Quantum Oscillations without a Fermi Surface and the Anomalous de Haas–van Alphen Effect

The de Haas–van Alphen effect (dHvAE), describing oscillations of the magnetization as a function of magnetic field, is commonly assumed to be a definite sign for the presence of a Fermi surface (FS). Indeed, the effect forms the basis of a well-established experimental procedure for accurately measuring FS topology and geometry of metallic systems, with parameters commonly extracted by fitting to the Lifshitz-Kosevich (LK) theory based on Fermi liquid theory. Here we show that, in contrast to this canonical situation, there can be quantum oscillations even for band insulators of certain types. We provide simple analytic formulas describing the temperature dependence of the quantum oscillations in this setting, showing strong deviations from LK theory. We draw connections to recent experiments and discuss how our results can be used in future experiments to accurately determine, e.g., hybridization gaps in heavy-fermion systems.

Figure 1

Main figure (i): Quantum oscillations of the magnetization $M$ as a function of $W/\hslash {\omega }_c\propto 1/B$. Inset (ii): Sketch of the band structure for our model (exaggerated hybridization gap for better visibility) and positions of the different chemical potentials $\mu$. If $\mu$ is far away from the gap (black dashed and dot dashed lines), which is opened by hybridizing a localized flat band with an itinerant band, the periodicity of standard QO is proportional to the extremal cross section of the Fermi surface (here directly related to $\mu =(S/2\pi m)$ with the area $S=\pi k_F^2$). We find that even if $\mu$ is inside the gap (blue dashed line) or in the flat band region (red line), there are well-defined QO that are directly proportional to the area picked out by the intersection of the unhybridized bands (here directly proportional to $W/\hslash {\omega }_c$).

Figure 2

Temperature dependence of the damping factor $R(T)$. In (i) we show, for $(\gamma /\hslash {\omega }_c)=0.7$ and for different values of the chemical potential, $\mu =W=\delta \mu$, parametrized by $\delta \mu /\gamma$. Dashed lines are calculated from the approximate $R_0(T)$ in which $\mathrm{\Gamma }$ is replaced by ${\mathrm{\Gamma }}_0=\mathrm{\Gamma }(n=0)$. The inset (ii) shows the same for a different value $(\gamma /\hslash {\omega }_c)=0.2$. In this case $R_0(T)$ always coincides with the exact $R(T)$. Note that for large values of $\delta \mu /\gamma$, the standard Lifshitz-Kosevich behavior $R_{\mathrm{LK}}$ (red dashed line) is quickly approached.

Figure 3

QO oscillations for a tight-binding lattice model with $W=-2.3$ and $\gamma =0.2$ (all energies in units of $t$). (i) The periodicity of the oscillations varies with chemical potential $\mu$. In the legend, we state the QO period $F$ in terms of an equivalent area in $k$ space, $S^{\mathrm{QO}}=2\pi eF/\hslash$, and also state the Fermi surface area, $S^{\mathrm{FS}}$. (These areas are expressed in units of the BZ area.) As long as $\mu$ is far from the gap (black dashed and dot dashed lines), these areas nicely coincide, indicating standard LK behavior. If $\mu$ is inside or close to the gap (green, blue, red dashed lines), we find anomalous QO as before, with the period $F_W$ set by the crossing with the flat band, for which $S^{\mathrm{QO}}=0.152$. In the inset (ii), we extracted the temperature dependence $R(T)$ by calculating the difference between a consecutive minimum and maximum of $M$ as a function of temperature, which confirms the analytical behavior of Eq. (2) (compare to Fig. 2).