Properties of topological insulators and superconductors under relativistic gravity

TL;DR

The authors examine how curved spacetime, via gravitational redshift, affects 1D topological phases realized by SSH and Kitaev chains. By deriving gravity-modified tight-binding parameters and constructing curved-space Dirac/BdG field theories, they show SSH edge states shift with the redshift and lose the protecting chiral symmetry, whereas Kitaev Majorana zero modes stay pinned at zero energy and can undergo a gravitationally induced bulk phase transition with domain-wall formation. Despite these energy shifts, real-space winding numbers remain quantized in gravity for both models, indicating robust topology under curved backgrounds albeit with model-dependent symmetry behavior. Overall, the work extends topological quantum matter into non-inertial frames and suggests gravitational effects may influence qubit stability differently in Majorana-supporting systems versus conventional symmetry-protected insulators.

Abstract

The interplay between the curved spacetimes of general relativity and quantum mechanical systems is an active field of research. However, analysis of relativistic gravitation on extended quantum systems remains understudied. To this end, we study here the effects of a general relativistic curved spacetime on the topological phases of the Su-Schrieffer-Heeger model and Kitaev superconducting wire. We find that the topological states remain robust and well localized. In the topological insulator we find that the energy level of the topological state becomes shifted away from zero according to the gravitational redshift, breaking the system's chiral symmetry. In contrast, the Majorana zero mode of the topological superconductor remains at zero energy. Furthermore, within the topological superconductor, we identify the possibility of a gravitationally induced topological phase transition leading to the formation of a domain wall, shifting one of the boundary Majorana zero modes into the bulk.

Figures (9)

• Figure 1: Diagram for the system geometry studied. The SSH and Kitaev models are $1d$ chains immersed radially in the gravitational field of a source $M$. One end of the chain is situated at the coordinate $r_0 > R_s$ near the source mass $M$. The terminus of the far end of the chain lies at the coordinate $r_L = r_0 + \ell L$ where $\ell$ is the lattice constant and $L$ is the number of sites in the tight-binding chain. The main quantities of interest are the local spectral functions on the two ends of the chain, $\mathcal{A}_1(\omega)$ and $\mathcal{A}_L(\omega)$. The redshift for a given site is calculated with respect to the gravitational potential at coordinate $r_0$.
• Figure 2: \ref{['fig:sshL']} Spectrum on site $L$, $\mathcal{A}_{L}(\omega)$, redshifted with respect to site $1$ of an SSH chain with $E5$ unit cells in presence of gravity with a source with Schwarzschild radius of $10^5 \ell$ for site $1$ positioned at $x_0 = 1.2 R_s ,\, 1.3 R_s ,\, 1.5 R_s ,\, 2 R_s ,\, 3 R_s ,\, 5 R_s ,\, 10 R_s$. The base band gap is $2\delta t = 0.3D$ in units of the band width $D = 2 t_0$. The SSH gap remains intact even as the shift in energy becomes on the order of relevant energy scales, such as the band width and band gap at site $1$. The topological state follows the observed redshift. The bold dashed line marks $\omega = 0$ where the pole of the gravity-free SSH model lies. The topological spectral pole is highlighted in red. \ref{['fig:ssh0']} Same sequence of spectral plots for site 1.
• Figure 3: \ref{['fig:kitaevL']} Spectrum of $\gamma_2$ on site $L$ redshifted with respect to site $1$ of a Kitaev wire with $E5$ Majorana unit cells in presence of gravity for site $1$ positioned at $x_0 = 1.2 R_s ,\, 1.3 R_s ,\, 1.5 R_s ,\, 2 R_s ,\, 3 R_s ,\, 5 R_s ,\, 10 R_s$, with $R_s = 10^{5}\ell$. The system parameters are $\mu_0/\Delta_0 = 0.3$, $t_0/\Delta_0 = 0.5$, $\varepsilon_F/\Delta_0 = 1$, $R_s = 10^{5}\ell$. The topological state, highlighted in red, is observed to remain at $\omega = 0$ for various values of $x_0$, in contrast to the SSH model where the topological state tracks with the observed redshift. \ref{['fig:kitaev0']} Same sequence of spectral plots for $\gamma_1$ on site 1.
• Figure 4: Gravitationally induced topological phase transition. Schematic demonstrating the localization of the Majorana zero modes (MZM) $\gamma_1$, $\gamma_2$ on the tight-binding chain. Filled sites represent the topological phase. \ref{['fig:dwschematic0']} In the flat spacetime case the MZM are localized on the end points of the chain and the entire system is in the topological phase. \ref{['fig:dwschematic1']} Under certain parameterization, in the presence of gravity the second Majorana $\gamma_2$ becomes localized on a domain wall within the bulk of the chain. The domain wall partitions the system into a part which is in the topological phase and a part in the trivial phase, illustrated by the open sites.
• Figure 5: Magnitude of the redshifted Kitaev wire parameters $\mu(x)$ and $t(x)$ along their position on the wire. System parameters are $\mu_0/\Delta_0 = 0.9$, $t_0/\Delta_0 = 0.5$, $\varepsilon_F / \Delta_0 = 1$, $R_s = 10\ell$, $x_0=5R_s$. Initially the system lies in the topological phase with $\mu(x_j) < 2 t(x_j)$, however their relative magnitude swaps at the domain wall transitioning the remainder of the system into the trivial phase. The domain wall occurs at $j_{\textsc{dw}} = 760$ as indicated by the arrow and vertical dashed line.