Quantum Landau Free Energy

Quantum Landau Free Energy is the Landau expansion of free energy in powers of the quantum order parameter, used to describe the phase transition in which a quantum system's collective coherence switches on below a quantum critical temperature. It is the quantum form of the Landau Free Energy in the book's harmonic threshold analysis. The order parameter it expands in measures how far the collective quantum state departs from a random ensemble; the sign of the quadratic term changes at the quantum critical temperature and the positive quartic term keeps the system stable, which is how the expansion captures critical behavior of collective coherence. The book presents the resulting quantum singularity as a thermodynamic phase transition once quantum systems phase-lock collectively to the substrate rhythm, and states that the expansion is valid only for small order parameter, |Φ_q| ≪ √(|a_q|/2b_q). F_q(Φ_q,T) = F₀,q + a_q(T − T_c,q)Φ_q² + b_qΦ_q⁴ + O(Φ_q⁶), where F_q and F₀,q are the quantum and reference free energies [𝕄·𝕃²·𝕋⁻²], a_q is the quantum temperature coefficient [𝕄·𝕃²·𝕋⁻²·Θ⁻¹], T and T_c,q are the temperature and quantum critical temperature [Θ], Φ_q is the quantum order parameter [∅] and b_q > 0 is the stability parameter [𝕄·𝕃²·𝕋⁻²]. It connects to Landau Free Energy, Landau Theory, Quantum Order Parameter, Order Parameter and Quantum Critical Exponents.