Modification of the Lifshitz-Kosevich formula for anomalous de Haas–van Alphen oscillations in inverted insulators

Traditionally, quantum oscillations were interpreted as the hallmark of a Fermi surface. Recently, it was understood that in order to have well defined de Haas–van Alphen oscillations the presence of a Fermi surface is not required. Following the Luttinger approach, we investigate the oscillations of an inverted insulator. Clear quantum oscillations are found and we provide a formula for the Lifshitz-Kosevich amplitude both in the clean as well as weakly disordered regime. We find that the LK formula is strongly modified compared to systems with a Fermi surface.

Figure 1

Sketch of the spectrum of an inverted insulator as described by Eq. (2). It shows a cut along an arbitrary direction in the two-dimensional momentum space. The energy $E$ is plotted on the vertical axis and the momentum on the horizontal axis. We furthermore allow for a chemical potential $\mu$ and all states below it are occupied at zero temperature. The system exhibits a gap of size $\delta$.

Figure 2

Amplitude function ${\overline{\mathrm{\Omega }}}_m$ as a function of the dimensionless temperature $\overline{T}$. We observe that the curves decay exponentially for $\overline{T}>\overline{\delta }$ and flatten out for $\overline{T}<\overline{\delta }$. Also, finite chemical potential $\overline{\mu }$ modifies the result only slightly.

Figure 3

Amplitude function ${\overline{\mathrm{\Omega }}}_m$ as a function of the dimensionless gap $\overline{\delta }$. We clearly see that for various values of temperature whenever $\overline{\delta }>\overline{T}$ the amplitude converges to the same curve characterized by the gap only (that part of the curve also does not depend on the chemical potential as long as $\overline{\mu }<\overline{\delta }$).

Figure 4

Contour plot of the amplitude ${\overline{\mathrm{\Omega }}}_m$ as a function of dimensionless gap $\overline{\delta }$ and dimensionless temperature $\overline{T}$ for $\overline{\mu }=0$. If $\overline{\mu }\ne 0$, the upper side of the graph would not be modified much, whereas the lower side of the graph would be.

Figure 5

Plots of ${\overline{\mathrm{\Omega }}}_m$ for different values of $\overline{T}$ but fixed $\overline{\delta }$. We have adjusted the scale to visualize all three functions in one plot because of the severe damping in the large-temperature case. We observe that all three functions oscillate with a period $2\pi$ for large $\overline{\mu }$; however, as $\overline{\mu }$ hits the gap $\overline{\delta }$, the function either keeps oscillating (high temperature $\overline{T}$ case) or goes to a constant (low temperature case).

Figure 6

Validities of formulas Eq. (23) (on the left) and Eq. (17) (on the right) as approximations to formula Eq. (15). The darker region represents a faithful approximation according to the $2\%$ criterion.

Figure 7

Validities of formulas Eq. (27) (on the left) and Eq. (28) (on the right) as approximations to formula Eq. (15).

Figure 8

Comparison of the amplitude function Eq. (15) as a function of temperature (setting the strength of disorder to zero) and as a function of disorder strength (setting the temperature to zero).