From WolfWikis

Continuous phase transitions: the Landau theory of symmetry change

Most phase transitions of condensed phase are sudden and involve a change to a totally unrelated structure with a totally unrelated symmetry. A good example is graphite and diamond. To make this transformation it is necessary to disassemble one structure pretty much totally and start all over putting together the new one. In such a case there is a jump in all first derivatives of G, such as the enthalpy H and the entropy S. These transitions are therefore called first order transitions.

Landau formulated a number of pretty stringent selection rules that need to be obeyed for it to be possible to make the transition continuous and his argument is rather similar to the Peierls distortion we discussed for our evaporating 1D chain of hydrogen atoms. For continuous transitions the first order derivatives H and S are continous but the second order derivatives of G, e.g. the C_p can be discontinuous. This is why continuous transitions are also often called 'second order phase transitions'.

Subgroups

The first requirement is that the resulting structure should have a symmetry that is a subgroup of the parent structure. What this means is that its elements are a smaller selection of the original ones but that they do form a group amongst themselves. When we were pairing up our atoms in the hydrogen chain that was certainly the case. The original 1D translation group looked like:

{E|0},{E|t}, {E|t²},{E|t³},{E|t⁴} ,{E|t⁵},{E|t⁶}

The pairing doubled the cell, so all odd translations were lost but the even ones kept:

{E|0},{E|t}, {E|t²},{E|t³},{E|t⁴} ,{E|t⁵},{E|t⁶}

And yes these remaining elements form a group, albeit with {E|t²}={E|t^new} as unit translation. (Notice that the new unit cell is bigger when we lose half the symmetry elements: small cells imply more symmetry!)

We might expect that we could play the same game at point X of the fcc lattice. If we were to let a phonon of one of its irreps go static, that would affect the translations in the a-direction, but the symmetry would no longer be cubic, because a,b and c would no longer be equivalent. Besides, there were three such k-vectors in the star of k. How do they fit in? And then there were multiple irreps in point X. Clearly things get a little more involved in 3D!

A single irrep

The second Landau requirement is that any continuous change should transform as a single irrep of the symmetry of the parent structure. Our Peierls distortion certainly fit that bill too. The argument behind this requirement is really a thermodynamic Occham's razor. While the parent structure is stable any distortion from its symmetry should be brief and fleeting, i.e. dynamic. Thermodynamically speaking G should go up if you distort the structure according to any of its irreps (except the totally symmetric one at Γ) otherwise the symmetry would simply not be what it is.

For reasons of symmetry the different irreps are independent from each other. This implies that the energy penalty paid for each one is different. If by changing thermodynamic variables (say composition, pressure, temperature, magnetic field etc.) a distortion becomes favorable it is therefore likely to correspond to a single irrep unless two go 'soft' simultaneously by chance rather than for reasons of symmetry. Implicitly we postulate that at the transition point it should be possible to describe the incipient distortion as infinitesimal.

If the transition cannot be described as infinitesimal this whole argument why only one mode should go soft at the time falls apart. That means that first order discontinuous transitions are quite possibly a mixture of a great many modes that come in with a vengeance all at once, or put differently: the change is too big to decompose it in an expansion involving normal modes or the original structure. Series expansions only work for small changes!

The higher order invariants

What happens when one irrep goes soft

We can therefore describe an incipient infinitesimal distortion as a single function η that starts to lower G at the transition point and as it is infinitesimal we can expand G in a power series at the transition point

G = G₀ + A₁.η + A₂.η² + A₃.η³+ A₄.η⁴+ ...

First order

The coefficients A_n should all be invariant under all elements of the parent group and should be of the order n of the term. A first order invariant A₁, i.e an odd term in the expression should not exist (be zero) because otherwise the parent structure would never be stable: we could always lower G by making η finite. (Fortunately only the totally symmetric irrep at Г posesses an A₁... For any other η A₁ is non-existent, i.e. always zero)

Second order

There is always only one second order invariant term A₂ whatever irrep we take for η. The value of A₂ is positive for all irreps until at a certain set of conditions one of them goes negative at the transition point. When that happens the shape of the Gibbs free energy curve as a function of η evolves from a single well into a 'Mexican hat' -shape, allowing the system to 'slither' into a new minimum. That is what makes it attractive to distort infinitesimally at the critical temperature T_c and later to a finite degree. Strictly speaking the theory only predicts what happens for A₂>0 and A₂=0, i.e. at the critical point T_c. Beyond that we have to describe the system with its new symmetry.

Whether there are higher order invariants A₃ and A₄ and how many of each strongly depends on what the irrep is that η transforms as and particularly its dimensionality. There are group theoretical procedures to determine how many of those terms can exit.

Third order

The presence of a third order term causes a discontinuous transition

Whether or not an irrep possesses a third order invariant term A₃ (such as e.g. the product φ₁φ₂φ₃ if the irrep contains three basis functions) can be computed by group theoretical methods. If it has a third order invariant a continuous transition is not possible because at some temperature it will always become favorable to increase η in a finite way (see jump in diagram) rather than infinitesimally. The presence of the third order term means that the potential well (G versus η) may have more than one inflection point and an asymmetric shape. Whether the A₃ term is positive or negative does not matter: in either case it is favorable to jump, albeit with opposite sign of η.

In such cases you often get a 'weakly first order' transition: the process seems to go continuously but at the critical temperature there is a small jump after all. A strictly continuous transition is only allowed for those irreps that do not possess a third order invariant and strictly speaking the whole Landau argument only applies for infinitesimal distortions not finite ones, but the 'jump' may be small enough that it may still be advantageous to think of the distortion as resulting from a single irrep of the parent phase as a reasonable approximation. It certainly shows the structures are related as opposed to totally unrelated like diamond and graphite.

The bigger the jump and the farther away from the transition point, the less likely that that is an adequate description. (In other words: what happens then may well require mixtures of a great many irreps if you -unwisely- insist on describing it in terms of the parent structure)

Fourth order

Tricritical points

G = G₀ + A₁.η + A₂.η² + A₃.η³+ A₄.η⁴+ ...

If there is no third order term a daughter symmetry can form by a continuous process from the parent, but the daughter does not have to be formed continuously. Whether it does or not depends on the sign of the fourth order invariant. A₄. If this sign is positive the transition can be continuous, if it is negative we can once again lower the value of G by making a finite rather than an infinitesimal distortion. The same arguments apply as shown above for the third order term.

This means that the same distortion can be continuous in one part of thermodynamic space and become discontinuous ('weakly' first order) in another, if A₄ changes sign. The transition from one regime to the other is called a tricritical point.

Weakly first order due to a negative forth order term

Multiple daughters

If the irrep is multidimensional we have to deal with more than one basis function. The basis functions are in principle thermodynamically equivalent (like degerate wave functions). Often this leads to a domain structure. E.g. if a cubic single crystal undergoes a lowering to tetragonal symmetry it often splits up in a threefold twin where either the x, the y or the z-direction of the cubic become the new fourfold axis.

However, it is a little more complicated than that because if the irrep is more than 1D, there can be more than one fourth order invariate term. These terms are thermodynamically independent and this means that which one dominates and lowers G more at a given point in thermodynamic space can differ in different regions of that space. This dictates what linear combination of basis functions of the active irrep will actually get expressed at the transition. One could say that the degeneracy is lifted because the system seeks to minimize the contributions of the fourth order terms and this leads to very specific linear combinations of the basis functions that minimize the sum of the fourth order terms. One can find these solutions mathematically.

For example if we take the X point of the fcc lattice we have a 3 (or 6) dimensional irrep. That means that we need three convenient basis functions φ₁, φ₂ and φ₃ for the irrep. They might look like Bloch functions (waves) in the a, b and c directions. Depending on the relative values of the fourth order derivatives we could get a distortion like η=φ₁ (or η=φ₂ or η=φ₃) leading to three domains each with a tetragonal symmetry or to η=[φ₁+φ₂+φ₃]/√3 leading to a cubic symmetry. (Roughly one could say that here the three possible distortions occur in 'overlay'). A change in e.g. pressure could very well convince the system to decide for the other 'sister' distortion. If all Landau requirements are fulfilled both siblings can arise in a continuous fashion. As the two different structures that result both transform as the same irrep of the parent structure this clearly puts a relationship between these structures.

Thus: a multidimensional irrep of a parent structure may give rise to more than one daughter symmetry.^[1]

Resulting lattices

We can determine which translations are lost and which are retained in the daughters by examining whether a translation element of the parent leaves the distortion η invariant or not. For example if η=φ₁ and φ₁ transforms as the k-vector (001)* all translations with integer shifts (the primitive ones) say {E|306} leave this function invariant as the inproduct of the vectors (001)* and (306)= 3*0+0*0+6*1 = 6 is integer. Thus this element is retained as it turns η into exp(2π.6).η= +η.

Also {E|½½0} is retained but {E|0½½} and {E|½0½} are lost, because the vector product k.t= (001)(½0½)^T= ½ and this results in turning η into exp(2πk.t).η=exp(2π½).η= -η. This is clearly not a case of invariance. In general:

Any translation element for which the t.k vector product is integer is retained, otherwise it is lost.

This leads to a loss of the centering in the A and B planes but not in the C plane. Clearly the c-direction behaves differently form the a and b ones and the rotation symmetry is no longer cubic but tetragonal as there is a unique axis. (We can then actually choose new axes in the C plane under 45 degrees with the old ones to make it a primitive tetragonal cell.)

For the sister-distortion η=[φ₁+φ₂+φ₃]/√3 an element like {E|½0½} will leave one of the φ's invariant but turn the other two into their negatives. Thus η will turn e.g. into [φ₁-φ₂-φ₃]/√3 and that's a different beast than η. This means that all centering operations turn η into something else and are therefore lost in the transition. However all primitive translations like {E|306} do survive as they turn all φ's into themselves. The result is a primitive cubic cell of the same size as the old one (but a P-cell, not an F-cell, so that only one fourth of all translations are still preserved!).

In order to narrow down which irrep might be responsible for a particular transition it is useful to compare the translation lattices of parent and daughter and figure out which k-vector may be responsible for any loss of translational elements. (If all translational elements are retained, the Γ point is the only candidate).

Incommensurate transitions

The status of 'general' points in the bulk of the BZ has been cause for discussion. Haas^[2] made an argument that general points do not produce allowed continuous transition because an infinitesimal change in thermodynamic conditions would shift the position of the 'soft' point around as there are no elements of symmetry in that point to pin it to its position. Thus he introduced a distinction between type I and type II transitions. The first of those have a 'fixed' k-vector (i.e. if we change the composition a little and adjust the temperature a bit we get the same kind of structure) and lead to a structure with clear translation symmetry. The loss in symmetry is typically a simple factor: 2,4,6 maybe more.

A distortion according to a general point would essentially destroy all translation symmetry as it would represent a modulation that is not a simple multiple of the unit cell in any direction. Such non-commensurate superstructures do exist. Landau theory would leave open the the possibility that the incommensurate modulation may differ (slightly) in its k-vector if the same transition is observed under somewhat different thermodynamic conditions.

Prior to Haas, Lifshitz had already considered the same problem from a different angle and in the most extensive monograph on the theory by Toledano and Toledano^[3] the problem is known as the Lifshitz criterion. Whether or not a point in the Brillouin zone fulfills this criterion determines whether a potential continuous transition is of a commensurate nature or not. Lyubarski worked out a group theoretical method to determine whether the condition is fulfilled or not. It involves the characters of the small representation but also the question whether k and -k are both in the star *k and whether they are related to each other by a whole vector Q.

Requirements for continuity

Summarizing the requirements for continuity we have:

1. there must be a supergroup-subgroup relation (parent-daughter)
2. the order parameter (distortion) transforms as a single irrep
3. the irrep must not have a third order invariant
4. its fourth order invariants must be positive
5. if there are multiple fourth order invariants, the order parameter (a linear combination of basis functions) must minimize the sum of the invariants
6. the irrep must fulfill the Lifshitz criterium for the transition to result in a commensurate phase

The group theoretical process to check these requirements is by no means simple, but some general rules do apply:

1. If the resulting space group has exactly half the elements of the parent one, continuity is always possible.
2. For a factor of three there is typically a third order term and thus a discontinuity

The isotropy website

Fortunately, Stokes, Hatch and Campbell have put together a computer program ISOTROPY that is freely available online (provided due credit is given to it in publication etc.). It makes it relatively easy to determine the irreps of a given space group in a given k-point and to compute whether this irrep allows any continuous transitions and, if so, to which symmetries this will lead. The website contains a tutorial on how to use the program.

An example

Let us take space group 217 (I-43m) and see which transitions are allowed in the H-point k=(100) or k=(111) (alias). There is ony one vector in the *k and the group of k is therefore the full -43m point group. There are five small irreps (cf. the irreps of the point group -43m: A,B,E,T₁,T₂) We can examine the possible transitions due to irrep H₅ (cf. T₂) with the following input:

• value parent 217
• value irrep h5
• show subgroup
• show landau
• show lifshitz
• show active
• show size
• show maximal
• show continuous
• d isotropy

The output is:

These five solutions all fulfill the Landau third order requirement (Lan=0) and the Lifshitz condition (Lif=0). They all lead to cells in which the centering translation {E|½½½} is lost, thus the size of the primitive cell doubles (Size=2), although the cell parameters of the conventional cell will remain similar. Only two of the solutions (marked by Cont=RG) also minimize the fourth order invariant so that this irrep can give rise to two daughter symmetries continuously: tetragonal cell P-42₁c and the rhombohedral R3c.

In the tetragonal case the unit cell will look like the cubic cell, but one direction may stretch or contract a bit compared to the other two. In the rhombohedral case we get a rhombohedral cell that looks like a 'squeezed' version of the old cube, with about the same value for a_o and an angle that may start to deviate from 90 once the symmetry has lowered. In both cases one would expect a domain structure, because the new unique axis (-4 in one case, 3 in the other) has three resp. four equivalent directions to choose from.

Kinds of 'distortions'

Above it was silently assumed that all distortions (read symmetry changes) are of the displacive type like a Peierls distortion. This is not really true. The Landau theory focuses on the symmetry change not on what phenomenon causes it. Other changes than displacive ('soft phonon') ones are certainly possible. One common type is order-disorder transitions where e.g. at higher temperatures two elements are randomly distributed over the same two sites, but at lower temperature one one site is mostly occupied by one and vice versa. In that case η corresponds to a function describing atomic occupancies rather than displacement. The symmetry math may well look the same. Another type of transition that is often continuous is encountered in ferroelectrics and in magnetic materials. In both cases η now represents an order parameter that describes whether dipoles (electrical or magnetic) are ordered although through striction there is usually also a displacive component.

The magnetic case is actually more complicated because we would have to include time-reversal symmetry (not just x,y,z but also t) in our consideration. This means we need the irreps of the color or double groups. They are no longer simple complex numbers but quaternions and we will not go into that.

The validity of Landau theory and the Ehrenfest classification

Since the emergence of Renormalization Group (RG) theory (outside the scope of this short course) it has become clear that Landau's theory has its shortcomings. The theory is a so-called mean field theory that does not take into account scaling and fluctuations. The latter do become important when approaching a critical point, like the transition point of a continuous transition. This is why the Landau prediction for exactly how an order parameter behaves when approaching the transition point often fails. It can even be shown that the mean field approach leads to lack of self-consistency.^[4] Better descriptions using RG theory are possible. However, the symmetry aspects of Landau theory -the main concern of this course- are still valid.

A similar caviat must be made about the Ehrenfest classification of transitions into first and second order, depending on which order of derivative of the Gibbs free energy first yields a discontinuity. Although useful to introduce people to the fact that there are various types of phase transitions, this classification is not correct. ^[5] It is preferable to speak of continuous rather than second order phase transitions. The fact that heat capacities show cusps rather than simple jumps is related to this fact.

References and notes

1. ↑ See e.g. Phase transition between nickel arsenide and manganese phosphide type phases. Franzen, H. F.; Haas, C.; Jellinek, F. Mater. Sci. Cent., Univ. Groningen, Groningen, Neth. Physical Review B: Solid State (1974), 10(4), 1248-51.
2. ↑ C. Haas. Phys. Rev. 140, A863 - A868 (1965)

   Phase Transitions in Ferroelectric and Antiferroelectric Crystals

3. ↑ Jean-Claude Toledano and Pierre Toledano The Landau theory of phase transitions World Scientific ISBN 9971-50-025-6 1987
4. ↑ Lectures on phase transitions and the renormalization group by Nigel Goldenfield, Addison-Wesley 1992 ISBN 0-201-55408-9
5. ↑ E.g. it implicitly assumes that there exist only derivatives of integral order (1st, 2nd). Mathematically fractional order derivatives do exist.