PHYS3036 · Condensed Matter and Particle Physics
Symmetry & Symmetry Breaking
Lectures 3–4 of University of Sydney PHYS3036 explain how a symmetric Hamiltonian can have a less-symmetric ground state — spontaneous symmetry breaking. Using the free energy F = U − TS, it shows why energy favours order while entropy favours disorder, so lowering the temperature condenses an ordered state, and it connects broken continuous symmetries to rigidity and gapless Goldstone modes. In the exam this appears both conceptually (which symmetry breaks, how many Goldstone modes) and as the setup for the Landau free-energy derivations that follow.
What this chapter covers
- 01Free energy F = U − TS: the energy term favours order/low symmetry, the entropy term favours disorder/high symmetry; minimising F selects the macrostate
- 02Spontaneous symmetry breaking: a symmetric Hamiltonian, a less-symmetric ground state (the ordered phase picks one of several degenerate minima)
- 03Discrete vs continuous symmetry: Ising up–down (Z₂) vs XY/Heisenberg rotational (continuous) order parameters
- 04The role of a symmetry-breaking field h conjugate to the order parameter, and the order of limits (h → 0 after N → ∞)
- 05Broken continuous symmetry ⇒ gapless Goldstone modes and generalised rigidity
- 06One-component (scalar) order parameters can only break discrete symmetries; multi-component order parameters can break continuous ones
- 07High-temperature states carry full symmetry; as T falls, lower-symmetry ordered states condense with distortions, defects and dynamical modes
Spontaneous symmetry breaking in a scalar order parameter
- +1(a) F(−m) = ½A t(−m)² + ¼B(−m)⁴ = ½A t m² + ¼B m⁴ = F(m), so the free energy is symmetric under m → −m. For t < 0 the minima sit at m = ±√(−A t/B): two degenerate but distinct states, each of which is NOT invariant under m → −m. [+1]
- +1(b) 'Spontaneous' means the Hamiltonian/free energy keeps the symmetry, yet the system must settle into one minimum and thereby picks a direction the symmetry does not — the choice is made by the system, not imposed by the equations. [+1]
- +1(c) A vanishingly small field h (adding −h m) tilts the double well and selects, say, the +m well; taking h → 0 after the thermodynamic limit N → ∞ leaves the system in that broken state rather than an equal mixture. The order of limits is essential. [+1]
- +1(d) The broken symmetry here is discrete (Z₂, m → −m), so there are NO Goldstone modes — Goldstone (gapless) modes accompany broken continuous symmetries such as the XY/Heisenberg rotational symmetry, not a discrete flip. [+1]
Key terms
- Free energy (F = U − TS)
- The thermodynamic potential minimised at fixed temperature; the internal-energy term U favours ordered, low-symmetry states while the entropy term −TS favours disordered, high-symmetry ones.
- Spontaneous symmetry breaking
- The situation in which a symmetric Hamiltonian (or free energy) has a ground/equilibrium state that does not share the full symmetry, because the system must select one of several degenerate minima.
- Symmetry-breaking field (h)
- An external field conjugate to the order parameter that explicitly favours one ordered state; the ordered phase survives as h → 0 provided the thermodynamic limit is taken first.
- Discrete vs continuous symmetry
- A discrete symmetry (e.g. Ising m → −m, Z₂) has isolated equivalent states; a continuous symmetry (e.g. rotating an XY spin angle) has a continuum of them.
- Goldstone mode
- A gapless (zero-energy at long wavelength) excitation that necessarily accompanies the breaking of a continuous symmetry; absent when only a discrete symmetry is broken.
- Generalised rigidity
- The resistance of an ordered phase to slow spatial variation of its order parameter — the stiffness (e.g. of a magnet or crystal) that emerges from broken symmetry.
Symmetry & Symmetry Breaking FAQ
What does 'spontaneous' actually mean?
It means the symmetry is broken by the state, not by the equations. The Hamiltonian and free energy stay fully symmetric, but the system cannot sit at the symmetric point (it is unstable) and must fall into one of the equivalent ordered minima — and in choosing one, it breaks the symmetry all by itself. An infinitesimal field only nudges which equivalent state it lands in.
Why does energy favour order and entropy favour disorder?
In F = U − TS the internal energy U is usually lowered by aligning or ordering the constituents (spins, dipoles), so the U term pulls toward order. The entropy S counts disordered configurations and is multiplied by T, so the −TS term rewards disorder more strongly at high temperature. Minimising F therefore gives a disordered phase when T is large and an ordered phase when T is small.
When do Goldstone modes appear?
Only when a continuous symmetry is broken. Breaking the rotational symmetry of an XY or Heisenberg magnet leaves gapless spin-wave (Goldstone) modes because you can slowly rotate the order parameter at almost no energy cost. Breaking a discrete symmetry like the Ising up–down flip costs a finite energy, so there is no Goldstone mode.
How is symmetry breaking examined in PHYS3036?
Both as short conceptual parts (name the broken symmetry, discrete or continuous, count Goldstone modes) and as the physical motivation for the Landau expansions you derive later. Since the condensed-matter paper provides no formula sheet, be ready to argue the F = U − TS competition and the degenerate-minima picture from memory. Confirm the assessment split on Canvas and the unit outline.
Exam move
Anchor this chapter on two sentences you can produce instantly: 'the free energy is symmetric but the ordered state is not' and 'energy favours order, entropy favours disorder, so lowering T condenses the ordered phase.' For every example, classify the broken symmetry as discrete or continuous and immediately state whether Goldstone modes appear. Practise sketching the symmetric double well and marking the degenerate minima, and rehearse the role of an infinitesimal field h and the order of limits. This material is the conceptual backbone for the Landau chapter, so keep it warm rather than leaving it to STUVAC; it recurs in weekly quizzes and the final. When the discrete-vs-continuous distinction blurs, ask Sia to walk the Ising-versus-XY comparison and check your Goldstone-mode counting.
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