**3. Two–dimensional d–Wave superconductivity**

## **3.1 2D–model of d–pairing in CuO2 planes of HTSC**

The existence of CuO2 planes — the common structural factor of HTSC suggests we consider two–dimensional (2D) models. For two–dimensional (2D) quantum antiferromagnet (AF) it was shown that only the *d*–channel provides an attractive interaction between fermions. The *d*– pairing arises also in symmetry classifications of CuO2 planes HTSC. In Sr2RuO4 where the p–pairing appears to have been precluded by recent NMR experiments, the two–component d–wave order parameters, namely {dxz,dyz} and even with admixture of g–wave {dx2 � y2,gxy (x2 � y2}, are now the prime candidates for the order parameter of the quasi–two– dimensional Sr2RuO4.

The two–dimensional (2D) model of *d*– pairing in the CuO2 planes of HTSC has been developed by Brusov and Brusova (BB) [9, 10, 13] and Brusov, Brusova and Brusov (BBB) [14] using a path integration technique. The hydrodynamic action functional, obtained by path integration over "fast" and "slow" Fermi–fields, has been used under construction of this model. This hydrodynamic action functional determines all properties of the CuO2 planes and, in particular, the spectrum of collective excitations.

the phases with matrices *A*<sup>6</sup> and *A*<sup>7</sup> and *A*<sup>9</sup> (last state has minimum energy in zero

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

The first two phases have been discovered by Brusov and Popov [7, 8] in the

have proved that the phases *a–* and *b–* are stabile relative to the small perturbations. Brusov and Popov [7, 8] have calculated the full collective mode spectrum for two these phases. Brusov et al. [9–14] have calculated the full collective mode spectrum

The full collective mode spectrum for each of these phases consists of 12 modes (the number of degrees of freedom). Among them we have found Goldstone modes as well as high frequency modes (with energy (frequency) which is proportional to

The results obtained by Brusov and Popov [7, 8] and Brusov et al. [9–14] are

shown below for collective mode spectrum for different two–dimensional

**The collective mode spectrum for** *a***–phase with order parameter**

<sup>1</sup> � <sup>5</sup>*c*<sup>2</sup> *Fk*2 96Δ<sup>2</sup>

> 2 *Fk*2

<sup>1</sup> � <sup>5</sup>*c*<sup>2</sup> *Fk*2 48Δ<sup>2</sup>

!

<sup>1</sup> � *<sup>c</sup>*<sup>2</sup> *Fk*2 72Δ<sup>2</sup>

<sup>1</sup> � *<sup>c</sup>*<sup>2</sup> *Fk*2 48Δ<sup>2</sup>

> 2 *Fk*2

It is seen that in *a*– and *b*–phases the so–called two–dimensional (2D) sound

phase is twice higher than in *a*–phase. We should remind that in bulk systems the

!

!

!

, 3 modes ð Þ

, 2 modes ð Þ

, 1 mode ð Þ

, 1 mode ð Þ

2 *Fk*2

2 *Fk*2

> 2 *Fk*2

<sup>p</sup> exists. Note that dispersion coefficient of 2D–sound in *<sup>b</sup>*–

2 *Fk*2

*=*2, 6 modes ð Þ (6)

*=*2, 4 modes ð Þ (7)

, 2 mode ð Þ

, 1 mode ð Þ

*:*ð Þ 1 mode

*:*ð Þ 3 modes

He. Authors [7, 8] have called them the *a* – and *b* –phases and

magnetic field (ν = 0)).

films of superfluid <sup>3</sup>

for *A*<sup>6</sup> and *A*<sup>7</sup> states.

1 2

1 2

100 *i* 0 0 � �

100 010 � �

*2.1.2 The collective mode spectrum*

energy of the gap in single–particle spectrum).

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> *<sup>c</sup>*<sup>2</sup> *Fk*2 2

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> *<sup>c</sup>*<sup>2</sup> *Fk*2 2

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>3</sup>*c*<sup>2</sup>

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> *<sup>c</sup>*<sup>2</sup> *Fk*2 4

2

*Fk*2 4

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>Δ<sup>2</sup> <sup>þ</sup> *<sup>c</sup>*

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>Δ<sup>2</sup> <sup>þ</sup> ð Þ <sup>0</sup>*:*<sup>500</sup> � *<sup>i</sup>*0*:*<sup>433</sup> *<sup>c</sup>*

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>Δ<sup>2</sup> <sup>þ</sup> ð Þ <sup>0</sup>*:*<sup>152</sup> � *<sup>i</sup>*0*:*<sup>218</sup> *<sup>c</sup>*

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>Δ<sup>2</sup> <sup>þ</sup> ð Þ <sup>0</sup>*:*<sup>849</sup> � *<sup>i</sup>*0*:*<sup>216</sup> *<sup>c</sup>*

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>Δ<sup>2</sup> <sup>þ</sup> *<sup>c</sup>*

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>Δ<sup>2</sup> <sup>þ</sup> ð Þ <sup>0</sup>*:*<sup>500</sup> <sup>þ</sup> *<sup>i</sup>*0*:*<sup>433</sup> *<sup>c</sup>*

**The collective mode spectrum for** *b***–phase with order parameter**

superconducting states under *p*–pairing.

:

:

with velocity *<sup>v</sup>*<sup>2</sup> <sup>¼</sup> *cF<sup>=</sup>* ffiffi

**70**

To develop the model of *d*–pairing in the two–dimensional (2D)–case we modify the three–dimensional (3D) considered by us in Ref. 1.

The main distinctions between 3D and 2D cases are as follows:


$$t = v\left(\hat{k}, \hat{k}'\right) = \sum\_{m=-2,2} \mathbf{g}\_m Y\_{2m}\left(\hat{k}\right) Y\_{2m}^\*\left(\hat{k}'\right) \tag{11}$$

where

*<sup>M</sup>*<sup>11</sup> <sup>¼</sup> *<sup>Z</sup>*�<sup>1</sup>

*<sup>M</sup>*<sup>22</sup> <sup>¼</sup> *<sup>Z</sup>*�<sup>1</sup>

<sup>21</sup> <sup>¼</sup> *<sup>σ</sup>*0*α β*ð Þ*<sup>S</sup>* �1*=*<sup>2</sup>

tive–mode spectrum, consisting of four collective–modes in each phase.

The effective functional *S*eff determines all properties of considering model system – superconducting CuO2 planes. It determines, in particular, the full collec-

*Path Integral Two Dimensional Models of P– and D–Wave Superconductors and Collective Modes*

Two SC states arise in the symmetry classification of CuO2 planes with OP which

Brusov and Brusova [9, 10] and Brusov, Brusova and Brusov [14] have calculated the collective– mode spectrum for both of these states. In the first approximation the collective excitations spectrum is determined by the quadratic part of

*<sup>j</sup>* þ *cj*ð Þ *p* in *S*eff. Here *c*

ð Þ 0

**The collective mode spectrum for the phases with order parameters**

The spectra in both phases turns out to be identical. Brusov and Brusova [9, 10] found two high frequency modes in each phase with following energies (frequencies):

*E*<sup>1</sup> ¼ Δ0ð Þ 1*:*42 � *i*0*:*65 ,

Note that the energies of both modes turn out to be complex. This results from

The other two modes are Goldstone or low–energy modes (with energy ≤ 0*:*1Δ0).

The made calculations of the collective mode spectrum are not completely self–-

consistent, because Brusov, Brusova and Brusov [14] working within spherical symmetry approximation, use the order parameters obtained with taking the lattice symmetry into account. The taking the lattice symmetry into account, as we mentioned above, requires a few coupling constants using instead of one. The number of collective excitations (collective modes) in superconducting state, which is equal to number of degrees of freedom, will change too (note, that in case of spherical

the *d*–pairing, or in other words, via the disappearance of a gap in the chosen directions. In this case the Bose–excitations decay into fermions. This leads to a damping of the collective modes. The value of imaginary part of energy is 23% for the second mode and 46% for first one. Thus both modes should be regarded as

resonances and the second mode is better defined than the first.

**4. Lattice symmetry and collective mode spectrum**

0 1

*M*<sup>12</sup> ¼ *M*<sup>þ</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.97041*

1 0 0 �1 and

**3.2 The collective mode spectrum**

gap is proportional to *<sup>Y</sup>*<sup>22</sup> <sup>þ</sup> *<sup>Y</sup>*<sup>2</sup>�<sup>2</sup> � sin <sup>2</sup>

is proportional to �*i Y*ð Þ� <sup>22</sup> � *<sup>Y</sup>*<sup>2</sup>�<sup>2</sup> sin <sup>2</sup>

*S*eff, obtained by the shift *cj*ð Þ! *p c*

values of the canonical Bose–fields *cj*ð Þ *p* .

0 1 1 0 .

are proportional to

*θ* ¼ *π=*2 and sin *θ* ¼ 1.

1 0 0 �1 **and**

**73**

½ � *iω* � *ξ* þ *μ*ð Þ **Hσ** *δp*1*p*<sup>2</sup>

½ � �*iω* þ *ξ* þ *μ*ð Þ **Hσ** *δp*1*p*<sup>2</sup> (16)

1 0 respectively. In the former phase the

*θ*j j cos 2*ϕ* � j j cos 2*ϕ* while in the later one

*θ*j j sin 2*ϕ* � j j sin 2*ϕ* . For 2D case we put

ð Þ 0

*E*<sup>2</sup> ¼ Δ0ð Þ 1*:*74 � *i*0*:*41 *:* (17)

*<sup>j</sup>* are the condensate

ð Þ *c*<sup>1</sup> cos 2*ϕ* þ *c*<sup>2</sup> sin 2*ϕ :*

We consider the case of circular symmetry *<sup>g</sup>*<sup>2</sup> <sup>¼</sup> *<sup>g</sup>*�<sup>2</sup> <sup>¼</sup> *<sup>g</sup>*, which is describes by one coupling constant *g*. Note, that less symmetric cases require both constants *<sup>g</sup>*<sup>2</sup> and *<sup>g</sup>*�<sup>2</sup>*.* We consider the circularly symmetric case where:

$$w\left(\hat{k},\hat{k}'\right) = \mathbf{g}\left[\mathbf{Y}\_{2-2}\left(\hat{k}\right)\mathbf{Y}\_{2-2}^\*\left(\hat{k}'\right) + \mathbf{Y}\_{22}\left(\hat{k}\right)\mathbf{Y}\_{22}^\*\left(\hat{k}'\right)\right] \tag{12}$$

c. **<sup>x</sup>** will be a 2D*–*vector and square "volume" will be *<sup>S</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> (instead of *<sup>V</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>3</sup> as in 3D case).

Account these distinctions between the two–dimensional (2D) and the three– dimensional (3D) cases we will describe our Fermi–system by the anticommuting functions *<sup>χ</sup>s*ð Þ **<sup>x</sup>**, *<sup>τ</sup>* , *<sup>χ</sup>s*ð Þ **<sup>x</sup>**, *<sup>τ</sup>* , defined in the square volume *<sup>S</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> and antiperiodic in "time" *<sup>τ</sup>* with period *<sup>β</sup>* <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> .

After path integrating over slow and fast Fermi–fields (which is a very similar to 3D one) one gets the effective action functional *S*eff, which takes (formally) the same form as in 3D case.

The number of degrees of freedom in the case of two–dimensional (2D) *d*– pairing is equal to 4. By the other words, one has two complex canonical variables. It is easy to see from non–diagonal elements of *M*^ matrix that the following canonical variables should be chosen:

$$
\mathcal{L}\_1 = \mathcal{c}\_{11} - \mathcal{c}\_{22}, \mathcal{c}\_2 = \mathcal{c}\_{12} + \mathcal{c}\_{21}.\tag{13}
$$

One has for the conjugate variables:

$$
\mathfrak{c}\_{1}^{+} = \mathfrak{c}\_{11}^{+} - \mathfrak{c}\_{22}^{+}, \mathfrak{c}\_{2}^{+} = \mathfrak{c}\_{12}^{+} + \mathfrak{c}\_{21}^{+}.\tag{14}
$$

Below we transform the effective action functional *S*eff to these new variables. One has:

$$\mathcal{S}\_{\text{eff}} = \left(\mathbf{2g}\right)^{-1} \sum\_{p,j} c\_j^+(p) c\_j(p) + \frac{\mathbf{1}}{2} \ln \det \frac{\hat{\mathcal{M}}\left(c\_j^+, c\_j\right)}{\hat{\mathcal{M}}\left(c\_j^{+\left(0\right)}, c\_j^{(0)}\right)} \tag{15}$$

*Path Integral Two Dimensional Models of P– and D–Wave Superconductors and Collective Modes DOI: http://dx.doi.org/10.5772/intechopen.97041*

where

To develop the model of *d*–pairing in the two–dimensional (2D)–case we modify

have only two projections on the ^*z*– axis: �2 instead of the three–dimensional (3D) case where the orbital moment can have five projections on the ^*z*– axis: �2; �1;0. Because the *d*– pairing is a singlet the total spin of the pair is equal zero, so in the case of the two–dimensional (2D) *d*– pairing one has 1 � 2 � 2 ¼ 4 degrees of freedom. Thus the superconductive state in this case can be described by complex symmetric traceless 2 � 2 matrices*cia*ð Þ *p ,* which have the same number of degrees of freedom (2 � 2 � 2 � 2 � 2 ¼ 4). This number is equal to the number of the CM in each phase. Note that in the three–dimensional (3D) case this number is equal to 10, as well as the number of the collective modes in each phase.

> <sup>¼</sup> <sup>X</sup> *m*¼�2, 2

*gmY*2*<sup>m</sup>* ^ *k* � � *Y* ∗ <sup>2</sup>*<sup>m</sup>* ^ *k* <sup>0</sup> � �

<sup>0</sup> h i � �

<sup>þ</sup> *<sup>Y</sup>*<sup>22</sup> ^ *k* � � *Y* ∗ <sup>22</sup> ^ *k*

*c*<sup>1</sup> ¼ *c*<sup>11</sup> � *c*22,*c*<sup>2</sup> ¼ *c*<sup>12</sup> þ *c*21*:* (13)

*M c* ^ <sup>þ</sup> *<sup>j</sup>* ,*cj* � �

*M c* ^ þð Þ <sup>0</sup> *<sup>j</sup>* ,*c* ð Þ 0 *j*

<sup>21</sup>*:* (14)

� � (15)

We consider the case of circular symmetry *<sup>g</sup>*<sup>2</sup> <sup>¼</sup> *<sup>g</sup>*�<sup>2</sup> <sup>¼</sup> *<sup>g</sup>*, which is describes by

c. **<sup>x</sup>** will be a 2D*–*vector and square "volume" will be *<sup>S</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> (instead of *<sup>V</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>3</sup>

Account these distinctions between the two–dimensional (2D) and the three– dimensional (3D) cases we will describe our Fermi–system by the anticommuting functions *<sup>χ</sup>s*ð Þ **<sup>x</sup>**, *<sup>τ</sup>* , *<sup>χ</sup>s*ð Þ **<sup>x</sup>**, *<sup>τ</sup>* , defined in the square volume *<sup>S</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> and antiperiodic in

After path integrating over slow and fast Fermi–fields (which is a very similar to 3D one) one gets the effective action functional *S*eff, which takes (formally) the

The number of degrees of freedom in the case of two–dimensional (2D) *d*– pairing is equal to 4. By the other words, one has two complex canonical variables. It is easy to see from non–diagonal elements of *M*^ matrix that the following canonical

Below we transform the effective action functional *S*eff to these new variables.

1 <sup>2</sup> ln det

*<sup>j</sup>* ð Þ *p cj*ð Þþ *p*

one coupling constant *g*. Note, that less symmetric cases require both constants *<sup>g</sup>*<sup>2</sup> and *<sup>g</sup>*�<sup>2</sup>*.* We consider the circularly symmetric case where:

should be perpendicular to the plane and can

(11)

(12)

the three–dimensional (3D) considered by us in Ref. 1.

! *l* � �! � � � � <sup>¼</sup> <sup>2</sup> � �

b. The pairing potential *t* is given by:

*v* ^ *k*, ^ *k* <sup>0</sup> � �

as in 3D case).

"time" *<sup>τ</sup>* with period *<sup>β</sup>* <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup>

same form as in 3D case.

variables should be chosen:

One has:

**72**

One has for the conjugate variables:

*<sup>S</sup>*eff <sup>¼</sup> ð Þ <sup>2</sup>*<sup>g</sup>* �<sup>1</sup>

*c* þ <sup>1</sup> ¼ *c* þ <sup>11</sup> � *c* þ 22,*c* þ <sup>2</sup> ¼ *c* þ <sup>12</sup> þ *c* þ

X *p*, *j c* þ

*<sup>t</sup>* <sup>¼</sup> *<sup>v</sup>* ^ *k*, ^ *k* <sup>0</sup> � �

<sup>¼</sup> *g Y*<sup>2</sup>�<sup>2</sup> ^

.

*k* � � *Y* ∗ <sup>2</sup>�<sup>2</sup> ^ *k* <sup>0</sup> � �

a. The orbital moment *l*

The main distinctions between 3D and 2D cases are as follows:

*Real Perspective of Fourier Transforms and Current Developments in Superconductivity*

$$M\_{11} = Z^{-1}[i\alpha - \xi + \mu(\mathbf{H}\sigma)]\delta\_{p\_1 p\_2}$$

$$M\_{22} = Z^{-1}[-i\alpha + \xi + \mu(\mathbf{H}\sigma)]\delta\_{p\_1 p\_2} \tag{16}$$

$$M\_{12} = M\_{21}^{+} = \sigma\_0 \alpha (\beta \mathbb{S})^{-1/2} (c\_1 \cos 2\phi + c\_2 \sin 2\phi).$$

The effective functional *S*eff determines all properties of considering model system – superconducting CuO2 planes. It determines, in particular, the full collective–mode spectrum, consisting of four collective–modes in each phase.

### **3.2 The collective mode spectrum**

Two SC states arise in the symmetry classification of CuO2 planes with OP which are proportional to 1 0 0 �1 and 0 1 1 0 respectively. In the former phase the gap is proportional to *<sup>Y</sup>*<sup>22</sup> <sup>þ</sup> *<sup>Y</sup>*<sup>2</sup>�<sup>2</sup> � sin <sup>2</sup> *θ*j j cos 2*ϕ* � j j cos 2*ϕ* while in the later one is proportional to �*i Y*ð Þ� <sup>22</sup> � *<sup>Y</sup>*<sup>2</sup>�<sup>2</sup> sin <sup>2</sup> *θ*j j sin 2*ϕ* � j j sin 2*ϕ* . For 2D case we put *θ* ¼ *π=*2 and sin *θ* ¼ 1.

Brusov and Brusova [9, 10] and Brusov, Brusova and Brusov [14] have calculated the collective– mode spectrum for both of these states. In the first approximation the collective excitations spectrum is determined by the quadratic part of *S*eff, obtained by the shift *cj*ð Þ! *p c* ð Þ 0 *<sup>j</sup>* þ *cj*ð Þ *p* in *S*eff. Here *c* ð Þ 0 *<sup>j</sup>* are the condensate values of the canonical Bose–fields *cj*ð Þ *p* .

**The collective mode spectrum for the phases with order parameters** 1 0 0 �1 **and** 0 1 1 0 .

The spectra in both phases turns out to be identical. Brusov and Brusova [9, 10] found two high frequency modes in each phase with following energies (frequencies):

$$E\_1 = \Delta\_0(\mathbf{1}.42 - i\mathbf{0}.65),$$

$$E\_2 = \Delta\_0(\mathbf{1}.74 - i\mathbf{0}.41). \tag{17}$$

Note that the energies of both modes turn out to be complex. This results from the *d*–pairing, or in other words, via the disappearance of a gap in the chosen directions. In this case the Bose–excitations decay into fermions. This leads to a damping of the collective modes. The value of imaginary part of energy is 23% for the second mode and 46% for first one. Thus both modes should be regarded as resonances and the second mode is better defined than the first.

The other two modes are Goldstone or low–energy modes (with energy ≤ 0*:*1Δ0).

### **4. Lattice symmetry and collective mode spectrum**

The made calculations of the collective mode spectrum are not completely self– consistent, because Brusov, Brusova and Brusov [14] working within spherical symmetry approximation, use the order parameters obtained with taking the lattice symmetry into account. The taking the lattice symmetry into account, as we mentioned above, requires a few coupling constants using instead of one. The number of collective excitations (collective modes) in superconducting state, which is equal to number of degrees of freedom, will change too (note, that in case of spherical

symmetry it is equal to 10). In case of the simple irreducible representation (IR) the number of collective modes is equal to twice number of irreducible representation dimensionality. For orthorhombic (OR) symmetry and singlet pairing all irreducible representations are one dimensional (1D), so in each superconducting state there are two modes corresponding to phase and amplitude variations. Amplitude mode is high frequency with *E*≈2Δ, where Δ is the gap in a single particle spectrum.

Among irreducible representations of tetragonal (TG) symmetry there are 1D as 2D (remind that we consider the singlet pairing). Thus in addition to the superconducting states, with two collective modes of conventional superconductors there are states which have four collective modes, none of which are Goldstone. We would like to mention, that for cylindrical Fermi–surface (*D*∞) among collective modes there is Goldstone mode in 1, 0 ð Þ and 1, 1 ð Þ states but there is not Goldstone mode in 1, ð Þ*i* state.

Because it looks like that there is a mixture of different irreducible representations (corresponding, for example, to *s*– and *d*–wave states or to two different *d*– wave states: *dx*<sup>2</sup>�*y*<sup>2</sup> and *dxy*; or *dxz* and *dyz*) it will be interesting to investigate the collective mode spectrum in this case for different admixture values of *s*–wave state (*dxy*– state). Considered by Brusov et al. particular case of *dx*<sup>2</sup>�*y*<sup>2</sup> þ *idxy* state [1] shows that such consideration leads to very interesting results. One more possibility is connected with the recent experiments in Sr2RuO4 where the p–pairing appears to have been precluded by recent NMR experiments, the two–component d–wave order parameters, namely {dxz,dyz} and even with admixture of g–wave {dx2�y2, gxy(x2�y2}, are now the prime candidates for the order parameter of the quasi–two– dimensional Sr2RuO4. So, it will be interesting to study the collective mode spectrum in such states.
