PACS: 74.20.De; 74.70.Rp; 74.70.Pq

1. Introduction

In a triplet superconductor the electrons in the Cooper pairs are bound with spins parallel
rather than antiparallel to one another, *i.e*. they are bound in
spin triplets^{5}^{,}^{7}^{,}^{13}. For this kind of superconductors, the spins are
lying on the basal plane, while the pair orbital momentum is directed along the
z-direction and their order parameter Ψ is represented by a three-dimensional vector
d(k). If Ψ is of the type *k*_{x
} ± *i k*_{y}, there is a Cooper pair
residual orbital magnetism, which gives place to an state of broken time reversal
symmetry, edge currents in the surface of the superconductor, and a tiny magnetic
field around non-magnetic impurities.

Based on the results of the Knight shift experiment performed through the superconducting
transition temperature *T*_{c}^{8}^{,}^{9}, it has been proposed that Sr_{2}RuO_{4} is
a triplet superconductor. These experiments showed that Pauli spin susceptibility of
the conduction electrons in the superconducting state remains unchanged respect to
its value in the normal state. Moreover, it has been reported^{10} that Ψ breaks time reversal symmetry, which
constitutes another key feature of unconventionality.

The Sr_{2}RuO_{4} elastic constants C_{ij} have been
measured as the temperature T is lowered through *T*_{c}. The
results show a discontinuity in one of the elastic constants^{2} . This implies that Ψ has two different components
with the time reversal symmetry broken. Similar conclusions from a muon spin
relaxation (*μ*SR) experiment were reported by Luke *et
al*.^{10} . Recently,
experiments on the effects of uniaxial stress σ_{i}, as a
symmetry-breaking field were performed by Clifford and collaborators^{3} , reporting that for
Sr_{2}RuO_{4} the symmetry-breaking field can be controlled
experimentally. Additionally, experiments by Lupien *et al*.^{2} showed the existence of small step
in the transverse sound mode T[100].

This body of results evidences the need of extending or developing theoretical models to
explain the changes occurring in *C*_{ij} at
*T*_{c}, which, as far as we know, has not been carried
out even in quite recent works^{3} .
Thus, the aim of our work is to extend an elasticity property phenomenological
theory to show that Sr_{2}RuO_{4} is an unconventional
superconductor with a two-component Ψ ^{4}^{,}^{11}. Here, let us mention that a different theory of
Sr_{2}RuO_{4} elastic properties was presented by Sigrist^{12} . However, unlike this paper,
Sigrist work does not take into account the splitting of
*T*_{c} due to σ_{i}, and
directly calculates the jumps at zero stress, where the derivative of
*T* with respect to σ_{i} doesn’t
exist.

In this work, we first perform an analysis based on a Ψ that transforms as one of the two
dimensional irreducible representations of the Sr_{2}RuO_{4} point
group^{4}^{,}^{13}. Subsequently, we construct the
Sr_{2}RuO_{4} superconducting phase diagram under an external
σ_{i} . This phase diagram is employed to develop a
complete theory of the elastic behavior of Sr_{2}RuO_{4}, based on a
two component Ginzburg-Landau (*GL*) model. This allows to properly
calculate the jumps in the components of the elastic compliances
S_{ij}. Finally, we propose that there are
significant advantages for using Sr_{2}RuO_{4} as a material for a
detailed study of symmetry-breaking effects in superconductivity described by a
two-component Ψ.

2.Ehrenfest relations for a uniaxial stress σ_{i}

Provided that σ_{i} does not split the phase transition , for applied
σ_{i}, Ehrenfest relations can be derived in
analogous manner to the case of applied hydrostatic pressure ^{11}^{,}^{14}, under the condition that T_{c} is known as a
function of σ_{i}. In order to simplify the calculations,
we make use of the Voigt notation *i* = *xx, yy, zz, yz, zx,
xy*.^{15}

For a second order phase transition, the Gibbs free energy *G* derivatives
respect to T, the entropy *S* =
-(∂*G*/∂*T*)_{σ}, and
respect to σ_{i}, the elastic strain
e_{i} = _{i} and T.
Therefore, at the transition line, *S*(*T*,
*σ*_{j}) = 0. From this, for
*S* and e_{i}, the boundary conditions
between the two phases are

By using the definitions of the thermal expansion *C*_{σ} = *T*(∂*S*/∂*T*)_{σ}, and the elastic compliances

From the first expression in Eq. (2), the relation for *α*_{i} is found to be

likewise, from the second expression of Eq. (2),
the relation for *S*_{ij} is obtained to
be,

It is important to distinguish that the print letter *S* denotes the entropy, while the symbol S_{ij} means the elastic compliances. In similar manner, the print letter *C* stands for the specific heat and the symbol *C*_{ij} for the elastic stiffness. Let us also point out that in deriving these expressions, we used the fact that for a given thermodynamic quantity *Q*, its discontinuity along the transition line points is obtained from Δ*Q* = *Q*(*T*_{c} + 0^{+})-Q(*T*_{c} - 0^{+}), where 0^{+} is a positive infinitesimal quantity. Finally, by combining Eqs. (3) and (4), the variation in S_{ij} is found to be:

Before continuing, it is interesting to mention that besides of our previous works^{4}^{,}^{11}, we are not aware of any other works that have
derived Ehrenfest relations for the case where applied
*σ*_{i} produces a phase transition
splitting.

3. Ginzburg-Landau model

In this section, a phenomenological model which takes into account the
Sr_{2}RuO_{4} crystallographic point group
D_{4h} is derived and employed. As we show, the
analysis of *G*, using an order parameter which belongs to any of the
one dimensional representations of D_{4h} is not able to
describe the splitting of T_{c} under an external stress
field. In order to account properly for the splitting, superconductivity in must be
described by a Ψ, transforming as one of the D_{4h} two
dimensional irreducible representations, E_{2g} or
E_{2u}, which at this level of theoretical
description render identical results^{4}^{,}^{11}.

3.1. Superconducting free energy

In order to derive a suitable *GL* free energy *G*^{Γ},
we first will suppose that the Sr_{2}RuO_{4} superconductivity
is described by an order parameter *ψ*^{Γ}, which
transforms according to one of the eight one-dimensional representations of
D_{4h}: *Γ* =
A_{1g}, A_{2g},
B_{1g}, B_{2g},
A_{1u}, A_{2u},
B_{1u}, or B_{2u}. Let
us notice that an analysis employing the D_{4} point group renders
similar results. Here we will analyze the terms in
*G*^{Γ} linear in
*σ*_{i} and quadratic in
*ψ*^{Γ}:

The terms proportional to *σ*_{xx},
*σ*_{yy} and
*σ*_{zz} in Eq. (6) give rise to discontinuities
in the elastic constants, evidenced from sound speed measurements^{17} . On the other hand,
discontinuities in the elastic compliance S_{66} and in the elastic
constant *C*_{66}^{i} arise from the linear coupling
with *σ*_{xy} . However, due to
symmetry, the later linear coupling does not exist for any *Γ*;
therefore, *S*_{66} and *C*_{66}
are expected to be continuous at *T*_{c}
for any of the one-dimensional irreducible representation that assumes a
one-dimensional *ψ*^{Γ},. Nevertheless, the results of
Lupien *et al*. experiments^{2} showed a discontinuity in
*C*_{66}. Hence, based exclusively on sound speed
measurements, we conclude that none of the one-dimensional irreducible
representations can provide an appropriate description of superconductivity in
Sr_{2}RuO_{4}. As far as we know, this conclusion has not
been previously established in the literature^{3} . Let us mention that for any one-dimensional
*Γ*, a detailed analysis of the calculation of the jumps in
*C*_{66} is presented in Ref. 11.

Due to the absence of discontinuity in *S*_{66} for any of the
one-dimensional *Γ*, the superconductivity in
Sr_{2}RuO_{4} must be described by an order parameter
*ψ*^{E} transforming as one of the two-dimensional
representations *GL* theory establishes that only the
parameters of one of the irreducible representations becomes non-zero at
*T*_{c}. Therefore, following the
evidence provided in Refs. 5 and 19, we choose the
*E*_{2u} spin-triplet state as
the correct representation for Sr_{2}RuO_{4}, and the speed
measurements are analyzed in terms of the model
*ψ*^{E} =
(*ψ*_{x},
*ψ*_{y}), with
*ψ*_{x} and
*ψ*_{y} transforming as the
components of a vector in the basal plane. The expression for *G*
is determined by symmetry arguments based on the analysis of the second and
fourth order invariants (real terms) of *G*^{Γ}. To
maintain gauge symmetry, only real and even products of *Ψ* can
occur in the expansion of *G*^{Γ}; thus, we find that all
real invariants should be formed by second and fourth order products of
*ψ’s*^{ii}. To obtain its expression, we use
the fact that *G* is invariant with respect to a transformation
by the generators *c*_{4z} and
*c*_{2x} of
*D*_{4h}. Applying the
generators to different second and fourth order combination of products of
*ψ*’s, we find only one second order invariant
|*ψ*_{x}|^{2} +
|*ψ*_{y}|^{2} and three
fourth order invariants, namely
|*ψ*_{x}|^{2}|*ψ*_{y}|^{2},
|*ψ*_{x}|^{4} +
|*ψ*_{y}|^{4}, and

For the zero *σ*_{i} case, the expansion of *G* gives place to:

where *α* = *α*'(*T* -
*T*_{c0}) and the coefficients
*b*_{1}, *b*_{2}, and
*b*_{3} are material-dependent real constants ^{20}^{,}^{21}. These coefficients have to
satisfy special conditions in order to maintain the free energy stability. The
analysis of *G* is accomplished by considering two component
(*ψ*_{x},
*ψ*_{y}) with the form:

where *η*_{x} and
*η*_{y} are both real and larger
than zero. After substitution of *ψ*_{x}
and *ψ*_{y} in Eq. (7), *G*
becomes:

For fixed values of the coefficients *b*_{1} and
*b*_{2}, if *b*_{3} > 0,
*G* will reach a minimal value if the last term vanishes,
*i.e*. if *φ* = 0. Moreover, if
*η*_{x} and
*η*_{y} have the form
*η*_{x} = *η*sin
*χ* and *η*_{x} =
*η*cos *χ*, *G* becomes

where *G* reaches its minimum value if *η*_{x} =
*η*_{y}. On the other hand, if *G* becomes minimal if *η*_{x} = 0 or
*η*_{y} = 0. Since for a
superconducting state (*ψ*_{x},
*ψ*_{y}) ∼ (1, ±
*i*), from the previous analysis, the lowest *G*
state corresponds to *b*_{3} -
*b*_{2} > 0. This thermodynamic state breaks
time-reversal-symmetry; and hence, it is believed to be the state describing
superconductivity in Sr_{2}RuO_{4}^{4}^{,}^{5}^{,}^{7}. In addition, it is found that for the phase
transition to be of second order, it is required that *b* ≡
*b*_{1} + *b*_{2} -
*b*_{3} > 0.

At this point it is important to understand why the state (*ψ*_{x}, *ψ*_{y}) ∼ (1, ± *iϵ*) has been chosen for the analysis of *σ*_{6} and why it gives rise to the discontinuity in *S*_{66}^{11}. Minimization of Eq. (7) with respect to *φ* and *χ*, and employing Eq. (8) renders a set of solutions for the two-component order parameter which depend on the relation between the coefficients *b*_{1}, *b*_{2}, and *b*_{3} and also on the value of the phases *φ* and *χ*. Thus, for the E representation, solutions of the form,

are obtained, which are very similar to those found for the *D*_{4} one-dimensional irreducible representation. Therefore, these solutions are not able to account for the jump in *C*_{66}. However, solutions with both components different than zero are also attained:

Each of these solutions corresponds to different relations for the *b*_{i}. This is illustrated by Fig. 1, which shows the phase diagram, displaying the domains of *ψ*_{1}, *ψ*_{2} and *ψ*_{3} as a function of *b*_{1}, *b*_{2}, and *b*_{3}. Now, if the jump in *C*_{66} corresponds to a *G* minimum, the coupling term with *σ*_{6} must be taken to be different from zero. If the solution *ψ*_{2} is considered, the term containing *σ*_{6} becomes zero; therefore it is not acceptable. On the other hand, this requirement is satisfied by *Ψ*_{3}, with the form (1, *i*)η. Hence, the *GL* analysis renders *Ψ*_{3} as the solution that breaks time reversal symmetry.

3.2. Coupling of the order parameter to an external stress

The transition to an unconventional superconducting state shows manifestations as the
breakdown of symmetries, such as the crystal point group or the time reversal
symmetry^{20}^{,}^{21}. This loss of symmetry has measurable
manifestations in observable phenomena, as the splitting of
*T*_{c} under an elastic
deformation. The coupling between the crystal lattice and the superconducting
state is described Refs. 20 and 21. As explained there, close to
*T*_{c}, a new term is added to
*G*, which couples in second order Ψ with
*e*_{ij} and in first order Ψ
with *σ*_{ij}. These couplings give
place to discontinuities in *S*_{ijkl}
at *T*_{c}.

3.3 Analysis of the phase diagram

An expression for *G* accounting for a phenomenological coupling to *C*_{66} in the basal plane is given by

Here, Λ_{i} are the temperature-dependent
*α*_{i},
d_{ij} are the coupling terms between Ψ and
S_{ij} and
*E*_{j} are the invariant
elastic compliance tensor components, defined below. In order to determine these
invariants describing the coupling of the order parameter to the stress tensor,
we construct the tensor *E*_{j} with
Voigt components *E*_{1} =
|*ψ*_{x}|^{2},
*E*_{2} =
|*ψ*_{y}|^{2} and *E*_{6} couples
*σ*_{6} and Ψ. The tensor
*d*_{ij} couples
*E*_{i} with
*σ*_{j} and has the same nonzero
components as S_{ij}. By applying symmetry
considerations^{4}, it is
shown that the only non-vanishing independent components of
*d*_{ij} are
*d*_{11}, *d*_{12} =
*d*_{21}, *d*_{31} =
*d*_{32}, and *d*_{66}.
Contributions to *G* that are quadratic in both, Ψ and
*σ*_{6} were neglected. Such terms would have given
an additional *T* dependence to the
S_{ij}^{17}. However, given the large number of independent
constants occurring in the associated sixth rank tensor, at this point, it is
not clear whether or not the explicit inclusion of such terms would be
productive.

Now, let us consider the case of uniaxial compression along the *a* axis (only
with *σ*_{1}*<* 0). If in Eq. (13), only quadratic terms in Ψ
are kept, this equation can be written as

here *T*_{c+} (*σ*_{1}) and
*T*_{cy}
(*σ*_{1}) are given by

In what follows, we assume that *d*_{11}
−*d*_{12}*>* 0, such that
*T*
_{c+}*>
T*_{cy} . Notice that this does not imply
any lost in generality, assuming *d*_{11} −
*d*_{12}*<* 0, would render an
identical model, simply by exchanging the *x* and
*y* indices. Here,
*T*_{c+} is the higher of the two
critical temperatures at which the initial transition occurs. As should be
expected, just below *T*_{c+}, only *ψ*_{
x} is non zero. As *T* is further
lowered, another phase transition happens at
*T*_{c-}, which is different
than *T*_{cy}. Below
*T*_{c-}, the
*ψ*_{y} is also different from
zero (see Fig. (2)). Thus, in the presence
of a non zero compressible *σ*_{1}, Ψ has the form
(*ψ*_{x},
*ψ*_{y}) ≈ *ψ*(1,
± *I ϵ*) where *ϵ* is real and equal to zero
between *T*_{c+} and
*T*_{c-} (phase 1), and
increases from *ϵ* = 0 to *ϵ* ≈ 1 as
*T* becomes smaller than
*T*_{c-} (phase 2), as
illustrated in Figs. (1) and (2).

The next step is finding *T*_{c-}. To achieve this goal, the equilibrium value of the non zero component of *ψ*_{x}, *T*_{c-} follows from

To obtain Eq. (16), it is assumed that *σ*_{1} are kept. The phase diagram for this system is shown in Fig. (3).

4. Calculation of the discontinuities

As discussed before, an external uniaxial stress acting on the Sr_{2}RuO_{4}
basal plane breaks the tetragonal symmetry of the crystal. As a consequence of this,
when a second order transition to the superconducting state occurs, it splits into
two transitions. For the case of applied *σ*_{1}, the
analysis of the behavior of the sound speed at
*T*_{c} requires a systematic study
of these second-order phase transitions. Moreover, thermodynamic quantities, such as
*dT*_{c}*/dσ*_{i},
C_{σ}, and
*α*_{σ}, which are needed in order
to calculate the components

As depicted in Fig. (3), for a given *σ*_{1} ≠ 0 as *T* is lowered below *T*_{c+}, a first discontinuity for a thermodynamical quantity *Q* is observed at the first line of transition temperatures. This discontinuity along the transition line, corresponding to the higher transition temperatures, *T* = *T*_{c+} (*σ*_{i}) is given by ∆*Q*^{+} = *Q*(*T*_{c+} + 0^{+}) − *Q*(*T*_{ c+} − 0^{+}), where 0^{+} is a positive infinitesimal number. If T is further dropped below *T*_{ c-}, a second discontinuity arises, and the lower line of transition temperatures appears. The discontinuity along this line, a *T* = *T*_{c-} (*σ*_{i}), is defined by ∆*Q*^{−} = *Q*(*T*_{c−} + 0^{+}) − *Q*(*T*_{c−} − 0^{+})^{18}. The sum of these two discontinuities

gives the correct expressions for the discontinuities at
*T*_{c0}, for the case with
*σ*_{i} = 0, where the Ehrenfest
relations do not hold directly ^{4}.
As an example of these discontinuities, the two jumps in
*C*_{σ} under an external
*σ*_{i} are sketched in Fig. (4).

**4.1. Jumps due to a uniaxial stress σ**

_{1}

The free energy, Eq. (13), for the cases where
both *σ*_{1} and *σ*_{6} are
nonzero is:

Here *α*_{x} =
*α*^{'}(*T*
−*T*_{c0}
)+*σ*_{1}*d*_{11} and
*α*_{y} =
*α*^{'}(*T*
−*T*_{c0} )+
*σ*_{1}*d*_{12}. If only
*σ*_{1} is applied, this equation becomes:

Where ∆*G* = *G* −
*G*_{0}(*T*). The nature of the
superconducting state that follows from Eq. (19), depends on the values of the coefficients
*b*_{1}, *b*_{2}, and
*b*_{3}. The analysis from Eq. (19) of the superconducting
part of *G* is performed by using, as was done previously, an
expression for Ψ given by Eq.
(8).

At *T*_{c+} and in the presence of *σ*_{1}, the second order terms in Eq. (19) dominate and *Ψ* has a single component *ψ*_{x}; whereas at *T*_{c-} a second component *ψ*_{y} appears. Thus, at very low *T*, the fourth order terms dominate the Eq. (19) behavior. Each of these two-component domains has the form of *ψ*_{2} given by Eq. (12). In this case, *G* can be written in terms of *η*_{x} and *η*_{y} as

The analysis of Eq. (20) depends on the relation between the coefficients *b*_{1}, *b*_{2}, and *b*_{3}. Assuming that *b*_{3} > 0, and *η*_{x} and *η*_{ y} are both different from zero, and following the procedure described after Eq. (9) one arrives to

where *ϵ* is real and grows from *ϵ* = 0 to *ϵ*
≈ 1 as *T* is reduced below
*T*_{c-}, while Eq. (20) becomes

To calculate the jumps at *T*_{c+}, we use
*α*_{x} =
*α'*(*T*
−*T*_{c+}) and
*α*_{y} =
*α'*(*T* −
*T*_{cy}), and assume that
*T*_{c+}*>
T*_{cy} . For the interval
*T*_{c+}*> T >
T*_{c−}, the equilibrium value for
*Ψ* satisfies *α*_{x}
*>* 0 and *α*_{y} = 0,
*i.e*. *η*_{x}
*>* 0 and *η*_{y} = 0,
with *T*
_{c+} and its derivative with respect to
*σ*_{1} are respectively,

The specific heat discontinuity at *T*_{c+}, relative to its normal state value, is calculated by using:

and renders the result

A schematic depiction of the *C*_{σ} discontinuities
below this transition temperature is exhibited in Fig. (4). At *T*_{c+}, the
discontinuity in *α*_{σ} is calculated
by applying the Ehrenfest relation of Eq.
(3), yielding:

The discontinuities in *S*_{ij} are obtained by using Eqs. (4) and (5), rendering the result,

In the previous expression a prime on an index (as in *i*' or *j*') indicates a Voigt index taking only the values 1,2, or 3. Thus, from Eq. (27) the change in *S*_{11} at *T*_{c+} can be calculated to be

To find the discontinuities at *T*_{c-}, the invariant *G* takes the form,

In this expression, the second order term in *η*_{y}
is renormalized by the square of
*η*_{x}. The second transition
temperature is determined from the zero of the total prefactor of *T*_{c-} and
its derivative with respect to *σ*_{1} are:

Below *T*_{c-} the superconducting free energy, Eq. (28) has to be minimized
respect to both components of Ψ. After doing so,
*η*_{x} and
*η*_{y} for this temperature
range are found to be

This analysis shows that the second superconducting phase is different in symmetry, and that time reversal symmetry is broken. The change in *T*_{c-}, with respect to its value in the normal phase, *T*_{c-} is,

which results in

The size of these jumps is complicated to infer, because it depends on the material parameters *b*_{1}, *b*_{2}, and *b*_{3}, and on the coupling constants *d*_{11} and *d*_{12}.

With the help of the Ehrenfest relation, Eq. (3), the discontinuity in *α*_{i} at *T*_{c-} is obtained to be

and after employing Eqs. (4) and (5), the discontinuity in *T*_{c-} is shown to be

Here *T* _{c0}, in the absence of uniaxial stress, can be obtained by adding the discontinuities occurring at *T*_{c+} and *T*_{c-}, yielding:

Before continuing, it is important to emphasize that at at zero stress, the derivative of *T*_{c} with respect to *σ*_{i} is not defined; therefore, there is no reason to expect any of the Ehrenfest relations to hold ^{4}^{,}^{11}.

**4.2Jumps due to a shear stress σ**

_{6}

When a shear stress *σ*_{6} is applied to the basal plane of
Sr_{2}RuO_{4}, the crystal tetragonal symmetry is broken,
and a second transition to a superconducting state occurs. Accordingly, for this
case the analysis of the sound speed behavior at
*T*_{c} also requires a
systematic study of the two successive second order phase transitions. Very
important to mention that the *C*_{66} discontinuity
observed by Lupien ^{2} at
*T*_{c}, can be explained in this context.

If there is a double transition, the derivative of *T*_{c} with
respect to *σ*_{6}
*i.e*.
*dT*_{c}*/dσ*_{6}
is different for each of the two transition lines. At each of these transitions,

The *T*_{c} −*σ*_{6} phase
diagram will be similar to that obtained for *σ*_{1};
therefore, the diagram in Fig. (3) also
qualitatively holds here. In the case of an applied
*σ*_{6}, Δ*G* is given by

Here *α* = *α'*(*T* −
*T*_{c0}), and the minimization
of Δ*G* is performed as in the *σ*_{1}
case, *i.e.* by substituting the general expression for Ψ given
in Eq. (8). After doing so,
Δ*G* becomes

In the presence of *σ*_{6}, the second order term determines the phase
below *T*_{c+}, which is characterized
by *ψ*_{x} and by
*ψ*_{y} = 0. As the temperature
is lowered below *T*_{c-} , depending of
the value of *b*_{3} a second component
*ψ*
_{y} may appear. If at
*T*_{c-} a second component
occurs, the fourth order terms in Eq.
(37) will be the dominant one. Thus for very low T’s, or for
*σ*_{6} → 0, a time-reversal symmetry-breaking
superconducting state emerges. The analysis of Eq. (37) depends on the relation between the coefficients
*b*_{2} and *b*
_{3}. It also depends on the values of the quantities *η*
_{
x} and
*η*_{y}, and of the phase
*φ*. If *b*_{3}*<*
0, and *η*_{x} and
*η*_{y} are both nonzero, the
state with minimum energy has a phase *φ* = π/2. The transition
temperature is obtained from Eq.
(37), by performing the canonical transformations:

If, as was done before, *η*_{ξ} = *η* sin*χ* and *η*_{µ} = *η* cos*χ*, Eq. (38) takes the form

Δ*G* is minimized if cos2*χ* = 1, this is, if
*χ* = 0. Also, in order for the phase transition to be of
second order, *b*', defined as *b*' ≡
*b*_{1} + *b*_{2} +
*b*_{3}, must be larger than zero. Therefore, if
*σ*_{6} is non zero, the state with the lowest free
energy corresponds to *b*_{3}*<* 0,
phase *φ* equal to *π/*2, and *Ψ*
of the form:

In phase 1 of Fig. (3), *φ* = 0, and as *T* is lowered below *T*_{c-}, phase 2, *φ* grows from 0 to approximately *π/*2. Again, following an analysis similar to that carried out for *σ*_{1}, the two transition temperatures *T*_{c+} and *T*_{c-} are obtained to be:

The derivative of *T*_{c+} with respect to *σ*_{6}, and the discontinuity in *T*_{c+} are respectively found to be:

After applying the Ehrenfest relations, Eqs. (4) and (5), the results for *T*_{c+} are:

For *T*_{c-}, the derivative of this transition temperature with respect to *σ*_{6}, and the discontinuities in the specific heat, thermal expansion and elastic stiffness respectively are:

Since for the case of *σ*_{6}, the derivative of
*T*_{c} with respect to
*σ*_{6} is not defined at zero stress point, the
Ehrenfest relations do not hold at
*T*_{c0}. Thus, the discontinuities
occurring at *T*_{c0}, in the absence of
*σ*_{6}, are calculated by adding the expressions
obtained for the discontinuities at
*T*_{c+} and
*T*_{c-},

Notice that in this case, there is no discontinuity for

Since the phase diagram was determined as a function of *σ*_{6},
rather than as a function of the strain, (see Fig.
(3)), in this work, as in Refs. 4 and 11, we make use of the 6 × 6
elastic compliance matrix *S*, whose matrix elements are
*S*_{ij}. However, the sound speed
measurements are best interpreted in terms of the elastic stiffness matrix
*C*, with matrix elements
*C*_{ij}, which is the inverse
of *S*
^{23}. Therefore, it is important
to be able to obtain the discontinuities in the elastic stiffness matrix in
terms of the elastic compliance matrix. Thus, close to the transition line,
*C*(*T*_{c}
+0^{+}) =
*C*(*T*_{c} −
0^{+}) + ∆ *C* and
S(*T*_{c} + 0^{+}) =
*S*(*T*_{c} −
0^{+}) + ∆*S*, where 0^{+} is positive and
infinitesimal. By making use of the fact that *C* ≈ - *C* Δ*S
C*. In this manner, it is found that, for instance at
*T*_{c+}, *T*_{c+},
*T*_{c-}, and
*T*_{c0}, the expressions that
define the jumps for the discontinuities in elastic stiffness and compliances,
due to an external stress, have either a positive or a negative value. In this
way, ∆*S*_{11}, ∆*S*_{22},
∆*S*_{33}, and ∆*S*_{66} are
all negative; while, the stiffness components ∆*S*_{11},
∆*S*_{22}, ∆*S*_{33}, and
∆*S*_{66} are all positive.

5. Final remarks

Since for Sr_{2}RuO_{4}, the symmetry-breaking field, due to
*σ*_{i}, is under experimental control,
states of zero symmetry-breaking stress and of
*σ*_{i} single direction can be a-
chieved^{1}^{,}^{2}^{,}^{3}. Hence, it has significant advantages the use of as
a material in detailed studies of superconductivity symmetry-breaking effects,
described by a two-component order parameter. Nevertheless, determining from
experimental measurements the magnitude of the parameters in the Ginzburg-Landau
model is complicated, because the number of independent parameters occurring for the
case of tetragonal symmetry is greater than for the case of hexagonal symmetry
(*i.e.* UP*t*_{3}) ^{24}^{,}^{25}^{,}^{26}. Thus for Sr_{2}RuO_{4}, three
linearly independent parameters *b*_{1},
*b*_{2}, and *b*_{3}, are required
to specify the fourth order terms in Ψ occurring in Eq. (1); whereas only two independent parameters,
*b*_{1} and *b*_{2}, are required
for UP*t*_{3}. For Sr_{2}RuO_{4}, two
independent ratios can be formed from the three independent
*b*_{i} parameters, and these two
independent ratios could be determined, for example, by experimentally determining
the ratios *σ*_{1} and
*σ*_{6}.

Measurements results for the Sr_{2}RuO_{4} elastic constants below
*T*_{c} are presented in Ref. 2. There,
it is concluded that the quantities *C*_{44} and
*C*_{11} − *C*_{12} follow the
same behavior as those of the BCS superconducting transition, which is evidenced by
a change in slope below *T*
_{c0}. On the other hand, a discontinuity is observed for
C_{66} below *T*_{c0}, without a
significant change in the sound speed slope as *T* goes below 1
Kelvin. It has been previously stated ^{2}^{,}^{11} that this kind of C_{66} changes can be understood
as a signature of an unconventional transition to a superconducting phase. Thus,
this set of results and others, as those of Clifford *et al*. ^{3}, lead to consider
Sr_{2}RuO_{4} as an excellent candidate for a detailed
experimental investigation of the effects of a symmetry-breaking field in
unconventional superconductors.