## Introduction

Magnetic fields are an important experimental tuning knob for dimensional reduction. This happens at sufficiently strong fields when all electrons are confined to the lowest Landau level—a regime known as the ultra-quantum limit (UQL)1. Given that the Landau bands in three-dimensional (3D) systems only disperse with the momentum parallel to the field, electrons within the UQL form a liquid occupying only a single quasi-1D band. Such liquids are known to be on the verge of instabilities and even small perturbations can cause them to undergo a phase transition2. It was consequently predicted that a 3D electron gas exposed to high enough magnetic fields can exhibit a plethora of exotic ground states such as Wigner crystals or exotic varieties of charge and spin density waves that could exhibit a quantised Hall effect3,4,5,6.

In most materials, the quantum limit cannot be reached since it requires too large magnetic fields. Certain low-density semimetals, however, have been shown to reach the UQL at experimentally accessible fields. With some of the the most prominent examples being graphite7, bismuth8, NbP9, and TaAs10 (quantum limits in the range 10–30 T), but a thorough understanding of their field-induced phases remains a difficult task due to the intrinsic complexities of those materials. Searching for a simpler material, ideally tuneable in situ, in which the UQL can be reached by available magnetic fields is, therefore, an important goal.

The recent surge in both experimental and theoretical studies of Dirac and Weyl semimetals has also sparked renewed interest in field-induced effects in those systems. One prominent example is the pentatelluride material family of ZrTe5 and HfTe5. These compounds are close to the transition between 3D strong and weak topological insulator (TI)11. ZrTe5 has, therefore, recently risen to prominence as an excellent solid-state realisation of the (weakly gapped) 3D Dirac Hamiltonian12,13,14,15,16,17. Because the chemical potential typically lies outside the gap, ZrTe5 in practice is a diamagnetic metal18. It shows no signatures of magnetic interactions and is in that sense a fairly “clean” electronic material.

In agreement with a Dirac model, at low temperatures our samples exhibit a single electron-type Fermi surface at the gamma point comprising of less than 1% of the Brillouin zone, and a charge-carrier concentration in the range 1016−1017 cm−3.15 Such small charge-carrier densities enable entry into the quantum limit at magnetic fields of the order of 1 T, enabling, to the best of our knowledge, the study of a system deeper in the quantum limit than in previously studied 3D materials. In addition, recent studies of ZrTe5 and HfTe5 have revealed that, due to the exceptional purity of the available samples (electron mobilities of the order of 400,000 cm2/Vs15) and simplicity of the band structure, most physical low-field (B < 2 T) properties can be described with simple linear-response calculations down to 2 K. As such, both ZrTe5 and HfTe5 are excellent model systems for the study of Dirac electrons deep in the ultra-quantum limit15.

Previous studies of interactions in ZrTe5 in magnetic fields, done by various groups, found indications of both interaction-driven gaps and samples that appear gapless. Tang et al., for example, identified a metal-insulator transition at about 4 T in samples with a small charge-carrier density with a quantum limit at 1.2 T19. This transition is associated with a quantisation of the Hall conductivity, which the authors identify as a possible realisation of the 3D quantum Hall effect4. Data with similar appearance, although with higher charge-carrier density and thus enhanced magnetic-field scales, have been interpreted by Liu et al. as evidence for an interaction-driven mass generation (density wave formation)20. In contrast, our own previous experiments have shown evidence for a gapless electronic system without density waves, but still exhibiting a quasi-quantised Hall conductivity15. One possible explanation for this discrepancy is that ZrTe5 might actually be close to interaction-driven instabilities, with sample and experiment-specific variations pushing the system to one state or the other.

This motivated us to further characterise the properties of ZrTe5 in high magnetic fields. To that end, we have performed magnetoresistance and ultrasound-propagation experiments in pulsed magnetic fields up to 64 T on two samples entering the quantum limit at 1.2 and 0.6 T, respectively, with the field applied along the crystallographic b-axis. As is well known from highly oriented graphite (HOPG), the combination of transport with ultrasound measurements is particularly helpful for identifying phase transitions. In HOPG, measurements of sound propagation and attenuation for example revealed a cascade of well-pronounced peaks signalling a series of field-induced electronic phase transitions, while the magnetoresistance of the same samples merely showed a broad anomaly with small kinks at the phase transitions - on the edge of the experimental resolution7.

We find that the presently analysed samples do not undergo any thermodynamic phase transition. Instead, the Zeeman effect in conjunction with the massive Dirac nature of electrons in ZrTe5 lead to a field-induced Lifshitz transition. The latter results in the formation of two 1D Weyl points. In addition, the Hall coefficient suggests that the system can be tuned to exhibit perfect charge neutrality at 25 and 8 T, respectively, in these samples.

## Results and discussion

Our study reports on ZrTe5 samples similar to those studied in refs. 15,16 (charge-carrier density n = 6.1 × 1016 cm−3), grown by the tellurium-flux method. In order to investigate possible magnetic field-induced phases we first performed temperature-dependent magnetoresistance and Hall measurements in pulsed fields up to 64 T at the Dresden High Magnetic Field Laboratory (HZDR). These measurements are shown in Fig. 1a. An important feature of our data is a rapid increase of the longitudinal resistance once the sample enters the quantum limit, which then almost saturates for fields above ca. 35 T. These observations agree with previous measurements and would also be consistent with the onset of a metal-insulator-transition. Indeed, the resistance at 60 T increases by almost 50.000% as compared to the zero-field value. Also, the Hall data seems to be consistent with the opening of a spectral gap due to the formation of a charge density wave (CDW). Our measurements of the high-field Hall effect, shown in Fig. 1a, exhibit a striking change of the Hall coefficient above 10 T, with the Hall resistivity beginning to decrease with increasing field to the point where it changes sign at about 25 T. Here we have taken the convention that the positive Hall resistance is electron-like for positive fields. Similar behaviour has been previously reported in a range of CDW materials with field-induced transitions21,22 and in the Weyl semimetal TaP where the change of sign of the Hall effect has been to attributed to annihilation of Weyl nodes and opening a spectral gap23.

However, as outlined above, transport alone can be insufficient to prove or disprove the existence of a density wave. Therefore, we have performed additional ultrasound-propagation experiments in pulsed magnetic fields up to 56 T. Such measurements provide an independent, and usually very sensitive, test for bulk phase transitions: the velocity of sound propagating through a crystal is proportional to the elastic modulus and thus a thermodynamic quantity24. Changes of the sound velocity reflect changes to the overall free energy of the coupled electron-phonon system. Sudden changes in the free energy, such as those caused by a gap opening related to a phase transition, should, therefore, have a significant effect on the sound velocity, especially when accompanied by breaking of translational invariance such as in the case of charge and spin density waves25,26.

Figure 1b shows the field dependence of the sound velocity of a transverse phonon mode polarised along the b-axis and of a longitudinal phonon, both propagating along the crystallographic a-axis with magnetic field parallel to the b-axis. The field dependence of the speed of sound exhibits a softening of both elastic modes at fields above 4 T and saturates above about 15 T, with a shallow minimum observable at about 10 T in the case of the longitudinal mode. Unlike what would be expected for a phase transition, however, both curves appear smooth without any sharp features that could be interpreted as the onset of such a transition.

This finding prompts us to reconsider our transport data measured at high fields. To avoid the complication that ρxx is a mixture of longitudinal and transverse conductivities, we have inverted the resistance tensor and extracted the longitudinal conductivity σxx(B) and Hall conductivity σxy(B). Figure 1c shows the longitudinal conductivity on 1/B. The data exhibits well-known quantum oscillations at low fields, with the material reaching the quantum limit at ca 1.2 T. This agrees with previous measurements15. Interestingly, at around 7 T, thus deep in the quantum limit, an unusual additional peak in the longitudinal conductivity occurs, suggesting a field-induced enhancement of the density of states. In addition, the occurrence of the peak coincides with the onset of a decrease in the Hall resistance. After continuing to decrease, σxy(B) eventually changes sign at B ≈ 25 T, which signals a change from electron-type charge carriers to hole-type carriers.

As a starting point to understand all of the above features, in particular the peak appearing in the high-field conductivity, we have investigated a toy model of the low energy spectrum of ZrTe5. The model is composed of all symmetry-allowed terms of the low-energy Hamiltonian up to quadratic order in momentum15,27,28. The parameters of this model—such as the Fermi velocity, effective mass, g-factor and Dirac mass gap—are set by the values extracted from our experiments whenever possible, while others are expected to be small in ZrTe5. In line with earlier studies, we assume the charge density in our samples is not conserved as magnetic field strength is altered15 and instead make the approximation that chemical potential is fixed (for a more detailed discussion, see ‘Methods’ and ref. 15).

Figure 2a shows the magnetic-field-induced evolution of the low-energy band structure upon including the two symmetry allowed Zeeman terms27. An important effect of the magnetic field is the closing of the Dirac gap, which we find happens already at a moderate field strength of about 2 T. Utilising the parameters relevant for our ZrTe5 samples15 and the sign of the g-factor found using ab initio techniques29, our model is consistent with a negative sign of the mass gap, which implies a strong topological insulator (TI) nature of ZrTe511 at low temperatures. Further increasing the magnetic field up to about 7 T enhances the energy of the hole band to a point where it crosses the chemical potential and introduces hole-like carriers. For the Hall conductivity, our theory therefore predicts a maximum around the same field strength, which is in good qualitative agreement with our experimental data. As the magnetic field is increased further, the system enters a regime with linearly crossing bands realising a 1D Weyl-band structure. The position of the chemical potential with respect to the 1D Weyl nodes is directly dependent on the applied magnetic field. As a result, the occupation of the hole band is enhanced while the occupation of the electron band decreases. Eventually the system enters a charge-neutral state at about 24 T, at this point the majority charge-carrier type changes from electrons to holes and the Hall resistance is predicted to change sign, which again qualitatively agrees with experiment. A comparison of the calculated and experimentally determined σxy(B) is shown in Fig. 2d. Most striking is the comparison of the calculated DOS and longitudinal conductivity from our experiment plotted as a function of 1/B together with the sound attenuation of the longitudinal phonon mode (Fig. 2b, c, e). At low fields below 2 T, clear signatures of quantum oscillations are visible in both the calculated DOS and measured quantities. Most importantly, the additional peak seen in σxx(B) at high fields is well reproduced by the calculated DOS, and also the change in sound attenuation Δα(B) shows a pronounced feature close to the same field. Within our theory, we can naturally interpret this as arising when the maximum of the hole Landau band crosses the Fermi energy. Full quantitative agreement is not expected since our simple single-particle theoretical model neglects parts of the magnetic field-dependence of our model parameters. In particular, we assume that chemical potential is entirely fixed for all field strengths, which likely is not true at fields above ca. 2 T (in this regime, both chemical potential and particle number should exhibit a non-trivial field dependence30). However, the strong qualitative agreement of our simple model, explaining all UQL features found in our experiment, strongly supports its relevance to the experiment. Future experiments should nevertheless aim at independently determining the particle density as a function of magnetic field in order to improve on our modelling.

To further test our picture, we have performed magnetotransport measurements on a sample (Sample C) with the lower charge-carrier density n = 3.8 × 1016 cm−3, entering the quantum limit at about 0.6 T. As a result of the lower Fermi level, one expects that the hole band should cross the chemical potential at lower fields and also that the sign change of the Hall coefficient at the charge neutrality point should occurs at lower fields than in our high-density sample. Results of our measurements on this low charge-carrier-density sample are shown in Fig. 3. Figure 3a displays the field dependence of the longitudinal magnetoresistance and Hall effect of Sample C measured at 2 K. In agreement with our expectation, we find the Hall coefficient is maximum already at a field of 5 T, and the charge neutrality point occurs at a field strength of just 8 T, a magnetic field that is easily accessible with in-house magnets. In addition, comparison of theoretical predictions of σxy(B) and the DOS are in good qualitative agreement with experiments, similar to the samples with higher charge-carrier density.

Our findings demonstrate that ZrTe5 is an excellent testbed for the largely unexplored regime of the ultra-quantum limit, which opens the door to a systematic analysis of interaction-driven instabilities at large magnetic fields and low temperatures. Our experiments, for example, reveal that a strong Zeeman splitting drives a field-induced Lifshitz transition that transforms the low-energy bands of ZrTe5 into the ones of a 1D Weyl Hamiltonian. In this regime, the distance between the chemical potential and the 1D Weyl nodes is directly tuneable by the magnetic field. This opens an avenue for the investigation of interactions of 1D Weyl Fermions in the vicinity of charge neutrality with band filling being controlled via the external magnetic field. Depending on the sample, this charge neutral state occurs already around 8 T and is therefore easily accessible with commercially available magnets.

## Methods

### Sample synthesis and preparation

High-quality single-crystal ZrTe5 samples were synthesised with high-purity elements (99.9% zirconium and 99.9999% tellurium), the needle-shaped crystals (about 0.1 × 0.3 × 20 mm3) were obtained by the tellurium-flux method. The lattice parameters of the crystals were confirmed by single-crystal X-ray diffraction. Prior to transport measurements, Pt contacts were sputter deposited on the sample surface to ensure low contact resistance. The contact geometry was defined using Al hard masks. Prior to Pt deposition, the sample surfaces were Argon etched and a 20-nm Ti buffer layer was deposited to ensure good adhesion of the contacts. Deposition was conducted using a BESTEC UHV sputtering system. This procedure allowed us to achieve contact resistance of the order of 1–2 Ohm.

### Sample environment

The pulsed magnetic field experiments up to 64 T were carried out at the Dresden High Magnetic Field Laboratory (HLD) at HZDR, a member of the European Magnetic Field Laboratory (EMFL). All transport measurements up to ±9 T were performed in a temperature-variable cryostat (PPMS Dynacool, Quantum Design).

### Electrical-transport measurements

The longitudinal ρxx and Hall resistivity ρxy were measured in a Hall-bar geometry with standard lock-in technique (Zurich Instruments MFLI and Stanford Research SR 830), with a frequency selected to ensure a phase shift below 1 degree—typically in the range between 10 and 1000 Hz across a 100 kΩ shunt resistor. In addition, some measurements were measured using a Keithley Delta-mode resistance measurement set-up for comparison. In both measurement modes, the electrical current was always applied along the a-axis of the crystal and never exceeded 100 μA in order to avoid self-heating.

### Ultrasound propagation measurements

Ultrasound measurements in pulsed magnetic fields up to 56 T were performed using a phase-sensitive pulse-echo technique. Two piezoelectric lithium niobate (LiNbO3) resonance transducers were glued to opposite parallel surfaces of the sample to excite and detect acoustic waves. The sample surfaces were polished using a focused ion beam in order to ensure that the transducers were attached to smooth and parallel surfaces. The longitudinal and transverse acoustic waves propagated along the a-axis with the transverse polarisation vector along the c-axis. Relative sound-velocity changes Δv/v, and the sound attenuation Δα, were measured for field applied along the b-axis. The longitudinal and transverse ultrasound propagation were measured at 28 and 313 MHz, respectively.

### Theoretical model for Hall effect and Landau quantisation in ZrTe5

To model our ZrTe5 samples we use a low-energy Hamiltonian that includes all symmetry-allowed terms of the Cmcm group up to quadratic order in momentum15,28. We do not consider the small additional symmetry-breaking terms, which can result in a non-centrosymmetric phase of ZrTe5 recently discovered in samples with low temperature resistivity maximum28. As such, near the Γ-point, our model reads:

$${{{{{\boldsymbol{H}}}}}}=\left({{{{{\boldsymbol{m}}}}}}+{\sum }_{{{{{{\boldsymbol{i}}}}}}={{{{{\boldsymbol{a}}}}}},{{{{{\boldsymbol{b}}}}}},{{{{{\boldsymbol{c}}}}}}}{{{{{{\boldsymbol{A}}}}}}}_{{{{{{\boldsymbol{i}}}}}}}{{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{i}}}}}}}^{{{{{{\bf{2}}}}}}}\right){{{{{{\boldsymbol{\tau }}}}}}}^{{{{{{\boldsymbol{z}}}}}}}+{{\hslash }}\left({{{{{{\boldsymbol{v}}}}}}}_{{{{{{\boldsymbol{a}}}}}}}{{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{a}}}}}}}{{{{{{\boldsymbol{\tau }}}}}}}^{{{{{{\boldsymbol{x}}}}}}}{{{{{{\boldsymbol{\sigma }}}}}}}^{{{{{{\boldsymbol{z}}}}}}}+{{{{{{\boldsymbol{v}}}}}}}_{{{{{{\boldsymbol{c}}}}}}}{{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{c}}}}}}}{{{{{{\boldsymbol{\tau }}}}}}}^{{{{{{\boldsymbol{y}}}}}}}+{{{{{{\boldsymbol{v}}}}}}}_{{{{{{\boldsymbol{b}}}}}}}{{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{b}}}}}}}{{{{{{\boldsymbol{\tau }}}}}}}^{{{{{{\boldsymbol{x}}}}}}}{{{{{{\boldsymbol{\sigma }}}}}}}^{{{{{{\boldsymbol{x}}}}}}}\right)+{\sum }_{{{{{{\boldsymbol{i}}}}}}={{{{{\boldsymbol{a}}}}}},{{{{{\boldsymbol{b}}}}}},{{{{{\boldsymbol{c}}}}}}}{{{{{{\boldsymbol{B}}}}}}}_{{{{{{\boldsymbol{i}}}}}}}{{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{i}}}}}}}^{{{{{{\bf{2}}}}}}}-{{{{{\boldsymbol{\mu }}}}}},$$
(1)

where a, b, c refers to crystalline directions, vi are the Dirac velocities in these directions, m is the Dirac mass, Ai and Bi are the prefactors for allowed terms quadratic in momentum, and the Pauli-matrices τi and σi represent the orbital and spin degrees of freedom, respectively. We neglect the Bi terms, which are expected to be small for ZrTe5 since it is a Dirac semimetal. When a magnetic field, B, is applied parallel to the b-axis the dispersion of the lowest Landau level is given by26

$${{{\boldsymbol{\varepsilon }}}}\left({{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{b}}}}}\right)=\pm \sqrt{{\left({\frac{{{{\bf{1}}}}}{{{{\bf{2}}}}}{{{\boldsymbol{\mu }}}}}_{{{{{\boldsymbol{B}}}}}}{{{\boldsymbol{gB}}}}+{{{{\boldsymbol{A}}}}}_{{{{\boldsymbol{b}}}}}{{{{{\bf{k}}}}}_{{{{\bf{b}}}}}}^{{{{\bf{2}}}}}+{{{\boldsymbol{m}}}}\right)}^{{{{\bf{2}}}}}+{{{{\boldsymbol{v}}}}}_{{{{\boldsymbol{b}}}}}^{{{{\bf{2}}}}}{{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{b}}}}}^{{{{\bf{2}}}}}}-{{{\boldsymbol{\mu }}}}-{\frac{{{{\bf{1}}}}}{{{{\bf{2}}}}}{{{\boldsymbol{\mu }}}}}_{{{{\boldsymbol{B}}}}}\bar{{{{\boldsymbol{g}}}}}{{{\boldsymbol{B}}}},$$
(2)

where g and $$\bar{{{{{{\boldsymbol{g}}}}}}}$$ are the g-factors corresponding to the two symmetry-allowed Zeeman contributions $${{{{{{\bf{H}}}}}}}_{{{{{{\bf{z}}}}}}}{{{{{\boldsymbol{=}}}}}}{{{{{\boldsymbol{-}}}}}}{\frac{{{{{{\bf{1}}}}}}}{{{{{{\bf{2}}}}}}}{{{{{\boldsymbol{\mu }}}}}}}_{{{{{{\boldsymbol{B}}}}}}}{{{{{\bf{g}}}}}}{{{{{\boldsymbol{B}}}}}}{{{{{{\boldsymbol{\sigma }}}}}}}_{{{{{{\boldsymbol{z}}}}}}}$$ and $${{{{{{\bf{H}}}}}}}_{\bar{{{{{{\bf{z}}}}}}}}{{{{{\boldsymbol{=}}}}}}{{{{{\boldsymbol{-}}}}}}{\frac{{{{{{\bf{1}}}}}}}{{{{{{\bf{2}}}}}}}{{{{{\boldsymbol{\mu }}}}}}}_{{{{{{\boldsymbol{B}}}}}}}\bar{{{{{{\bf{g}}}}}}}{{{{{\boldsymbol{B}}}}}}{{{{{{\boldsymbol{\sigma }}}}}}}_{{{{{{\boldsymbol{z}}}}}}}{{{{{{\boldsymbol{\tau }}}}}}}_{{{{{{\boldsymbol{z}}}}}}}$$, respectively. Previous experiments15,27 on ZrTe5 have shown that vb is very small and so we neglect this term; including such a term would generate a small mass gap in the 1D Weyl dispersion. Further, we assume that Ab < 0, and μ > 0, the evolution of the dispersion described by Eq. (2) is shown in Fig. 2a. We further take the mass gap m < 0, which implies ZrTe5 is in a strong TI phase. the corresponding density of states ρ and Hall conductivity as a function of magnetic field are shown in Fig. 2b. In fact, we can describe all experimentally observed transport features in the ultra-quantum limit by the scenario depicted in Fig. 2. In particular, for fixed chemical potential, at a magnetic field strength $${{{{{{\boldsymbol{B}}}}}}}_{{{{{{\bf{1}}}}}}}{{{{{\boldsymbol{=}}}}}}\frac{{{{{{\bf{2}}}}}}{{{{{\bf{|m|}}}}}}}{{{{{{\bf{g}}}}}}{{{{{{\boldsymbol{\mu }}}}}}}_{{{{{{\bf{B}}}}}}}}$$ the gap at k = 0 closes and for all fields B > B1 the dispersion consists of two 1D Weyl points at $${{{{{{\boldsymbol{k}}}}}}}_{{{{{{\boldsymbol{b}}}}}}}{{{{{\boldsymbol{=}}}}}}{{{{{\boldsymbol{\pm }}}}}}\sqrt{{{{{{\boldsymbol{(}}}}}}{{{{{\boldsymbol{m}}}}}}{{{{{\boldsymbol{+}}}}}}{\frac{{{{{{\bf{1}}}}}}}{{{{{{\bf{2}}}}}}}{{{{{\boldsymbol{\mu }}}}}}}_{{{{{{\boldsymbol{B}}}}}}}{{{{{\boldsymbol{g|B|}}}}}}{{{{{\boldsymbol{)}}}}}}{{{{{\boldsymbol{/}}}}}}{{{{{{\boldsymbol{|A}}}}}}}_{{{{{{\boldsymbol{b}}}}}}}{{{{{\boldsymbol{|}}}}}}}$$. As the magnetic field is further increased the hole band intersects the Fermi level for $${{{{{{\boldsymbol{B}}}}}}}_{{{{{{\bf{2}}}}}}}{{{{{\boldsymbol{=}}}}}}{{{{{\bf{2}}}}}}\frac{{{{{{\boldsymbol{|}}}}}}{{{{{\boldsymbol{-}}}}}}{{{{{\bf{m}}}}}}{{{{{\boldsymbol{+}}}}}}{{{{{\boldsymbol{\mu }}}}}}{{{{{\boldsymbol{|}}}}}}}{\left({{{{{\boldsymbol{g}}}}}}{{{{{\boldsymbol{-}}}}}}\bar{{{{{{\boldsymbol{g}}}}}}}\right){{{{{{\boldsymbol{\mu }}}}}}}_{{{{{{\boldsymbol{B}}}}}}}}$$ resulting in a sharp increase in the density of states—associated with an increase in the longitudinal conductivity—as well as the start of a decrease in the charge-carrier density n. The decreasing charge-carrier density due to the occupation of the hole band eventually results in a sign change of n and, correspondingly, zero Hall conductivity at the field $${{{{{{\boldsymbol{B}}}}}}}_{{{{{{\bf{3}}}}}}}{{{{{\boldsymbol{=}}}}}}\frac{{{{{{\bf{2}}}}}}{{{{{\boldsymbol{\mu }}}}}}}{\overline{{{{{{\boldsymbol{|g|}}}}}}}{{{{{{\boldsymbol{\mu }}}}}}}_{{{{{{\boldsymbol{B}}}}}}}}$$, which is the same field at which the chemical potential intersects the Weyl points of the 1D Weyl dispersion. Using parameters consistent with those found in found in ref. 15, g = 18, m = −1 meV, Ab = −25.6 meV nm2 as well as the new fit parameter g = −10 and μ = 6.8 meV for the sample with high charge-carrier density and μ = 4.5 meV for the sample with low n, which are both consistent with the corresponding sample density as measured via the Hall effect, we can calculate the following fields: For the large-n sample, we estimate BQL ≈ 1.4 T, B1 ≈ 2 T, B2 ≈ 10 T, and B3 ≈ 24 T, whereas for the low-n sample BQL ≈ 0.6 T, B1 ≈ 2 T, B2 ≈ 6.5 T, and B3 ≈ 14 T. We also include a 0.5 meV Lorentzian broadening of the chemical potential. These values are all broadly consistent with the features observed experimentally.