Unconventional Hall response in the quantum limit of HfTe5

Interacting electrons confined to their lowest Landau level in a high magnetic field can form a variety of correlated states, some of which manifest themselves in a Hall effect. Although such states have been predicted to occur in three-dimensional semimetals, a corresponding Hall response has not yet been experimentally observed. Here, we report the observation of an unconventional Hall response in the quantum limit of the bulk semimetal HfTe5, adjacent to the three-dimensional quantum Hall effect of a single electron band at low magnetic fields. The additional plateau-like feature in the Hall conductivity of the lowest Landau level is accompanied by a Shubnikov-de Haas minimum in the longitudinal electrical resistivity and its magnitude relates as 3/5 to the height of the last plateau of the three-dimensional quantum Hall effect. Our findings are consistent with strong electron-electron interactions, stabilizing an unconventional variant of the Hall effect in a three-dimensional material in the quantum limit.

In these measurements, the electrical current is applied along the x-axis with the magnetic field set along the x, y, and z directions. The results of our analysis are summarized in Table S1. For all directions, we observe single frequencies BF,i, as shown in Fig. 1 Assuming that the massive Dirac band exhibits a linear dispersion at low energies, we finally can obtain the effective masses m * from the cyclotron masses in the x, y and z direction: mc,x = * * , mc,y = * * and mc,z = * * , respectively. The Fermi velocities vF,i can be further obtained with vF,i m * i = ℏ kF,i. Eventually, the average Fermi energy can be estimated using EF = (vF,x 2 ℏ 2 kF,x 2 + vF,y 2 ℏ 2 kF,y 2 + vF,z 2 ℏ 2 kF,z 2 ) 0.5 . For sample A we obtain EF = (9 ± 2) meV, where the deviation is obtained from the error of the fits in kF,i and vF,i.

Supplementary note 3: Comment on the calculation of the Hall conductivity tensor element
We calculate the Hall conductivity tensor element xy using xy = xy/(xx 2 +xy 2 ), assuming that xx = yy. However, in general xy = xy/(xxyy +xy 2 ) with a magnetic field in z-direction. Due to the geometry of the HfTe5 crystals (elongated needles) and its mechanical fragility, performing reliable measurements of yy on our samples is not possible. Instead, we estimate the error of the xy using the ratio of Drude resistivities yy/xx = (nxye 2 x/ * )/ (nxye 2 y/ * ) estimated from the quantum lifetimes and effective masses obtained from Shubnikov-de Haas 5 oscillations on sample A, given in Supplementary Table S1. nxy is the charge-carrier concentration in the x-y-plane. Based on this analysis we find yy/xx ≈ 0.4, which results in an error of 2 % in the estimated xy at the 3/5 plateau, owing to xx (B) < xy (B). Both these errors lay well within the estimated error of kF,z of 10 %. Therefore, the Hall plateaus in xy are expected to be observable in the Hall plateaus in xy. Fixed chemical potential vs. fixed particle number The electric response of a Hall sample can take qualitatively different forms depending on whether an experiment is performed at fixed particle number or at fixed chemical potential, i.e. S6a-c, the quantum limit approximately corresponds to 3 T < |B| < 9 T. Above 9 T, also the zeroth Landau-level shifts above the chemical potential (its low-energy tail results in a small residual electron density that disappears when |B| increases further).
8 (ii) Fixed particle number In samples containing many electrons, a change of the electron density costs a large charging energy. 4 If such a large sample is only weakly coupled to leads, the particle number rather than the chemical potential is kept fixed when the magnetic field is varied. This in turn requires the chemical potential to vary as a function of magnetic field. Each time a Landau-level bottom crosses the varying chemical potential, the chemical potential exhibits a peak. Using that the Landau-level degeneracy is given by , one furthermore finds that the Hall conductivity is given by In a system with fixed particle number, the inclusion of level broadening smoothens out the dependence of the chemical potential on the magnetic field. As a result, the Hall response is slightly closer to the Hall response of a system with fixed chemical potential: we find that the Hall conductivity exhibits small kinks once level broadening is considered.
(iv) Specifics of our samples In general, a real sample will be in between the two extreme cases of fixed conduction electron (charge carrier) number and fixed chemical potential (Fermi level). This is also apparent from the behavior our samples. At small fields, the Hall response shows slightly smoothened kinks, followed by plateau-like features. This behavior is similar 9 to the theoretical expectations for a system at fixed chemical potential. We note that our samples are not very large, and that the Dirac pocket is comparably small. Furthermore, at magnetic fields below the quantum limit, the variation of particle density in our toy model at exactly fixed chemical potential is at most about 20%. Finally, the chemical potential does not have to be perfectly conserved for the Hall data to exhibit kinks and plateau-like features: an ideal 1/|B|-behavior arises only if the chemical potential adjusts perfectly, and if the level broadening is small. Our data, therefore, suggests the presence of localized states that can soak up/release some amount of conduction electrons in order to keep the chemical potential of the sample closer to the chemical potential in the leads.
At fields beyond the quantum limit and temperatures at which the = 3/5-plateau has not yet developed; the Hall conductivity instead follows a 1/|B|-behavior. We interpret this behavior as the system's tendency to avoid large changes of the conduction electron density. Finally, the electrons in our samples form a (gapped) Dirac semimetal rather than a quadratic dispersion. As a consequence, the spectrum is particle-hole symmetric.