Phase transitions, order parameters and broken symmetry
A thermodynamic phase transition makes the free energy nonanalytic. First-order transitions have a jump in a first derivative; continuous transitions do not. For symmetry-breaking transitions, an order parameter distinguishes a selected ordered state from the symmetric phase.
What a learner can do afterwards
- Chooses an order parameter for a named transition and says what it measures
- Separates first-order from continuous transitions by the behaviour of free energy derivatives
- Explains which symmetry the ordered phase breaks and why the disordered one has it
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In the thermodynamic limit, a phase transition makes free energy nonanalytic: no convergent local power series represents it at the transition. A jump in a first derivative produces a kink and defines a first-order transition, which can absorb latent heat on heating. A higher-derivative singularity can occur without a kink. An order parameter distinguishes phases. For symmetry-breaking order, it is zero in the symmetric phase and nonzero in a selected ordered phase.
For an ideal isotropic ferromagnet at zero applied field, magnetisation measures spin alignment. Above its ordering temperature the mean magnetisation is zero, although spins still fluctuate. Below it, a selected ordered state has nonzero magnetisation pointing in one direction. The interactions treat all directions alike, but rotating this state changes it: the state breaks rotational symmetry.
With other variables fixed, entropy and magnetisation are minus first derivatives of free energy with respect to temperature and conjugate field. A jump in either is first-order. Continuous transitions keep first derivatives continuous but have a nonanalytic higher derivative. Heat capacity may jump finitely rather than diverge. Continuous transitions have no latent heat. Zero latent heat alone does not prove continuity: a field-driven magnetisation jump can occur without an entropy jump.
A nonzero external field favours one Ising spin sign in the energy H. It breaks spin-reversal symmetry explicitly, independent of the sample's state. Any magnetisation it produces can be a field response. Spontaneous symmetry breaking instead means selecting an ordered state at zero field that has less symmetry than the interactions. Magnetisation in a field alone does not establish spontaneous breaking.
In a finite zero-field Ising sample, an equilibrium average over equally likely positive and negative ordered configurations can be zero below the ordering temperature. Zero mean does not rule out order or correlations. Spontaneous order is defined by selecting one phase in the thermodynamic limit, for example by taking system size to infinity before letting a small selecting field go to zero.
Classify transitions by free-energy derivatives. State the symmetry and field conditions, and distinguish a selected ordered phase from an average over phases.
Original teaching text, questions and flashcards for phase transitions and symmetry breaking, authored for this learning dataset.
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