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NL2674 Order Parameters

The concept of an order parameter was introduced by Landau in the 1930s and until now it does not have a

precise definition. It is a thermodynamic quantity that is invariant with respect to the symmetry group of the low-

temperature phase and is zero above the transition temperature while nonzero below. It is a measure of the amount of

order that is built up in the system in the neighborhood of the critical point. In general, it has both an amplitude and a

phase. To find the equation of state, a minimization procedure has to be followed for an appropriate thermodynamicpotential. From its original application to second order phase transitions where it changes continuously from Tc to lower

temperatures, the idea of an order parameter has been extended to first order transitions where it undergoes a sudden

change at Tc (Fig. 1). It has been generalized from a scalar to a time- and space-dependent function. Note that first order

phase transitions are associated with discontinuities of the order parameter and thermal hysteresis effects. Second order

phase transitions have a continuous order parameter and show field-induced hysteresis.

Figure 1 A typical plot of the scalar order parameter as a function of temperature.

Diverse applications of the order parameter concept to both equilibrium and nonequilibrium critical phenomena are listed

in Table 1.

Table 1 Examples of Order Parameters

PHENOMENON DISORDERED

PHASE

ORDERED PHASE ORDER PARAMETER

EQUILIBRIUM

Condensation Gas Liquid Density difference

L -GSpontaneous

Magnetization

Paramagnet Ferromagnet Net magnetization

Mr

Antiferromagnetism Paramagnet Antiferromagnet Staggered

Magnetization

21 MMrr

Superconductivity Conductor Superconductor Cooper pair

Wavefunction

Alloy Ordering Disordered Mixture Sublattice Ordered

Alloy

Sublattice

Concentration

Ferroelectricity Paraelectric Ferroelectric Polarization

Superfluidity Fluid Superfluid Condensate

Wavefunction

NONEQUILIBRIUM

Tunnel Diode Insulator Conductor Capacitance Charge

Laser Action Lamp (Incoherent) Laser (Coherent) Electric Field Intensity

Super-radiant Source Noncoherent

Polarisation

Coherent

Polarisation Atomic Polarization

Fluid Convection Turbulent Flow Bnard Cells Amplitude of Mode

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NL2674 Order Parameters 2

Crystal formation is demonstrated by the existence of a regular diffraction pattern associated with the Fourier

components of the mass density distribution )(rr

=G

rGi

Ger

r

r

r

r

)( , (1)

where G

r

are vectors in the reciprocal space. The set of numbers

G

r

can be used as order parameters characterizingthe low-temperature (crystal) phase. The nonzero coefficients

Gr in (1) define a multi-component order parameter.

The order parameter field for a magnet is defined at each position xr

by a direction of the local

magnetization )(xMr

r

whose length is fixed. By becoming a magnet, this material has broken the rotational

symmetry and its order parameter field Mr

defines the broken symmetry directions chosen in the material.

A number of metals, alloys and ceramics below their critical temperature Tc exhibit an ordered state in

the conduction electron degrees of freedom manifested by zero resistance. The order parameter for superconductors is

the wavefunction of the Cooper pair condensate )(rr and it exhibits a Hopf bifurcation at T = Tc.The superfluidproperties in 4He and 3He are manifested by the absence of viscosity. The 4He atoms are

Bosons that, below a transition temperature T, undergo the so-called Bose condensation into a k = 0 mode. The

associated order parameter is the condensate's quantum wavefunction. Since 3He atoms are Fermions, below T they

form Cooper pairs. Liquid crystals are anisotropic fluids composed of strongly elongated molecules. The nematic phase is

characterized by the existence of a direction to which most of the molecules are parallel, so the order parameter is a

second rank tensor describing correlations along that direction. Numerous other examples of critical phenomena such as:

binary fluids, the metal-insulator transition, polymer transitions, spin- and charge-density waves, have their own order

parameters.

Landau deduced that second order phase transitions are associated with symmetry breaking and can be

qualitatively described by an order parameter . Assuming that the free energy F depends on V,T and , theequilibrium conditions are:

0),,(

0),,(

00

2

2

>

=

==

VTFand

VTF, (2)

where 0 is the equilibrium value of .The universality hypothesis states that any two physical systems with the same spatial dimensionality, d, and

the same number of order parameter components, n, belong to the same universality class having identical critical

exponents (see Table 2).

Table 2 Examples of Universality Classes.

Universality Class System Order Parameter

d = 2 n = 1 Absorbed films Surface density

d = 2 n = 2 Superfluid He4 film Superfluid wave function

d = 3 n = 1 Uniaxial ferromagnets Magnetization

d = 3 n = 1 Fluids Density difference

d = 3 n = 1 Mixtures, alloys Concentration difference

d = 3 n = 2 Planar ferromagnets Magnetization

d = 3 n = 2 Superfluids Superfluid wave function

d = 3 n = 3 Isotropic ferromagnets Magnetization

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Order parameters accompany broken symmetry phenomena where the new ground state of the system does not

possess the full symmetry of the Hamiltonian. A classic example is the ferromagnetic-to-paramagnetic phase transition at

Tc where the full rotational symmetry of the paramagnetic phase is broken by the axiality of the ground ferromagnetic

state below Tc. When a symmetry that is broken is continuous, a massless Goldstone Boson excitation appears whose

frequency vanishes at long wavelengths. Examples of Goldstone Bosons include ferromagnetic domain walls and

acoustic soft modes in structural phase transitions.

There exist several different types of broken symmetries: (a) translational (crystal formation, structural

transitions), (b) gauge (superfluidity, superconductivity), (c) time reversal (ferromagnets), (d) local rotational (liquid

crystals), (e) rotational (some structural phase transitions), (f) space inversion (ferroelectricity). Gauge symmetry is a

universal property of Hamiltonians whenever the total number of particles is conserved or a generalized charge-like

conserved quantity exists. Then, the order parameter is complex and its local density is so that aphase shift leaves the Hamiltonian invariant.

)(*

r =

ie

Defects in the order parameter space occur naturally as:

a) topological (i.e. kinks also referred to as domain walls)b) non-topological (i.e. solitons, also called nucleation centers)

and they can be obtained as solutions to the equations of motion for the order parameter field. Also, point defects,

line defects, vortices, dislocations, vacancies, interstitials with attendant singularities can be experimentally seen incritical systems.

Hakens separation of modes in synergetic systems into masters (order parameters) and slaves has been

influenced by Landaus theory of phase transitions.

JACK A. TUSZYNSKI

See also critical phenomena; domain walls; ferromagnetism and ferroelectricity; hysteresis, Bose-Einstein

condensation, liquid crystals, solitons, synergetics

Further Reading

Anderson, P.W. 1984. Basic Notions of Condensed Matter Physics. Menlo Park, California: Benjamin/Cummings

Haken, H. 1980. Synergetics. Berlin: Springer-Verlag

Landau, L.D. and Lifshitz, E.M. 1959. Statistical Physics. London: Pergamon

Ma, S.-K. 1976. Modern Theory of Critical Phenomena. New York: Benjamin

Reichl, L.E. 1979. A Modern Course in Statistical Physics. Austin, Texas: Univ. of Texas Press

White, R.H. and Geballe, T. 1979. Long Range Order in Solids. New York: Academic Press

Yeomans, J.M. 1992. Statistical Mechanics of Phase Transitions. Oxford: Oxford University Press