What is the spontaneously broken symmetry group for the liquid-gasphase transition?

Max

For phase transitions, symmetry is spontaneously broken. According to
Landau, the order parameter acquires a non-zero value. For eg, in a
superconductive system the spontaneously broken symmetry is the U(1)
symmetry group of the order parameter.

My question is:
What is the spontaneously broken symmetry group for the liquid-gas
phase transition?

Max

Igor Khavkine

Max wrote:
> For phase transitions, symmetry is spontaneously broken. According to
> Landau, the order parameter acquires a non-zero value. For eg, in a
> superconductive system the spontaneously broken symmetry is the U(1)
> symmetry group of the order parameter.
>
> My question is:
> What is the spontaneously broken symmetry group for the liquid-gas
> phase transition?

Landau's phenomenology of spontaneous symmetry breaking is applicable
when the phase transition is continuous (of second order). The liquid
gas transition is of first order (both liquid and gaseous states can
coexist right at the critical point, and are metastable in a finite
region around the transition; think of vapor bubbles in a boiling pot
of water).

A first order transition is noticeable not because there is a
structural change in the medium, but because there is a sudden change
in the medium. The macroscopic property that changes can still play the
role of an order parameter, though. For the liquid gas transition, it
is the density. As a vapor is cooled down or put under increasing
pressure, at some point it becomes more energetically favorable for it
to be at a higher density. The finite jump in the density gives the
transition its discontinuous (first order) character. However, at
higher temperatures and pressures, this density jump disapears and so
does the actual transition.

It is worth mentioning that at zero temperature, a continuous
transition is possible without any symmetry breaking. In such a
situation, one is said to have found a Quantum Critical Point. It's a
hot reasearch topic in current condensed matter physics. Some people
believe that superconductivity is hiding such a quantum critical point
on the phase diagram of the cuprate high temperature superconductors.

Hope this helps.

Igor

Max

Igor wrote:
>Landau's phenomenology of spontaneous symmetry breaking is applicable
>when the phase transition is continuous (of second order). The liquid
>gas transition is of first order.

The water-gas critical point at Tc = 647K, Pc = 2.2*10^8Pa is a second
order phase transition.

Actually there is a lot in common between the water-gas phase
transition and the ferromagnetic-paramagnetic phase transition, if one
regards the water-gas phase separation line in the pressure-temperature
coordinate as the zero external magnetic field line in the magnetic
field-temperature coordinate. Both lines end at a second order critical
point.

We know the symmetry involved with the ferromagnetic-paramagnetic phase
transition, which is O(3). What is the symmetry involved with the
water-gas phase transition?

Conventional wisdom would say that there is no symmetry difference
between the water and gas phases, because one can continuously link the
water and gas phases bypassing the critical point in the
pressure-temperature coordinate. However, the same argument can be made
for the ferromagnetic and paramagnetic phases in the magnetic
field-temperature coordinate.

>It is worth mentioning that at zero temperature, a continuous
>transition is possible without any symmetry breaking. In such a
>situation, one is said to have found a Quantum Critical Point.

There IS symmetry breaking around the Quantum Critical Point. For 2+1D
nonlinear sigma model phase transition is controlled by varying the
spin stiffness at zero temperature.

The real example of departure from the conventional Landau's paradigm
is the fractional quantum hall effect, where the ground state is the
Laughlin state. There is a gap between the ground state and the excited
states, without the typical gapless Goldstone modes.

-Max