/flux noise source in superconducting qubits).