Phase Transitions of Dirac Electrons in Bismuth

The Fermi Surface (FS) in elemental bismuth consists of 3 electron ellipsoids and one hole ellipsoid [1]. The accidental coincidence of the hole and electron caliper areas when the field $\bf H$ is aligned with the trigonal axis $\bf Z$ has long stymied analyses of the quantum oscillations. Because of current strong interest in how electrons with Dirac dispersion behave in intense fields, we have renewed attack on this problem [2] using high-resolution torque magnetometry in fields up to 31 T and at temperatures $T$ down to 300 mK. When $\bf H$ is tilted with respect to $\bf Z$ by a slight angle $\theta$, the torque $\vec{\tau}$ on the sample derived from the 3 electron ellipsoids dominates the torque from the hole FS, allowing the Landau Level crossings of the Dirac electron to be resolved. By measuring the curves of $\vec{\tau}$ vs. $H$ at 19 values of $\theta$ straddling the trigonal axes, we completely resolve the Landau Levels of the Dirac electrons. A new result is the detection of jumps in the transverse magnetization when $H$ exceeds the quantum limit of the electron pockets. By tracing the jumps in the plane of $H$ vs. $\theta$, we uncover a region in which the Dirac electrons enter a new ground state. Within this cone-shaped region, Landau Level anomalies are severely suppressed. We interpret the state as one in which the 3-fold valley degeneracy of the Dirac gas is lifted to form a many-body state. The unusual nature of the magnetization within this region will be described. \\[4pt] [1] M. H. Cohen and E. I. Blount, Phil. Mag. {\bf 5}, 115 (1960). \\[0pt] [2] Lu Li {\it et al.}, Science {\bf 321}, 547 (2008).