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## "The theory of phase transitions", Lev Landau, 1936

*Reviewed November 16, 2023*

*Citation:* Landau, Lev. "The theory of phase transitions." Nature 138.3498 (1936): 840-841.

*Web:* https://www.nature.com/articles/138840a0

*Tags:* Physical, Foundational

This paper was Lev Landau's introduction of his theory of phase transitions to the world.
In his own words: "the only phase transitions which have up to the present time been
thoroughly investigated are transitions between the liquid and gaseous states".
Landau wanted a general theory which also covered other types of transitions, like the
liquid-solid transition.

This paper is extremely short, and presents only an essential image. He defines the function
rho(x,y,z) as the probability density for an atom or electron to be found at a given
spacial coordinate (x,y,z). He posits that the difference between the liquid and solid
phase is the difference in symmetry in rho(x,y,z): the solid has some non-trivial symmetry group,
whereas the liquid will have none. This theory also coverers transitions between different crystals,
because the symmetry groups of the various crystals will be different.

While this idea of "phase transition = symmetry breaking" is still the core of Landau's theory,
it has now evolved into a well-oiled machine with lots of formulas and established techniques:

> Toledano, Pierre, and Jean-claude Toledano. Landau Theory Of Phase Transitions, The: Application To Structural, Incommensurate, Magnetic And Liquid Crystal Systems. Vol. 3. World Scientific Publishing Company, 1987.

Typically, the reason that Landau theory is brought up in the context of
topological quantum computing because topological phase transitions
are *not* governed by symmetry breaking, and require a new explanation.
It's partially for this reason that people were so surprised when
topological order was discovered.