Questions: Symmetry Breaking and Phase Transitions

Question 1

A ferromagnet's Hamiltonian is invariant under rotating all spins simultaneously. Below the Curie temperature, the magnet develops a net magnetization pointing north. What does this imply?

• A. The Hamiltonian has changed — it now favors the north direction over others
• B. The ground state has broken the rotational symmetry that the Hamiltonian still respects
• C. The order parameter enforces the symmetry of the Hamiltonian onto the ground state
• D. The symmetry is broken because the critical temperature removes the rotational invariance from the equations

Answer: (B)

This is the core of spontaneous symmetry breaking: the Hamiltonian remains rotationally invariant — it still treats all spin directions equally. The ground state, however, has 'chosen' one direction from the family of equivalent low-energy configurations. Tiny perturbations during cooling determined which direction was chosen, but the laws themselves did not change. The symmetry is hidden in the state, not broken in the physics.

Question 2

At a continuous (second-order) phase transition, how does the order parameter behave as temperature passes through T_c?

• A. It jumps discontinuously from zero to a finite value at T_c
• B. It remains zero above T_c and grows continuously from zero below T_c
• C. It is nonzero on both sides of T_c but changes sign at T_c
• D. It diverges to infinity at T_c, then decreases below it

Answer: (B)

The order parameter is the signature of symmetry breaking: it is exactly zero in the disordered phase (T > T_c, where the full symmetry is intact) and grows continuously from zero below T_c, typically as M ~ (T_c − T)^β. The discontinuous jump describes a first-order transition, not a continuous one. Divergence at T_c is characteristic of susceptibility or correlation length, not the order parameter itself.

Question 3

In spontaneous symmetry breaking, the Hamiltonian retains the symmetry that the ground state breaks.

• T. True
• F. False

Answer: True

This is the defining feature of spontaneous symmetry breaking: the laws (Hamiltonian) are symmetric, but the equilibrium state the system occupies is not. A ferromagnet's Hamiltonian treats all spin directions equally, yet below T_c the magnet points in a definite direction. The symmetry is spontaneously broken by the state, not by the forces.

Question 4

The 'Mexican hat' free energy landscape of Landau theory means the system below T_c is trapped in a unique, stable minimum with no symmetry.

• T. True
• F. False

Answer: False

The Mexican hat potential has a continuous ring of degenerate minima at the bottom — every point on the ring is equally valid. The system sits at one point on the ring, which breaks the symmetry, but that point is not unique: all points on the ring are equivalent ground states related by the symmetry that was broken. This continuous family of degenerate ground states is exactly what implies the existence of Goldstone modes (massless excitations) through Goldstone's theorem.

Question 5

Why does a ferromagnet below its Curie temperature spontaneously magnetize in a particular direction, even though the Hamiltonian treats all directions equally?

Model answer: Because the symmetric Hamiltonian has a family of energetically equivalent ordered states (all pointing in different directions), and the system must occupy one of them. Infinitesimal perturbations — the Earth's magnetic field, microscopic imperfections — select which of these degenerate minima the system falls into during cooling. The symmetry is broken by the state the system occupies, not by the laws governing it.

This question targets the key conceptual move: the distinction between symmetry of the laws and symmetry of the state. Many students conflate the two, thinking that if a magnet points north, the physics must somehow favor north. Instead, the physics is indifferent — but the system must pick one direction, and it does so through the amplification of microscopic fluctuations. Understanding this prepares students for Goldstone's theorem, where the continuous degeneracy of ground states is the direct cause of massless excitations.