# The Casimir Effect in Topological Matter

## Abstract

**:**

## 1. Introduction

## 2. Axionic Topological Insulators

#### 2.1. Equations of Axion Electrodynamics

#### 2.2. Casimir Force Behavior

#### 2.3. Van der Waals Torque between Birefringent Topological Insulators

#### 2.4. Casimir–Polder Interaction between an Atom and a Topological Insulator

#### 2.5. Axion Dispersion

## 3. Chern Insulators

#### 3.1. Surface Conductivity via the Kubo Formula

#### 3.2. Lattice Models for Higher Chern Numbers

#### 3.3. Casimir Repulsion between Chern Insulators

#### 3.4. Nondispersive Chern Insulator: Chern–Simons Surface

## 4. Relativistic Quantum Hall Effect (RQHE) in Graphene

#### 4.1. Magneto-Optical Conductivity

#### 4.2. Casimir–Polder Interaction between Rubidium Atom and RQHE Graphene

#### 4.3. Casimir Interaction between Two RQHE Graphene Monolayers

## 5. Short Skit on Weyl Semimetals, and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Kubo Formula

## Appendix B. Fresnel Coefficients for a Pair of Chern Insulators

**Figure A1.**Boundary-value problem for a pair of coplanar Chern insulators (light blue and labeled “CI 1” and “CI 2”) in vacuum: (

**a**) ray incident (blue solid arrow) on the left Chern insulator, with wavevector ${\mathbf{k}}^{-}={({k}_{x},0,-{k}_{z})}^{\mathrm{T}}$; the transmitted ray (orange dashed arrow) has the same wavevector ${\mathbf{k}}_{-}$ and the reflected ray (blue dashed arrow) has the wavevector ${\mathbf{k}}^{+}={({k}_{x},0,{k}_{z})}^{\mathrm{T}}$. (

**b**) Ray incident on the right Chern insulator, with reflected and transmitted rays. This is a mirror image of the rays in (

**a**). ${\mathbf{E}}^{<},{\mathbf{H}}^{<}$ (${\mathbf{E}}^{>},{\mathbf{H}}^{>}$) denote the electric and magnetic fields to the left (right) of a Chern insulator. The polarization vectors (red solid arrows) are ${\widehat{e}}_{p}^{\pm}={\widehat{e}}_{s}^{\pm}\times {\widehat{k}}^{\pm}$.

## Notes

1 | Time reversal symmetry-breaking quantities are representable by antisymmetric tensors, and such tensors in turn can be represented by axial vectors. For an antisymmetric $3\times 3$ matrix ${A}_{ij}$ ($i,j=1,2,3$), an axial vector $\mathbf{a}={({a}_{1},{a}_{2},{a}_{3})}^{\mathrm{T}}$ is constructed by assigning ${a}_{1}={A}_{2}3$, ${a}_{2}={A}_{3}1$ and ${a}_{3}={A}_{1}2$ [H. B. G. Casimir and A. N. Gerritsen, Physica 1941, 8, 1107]. The sign of the axial vector is related to the circulation direction of the TRSB current. |

2 | A two-dimensional topological insulator is characterized by one ${\mathbb{Z}}_{2}$ invariant, and a three-dimensional topological insulator is characterized by four such invariants. The difference is explained in Ref. [31]. |

3 | Here it may appear that the result for ${r}_{\alpha \beta}^{\prime}$ contradicts that obtained later for Chern insulators in Appendix B, where ${r}_{ss}^{\prime}={r}_{ss},{r}_{pp}^{\prime}={r}_{pp}$ but ${r}_{ps}^{\prime}=-{r}_{ps},{r}_{sp}^{\prime}=-{r}_{sp}$. The latter result is obtained on the assumption that ${\sigma}_{xy}$ has the same sign/orientation for both surfaces. There is no contradiction, because for the oppositely facing topological insulator surfaces that we presently consider, to keep the sign of both surface Hall conductivities the same one has to change the relative sign of ${\overline{\alpha}}_{L}$ and ${\overline{\alpha}}_{R}$, and correspondingly ${r}_{ps}^{\prime}={r}_{ps}$ and ${r}_{sp}^{\prime}={r}_{sp}$ for topological insulators. |

4 | For larger values of $\overline{\alpha}$ the force reverts to attraction. This is analogous to what happens for Chern insulators that have large values of the Chern number, which we discuss in Section 3.3. |

5 | Note that the result mentioned here, i.e., the Chern insulators attract each other as perfect conductors in the limit that $|C|\alpha \to \infty $, is correct. It is different from that in Ref. [86], where they found that the energy is equal to $0.92$ times that of a pair of perfect conductors. The author thanks an anonymous referee for pointing this out. |

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**Figure 1.**A pair of flat, coplanar topological insulator slabs (labeled 1 and 3) separated by a vacuum gap (labeled 2) of width d. The optic axis of slab 1 (3) is colored cyan (green), and the orientations of the optic axes measured with respect to a reference axis (in this case the y axis) are ${\varphi}_{1}$ and ${\varphi}_{3}$. In the figure, we only show a finite slice of the slabs, which are assumed to have infinitely large thicknesses and cross-sectional areas. Figure adapted from Ref. [56].

**Figure 2.**(

**a**) Schematic depiction of an atom (modeled by a two-level system with ground state energy ${E}_{0}$ and excited state energy ${E}_{1}$) in vacuum, interacting with an axionic topological insulator (represented by the gold surface). The interaction is via dipole radiation coupling; in vacuum, this gives rise to the Lamb shift $\delta {E}_{a}^{Lamb}$. As an atom in state $|a\rangle $ is brought near the surface, the surface induces an additional shift $\delta {E}_{a}^{CP}$; this is the Casimir–Polder energy for the atomic state $|a\rangle $. (

**b**) s and p polarizations denoted, respectively, by ${e}_{s}^{\pm}$ (blue dot, directed along the negative y-direction) and ${e}_{p}^{\pm}$ (green arrow), for the case where the plane of incidence is the xz plane. For this case, the transverse wave vector lies entirely along the x direction, ${k}_{y}=0$, and ${e}_{s}^{\pm}=(0,-1,0)$ and ${e}_{p}^{\pm}=(1/k)(\mp {k}_{z},0,{k}_{x})$. The + (−) superscript refers to a wave propagating in the positive (negative) z-direction, $k=\omega /c$, and red arrows denote the propagation directions of incident and reflected waves.

**Figure 3.**(Color online.) Real-frequency dispersion behavior, shown by curves with blue squares (red circles), of the Chern insulator’s conductivity tensor for $u/t=1$ ($u/t=-1$) in Equation (29): (

**a**) $\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{xx}(\omega )/(\alpha c)$, (

**b**) $\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{xx}(\omega )/(\alpha c)$, (

**c**) $\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{xy}(\omega )/(\alpha c)$, and (

**d**) $\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}{\sigma}_{xy}(\omega )/(\alpha c)$ as functions of $\hslash \omega /t$ (horizontal axis). Figure reproduced from Ref. [70].