Norma ISO 45001:2018

In this section we will briefly discuss the theory of second order phase transitions by Landau and the applications of group theory to this theory, the default reference for which is [55]. The reason we discuss this, is that the transition from the normal to the superconducting state is such a transition and this theory allows us to gain further insight in the properties of the gap matrix ∆(k) introduced in section5.3.3. Roughly speaking, second order phase transitions are transitions where the symmetry of the system undergoes a discontinuous change that can be described by an appropriate continuous function, called the order parameter. For example it could be that in a crystal constituted by different atoms, all atoms of a certain kind are displaced from their original position. This displacement is of course continuous, but as soon as it is nonzero, the symmetry of the crystal is different from its original symmetry. At any instant we can say whether the crystal is in its original state or in its state with lower symmetry.

This should be seen as opposed to the more familiar phase transitions of the first kind, like the liquid-gas transition. In such a transition both phases can, and often do, coexist during the transition progress, making it a continuous transition.

It was shown by Landau that the symmetry of one phase must be a subgroup of the symmetry of the other phase. We will refer to the two phases as the one with higher symmetry and the one with lower symmetry. Most often the phase with lower symmetry corresponds to temperatures below the transition temperature. Therefore we will also call the phase with lower symmetry the phase below the transition point.

6.2.1 Expansion of the Free Energy

As mentioned above, second order phase transitions are described by an order parameter, which represents the deviation from the normal, higher symmetry state. In the case of crystal lattice deformation for example, it would be the displacement of the atoms from their original position. We therefore assume that it is zero in the higher symmetry phase and by definition of a second order phase transition it evolves continuously below the transition point. Let us consider the free energyF of the system. Sinceη = 0 at the transition point, we can expand the free energy

−1 −0.5 0 0.5 1 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 O r d e r p a r a m e t e rη F r e e E n e r g y F

Figure 6.1 – Free energy as function of the order parameter η for a2(T) ≥ 0 (blue curve) and

a2(T)<0 (orange curve).

in a power series expansion inη around the transition point18

F(T, η) =F0+a1(T)η+a2(T)η2+a3(T)η3+a4(T)η4+... (6.8)

We assume that an expansion to fourth order will be sufficient to decribe the system around the transition point. Since the minimum of the free energy determines the stable state of a system, we can draw a few conclusions about the coefficients. First of all, it must hold thata4(T)>0,

since otherwise the system would diverge to infinite η. Second, a1(T) anda3(T) must be zero

because otherwise the minimum of the free energy corresponds toη 6= 0 for any temperatureT. Finally, above the transition it must hold that a2(T)≥0, because otherwise the energy would

have a minimum for η 6= 0. Similarly, we must havea2(T) <0 below the transition for η 6= 0

to be the stable phase. This is illustrated in figure6.1. This invites us to expanda2(T) around

the transition temperatureTcas

a2(T) =a2 T−Tc

Tc

(6.9) for some constanta2.

6.2.2 The Order Parameter and Irreducible Representations

Now let us set a step towards superconductivity and assume we are dealing with a system with a Hamiltonian H, and that this Hamiltonian has symmetry group G with irreducible representations ρl with basis functions φl,n. Also assume that the order parameter η is a complex-valued function of position. We know that the eigenfunctions of the Hamiltonian

18

In general the free energy will depend on more factors, pressure for example, but let us assume for now it is a function of the temperature T and the order parameter η only. For simplicity we also assume the order parameter to be a scalar, which in general it is not. We will come back to this point later.

form a complete basis of this function space, and that they form basis functions of irreducible representations of the symmetry group of the Hamiltonian. Conversely, we can express the order parameterη in terms of basis functions of the irreducible representations:

η=X

l,n

ηl,nφl,n (6.10)

whereηl,n are coefficients. Note that since the functionsφl,n determine the symmetry properties of the order parameter, all symmetry information ofηis equivalently encoded in the coefficients ηl,n. So we could equally well agree that the coefficients transform into each other under symmetry operations as the basis functions do, and use these functions in the expansion. Also observe that the expression for the free energy must be invariant under the total symmetry group of the system, hence that of the phase with higher symmetry. Combining these observations it follows that we can expand the free energyF in terms of theηl,n, keeping only combinations of terms that are invariant under the full symmtry group. For the second order terms these are terms of the form Al

P

n|ηl,n|2, while for the fourth order terms these depend on the specific representations. In general we can then write the free energy as

F =F0+ T−Tc Tc X l AlX n |ηl,n|2+ X l fl(η4l), (6.11)

wherefl(η4l) denotes the fourth order terms of the ηl,n for given l that are invariant under the total group.

It turns out that this expansion of the order parameter in terms of the basis functions of irreducible representations is valid in general and does not depend on the specific type of order parameter.