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2 LANDAU FREE ENERGY Consider (as a simple example) a magnetic system governed by the Hamiltonian In mean field theory we may write S z = coordination # W = number of microstates w We study the magnetic free energy (per spin) S 2

3 LANDAU FREE ENERGY The number of microstates decreases as the spin ordering increases. The sign of the spin does not matter. So Thus (S) SS T = T c -1T = T c -2 Free energy Spin S So T/T c Spin S According to the renormalization group, fluctuations (neglected by mean field theory) cause this exponent to assume a modified value often close to 1/3. 3 T = T c +1

4 WAVE VECTOR SELECTION Consider mean field theory for a one dimensional system * 4 FERRO ANTI- FERRO Note: the paramagnetic phase becomes unstable with respect to the formation of long-range order. For FERRO this happens at q=0. For ANTIFERRO it happens at q =  /a, as expected. J < 0 J > 0  (q) -1 aq/ 

5 WAVE VECTOR SELECTION T = T c T = T c + 2T = T c + 1 Lifshitz criterion: if q at T=T c is incom- mensurate (q=q*), then q is temperature dependent in the ordered phase. q at T { "@context": "", "@type": "ImageObject", "contentUrl": "", "name": "WAVE VECTOR SELECTION T = T c T = T c + 2T = T c + 1 Lifshitz criterion: if q at T=T c is incom- mensurate (q=q*), then q is temperature dependent in the ordered phase.", "description": "q at T

6 WAVE VECTOR SELECTION In the above discussion it was not said whether several values of wave vector could be simultaneously selected. This brings up the whole question of accidental degeneracy as it becomes relevant in its many guises. For this discussion it is useful to consider the “vacuum” as being the disordered phase when all the order parameters are zero. The symmetry of the vacuum, is the symmetry of this paramagnetic state. For any excitation relative to the vacuum, i. e. for any set of values of the order parameters, there will be a family of states with identical free energy which are generated by applying all the symmetry operators of the vacuum to that excitation. 6 A simple application of this principle is that if the free energy has an instability at a wave vector q, it will also have symmetry-related instabilities at all wave vectors in the STAR of q. (The STAR of q consists of all wave vectors which are generated by symmetry operators of the vacuum providing that we do not accept wave vectors which are equivalent in the sense that their difference is a reciprocal lattice vector, a RLV.) An important STAR includes q and –q if q and –q do not differ by an RLV. This is not all!!

7 WAVE VECTOR SELECTION Can I possibly find more to say under this heading?? YES. You may ask: is it possible that two wave vectors not related by symmetry could simultaneously become unstable? This is an accidental degeneracy of the type theoreticians excluded two centuries ago in the development of thermodynamics. 7 I will give an example of this discussion later. But for now I call your attention to the phase rule. Consider a system (like water, see right) that has gas, liquid, and solid phases all of which are characterized by specifying two intensive variables, the pressure P and the temperature T. If you vary one of them, say, the temperature, you can reach exceptional points where two phases coexist. This does not happen at a generic point in phase space: it requires adjusting one parameter to its special value. Is it possible that by varying the temperature you can reach the triple point where all three phases coexist? Absolutely not: to reach that point you need to adjust the temperature AND pressure. G L S p T

8 LOCK-IN LOCK-IN: Consider the competition between this q* near aq/  =1/4 and aq/  =1/4: Then, when S 4 is large enough, the Umklapp term in d can lock in aq=1/4. If d were zero, then F would be minimal for q=q*. However, by proper choice of  in S(1/4)=|S(1/4)|exp(i  ), the term in d can be made negative. 8 So, as the temperature is lowered further into the ordered phase, the range over q for which phase locking occurs grows. I should have mentioned that once on order parameter condenses, entropic effects prevent another order parameter of the same symmetry from condensing.

9 PHASE LOCKING J 1 /J 2 qa/  When J 1 and J 2 compete (both antiferromagnetic) AF Experimental trajectory If there are several J’s, then when such a system is mapped onto the model system with two J’s, the ratio J 1 /J 2 can be temperature dependent. 9 Usually, one ignores phase locking. It is very difficult to really establish experimentally discontinuities in slope.  T lock (Nature 2003)

10 A BIT OF THEORY When we consider a system with several vector spins per unit cell, we need a more formal approach. One introduces the susceptibility i. e. the spin correlation function: where R labels the unit cell and  labels sites 1, 2, … n in the unit cell. This is a 3n x 3n matrix. This matrix is a function of wave vector q. Stat mech tells us that the leading term in the Landau expansion is where Warning: many people use e iqR 10

11 COMPLEX UNIT CELL You can think of the inverse susceptibility matrix as being This is the generalization of 11 temperature independent interaction To repeat: Imagine diagonalizing the quadratic form F. In terms of its eigenvectors the free energy is The mode  with the highest T n is the critical mode and its eigenvalue k(T-T n ) is the critical eigenvalue. This mode determines the structure of the ordered phase. Its amplitude is fixed by the terms of order S 4 (which we have not shown explicitly.)

12 COMPLEX UNIT CELL 12 This is reminiscent of phonons, where    plays the role of the dynamical matrix. The eigenvectors  of this matrix are characterized by symmetry labels. The critical eigenvector gives the distribution of spin components over the unit cell when the system orders and within mean field theory the amplitude of the ordering is governed by the terms of order S 4. To repeat: NOTE: if the ground state does not have the same symmetry as the state just below the ordering temperature T c, then we can be sure that there is an additional phase transition below T c. As the temperature decreases the eigenvector will have admixtures from other eigenvectors of the same symmetry. If Q is the critical eigenvector and X another eigenvector of the same symmetry as Q, then

13 SYMMETRY CONSIDERATIONS 13 Here I am not going to discuss F from the formal point of view of group theory. As it happens, there are canned programs (which I will refer to later) which do the group theory analysis. My aim here is to treat some simple examples where a knowledge of group theory is not necessary. The point of this discussion is to be sure that we understand the basic language of group theory so that we can check that we know how to use the output of such canned programs. In these simple examples we will invoke the well-known principle for the eigenvectors of a matrix M. Namely, if we have a set of mutually commuting operators which commute with M, the eigenvectors of M are also simultaneously eigenvectors of the commuting operators. (Group theory was invented to treat the case when the operators which commute with M do not commute with each other.) The point is that we definitely do not want to try to construct the inverse susceptibility and then diagonalize it. However, we will rely on the unambiguous experimental determination of the selected wave vector q.

14 DETERMINING q Time reversal says that the total number of powers of Q is even. We have wave vector conservation because Q(q)* = Q(-q). 14 Kenzelmann et al, PRB 76, 014429 (’06)Kenzelmann et al, PRL 95, 087206 (’05)

15 MULTICRITICAL POINT Look at the phase diagram for an antiferromagnet with the z-axis being the easy axis in a magnetic field H along the z axis. If we lower the temperature ordering will be into either the low field state or high field state. Only if the field is adjusted to be exactly at the critical value will one access the bicritical (magenta) point. This example indicates that if we consider a phase transition when the temperature is lowered, we ought not allow an accidental degeneracy in which two different irreducible representations condense. In simpler language: we are only allowed to break one symmetry at a time. 15 “SPIN FLOP” Case in point: in a Coulomb potential the 2s and 2p orbitals (which have different symmetry) have the same energy. This is explained by identifying the hidden Laplace-Runge-Lenz symmetry vector. (Wikipedia) If, unusually, two irreps condense simultaneously in a continuous Transition, then what? The, the most probable explanation is that There is a hidden symmetry that you did not identify.

16 SIMPLE ILLUSTRATION Finding the eigenvectors of the commuting operators which leave  -1 invariant will reproduce the results of group theory. Accordingly, let us consider the magnetic ordering of TbMnO 3 (TMO). = Mn= Tb The ordering wave vector is along b which is called y here. q=(0.52  /b) y 16 There are 8 spins per unit cell.

17 PHASE DIAGRAM OF TMO At right I show a schematic phase diagram of TMO. The structure of the HTI phase is a predominantly magnetically ordered collinear incommensurate phase described by irrep  3 (explained below). The low field LTI phase has additional contributions from  2, so that a magnetic spiral is formed which gives rise to a ferroelectric phase. Paramagnetic Paraelectric 17 P = spontaneous polarization HTI high LTI low =temperature incom- mensurate phase } { { }

18 EIGENVECTORS FOR TMO  (m x ) (m z ) x y z x y z x y z x y z  1  1 a b c a b c a b c a b c  2 -  -1 a b c a b c a b c a b c  3 -  1 a b c a b c a b c a b c  4  -1 a b c a b c a b c a b c S(1)S(2)S(3) S(4)  (m x ) (m z ) x y z x y z x y z x y z  1  1 0 0 C 0 0 F 0 0 F 0 0 C  2 -  -1 A B 0 C D 0 C D 0 A B 0  3 -  1 0 0 C 0 0 F 0 0 F 0 0 C  4  -1 A B 0 D E 0 D E 0 A B 0 S(5)S(6)S(7)S(8) NOTES: All parameters are complex valued. The constants for different irreps are unrelated. So for  1 a denotes a 1, for  2 a denotes a 2, etc.  1 has 5 constants:  1 is contained 5 times in the original reducible representation.      and   are contained 7, 5, and 7 times respectively.  = exp(i  q). 18 If  is an eigenvector, irrep spin component labels x denotes -x

19 SPATIAL INVERSION?? What about spatial inversion? This operation is not in the group of the wave vector because inversion takes q into –q. However, the free energy is invariant under inversion. It is amazing that the world wide neutron scattering community (including many famous people!) didn’t know how to take this into account. As a relative novice I uncovered this problem. In hindsight this problem had been solved long ago. But this important work was either forgotten or overlooked. There are several ways to take account of inversion. One way is to use the rather arcane formalism of corepresentations. Another is to use the full subgroups of the entire space group (rather than the group of the wave vector). This latter method has been used to develop a canned program which I will refer to in a moment. Meanwhile I continue with the “poor man’s” approach to symmetry which does not invoke corepresentations. To illustrate the simple approach, let us consider irrep  3. I consider this irrep because it is the one that experiment identifies as the first magnetic irrep to condense as the temerature is lowered. 19

20 INVERSION For notational simplicity, let a, b, c, C, and F be denoted x 1, x 2, x 3, x 4, and x 5, respectively. Because we are considering a wave vector having only a y component, we only need keep track of how the y coordinate of position transforms. (Because spin is a pseudo vector its orientation is invariant under inversion: the only effect of inversion is to relocate the spin.) If we invert spin #1 (or #2, #3, or #4), we change the sign of its y component of position which takes exp(iqy) into its complex conjugate. Thus If we invert spin #5, it goes into spin #7 and #6 goes into #8. So Invariance under inversion yields the relation This has to be true no matter what we choose for the x’s. Suppose only x 1 and x 2 are nonzero. Then 20

21 INVERSION (continued) Note that G is Hermitian: G 21 = G 12 *. So if G 12 = G 21, then G 21 =G 12 is real. Proceeding similarly we find that G is of the form a b c   b d e   c e f       g     g Where a, b, c, d, e, f, and g are real valued and the Greek letters are complex valued. The eigenvectors of this form of matrix must be of the form We included a phase factor because we are working in a complex vector space. In comparison with what we had before inversion: a, b, and c were complex quantities. But now we see that they must all have the same phase. The Tb amplitudes C and F, which were arbitrary complex numbers, now must be complex conjugates of one another. If one ignores the consequences of inversion symmetry, one has these phases to determine from the data and that proved to be a very difficult task, so invariably some guesswork was performed. Furthermore, when these constraints are ignored, the symmetry of the structure is wrong: a single irrep would then possibly induce ferroelectricity, a result we know to be incorrect. where r, s, and t are real and  is complex. 21

22 ISODISTORT 2.Construct crystallographic input file (ISODISTORT will prompt you) I upload the cif file prepared previously. 1. Search for ISODISTORT and click on the entry 3. space-group preferences: I click OK, but maybe you can do better. 4.Types of distortions to be considered. Magnetic Tb and Mn. Then OK. 5. Method 2: SM (0,a,0) a=0.52. I think # of inc. mod. = 1. Then OK 6.Pick IR. As Wills says, we may not be sure of the convention. I took mSM3 because I thought that Kenzelmann took the best reference. Then click OK. 7. Finish selecting: I chose P1Z. The OK. 8. Click on Complete modes details. Then OK 9. Then “Magnetic mode definitions” gives the IR basis vectors. 22

23 COMPARE TO ISODISTORT atom x y z M z M x M y 2 3 4 1 2 3 4 1 2 3 4 1 a tom s s s s t t t t r r r r ME 23

24 Tb Spin Function for  3 z x y Mag. mom. 5 8 7 6 5 8 7 6      ISODISTORT ME 24

25 TWO IRREPS AT THE SAME TIME? We argued against having an accidental degeneracy so that the disordered phase would have simulataneous instabilities to ordering. In what I call “the Green Bible” all `possible’ subgroups which can arise are tabulated.* (This is actually too bold a claim! If you have a first order transition, I think anything is possible, at least in principle.) Here I give a simple mechanism for having two different irreps simultaneously appear at a first order phase transition. 25 * H. T. Stokes and D. M. Hatch “Isotropy Subgroups of the 230 Crystallographic Space Groups” A first order transition arises from At right I show curves for F vs  for a sequence of temperatures (black highest, blue lowest) F

26 TWO IRREPS?? Now suppose we have two different symmetry order parameters, Q 1 and Q 2, each of which, if they were uncoupled, would have a first order phase transition. With T 1 > T 2 and all parameters >0 It can be shown rigorously that if w is large enough this model has a single phase transition in which both order parameters simultaneously condense. To see that this is reasonable, suppose that Q 1 condenses. If w is large enough, then the coefficient of Q 2 2, which is can become negative, so that Q 2 is sure to also order. 26

27 ORDER PARAMETERS In TMO we have an incommensurate colinear phase at 42K and then a spiral at 32K at which point one has a ferroelectric. In the Nature paper this was attributed to the hypothetical lock-in phenomena. Nearly at the Same time we similarly analyzed Ni 3 V 2 O 8. To discuss the symmetry of this situation it is helpful to introduce order parameters. 27

28 ORDER PARAMETERS Landau theory for the higher T transition is as we have described it above. For the lower T transition the vacuum is no longer the trivial state with zero ordering. In principle we need to discuss the symmetry of the second ordering in the presence of the first ordering. This is a nasty project!! Consider the hypothetical phase diagram shown below. T T>T> T { "@context": "", "@type": "ImageObject", "contentUrl": "", "name": "ORDER PARAMETERS Landau theory for the higher T transition is as we have described it above.", "description": "For the lower T transition the vacuum is no longer the trivial state with zero ordering. In principle we need to discuss the symmetry of the second ordering in the presence of the first ordering. This is a nasty project!. Consider the hypothetical phase diagram shown below. T T>T> T

29 ORDER PARAMETERS The spin wave function for the unit cell (which is just the Fourier coefficient At the wave vector q), has the form of the basis function whose coefficients may or may not be determined from a refinement of scattering data. What we can say is that where  n with n=1,2 are the basis vector for the two irreps. Note that of course the Q’s are complex because their phase regulates where the origins of the waves are placed. If we had only one wave, this phase would be unobservable. So really what is important is the relative phase. Initially we do not treat the interactions between irreps and write 29 There is a problem with this formulation: why should the same value of the wave vector be selected for the two irreps (which predominantly describe ordering in different directions??? Surely the exchange interactions for, say, S x are different, if only slightly, than those for S y. Solution: only if q > =q <

30 LOOK AGAIN AT LOCKING The two irreps are out of phase – this is what is needed for a spiral. But this actually follows from simple hand-waving arguments: As the temperature is lowered, the spins more and more want to have fixed length (as they do at T=0). Thus when spin ordering from one irrep is large, the spin ordering from the other irrep must be small. 30 In phase Out of phase

31 MAGNETOELECTRICITY  (m x ) (m z )  1  1  2 -  -1  3 -  1  4  -1 31 If  is an eigenvector,  =exp(ipq) Let Q 3 be the complex amplitude (order parameter) for  3 and Q 2 that for  2. then Q 3 * will be that for  3 * and Q 2 * will be that for  2 *. Note also I Q n = Q n *. We want to see how the Q’s can be combine with the spontaneous polarization P to induce a nonzero value of P. To induce a nonzero value of P the interaction should be linear in P. For time reversal invariance we must have an even number of Q’s To conserve wave vector one will be Q and the other will be Q*=Q(-q). We can not have Q n Q n * because this transforms like unity. So F is real, so c  =d  *. Under inversion Q 2 *Q 3 -> Q 2 Q 3 * and P -> -P, so c  = -d  = i r , where r  is real. The square bracket is even under m x, odd under m z and odd under inversion. Therefore only r z can be nonzero. The polarization is found to lie along z!!

32 CONCLUSION 1.Landau theory provides a natural way to understand the phenomenology of phase transitions. 2. An alternative to group theory can provide a convenient route to categorizing the symmetry of system whose symmetry is not too complicated. Of course, group theory provides the definitive answers concerning symmetry. 3. Where Landau theory becomes essential is when one Investigates coupling between several different symmetries. 4. The principal of “no accidental degeneracy” has wide application. 5 A corollary: “if it allowed by symmetry, it happens.”


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