**1. Introduction**

The main parameter, which describes superfluids and superconductors and all their main properties is the order parameter, which is equal to zero above transition temperature Tc into superconducting (superfluid) state and becomes nonzero below Tc. In Bose–systems the transition is caused by Bose–Einstein condensation of bosons while in case of Fermi–systems first the pairing of fermions with creation of Bose particles (Cooper pairs) takes place with their subsequent condensation. Besides the ordinary superconductors (where traditional s–pairing takes place) after discovery the high temperature superconductors (HTSC) and heavy fermion superconductors (HFSC) the unconventional pairing in different superconductors is studied very intensively [1–5]. The main problem here is the type of pairing: singlet or triplet, orbital moment of Cooper pair value L, symmetry of the order parameter etc.

Recent experiments in Sr2RuO4 [2–4] renewed interest in the problem of the symmetry of superconducting order parameters of the high temperature superconductors (HTSC).

*2.1.1 Two–dimensional* p*–wave superconducting states*

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

XIX of Ref. 1)

where

minimizing *F*:

*<sup>A</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> 2

*<sup>A</sup>*<sup>8</sup> <sup>¼</sup> <sup>1</sup> � *<sup>ν</sup>* 3

*<sup>F</sup>*<sup>1</sup> ¼ � <sup>1</sup>

*<sup>F</sup>*<sup>6</sup> ¼ � <sup>1</sup>

*<sup>A</sup>*<sup>5</sup> <sup>¼</sup> <sup>1</sup> ffiffiffi 3 p

**69**

Effective action In case of two–dimensional *p*–wave superconductivity effective

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

7*ζ*ð Þ3

*Seff* ¼ �*β<sup>V</sup>* <sup>16</sup>*π*<sup>2</sup>*TC*Δ*<sup>T</sup>*

*<sup>F</sup>* ¼ �*trAA*<sup>þ</sup> <sup>þ</sup> *<sup>ν</sup>trA*þ*AP* <sup>þ</sup> *trA*<sup>þ</sup> ð Þ *<sup>A</sup>* <sup>2</sup> <sup>þ</sup> *trAA*þ*AA*<sup>þ</sup> <sup>þ</sup> *trAA*þ*A*<sup>∗</sup> *AT*�

*H*2 *=*4*π*<sup>2</sup>

The effective action F is identical in form with that arising in the case of three– dimensional (3D) superconducting system. The difference is connected with the fact that the matrix *A* with elements *aia* for the two–dimensional system is а 2 � 3

> 000 000 001

1

CA

superconducting system. The matrix *P* is the projector оn the third axis: P = δi3δj3

0

B@

The following equation for the condensate matrix *А* could be obtained by

�*<sup>A</sup>* <sup>þ</sup> *<sup>ν</sup>AP* <sup>þ</sup> <sup>2</sup> *trAA*<sup>þ</sup> ð Þ*<sup>A</sup>* <sup>þ</sup> <sup>2</sup>*AA*þ*<sup>A</sup>* <sup>þ</sup> <sup>2</sup>*A*<sup>∗</sup> *ATA* � <sup>2</sup>*AATA*<sup>∗</sup> �

There are several solutions of Eq. (3), corresponding to the different superfluid

100 <sup>010</sup> � �, *<sup>A</sup>*<sup>3</sup> <sup>¼</sup> <sup>1</sup>

100

<sup>0</sup> �1 0 � �, *<sup>A</sup>*<sup>7</sup> <sup>¼</sup> <sup>1</sup>

4

<sup>8</sup> , *<sup>F</sup>*<sup>4</sup> ¼ � <sup>1</sup>

2

4

� �<sup>1</sup>*=*<sup>2</sup> 001

<sup>6</sup> , *<sup>F</sup>*<sup>5</sup> ¼ � <sup>1</sup>

*=*6, *F*<sup>9</sup> ¼ �ð Þ 1 � *ν*

2

0 0 *i*

6 ,

2 *=*4*:* (5)

1 *i* 0 *<sup>i</sup>* �1 0 � �,

> 0 �1 0 <sup>100</sup> � �,

� �*:* (4)

2

2

The corresponding values of the effective action *F* are equal to:

<sup>4</sup> , *<sup>F</sup>*<sup>3</sup> ¼ � <sup>1</sup>

<sup>4</sup> , *<sup>F</sup>*<sup>8</sup> ¼ �ð Þ <sup>1</sup> � *<sup>ν</sup>*

The quantity of the effective action *F* for the first eight phases does not depend оn *H*. The minimum value of *F* = �1/4 is reached for phases *A*<sup>1</sup> and *A*<sup>2</sup> as well as for

<sup>000</sup> � �, *<sup>A</sup>*<sup>9</sup> <sup>¼</sup> <sup>1</sup> � *<sup>ν</sup>*

� *trAA<sup>T</sup>* � �*A*<sup>∗</sup> <sup>¼</sup> <sup>0</sup>*:* (3)

He in Chapter

*F* (1)

*TC*Δ*T* (2)

action takes a form (see the case of two–dimensional superfluidity of <sup>3</sup>

�*trAATA*<sup>∗</sup> *<sup>A</sup>*<sup>þ</sup> � ð Þ <sup>1</sup>*=*<sup>2</sup> *trAATtrA*<sup>∗</sup> *<sup>A</sup>*þ,

*<sup>ν</sup>* <sup>¼</sup> <sup>7</sup>*ζ*ð Þ<sup>3</sup> *<sup>μ</sup>*<sup>2</sup>

matrix instead of 3 � 3 matrix in the case of three–dimensional (3D)

*P* ¼

phases. Let us consider the following possibilities:

*<sup>i</sup>* 0 0 � �, *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>

100

000 <sup>010</sup> � �, *<sup>A</sup>*<sup>6</sup> <sup>¼</sup> <sup>1</sup>

� �<sup>1</sup>*=*<sup>2</sup> 001

<sup>4</sup> , *<sup>F</sup>*<sup>2</sup> ¼ � <sup>1</sup>

<sup>4</sup> , *<sup>F</sup>*<sup>7</sup> ¼ � <sup>1</sup>

Sr2RuO4 has been the candidate for a spin–triplet superconductor for more than 25 years. Recent NMR experiments have cast doubt on this candidacy. Symmetry– based experiments are needed that can rule out broad classes of possible superconducting order parameters. In Ref. 3 authors use the resonant ultrasound spectroscopy to measure the entire symmetry–resolved elastic tensor of Sr2RuO4 through the superconducting transition. They observe a thermodynamic discontinuity in the shear elastic modulus c66, which implies that the superconducting order parameter has two components. A two–component p–wave order parameter, such as px + ipy, satisfies this requirement. As this order parameter appears to have been precluded by recent NMR experiments, the alternative two–component order parameters of Sr2RuO4 are as following {dxz,dyz} and {dx2�y2,gxy(x2�y2)}.

Authors of Ref. 4 have come to similar conclusions. They use ultrasound velocity to probe the superconducting state of Sr2RuO4. This thermodynamic probe is sensitive to the symmetry of the superconducting order parameter. Authors observe a sharp jump in the shear elastic constant *c*<sup>66</sup> as the temperature is increased across the superconducting transition. This supposes that the superconducting order parameter is of a two–component nature.

The existence of CuO2 planes [6] – the common structural factor of HTSC – suggests we consider 2D models. A 2D– model of *p*–pairing using a path integration technique has been developed by Brusov and Popov [7, 8]. A 2D model of *d*–pairing within the same technique has been developed by Brusov et al. [9–14]. The models use the hydrodynamic action functionals, which have been obtained by path integration over "fast" and "slow" Fermi–fields. All properties of 2D–superconductors (for example, of CuO2 planes of HTSC) and, in particular, the collective excitations spectrum, are determined by these functionals. We consider all superconducting states, arising in symmetry classification of 2D–superconductors and calculate the full collective modes spectrum for each of these states. Current study continue our previous investigation [5], where we consider the problem of distinguish the mixture of two d–wave states from pure d–wave state of HTSC.
