Quantum Oscillation from In-Gap States and a Non-Hermitian Landau Level Problem

Motivated by recent experiments on Kondo insulators, we theoretically study quantum oscillations from disorder-induced in-gap states in small-gap insulators. By solving a non-Hermitian Landau level problem that incorporates the imaginary part of electron’s self-energy, we show that the oscillation period is determined by the Fermi surface area in the absence of the hybridization gap, and we derive an analytical formula for the oscillation amplitude as a function of the indirect band gap, scattering rates, and temperature. Over a wide parameter range, we find that the effective mass is controlled by scattering rates, while the Dingle factor is controlled by the indirect band gap. We also show the important effect of scattering rates in reshaping the quasiparticle dispersion in connection with angle-resolved photoemission measurements on heavy fermion materials.

Figure 1

Schematic band structure of a Kondo insulator. (Inset) Enlargement of the hybridization gap near $k=k_F.\delta (k_F)$ is the hybridization gap defined as the energy difference of the two bands at $k=k_F$. $\delta$ is the indirect band gap.

Figure 2

(a) Spectral function $A(k,\omega )$ for model ${\epsilon }_i=k^2/(2m_i)-{\mu }_i$ ($i=1$, 2), $m_2/m_1=50$, $\delta (k_F)/({\mu }_2-{\mu }_1)=0.02$, ${\mathrm{\Gamma }}_2/\delta (k_F)=0.1$, ${\mathrm{\Gamma }}_1/\delta (k_F)=0.7$, where the electron is spinless. The unit for the color bar is $1/\delta (k_F)$. (b) Same as (a) but with ${\mathrm{\Gamma }}_1/\delta (k_F)=1.7$. (c) Momentum-integrated spectral function $A(\omega )$ for models in (a) (blue, solid) and (b) (red, dashed), and in the clean limit ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2=0$ (black, dotted). The unit for $A(\omega )$ is $m_2$.

Figure 3

(a) Real part of the Landau level spectrum $\mathrm{Re}[{\mathcal{E}}_{n,\pm }^{\downarrow }]$ as a function of magnetic field, with the same model in Fig. 2. The $s=\uparrow$ sector is similar. (b) Same as (a) but with the same model as Fig. 2. (c) Exact numerics of spectral function $A(\omega =0)$ as a function of $1/B$ and $B$ (inset) with the model in (a). Here, both spin sectors are taken into account. The unit for $A(\omega )$ is $m_2$. $m_1=m_e$ is the free electron mass, and the hybridization gap is $\delta (k_F)=2\mathrm{meV}$. The resulting indirect band gap is $\delta =0.56\mathrm{meV}$, and the oscillation frequency is $F=800\mathrm{T}$.