# Topological Structure of the Order Parameter of Unconventional Superconductors Based on d- and f- Elements

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{~}1.3 T) [11]. The anisotropy and temperature dependence of the magnetic field penetration in the B- phase of UPt${}_{3}$ measured by μSR (muon spin relaxation) was accounted for by a superconducting gap function with a line of nodes in the basal plane and axial point nodes [12]. Neutron scattering experiments showed that the superconducting gap has a lower rotational symmetry than crystal symmetry [13]. The results of Kerr effect [14] and that of Josephson interferometry [15,16] manifest a transition between real and complex SOP, corresponding to A and B phases, respectively, which are consistent with the spatial symmetries of the ${E}_{2u}$ order parameter written as:

## 2. Preliminaries

#### Paper Construction

## 3. Space Group Approach to the Wavefunction of a Cooper Pair

## 4. Coupling with Larger Total Angular Momentum

## 5. Phase Winding and Group Theory

## 6. Order Parameters of Sr${}_{2}$RuO${}_{4}$ and UPt${}_{3}$

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Nodal structure and phase winding of a SOP in ${D}_{2h}$ symmetry for odd IRs (

**a**) ${A}_{u}$, (

**b**) ${B}_{1u}$, (

**c**) ${B}_{2u}$, and (

**d**) ${B}_{3u}$. Bold red lines denote vertical nodal planes and red circles denote nodal basal planes.

**Figure 2.**Nodal structure and phase winding of SOP in ${m}^{\prime}{m}^{\prime}m$ symmetry for odd ICRs (

**a**) ${A}_{u}^{+}$, (

**b**) ${A}_{u}^{-}$, (

**c**) ${B}_{u}^{+}$, and (

**d**) ${B}_{u}^{-}$. Bold red lines denote vertical nodal planes and red circles denote nodal basal planes.

**Figure 3.**Nodal structures of real SOP (

**a**) ${E}_{g\left(u\right)}$ and (

**b**) ${E}_{g\left(u\right)}\times {A}_{2g}$ in ${D}_{4h}$ symmetry. In both cases, ${E}_{g}$ is nodal and ${E}_{u}$ in nodeless in the basal plane.

**Figure 4.**Nodal structures of complex SOPs (

**a**) ${E}_{g\left(u\right)}$ and (

**b**) ${E}_{g\left(u\right)}\times {A}_{2g}$ in ${D}_{4h}$ symmetry. In both cases, ${E}_{g}$ is nodal and ${E}_{u}$ in nodeless in the basal plane.

**Figure 5.**Nodal structures of complex IRs of the ${D}_{6h}$ group. Winding direction changes when passing via vertical symmetry planes. (

**a**) ${E}_{1u}$ and (

**b**) ${E}_{2u}$.

**Figure 6.**Nodal structures of SOP for the ${D}_{6h}$ group. (

**a**) Unitary structure, i.e., the difference of two chiral pair functions $1/2({E}_{2u}-{E}_{2u}^{*})$ and (

**b**) non-unitary structure, corresponding to magnetic group $6{m}^{\prime}{m}^{\prime}m$. Winding direction is the same in all sectors.

IR | Basis Functions |
---|---|

${A}_{u}$ | ${k}_{x}\widehat{x}$, ${k}_{y}\widehat{y}$, ${k}_{z}\widehat{z}$ |

${B}_{1u}$ | ${k}_{y}\widehat{x}$, ${k}_{x}\widehat{y}$, ${k}_{x}{k}_{y}{k}_{z}\widehat{z}$ |

${B}_{2u}$ | ${k}_{x}\widehat{z}$, ${k}_{z}\widehat{x}$, ${k}_{x}{k}_{y}{k}_{z}\widehat{y}$ |

${B}_{3u}$ | ${k}_{z}\widehat{y}$, ${k}_{y}\widehat{z}$, ${k}_{x}{k}_{y}{k}_{z}\widehat{x}$ |

H | IRs |
---|---|

${H}_{x}$ | ${A}_{u}+i{B}_{3u}$, ${B}_{1u}+i{B}_{2u}$ |

${H}_{y}$ | ${A}_{u}+i{B}_{2u}$, ${B}_{1u}+i{B}_{3u}$ |

${H}_{z}$ | ${A}_{u}+i{B}_{1u}$, ${B}_{2u}+i{B}_{3u}$ |

Singlet Pair | Triplet Pair | ||
---|---|---|---|

IR | Basis function | IR | Basis function |

${A}_{1g}$ | ${k}_{x}^{2}+{k}_{y}^{2}$ | ${A}_{1u}$ | $\widehat{x}{k}_{x}+\widehat{y}{k}_{y}$ |

${A}_{1g}$ | ${k}_{z}^{2}$ | ${A}_{1u}$ | $\widehat{z}{k}_{z}$ |

${A}_{2g}$ | ${k}_{x}{k}_{y}\left({k}_{x}^{2}-{k}_{y}^{2}\right)$ | ${A}_{2u}$ | $\widehat{x}{k}_{y}-\widehat{y}{k}_{x}$ |

${B}_{1g}$ | ${k}_{x}^{2}-{k}_{y}^{2}$ | ${B}_{1u}$ | $\widehat{x}{k}_{x}-\widehat{y}{k}_{y}$ |

${B}_{2g}$ | ${k}_{x}{k}_{y}$ | ${B}_{2u}$ | $\widehat{x}{k}_{y}+\widehat{y}{k}_{x}$ |

${E}_{g}$ | ${k}_{z}\left({k}_{x},{k}_{y}\right),$ | ${E}_{u}$ | $\widehat{z}\left({k}_{x},{k}_{y}\right),$${k}_{z}\left(\widehat{x},\widehat{y}\right),$$\widehat{z}({k}_{x}\pm i{k}_{y})$ |

**Table 4.**Symmetry classification of BdG systems. The symbols (+) and (−) denote the presence and the absence, respectively, of $SU\left(2\right)$ spin-rotation symmetry and TRS. In classes A and AIII, Hamiltonians are invariant under rotations about the z or any fixed axis in spin space, but not under full $SU\left(2\right)$ rotations, as denoted by R [58].

AZ Class | TRS | $\mathit{SU}\left(2\right)$ | Examples in $2\mathit{d}$ |
---|---|---|---|

D | − | − | Spinless chiral $(p\pm ip)$ wave |

DIII | + | − | Superposition of $(p+ip)$ and $(p-ip)$ waves |

A | − | R | Spinfull chiral $(p\pm ip)$ wave |

AIII | + | R | Spinfull ${p}_{x}$ or ${p}_{y}$ wave |

C | − | + | $(d\pm id)$ wave |

CI | + | + | ${d}_{{x}^{2}-{y}^{2}}$ or ${d}_{xy}$ wave |

J | IR of ${\mathit{O}}_{\mathit{h}}$ | |
---|---|---|

0 | Odd | ${A}_{1g}$ |

2 | Odd | ${E}_{g}+{T}_{2g}$ |

1 | Even | ${T}_{1g}$ |

3 | Even | ${A}_{2g}+{T}_{1g}+{T}_{2g}$ |

**Table 6.**Two-electron wavefunctions constructed from function ${j}_{3/2}$. Normalization factors are omitted. Functions for $M=-1$ and $M=-2$ may be obtained from functions $M=1$ and $M=2$ by changing the signs of magnetic quantum numbers. Note that functions $J=1,3$ are forbidden by the Pauli exclusion principle.

J | M | Function |
---|---|---|

0 | 0 | $\left\{\left(\frac{1}{2},-\frac{1}{2}\right)-\left(-\frac{1}{2},\frac{1}{2}\right)\right\}$$+\left\{\left(\frac{3}{2},-\frac{3}{2}\right)-\left(-\frac{3}{2},\frac{3}{2}\right)\right\}$ |

2 | 2 | $\left\{\left(\frac{3}{2},\frac{1}{2}\right)-\left(\frac{1}{2},\frac{3}{2}\right)\right\}$ |

2 | 1 | $\left\{\left(\frac{3}{2},\frac{-1}{2}\right)-\left(\frac{-1}{2},\frac{3}{2}\right)\right\}$ |

2 | 0 | $\left\{\left(\frac{1}{2},-\frac{1}{2}\right)-\left(-\frac{1}{2},\frac{1}{2}\right)\right\}$$-\left\{\left(\frac{3}{2},-\frac{3}{2}\right)+\left(-\frac{3}{2},\frac{3}{2}\right)\right\}$ |

**Table 7.**Odd IRs of point group ${D}_{2h}$ and magnetic group ${m}^{\prime}{m}^{\prime}m$. Non-unitary element is $A=\theta {\sigma}_{y}$.

E | ${\mathit{C}}_{2\mathit{z}}$ | I | ${\mathit{\sigma}}_{\mathit{z}}$ | $\mathit{\theta}{\mathit{\sigma}}_{\mathit{y}}$ | $\mathit{\theta}{\mathit{\sigma}}_{\mathit{x}}$ | $\mathit{\theta}{\mathit{C}}_{2\mathit{y}}$ | $\mathit{\theta}{\mathit{C}}_{2\mathit{x}}$ | |
---|---|---|---|---|---|---|---|---|

${A}_{u}^{\pm}$ | 1 | 1 | $-1$ | $-1$ | $\pm 1$ | $\pm 1$ | $\mp 1$ | $\mp 1$ |

${B}_{u}^{\pm}$ | 1 | $-1$ | $-1$ | 1 | $\pm 1$ | $\mp 1$ | $\mp 1$ | $\pm 1$ |

${A}_{u}$ | 1 | 1 | $-1$ | $-1$ | $-1$ | $-1$ | 1 | 1 |

${B}_{1u}$ | 1 | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ |

${B}_{2u}$ | 1 | $-1$ | $-1$ | 1 | $-1$ | 1 | 1 | $-1$ |

${B}_{3u}$ | 1 | $-1$ | $-1$ | 1 | 1 | $-1$ | $-1$ | 1 |

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**MDPI and ACS Style**

Yarzhemsky, V.G.; Teplyakov, E.A.
Topological Structure of the Order Parameter of Unconventional Superconductors Based on *d*- and *f*- Elements. *Symmetry* **2023**, *15*, 376.
https://doi.org/10.3390/sym15020376

**AMA Style**

Yarzhemsky VG, Teplyakov EA.
Topological Structure of the Order Parameter of Unconventional Superconductors Based on *d*- and *f*- Elements. *Symmetry*. 2023; 15(2):376.
https://doi.org/10.3390/sym15020376

**Chicago/Turabian Style**

Yarzhemsky, Victor G., and Egor A. Teplyakov.
2023. "Topological Structure of the Order Parameter of Unconventional Superconductors Based on *d*- and *f*- Elements" *Symmetry* 15, no. 2: 376.
https://doi.org/10.3390/sym15020376