Extensions and some recent applications of the Landau theory of phase transitions

The basic concepts of the Landau theory of phase transitions are introduced through working examples. A number of rules governing the application of the theory to second and first-order transitions are given. The approach to magnetostructural transitions in magnetic mulltiferroic materials is detailed. Examples of transitions induced by replica of the same order-parameter in superconducting and liquid crystal systems, are described. The theory of reconstructive transitions is outlined and illustrated by the examples of the graphite-diamond transition and by the phase diagram of iron. An extention of the phenomenological approach to incommensurate structures to the crystal-amorphous transition is proposed (see also the chapter by Perez-Mato et al. for further discussion and applications).


INTRODUCTION
In the pioneering article of Landau [1] the phenomenological theory of phase transitions intended to establish the mutual compatibility of the symmetry and physical characteristics of a phase transition: Relationship between the symmetry of the phases, consistency between the nature of the symmetry change and the nature of the physical quantities behaving anomalously across the transition.This aim was achieved by means of introducing the concepts of order-parameter and Landau free-energy.Although in its initial formulation the theory considered the simple case of continuous (second-order) transitions induced by a single irreducible representation between group-subgroup related phases, it proved subsequently to apply, providing suitable extensions, to discontinuous (first-order) transitions [2], to transitions associated with several order-parameters [2], or to reconstructive transitions involving a loss of the group-subgroup relationship [3].The theory was used to describe almost all types of phase transitions occurring in hard or soft condensed matter physics, such as structural and magnetic transitions in crystalline materials [4], the superconducting [5] and superfluid [6] transitions, the liquidsolid [7] and liquid-vapour [8] transitions, and the variety of transitions disclosed in liquid crystals [9] and complex fluids [10].After introducing the basic concepts of the Landau theory (Sec.2) and some rules governing its application (Sec.3), we describe recent extensions and selected applications of the theory: Application to multiferroic materials (Sec.4), to unconventional superconductors and liquidcrystal systems (Sec.5), to reconstructive transitions (Sec.6), and to the formation of the amorphous state (Sec.7).

Non-equilibrium free-energy and order-parameter
Let us introduce the basic concepts of the Landau theory through the example of a ferroelectric phase transition occurring at the critical temperature T c in a crystal having a tetragonal paraelectric structure, Table 1.Transformation of the polarization components under the symmetry operations of the crystallographic point-group 4/mmm.denoted I, with 4/mmm point-group symmetry (Figure 1).Its vertices and its centre are respectively occupied by negative (M − ) and positive (M + ) ions.The ferroelectric phase II, stable below T c , results from the displacement of the M + ions out of the centre of the cell, in an unspecified direction.As this displacement generates an electric dipole p, phases I and II are characterized by p = 0 and p = 0, respectively, and at the macroscopic level by P = 0 and P = 0, where P is the electric polarization representing the sum of the dipoles over the unit volume.
The first concept of the Landau theory is that the I → II phase transition can be described by a non-equilibrium free-energy F (T , P ) expressed as a truncated Taylor expansion of the P components, the equilibrium values of the P components in the I and II phases corresponding to absolute minima of F with respect to the (P x , P y , P z ) variables.Such expansion is justified by the assumption of continuity of the phase transition in two different meanings: 1) The components vary continuously across T c and therefore take small values close below T c ; 2) F is a continuous and derivable function of T and P .The new idea is that F is invariant by the symmetry operations of the parent phase I and is therefore formed by invariant monomials of the P components.Table 1 shows the transformation of (P x , P y , P z ) by the symmetry operations of 4/mmm.One can verify that only quadratic (P 2 z , P 2 x + P 2 y ) and higher even-degree monomials (P 4 z , P 6 z , P 4 x + P 4 y , P 2 x P 2 y ) remain invariants.Minimization of F (T , P ) = F 0 (T ) + a 1 P 2 z + a 2 (P 2 x + P 2 y ) + ...
with respect to P x , P y and P z yields the equations of state that one of these coefficients has to change its sign at the transition, i.e., a 1 or a 2 is equal to a 0 (T − T c ) with a 0 > 0.Here comes an important result of the Landau model: Either a 1 or a 2 changes its sign at T = T c , but both coefficients cannot change their sign simultaneously.It means that below T c the M + ion can move either along the z axis or in the (x, y) plane but not in a general direction.This result stems from the property, illustrated by Figures 2(a) and 2(b), that a simultaneous vanishing at T c of a 1 and a 2 should be observable at a single point of a two-dimensional (e.g.Temperature-Pressure) phase diagram (Fig. 2 (a)) and would lead to a ferroelectric state II, the symmetry of which would be modified by a vanishingly small variation of the external variables (temperature or pressure).Since the Landau model aims to describe a ferroelectric state stable within a finite range of temperature and pressure, one has to conclude that only one coefficient, let say a 1 , changes it sign at T c (Fig. 2 (b)), the other coefficient (a 2 ) remaining positive within a certain interval close to T c .In this interval one has a 2 = 0 above and below T c .Therefore the useful form of F reduces to Minimization of F truncated at the fourth-degree is expressed by the equations: P z (a 1 + 2b 1 P 2 z ) = 0 and a 1 + 6b 1 P 2 z ≥ 0 ( 5 ) which give the equilibrium values of P z indicated in Fig. 3: P e z = 0 for T ≥ T c and for T < T c , under the condition b 1 > 0. The variable P z whose equilibrium is zero above T c and becomes non-zero below T c is the transition order-parameter.The property that a single quadratic invariant coefficient (a 1 ) becomes negative at the transition can be formulated in group-theoretical terms, by considering the set of one and two-dimensional matrices given in Table 1, which transform as P z and (P x , P y ), respectively.These matrices form two among the irreducible representations of the 4/mmm point-group.Therefore, the Landau theory shows that a continuous transition is induced by a single  irreducible representation of the parent symmetry group.The dimension of the matrices (here one or two) determines the order-parameter dimensionality.

Low-symmetry phases. Domains. Critical behaviour
A number of classical results of the Landau theory can be deduced from the preceding model: 1) Symmetry of the low-temperature phase is a subgroup G of the parent symmetry-group G 0 = 4/mmm, formed by the symmetry operations leaving invariant the order-parameter considered as a direction in a vector space.In the one-dimensional vector space P z , Table 1 shows that G = 4 mm.In the two-dimensional vector space (P x , P y ) the invariant directions are (P x , 0), (P y , 0), and (P x , P y ), (−P x , P y ) with P x = P y , corresponding to absolute minima of the fourth-degree expansion: and to the G-groups 2 x mm, 2 y mm, 2 xy mm and 2 xy mm.
2) The low-temperature states split into energetically equivalent domains of identical symmetries, the number of which is determined by the ratio between the order of G 0 (13) and the order of G (8 or 4).Therefore, two ferroelectric domains (±P z ) coexist in the G = 4mm state (Fig. 4 3) The critical behaviour of measurable physical properties related to first or second derivatives of F , can be obtained, namely (i) the order-parameter P z given by Eq. ( 6), varying as ≈ (T c − T ) 1/2 (Fig. 3), (ii) the entropy S = −( *F *T ) P z which displays no discontinuity at T c (Fig. 5 (a)), implying zero latent heat L = T c S(T c ) at the transition, (iii) the specific heat C E z = T *S *T E z at constant field E z conjugated to the order-parameter P z , which undergoes an upward discontinuity at T c (Fig. 5(b)), and (iv) the dielectric susceptibility = Lim E z →0 *P z *E z which diverges at T c following the Curie-Weiss law (Fig. 5 (c)).The continuity across T c of the physical quantities P z and S, related to first derivatives of F , and the discontinuity of C E z and , corresponding to second derivatives of F , characterize second-order phase transitions, as defined by Ehrenfest [11].

Secondary order-parameters. Improper ferroelectric transitions
The preceding considerations have been restricted to the "primary" order-parameters, P z or (P x , P y ), the symmetry-breaking of which determines the essential properties of the phase transition at T c , such as the symmetry of the stable state(s) which may arise below T c , the domain pattern and the shape of the critical anomalies at the transition.Other physical quantities may arise below T c , which do not modify the preceding properties but may influence secondary features of the phase diagram.In our introductory example, an eventual deformation of the unit-cell along z induced by the displacement of the M + ion, which is expressed by the strain-tensor component e zz = (z − z 0 )/z 0 , constitutes an example of secondary (non-symmetry-breaking) order-parameter.In order to determine the influence of e zz on the transition behaviour, one has to include in the free-energy F the lowest degree coupling invariant between e zz and the primary order-parameter P z as well as the elastic energy 1  2 C 33 e 2 zz .F reads: Minimizing with respect to e zz gives: below T c , and e e zz = 0 for T ≥ T c (Fig. 6. (a)).Replacing in Eq. ( 8) e zz by its equilibrium value given by Eq. ( 9), one gets the renormalized free energy It shows that the secondary order-parameter e zz renormalizes the fourth-degree coefficient.It therefore influences the limit of stability of the phases and the amplitude of the critical anomalies, but the phase diagram remains unchanged.The temperature dependences of e e zz , which varies linearly with (T c − T ) (Fig. 6 In our introductory example, the electric polarization is the primary order-parameter since it is directly correlated to the symmetry-breaking transition mechanism consisting of the displacement of the M + ions.One deals here with a proper ferroelectric transition.However, the spontaneous polarization may also be a secondary order-parameter, the primary symmetry-breaking mechanism corresponding to a set of displacements inducing indirectly the formation of electric dipoles.In this case one speaks of an improper ferroelectric transition.This denomination is also used for ferroelastic transitions in which a spontaneous strain may constitute either the primary or the secondary order-parameter.

First-order phase transitions
A large majority of the experimentally observed structural phase transitions are discontinuous [2].When a group-subgroup relationship is preserved between the phases, discontinuous transitions can be nevertheless described theoretically, using the concepts introduced in Sec. 2 for continuous transitions.As a first example let us consider the free-energy given by Eq. (4), describing the paraelectricferroelectric transition I → II, in which the stability condition b 1 > 0 was assumed.For b 1 < 0 there exists for T < T c , no stable minimum of F truncated at the fourth degree.Expanding F at the sixth degree and assuming a 1 (T − T c ), b 1 < 0 and c 1 > 0, the equations minimizing F , which are provide the following description, summarized in Fig. 7, for the transition between phases I and II.For 3a 0 c 1 phase I (P e z = 0) is the only stable phase.At T = T 2 phase II, corresponding to arises as a metastable state, associated with a secondary minimum in the non-equilibrium curve of F (T , P z ) − F 0 (T ).On cooling the stability increases and phases I and II become equally stable at Below T 1 phase I becomes metastable down to T c where it becomes unstable.Therefore, discontinuous transitions are characterized by the following properties: 1) A region of coexistence between phases I and II in the temperature interval T = T 2 − T c = b 2 1 3a 0 c 1 in which the two phases are alternately metastable.The transition may occur at any temperature within the interval.However, the most probable transition temperature is T 1 , corresponding to an equal stability for the two phases.The  existence of a region of coexistence for the two phases implies a thermal hysteresis on cycling across the transition, i.e, the transition does not take place at the same temperature on cooling and heating.2) A discontinuous jump P e z (T 1 ) = (− b 1 6c 1 ) 1/2 of the equilibrium order-parameter P e z at the transition (Fig. 8(a)).

Contribution of Symmetries in Condensed Matter
3) A latent heat at T 1 (Fig. 8(b)) which results from the order-parameter discontinuity, given by As for P e z and S, a discontinuity will be also observed for all physical quantities related to first derivatives of F , such as the volume or the spontaneous strains.Therefore, discontinuous transitions are first-order transitions following Ehrenfest classification [11].
4) Specific discontinuities at T 1 for the physical quantities related to second derivatives of F .Figures 8(c) shows, for example, the temperature-dependence of the dielectric susceptibility, which undergoes a finite upward jump on cooling across T 1 .
Figure 9 summarizes the phase diagram involving a second-order (b 1 > 0) and first-order (b 1 < 0) transitions.The cross-over between the two regimes occurs at a tricritical point corresponding to b 1 (T c ) = 0 and a 1 (T c ) = 0.

Integrity basis. Rule for truncating the Landau free-energy. Number of low-symmetry stable states
The Landau free-energy given by Eq. (4) or Eq. ( 11) can be expanded using a single invariant of the order-parameter = P 2 z and its successive powers 2 , 3 , . . .By contrast the free-energy defined by Eq. ( 7) requires two independent invariants 1 = P 2 x + P 2 y and 2 = P 2 x P 2 y .and ( 1 , 2 ) form the integrity basis associated with the order-parameters P z and (P x , P y ), respectively.In a more general way, the integrity basis associated with a given order-parameter is the set of independent invariant monomials which can be formed from the order-parameter components.The concept of integrity basis  allows determining the necessary degree at which one has to expand a free-energy F in order to disclose, from the minimization of F , the full set of stable states associated with a given order-parameter.

Rule 1:
If n is the highest-degree of the monomials forming the integrity basis, one has to truncate the free-energy at the degree 2n for a second-order transition from the parent phase, and at the degree 2n+2 for a first-order transition.
In the case of F (P z ), n = 2 and F has to be expanded at the fourth-degree for a second-order transition (Eq.( 4)), and at the sixth-degree for a first-order transition (Eq.( 11)).For F (P x , P y ), n = 4 and F should be expanded at not less than the eighth-degree.Therefore, the fourth-degree at which the free-energy given by Eq. ( 7) is expanded does not account for the full set of stable states associated with the order-parameter (P x , P y ).One can show that expanding F (P x , P y ) at the eighth-degree stabilizes an additional phase, corresponding to P x = P y and to the lower symmetry m z , associated with a general displacement of the M + ion in the (x, y) plane.
In order to predict the number of stable states N associated with a given order-parameter one can construct the phase diagram in the space of the order-parameter invariants (the orbit-space).Figures 10(a) and 10(b) show such phase diagrams for P z and (P x , P y ).One can verify that the stable states correspond to the origin (the parent phase), and to curves and surfaces of the orbit space (the low symmetry states) merging at the origin.

Rule 2:
The low-symmetry stable state associated with a given order-parameter corresponds to curves merging at the origin of the orbit-space, to surfaces limited by the curves, and to volumes limited by the 00007-p.8

Contribution of Symmetries in Condensed Matter
surfaces.Surfaces are associated with subgroups of the limiting curves, and volumes to subgroups of the limiting surfaces.
The preceding rule applies straightforwardly when the orbit space has a dimension m ≤ 3.For m > 3 one deals with hyper-surfaces and hyper-volumes and it is more practical to consider projections (cuts) of the orbit-space on two-or-three-dimensional spaces.In the particular case where m is equal to the order-parameter dimensionality n (m = n), the total number of stable states for groups generated by reflexions is N = 2 n .This is the case for theP z and (P x , P y ) order parameters for which one has, respectively, m = n = 1, and m = n = 2. Therefore the number of stable states are respectively N = 2 and N = 2 2 = 4.

Dependence of the phenomenological coefficients on external variables
In the free-energies F (T , P z ) and F (T , P x , P y ) only one phenomenological coefficient, a 1 or a 2 corresponding to the quadratic invariants P 2 z or P 2 x + P 2 y , has been assumed to vary linearly and critically with temperature.The theoretical phase diagram resulting from a Landau model, shown for example in Fig. 9, is drawn in the space of the free-energy coefficients.In order that the theoretical phase diagram reflects the experimental phase diagram, one has to follow precise rules, deduced from the theory of Singularities [12], for determining the coefficients varying with external variables (T , P , x, ...) and the form of their dependence on the external variables.1) The number of varying coefficients should be equal to the number of external variables, i.e., one-two-or-three dimensional phase diagrams require a dependence of one-two-or-three coefficients, respectively, on the external variables.2) Only the coefficients, whose sign determines the stability of the phases can be assumed to depend on the external variables.The remaining coefficients remain constant.3) The coefficient of the highest degree invariant in the free-energy is always a positive constant.4) The coefficient of the quadratic invariant depends linearly and critically on the external variable inducing the transition (e.g; the temperature).Such linear dependence ensures that the experimental phase diagram can be deduced from the theoretical phase diagram by a linear transformation preserving its singularities.However, the preceding coefficient, as well as coefficients of higher power invariants may depend nonlinearly on the other external variables (e.g.pressure or concentration).The non-linear dependence may involve integer or fractional powers of the external variables.
According to the preceding rules, the phase diagram associated with F (T , P z ) expanded at the fourth-degree (Eq.( 4)) is necessarily one dimensional, since b 1 is a positive constant, and a 1 can be assumed to vary in function of the single external variable T .Expanding F (T , P z ) at the sixthdegree (Eq.( 11)) allows constructing the two-dimensional phase diagram of Fig. 9. Assuming c 1 > 0, one can simply take a 1 = a 0 (T − T c ) and b 1 = b 0 (P − P c ) (a 0 > 0, b 0 > 0) for preserving the main topological features of the theoretical phase diagram in the Temperature-Pressure phase diagram, namely, a parabolic shape for the first-order transition curve T 1 (P ) = T c + b 2 0 (P −P c ) 2 4a 0 c 1 and a tricritical point at T 1 (P c ) = T c .
Figures 11(a)-(c) show different two-dimensional phase diagrams associated with the (P x , P y ) orderparameter for an eighth-degree expansion: where 1 = P 2 x + P 2 y and 3 = 2 1 − 8 2 , with 2 = P 2 x P 2 y .One can verify that the topological features of the phase diagram associated with a given order-parameter can be modified depending on the signs and values of the phenomenological coefficients.However, a number of features are preserved, which can be foreseen from the topology of the phase-diagram in the orbit space (Fig. 10(b)), as for example the existence of four distinct phases, the region of stability of the lower symmetry phase IV, which is always found between phases II and III, or the property of phases II and III to be reached directly from phase I across a second-or-first-order transition line.Note that in the free-energy expansion, given by 13 .Dashed, full, and dashed-dotted curves correspond, respectively, to second-order, first-order, and limit of stability curves.
Table 2. Set of matrices associated with the three-dimensional irreducible representation T 2u of the P m 3m spacegroup at the centre of the primitive-cubic Brillouin-zone.
Eq. ( 14), a number of high-degree invariants have been neglected, such as 3  1 or 2 1 3 , which can be shown to modify non-essential aspects of the phase diagrams, as the shape of the transition lines or the width of the metastability regions.Here, Pressure-Temperature phase diagrams can be deduced from the theoretical phase diagrams by assuming a linear temperature and pressure dependence for a 1 and b 2 , while the other coefficients remain constant.

Application to Barium titanate
Let us illustrate the theoretical considerations presented in the preceding sections by the concrete example of Barium titanate.The parent phase of BaTiO 3 corresponds to the cubic perovskite structure of symmetry P m 3m.The order-parameter associated to the sequence of ferroelectric phases [13] is formed by the three components (P x , P y , P z ) of the polarization, which span a three-dimensional irreducible representation (denoted T 2u ) of the P m 3m space-group, at the centre of the primitive-cubic Brillouin-zone.Table 2 gives the 48 matrices corresponding to the image-group (the set of distinct matrices) of T 2u .From the transformation 00007-p.10  properties of these matrices, the following independent invariants, forming the integrity basis of T 2u , can be constructed: 1 = P 2

Contribution of Symmetries in Condensed Matter
x + P 2 y + P 2 z , 2 = P 4 x + P 4 y + P 4 z and 3 = P 2 x P 2 y P 2 z .Since the number of invariants m=3 is equal to the dimensionality n = 3 of the order-parameter, the number of distinct stable states permitted by the symmetry of the order-parameter is 2 3 = 8. Figure 12 shows the location of the phases in the orbit spaces ( 1 , 2 , 3 ) and ( 2 , 3 ).Figure 13 summarizes the space-group symmetries of the phases, the corresponding invariant directions in the three-dimensional order-parameter space, which can be obtained from the matrices of T 2u , and the number of ferroelectric domains for each lowsymmetry phase.In addition to the three ferroelectric phases reported in BaTiO 3 , four lower symmetry phases having the symmetries P m, Cm or C1 display regions of stability in the phase diagram.In order to obtain the full set of stable phases, the free-energy should be expanded at not less than the twelfth degree.Neglecting a number of high-degree invariants, one can take: When F is truncated at the sixth degree, the theoretical phase diagram contains the three ferroelectric phases actually found in BaTiO 3 (Fig. 14(a)), or only two of them (Fig. 14(b)).An eighth-degree expansion allows stabilization of one more phase of Cm (Fig. 14(c)) or P m (Fig. 14(d)) symmetry.Only a twelfth-degree expansion permits the full set of low-symmetry phases to be stabilized.The corresponding three-dimensional phase diagram, shown in Fig. 15, requires three varying external variables.
The proper paraelectric-ferroelectric transitions in Barium titanate are also improper ferroelastic transitions, the spontaneous strains corresponding to secondary order-parameters.This can be foreseen by the number of ferroelastic domains arising in each phase, which corresponds to half of the number

Phase diagrams associated with a reducible order-parameter
The transitions examined in the preceding sections were related to a single irreducible order-parameter, in agreement with the essential statement of the Landau theory which associates a phase transition to a single irreducible representation.There exist, however, experimentally observed phase transitions which 00007-p.12 have to be connected to more than one irreducible order-parameter.Referring to Fig. 2(b), two main situations can be met: 1) The transition associated with the P z order-parameter occurs discontinuously at T 1 > T c and triggers the vanishing of the a 2 coefficient at T 1 , giving rise below T 1 to a phase displaying non-zero equilibrium values of P z and (P x , P y ).The symmetry of this phase corresponds to the intersection of the symmetries induced by both order-parameters.Such situation, first described by Holakovsky [14], is found in a number of structural transitions.2) Another situation, described by Lifshitz [15], corresponds to a sequence of two successive second-order transitions taking place consecutively at T c1 and T c2 .The first transition at T c1 , corresponds to the vanishing of a 1 , with the spontaneous onset of a P z component, whereas the second transition at T c2 gives rise to spontaneous values of (P x , P y ).The symmetry of the phase arising below T c2 will reflect the coupling of both order-parameters P z and (P x , P y ), although only one symmetry breaking mechanism associated with (P x , P y ) is involved.This situation, which is realized in a number of magnetic multiferroic compounds, is described in the following subsections.

Phenomenological description at zero magnetic field
As a first illustrative example of transition induced by coupled order-parameters let us describe the sequence of magnetic phases occurring in MnWO 4 .Below its wolframite-type structure of monoclinic P 2/c symmetry, MnWO 4 undergoes three successive magnetic transitions [16,17] at 13.5 K (T N ), 12.7 K (T 2 ) and 7.6 K (T 1 ).They lead, respectively, to an incommensurate antiferromagnetic phase (AF3), an incommensurate elliptical spiral phase (AF2) displaying an electric polarization P //b, and a commensurate antiferromagnetic phase (AF1).The transition wave-vectors are k inc = (−0.214, 1 2 , 0.457) for AF2 and AF3, and At variance with structural transitions which are induced by irreducible representations of crystallographic space-groups, magnetic transitions are induced by irreducible corepresentations of magnetic space-groups [18].In the present case, the parent phase is the paramagnetic gray spacegroup P 2/c1 , which is formed by the symmetry operations of P 2/c plus their combination with the time-reversal symmetry operation T .Since the spin-densities considered in the Landau approach to magnetic phase transitions are macroscopic (average) non-quantum objects, T is simply represented by Table 3. Generators of the irreducible corepresentations k1 and k2 of the P 2/c1' paramagnetic space-group, deduced from the irreducible representations 1 and 2 of G k = m y , given in Kovalev's tables [19], for k inc and k com .
the matrix −I , where I is the identity matrix.Therefore the irreducible corepresentations of P 2/c1 are simply deduced from the corresponding irreducible representations of P 2/c, as shown in Table 3, by adding a set of matrices obtained by multiplying the matrices of the relevant irreducible representations of P 2/c by −I .Taking into account the symmetry group G k = m y of kinc and k com one gets two relevant two-dimensional irreducible corepresentations [19], denoted k1 and k2 , whose generators are given in Table 3.The complex amplitudes transforming according to k1 and k2 are denoted S1 = S 1 e i 1 , S * 1 = S 1 e −i 1 and S2 = S 2 e i 2 , S * 2 = S 2 e −i 2 .For k = k inc the order-parameter invariants 1 = S 2 1 , 2 = S 2 2 and 3 = S 2 1 S 2 2 cos 2 , with = 1 − 2 yield the Landau expansion which has to be expanded up to the degree 2n = 8 since the degree of 3 is n = 4. Minimizing with respect to gives Eq. ( 17) and the minimization of 1 with respect to S 1 and S 2 show that five distinct stable states, denoted I-V, may arise below the paramagnetic (P ) phase for different equilibrium values of S 1 , S 2 and .This can be foreseen from the phase diagram in the orbit-space shown in Fig. 17(a) which contains two curves, two surfaces limited by the curves and the volume limited by the surfaces.Theoretical phase diagrams showing the location of the phases can be constructed in the two dimensional spaces ( 1 , 2 ) and ( 2 − 1 , 1 ) (Figs. 17(b) and 17(c)), and in the three-dimensional space ( 1 , 2 , 1 ) (Fig. 17(d)).Since the AF2 and AF3 phases possess incommensurate structures one can only determine their magnetic point-groups.On that goal one cannot use, as for commensurate structures, the matrices of the irreducible corepresentations for disclosing the invariant directions in the order-parameter space.
The procedure for determining the macroscopic (point-group) symmetries of incommensurate phases was given by Dvorak et al [20].In particular, one can use here the following useful rules: 1) When the phase of the order-parameter (here 1 , 2 or ) can be shifted without modifying the stability of the incommensurate structure, its point-group coincides with the point-group G P of the parent structure (here the paramagnetic point-group 2/m1 ), 2) If the phase of the order parameter is fixed, the point-group symmetry of the incommensurate structure is a subgroup of G P formed by the symmetry elements preserving the spontaneous macroscopic physical tensor components induced by a coupling with the order-parameter (i.e.here, components of the magnetization M and electric polarization P ). Figure 18 gives the magnetic point-group symmetries of the five phases which may stabilize below the paramagnetic phase.They correspond to gray magnetic point-groups associated with antiferromagnetic structures.Neutron diffraction data [21] show that the AF3 phase is induced by k2 , coinciding with phase I in Fig. 17, with S 1 = 0 and S 2 = 0.It has the magnetic point-group 2 y /m y 1 involving a doubling of the b-lattice parameter and an incommensurate modulation of the spin-density in the (x, z) plane.The AF2 spiral phase, induced by k1 + k2 identifies to phase II in Fig. 17, corresponding to S 1 = 0, S 2 = 0, which changes its sign for opposite senses of the spiral configuration, as observed experimentally [22].The order-parameter S2 , activated at the paramagnetic→AF3 transition, is frozen at the AF3→AF2 transition.Therefore, Eq. ( 3) expresses a linear dependence of P y on S1 .In the AF3 and AF2 phases 1 can be truncated at the fourth-degree.Putting 1 = a 1 (T − T 0 ) and 2 = a 2 (T − T N ) one finds that P y varies in the AF2 phase as . The dielectric permittivity follows a Curie-Weiss-type law . Figure 19 shows the excellent fit of P e y (T ) and yy (T ) with the experimental curves.The low value of ≈ 40 C/m 2 found for P y at 10 K, reflects the hybrid character of the ferroelectricity in spiral magnets: The critical anomalies are typical of a proper ferroelectric transition, while the magnitude of the polarization is of the order found in improper ferroelectrics.
Neutron diffraction results [21] indicate that the transition to the commensurate AF1 phase triggers a decoupling of S1 and S2 .The lock-in at T 1 , induced by k2 , gives rise to the invariant 4 = S 4 2 cos 4 2 .19)) and dielectric permittivity yy (T ) (Eq. ( 20)) with the experimental points from Taniguchi et al [16].

00007-p.15 EPJ Web of Conferences
The Landau expansion associated with the transition to AF1 is The equations of state show that three commensurate phases, denoted I'-to III', displaying a fourfold unit-cell ( b + c, c − b, 2 a + c), may appear below T 1 .The AF1 phase identifies to phases I' or II', stable for cos 4 e 2 = ±1, both described by the magnetic space group C a 2/c (see the chapter by Rodríguez-Carvajal and Bourrée for the notation of magnetic space groups).Phase III', stable for cos 4 e 2 = 1 2 2 2 , has the symmetry C a c.

Magnetic field induced effects
The AF1, AF2 and AF3 order-parameter symmetries allow describing the magnetoelectric effects [16,17,22] observed in MnWO 4 .Let us first note the following: 1) Since the magnetic symmetries of the 00007-p.16ordered phases correspond to gray point-groups one can expect magnetic-field induced magnetoelectric effects, but not the reciprocal electric-field induced magnetoelectric effects, since the time-reversal symmetry forbids the existence of non-zero macroscopic components of the magnetization M under applied electric field.2) When applying a magnetic field B to a phase, the magnetic symmetry of the phase may be lowered as it corresponds to the intersection of its symmetry group at B = 0 with the symmetry group of the magnetic field.

Contribution of Symmetries in Condensed Matter
Various magnetic field induced effects are observed in MnWO 4 : 1) The region of stability of the AF2 phase decreases with respect to AF3 and AF1 with increasing B y field (Fig. 20) and depends on the orientation of B y with respect to the magnetic easy axis; 2) Above a threshold field B th y of about 10 T, the AF2 phase undergoes a polarization-flop transition P y → P x ; 3) Cycling across B th y and canting oppositely B with respect to the y axis, yields an opposite sign for P x .The reduction of the stability range of AF2 with respect to AF3 and AF1 can be deduced from the equation expressing the stability of the AF2 phase under B y field.Taking into account the magnetic free-energy M 1 = M 2 2 − B. M and coupling invariants 1 M 2 y S 2 1 and 2 M 2 y S 2 2 , the stability condition where For < 0 the stability range of the AF2 phase decreases with respect to AF3 under increasing B y field.In a similar way the AF2 stability range reduces with respect to AF1, under B y field.The polarization flop corresponds to a first-order transition to phase III', which displays a polarization component P x and no P y component.Therefore, the threshold field B y is given by the equality of the expansions Canting oppositely B from the b-axis ( → −) reverses P e x .Increasing again B y above B th y yields an opposite sign for P III x in phase III', as observed by Taniguchi et al [23].The AF2 stability range depends on the angle 0 of B and magnetic easy axis in the paramagnetic phase.This property relates to the anisotropy of the magnetic free-energy , where ˆ is the paramagnetic susceptibility tensor, with tan 2 0 = 2 xz ( xx − zz ) −1 .If B is at an angle with the x-axis in the (x, z) plane, one finds Where t = xx + zz and d = xx zz − 2 xz .For < 0 the AF2 stability range is maximum along the easy axis ( = 0 ).It decreases when increases from 0 to 0 + 2 , reducing to the stability range at zero field if xz , xx << zz .Such variation has been observed in the AF2 phase, in which 0 ∼ = 35 • coincides with the direction of the spins in the (x, z) plane.When B is at an azimuthal angle with the (x, z) plane, the AF2 stability range decreases when increases from = 0 to = /2, as reported experimentally.

Microscopic origin of the unconventional ferroelectricity in magnetic multiferroics: The Dzialoshinskii-Moriya and symmetric interactions
Let us express the phenomenological order-parameters S1 and S2 in function of the magnetic spins in the commensurate phases I'-III'.Denoting s 1 -s 8 the spins associated with the eight Mn 2+ ions of the fourfold primitive monoclinic unit-cell (Fig. 21 Figures 21(a) and 21(b) represent the magnetic structures in the (x, z) plane for phases I' and II'.It shows that structure of phase II' coincides with the antiferromagnetic order reported in the AF1 phase of MnWO 4 .The lack of spin components along b assumed for this phase may be due to their relativistic origin as suggested in Ref. [21].The equilibrium values given for the spin components in Table 4, allow to determine the magnetic structure of Phase III', which identifies to the high-field phase of MnWO 4 .
Other commensurate structures can take place for k = k com when the spin components associated with S1 and S2 order simultaneously.Two among these structures, denoted IV' ( e 1 = 0, e 2 = /2) and V' ( e 1 = /2, e 2 = 0), display the symmetry 2 y and the same form of P y = 1 2i ( S1 S * 2 − S2 S * 1 ) = S 1 S 2 sin as in the AF2 phase.Although these structures are not stabilized in MnWO 4 they represent lock-in limits of the incommensurate AF2 structure, and can be used for investigating the microscopic origin of ferroelectricity in the spiral magnetic structure.Since S1 and S2 are both realized by three Table 4. Equilibrium values of the magnetic spins for phases I'-to-V' considered in the text.e ij × s i × s j (where e ij is the distance between neighbouring Mn atoms i and j), cancel in Eq. ( 28) when all spin components are cancelled except s a 1 and s a 7 in phase IV', or s a 3 and s a 5 in phase V', while keeping a finite value P y = 1 (s a 2 1 − s a 2 7 ) and P y = 1 (s a 2 3 − s a 2 5 ) in phases IV' and V'.Therefore symmetry considerations predict the existence of a polarization induced by symmetric interactions, although such effect may encounter considerable restrictions at the microscopic level.

Survey of the theoretical description of RMnO 3 and RMn 2 O 5 multiferroics
A similar theoretical description than for MnWO 4 can be proposed for the sequence of three magnetic phases, denoted I-III, observed below T 1 = 41 K in TbMnO 3 [27][28][29], although the lower-temperature non-polar phase III arising below T 3 = 7 K is more complex than a simple lock-in structure of the ferroelectric spiral structure of phase II, stabilized below T 2 = 28 K.The similarity with MnWO 4 stems from the property that the two irreducible corepresentations involved in the phase sequence of TbMnO 3 , denoted 2 and 3 , have the same image-group than k1 and k2 , in spite of a different paramagnetic symmetry , and transition wave-vector k = (0, 0.28, 0).Therefore, the coupling of 2 and 3 yields for TbMnO 3 the same Landau expansion, given by Eq. ( 16), the same phase diagrams shown in Fig. 17, and an analogous hybrid "pseudo-proper" behaviour for the ferroelectric spiral phase.However, due to the higher Pbnm1' symmetry of the paramagnetic phase, the magnetic symmetries of the antiferromagnetic phases I an II display higher symmetric gray magnetic point groups, given in Fig. 22. Besides, the polarization-flop process in TbMnO 3 shows more complex features than in MnWO 4 .Under applied magnetic field along the x or y directions, the spontaneous P z polarization existing at zero field in the spiral phase of TbMnO 3 is switched along the x direction above threshold fields B th x ≈ 9 T and B th y ≈ 4.5 T, respectively, at a first-order lock-in transition to a commensurate phase with k = (0, 1 4 , 0).The first-order transition process can be decomposed in two stages: Under B x or B y field the spiral phase II is switched to a lower symmetric ferroelectric phase, above threshold fields given by the stability condition of phase II under B x or B y field, which triggers a decoupling of the two order-parameters involved in phase II, and a lock-in transition to a ferroelectric phase, associated with a single orderparameter.This phase has the symmetry 2mm, which cancels P z and gives rise to P x .Such interpretation in terms of a two-stage process is supported by the observation in TbMnO 3 of the coexistence of distinct structural phases in the intermediate field regime [30].Another support is provided by the observation in DyMnO 3 of similar field-induced flops [31], not associated with a high-field commensurate lock-in phase.It implies the stabilization of an intermediate phase with 2 x symmetry induced by magnetoelectric 1 P x M x M z S 2 S 3 sin or 2 P x M x M y S 2 S 3 sin under B x or B y fields, where S2 and S3 are the order-parameters transforming as 2 and 3 .
A different situation is found in TbMn 2 O 5 [32] due to the property that a single four-dimensional irreducible corepresentation induces the spiral phase observed in this compound.On cooling below the paramagnetic (P) phase, of symmetry Pbam1', TbMn 2 O 5 undergoes five phase transitions taking place successively at T 1 = 43 K, T 2 = 38 K, T 3 = 33 K, T 4 = 24 K and T 5 = 10 K [33,34], the corresponding phases being denoted I-V.The P → I → II sequence of incommensurate phases is associated with a four-component order-parameter transforming as an irreducible corepresentation, denoted G 1 , with the wave-vector k = (1/2, 0, k z ) At T 3 = 33 K the commensurate phase III takes place for k = (1/2, 0, 1/4) preserving the dimensionality, but not the symmetry of the order-parameter.By contrast the incommensurate phases IV and V correspond to the coupling of two four-dimensional orderparameters associated with two irreducible corepresentations, denoted 1 and 2 , at k = (0.48, 0, 0.32).
The four-component order-parameter associated with the P → I → II transitions can be denoted Minimizing 1 shows that seven phases can be stabilized below the P phase for different equilibrium values of 1 , 2 and , summarized in Fig. 23.The corresponding phase diagrams in the orbit space   corresponding to an effective linear dependence of P e y on 2 , since 1 has been already activated in phase I. Therefore, below T 2 , in the ferroelectric phase II, one has P e y (T ) ∝ (T 2 − T ) 1/2 (Fig. 25(a)), and the same temperature dependence, given by Eq. ( 20), for the dielectric permittivity yy (T ) (Fig. 25(b)).Therefore the transition to phase II shows the same hybrid behaviour as the analogous transitions in TbMnO 3 and MnWO 4 , i.e. it behaves critically as a proper ferroelectric transition, although the value of the polarization (about 40 nC/cm 2 ) in phase II is that of an improper ferroelectric.
At T 3 = 33 K the wave-vector locks into the commensurate value k = (1/2, 0, 1/4) giving rise to the (umklapp) invariants 4  1 cos 4 1 + 4 2 cos 4 2 and 2 1 2 2 cos 2( 1 + 2 ).The symmetry of the orderparameter is changed but not its dimensionality.Taking into account these additional invariants yields an equilibrium ferroelectric phase for 1 = ± 2 = n /4, having the same magnetic point-group symmetry m2m as phase II, with a commensurate lattice parameter 4c.The transition coincides with a slight change in the slope of P e y (T ) (Fig. 25(a)), and no noticeable anomaly of the dielectric permittivity.In contrast the transition occurring at T 4 = 24K to phase IV exhibits a finite jump of e yy (T ) which reflects a typical improper ferroelectric transition.The improper character of the transition is consistent with the form of the polarization in phase IV which is: where 1 and 3 , which are amplitudes of two distinct four-dimensional order-parameters which are both activated at the III → IV transition mechanism, vary as ∝ (T 4 − T ) 1/2 for T < T 4 .Therefore, P y  Below T 4 , P e y (T ) is the sum of two distinct contributions given by Eqs. ( 30) and (31): where 1 ∝ (T 1 − T 2 ) 1/2 .Assuming 1 > 0 and 2 < 0, one can verify that P e y (T ) decreases for The strong increase observed below T 5 reflects a positive contribution to the total polarization of the electric polarization in phase V (Fig. 26).Under applied magnetic field H x the spontaneous polarization in phase IV becomes: due to the magnetoelectric coupling ν 1 3 P Y M 2 x .For ν < 0 the application of a H x field enhances the negative contribution to the temperature dependence of the total polarization, leading to: Therefore, P e y (T , H x ) changes sign for the temperature dependent threshold field as shown in Fig. 26.

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Similar sequences of ferroelectric phases are found in other RMn 2 O 5 compounds [35][36][37], where R = Bi, Y or a rare-earth heavier than Nd.In these compounds, at variance with TbMn 2 O 5 in which the pseudo-proper coupling occurs between distinct components of the same order-parameter, the pseudo-proper coupling is created between two distinct four-component order-parameters associated with the wave-vector k = (k x , 0, k z ).However, with the exception of DyMn 2 O 5 and BiMn 2 O 5 , the orderparameters involved in the RMn 2 O 5 family correspond to the symmetries disclosed for R = Tb, i.e., to k = (1/2, 0, k z ), (k x , 0, k z ) and (1/2, 0, 1/4).

Specific property of phase transitions associated with continuous symmetry groups
In the preceding sections the parent structure was described by a discrete, crystallographic or magnetic, symmetry group.When continuous groups are involved, the lowest symmetry group induced by an irreducible representation of dimension n ≥ 2 of the parent symmetry group G 0 , can only be obtained by considering the reducible representation + + + ..., the number of s figuring in the sum being smaller or equal to n.This result stems from the specific property of continuous groups that the minimal symmetry group G m induced by (defined as the invariance group of the general direction in the representation space) does not coincide with the lowest symmetry group G k associated with (corresponding to the kernel of the homomorphism of G 0 on (G 0 )).Therefore, in order to obtain the full set of broken symmetry phases associated with a given irreducible representation of a continuous group, one has to consider different replica of .We will illustrate this property by considering examples of phase transitions in superconducting and liquid crystal systems, which are both described by continuous symmetry groups.

Unconventional s-pairing wave superconductivity
The parent (Normal) state of a superconductor is generated by four groups: Where U (1) is the continuous group generated by gauge operations g .T is the group formed by the time-reversal operation T and the identity.O R (3)and O S (3) are, respectively, the full orthogonal group in real and spin spaces.In the Ginzburg-Landau theory of superconductivity [5], the transition to the superconducting state is associated with a two-component s-pairing wave order-parameter corresponding to a complex wave function 1 = 1 e i 1 and its complex conjugate * 1 = 1 e −i 1 .It transforms as a two-dimensional irreducible representation of G 0 which can be written: where 1 is the identity irreducible representation of O R (3) ⊗O S (3) and V is the vector representation of U (1) ×T .The single superconducting state induced by has the same structural symmetry as the normal state and is time-reversal invariant: Only the gauge-symmetry is broken at the Normal-tosuperconducting transition.Let us show that the phase diagram associated with the four-component reducible order-parameter ( 1 , * 1 , 2 = 2 e i 2 , * 2 = 2 e −i 2 ) transforming as + contains two different superconducting states, which possess respectively the minimal symmetry G m and the lowest symmetry G k associated with [38].From the transformation properties of the order-parameter components under the symmetry operations of G 0 one finds for any continuous or discrete symmetries of O R (3) ×O S (3), the following invariant monomials: 1 = 2 1 , 2 = 2 2 and 3 = 1 2 cos( 1 − 2 ), which give the Landau expansion:  Minimization of yields three possible stable states, which are represented in the phase diagrams of Fig. 27.In addition to the Normal state of symmetry G 0 , which is stable for 1 = 2 = 0, two different superconducting states can be stabilized:
2) State II, which is stabilized for 1 = 0, 2 = 0 and 1 − 2 = 0, .It has the lowest symmetry 3) associated with the irreducible representation (Fig. 28(c)).Accordingly, at the Normal→State II and State I →State II transitions, the time reversal symmetry is broken even when assuming an isotropic symmetry in real and spin spaces.The spontaneous physical quantity arising in phase II transforms as: A non vanishing macroscopic physical quantity corresponding to is ( E 1 × E 2 ).H , where E 1 and E 2 are differently oriented electric fields, and H is a magnetic field.In this case identifies to a combination of components of the third-rank tensor ij k characterizing a second-order magnetoelectric effect expressed by: where P is the polarization induced by the conjuncted application of E and H , and is the dielectric permittivity.Note that P will appear perpendicularly to E under application of H .Note also that the reciprocal magnetoelectric effect, corresponding to the equation of state M i = ij k E j E k will not be observable, since the induced magnetization vanishes identically.

Unconventional chiral ferroelectric liquid crystal phase
A close analogy with the s-pairing wave superconducting states can be found in the ferroelectric phases of chiral liquid crystals formed from rod-like molecules [39,40].The order-parameter of the Sm − A * → Sm − C * phase transition is a two-component tilt vector = ( 1 , 2 ) describing the magnitude and the tilt of the long molecular axis n from the normal z to the smectic layers, with a tilt angle .Because of the chirality of the molecules, a linear coupling between the tilt and the inplane polarization exists, leading to a spontaneous transverse polarization P t = (P x , P y ) perpendicular to n and (Fig. 29(a)).The precession of n around the z direction, results in a helical ferroelectric Sm − C * phase, having a zero spontaneous macroscopic polarization.Considering a bilinear coupling between the components of and P t implies assuming a four-component reducible order-parameter = ( 1 , 2 , P x , P y ) which transforms as the sum E 1 + E 2 of two identical irreducible representations associated with the D ∞ symmetry-group of the Sm − A * phase.In this case the monoclinic C 2 symmetry of the Sm − C * phase, which corresponds to orthogonal directions for and P t , is not the lowest symmetry involved by the E 1 + E 2 representation.The lowest symmetry is triclinic C 1 , and coincides with an angle = /2, between and P t .Let us show that the corresponding triclinic smectic configuration, which displays a spontaneous polarization component P z (Fig. 29(b)), can be stable.
From the transformation property of the four-component order-parameter under the symmetry operations of D ∞ one finds the following independent invariants 1 = 2 1 + 2 2 = 2 , 2 = P 2 x + P 2 y = P 2 t and the piezoelectric coupling invariants 3 = 1 P y − 2 P x = P t sin and 4 = P z ( 1 P x + 2 P y ) = P z P t cos .The corresponding Landau free-energy density reads: Minimization of shows that, in addition to the parent Sm − A * phase ( e = P e t = 0), two distinct tilted smectic configurations with = 0 and P t = 0 can be stabilised: 1) The standard Sm − C * configuration shown in Fig. 29(a), for e = (2n + 1) /2, which has the monoclinic C 2 symmetry, and the spontaneous polarization second-order piezoelectric effect, i.e. from the simultaneous existence of tilt-angle and in-layer polarization, under the condition that the scalar product .P t = 0.The phase diagrams involving the Sm − C * and Sm − C * P phases are shown in Fig. 30.They are topologically identical to the phase diagrams of Fig. 27.

Phase transitions between group-subgroup unrelated structures
Continuous or slightly discontinuous structural transitions have been the almost exclusive subject of the theoretical approaches to phase transitions: From the initial work of Landau (in which the author assumes a group-subgroup relationship between the symmetries of the phases) to the different statistical physics approaches, such as the Ising model (which leads to a lowering of symmetry with respect to the initial structure), and the sophisticated theory of critical phenomena.However, the most widespread and fundamental types of structural transitions found in nature involve no group-subgroup relationship between the phase symmetries.In this respect, two different situations should be distinguished.The phases may correspond to different equilibrium components of the same order-parameter and display different subgroups of a common parent symmetry group.This is, for example, the case of the low temperature phases of Barium titanate, corresponding to subgroups of the P m 3m perovskite symmetry.Here, the structures of the phases are closely related between them by small structural modifications, across slightly discontinuous transitions.Reconstructive phase transitions [3,[41][42][43][44] belong to another class in which the transition mechanisms exhibit drastic changes, with critical displacements of the order of magnitude of the lattice parameters, large latent heat and thermal hysteresis, and an eventual breaking of part of the chemical bonds.In the following subsections we show that despite the preceding features, a theoretical description of reconstructive transitions can be performed, preserving the main concepts of the Landau approach.

Basic ideas of the theory of reconstructive transitions
We summarize hereafter the basic ideas of the theory of reconstructive phase transitions, assuming that the transition mechanism can be formulated in terms of atomic displacements.Let us consider two phases A and B having no group-subgroup relationship between the space-groups G A and G B describing their structures.G A and G B possess common subgroups G 1 , G 2 , G 3 . ...corresponding to common stable substructures of the A and B structures.Starting from the A structure, the reconstructive mechanism transforming A in B proceeds via a critical displacement field i going across one of the substructures of A and B associated with one of the G i subgroups denoted G AB , which fulfils the following constraints: 1) It corresponds to the set of minimal critical displacements permitting to reach the B structure.2) It is compatible with the observed epitaxial relationships between A and B and with the various experimental data reported for the two phases (domains, strains, vibration spectra...).
3) It is consistent with the transition mechanisms leading to other stable phases having regions of stability in the phase diagram.Accordingly, the determination of the transition path transforming A into B cannot be simply based on symmetry (group-theoretical) considerations but requires a careful analysis of the real structures, and of the available experimental observations on the transition.
For vanishingly small or non specific critical displacements, the symmetry of the initial A structure is lowered to G AB .The B structure is obtained for specific large fractional displacements i at which the new symmetry elements pertaining to the B structure arise.Furthermore, determining the orderparameter symmetry from the irreducible representations 1 and 2 associated, respectively, with the hypothetical G A → G AB and G B → G AB transitions, one can show that the order-parameter modulus of the G A → G B reconstructive transition is a periodic function ( i ) of the critical displacements, whose minima and maxima coincide with G B , whereas ( i ) = 0 corresponds to G A .Therefore, the Landau expansion ( ( i )) has to be minimized with respect to the i variables, the equations of state * * i = * * .* i = 0 showing that two types of equilibrium states can be distinguished: 1) Landau states obtained for * * = 0, which are group-subgroup related to the A and B structures; 2) Limit (non-Landau) states, given by * * i = 0, for which the group-subgroup relationship is lost.As a consequence of the periodic-dependence on the critical displacements of the order-parameter, the order-parameter space becomes a fibre bundle instead of a Hilbert space for group-subgroup related transitions.Another consequence is the specific critical properties of reconstructive phase transitions, such as i) the steplike temperature dependence of the order-parameter, since the initial and final structures correspond to fixed displacements (Fig. 31(a)); ii) the delta-shaped specific heat at the transition (Fig. 31(b)), and iii) the finite discontinuity of the susceptibility associated with the field-conjugate to the order-parameter (Fig. 31(c)).Besides, since symmetry operations are lost at both the A → B and B → A transitions, domains are formed in both phases, in contrast to group-subgroup related phases where domains are observed only in the "daughter" phase.

The graphite-diamond reconstructive transition
As a first illustrative example let us consider the reconstructive transition mechanism between the graphite and diamond structures [45,46].Figure 32 represents the assumed displacive mechanisms    43), associated with the transition from hexagonal graphite (HG) to rhombohedral graphite (RG), cubic diamond (CD) and hexagonal diamond (HD), in function of the critical displacements along [120] H G .The curve in black represents the periodicity of the displacements giving rise to the RG structure (first term in Eq. ( 43)).The light-grey curve is a higher harmonic (second term in Eq. ( 43)) whose periodicity reflects the onset of the HD and CD structures from HG and RG, respectively.
The first term in Eq. ( 43) expresses the "primary" symmetry-breaking displacements giving rise to the RG structure from HG, and the second term is a higher harmonic related to secondary displacements leading to the HD and CD structures from HG and RG, respectively.Taking into account the transformation properties of the displacements by the symmetry operations of the HG spacegroup, yields two distinct effective free-energies F 1 ( ( )) = 2 2 + 4 4 + 6 6 , for the HG →(RG, 00007-p.30[47].The anti-isostructural CD variant may coincide with the cubic n-diamond structure [48], which is composed of hexagonal-ring planes puckered in the opposite direction with respect to CD.

The phase diagram of iron
Two of the most commonly encountered reconstructive mechanisms are found in the phase diagram of iron: the bcc → hcp Burgers mechanism [49], and the bcc → fcc Bain deformation mechanism [50].The three -bcc, -fcc and -hcp phases merge at a triple point and are separated by transition curves which are almost straight lines.The temperature-pressure phase diagram of iron, shown in Fig. 35, contains an additional -bcc phase in the low pressure-high temperature region below the melt, as well as a more recently discovered high-pressure -phase, for which two different structures have been proposed: a double-hcp structure and an orthorhombic structure of symmetry Pbcm [51,52].The reconstructive mechanisms relating the different phases are shown in Fig. 36.They can be described as occurring via a common substructure having the monoclinic symmetry C2/m (Z = 4).All the mechanisms can be expressed in terms of critical strains and fractional displacements [53].The corresponding order-parameters are periodic functions of the critical displacements as represented in Fig. 37.

Landau theory of incommensurate structures
In the Landau theory the increment of the probability density describing the change in the spatial distribution of atoms in a crystal undergoing a phase transition reads: where the k i ( r) are normalized functions transforming as the basis of an irreducible representation of the parent space-group G 0 , the k-vector corresponding to a point of the Brillouin zone of the parent structure.The stable states of the crystal are characterized by definite values k i of the order-parameter components associated with an absolute minimum of the transition free-energy ( k i ), expressed as a polynomial expansion of the k i .Lifshitz has noted that when the transition leads to an inhomogeneous low-symmetry structure, the preceding condition does not test completely the stability of the phases 00007-p.32 at the macroscopic level, as ( k i ) has also to be minimum with respect to variations of the k i , which vary continuously from one point to another in the low symmetry phase.As a consequence, instead of a homogeneous polynomial expansion for the free-energy, one has to consider a functional

Contribution of Symmetries in Condensed Matter
where the sum runs over the volume of the system, the free-energy density depending on the k i and on their spatial derivatives.If the k i vary slowly at the microscopic level, Lifshitz shows [54] that to determine the eventual stability of an inhomogeneous structure, it is sufficient to consider antisymmetric gradient invariants of the form k i * k j *x l − k j * k i *x l with i = j , (x l = x, y, z).If such terms (called Lifshitz invariants) are allowed by the parent symmetry, an inhomogeneous phase may stabilize across a secondorder transition from the parent phase.These ideas were used by Dzialoshinskii [55] in the description of incommensurate helimagnetic structures.In incommensurate structures the transition wave-vector varies with temperature (or pressure, or concentration) within an interval k, and therefore one has to consider the variation of k ( r) in this interval.Dzialoshinskii proposed to fix the value k = k c , where k c represents a rational vector associated with the lock-in transition, which generally takes place below the range of stability of the incommensurate phase.The physical assumption underlying this choice is that, within the incommensurate phase, the system is asymptotically governed by a periodic potential possessing the symmetry of the lock-in phase, i.e., the incommensurate structure is treated as a spatial modulation of the lock-in structure.As a consequence the k i functions form the basis of a single irreducible representation associated with k c , but the order-parameter components k c i appear as slowly modulated functions of the actual order-parameter components k i .As a consequence is a continuous function of k through the sequence of parent-incommensurate-lock-in phases.

Basic assumptions of the model: The gauge invariant free-energy
An amorphous solid phase can be obtained by overcooling a liquid [56], by irradiating a crystal with different types of particles [57], or by pressurizing a crystal phase [58].Although the microscopic mechanism and transformation kinetics depend on the amorphization process and class of materials, all amorphous phases show no long-range order and physical properties (solidity, high viscosity, small diffusion) which make the amorphous state essentially different from the liquid state [56].An amorphous phase can be assumed to result from a parent crystal phase.This assumption is obviously justified for irradiation-induced and pressure-induced amorphizations.For glasses formed from a liquid one can consider as the parent phase the crystal phase bypassed on overcooling, since the local glass structure reflects the chemical bonding of the bypassed crystal.For introducing the model [59], let us assume that the order-parameter of the crystal amorphous transition has two complex components = e i and * = e −i , where and depend on the space coordinates x j (j = 1, 2, 3).At variance with the approach to incommensurate structures, the critical wave vector k associated with the crystalamorphous transition is assumed to vary continuously from one point to another of the amorphous phase: where the t * k are the reciprocal lattice vectors of the parent crystal phase, the k coefficients varying continuously in space.This is consistent with assumed geometrical topology of amorphous structures in which each two neighbouring cells display different basic translations.
The primitive translation operators ti of the parent crystal phase transform the order-parameter as: ti = e i k.t ic and ti * = e −i k c .t i * .Since k c .t i = 2 i , one has: ti = e 2i i and ti * = e −2i i * (47) showing that and * are eigen functions of t i and form a continuous spectrum, the free-energy density of the crystal-amorphous transition containing the invariants * , ( * ) 2 .By contrast one has: expressing that the order-parameter derivatives are not eigen functions of ti .However, one can use covariant derivatives, which are invariant under ti .One can take here: where is a coupling constant, and the A ji are components of a second-rank tensor transforming as: The physical meaning of the space-dependent (A ji ) tensor can be anticipated as representing a compensating stress-field potential, i.e., a gauge-field potential which ensures the local invariance under translation of the order-parameter derivatives.One has: ti D i = e are internal stress-field components, and kj l is the Levi-Civita tensor.Therefore the free-energy associated with he crystal-to-amorphous transition has the form: = , * , D i , D * i * , *A il *x k dV and can be minimized independently with respect to , * and A ji .For the purpose of our demonstration we take: where a, b, c and d are phenomenological coefficients.

Analogy with Ginzburg-Landau theory of superconductivity
We now refer to the formal analogy existing between the equations describing the deformed states of a solid and Maxwell's equations of magnetostatics [60].In this analogy the internal stress-field potential (A ji ) coincides with the vector potential A, and the internal stress-field with the magnetic induction B, Eq. ( 52) corresponding to B = curl A. It leads to another analogy between: 1) Our description of the amorphous state resulting from the breaking of the translational symmetry of the parent crystal phase, where the spatial dependence of the order-parameter gradient is compensated by a stress-field potential, and 2) the Ginzburg-Landau time independent description of the superconducting state [5], in which the gauge symmetry of the normal state is broken, and the spatial dependence of the gradient of the wave function = | |e i is compensated by the vector potential A. Substituting in Eq. ( 53) by , D i by ( ∇ − 4i he A), and ik by B, one gets the free-energy density used in the Ginzburg-Landau model [5].Accordingly, minimization of should yield equations, which are analogue to the Ginzburg-Landau equations, and constitute the equations of state of an amorphous solid.Minimizing with respect to A ji gives the analogue of j = curl H , relating the current density to the magnetic field.It reads: and a similar equation of state for * .Equations ( 54) to (56) show that the local state of an amorphous solid phase is stabilized by a stress-field deriving from a stress-field potential coupled to the crystalamorphous order-parameter, which compensates the structural distortions and creates a stationary dislocation flux.In the absence of an internal stress-field the amorphous is unstable.This is in contrast to the property of the superconducting state to be stable in the absence of an applied magnetic field,

00007-p.35
EPJ Web of Conferences corresponding to a homogeneous state with constant amplitude and phase of the superconducting order-parameter.Therefore the amorphous state appears as the analogue of the mixed state of type II superconductors [61], the amorphous analogue of the Meissner effect consisting of a partial extrusion of the internal stress-field which lowers the local distortion and preserves a finite density of dislocations.Within this interpretation the homogeneous state with constant and is a superhard crystal phase from which the stress-field is extruded, i.e., its structure remains undeformed under applied stress except close to its surface.One can show that: where = − bd 2 2 ac 1/2 represents the penetration length of the stress-field inside the superhard crystal in the direction normal to the surface of the sample.Assuming a = a 0 ( S − ) (a 0 > 0), where is an applied stress (or pressure), and S the critical stress at which the amorphous-to-superhard crystal transition takes place, Eq. ( 57) shows that with increasing applied stress > S the internal stress-field decreases exponentially, i. e. the penetration depth is limited to a near surface layer of thickness ( ).Introducing the correlation length of the order-parameter fluctuations ( ) = − c , which for a type II superconductor fulfils the condition > 1/ √ 2. It yields the relationship between S and the critical stress A at which the amorphous state occurs: where the condition bd 2 2 c 2 > 1 4 is required for the amorphous phase to be stable.Recrystallization of amorphous phases under pressure has been observed in amorphous − quartz [62], which transforms into stishovite at about 60 GPa, and for the high-density amorphous phase of ice [63], which recovers the ice VII or ice VIII structure at high pressure.Recrystallization under pressure or stress is also observed in a number of glasses [64].On the other hand, superhard properties, i.e. extremely low compressibility of the same order of magnitude as in diamond, have been reported for cubic boron nitride [65], the tridymite-like phase of carbon dioxide [66], or superhard graphite [67].
The most typical effect characterizing the mixed state of type II superconductors is the existence of vortices oriented in the direction of the applied magnetic field [61].Their amorphous elastic analogs should be formed by defect lines, about which the wave-vector k c (x i ) turns continuously due to the non-quantization of the stress-flux.At variance with superconducting vortices, amorphous elastic vortices are non-directional due to the tensorial character of the stress-field, and to the property of the applied and internal stresses to have different proper systems of coordinates.The defect lines can be formed, for example, by Somigliana-type dislocations [68], as suggested for glassy polymers [69], or by screw dislocations with variable Burgers vectors [70].

CONCLUSION
In this article the flexibility of the Landau theory for investigating different fields of condensed matter physics has been illustrated.The initial Landau approach, restricted to continuous phase transitions between strictly periodic structures, induced by a single irreducible representation of the parent space-group, has been shown to adapt to the description of first-order transitions, eventually associated with several order-parameters, or leading to incommensurate or amorphous phases, and to reconstructive transitions for which the symmetry relationship between the phases is apparently lost.Each extention requires some modification of the initial Landau model, but preserves the basic concepts of the model, i.e. the order-parameter and Landau expansion concepts.Symmetry and related grouptheoretical considerations, as well as thermodynamics form the essential ingredients of the Landau

Figure 1 .
Figure 1.Unit-cell of the parent structure with tetragonal 4/mmm symmetry.p shows the displacement of the M + ion at the transition, while the M − ions are not shifted.

Figure 2 .
Figure 2. (a) Regions of the pressure-temperature plane corresponding to different signs of the phenomenological coefficients a 1 and a 2 .(b) Figure illustrating the vanishing of the coefficient a 1 at a continuous transition while a 2 remains positive.
(a)), whereas four domains (Fig. 4(b)) are found for each of the two classes of mm2 states [(±P x , 0), (0, ±P y )] and [±P xy , ±P xy ].The domains transform into one another by the symmetry operations lost at the G 0 → G symmetry change.

Figure 5 .
Figure 5. Critical behaviour at a second-order phase transition associated with the free-energy F (T , P z ) for: (a) the entropy S(T ), (b) the specific heat C Ez (T ) and (c) the dielectric susceptibility E (T ).

Figure 6 .
Figure 6.Temperature dependence of (a) the secondary order-parameter e zz given by Eq. (9); (b) the corresponding elastic susceptibility e .

Figure 8 .
Figure 8. Temperature dependence of (a) the order-parameter, (b) the entropy and (c) the susceptibility associated with the order-parameter, across a first-order transition.

Figure 9 .
Figure 9. Phase-diagram in the (a 1 , b 1 ) plane, corresponding to the Landau expansion given by Eq. (11).The tricritical point Tc separates the second-order transition line T = Tc from the first-order transition curve T = T 1 .The temperature dependence of F − F 0 is shown in significant regions of the phase-diagram.

Figure 10 .
Figure 10.Orbit spaces associated with the order-parameters (a) P z and (b) (P x , P y ).

Figure 13 .
Figure 13.Symmetry relationship between the possible stable states associated with the three-component orderparameter inducing the ferroelectric transitions in BaTiO 3 .The equilibrium values of the polarization components and the number of ferroelectric domains are indicated for each stable state.

Figure 14 .
Figure 14.Selected phase diagrams in the (a 1 , b 1 ) plane, associated with the Landau expansion given by Eq. (15) truncated at the sixth degree (Figures (a) and (b)) and at the eighth degree (Figures (c) and (d)).

Figure 15 .
Figure 15.Three-dimensional phase-diagram in (a 1 , b 1 , d 1 ) space, corresponding to the twelve-degree expansion given by Eq. (15). of ferroelectric domains.For example, the tetragonal P 4mm phase displays six ferroelectric domains, corresponding to ±P x , ±P y and ±P z .Each pair of ferroelectric domains is coupled to a single ferroelastic domain, as shown in Fig. 16, via the coupling invariants: P 2 z (2e zz − e xx − e yy ), P 2 y (2e yy − e zz − e xx ) and P 2 x (2e xx − e yy − e zz ).

Figure 16 .
Figure 16.Configuration of the unit-cell for the six ferroelectric domains and three ferroelastic domains associated with the P 4mm phase of BaTiO 3 .Ps and es denote the corresponding spontaneous polarization and strain.

Figure 18 .
Figure 18.Connections between the magnetic point-groups of phases I-V induced by k1 + k2 and equilibrium conditions fulfilled by the order-parameters involved in MnWO 4 , in each phase.

Figure 20 .
Figure 20.Magnetoelectric phase diagram of MnWO 4 in magnetic field parallel to the b axis, from Taniguchi et al [16].

.
y the AF2 phase switches to the high-field phase III', which symmetry C a c cancels P y and gives rise to the polarization Decreasing B y below B th y switches back phase III' to AF2.If B is canted at an angle with respect to the b-axis in the (x, z) plane, the magnetoelectric coupling νP x M x M y S 1 S 2 sin induces the polarization

Figure 22 .
Figure 22.Connections between the magnetic point-groups I-V and corresponding equilibrium conditions fulfilled by the order-parameters inducing the sequence of phases observed in TbMnO 3 .

Figure 23 .
Figure 23.Connections between the magnetic point-groups of phases 1-7 induced by the irreducible corepresentation, denoted G 1 in ref. [32], of the P bam1 paramagnetic structure of TbMn 2 O 5 , and equilibrium conditions fulfilled by the order-parameter in each phase.

Figure 27 .
Figure 27.Phase diagrams in the (a 1 , b 1 ) plane, corresponding to the Landau expansion given by Eq. (38), and truncated either at the fourth degree ((a)), or at the sixth degree ((b)).Dashed and full lines are, respectively, second-order and first-order transition lines.N 1 and N 2 are three-phase points; T 1 and T 2 are tricritical points.

Figure 28 .
Figure 28.Difference between the minimal (G m ) and lowest (G k ) symmetry groups describing the superconducting states associated with the representation defined by Eq. (37).The order-parameters 1 and 2 , transforming as , are represented in the complex plane schematized by a circle.The gauge (g ) and time-reversal (T ) operations act as rotations and reflections ( v ) in this plane.When only one (a) or two collinear order-parameters (b) are involved a single superconducting state (I) is stabilized, which possesses the minimal symmetry Gm generated by the reflection 1 v = T g −2 1 .At variance, when 1 and 2 are dephased (c) and additional superconducting state (II) can be stabilized, which possesses the lowest symmetry group Gk reduced to the identity: the time reversal symmetry T is lost.

Figure 29 .
Figure 29.In-plane configurations of (a) the Sm-C * phase, and (b) the Sm-C * P phase.

P e t = c 1 0Figure 30 .
Figure 30.Phase diagrams corresponding to the Landau expansion defined by Eq. (41), assuming (a) a fourth degree expansion in and P t , (b) a sixth degree expansion.

Figure 31 .
Figure 31.(a) Step-like dependence on temperature for the order-parameter associated with a reconstructive transition.Temperature dependence of the specific heat (b) and susceptibility (c) for a reconstructive transition.

Figure 32 .
Figure 32.Structural mechanisms for the reconstructive phase transitions between carbon polymorphs via a common substructure.The conventional hexagonal and cubic unit-cells are represented by thick lines.The unitcells of their common substructures are shown by thin lines.The small arrows represent the real magnitudes of the atomic displacements.
c) CD In agreement with the observed orientational relationships [001] H G //[120] H D , [100] H G //[100] H D and [120] H G //[001] H D , the atomic displacements transforming HG into HD is shown in Figs.32(a) → (d) → (h).It consists of antiparallel fractional displacements ± 1 16 b and ± 1 12 c combined with ± 5 48 b and ± 112 c.The HG → HD transition involves a drastic compression of about 35% along c and small decompression (2%) and compression (6%) along a and b, respectively.A similar mechanism, represented in Figs.32(a)→ (e) → (i), holds for the HG → CD transition.Starting from the transformed uni-cell of HG, the CD structure results from a reduction of the angle from 90 • in HG

Figure 33 .
Figure 33.Periodic dependence (dark grey curve) of the effective transition order-parameter ( ), defined by Eq. (43), associated with the transition from hexagonal graphite (HG) to rhombohedral graphite (RG), cubic diamond (CD) and hexagonal diamond (HD), in function of the critical displacements along [120] H G .The curve in black represents the periodicity of the displacements giving rise to the RG structure (first term in Eq. (43)).The light-grey curve is a higher harmonic (second term in Eq. (43)) whose periodicity reflects the onset of the HD and CD structures from HG and RG, respectively.

to 70 .
529 • in CD, and a set of antiparallel fractional displacements: ± 1 16 b and ± 1 16 c, combined with ± 5 48 b and ± 1 16 c.The transition involves a large compression of 35% along c and small decompressions (2%) along a and b.The HG→RG transition mechanism shown in Figs.32(a) → (b) → (c), consists of a reduction of the angle from 90 • in HG to 78.040 • in RG, and of fractional displacements of the carbon atoms by ± 1 12 b.The order-parameters associated with the transitions from HG to the other carbon polymorphs are of two types.(1) Macroscopic strains which consist of (i) the shear strain e yz expressing the changes in the angle and reduces the hexagonal HG symmetry to monoclinic, (ii) the tensile strain e xx -e yy corresponding to an orthorhombic deformation of the hexagonal cell, (iii) the compression e zz along c. (2) A cooperative set of fractional displacements along b and c.Both types of orderparameters are activated for obtaining the polymorphic structures, which appear as limit states, arising for definite atomic displacements and critical strains.The periodic character of the reconstructive transition mechanisms is illustrated in Fig.33, in which the order-parameter is plotted as a function of the atomic displacements along the [120] direction in HG.The periodic function ( ) representing the effective order-parameter associated with the HG →(HD, CD, RG) transitions is the truncated Fourier series:( ) = 0.958 sin(6 ) + 0.125 sin(42 )

Figure 34 .
Figure 34.Theoretical phase diagrams corresponding to the minimization of the (a) F 1 ( ( )) and (b) F 2 ( ( ))Landau expansions, given in the text, with respect to the displacements .Solid, dashed and dashed-dotted lines represent, respectively, first-order transition, second-order transition and limit of stability lines.The hatched areas are regions of phase coexistence.N is a three-phase point.T r is a tricritical point.The arrows suggest the thermodynamic path followed in carbon.

Figure 35 .
Figure 35.Schematic representation of the pressure-temperature phase-diagram of iron.

Figure 36 .
Figure 36.Structural mechanisms associated with the reconstructive transitions between the fcc -phase of iron and bcc −Fe or hcp −Fe.The intermediate substructure common to the three phases, represented in all the figures, has the symmetry I 4/mmm (Z = 1) for the → transition, and Cmcm (Z = 2) for the → transition.The (a)→(d)→(i) path represents the reconstructive mechanism from −Fe to an eventual high-pressure modification [52], the existence of which remains to be confirmed.

Figure 37 .
Figure 37. (a) and (b): Periodic dependence of the order-parameter for the → → phase sequence in function (a) of the angle between the diagonals in the (110) cubic plane, and (b) of the displacements along the [011] cubic direction.(c) and (d): Periodic dependence of the order-parameter at the → transition in function of the displacements along the [1 10] bcc direction.In (d) the geometrical constraints imposed by the size of the atoms is also taken into account.

2 ik
2i i D i and t * i D * i * = e −2i i D * i *(51)Thus, the invariants D i D * i * , D * i * , * D i can be included into the free-energy density , which is gauge invariant under the phase change → e 2i i .From Eq. (50) one finds the quadratic sums 00007-p.34Contribution of Symmetries in Condensed Matterk to be also gauge invariant, where: equation[60] relating the dislocation density tensor ij = * *A ij to the distortion tensor ij = * * ij , the conservation of the dislocation current j * ij *x j = 0 corresponding to div j = 0.The ij components depend on the transition order-parameter and stress-field potential as: