# Constraining Forces Stabilizing Superconductivity in Bismuth

## Abstract

**:**

## 1. Introduction

## 2. Group-Theoretical and Computational Methods Used in the Paper

## 3. Superconducting Bands in the Band Structure of Bismuth

#### 3.1. Band Structure of Bi–I

#### 3.2. Band Structure of Bi–V

## 4. Results

- is one of the narrowest bands in the band structure,
- is nearly half filled,
- and comprises a great part of the electrons at the Fermi level.

## 5. Discussion

- The associated localized states are not represented by (hybrid) atomic orbitals but consistently by symmetry-adapted optimally-localized Wannier states.
- The postulates of the NHM are physically evident and require the introduction of nonadiabatic localized states of well-defined symmetry emphasizing the correlated nature of any atomic-like motion.
- The atomic-like motion is determined by the conservation of the total crystal–spin angular momentum, which must be satisfied in the nonadiabatic system. In a narrow, roughly half-filled superconducting band, this conservation law plays a crucial role because the localized (Wannier) states are spin-dependent.
- The strongly correlated atomic-like motion in a narrow, roughly half-filled superconducting band produces an interaction between the electron spins and “crystal–spin-1 bosons”: at any electronic scattering process, two crystal–spin-1 bosons are excited or absorbed in order that the total crystal–spin angular momentum stays conserved.
- Crystal-spin-1 bosons are the energetically lowest localized boson excitations of the crystal that possess the crystal–spin angular momentum $1\xb7\hslash $ and are sufficiently stable to transport it (as Bloch waves) through the crystal.
- The spin–boson interaction in a narrow, roughly half-filled superconducting band leads to the formation of Cooper pairs below a transition temperature ${T}_{c}$.
- The Cooper pairs arise inevitably since any electron state in which the electrons possess their full degrees of freedom violates the conservation of crystal–spin angular momentum.
- This influence of the crystal–spin angular momentum may be described in terms of constraining forces that constrain the electrons to form Cooper pairs. This feature distinguishes the present concept from the standard theory of superconductivity.
- As already mentioned in Section 1, there is evidence that only these constraining forces may produce superconducting eigenstates.
- Hence, the constraining forces are responsible for all types of superconductivity, i.e., conventional, high-${T}_{c}$ and other superconductivity.
- Crystal-spin-1 bosons are coupled phonon–plasmon modes that determine the type of the superconductor.
- In the isotropic lattices of the transition elements, crystal–spin-1 bosons have dominant phonon character and confirm the electron–phonon mechanism that enters the BCS theory [13] in these materials.
- Phonon-like excitations are not able to transport crystal–spin angular-momenta within the anisotropic materials of the high-${T}_{c}$ superconductors [17], often containing two-dimensional layers. Within these anisotropic materials, the crystal–spin-1 bosons are energetically higher lying excitations of dominant plasmon character leading to higher Debye temperatures and, hence, to higher superconducting transition temperatures [13].
- The theory of superconductivity as developed so far is valid without any restrictions in narrow, roughly half-filled superconducting bands because constraining forces do not alter the energy of the electron system.
- However, the standard theory may furnish inaccurate information if no narrow, roughly half-filled superconducting band exists in the band structure of the material under consideration.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NHM | Nonadiabatic Heisenberg model |

## A. Appendix A. Group-Theoretical Tables for the Trigonal Space Group $\mathit{R}\overline{\mathbf{3}}\mathit{m}$ (166) of Bi–I

- The letter R stands for the letter denoting the relevant point of symmetry. For example, at point F, the representations ${R}_{1}^{+},{R}_{2}^{+},\dots $ stand for ${F}_{1}^{+},{F}_{2}^{+},\dots $.
- Each column lists the double-valued representation ${R}_{i}\times {\mathbf{d}}_{1/2}$ below the single-valued representation ${R}_{i}$, where ${\mathbf{d}}_{1/2}$ denotes the two-dimensional double-valued representation of the three-dimensional rotation group $O\left(3\right)$ given, e.g., in Table 6.1 of Ref. [12].
- The single-valued representations are defined in Table A1.
- The notations of double-valued representations follow strictly Table 6.13 (and Table 6.14) of Ref. [12]. In this paper, the double-valued representations are not explicitly given but are sufficiently defined by this table.

- $z=0.23\dots $ [1]; the exact value of z is immaterial in this table. In the hexagonal unit cell, the Bi atoms lie at the Wyckoff positions $6c(00\pm z)$ [1]. In the trigonal system, their positions in the unit cell are $\mathbf{\rho}=\pm (z{\mathbf{T}}_{1}+z{\mathbf{T}}_{2}+z{\mathbf{T}}_{3})$, where the vectors ${\mathbf{T}}_{1},{\mathbf{T}}_{2},$ and ${\mathbf{T}}_{3}$ denote the basic vectors of the trigonal lattice as given, e.g., in Table 3.1 of Ref. [12].
- The notations of the representations are defined in Table A1.
- Assume a closed band of the symmetry in one of the two rows of this table to exist in the band structure of Bi–I. Then, the Bloch functions of this band can be unitarily transformed into Wannier functions that are
- localized as well as possible,
- centered at the Bi atoms, and
- symmetry-adapted to the space group $R\overline{3}m$ (166) [6].

The entry “OK” below the time-inversion operator K indicates that the Wannier functions may even be chosen symmetry-adapted to the magnetic group$$M=R\overline{3}m+K\xb7R\overline{3}m.$$ - The bands are determined following Theorem 5 of Ref. [6].
- The Wannier functions at the Bi atoms listed in the upper row belong to the representation ${\mathit{d}}_{i}$ of ${C}_{3v}$ included below the atom. These representations are defined in Table A2.
- Each row defines one band consisting of two branches because there are two Bi atoms in the unit cell.

- $z=0.23\dots $ [1]; the exact value of z is immaterial in this table. In the hexagonal unit cell, the Bi atoms lie at the Wyckoff positions $6c(00\pm z)$ [1]. In the trigonal system, their positions in the unit cell are $\mathit{\rho}=\pm (z{\mathit{T}}_{1}+z{\mathit{T}}_{2}+z{\mathit{T}}_{3})$, where the vectors ${\mathit{T}}_{1},{\mathit{T}}_{2},$ and ${\mathit{T}}_{3}$ denote the basic vectors of the trigonal lattice as given, e.g., in Table 3.1 of Ref. [12].
- Assume an isolated band of the symmetry listed in this table to exist in the band structure of Bi–I. Then, the Bloch functions of this band can be unitarily transformed into spin-dependent Wannier functions that are
- localized as well as possible,
- centered at the Bi atoms, and
- symmetry-adapted to the space group $R\overline{3}m$ (166) [6].

The entry “OK” below the time-inversion operator K indicates that the spin-dependent Wannier functions may even be chosen symmetry-adapted to the magnetic group$$M=R\overline{3}m+K\xb7R\overline{3}m.$$ - The listed band is the only superconducting band of Bi–I.
- The notations of the double-valued representations are (indirectly) defined by Table A3.
- Following Theorem 9 of Ref. [6], the superconducting band is simply determined from one of the two single-valued bands listed in Table A4 by means of Equation (97) of Ref. [6]. (According to Definition 20 of Ref. [6], both single-valued bands in Table A4 are affiliated bands of the superconducting band.)
- The superconducting band consists of two branches because there are two Bi atoms in the unit cell.
- The point group of the positions of the Bi atoms (Definitions 11 and 12 of Ref. [6]) is the group ${C}_{3v}$. The Wannier functions at the Bi atoms belong to the double-valued representation$$\mathit{d}={\mathit{d}}_{1}\otimes {\mathit{d}}_{1/2}={\mathit{d}}_{2}\otimes {\mathit{d}}_{1/2}$$

## B. Appendix B. Group-Theoretical Tables for the Cubic Space Group $\mathit{Im}\mathbf{3}\mathit{m}$ (229) of Bi–V

- $m=x,y,z;\phantom{\rule{1.em}{0ex}}p=a,b,c,d,e,f;\phantom{\rule{1.em}{0ex}}j=1,2,3,4.$
- The symmetry elements are labeled in the Schönflies notation as illustrated, e.g., in Table 1.2 of Ref. [12].
- The character tables are determined from Table 5.7 of Ref. [12].
- The notations of the points of symmetry follow Figure 3.15 of Ref. [12].

- In the table for $\Gamma $ and H, the letter R stands for the letter denoting the point of symmetry. For example, at point H, the representations ${R}_{1}^{+},{R}_{2}^{+},\dots $ stand for ${H}_{1}^{+},{H}_{2}^{+},\dots $.
- Each column lists the double-valued representation ${R}_{i}\times {\mathbf{d}}_{1/2}$ below the single-valued representation ${R}_{i}$, where ${\mathbf{d}}_{1/2}$ denotes the two-dimensional double-valued representation of the three-dimensional rotation group $O\left(3\right)$ given, e.g., in Table 6.1 of Ref. [12].
- The single-valued representations are defined in Table A6.
- The notations of double-valued representations follow strictly Table 6.13 (and Table 6.14) of Ref. [12]. In this paper, the double-valued representations are not explicitly given but are sufficiently defined by this table.

- The notations of the representations are defined in Table A6.
- Assume a closed band of the symmetry in any row of this table to exist in the band structure of Bi–V. Then, the Bloch functions of this band can be unitarily transformed into Wannier functions that are
- localized as well as possible,
- centered at the Bi atoms, and
- symmetry-adapted to the space group $Im3m$ (229) [6].

The entry “OK” below the time-inversion operator K indicates that the Wannier functions may even be chosen symmetry-adapted to the magnetic group$$M=Im3m+K\xb7Im3m.$$ - The bands are determined following Theorem 5 of Ref. [6].

- Assume an isolated band of the symmetry listed in any row of this table to exist in the band structure of Bi–V. Then, the Bloch functions of this band can be unitarily transformed into spin-dependent Wannier functions that are
- localized as well as possible,
- centered at the Bi atoms, and
- symmetry-adapted to the space group $Im3m$ (229) [6].

- The notations of the double-valued representations are (indirectly) defined in Table A7.
- Following Theorem 9 of Ref. [6], the superconducting bands are simply determined from the single-valued bands listed in Table A8 by means of Equation (97) of Ref. [6]. (According to Definition 20 of Ref. [6], each single-valued band in Table A8 is an affiliated band of one of the superconducting bands.)
- The superconducting bands consists of one branch each because there is one Bi atom in the unit cell.
- The point group of the positions of the Bi atoms (Definitions 11 and 12 of Ref. [6]) is the full cubic point group ${O}_{h}$. The Wannier functions at the Bi atoms belong to the double-valued representations of ${O}_{h}$ listed in the second column, where the single-valued representations ${\Gamma}_{1}^{\pm}$ and ${\Gamma}_{2}^{\pm}$ are defined by Table A6, and ${\mathit{d}}_{1/2}$ denotes the two-dimensional double-valued representation of $O\left(3\right)$ as given, e.g., in Table 6.1 of Ref. [12].

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**Figure 1.**Band structure of Bi–I calculated by the “Fritz Haber Institute ab initio molecular simulations” (FHI-aims) program [10,11], using the structure parameters given by Degtyareva et al. [1]. The symmetry labels are determined by the author. Bi–I has the trigonal space group $R\overline{3}m$ [2] (international number 166), the notations of the points and lines of symmetry in the Brillouin zone for ${\Gamma}_{rh}$ follow Figure 3.11b of Ref. [12], and the symmetry labels are defined in Table A1. ${E}_{F}$ denotes the Fermi level. The band highlighted in red is the superconducting band.

**Figure 2.**Band structure of Bi–V at the pressure of 13.5 GPa calculated by the FHI-aims program [10,11], using the structure parameters at this pressure as given by Degtyareva et al. [1]. The symmetry labels are determined by the author. Bi–V has the cubic space group $Im3m$ [1] (international number 229), the notations of the points and lines of symmetry in the Brillouin zone for ${\Gamma}_{c}^{v}$ follow 3.15 of Ref. [12], and the symmetry labels are defined in Table A6. ${E}_{F}$ denotes the Fermi level. The band highlighted in red forms the superconducting band.

**Table A1.**Character tables of the single-valued irreducible representations of the trigonal space group $R\overline{3}m={\Gamma}_{rh}{D}_{3d}^{5}$ (166) of Bi–I.

$\mathbf{\Gamma}\left(\mathbf{000}\right)$, $\mathit{Z}\left(\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$ | ||||||

$\mathit{E}$ | $\mathit{I}$ | ${\mathit{S}}_{\mathbf{6}}^{\pm}$ | ${\mathit{C}}_{\mathbf{3}}^{\pm}$ | ${\mathit{C}}_{\mathbf{2}\mathit{i}}^{\prime}$ | ${\mathit{\sigma}}_{\mathit{di}}$ | |

${\Gamma}_{1}^{+}$, ${Z}_{1}^{+}$ | 1 | 1 | 1 | 1 | 1 | 1 |

${\Gamma}_{2}^{+}$, ${Z}_{2}^{+}$ | 1 | 1 | 1 | 1 | −1 | −1 |

${\Gamma}_{1}^{-}$, ${Z}_{1}^{-}$ | 1 | −1 | −1 | 1 | 1 | −1 |

${\Gamma}_{2}^{-}$, ${Z}_{2}^{-}$ | 1 | −1 | −1 | 1 | −1 | 1 |

${\Gamma}_{3}^{+}$, ${Z}_{3}^{+}$ | 2 | 2 | −1 | −1 | 0 | 0 |

${\Gamma}_{3}^{-}$, ${Z}_{3}^{-}$ | 2 | −2 | 1 | −1 | 0 | 0 |

$\mathit{L}\left(\mathbf{0}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{0}\right)$ | ||||||

$\mathit{E}$ | ${\mathit{C}}_{\mathbf{22}}^{\prime}$ | $\mathit{I}$ | ${\mathit{\sigma}}_{\mathit{d}\mathbf{2}}$ | |||

${L}_{1}^{+}$ | 1 | 1 | 1 | 1 | ||

${L}_{1}^{-}$ | 1 | 1 | −1 | −1 | ||

${L}_{2}^{+}$ | 1 | −1 | 1 | −1 | ||

${L}_{2}^{-}$ | 1 | −1 | −1 | 1 | ||

$\mathit{F}\left(\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{0}\right)$ | ||||||

$\mathit{E}$ | ${\mathit{C}}_{\mathbf{23}}^{\prime}$ | $\mathit{I}$ | ${\mathit{\sigma}}_{\mathit{d}\mathbf{3}}$ | |||

${F}_{1}^{+}$ | 1 | 1 | 1 | 1 | ||

${F}_{1}^{-}$ | 1 | 1 | −1 | −1 | ||

${F}_{2}^{+}$ | 1 | −1 | 1 | −1 | ||

${F}_{2}^{-}$ | 1 | −1 | −1 | 1 |

**Table A2.**Character tables of the single-valued irreducible representations of the point group ${C}_{3v}$ of the positions of the Bi atoms (Definitions 11 and 12 of Ref. [6]) in Bi–I.

E | ${\mathit{C}}_{3}^{\pm}$ | ${\mathit{\sigma}}_{\mathit{di}}$ | |
---|---|---|---|

${\mathit{d}}_{1}$ | 1 | 1 | 1 |

${\mathit{d}}_{2}$ | 1 | 1 | −1 |

${\mathit{d}}_{3}$ | 2 | −1 | 0 |

**Table A3.**Compatibility relations between the single-valued (upper row) and double-valued (lower row) representations of the space group $R\overline{3}m$.

$\mathbf{\Gamma}\left(\mathbf{000}\right)$, $\mathit{Z}\left(\frac{1}{2}\frac{1}{2}\frac{1}{2}\right)$ | |||||

${R}_{1}^{+}$ | ${R}_{2}^{+}$ | ${R}_{1}^{-}$ | ${R}_{2}^{-}$ | ${R}_{3}^{+}$ | ${R}_{3}^{-}$ |

${R}_{4}^{+}$ | ${R}_{4}^{+}$ | ${R}_{4}^{-}$ | ${R}_{4}^{-}$ | ${R}_{5}^{+}$ + ${R}_{6}^{+}$ + ${R}_{4}^{+}$ | ${R}_{5}^{-}$ + ${R}_{6}^{-}$ + ${R}_{4}^{-}$ |

$\mathit{L}\left(\mathbf{0}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{0}\right)$, $\mathit{F}\left(\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{0}\right)$ | |||||

${R}_{1}^{+}$ | ${R}_{1}^{-}$ | ${R}_{2}^{+}$ | ${R}_{2}^{-}$ | ||

${R}_{3}^{+}$ + ${R}_{4}^{+}$ | ${R}_{3}^{-}$ + ${R}_{4}^{-}$ | ${R}_{3}^{+}$ + ${R}_{4}^{+}$ | ${R}_{3}^{-}$ + ${R}_{4}^{-}$ |

**Table A4.**Single-valued representations of all the energy bands in the space group $R\overline{3}m$ of Bi–I with symmetry-adapted and optimally localized usual (i.e., spin-independent) Wannier functions centered at the Bi atoms.

Bi($\mathit{zzz}$) | Bi($\overline{\mathit{z}}\overline{\mathit{z}}\overline{\mathit{z}}$) | K | $\mathbf{\Gamma}$ | Z | L | F | |
---|---|---|---|---|---|---|---|

Band 1 | ${\mathit{d}}_{1}$ | ${\mathit{d}}_{1}$ | OK | ${\Gamma}_{1}^{+}$ + ${\Gamma}_{2}^{-}$ | ${Z}_{1}^{+}$ + ${Z}_{2}^{-}$ | ${L}_{1}^{+}$ + ${L}_{2}^{-}$ | ${F}_{1}^{+}$ + ${F}_{2}^{-}$ |

Band 2 | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{2}$ | OK | ${\Gamma}_{2}^{+}$ + ${\Gamma}_{1}^{-}$ | ${Z}_{2}^{+}$ + ${Z}_{1}^{-}$ | ${L}_{1}^{-}$ + ${L}_{2}^{+}$ | ${F}_{1}^{-}$ + ${F}_{2}^{+}$ |

**Table A5.**Double-valued representations of the superconducting band in the space group $R\overline{3}m$ of Bi–I.

Bi($\mathit{zzz}$) | Bi($\overline{\mathit{z}}\overline{\mathit{z}}\overline{\mathit{z}}$) | K | $\mathbf{\Gamma}$ | Z | L | F | |
---|---|---|---|---|---|---|---|

Band 1 | $\mathit{d}$ | $\mathit{d}$ | OK | ${\Gamma}_{4}^{+}$ + ${\Gamma}_{4}^{-}$ | ${Z}_{4}^{+}$ + ${Z}_{4}^{-}$ | ${L}_{3}^{+}$ + ${L}_{4}^{+}$ + ${L}_{3}^{-}$ + ${L}_{4}^{-}$ | ${F}_{3}^{+}$ + ${F}_{4}^{+}$ + ${F}_{3}^{-}$ + ${F}_{4}^{-}$ |

**Table A6.**Character tables of the single-valued irreducible representations of the space group $Im3m$ = ${\Gamma}_{c}^{v}{O}_{h}^{9}$ of Bi–V.

$\mathbf{\Gamma}\left(000\right)$, $\mathit{H}\left(\frac{1}{2}\overline{\frac{1}{2}}\frac{1}{2}\right)$ | ||||||||||

$\mathit{E}$ | $\mathit{I}$ | ${\mathit{\sigma}}_{\mathit{m}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{m}}$ | ${\mathit{C}}_{\mathbf{3}\mathit{j}}^{\pm}$ | ${\mathit{S}}_{\mathbf{6}\mathit{j}}^{\pm}$ | ${\mathit{C}}_{\mathbf{4}\mathit{m}}^{\pm}$ | ${\mathit{S}}_{\mathbf{4}\mathit{m}}^{\pm}$ | ${\mathit{C}}_{\mathbf{2}\mathit{p}}$ | ${\mathit{\sigma}}_{\mathit{dp}}$ | |

${\Gamma}_{1}^{+}$, ${H}_{1}^{+}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

${\Gamma}_{2}^{+}$, ${H}_{2}^{+}$ | 1 | 1 | 1 | 1 | 1 | 1 | −1 | −1 | −1 | −1 |

${\Gamma}_{2}^{-}$, ${H}_{2}^{-}$ | 1 | −1 | −1 | 1 | 1 | −1 | −1 | 1 | −1 | 1 |

${\Gamma}_{1}^{-}$, ${H}_{1}^{-}$ | 1 | −1 | −1 | 1 | 1 | −1 | 1 | −1 | 1 | −1 |

${\Gamma}_{3}^{+}$, ${H}_{3}^{+}$ | 2 | 2 | 2 | 2 | −1 | −1 | 0 | 0 | 0 | 0 |

${\Gamma}_{3}^{-}$, ${H}_{3}^{-}$ | 2 | −2 | −2 | 2 | −1 | 1 | 0 | 0 | 0 | 0 |

${\Gamma}_{4}^{+}$, ${H}_{4}^{+}$ | 3 | 3 | −1 | −1 | 0 | 0 | 1 | 1 | −1 | −1 |

${\Gamma}_{5}^{+}$, ${H}_{5}^{+}$ | 3 | 3 | −1 | −1 | 0 | 0 | −1 | −1 | 1 | 1 |

${\Gamma}_{4}^{-}$, ${H}_{4}^{-}$ | 3 | −3 | 1 | −1 | 0 | 0 | 1 | −1 | −1 | 1 |

${\Gamma}_{5}^{-}$, ${H}_{5}^{-}$ | 3 | −3 | 1 | −1 | 0 | 0 | −1 | 1 | 1 | −1 |

$\mathit{P}\left(\frac{\mathbf{1}}{\mathbf{4}}\frac{\mathbf{1}}{\mathbf{4}}\frac{\mathbf{1}}{\mathbf{4}}\right)$ | ||||||||||

$\mathit{E}$ | ${\mathit{C}}_{\mathbf{2}\mathit{m}}$ | ${\mathit{S}}_{\mathbf{4}\mathit{m}}^{\pm}$ | ${\mathit{\sigma}}_{\mathit{dp}}$ | ${\mathit{C}}_{\mathbf{3}\mathit{j}}^{\pm}$ | ||||||

${P}_{1}$ | 1 | 1 | 1 | 1 | 1 | |||||

${P}_{2}$ | 1 | 1 | −1 | −1 | 1 | |||||

${P}_{3}$ | 2 | 2 | 0 | 0 | −1 | |||||

${P}_{4}$ | 3 | −1 | 1 | −1 | 0 | |||||

${P}_{5}$ | 3 | −1 | −1 | 1 | 0 | |||||

$\mathbf{N}\left(\mathbf{00}\frac{\mathbf{1}}{\mathbf{2}}\right)$ | ||||||||||

$\mathit{E}$ | ${\mathit{C}}_{\mathbf{2}\mathit{z}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{b}}$ | ${\mathit{C}}_{\mathbf{2}\mathit{a}}$ | $\mathit{I}$ | ${\mathit{\sigma}}_{\mathit{z}}$ | ${\mathit{\sigma}}_{\mathit{db}}$ | ${\mathit{\sigma}}_{\mathit{da}}$ | |||

${N}_{1}^{+}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||

${N}_{2}^{+}$ | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 | ||

${N}_{3}^{+}$ | 1 | 1 | −1 | −1 | 1 | 1 | −1 | −1 | ||

${N}_{4}^{+}$ | 1 | −1 | −1 | 1 | 1 | −1 | −1 | 1 | ||

${N}_{1}^{-}$ | 1 | 1 | 1 | 1 | −1 | −1 | −1 | −1 | ||

${N}_{2}^{-}$ | 1 | −1 | 1 | −1 | −1 | 1 | −1 | 1 | ||

${N}_{3}^{-}$ | 1 | 1 | −1 | −1 | −1 | −1 | 1 | 1 | ||

${N}_{4}^{-}$ | 1 | −1 | −1 | 1 | −1 | 1 | 1 | −1 |

**Table A7.**Compatibility relations between the single-valued (upper row) and double-valued (lower row) representations of the space group $Im3m$.

$\mathbf{\Gamma}\left(000\right)$, $\mathit{H}\left(\frac{1}{2}\overline{\frac{1}{2}}\frac{1}{2}\right)$ | |||||||||

${R}_{1}^{+}$ | ${R}_{2}^{+}$ | ${R}_{2}^{-}$ | ${R}_{1}^{-}$ | ${R}_{3}^{+}$ | ${R}_{3}^{-}$ | ${R}_{4}^{+}$ | ${R}_{5}^{+}$ | ${R}_{4}^{-}$ | ${R}_{5}^{-}$ |

${R}_{6}^{+}$ | ${R}_{7}^{+}$ | ${R}_{7}^{-}$ | ${R}_{6}^{-}$ | ${R}_{8}^{+}$ | ${R}_{8}^{-}$ | ${R}_{6}^{+}$ + ${R}_{8}^{+}$ | ${R}_{7}^{+}$ + ${R}_{8}^{+}$ | ${R}_{6}^{-}$ + ${R}_{8}^{-}$ | ${R}_{7}^{-}$ + ${R}_{8}^{-}$ |

$\mathit{P}\left(\frac{\mathbf{1}}{\mathbf{4}}\frac{\mathbf{1}}{\mathbf{4}}\frac{\mathbf{1}}{\mathbf{4}}\right)$ | |||||||||

${P}_{1}$ | ${P}_{2}$ | ${P}_{3}$ | ${P}_{4}$ | ${P}_{5}$ | |||||

${P}_{6}$ | ${P}_{7}$ | ${P}_{8}$ | ${P}_{6}$ + ${P}_{8}$ | ${P}_{7}$ + ${P}_{8}$ | |||||

$\mathit{N}\left(\mathbf{00}\frac{\mathbf{1}}{\mathbf{2}}\right)$ | |||||||||

${N}_{1}^{+}$ | ${N}_{2}^{+}$ | ${N}_{3}^{+}$ | ${N}_{4}^{+}$ | ${N}_{1}^{-}$ | ${N}_{2}^{-}$ | ${N}_{3}^{-}$ | ${N}_{4}^{-}$ | ||

${N}_{5}^{+}$ | ${N}_{5}^{+}$ | ${N}_{5}^{+}$ | ${N}_{5}^{+}$ | ${N}_{5}^{-}$ | ${N}_{5}^{-}$ | ${N}_{5}^{-}$ | ${N}_{5}^{-}$ |

**Table A8.**Single-valued representations of the space group $Im3m$ of all the energy bands of Bi–V with symmetry-adapted and optimally localized usual (i.e., spin-independent) Wannier functions centered at the Bi atoms.

Bi(000) | K | $\mathbf{\Gamma}$ | H | P | N | |
---|---|---|---|---|---|---|

Band 1 | ${\Gamma}_{1}^{+}$ | OK | ${\Gamma}_{1}^{+}$ | ${H}_{1}^{+}$ | ${P}_{1}$ | ${N}_{1}^{+}$ |

Band 2 | ${\Gamma}_{2}^{+}$ | OK | ${\Gamma}_{2}^{+}$ | ${H}_{2}^{+}$ | ${P}_{2}$ | ${N}_{3}^{+}$ |

Band 3 | ${\Gamma}_{2}^{-}$ | OK | ${\Gamma}_{2}^{-}$ | ${H}_{2}^{-}$ | ${P}_{1}$ | ${N}_{3}^{-}$ |

Band 4 | ${\Gamma}_{1}^{-}$ | OK | ${\Gamma}_{1}^{-}$ | ${H}_{1}^{-}$ | ${P}_{2}$ | ${N}_{1}^{-}$ |

**Table A9.**Double-valued representations of the space group $Im3m$ of all the energy bands of Bi–V with symmetry-adapted and optimally localized spin-dependent Wannier functions centered at the Bi atoms.

Bi(000) | K | $\mathbf{\Gamma}$ | H | P | N | |
---|---|---|---|---|---|---|

Band 1 | ${\Gamma}_{1}^{+}\otimes {\mathit{d}}_{1/2}={\Gamma}_{6}^{+}$ | OK | ${\Gamma}_{6}^{+}$ | ${H}_{6}^{+}$ | ${P}_{6}$ | ${N}_{5}^{+}$ |

Band 2 | ${\Gamma}_{2}^{+}\otimes {\mathit{d}}_{1/2}={\Gamma}_{7}^{+}$ | OK | ${\Gamma}_{7}^{+}$ | ${H}_{7}^{+}$ | ${P}_{7}$ | ${N}_{5}^{+}$ |

Band 3 | ${\Gamma}_{2}^{-}\otimes {\mathit{d}}_{1/2}={\Gamma}_{7}^{-}$ | OK | ${\Gamma}_{7}^{-}$ | ${H}_{7}^{-}$ | ${P}_{6}$ | ${N}_{5}^{-}$ |

Band 4 | ${\Gamma}_{1}^{-}\otimes {\mathit{d}}_{1/2}={\Gamma}_{6}^{-}$ | OK | ${\Gamma}_{6}^{-}$ | ${H}_{6}^{-}$ | ${P}_{7}$ | ${N}_{5}^{-}$ |

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Krüger, E.
Constraining Forces Stabilizing Superconductivity in Bismuth. *Symmetry* **2018**, *10*, 44.
https://doi.org/10.3390/sym10020044

**AMA Style**

Krüger E.
Constraining Forces Stabilizing Superconductivity in Bismuth. *Symmetry*. 2018; 10(2):44.
https://doi.org/10.3390/sym10020044

**Chicago/Turabian Style**

Krüger, Ekkehard.
2018. "Constraining Forces Stabilizing Superconductivity in Bismuth" *Symmetry* 10, no. 2: 44.
https://doi.org/10.3390/sym10020044