In the Landau free energy F = f₀ + αm² + βm⁴, why are odd-power terms like m³ excluded from the expansion?
AOdd powers would make the free energy unbounded below for large |m|, causing the model to predict unphysical infinite order
BThe free energy must be symmetric under m → −m because the system looks the same in states of equal and opposite order — the symmetry of the problem forbids odd powers
COdd powers are negligible near the critical point where m is small, so they are dropped as an approximation
DLandau theory only applies to magnetic systems where m is a magnetization, which is always positive
The exclusion of odd powers is a symmetry argument, not an approximation. If a system can be in a state with order parameter +m, an equally valid state −m must exist by symmetry (e.g., a magnet can point up or down; a binary alloy can be rich in either component). The free energy must therefore satisfy F(m) = F(−m), which forces all odd-power terms to zero. This is not about m being small near T_c — it is a fundamental constraint from the physical symmetry of the problem, and it applies at all temperatures.
Question 2 Multiple Choice
Landau theory predicts the same critical exponents (e.g., m ∝ (T_c − T)^{1/2} below T_c) for magnets, superfluids, and binary alloys. What drives this universality within the theory?
AThe microscopic interactions in these systems are identical at sufficiently high temperatures
BThe Landau free energy expansion in powers of m has the same mathematical structure for all systems with the same symmetry, so minimizing it gives identical scaling regardless of microscopic details
CLandau used empirical critical exponents measured in magnets and applied them to all other systems by analogy
DUniversality is an approximation that holds far from T_c but breaks down close to the critical point
Universality in Landau theory arises because the free energy F = f₀ + αm² + βm⁴ (with α changing sign at T_c) is the same mathematical object for any system with this symmetry pattern — regardless of whether m is magnetization, superfluid density, or concentration difference. Minimizing dF/dm = 0 gives m ∝ (T_c − T)^{1/2} by pure algebra, independent of the physical system. This is why systems with the same symmetry share critical exponents: the mathematical structure of the free energy, not the microscopic Hamiltonian, determines the scaling.
Question 3 True / False
Landau theory becomes more accurate as temperature approaches T_c because the order parameter m becomes small, validating the power-series expansion.
TTrue
FFalse
Answer: False
This is precisely backwards. Although m becomes small near T_c (validating the power-series expansion of F in m), fluctuations in m become large and spatially correlated on long length scales near T_c. Landau theory ignores these fluctuations — it assumes m is uniform in space and uses only the mean value. The Ginzburg criterion quantifies when fluctuations dominate: close to T_c, fluctuations overwhelm the mean-field prediction, and the Landau critical exponents (β = 1/2, γ = 1) are replaced by non-classical values. Landau theory is accurate away from T_c, not at it.
Question 4 True / False
In Landau theory, the phase transition occurs when the coefficient α(T) changes sign from positive to negative as temperature decreases through T_c, causing two new free-energy minima with nonzero order parameter to appear.
TTrue
FFalse
Answer: True
This is the mathematical heart of the theory. For α > 0, dF/dm = 2αm + 4βm³ = 0 has only the solution m = 0 — a single minimum at the disordered state. When α < 0, the second derivative at m = 0 becomes negative (it is a local maximum), and two symmetric minima appear at m = ±√(−α/2β). Landau assumes α(T) = a(T − T_c), so α changes sign exactly at T = T_c. The system spontaneously breaks symmetry by choosing one of the two minima — this is the phase transition, and it is entirely encoded in the sign change of α.
Question 5 Short Answer
What is the 'order parameter' in Landau theory, and how does its behavior across the phase transition encode the physics of spontaneous symmetry breaking?
Think about your answer, then reveal below.
Model answer: The order parameter m is a quantity that is zero in the disordered phase and takes a nonzero value in the ordered phase. For a ferromagnet it is average magnetization; for a liquid-gas transition near the critical point it is the density difference; for a superconductor it is the Cooper pair amplitude. Above T_c, α > 0 and the free energy has a single minimum at m = 0 — no preferred ordering. As T decreases through T_c, α changes sign and the potential well at m = 0 becomes a local maximum while two new symmetric minima appear at ±m₀. The system must 'choose' one — this spontaneous symmetry breaking is the transition. Below T_c, m ∝ (T_c − T)^{1/2}, growing continuously from zero and encoding both the fact and degree of ordering.
The order parameter is Landau's key abstraction: it reduces the enormous complexity of a phase transition (involving 10²³ interacting particles) to the behavior of a single scalar field whose equilibrium value encodes the thermodynamic phase. The same mathematical object works for magnets, superfluids, and liquid crystals because the symmetry structure, not the microscopic details, determines the form of the free energy expansion.