When all the carriers were confined to the lowest Landau level, the QL is reached. In recent work on ZrTe5 crystals grown via chemical vapor transport using iodine as the transport agent, the carrier density is reported to be 1017 to 1018/cm3 with an SdH period of 3 to 5 T, which means that the QL can be reached in 3 to 5 T (23, 24). Using Te-flux method, a lower magnetic field (about 1 T) is needed to drive the compound into its QL (31). In our growth conditions, the ZrTe5 crystals have the desired stoichiometry and show very low carrier densities (22). According to the Onsager relation, our high-quality ZrTe5 samples with the much lower densities should show a smaller SdH period and, simultaneously, the critical field when the system enters the QL is smaller. We usually judged whether a system enters the QL by analyzing its SdH effect. In our samples, it is hard to extract the SdH oscillations that are merged into the sharp increase of MR around 0 T. However, we could estimate the QL magnetic field Bc for our ZrTe5 crystal based on the carrier density. The critical field at which the system enters the QL field can be estimated by the formula (44) Embedded Image, where the carrier density is in cm−3, ma and mc are masses perpendicular to the magnetic field, and mb is the mass along the field.

The very low carrier density and the strong anisotropic property (fig. S5) of our bulk ZrTe5 crystals indicate a very small value of the critical field. Assuming the anisotropy of the carrier mass is constant for a specific material, one obtains the critical field Embedded Image. In our calculation, the anisotropic masses for carriers reported in previous literatures are referred. It is estimated that the QL magnetic field for our crystals is about 0.2 T, which is rather small. Thus, the observed oscillations are beyond the QL, which also excludes the SdH effect as the underlying mechanism. Besides, the investigation on the ground state of a 3D electronic gas system beyond the QL is a long-standing research subject (30, 3841). Our work also provides an exciting playground to explore new physics beyond the QL.

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