Avoided level crossing at the magnetic field induced topological phase transition due to spin-orbital mixing

In three-dimensional topological insulators, an effective closure of the bulk energy gap with increasing magnetic field expected at a critical point can yield a band crossing at a gapless Dirac node. Using high-field magneto-optical Landau-level spectroscopy on the topological crystalline insulator $\mathrm{P}{{\mathrm{b}}_1}_{-x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$, we demonstrate that such a gap closure does not occur, and an avoided crossing is observed as the magnetic field is swept through the critical field. We attribute this anticrossing to orbital parity and spin mixing of the $N=0$ levels. Concurrently, we observe no gap closure at the topological phase transition versus temperature suggesting that both are due to a single, likely extrinsic, mechanism.

Figure 1

(a) Topological phase transition from trivial to topological with the critical gapless state shown in gray. The blue shade indicates the topological regime. (b) Evolution of the bulk $N$ = 0 Landau levels as a function of magnetic field for a topological system. The color indicates the $L_6^{\pm ,\downarrow ,\uparrow }$ orbital character of the $N$ = 0 levels. (c) Zoom-in on the region at $k_z=0$ showing crossing LL with pure $L_6^{\pm }$ character in blue and red and the anticrossing LL with mixed $L_6^{\pm }$ character in black. (d) Magneto-optical transmission [$T$($B$)/$T$(0)] spectra measured in $\mathrm{P}{{\mathrm{b}}_1}_{-x}\mathrm{S}{\mathrm{n}}_x\mathrm{Se}$ ($x=$ 0.19) versus magnetic field between 2 and 34 T at 1.6 K. Red curves mark magnetic fields that are multiple of 5 T and the curves are arbitrarily shifted vertically for clarity. (e) Zoom-in on the low-energy region between 20 and 34 T highlighting the behavior of three particular transitions labeled $A_1, A_2$, and ${A_2}^{\prime }$. A multipeak fit shown in green is used to separate ${A_2}^{\prime }$ and $A_2$ at high fields.

Figure 2

(a) LL energy versus magnetic field computed using Eq. (1). The effective spin-“up” (-“down”) levels are plotted in red (blue). The hybridized $N$ = 0 levels are plotted with a color gradient that illustrates their spin-orbital mixing. $B_c$ is the critical field. The $A_1,A_2,{A_1}^{\prime }$, and ${A_2}^{\prime },$ transitions are shown as full or broken arrows in red and blue. The blue shade indicates the topological regime. (b) Magneto-optical transition fan chart at low energies up to $B$ = 34 T. Red dots correspond to the $A_1$ absorption data points, the full black ones to $A_2$, and the empty circles to ${A_2}^{\prime }$. The solid red and blue curves correspond to the calculated variation of $A_1$ and ${A_1}^{\prime }$ and the dashed red and blue curves to that of $A_2$ and ${A_2}^{\prime }$, respectively. The gray dots and lines are data points and curve fits using Eq. (2) for transitions not involving the $N$ = 0 levels ($N>1$). The green line represents the cutoff of the ZnSe Grenoble probe window.

Figure 3

(a) Relative amplitude of $A_2$ to $A_1$ (empty circles) compared with the relative oscillator strength (solid lines) calculated using Eq. (5) for $T$ = 1.6, 40, and 60 K for $\mathrm{P}{\mathrm{b}}_{0.81}\mathrm{S}{\mathrm{n}}_{0.19}\mathrm{Se}$. (b) Energy gap versus temperature for $\mathrm{P}{\mathrm{b}}_{0.81}\mathrm{S}{\mathrm{n}}_{0.19}\mathrm{Se}$. Empty circles are data points and the solid line is calculated using Eq. (6). (c),(d) Magneto-optical spectra measured between 15 and 16 T for $T$ = 40 and 60 K, respectively. The $A_1$ and $A_2$ transitions are highlighted. A Gaussian-broadened multipeak fit to the data is shown in black.