Interaction effects in ultra-quantum bismuth

Bismuth is an unusual material in elemental form. It is a semi-metal, with very small electron and hole Fermi surfaces occupying a fraction 10^-5 of the Brillouin zone - an order of magnitude smaller even than graphite. Though it has a long history (for instance, the Seebeck, Nernst, Shubnikov-de Haas, and de Haas-van Alphen effects we all discovered in Bi), it still is a fascinating subject of study. Recently, experimentalists have return to the study of Bi in high magnetic fields, strong enough that the carriers become confined to the first few Landau levels. The electron and hole motions then become quasi-one-dimensional, with particles moving freely only along the field direction. Indeed, a resistivity anisotropy of 10⁶ is observed between the field direction and perpendicular to it, despite the fact that the crystal is nearly cubic (it has a rhombohedral crystal structure very close to diamond).

In this ultra-quantum regime, anomalies have been observed in the Nernst coefficient and Hall resistivity by Kamran Behnia's group in Paris, and in torque magnetometry by N. Phuan Ong's group at Princeton. The latter features form a complex network of curves in the plane of field magnitude and angle of the field with respect to the trigonal axis. Moreover, at the highest fields studied, hysteretic apparently first order behavior occurs along these curves.

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Our work:

Jason Alicea (a postdoc at Caltech) and I have derived an effective low energy model for the electrons and holes in high fields, based on an established tight-binding Hamiltonian for Bi. Notably, the electrons are described by a massive 3+1-dimensional Dirac Hamiltonian, with a non-trivial Zeeman coupling. The Landau level structure of our model agrees well with the field-angle diagram observed by the Ong group. Introducing interactions, we argue that instabilities toward charge density wave and Wigner crytal ordering occur at high fields, and are especially strong along the curves in the Ong diagram, at which a Landau level is becoming depopulated. Analytic calculations are carried out using functional renormalization group techniques. We propose that the hysteretic features in the experiments are related to first order Wigner solid transitions at low temperatures.

Read the paper.