Finite temperature fluctuation-induced order and responses in magnetic topological insulators

We derive an effective field theory model for magnetic topological insulators and predict that a magnetic electronic gap persists on the surface for temperatures above the ordering temperature of the bulk. Our analysis also applies to interfaces of heterostructures consisting of a ferromagnetic and a topological insulator. In order to make quantitative predictions for ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ and for EuS-${\mathrm{Bi}}_2{\mathrm{Se}}_3$ heterostructures, we combine the effective field theory method with density functional theory and Monte Carlo simulations. For ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ we predict an upwards Néel temperature shift at the surface up to $15\%$, while the EuS-${\mathrm{Bi}}_2{\mathrm{Se}}_3$ interface exhibits a smaller relative shift. The effective theory also predicts induced Dzyaloshinskii-Moriya interactions and a topological magnetoelectric effect, both of which feature a finite temperature and chemical potential dependence.

Figure 1

Left: Symmetry breaking induced by the proximity effect. An exchange coupling is induced across the interface between a FMI and a TI. The FMI polarizes the TI surface by the proximity effect and gaps the surface spectrum as, e.g., EuS-${\mathrm{Bi}}_2{\mathrm{Se}}_3$ heterostructures [13]. Right: Intrinsic spontaneous symmetry breaking. Here, the TI is itself a magnetic insulator such as, e.g., ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$ [17].

Figure 2

(a) Normalized massive DMI coupling strength $\tilde{D}=D/\overline{D}$ and (b) normalized topological mass $\tilde{\sigma }=\sigma /\overline{\sigma }$, both as a function of temperature and chemical potential. The normalizations are set to $\overline{D}=-J_0^2/4\pi \hslash v_{\text{F}}$ and $\overline{\sigma }=e^2/2h$, respectively. The DMI is only present when the chemical potential exceeds the gap (metallic regime). On the contrary the topological mass only exists for a chemical potential inside the gap (insulating regime).

Figure 3

Crossover of the normalized topological mass $\tilde{\sigma }$ and DMI coupling strength $\tilde{D}$ as a function of the chemical potential at different temperatures. For a vanishing temperature both quantities are step functions. Increasing the temperature causes the step functions to smear out, making coexistence possible. The two quantities always sum up to unity at a given value for the chemical potential.