Unveiling the double-peak structure of quantum oscillations in the specific heat

Quantum oscillation phenomenon is an essential tool to understand the electronic structure of quantum matter. Here we report a systematic study of quantum oscillations in the electronic specific heat Cel in natural graphite. We show that the crossing of a single spin Landau level and the Fermi energy give rise to a double-peak structure, in striking contrast to the single peak expected from Lifshitz-Kosevich theory. Intriguingly, the double-peak structure is predicted by the kernel term for Cel/T in the free electron theory. The Cel/T represents a spectroscopic tuning fork of width 4.8kBT which can be tuned at will to resonance. Using a coincidence method, the double-peak structure can be used to accurately determine the Landé g-factors of quantum materials. More generally, the tuning fork can be used to reveal any peak in fermionic density of states tuned by magnetic field, such as Lifshitz transition in heavy-fermion compounds.


Introduction
Oscillations of physical properties of materials with magnetic field are powerful tools to reveal the electronic properties of quantum matter.They range from Aharonov-Bohm oscillations [1] in mesoscopic rings, which provide a direct measure of the electron coherence, to quantum oscillations which provide a sensitive and incisive probe of the Fermi surface.
In the latter case, with increasing the magnetic field, the Landau quantisation of the carrier motion gives rise to a series of quantized singularities in the density of states (DOS) that cross the Fermi level, resulting in the oscillatory behaviour of various of physical quantities, such as resistivity (Shubnikov-de Haas effect), magnetic susceptibility (de Haas-van Alphen effect), thermopower and specific heat.
Lifshitz-Kosevich (LK) theory has been widely used to describe these oscillatory phenomena [2][3][4], notably to extract parameters such as the effective mass and Landé g-factor.
Although the theory is remarkably successful in describing quantum oscillations in metals over a wide range of magnetic fields and temperatures, there is growing evidence to suggest that experiment often deviates from the predicted LK behaviour [5][6][7][8][9].At high magnetic fields, the oscillatory magnetoresistance [5][6][7], magnetization [9] and thermopower [8] exhibit a clear departure from LK theory when the systems are pushed towards the quantum limit.
It is natural to expect that a similar departure is also observed in specific heat.However, the oscillatory behaviour of the specific heat in the quantum limit has yet to be fully explored.
In this respect, graphite, in which the quantum limit is reached already at fields as low as 7 T [10], is almost an ideal system for this purpose.
In this study, we report the quantum oscillations of specific heat in natural graphite with temperatures down to 90 mK.Intriguingly, as the field increases and the system approaches the quantum limit, a characteristic double-peak structure appears in the specific heat for magnetic fields corresponding to the expected crossing of an individual spin Landau level and the Fermi energy.This result is in striking contract to the single peak feature predicted in LK theory for the quantum oscillations of specific heat, which is widely used in the literature [11][12][13] (see also Supplementary Note 1).The double-peak structure, which unexpectedly vanishes as T → 0, occurs when a narrow Landau level crosses the thermally broadened edge of the Fermi-Dirac distribution in the vicinity of the Fermi energy.We demonstrate that the double-peak structure in the oscillatory specific heat originates from the kernel term in the detailed functional form of the free electron theory expression for the specific heat [14].A quantitative understanding of the double-peak structure is achieved by the comparison of a DOS model and the Slonczewski-Weiss-McCure (SWM) tight binding Hamiltonian for graphite [15,16].Using graphite as an example, we demonstrate that the double-peak structure provides a new way to accurately determine the g-factor of charge carriers without any assumptions concerning the Landau index or Fermi energy shift, and it can also be extended to other Dirac materials which is crucial in the determination of the Berry phase.Furthermore, the double-peak structure detected here is not restricted to C el in presence of Landau quantisation.It can occur in other probes related to specific heat, such as thermal conductivity, and in any system where a fermionic sea is tuned by the magnetic field such as a Lifshitz transition or in frustrated magnetic materials with fermionic like excitations.

Experimental results
When a Landau level crosses the Fermi energy, the occupation of the Landau level changes rapidly, inducing large changes in the entropy of the system, which can be probed using thermodynamic measurements.The magnetocaloric effect (MCE) measures the sample temperature as a function of applied magnetic field under quasi-adiabatic conditions.In this case, the absolute value of entropy is roughly proportional to the reciprocal sample temperature.
To follow the evolution of the entropy in our graphite sample, we show in Fig. 1a the measured reciprocal sample temperature (1/T ) as a function of the magnetic field taken at an initial temperature of 0.7 K.The field was applied along the c-axis of the graphite crystal for all the measurements presented in this paper.The entropy is proportional to the logarithm of the number of states within the Fermi edge, and therefore shows a maximum when a Landau level is located at the Fermi level, resulting in a series of well-defined single peaks (Supplementary Note 7) in the reciprocal sample temperature labeled as N ± e/h in Fig. 1a.Here, N is the Landau index, e/h indicates if the Landau level originates from the electron or hole pocket, and ± indicate the spin up/down levels.For better comparison, Fig. 1b shows background removed magnetoresistance ∆R xx on natural graphite at 0.5 K.
These results are in stark contrast to the electronic specific heat divided by temperature C el /T which is proportional to the temperature derivative of entropy.The C el /T of the graphite sample taken at a similar temperature (T = 0.5 K) is shown as a function of magnetic field in Fig. 1c.The electronic specific heat C el was obtained by subtracting the phonon contribution from the total specific heat (Supplementary Note 2).Crucially, when low-index Landau levels (N e/h < 3) cross the Fermi energy, C el /T exhibits a series of doublepeak structure, as indicated by the double arrows in Fig. 1c.Our observations demonstrate that as we approach the quantum limit, Landau levels crossing the Fermi energy give rise to single features in MCE and magnetoresistance, while simultaneously a novel double-peak structure is observed in the specific heat C el /T .To verify that the double-peak structure in C el /T is an intrinsic effect, we have measured C el /T versus magnetic field for three different samples, together with an angle-dependence C el /T (Supplementary Note 3).The doublepeak structure in C el /T shows good reproducibility and follows the expected (for graphite) quasi-2D behavior in magnetic field, allowing us to conclude that the double-peak structure is an intrinsic effect.
Strikingly, the magnetic field splitting ∆B = B 2 − B 1 of the double-peak structure in C el /T is strongly temperature dependent and vanishes as T → 0. Fig. 2a shows the doublepeak structure in C el /T as a function of magnetic field for the N = 1 ± h Landau levels measured for different temperatures from 90 mK to 1.5 K.The C el /T curves are vertically shifted for clarity.Symbols indicate the peak positions (corresponding to fields B 1 and B 2 ) for 1 + h and 1 − h levels, respectively.At high temperature, the splitting is clearly resolved.At lower temperatures, the splitting decreases, and the double-peak structure eventually merges into a single peak at T = 90 mK.To analyse the evolution of the splitting, we plot the magnetic field position of the double-peak structure as a function of temperature in Fig. 2b for various spin up/spin down hole and electron Landau levels.The peak positions B 1 , B 2 scale linearly with the temperature and the extrapolated splitting vanishes at T = 0 K for all Landau levels.The T -linear dependence of the peak positions B 1 , B 2 is a characteristic feature for the double-peak structure presented in this study.
The SWM Hamiltonian [15,16] with its seven tight binding parameters γ 0 , ..., γ 5 , ∆ provides a remarkably accurate description of the band structure of graphite [17,18].In a first approach, we use the SWM-model to understand the observed Landau level crossings with the Fermi energy.The Landau levels were calculated by finding the local extrema for each Landau band (dE N /dk z = 0), where a saw-tooth-like singularity in the DOS is located.Moreover, as we approach the quantum limit, the movement of the Fermi energy to keep the charge neutrality is non-negligible, and inevitably influences the magnetic field at which a given Landau level crosses the Fermi level [18][19][20].For this reason, the Fermi level movement was calculated based on the principle of charge neutrality, that is, the difference of the electron (n) and hole (p) carrier concentration is a constant: n − p = n 0 .
For the SWM parameters, we used the values which were fine-tuned to correctly reproduce de Haas-van Alphen measurements in natural graphite [21] (Supplementary Table 1).The In order to elucidate the origin of the double-peak structure in C el /T versus B, it is necessary to consider the exact form of the expression for the specific heat.For electronic quasiparticles, C el /T is given by [14], where , actually suppresses the double-peak structure in C el /T [22].As we will see, the double-peak structure in C el /T originates in the temperature dependent splitting of the double maxima in the kernel term −x 2 dF (x)/dx (when plotted versus E = xk B T ).To illustrate this, in Fig. 3a we plot the kernel term −x 2 dF (x)/dx in the vicinity of the Fermi energy (x = 0) at different temperatures.This function shows a distinctly non-monotonic behaviour with maxima located at x = ±2.4.The maxima on either side of the Fermi energy occur at an energy E = ±2.4kB T (Supplementary Note 8), so that the splitting of the maxima ∆E = 4.8k B T varies linearly with temperature and vanishes as T → 0. Qualitatively, this exactly predicts the temperature dependence exhibited by the double-peak structure in C el /T .The double-peak structure in quantum oscillations was also predicted in earlier theoretical calculations using an explicit expression for the specific heat [23], however, a quantitative comparison between experiment and theory is still missing.
To obtain a quantitative comparison, we use a model DOS, calculating the specific heat C el /T as the Landau level crosses the Fermi energy using Eq. ( 16).The shape of Landau quantized three-dimensional DOS is saw-tooth-like, resulting from the superposition of the quantized DOS of a two dimensional system perpendicular to the field direction (delta function) and the density of states due to the dispersion along k z (DOS ∝ 1/ √ E) [24].We approximate the DOS for a single Landau level, with its "singularity" at E = E 0 , by the following rigid expression for energies E ≥ E 0 , The one in the denominator prevents the unphysical (in a real system) divergence of the DOS which has a maximum amplitude of A at E = E 0 .The parameter Γ is the full width at half maximum (FWHM) of the Landau level.For simplicity all energies are calculated relative to the Fermi energy E F = 0.The position of the Landau level at a given magnetic field is dB, B 0 is the magnetic field at which the Landau level crosses the Fermi energy at T = 0.In this model, the magnetic field dependence of the Landau level energy dE/dB measured relative to the Fermi energy is a fitting parameter, and thus includes the cyclotron energy and Zeeman energy, together with any movement of the Fermi energy in magnetic field.We stress that the double-peak structure, is the result of a single spin Landau level crossing the Fermi energy.However, since the spin splitting is small, the spin up/down Landau levels cross E F in quick succession, generally producing quadruple peaks.In order In this approximation the spin split Landau level rigidly shifts through the Fermi energy i.e. the spin gap remains constant over the limited magnetic field range involved.This approximation is justified by the fact that the extracted dE/dB values for a given spin up/down Landau level are quasi-identical, and for simplicity we force them to be identical in the final fit.Fig. 3b shows the magnetic field dependence of C el /T data in the magnetic field region where the 1 ± e spin Landau levels cross the Fermi energy, together with the results of the fit.The fitting parameters used are Γ = 0.21 meV, B ↑ 0 = 7.05 T, B ↓ 0 = 7.80 T, and dE/dB = 1.04 meV/T.Note that the FWHM Γ, obtained here by fitting C el /T versus B, is very close to the Landau level broadening Γ q = h/τ q determined from the magnetic field for the onset of the Shubnikov-de Haas oscillations (ω c τ q = 1) in natural graphite at mK temperatures reported in a previous study [25] (see Supplementary Table 2).The calculated curve is in excellent agreement with the C el /T data, reproducing correctly the position, and the amplitude of each double-peak structure feature, together with the asymmetric lineshape, which naturally arises due to the asymmetric nature of the Landau level DOS.Fig. 3c schematically shows the DOS for the 1 ± e spin-split Landau level used to calculate C el /T at the four magnetic fields corresponding to maxima in C el /T .For comparison, we also plot the kernel term −x 2 dF (x)/dx calculated for the measurement temperature of T = 0.5 K.
The peaks in C el /T appear at certain magnetic field when the DOS peak is tuned to the  A crucial test of our model for the origin of the double-peak structure is shown in Fig. 4.
For each Landau level and each temperature we can compare the energy through which the Landau level moves (from field B 1 to B 2 ) with the energy separation of the maxima in −x 2 dF (x)/dx which depends only on the temperature.The energy shift, as the magnetic field changes by ∆B = B 2 −B 1 , can be calculated provided we know the slope of the Landau levels (movement relative to Fermi energy).In Fig. 4 we plot the energy shift of the Landau levels ∆B (S N − S F ) versus temperature T using the SWM values of (S N − S F ) summarized in Table 1.Plotted in this manner all of the data collapse onto a single straight line through the origin.The solid line is the expected splitting of the maxima in −x 2 dF (x)/dx, namely Estimate of the dE/dB from the double-peak structure The comparison between the DOS model and SWM-model allows us to derive the following relation for a quantitive charaterization of the double-peak structure, Intriguingly, Eq. ( 3) implies that the slope of the Landau level dE/dB can be estimated based on the magnetic positions B 1 , B 2 of the double peaks (note that ∆B = B 2 − B 1 ).This is apparently useful for a new system with unknown shape of DOS peak, when the DOS model fitting is not applicable.Here, it is important to note that Eq. ( 3) is accurate provided the DOS peak is symmetric.However, in the case of asymmteric DOS peak, our simulations (Supplementary Fig. 9) show that ∆B(dE/dB) can be 10 − 20% larger than the 4.8k B T splitting of −x 2 dF (x)/dx depending on the Landau level width Γ.In addition, the asymmetric DOS peak also induces deviation between the peak position in 1/T and the center of double-peak structure (Supplementary Fig. 10).
As T → 0, we expect the double-peak structure to merge into a single peak (as seen in 90 mK data in Fig. 2a), when the splitting (4.8 k B T ) of maxima of kernel term −x 2 dF (x)/dx is smaller than linewidth of Landau level DOS.Due to the highly asymmetric nature of the Landau level DOS, this condition is fulfilled when the splitting of the maxima 4.8k B T ≃ Γ/2, i.e. half the FWHM of D(E).Applying this condition, the values of Γ extracted from the simple DOS model in Table 1 provide a reasonable estimate of the temperature below which the double-peak structure is quenched in the experimental C el /T data.For example, the double-peak structure disappears between 0.3 K and 0.09 K for the 1 h Landau level in Fig. 2, while the predicted quench temperature Γ/9.6k B ≃ 0.2 K.
Estimate of the g-factor from the double-peak structure In general, to extract the g-factor using techniques such as SdHs, dHvA, MCE etc, one has to know the Landau index (orbital quantum number) for each peak, and the system dependent Fermi energy shift [26].While the double-peak feature observed in specific heat allows us to estimate the g-factor, without having to make any assumptions concerning the Landau index or Fermi energy shift.As a first approach, it is possible to estimate the electron and hole g-factors, implicitly involved in the DOS model, from the magnetic fields (B ↑↓ 0 ) at which the spin Landau levels cross E F .The crossing condition gives . Using the values in Table 1, we obtain g = 2.0, 2.2 and 2.0 for the 1 e , 2 e and 1 h Landau levels respectively.These values are close to free electron g-factor due to the small spin-orbit coupling of the carbon atom [27], and in good agreement with electron-spin-resonance measurements in graphite [28][29][30][31].
It is clear that our simple DOS model provides a reasonable estimate of the g-factor.
However, in most cases of quantum oscillations, the exact shape of the DOS is unknown, making it difficult to fit the data in order to extract the g-factor.Alternatively, this limitation can be overcome by using a coincidence method based on the magnetic field positions of the double peaks.The specific heat which depends on an integral involving the kernel term −x 2 dF (x)/dx represents a spectroscopic tuning fork of width 4.8k B T which can be tuned at will to resonance.For example, the observed coincidence of the B 1 and B 2 features of the 1 ± h spin Landau levels at T = 1.09K and B = 3.75 T (marked as orange arrow in Fig. 2b), corresponds to the condition where the spin-split Landau levels simultaneously cross the maxima in −x 2 dF (x)/dx, i.e. g h µ B B = 5.8k B T (here we use the apparent splitting in C el /T due to Landau level width -see Supplementary Note 10 for details) allowing us to extract the hole g-factor g h = 2.49.Likewise, the extrapolated crossing of the 1 ± e spin Landau levels at T = 2.05 K and B = 7.42 T provides an estimate for the electron g-factor g e µ B B = 5.9k B T (apparent splitting in C el /T ) gives g e = 2.42.These values compare well with the accepted value of the electron/hole g-factor g s = 2.50 used to fit de Haas van-Alphen data using the SWM Hamiltonian in natural graphite [21].Note, g-factors measured by electron spin resonance [28][29][30][31] are smaller (g = 2.15) as they measure the single particle spin gap, while transport techniques measure the exchanged-enhanced spin gap.We emphasis that extracting the g-factor using both DOS model and coincidence method does not require the knowledge of Landau index and Fermi energy shift, which is an advantage beyond other techniques (see more discussion in Supplementary Note 14 The origin of double-peak structure reported here is not restricted to the Landau quantisation, but also applies to any system where a femionic sharp DOS peak is tuned by magnetic field.For example, the double-peak structure was observed in the vicinity of Lifshitz transition for heavy fermion compounds CeRu 2 Si 2 [36] and UCoGe (Supplementary Note 12-13).
In graphite, dE/dB of Landau levels extracted from the double-peak structure originates from the cyclotron/Zeeman energies corrected for the Fermi energy shift in magnetic field.
To understand what drives the double-peak structure near the Lifshitz transition, we compare the measured dE/dB with the field dependence of the cyclotron/Zeeman energies in Table 2. Clearly, the dE/dB values extracted from the double-peak structure in both UCoGe and CeRu 2 Si 2 are too large to be explained by the Zeeman/cyclotron energy of the heavy quasiparticles (dE/dB ≫ he/m * , g j m j µ B ).We conclude that the main contribution to the dE/dB near the Lifshitz transition is the shift of the Fermi energy.For example, in the case of CeRu 2 Si 2 , the field-induced valence instability is expected to induce the large shift of the Fermi energy [37].Therefore, the double-peak structure in C el /T can potentially be used to determine the Fermi energy shift in the vicinity of Lifshitz transition, a physical quantity that is not easy to access using other probes.

Kernel term for different probes
It is interesting to consider in what other thermodynamic and transport properties a double-peak structure is to be expected within the free electron theory.The exact form is an experimentally verified [8,18,38].The presence of a double peak structure in thermal transport is currently a theoretical prediction.
integral involving a convolution of the density of states D(E) and a kernel term −x n dF (x)/dx with n = 0, 1, 2 depending upon the probe considered (see Supplementary Note 6) [39,40].
Fig. 5 shows the three different kernel terms and the corresponding techniques.Although the shape of experimental data for different probes can be influenced by the D(E) or other factors (e.g.scattering time for transport probes), it is the kernel term that determines the shape of the experimental data to be single-or double-peak feature.The single peak feature predicted for conductance and magnetization is well known [18,38].The predicted positive and negative peaks in thermopower has also been observed experimentally in graphite (see Supplementary Fig. 7) [8,41].With the single-and double-peak features of MCE and specific heat reported in present study, the currently only unverified probe is thermal transport.As seen in Fig. 5, as thermal conductance has the same kernel term as the specific heat, simple theory predicts a similar double-peak structure in thermal transport.To the best of our knowledge, a double peak feature has yet to be observed in thermal transport, so we expect our results should stimulate further research in this direction.
We have shown that, as the quantum limit is approached in high quality graphite, the electronic specific heat divided by temperature C el /T exhibits a double-peak structure when a single spin Landau level crosses the Fermi energy that vanishes as T → 0. A simple DOS model, combined with the predictions of the SWM Hamiltonian, successfully reproduces the double-peak structure, which can be understood with the exact form of the free electron expression for C el /T .The specific heat, which depends on an integral involving the kernel term (−x 2 dF (x)/dx), represents a spectroscopic tuning fork of width 4.8k B T that can be tuned at will to resonance.Using a coincidence method, the double-peak structure provides a reliable estimate of the exchange enhanced g-factor.Crucially, the double-peak structure is also observed in the specific heat of heavy-fermion compounds in the vicinity of the Lifshitz transition, potentially providing direct access to the Fermi energy shift at the Lifshiz transition.

Sample description
The measurements were performed on high quality natural graphite samples.The graphite flakes have a typical length of ≃ 1 mm and thickness of ≃ 0.1 mm.The weight of Sample#1 -Sample#3 are 0.96 mg, 0.18 mg, 0.23 mg respectively.

Experimental setup
AC specific heat measurements were performed in a static magnetic field on natural graphite samples.During the experiment, the specimen was attached to the backside of a bare CERNOX resistive chip by a minute amount of Apiezon grease.The resistive chip was split into heater and thermometer part by artificially making a notch along the middle line of the chip.The heater part was used to generate a periodically modulated heating power P ac with a frequency of 2ω, which can be described as the following relation, where R H is the resistance of heater part, i ac is a modulating current with a frequency of ω.
The induced oscillating temperature T ac of the sample was monitored by the thermometer part of the resistive chip.To do so, we applied a DC reading current i DC and monitored the induced AC voltage V ac .Based on a precise calibration of the thermometer(R-T relation), T ac can be calculated from, Knowing P ac and T ac , specific heat can be calculated by [42], Here, ϕ stands for the phase shift between P ac and T ac .By properly choosing the measurement frequency (ω), ϕ is close to -90 o (|sin(ϕ)| ≃ 1).
To measure the angle-dependence of the specific heat in magnetic field, a CERNOX resistive chip is mounted on a copper ring attached to an attocube rotator.On the back of the copper ring, a Hall probe allows to measure the angle with the magnetic field.The misalignment between the sample and the Hall probe is estimated to be within ±2 degrees.
MCE measurement was carried out in long pulsed fields with duration of 1.2 s.The temperature of the natural graphite sample was read by the home-made RuO 2 thermometer, which was calibrated in temperature and magnetic field [43].The temperature of the sample was monitored and recorded during the pulse field sweeps.For both measurements, the magnetic field was applied along c-axis.

SWM Hamiltonian
Graphite is a semi-metal with the carriers occupying a small region along the H − K − H edge of the hexagonal Brillouin zone.The SWM Hamiltonian [15,16] with its seven tight binding parameters γ 0 , ..., γ 5 , ∆ provides a remarkably accurate description of the band structure of graphite [17,18].In a magnetic field, when trigonal warping is included (γ 3 ̸ = 0) levels with orbital quantum number N couple to levels with orbital quantum number N + 3 and the Hamiltonian has infinite order.Nevertheless, the infinite matrix can be truncated and numerically diagonalized, as the eigen-values converge rapidly [44].
The values of SWM parameters that are used in this study are shown in Supplementary Table 1, taken from the SWM parameter set optimized to fit de Haas-van Alphen measurements in natural graphite [21].They vary very little from the published values in other reports, e.g.[10,18,19].
Supplementary Note 1: Deviation from LK theory in quantum oscillation of specific heat in graphite In this section, we briefly introduce the expression of quantum oscillations in specific heat predicted in LK theory, and compare it with the double-peak structure observed in experimental results.The oscillatory component of specific heat due to Landau level quantization of the orbits is given by the extended Lifshitz-Kosevich formula [45], where A is a constant, R D = exp(−2π 2 pk B T D /(hω c )) is the Dingle term with dingle temperature T D , J 0 is a Bessel function of the first kind, t w is the c-axis hopping energy, µ is changes sign at z = 1.6 which induces a "π-phase shift" for ∆C el (T, B).When T and B satisfy the condition of z = 1.6, the amplitude of ∆C el (T, B) drops to zero, which can be used to determine the effective mass m * [11,13,46].This kind of π phase shift in specific heat quantum oscillations has been reported in the organic superconductor (BEDT-TTF) 2 Cu(NCS) 2 [46] and unusual interplay between superconductivity and field-induced charge order in YBa 2 Cu 3 O y [11][12][13].
The π-phase shift of the oscillatory components is indeed observed in graphite at low magnetic field (B < 0.9 T) with Landau indexes N > 6, which is consistent with the prediction of the extended LK formula [4].The quantum oscillation of specific heat in graphite is a mixture of oscillatory component from both electron and hole pockets, which make it difficult to find the zero amplitude of ∆C el (T, B).Therefore, we first calculated the temperature-field boundary of the π-phase shift based on the known effective mass for electron and holes in graphite, and compare it with our experimental data.Supplementary Fig. 6a shows the calculated boundary, T b , for the π-phase shift (z e/h = 1.6) using 0.056 m e and 0.067 m e as the electron and hole effective mass [10].The π-phase shift occurs around 1-1.5 K in the field range from 0.55 T to 0.85 T. Supplementary Fig. 6b shows the field sweep of C el /T taken in the temperature range of 0.3 -2 K.The Landau index for each oscillatory component are marked.It is clear that the C el /T has a π-phase shift between the 0.5 K and 1.5 K data, which is consistent with our calculation in Supplementary Fig. 6a.It has been shown in the transport measurement that the movement of Fermi energy of graphite is negligibly small at B < 2 T, but becomes significant at B > 2 T [18].The fundamental principle behind this phenomenon is the "charge neutrality" condition, that is, the difference of the electron and hole concentration should be a constant value, where n e and n h are the electron and hole concentration, n 0 is a constant value representing the difference.Considering the degeneracy of the Landau bands, the electron and hole concentrations are given by [24], here, k e,i and k h,j are the k z distance for the occupied i-th electron and j-th hole Landau The maximum and minimum in z correspond to the roots of the function Using Newton's method the roots occur at x = ±1.543404638418208,so that the expected splitting of the maximum and minimum in thermopower, when a DOS singularity crosses the Fermi energy is ≃ 3.09k B T .
features occur at magnetic fields slight below the B = 5 T crossing of the Fermi energy.
passes through the Fermi energy at the Lifshitz transition of CeRu 2 Si 2 .Moreover, the MCE curve of CeRu 2 Si 2 only exhibit a single-peak feature [36], which is fully consistent with our model.The simultaneous occurrence of single-and double-peak features in MCE and specific heat are smoking gun thermodynamic signatures of a single DOS peak crossing the Fermi level.Moreover, the magneto-resistance, Hall resistivity and thermopower measurements on CeRu 2 Si 2 also suggest the same scenario [50][51], that is, a single spin subband crosses the Fermi energy at the Lifshitz transition of CeRu 2 Si 2 .
In heavy fermion system, it is known that a 'logarithmic divergence' feature in temperature sweep of specific heat (C/T ∝ log (T 0 /T )) is a signature of non-Fermi Liquid behaviour [52].Here, it is interesting to note that such logarithmic divergence feature can also be observed in C/T versus T for a fermionic singularity in DOS that passes over the Fermi energy.In Supplementary Fig. 17b, we show the semi-log plot of calculated C/T as a function of temperature using single symmetric DOS peak at constant magnetic fields.If the system has a logarithmic divergence behaviour, the temperature-sweep of C/T exhibits a straight line in the semi-log plot.As seen in Supplementary Fig. 17b, C/T versus T exhibits as a straight line at the critical field B = 7.7 T. Therefore, the logarithmic divergence in C/T is not only due to an existence of a canonical quantum critical point, but can also be attributed to the formation of a fermionic DOS singularity.

Fig. 2 .
Fig. 2. Temperature dependence of the double-peak structures in C el /T .a Electronic specific heat divided by temperature C el /T in graphite as a function of magnetic field in the vicinity of the 1 ± h Landau level/Fermi energy crossing.Measurements at different temperatures are vertically offset for clarity.At higher temperature each spin Landau level gives rise to a doublepeak structure (peaks indicated by symbols) as it crosses the Fermi energy.b Measured magnetic field position of the double-peak structure B 1 , B 2 as a function of temperature for various spin up/spin down electron and hole Landau levels.The error bars represent one standard deviation of uncertainty.
calculated results are shown in Fig 1d.Solid lines show the evolution of the lowest electron and hole Landau levels with magnetic field, while the green dashed line shows the calculated evolution of Fermi level.To facilitate the comparison of theory and experiment, we draw a series of vertical dashed lines in Fig 1d to indicate magnetic fields corresponding to the crossing of electron/hole Landau levels with the Fermi energy.The positions of the dashed lines are in near perfect agreement with the magnetic fields of the observed peaks in MCE and the double-peak structure in C el /T .Discussion Origin of double-peak structure
and k B is the Boltzmann constant.The specific heat depends on the convolution of the Landau level DOS D(E) and a kernel term −x 2 dF (x)/dx which involves the first derivative of the Fermi-Dirac distribution function.The usual approximation, removing D(E) from the integral, and replacing it with D(E F ), to obtain the well know formula

Fig. 4 .
Fig. 4. Consistency between DOS model and SWM-Hamiltonian.Temperature dependence of the energy through which the different Landau levels move ∆B (S N −S F ) calculated using the movement of the Landau level relative to the Fermi energy extracted from the SWM model (see Table1).All data collapses onto a single straight line through the origin.The solid line is the ).All data collapses onto a single straight line through the origin.The solid line is the calculated E = 4.8k B T temperature dependent splitting of the two maxima in −x 2 dF (x)/dx.maxima of the kernel term −x 2 dF (x)/dx.In Table1, we summarize parameters extracted from the simple DOS model for all the Landau levels.In order to compare the values of dE/dB with the predictions of the SWM Hamiltonian, we calculate the slope of SWM Landau level energy with respect to the Fermi energy in the vicinity of the crossings i.e. S N −S F .Here, S N is the field dependence of the Nth SWM Landau level energy, while S F is the field dependence of the Fermi energy originated from charge neutrality condition.We see that the DOS model and SWM Landau level slopes agree to within 10% (dE/dB ≃ |S N −S F |), which is very reasonable given the approximations involved.The good agreement between the predicted and experimental values lends further strong support to our model, and indicates that the double-peak structure in C el /T is a new way to access the Landau level dispersion.

Fig. 5 .
Fig. 5. Kernel terms for various thermodynamic and transport probes.Fermi-Dirac distribution function F (x) with x = E/k B T , and the kernel terms −x n dF (x)/dx of the exact formula predicting the behaviour of different thermodynamic and transport probes.All the curves were calculated at T = 0.5 K.The features of conductance, magnetization and thermopower have been Fig. 10.Then, by calculating the energy of local extreme at various magnetic field, we obtained the Landau level for electrons and holes in Manuscript Fig. 1d.

dx 2 =
bands, as shown by blue and red arrows in Supplementary Fig. 10.At low magnetic field, many Landau bands are occupied, adding electrons or holes does not require significant changes of the Fermi level to fulfill the n e − n h = n 0 condition.While at high magnetic field, when only one or two Landau bands are occupied, the Fermi energy has to shift to fulfill this condition.maximum at x = 0.The kernel function for thermopower −xdF/dx is an odd function of energy around the Fermi energy, as can be seen in Supplementary Fig. 13.We define the kernel function z = −xdy/dx with the Fermi-Dirac distribution function y = (1 + e x ) −1 .We find the maximum and minimum in the kernel function by looking for zeros in the first derivative, y 3 e x ((1 + x) + (1 − x)e x ),

Table 1 .
Summary of the parameters obtained from the simple DOS model and the SWM-Hamiltonian close to where the Landau levels cross the Fermi energy.The tight binding parameters of the SWM Hamiltonian can be found in Supplementary Table1.
to locally fit the C el /T data we use D ↑↓