Introduction to phase transitions

Phase of matter: a macroscopic portion of space which presents the same properties and characteristics

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🗒️ Note: the definition is intentionally very general to allow it to accommodate everything

Phase transitions and symmetry

Phase transition: the act of crossing from one phase to another

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🗒️ Note: typically these are driven by temperature but not necessarily

Landau insight: different phases can be distinguishes based on their symmetry properties*

💃 Example: liquid (treated as a continuous medium) is symmetric under arbitrary translations and rotations but solids only have discrete translational and rotational symmetries

This statement only applies to the difference between “ordered” and “disordered” phases, because if we take the example of 2 disordered phases, ie liquid and gas they have the same symmetries

🗒️ Note: symmetry breaking for first order phase transitions is abrupt but not for higher-order, for these the change is continuous

💃 Example: transitioning from a paramagnet to a ferromagnet, magnetization grows continuously from zero to a finite value, once reached the symmetry changes.

Phase transitions and thermodynamics

Putting the above examples between continuous and first order into words we have

🔍 Classification:

First order phase transition

Occurs when the free energy (or its derivatives) become singular (discontinuity)

💃 Example: discontinuity in entropy leads to sudden heat release

Continuous (higher order) phase transitions

Occurs when one or more of the second or higher-order derivatives are singular

💃 Example: heat capacity with no latent heat released

The phase diagram

🗺️ Phase diagram:

The axis represent two changing parameters (here temperature pressure)
The lines are phase boundaries, either side of the boundary the phase is stable

🗒️ Note: the boundaries are not actually lines but have a width allowing for edge states

Where the liquid phase ends is known as a critical point (expand for a closer look)

Order parameters and critical points

Order parameter: a quantity used to determine if a system is in an ordered phase

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The phase diagram of a simple ferromagnet

There is a first-order phase transition until the critical point.

$T<T_C$ here the magnetization jumps from negative to positive (discontinuous)

$T>T_C$ continuous magnetization from negative to positive

🗒️ Note: this is a simplification

Looking at the magnetization $M$ at $H=0$:

The transition between discontinuous and continuous is gradual with temperature this makes the discontinuity explicit
In this case the magnetization $M$ is continuous but its derivatives $H$ are not

💎 Conclusion: $M$ is the order parameter

🗒️ Note: here is my guess at a representation of all 3 parameters at once

Here $X,Y,Z$-axis are $T,H,M$ respectively and the $T_C$ is at the center of the axis