**2. Two–dimensional models of p– and d–pairing in unconventional superconductors**

### **2.1** *p***–Pairing**

Below we develop 2D–model of *p*–pairing starting with the 3D scheme considered by Brusov et al. [9–14].

Two main distinctions between 3D–case and 2D–case are as follows:


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

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

Effective action In case of two–dimensional *p*–wave superconductivity effective action takes a form (see the case of two–dimensional superfluidity of <sup>3</sup> He in Chapter XIX of Ref. 1)

$$S\_{\rm eff} = -\beta V \frac{16\pi^2 T\_C \Delta T}{7\zeta(3)} F \tag{1}$$

where

Recent experiments in Sr2RuO4 [2–4] renewed interest in the problem of the

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–

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

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 mix-

**2. Two–dimensional models of p– and d–pairing in unconventional**

Below we develop 2D–model of *p*–pairing starting with the 3D scheme

a. The Cooper pair orbital moment *l* (*l* = 1) should be perpendicular to the plane and can have only two projections on the ^*z*–axis: �1. The *p*– pairing is a triplet, thus the total spin of the pair is equal to 1, and in the case of 2D *p*–pairing we have 3 � 2 � 2 ¼ 12 degrees of freedom. In this case one can describe the superconducting state by complex 2 � 3 matrices *cia*ð Þ *p* . The number of the collective modes in each phase is equal to the number of degrees of freedom. Just remind that in the 3D case this number is equal to 18.

b. Vector **<sup>x</sup>** is 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>

Two main distinctions between 3D–case and 2D–case are as follows:

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)}.

symmetry of superconducting order parameters of the high temperature

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

based experiments are needed that can rule out broad classes of possible

superconductors (HTSC).

parameter is of a two–component nature.

**superconductors**

in 3D case).

**68**

considered by Brusov et al. [9–14].

**2.1** *p***–Pairing**

ture of two d–wave states from pure d–wave state of HTSC.

$$F = -trAA^{+} + \nu trA^{+}AP + \left(trA^{+}A\right)^{2} + trAA^{+}AA^{+} + trAA^{+}A^{\*}A^{T} - \nu - \frac{1}{2}trAA^{T}A^{\*}A^{+} - \frac{1}{2}trAA^{T}A^{\*}A^{+} - (\mathbf{1} \cdot \mathbf{2})\mathbf{T} + \mathbf{A}^{\*}\mathbf{A}^{-} + \mathbf{A}^{\*} + \mathbf{A}^{\*} = \mathbf{0}, \tag{2}$$

$$\nu = 7\zeta(3)\mu^{2}H^{2}/4\pi^{2}T\_{C}\Delta T\tag{3}$$

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 matrix instead of 3 � 3 matrix in the case of three–dimensional (3D) superconducting system. The matrix *P* is the projector оn the third axis: P = δi3δj3

$$P = \begin{pmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & 1 \end{pmatrix}.$$

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

$$\begin{cases} -\mathbf{A} + \nu \mathbf{A} \mathbf{P} + 2(\text{tr} \mathbf{A} \mathbf{A}^{+}) \mathbf{A} + 2 \mathbf{A} \mathbf{A}^{+} \mathbf{A} + 2 \mathbf{A}^{\*} \mathbf{A}^{T} \mathbf{A} - 2 \mathbf{A} \mathbf{A}^{T} \mathbf{A}^{\*} - \\ -(\text{tr} \mathbf{A} \mathbf{A}^{T}) \mathbf{A}^{\*} = \mathbf{0}. \end{cases} \tag{3}$$

There are several solutions of Eq. (3), corresponding to the different superfluid phases. Let us consider the following possibilities:

$$A\_1 = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}, A\_2 = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, A\_3 = \frac{1}{4} \begin{pmatrix} 1 & i & 0 \\ i & -1 & 0 \end{pmatrix},$$

$$A\_5 = \frac{1}{\sqrt{3}} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, A\_6 = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \end{pmatrix}, A\_7 = \frac{1}{2} \begin{pmatrix} 0 & \pm 1 & 0 \\ 1 & 0 & 0 \end{pmatrix},$$

$$A\_8 = \left(\frac{1 - \nu}{3}\right)^{1/2} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, A\_9 = \left(\frac{1 - \nu}{4}\right)^{1/2} \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & i \end{pmatrix}.\tag{4}$$

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

$$\begin{aligned} F\_1 &= -\frac{1}{4}, F\_2 = -\frac{1}{4}, F\_3 = -\frac{1}{8}, F\_4 = -\frac{1}{6}, F\_5 = -\frac{1}{6}, \\ F\_6 &= -\frac{1}{4}, F\_7 = -\frac{1}{4}, F\_8 = -(1 - \nu)^2 / 6, F\_9 = -(1 - \nu)^2 / 4. \end{aligned} \tag{5}$$

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 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 magnetic field (ν = 0)).

three–dimensional sound with velocity *<sup>v</sup>*<sup>3</sup> <sup>¼</sup> *cF<sup>=</sup>* ffiffiffi

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

Tewordt [16] etc.).

001 0 0 *i* � �:

0 �1 0 <sup>100</sup> � �:

100 <sup>0</sup> �1 0 � �:

dimensional Sr2RuO4.

collective excitations.

**71**

1 2

1 2

1 2

et al. [7, 8] this result has been reproduced by a number of authors (Nagai [15],

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

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> 0, 3 modes ð Þ;

, 6 modes ð Þ;

, 4 modes ð Þ;

, 4 modes ð Þ;

**The collective mode spectrum for the phase with order parameter**

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

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>Δ<sup>2</sup>

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

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>Δ<sup>2</sup>

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

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>Δ<sup>2</sup>

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

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

**The collective mode spectrum for the phase with order parameter**

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

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> 0, 4 modes ð Þ;

*<sup>E</sup>*<sup>2</sup> <sup>¼</sup> 0, 4 modes ð Þ;

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–

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

<sup>3</sup> <sup>p</sup> is well known). After Brusov

*:*ð Þ 3 modes (8)

*:*ð Þ 4 modes (9)

*:*ð Þ 4 modes (10)

The first two phases have been discovered by Brusov and Popov [7, 8] in the films of superfluid <sup>3</sup> He. Authors [7, 8] have called them the *a* – and *b* –phases and 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 for *A*<sup>6</sup> and *A*<sup>7</sup> states.

### *2.1.2 The 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 energy of the gap in single–particle spectrum).

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 superconducting states under *p*–pairing.

**The collective mode spectrum for** *a***–phase with order parameter** 1 2 100 *<sup>i</sup>* 0 0 � �: *<sup>E</sup>*<sup>2</sup> <sup>¼</sup> *<sup>c</sup>*<sup>2</sup> *Fk*2 2 <sup>1</sup> � <sup>5</sup>*c*<sup>2</sup> *Fk*2 96Δ<sup>2</sup> !, 3 modes ð Þ *<sup>E</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>Δ<sup>2</sup> <sup>þ</sup> *<sup>c</sup>* 2 *Fk*2 *=*2, 6 modes ð Þ (6) *<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>* 2 *Fk*2 *:*ð Þ 3 modes

**The collective mode spectrum for** *b***–phase with order parameter** 1 2 100 <sup>010</sup> � �:

$$E^2 = \frac{c\_F^2 k^2}{2} \left( 1 - \frac{5c\_F^2 k^2}{48\Delta^2} \right), (2 \text{ modes})$$

$$E^2 = \frac{3c\_F^2 k^2}{4} \left( 1 - \frac{c\_F^2 k^2}{72\Delta^2} \right), (1 \text{ mode})$$

$$E^2 = \frac{c\_F^2 k^2}{4} \left( 1 - \frac{c\_F^2 k^2}{48\Delta^2} \right), (1 \text{ mode})$$

$$E^2 = 2\Delta^2 + c\_F^2 k^2 / 2, (4 \text{ modes}) \tag{7}$$

$$E^2 = 4\Delta^2 + (0.500 - i0.433)c\_F^2 k^2, (2 \text{ mode})$$

$$E^2 = 4\Delta^2 + (0.152 - i0.218)c\_F^2 k^2, (1 \text{ mode})$$

$$E^2 = 4\Delta^2 + (0.849 - i0.216)c\_F^2 k^2. (1 \text{ mode})$$

It is seen that in *a*– and *b*–phases the so–called two–dimensional (2D) sound with velocity *<sup>v</sup>*<sup>2</sup> <sup>¼</sup> *cF<sup>=</sup>* ffiffi 2 <sup>p</sup> exists. Note that dispersion coefficient of 2D–sound in *<sup>b</sup>*– phase is twice higher than in *a*–phase. We should remind that in bulk systems the

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

three–dimensional sound with velocity *<sup>v</sup>*<sup>3</sup> <sup>¼</sup> *cF<sup>=</sup>* ffiffiffi <sup>3</sup> <sup>p</sup> is well known). After Brusov et al. [7, 8] this result has been reproduced by a number of authors (Nagai [15], Tewordt [16] etc.).

**The collective mode spectrum for the phase with order parameter** 1 2 001 � �:

0 0 *i*

1 2

$$E^2 = \mathbf{0}, (\text{3 modes});$$

$$E^2 = 2\boldsymbol{\Delta}^2, (\text{6 modes});$$

$$E^2 = 4\boldsymbol{\Delta}^2. (\text{3 modes})\tag{8}$$

**The collective mode spectrum for two phases with order parameters** 0 �1 0 <sup>100</sup> � �:

$$E^2 = 0, (4 \text{ modes});$$

$$E^2 = 2\Delta^2, (4 \text{ modes});$$

$$E^2 = 4\Delta^2. (4 \text{ modes})\tag{9}$$

**The collective mode spectrum for the phase with order parameter** 1 2 100 <sup>0</sup> �1 0 � �:

$$E^2 = 0, (4 \text{ modes});$$

$$E^2 = 2\Delta^2, (4 \text{ modes});$$

$$E^2 = 4\Delta^2. (4 \text{ modes}) \tag{10}$$
