# Anisotropic Magnetic Responses of Topological Crystalline Superconductors

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary

## 3. Topological Invariants for Irreducible Representations

#### 3.1. Symmetry of Crystalline Systems Including a Surface

#### 3.2. Symmetry Operations in Superconducting States

#### 3.3. Topological Invariant

## 4. Winding Number Protected by n-Fold Rotational Symmetry

#### 4.1. Definition

#### 4.2. Time-Reversal Symmetry

#### 4.3. Spatial Symmetry

## 5. Example: Bilayer Rashba System

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

BdG | Bogoliubov-de Gennes |

## Appendix A. Bulk-Edge Correspondence in a Lattice Model

#### Appendix A.1. Number of Edge States

#### Appendix A.2. Bulk-Edge Correspondence

## Appendix B. Representation of Symmetry Operation

## Appendix C. Tables for Irreducible Representations and Majorana Ising Spins

**Table A1.**Possible topological invariants $W[{\tilde{U}}_{i},{\mathit{x}}_{\perp}]$ and the direction of the Majorana Ising spin on the given surfaces for each irreducible representation of ${D}_{4h}$, ${C}_{4v}$, and ${C}_{2v}$. The surface is denoted by the mirror index $(hkl)$ or the Cartesian coordinate $({x}_{i}{x}_{j})$. ${\mathit{x}}_{\sigma}$ of $W[\sigma ,{\mathit{x}}_{\sigma}]$ is the direction normal to the surface and is on the σ mirror plane. The ${C}_{2}^{\prime}$ axis is set to the x axis. The representations ${E}_{u}(x)$ and ${E}_{u}(y)$ are defined in the bottom table.

${\mathit{D}}_{4\mathit{h}}$ | $\mathit{W}[\mathit{U},{\mathit{x}}_{\perp}]$ | (001) | (100) | (110) | ||

${A}_{1u}$ | $W[{C}_{n}]$, $W[{C}_{2}^{\prime}]$, $W[{C}_{2}^{\u2033}]$ | $[001]$ | $[100]$ | $[110]$ | ||

${A}_{2u}$ | $W[{\sigma}_{v},{\mathit{x}}_{{\sigma}_{v}}\ne [001]]$, $W[{\sigma}_{d},{\mathit{x}}_{{\sigma}_{d}}\ne [001]]$ | $\mathbf{0}$ | $[010]$ | $[1\overline{1}0]$ | ||

${B}_{1u}$ | $W[{C}_{2}^{\prime}]$, $W[{\sigma}_{d},{\mathit{x}}_{{\sigma}_{d}}\ne [001]]$ | $\mathbf{0}$ | $[100]$ | $[1\overline{1}0]$ | ||

${B}_{2u}$ | $W[{C}_{2}^{\u2033}]$, $W[{\sigma}_{v},{\mathit{x}}_{{\sigma}_{v}}\ne [001]]$ | $\mathbf{0}$ | $[010]$ | $[110]$ | ||

${\mathit{D}}_{\mathbf{4}\mathit{h}}$ | $\mathit{W}[\mathit{U},{\mathit{x}}_{\perp}]$ | (001) | (100) | (010) | (110) | |

${E}_{u}(x)$ | $W[{\sigma}_{v}(010),{\mathit{x}}_{{\sigma}_{v}(010)}\ne [100]]$, $W[{\sigma}_{h},{\mathit{x}}_{{\sigma}_{h}}\ne [100]]$ | $[010]$ | $\mathbf{0}$ | $[001]$ | $[001]$ | |

${E}_{u}(y)$ | $W[{\sigma}_{v}(100),{\mathit{x}}_{{\sigma}_{v}(100)}\ne [010]]$, $W[{\sigma}_{h},{\mathit{x}}_{{\sigma}_{h}}\ne [010]]$ | $[100]$ | $[001]$ | $\mathbf{0}$ | $[001]$ | |

${\mathit{C}}_{\mathbf{4}\mathit{v}}$ | $\mathit{W}[\mathit{U},{\mathit{x}}_{\perp}]$ | (001) | (100) | (110) | ||

${A}_{1}$ | $W[{\sigma}_{v},{\mathit{x}}_{{\sigma}_{v}}\ne \mathit{z}]$, $W[{\sigma}_{d},{\mathit{x}}_{{\sigma}_{d}}\ne \mathit{z}]$ | $\mathbf{0}$ | $[010]$ | $[1\overline{1}0]$ | ||

${A}_{2}$ | $W[{C}_{2}]$ | $[001]$ | $\mathbf{0}$ | $\mathbf{0}$ | ||

${B}_{1}$ | $W[{\sigma}_{v},{\mathit{x}}_{{\sigma}_{v}}\ne \mathit{z}]$ | $\mathbf{0}$ | $[010]$ | $\mathbf{0}$ | ||

${B}_{2}$ | $W[{\sigma}_{d},{\mathit{x}}_{{\sigma}_{d}}\ne \mathit{z}]$ | $\mathbf{0}$ | $\mathbf{0}$ | $[1\overline{1}0]$ | ||

$E(x)$ | $W[{\sigma}_{v}(zx),{\mathit{x}}_{{\sigma}_{v}(zx)}\ne \mathit{x}]$ | $[010]$ | $\mathbf{0}$ | $\mathbf{0}$ | ||

$E(y)$ | $W[{\sigma}_{v}(yz),{\mathit{x}}_{{\sigma}_{v}(yz)}\ne \mathit{y}]$ | $[100]$ | $\mathbf{0}$ | $\mathbf{0}$ | ||

${\mathit{C}}_{\mathbf{2}\mathit{v}}$ | $\mathit{W}[\mathit{U},{\mathit{x}}_{\perp}]$ | $(\mathit{xy})$ | $(\mathit{yz})$ | $(\mathit{zx})$ | ||

${A}_{1}$ | $W[{\sigma}_{v}(zx),{\mathit{x}}_{{\sigma}_{v}(zx)}\ne \mathit{z}]$, $W[{\sigma}_{v}(yz),{\mathit{x}}_{{\sigma}_{v}(yz)}\ne \mathit{z}]$ | $\mathbf{0}$ | $\mathit{y}$ | $\mathit{x}$ | ||

${A}_{2}$ | $W[{C}_{2}]$ | $\mathit{z}$ | $\mathbf{0}$ | $\mathbf{0}$ | ||

${B}_{1}$ | $W[{\sigma}_{v}(zx),{\mathit{x}}_{{\sigma}_{v}(zx)}\ne \mathit{x}]$ | $\mathit{y}$ | $\mathbf{0}$ | $\mathbf{0}$ | ||

${B}_{2}$ | $W[{\sigma}_{v}(yz),{\mathit{x}}_{{\sigma}_{v}(yz)}\ne \mathit{y}]$ | $\mathit{x}$ | $\mathbf{0}$ | $\mathbf{0}$ | ||

${\mathit{D}}_{\mathbf{4}\mathit{h}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{2}}^{\prime}(\mathit{x})$ | ${\mathit{C}}_{\mathbf{2}}^{\prime}(\mathit{y})$ | ${\mathit{\sigma}}_{\mathit{h}}$ | ${\mathit{\sigma}}_{\mathit{v}}(\mathit{yz})$ | ${\mathit{\sigma}}_{\mathit{v}}(\mathit{xz})$ |

${E}_{u}(x)$ | − | + | − | + | − | + |

${E}_{u}(y)$ | − | − | + | + | + | − |

**Table A2.**Possible topological invariants $W[{\tilde{U}}_{i},{\mathit{x}}_{\perp}]$ in the $Pmma$ space group for ${k}_{x}{a}_{x}=0$ (upper) and ${k}_{x}{a}_{x}=\pi $ (lower).

$\mathit{Pmma}$ | $\mathit{W}[\mathit{U},{\mathit{x}}_{\perp}]({\mathit{k}}_{\mathit{x}}=\mathbf{0})$ | $(\mathit{xy})$ | $(\mathit{yz})$ | $(\mathit{xz})$ |

${A}_{g}$ | 0 | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ |

${B}_{1g}$ | 0 | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ |

${B}_{2g}$ | 0 | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ |

${B}_{3g}$ | 0 | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ |

${A}_{u}$ | $W[\{{C}_{2}(z)|\mathit{a}/2\}]$, $W[\{{C}_{2}(x)|\mathit{a}/2\}]$, $W[\{{C}_{2}(y)|\mathbf{0}\}]$ | $\mathit{z}$ | $\mathbf{0}$ | $\mathit{x}$ |

$W[\{\sigma (xy)|\mathit{a}/2\},{\mathit{x}}_{\sigma (xy)}\ne \mathit{x},\mathit{y}\}]$ | ||||

$W[\{\sigma (yz)|\mathit{a}/2\},{\mathit{x}}_{\sigma (yz)}\ne \mathit{y},\mathit{z}\}]$ | ||||

$W[\{\sigma (xz)|\mathbf{0}\},{\mathit{x}}_{\sigma (xz)}\ne \mathit{x},\mathit{z}\}]$ | ||||

${B}_{1u}$ | $W[\{\sigma (xy)|\mathit{a}/2\},{\mathit{x}}_{\sigma (xy)}\ne \mathit{x},\mathit{y}\}]$ | $\mathbf{0}$ | $\mathit{y}$ | $\mathit{x}$ |

$W[\{\sigma (yz)|\mathit{a}/2\},{\mathit{x}}_{\sigma (yz)}\ne \mathit{z}\}]$ | ||||

$W[\{\sigma (xz)|\mathbf{0}\},{\mathit{x}}_{\sigma (xz)}\ne \mathit{z}\}]$ | ||||

${B}_{2u}$ | $W[\{\sigma (xy)|\mathit{a}/2\},{\mathit{x}}_{\sigma (xy)}\ne \mathit{y}\}]$ | $\mathit{x}$ | $\mathbf{0}$ | $\mathbf{0}$ |

$W[\{\sigma (yz)|\mathit{a}/2\},{\mathit{x}}_{\sigma (yz)}\ne \mathit{y}\}]$ | ||||

$W[\{\sigma (xz)|\mathbf{0}\},{\mathit{x}}_{\sigma (xz)}\ne \mathit{x},\mathit{z}\}]$ | ||||

${B}_{3u}$ | $W[\{\sigma (xy)|\mathit{a}/2\},{\mathit{x}}_{\sigma (xy)}\ne \mathit{x}\}]$ | $\mathit{y}$ | $\mathbf{0}$ | $\mathit{z}$ |

$W[\{\sigma (yz)|\mathit{a}/2\},{\mathit{x}}_{\sigma (yz)}\ne \mathit{y},\mathit{z}\}]$ | ||||

$W[\{\sigma (xz)|\mathbf{0}\},{\mathit{x}}_{\sigma (xz)}\ne \mathit{x}\}]$ | ||||

$\mathit{Pmma}$ | $\mathit{W}[\mathit{U},{\mathit{x}}_{\perp}]({\mathit{k}}_{\mathit{x}}{\mathit{a}}_{\mathit{x}}=\mathit{\pi})$ | $(\mathit{xy})$ | $(\mathit{yz})$ | $(\mathit{xz})$ |

${A}_{g}$ | $W[\{\sigma (yz)|\mathit{a}/2\},{\mathit{x}}_{\sigma (yz)}\ne \mathit{z}]$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathit{x}$ |

${B}_{1g}$ | $W[\{{C}_{2}(z)|\mathit{a}/2\}]$ | $\mathit{z}$ | $\mathbf{0}$ | $\mathbf{0}$ |

$W[\{\sigma (xy)|\mathit{a}/2\},{\mathit{x}}_{\sigma (xy)}\ne \mathit{y}]$ | ||||

$W[\{\sigma (yz)|\mathit{a}/2\},{\mathit{x}}_{\sigma (yz)}\ne \mathit{y},\mathit{z}]$ | ||||

${B}_{2g}$ | $W[\{\sigma (xy)|\mathit{a}/2\},{\mathit{x}}_{\sigma (xy)}\ne \mathit{x},\mathit{y}]$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ |

$W[\{\sigma (yz)|\mathit{a}/2\},{\mathit{x}}_{\sigma (yz)}\ne \mathit{y},\mathit{z}]$ | ||||

${B}_{3g}$ | $W[\{\sigma (xy)|\mathit{a}/2\},{\mathit{x}}_{\sigma (xy)}\ne \mathit{x},\mathit{y}]$ | $\mathit{x}$ | $\mathbf{0}$ | $\mathbf{0}$ |

$W[\{\sigma (yz)|\mathit{a}/2\},{\mathit{x}}_{\sigma (yz)}\ne \mathit{y}]$ | ||||

${A}_{u}$ | $W[\{\sigma (xz)|\mathbf{0}\},{\mathit{x}}_{\sigma (xz)}\ne \mathit{x},\mathit{z}]$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ |

${B}_{1u}$ | $W[\{\sigma (xz)|\mathbf{0}\},{\mathit{x}}_{\sigma (xz)}\ne \mathit{z}]$ | $\mathbf{0}$ | $\mathit{y}$ | $\mathbf{0}$ |

${B}_{2u}$ | $W[\{{C}_{2}(y)|\mathbf{0}\}]$, $W[\{\sigma (xz)|\mathbf{0}\},{\mathit{x}}_{\sigma (xz)}\ne \mathit{x},\mathit{z}]$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathit{y}$ |

${B}_{3u}$ | $W[\{{C}_{2}(x)|\mathit{a}/2\}]$, $W[\{\sigma (xz)|\mathbf{0}\},{\mathit{x}}_{\sigma (xz)}\ne \mathit{x}]$ | $\mathit{y}$ | $\mathbf{0}$ | $\mathbf{0}$ |

**Table A3.**Commutator $p({U}_{j},{U}_{l})={U}_{j}^{-1}{U}_{l}^{-1}{U}_{j}{U}_{l}$ for the $Pmma$ group. A row and column correspond to ${U}_{j}$ and ${U}_{l}$, respectively.

$\{{\mathit{C}}_{2}(\mathit{z})|\mathit{a}/2\}$ | $\{{\mathit{C}}_{2}(\mathit{x})|\mathit{a}/2\}$ | $\{{\mathit{C}}_{2}(\mathit{y})|0\}$ | $\{\mathit{\sigma}(\mathit{xy})|\mathit{a}/2\}$ | $\{\mathit{\sigma}(\mathit{yz})|\mathit{a}/2\}$ | $\{\mathit{\sigma}(\mathit{xz})|0\}$ | |
---|---|---|---|---|---|---|

$\{{C}_{2}(z)|\mathit{a}/2\}$ | 1 | $-{e}^{i{k}_{x}{a}_{x}}$ | $-{e}^{i{k}_{x}{a}_{x}}$ | ${e}^{i{k}_{x}{a}_{x}}$ | $-1$ | $-1$ |

$\{{C}_{2}(x)|\mathit{a}/2\}$ | $-{e}^{-i{k}_{x}{a}_{x}}$ | 1 | $-{e}^{-i{k}_{x}{a}_{x}}$ | $-1$ | ${e}^{i{k}_{x}{a}_{x}}$ | $-1$ |

$\{{C}_{2}(y)|\mathbf{0}\}$ | $-{e}^{-i{k}_{x}{a}_{x}}$ | $-{e}^{i{k}_{x}{a}_{x}}$ | 1 | $-{e}^{i{k}_{x}{a}_{x}}$ | $-{e}^{i{k}_{x}{a}_{x}}$ | 1 |

$\{\sigma (xy)|\mathit{a}/2\}$ | ${e}^{-i{k}_{x}{a}_{x}}$ | $-1$ | $-{e}^{-i{k}_{x}{a}_{x}}$ | 1 | $-{e}^{-i{k}_{x}{a}_{x}}$ | $-1$ |

$\{\sigma (yz)|\mathit{a}/2\}$ | $-1$ | ${e}^{-i{k}_{x}{a}_{x}}$ | $-{e}^{-i{k}_{x}{a}_{x}}$ | $-{e}^{i{k}_{x}{a}_{x}}$ | 1 | $-1$ |

$\{\sigma (xz)|\mathbf{0}\}$ | $-1$ | $-1$ | 1 | $-1$ | $-1$ | 1 |

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**Figure 1.**(

**a**) Topological superconductor (TSC) with the surface on ${x}_{\perp}=0$; (

**b**) Energy dispersion of the surface Majorana zero modes located at the time-reversal-invariant momentum ${\mathit{k}}_{\parallel}=\mathbf{\Gamma}$.

**Figure 2.**Point-group symmetry operations which preserve (

**a**) and invert (

**b**) the surface of ${x}_{\perp}=0$.

**Figure 3.**Schematic view of the Rashba bilayer between LaAlO${}_{3}$ and SrTiO${}_{3}$ [34]. (

**a**) Two-dimensional electron gases are formed in the interfaces. The energy dispersions of the finite system with the edge normal to the x axis are shown in Figure 5 for possible pair potentials; (

**b**) The energy dispersion of the Hamiltonian Equation (38) in the normal state. The Fermi energy is located at $E=0$. The parameters are taken as $m=0.5$, $\epsilon =0.5$, $\alpha =2$.

**Figure 4.**Winding of the phase (integrand of $W[U,\mathit{x}]$) for the (

**a**) ${A}_{1u}$; (

**b**) ${A}_{2u}$; (

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

**d**) ${E}_{u}(y)$ pairings. The winding numbers are obtained to be $W[{C}_{2}^{\prime}(\mathit{x})]=-2$, $W[{\sigma}_{v}(xz),\mathit{x}]=-2$, $W[{\sigma}_{v}(xz),\mathit{x}]\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2$, and $W[{\sigma}_{h},\mathit{x}]=2$, for cases (

**a**–

**d**), respectively.

**Figure 5.**Energy spectra for odd parity pair potentials for $\Delta =0.1$ and $h=0.03$. The parameters are the same as in Figure 3b.

**Table 1.**Symmetry of magnetic field $\mathit{B}$ applied along the ${x}_{\perp}$, ${x}_{\parallel 1}$, and ${x}_{\parallel 2}$ directions, which are depicted in Figure 1. ${C}_{2}({x}_{\perp})$ is the two-fold rotation along the ${x}_{\perp}$ axis. $\sigma ({x}_{i}{x}_{j})$ is the mirror reflection with respect to the ${x}_{i}{x}_{j}$ plane. These are symmetry operations of the semi-infinite system with the surface of ${x}_{\perp}=0$. − (+) indicates that the magnetic field is (not) flipped by the symmetry operation. $\mathit{S}$ denotes the direction of Majorana Ising spin protected by the topological invariant $W[U]$ for $U={C}_{2}({x}_{\perp})$, $\sigma ({x}_{\perp}{x}_{\parallel 1})$, and $\sigma ({x}_{\perp}{x}_{\parallel 2})$.

U | $\mathit{B}\parallel {\mathit{x}}_{\perp}$ | $\mathit{B}\parallel {\mathit{x}}_{\parallel 1}$ | $\mathit{B}\parallel {\mathit{x}}_{\parallel 2}$ | S |
---|---|---|---|---|

${C}_{2}({x}_{\perp})$ | + | − | − | ${x}_{\perp}$ |

$\sigma ({x}_{\perp}{x}_{\parallel 1})$ | − | − | + | ${x}_{\parallel 2}$ |

$\sigma ({x}_{\perp}{x}_{\parallel 2})$ | − | + | − | ${x}_{\parallel 1}$ |

**Table 2.**Irreducible representations Γ of odd-parity pair potentials Δ in the bilayer Rashba superconductor under the ${D}_{4h}$ symmetry. $W[U,\mathit{x}]$ is the winding number related to Majorana fermions on the (100) edge and $\mathit{S}$ denotes the direction of its Ising spin. These are taken from Table A1. $\mathbf{0}$ denotes the absence of Majorana fermion.

Γ | $\mathit{W}[\mathit{U},\mathit{x}]$ | Δ | S |
---|---|---|---|

${A}_{1u}$ | $W[{C}_{2}^{\prime}(x)]$ | ${\sigma}_{y}{s}_{z}$ | $\mathit{x}$ |

${A}_{2u}$ | $W[{\sigma}_{v}(xz),\mathit{x}]$ | ${\sigma}_{z}{s}_{0}$ | $\mathit{y}$ |

${B}_{1u}$ | $W[{C}_{2}^{\prime}(x)]$ | $\mathrm{sin}{k}_{x}\mathrm{sin}{k}_{y}{\sigma}_{z}{s}_{0}$ | $\mathit{x}$ |

${B}_{2u}$ | $W[{\sigma}_{v}(xz),\mathit{x}]$ | $(\mathrm{cos}{k}_{x}-\mathrm{cos}{k}_{y}){\sigma}_{z}{s}_{0}$ | $\mathit{y}$ |

${E}_{u}(x)$ | 0 | ${\sigma}_{y}{s}_{y}$ | $\mathbf{0}$ |

${E}_{u}(y)$ | $W[{\sigma}_{h},\mathit{x}]$ | ${\sigma}_{y}{s}_{x}$ | $\mathit{z}$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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Xiong, Y.; Yamakage, A.; Kobayashi, S.; Sato, M.; Tanaka, Y.
Anisotropic Magnetic Responses of Topological Crystalline Superconductors. *Crystals* **2017**, *7*, 58.
https://doi.org/10.3390/cryst7020058

**AMA Style**

Xiong Y, Yamakage A, Kobayashi S, Sato M, Tanaka Y.
Anisotropic Magnetic Responses of Topological Crystalline Superconductors. *Crystals*. 2017; 7(2):58.
https://doi.org/10.3390/cryst7020058

**Chicago/Turabian Style**

Xiong, Yuansen, Ai Yamakage, Shingo Kobayashi, Masatoshi Sato, and Yukio Tanaka.
2017. "Anisotropic Magnetic Responses of Topological Crystalline Superconductors" *Crystals* 7, no. 2: 58.
https://doi.org/10.3390/cryst7020058