A ferromagnet is cooled below its Curie temperature with no external magnetic field applied at any point. It develops a net magnetization pointing in one direction. Which statement best explains this outcome?
AThe Hamiltonian changes at Tc to favor one spin orientation over the other
BAlthough the Hamiltonian remains spin-flip symmetric, the system must choose one of two degenerate free-energy minima, and even infinitesimal fluctuations select one
CThermal fluctuations above Tc permanently break the spin-flip symmetry before cooling begins
DQuantum mechanical effects in the spin Hamiltonian force a preferred orientation below Tc
The key is that the Hamiltonian is always symmetric — flipping all spins leaves the energy unchanged. But below Tc, the free energy develops two minima (±M₀). The system cannot sit at M=0 (that becomes a local maximum) and must choose one minimum. The selection is made by infinitesimal asymmetries — a stray field, a fluctuation, a boundary effect — but the symmetry-breaking state persists even after these perturbations vanish. The symmetry of the equations does not prevent the state from being asymmetric.
Question 2 Multiple Choice
Breaking a continuous symmetry (e.g., the full rotational symmetry of magnetization direction in a Heisenberg ferromagnet) has a consequence that breaking a discrete symmetry (e.g., spin-flip in an Ising model) does NOT. What is it?
AThe ordered phase is thermodynamically stable only for continuous symmetries
BGapless Goldstone modes appear — low-energy collective excitations (like magnons) that cost zero energy in the long-wavelength limit
CThe phase transition occurs at a uniquely defined critical temperature only for continuous symmetries
DMultiple degenerate ground states exist only when a continuous symmetry is broken
Goldstone's theorem applies specifically to continuous symmetries: whenever a continuous symmetry is spontaneously broken, gapless excitations appear corresponding to slow spatial rotations of the order parameter. Magnons in ferromagnets, phonons in crystals, and pions in nuclear physics are all Goldstone modes. Discrete symmetry breaking (like the Ising model's Z₂ spin flip) does not produce Goldstone modes — there is no continuous direction to rotate the order parameter.
Question 3 True / False
A system can occupy a ground state with lower symmetry than its own Hamiltonian.
TTrue
FFalse
Answer: True
This is precisely what spontaneous symmetry breaking means. The Hamiltonian (and all the laws governing the system) may be fully symmetric, yet the actual state the system occupies — the ground state selected from among degenerate minima — can break that symmetry. The ferromagnet is the textbook example: H is spin-flip symmetric, but the ground state has ⟨M⟩ ≠ 0.
Question 4 True / False
Spontaneous symmetry breaking requires a finite external symmetry-breaking field to be permanently applied in order to maintain the ordered state below Tc.
TTrue
FFalse
Answer: False
The word 'spontaneous' means exactly that no sustained external field is required. An infinitesimal perturbation (a tiny field, a fluctuation, a boundary condition) can select which minimum the system falls into, but once there, the system stays even after the perturbation is removed. This is the thermodynamic limit effect: in a finite system, quantum or thermal tunneling between minima is possible, but in the thermodynamic limit (N → ∞), the barrier becomes infinite and the broken-symmetry state is stable indefinitely.
Question 5 Short Answer
Why is the term 'spontaneous' essential in 'spontaneous symmetry breaking'? How does it differ from explicit symmetry breaking?
Think about your answer, then reveal below.
Model answer: Spontaneous symmetry breaking occurs when the Hamiltonian (the fundamental equations) retains full symmetry, but the system's actual state does not share that symmetry — it has selected one of several equivalent ground states. No asymmetric term appears in the equations. Explicit symmetry breaking, by contrast, occurs when the Hamiltonian itself is modified by an asymmetric term (e.g., adding an external field H·M), so the equations themselves are no longer symmetric.
The distinction matters because spontaneous breaking is an emergent property of the many-body system, not a feature of the fundamental laws. It explains how ordered phases arise in systems governed by symmetric physics. In the Mexican-hat free-energy picture: spontaneous breaking means the hat is symmetric, but the ball rolls to the rim and stays there; explicit breaking would mean the hat is tilted from the start, with one part of the rim lower than the rest.