Quantum oscillations from networked topological interfaces in a Weyl semimetal

Layered transition metal chalcogenides are promising hosts of electronic Weyl nodes and topological superconductivity. MoTe$_2$ is a striking example that harbors both noncentrosymmetric T$_d$ and centrosymmetric T' phases, both of which have been identified as topologically nontrivial. Applied pressure tunes the structural transition separating these phases to zero temperature, stabilizing a mixed T$_d$-T' matrix that entails a unique network of interfaces between the two non-trivial topological phases. Here, we show that this critical pressure range is characterized by unique coherent quantum oscillations, indicating that the change in topology between two phases give rise to a new topological interface state. A rare combination of topologically nontrivial electronic structures and locked-in transformation barriers leads to this counterintuitive situation wherein quantum oscillations can be observed in a structurally inhomogeneous material. These results open the possibility of stabilizing multiple topological superconducting phases, which are important for solving the decoherence problem in quantum computers.


INTRODUCTION
Topologically protected electronic states at material interfaces are attractive because they cannot be destroyed by many types of perturbations 1,2 . An especially fruitful host of such exotic states is MoTe 2 , which has had both its bulk orthorhombic T d phase, and hole-doped monolayer specimens identified as possible topological superconductors [3][4][5] . In addition, a topological superconducting phase was recently discovered in sulfur-substituted samples, with S +− -wave pairing 6 . MoTe 2 also features topologically nontrivial normal states: the T d phase has been identified as a type II Weyl semimetal 7-10 , whereas the monoclinic T' phase is predicted to be a higher-order topological material 11 ; the latter is also found to describe the T d phase in some calculations 11 . In this work, we demonstrate via quantum oscillations and neutron scattering measurements, and first-principles calculations, how pressure drives MoTe 2 between the T d and T' phases, through an intermediate mixed-phase region. The mixed-phase region gives rise to a network of topological interface states that yield quantum oscillations that survive despite the strong structural disorder.

Orthorhombic T d -MoTe 2
The first-order structural transition separating the T' and T d phases in MoTe 2 has a distinct pressure dependence (Fig. 1a). At ambient pressure, the inversion-symmetric T' phase is stable at room temperature, only transforming into the noncentrosymmetric T d phase when cooled below roughly 250 K 3,12 . Neutron diffraction allows the determination of the relative volume fraction of these phases under different conditions 12 . As pressure increases, the transition temperature decreases. At pressures higher than 0.8 GPa, a completely different phenomenon emerges, where a roughly balanced mixture of the T' and T d phases stabilizes over an appreciable temperature range, and crucially, extends to the lowest measured temperatures. The existence of this frozen mixed-phase region is stabilized by the lack of sufficient entropy at these suppressed temperatures for atoms to move to their lowest-energy configuration, implying that there is a dominant extrinsic transformation energy barrier between two energetically nearly degenerate structures 12 .
The basic components underlying the Weyl semimetallic state of the low-pressure, low-temperature T d phase are a large hole pocket centered on the Brillouin zone and two neighboring electron pockets along the Γ -X direction [8][9][10]13 . The hole pocket is observed in angle-resolved photoemission spectroscopy (ARPES) 8-10 , but is not apparent in SdH measurements 14 . Prominent quantum oscillations observed in the T d phase arise from orbits associated with the electron pocket 14 . Figure 2a, b shows magnetoresistance and SdH oscillations at ambient pressure, in which these are clearly seen. As the fast Fourier transform (FFT) explicitly shows (Fig. 2c), the beating seen in Fig. 2b is due to two similar frequencies, F α = 240.5 T and F β = 258 T, the result of symmetry-allowed spin-dorbit splitting. First-principles calculations identify these frequencies with the larger extremal k z = 0 cross sections of the electron pocket.
Modeling of the SdH oscillations yields a remarkably good fit (Fig. 2d) to the experimental SdH by the Bumps program 15 Supplementary Table 5, Supplementary Notes 9 and 10). Notably, all of the oscillations feature a π Berry's phase 16 , consistent with a Weyl topology 14, and Supplementary Note 2). The effective band masses are light and slightly less than previously reported 14,17 , as shown in Fig. 2e. As a function of pressure, the electron pockets increase modestly in size due to lattice compression, but the nontrivial phase shift is maintained throughout the T d phase (Supplementary Tables 1-4). This trend is consistent with firstprinciples calculations as shown in Fig. 3a and b, which indicates the persistence of Weyl nodes up to 1.4 GPa (Supplementary Fig. 6 and Supplementary Note 2).

Monoclinic T'-MoTe 2
In the T' phase at pressures of 1.2 GPa and greater, one main frequency F η replaces the two frequencies F α and F β . The Fermi surface in the T' phase closely resembles that of the T d phase, with the exception that the centrosymmetry of the T' phase nullifies the spin-orbit splitting of the bands that contribute to this electron pocket (Fig. 3d, e). This similarity leads to a common pressure dependence of the measured electron pocket frequency in both phases, consistent with the pressure evolution of the calculated Fermi surface (Fig. 3c, Supplementary Fig. 8, and Supplementary Tables 1-4). The main feature not captured by the calculations is a pronounced discontinuity in band structure between T d and T' phases due to the discontinuous c axis dilation at the structural transition, which is not modeled (Fig. 3c). Unexpectedly, our SdH oscillations analysis shows that a π Berry's phase also exists in the centrosymmetric T' phase (Supplementary  Table S6). After an ARPES study identified surface states in this phase 13 , it was identified theoretically with an unusual type of nontrivial topological state 11 . Our results are consistent with this prediction, and further have the exciting implication that the high-pressure superconductivity in the T' phase may be inherently topologically nontrivial, as suggested by μSR measurements at these pressures 18 .
Natural topological interface network (TIN) A structural mixed region exists over a range of pressures and temperatures in between the bulk T d and T' phases ( Fig. 1a). It consists of an approximately balanced partial volume fraction of T d and T'. We emphasize that no other structural phases or ordered superstructures are apparent from neutron diffraction measurements (Supplementary Figs. 1 and 2). It would, therefore, be expected that any measured SdH oscillations in the mixed region would consist of a superposition of T d and T' signals, but we do not observe oscillations from either phase. We conclude that the mixed region is sufficiently disordered that SdH oscillations from both the T d and T' phases are suppressed due to increased electron scattering.
In light of this, it is completely unexpected that a distinct set of SdH oscillations appears (Fig. 1b, d). This is a robust effect; the oscillations in the mixed phase always appear on both increasing and decreasing applied pressures through the critical range, over multiple cycles, confirming their intrinsic nature. A different band structure in the mixed region is inferred from the presence of frequencies corresponding to changed Fermi surfaces, and a change in effective mass and much weaker oscillation amplitude relative to bulk T d and T', as shown in Fig. 4a-f. These quantum oscillations reflect features typical of topologically protected states, namely, persistence in the presence of strong disorder, as The green and blue symbols delineate the extent of 100% volume fraction of T' and T d phases, respectively, determined through neutron scattering. The onset of full volume superconductivity coincides with the end of the T d phase at 0.8 GPa 12 , where a topological interface network (TIN) is observed. b Pressure dependence of the quantum oscillation frequency, with different branches labeled by Greek letters. The numbers correspond to the effective mass, which changes slightly with pressure. The quantum oscillations with the strongest pressure dependence, α and β in the T d phase, and η in the T' phase, correspond to extremal orbits on the large electron pockets. In the TIN region, these disappear and are replaced by a completely distinct set of oscillations arising from topological interface states. Representative quantum oscillations (c) at ambient pressure (Weyl semimetal in T d ), d 0.9 GPa (TIN), and e 1.8 GPa (higher-order topology in T'). Clear changes in the quantum oscillations reflect significant changes in the electronic structure. well as π Berry's phases (Supplementary Tables 5 and 6). Because there are no additional structural phases in the mixed region, only T d and T', these topological states must be surface states of the bulk phases, which in this case, exist at their interfaces. Thus, the multiple natural interfaces stabilized in MoTe 2 by the first-order nature of the structural transition serve as the foundation for a completely different type of electronic system in MoTe 2 : a natural topological interface network (TIN).
In the TIN, due to the layered structure of both T d and T' phases, the ab plane is preserved, and the largest grain boundaries fall along the ab plane, which is the orientation probed by the SdH measurements (Fig. 4g). In MoTe 2 , a naturally generated heterostructure provides an interesting demonstration of topological transport protection. The lateral dimensions of the interfaces are the same as those of the bulk grains, based upon which one naively expects similar damping of the SdH oscillations from the interfaces. The absence of quantum oscillations from bulk T d and T' phases illustrates that the scattering from the TIN microstructure is significant for the bulk bands. Yet the clear SdH oscillations from the interfaces prove that the interfacial states have lower scattering than the bulk, and are a sign of their topologically nontrivial nature. In other words, the interfacial signal has been amplified by suppressing the bulk SdH oscillations through grain boundary scattering, and increasing the interface volume.
Density-functional theory (DFT) calculation of T d -T' slab model First-principles calculations offer additional insight into the electronic structure of the TIN and its stability. We investigated a variety of possible mixed phases that might describe the mixed region, including T d -like phases with different MoTe 2 stacking orders, periodic superstructures, and finite slabs of T d and T' phases and their interfaces as shown in Fig. 5a and b (detailed calculations are discussed in the Supplementary Figs. 9-13 and Supplementary Notes 4-6). Calculations demonstrate that the only model consistent with the experimentally determined highfrequency oscillations of~1 kT (Fig. 4d) is a system consisting of only T d -T' interfaces, the TIN (Fig. 5d).
The Fermi-surface pockets associated with the interface are identified in two different but complementary calculations. Consider a periodic superstructure of T d and T' phases, as shown in Fig. 5a. Surprisingly, the T d -T' interface has lower energy if the MoTe 2 layers at the interface have the same type of Te distortion, labeled as L or R, compared to the L-R-L-R-type stacking order found in bulk T d and T' phases. Since L-or R-type planes cannot be The corresponding SdH oscillations after subtraction of a second-order polynomial background. c The fast Fourier transform (FFT) spectra show three Fermi surfaces with oscillation frequencies at F γ = 32.5 T, F α = 240.5 T, and F β = 258 T. The inset shows the α and β orbits on the calculated electron pocket. d Fit to the SdH oscillation at 1.8 K yield nontrivial Berry's phases ϕ γ = π, ϕ α = 0.88π, and ϕ β = 0.88π. Details of the fit are discussed in the Supplementary Information. e The effective masses of the carriers are obtained through temperature-dependent Lifshitz-Kosevich fits. converted to each other by simply sliding the planes, this sort of interface presents a large energy barrier to the removal of stacking faults. The fact that this LL-type interface has lower energy than the interface with L-R stacking explains the stability of the mixed intermediate phase. Comparing the Fermi surfaces of the pure T d and T' phases with the T d -T' superstructure yields distinct Fermisurface pockets with quantum oscillation orbits of 0.16 kT and 1.0 kT (bottom panel of Fig. 5d) in excellent agreement with our measurements.
Even though the periodic superstructure calculations of the T d -T' interface are consistent with the TIN model, the interface pockets have some dispersion due to interactions between the periodic images of the interface. To eliminate this effect, consider two slabs of T d and T' initially separated by about 9.5 Å (Fig.5b). These two slabs are brought together to form the T d -T' interface in a supercell where the combined T d -T' slab is separated bỹ 16.3 Å, large enough to avoid any image interactions (Fig. 5b, bottom). Comparing the Fermi surfaces of the isolated slabs ( Fig. 5b) and combined slabs with the interface (shown in Fig. 5c), yields Fermi-surface pockets associated with the interface (Fig. 5d, top).
The agreement is excellent between the surface states from the periodic and finite slab calculations. These 2D quantum oscillation orbits are in impressive agreement with the experimental results (middle panel of Fig. 5d). Thus, the slab calculations directly link the observed high-frequency oscillation in the mixed region to interface states, as a TIN would produce.

DISCUSSION
Essential to any topological surface state is the meeting of two bulk phases with different topological invariants. A typical example is that of a Weyl semimetal interfaced with a topologically trivial vacuum, on whose boundary Fermi arcs are observed, as in T d -MoTe 2 8 . In contrast, the second phase of the TIN is the topologically nontrivial T' phase, which forms a large number of T d /T' interfaces due to the structural disorder. Then, for surface states to exist at these interfaces, there must be a change in topology between T d and T' phase, as expected for phases of Weyl and higher-order topological states. Although such interfaces have until now not received much theoretical attention, recent scanning tunneling microscopy measurements on MoTe 2 have suggested the existence of conducting hinge states in the T'/ T d domain wall at ambient pressure 19 . Our results did not directly observe 1D-hinge state, but cannot completely exclude the possibility of 1D paths. Our measurements and calculations show that 2D topological interfaces are also possible. Our discovery suggests that topological interfaces in 3D materials are a rich area for future exploration and exploitation.
Finally, we briefly discuss the relationship between the TIN and superconductivity in MoTe 2 . Both the superconducting volume fraction and T c increase dramatically at the same pressure at which the TIN is stabilized (Fig. 1a). Taking into further consideration the 2D nature of superconductivity, based on our previous calculations 12 and recent upper critical field measurements under pressure 20 , a strong case is made that superconductivity is topologically nontrivial, consistent with previous claims from  μSR measurements 18 . The combination of pressure-sensitive structure and topology in MoTe 2 makes this an exciting playground for the use of topological interfaces, with potential applications in a variety of quantum devices.

First-principles calculations
The total energy, structure optimization under pressure, and band structure and Fermi-surface calculations were performed by Quantum Espresso 21 , which is based on DFT, using a plane-wave basis set and fully relativistic all-electron projected augmented wave (PAW) potentials 22 . The 4s, 4p, 4d, and 5s electrons of Mo and the 4d, 5s, and 5p electrons of Te were treated as valence. We used 0.02 Ry Methfessel-Paxton smearing with wavefunction and charge density cut-off energies of 100 Ry and 800 Ry, respectively. The exchange-correlation interactions were described by the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof exchange-correlation functional 23 . The Brillouinzone integration was performed using Monkhorst-Pack grids of special points with 16 × 8 × 4 for total energy and structure optimizations and 32 × 16 × 8 with tetrahedra method for electronic density of states and Fermi-surface calculations. The spin-orbit (SO) interactions and the weak inter-layer van der Waals (vdW) interactions were all included in our calculations. We used grimme-d2 24 vdW correction with parameter london -s6 = 0.6. The effect of electron correlations is included within DFT+U method with U = 3.0 eV for the Mo 4d-states. Including electron correlation brings the calculated band structure and Fermi surface into an excellent agreement with quantum oscillation and ARPES measurements, as discussed in Supplementary Information and also found in other very recent studies 25,26 . Fermi-surface sheets and SdH orbits are visualized by our custom python code using Mayavi 27 . The quantum oscillation frequencies/orbits and their angle dependence were calculated using the skeaf code 28 .

Crystal synthesis
Large single crystals were grown using the Te self flux method described elsewhere 12 using high purity Mo powder (5N metals basis excluding W, Alpha Aesar), and Te shot (6N, Alpha Aesar). High sample quality has been confirmed by X-ray and neutron diffraction, stoichiometry has been confirmed by wavelength dispersive spectroscopy, and the samples measured have residual resistivity ratios >1000.

Structural measurements
Determinations of the temperature and pressure-dependent crystal structure were made using elastic neutron scattering measurements at 14.7 meV on the BT-4 triple-axis spectrometer at the NIST Center for Neutron Research using a collimation and filter setup of open-pg-40'-pg-spg-40'-120', where pg refers to pyrolytic graphite. Single crystals were mounted in a steel measurement cell aligned in the H0L M zone, and He was supplied as a pressure medium to maintain hydrostatic pressure conditions as described elsewhere 12 . The T d and T' phases and their volume fractions were identified from the position and intensity of (201) M reflections, which both split in 2θ and shift in ω in the T' phase. Rocking curves and ω-2θ scans were taken at each pressure and temperature. Scans along (00L) from (2 0 0.5) to (2 0 4.5) were also obtained at 0.8 GPa in all T d or T' condition, as well as in the mixed region at both 0.8 and 1 GPa to look for possible superstructure reflections. None were observed.

Transport measurements
A nonmagnetic piston-cylinder pressure cell was used for transport measurements under pressure up to 1.8 GPa, using a 1:1 ratio of n-Pentane to 1-methyl-3-butanol as the pressure medium and superconducting temperature of lead as pressure gauge at base temperature.
All transport measurements were performed on the same MoTe 2 crystal with 110-μm thickness using four-point contacts made with silver epoxy. The pressure dependence of the superconducting transition and structural

ACKNOWLEDGEMENTS
We thank Jay D. Sau, Paul Kienzle, and Daniel J. Campbell for helpful discussions. We specially appreciate John Singleton for the thorough discussion on the analysis of the g factor and Berry phase in Lifshitz-Kosevich (LK) formula. The computational work was performed using the NIST Enki high-performance computer cluster. This work utilized

COMPETING INTERESTS
The authors declare no competing interests.
Correspondence and requests for materials should be addressed to IL.L. or N.P.B.

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Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.  The stated relationship between quantum oscillations and crystal structure is based on our pressure and temperature dependent elastic neutron scattering measurements. As described in our previous work 1 , we determine the T d and T' phase fractions from the scattering intensity at the (201) reflections of the monoclinic unit cell for the T' phase. For the T d phase, this is for a unit cell with a β-angle of 90 0 . The scattered intensity is determined from the integrated intensity of transverse scans centered at the nominal peak positions for each phase. We used the BT-4 triple-axis spectrometer at the NIST Center for Neutron Research with a monochromatic 14.7 meV neutron beam and a collimation and filter geometry of Open-pg-40'-pg-s-pg-40'-120 where pg refers to an oriented pyrolytic graphite filter used to remove higher order neutrons. The temperature dependent integrated intensity of each phase, taken upon warming, is given in Fig. 1(a)-(c) at 0.3 GPa, 0.8 GPa, and 1 GPa respectively. Figs. 1(d)-(f) show the raw scattering data fit to a Gaussian peak shape used to extract the integrated intensity for the three different temperature regimes seen at 0.8 GPa. In the mixed region, the temperature dependent isobars show a clear deviation from normal mean field-like behavior expected for a first order phase transition. We attribute this to kinetic trapping of the structural phase transition.
Critical to our discussion of phase coexistence and our interpretation of the cause of the arrested transition is that despite an apparent suppression of a phase transition temperature to zero temperature at a specific pressure, we do not believe this should be described as a quantum structural phase transition. We attribute the effective "freezing" of the transition to the reduction of the structural transition temperature into a temperature regime where thermal fluctuations cannot overcome the energy barrier between the phases. We can draw an analogy between this behavior and the kinetic freezing observed in the low temperature first order magnetic transitions in (La,Pr,Ca)MnO 3 between a ferromagnetic and charge-ordered antiferromagnetic state 2 . In the magnetic case the complex energy landscape is attributed to competition between magnetic order, structural and chemical disorder, and inter-grain strain, which is consistent with other observations of similar transistions in magnetocaloric materials with magnetostrictive coupling 3 . Given the previously established sensitivity of the MoTe 2 structural phase transition to pressure and strain as well as the known structural disorder in van der Waals materials, it is reasonable to assume that inter-grain strain and disorder may dominate the phase transition kinetics in this system as well 1,4 . At 0.8 GPa where we first observe a broad mixed region, the transition temperature is still high enough that it is possible to undercool the sample such that it eventually completes the phase transition, but at 1 GPa the transition temperature is further suppressed and no sufficient undercooling is possible leading to the mixed region as the ground state structure.
Given the coherent oscillations in the mixed region, we considered whether this could be evidence of a new bulk structural phase driven by stacking disorder during the phase transition. If this were the case, we would expect to observe new reflections along the stacking direction. We performed [00L] scans along [20L] at 0.8 GPa at temperatures corresponding to the T d phase and the mixed region. These are shown in figure 1. Clearly we see no evidence of new reflections that would indicate a new phase, instead observing only integer L peaks for the T d phase, split integer peaks from both monoclinic twins in the T' phase, and scattering attributable to the mixed phase in the mixed region. While we cannot completely rule out a new intermediate meta-stable phase, we see no indication of this either from superlattice reflections along [00L] or from missing intensity in the mixed region. The same is seen at 1.5 K and 1 GPa, with a slightly different ratio of T d to T' phase, consistent with the volume fractions extracted from the (201) scans used for the phase diagram in figure 2. In this section, we discuss the details of our first-principles electronic structure calculations of MoTe 2 as a function of pressure for both 1T'-and T d -phases, respectively. The total energy, structure optimization under pressure, and band structure and Fermi surface calculations were performed using Quantum Espresso, 5 , which is based on density-functional theory (DFT), using a plane wave basis set and fully relativistic all-electron projected augmented wave (PAW) potentials 6,7 . The 4s, 4p, 4d, and 5s electrons of Mo and the 4d, 5s, and 5p electrons of Te were treated as valence electrons. We used 0.02 Ry Methfessel-Paxton smearing, with wavefunction and charge density cut-off energies of 100 Ry and 800 Ry, respectively. The exchange-correlation interactions were described by the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof exchangecorrelation functional 8 . The Brillouin-zone integration were performed using Monkhorst-Pack grids of special points with dimensions of 16 × 8 × 4 for total energy and structure optimizations and 32 × 16 × 8 for electronic density of states and Fermi surface calculations using the tetrahedron method. The spin-orbit (SO) interactions and the weak inter-layer van der Waals (vdW) interactions were all included in our calculations. We used the grimme-d2 9 vdW correction with parameter london -s6 = 0.6. The effects of electron correlations are included within the DFT+U method with U =3.0 eV for the Mo 4d-states. Including electron-correlation brings the calculated band structure and Fermi-surface into excellent agreement with Quantum Oscillation and ARPES measurements as found in very recent studies 10,11 . Fermi-surface sheets and SdH orbits are visualized by our custom python code using Mayavi 13 . The quantum oscillation frequencies/orbits and their angle dependence were calculated using the skeaf code 14 .

C. Supplementary Note 2: Effect of Electron Correlations on the Band Structure and Fermi Surface of MoTe2
Recent studies 10,11 found that electron correlations are essential for a precise description of the bulk electronic structure of T d -MoTe 2 as reFealed by angular resolved photoemission spectroscopy (ARPES) 10 and the angular dependence of the Fermi surface by quantum oscillation (QO) experiments 11 . Hence, in our study we adopted DFT+U scheme to describe the electron correlations within the Mo 4d-states. The overall best agreement with ARPES and QO data is obtained for U =3 eV 10,11 , which was also used in our calculations in this study. Figures 3 and 4 show the effect of the Hubbard U (taken as 3.0 eV) on the band structure and the Weyl-points in the T d -and 1T'-phases of MoTe 2 , respectively. A detailed study of the effect of Hubbard U on the Weyl point (WP) in MoTe 2 can be found in Ref 11 . Briefly, in total there are four pairs of WPs, each in one quadrant, all of which are located on the k z = 0 plane and are related by the crystal symmetry. In our fully self-consistent calculations where both the lattice parameters and atomic positions were obtained self consistently from structure optimization, we usually get two Weyl points at (±k x , ±k y ,0) with very small k y .
We note that the biggest effect of including U is to shift the bands near the Y -point so that there is no longer a small electron pocket at the Fermi surface as shown in Fig. 5. The other main effect is to lower some of the bands further below the Fermi level  which does not have any effect on the Fermi surface. It is important to note that the Weyl point near the Fermi level remains unaffected by the inclusion of the Hubbard correlation term U in our calculations. However, as we shall see below, including U is critical to explain the pressure dependence of the QO frequencies that we have measured in this study. Finally, Figure 6 shows the effect of pressure on the Weyl point. The main effect of pressure is to shift up the hole pocket while shifting down the electron pocket near the Fermi level such that the Weyl point roughly does not change with initial applied pressure, at least up to 14 kbar. At higher pressure, it seems that the Weyl point moves slightly above the Fermi level (Fig. 6). A HOTI in the T d phase requires the annihilation of the Weyl nodes 12 , therefore, we would not expect HOTI in T d in our system here.
In our calculations, besides the Hubbard term U , there are other parameters such as lattice constants and atomic positions that we need to determine. Calculations can be based on the experimental parameters or the parameters can be determined self consistently within the DFT+U structural optimization at any given pressure. Figure 5 shows the Fermi surface of T d -and 1T' phases of MoTe 2 for both experimental and DFT optimized structures with and without electron correlation effects (i.e. U ). We note that the Fermi surface is very sensitive to the lattice parameters and the atomic positions. Our optimized lattice parameters and atomic positions are within 1% of the experimental values as shown in Tables 1 and 2. The biggest variation is in the a-axis for the T d phase and it is 1.4%. Despite this excellent agreement between experiment and calculations, the difference in the Fermi surface between experimental and the optimized structures is quite large. In order to be self-consistent, we decided to use optimized lattice parameters and atomic positions for a given pressure as obtained from our DFT+U calculations. In this way, we are able to determine the pressure dependence of the Fermi Surface and determine the quantum oscillation orbits and frequencies. The only free-parameter in our DFT+U calculations is the Hubbard U , which was shown to be around U =3 eV for MoTe 2 to match the ARPES measurements as well as the angle dependence of the QO frequencies 10,11 .

D. Supplementary Note 3: Pressure Dependence of the Fermi Surface and Quantum Oscillations
In this section, we present our results related to the pressure dependence of the Fermi surface and quantum oscillation orbits as a function of applied pressure for both phases of MoTe 2 . In both phases, we calculate similar Fermi surfaces and orbits which are summarized in Figure 7. Near zone center Γ, we have a square-box hole-like Fermi surface (red). The orbit around this Fermi surface is shown in Fig.7 as "s". Then, we have an electron-like Fermi surface with a shape of a coffee mug (light blue). This shape has basically three types of extremal orbits as shown in the Figure 7. We label the orbits at the opening as "op". Then, the orbit near the handle like surface as "h". And, finally we have the orbits near the cup like surface denoted as "c". As we shall see below, this orbit is sensitive to pressure and we identified this orbit as in our quantum oscillation measurements. Finally, we have a small pocket of squashed elliptical surface (dark blue), which we call the "e" orbit. In the case of T d -phase, these orbits have two counterparts due to SO-splitting.
We have carried out full structure optimization at a given pressure and then calculated the Fermi surface over a dense k-grid to determine the orbit frequencies using skeaf code 14 . Our results are summarized in Tables 3 and 4. We note that most of the orbit-frequencies do not change much with applied pressure but the cup-orbit frequency increases with increasing pressure. As discussed in the main text, the slope of the frequency increase with pressure is in excellent agreement with the observed shifts Supplementary in the experimental measurements. Hence, we identified this orbit as the one probed in our quantum oscillation measurements. The topology of the Fermi surface pretty much stays the same with applied pressure up to 16 kbar (i.e. 1.6 GPa). Due to the reduction of the lattice constants the band overlap gets larger with increasing pressure, which in turn increases the orbit frequencies. However, at pressures larger than 1.6 GPa, due to strong inter-layer interaction, the hole-band with the square box shape starts to have an opening at the top/bottom of the box-surface as shown in Figure 8. For comparison, we show the Fermi surface at 20 kbar for both phases in Fig. 8 but we note that at these pressures, the main phase is the 1T' phase which is inversion symmetric. Interestingly the new orbit at the top of the hole-square band has about the same oscillation frequency as the cup-orbit near 0.65 kT.  Figure 5: Fermi surface plots of MoTe 2 . Fermi surface plots of MoTe 2 for various cases; (a) Experimental structure with and without a U term; (b) Experimental structure with U = 3 eV; the main effect of which is to remove the (cyan) electron pockets near the Y -point; (c) Fully optimized structure with U =3 eV; Note that it is quite different than the one from the experimental structure shown in b; (d) Fully optimized structure with U = 3 eV and also with spin-orbit (SO) coupling; The main effect of spin-orbit coupling is to shrink and expand the surfaces so that they split; Note the significant shrinkage of the surface shown as dark blue color;

Supplementary
(e-f) shows the Fermi surface in the 1T' phase with optimized structure and U =3 eV. Due to inversion symmetry, there is no splitting of the Fermi surface in the 1T'-phase but the bands are shifted around and the resulting Fermi surface is different. Also note that the small electron packet (dark blue in (e)) is removed with the inclusion of SO coupling (f). The red one is a square like box shape with mainly hole-character. We denote the orbits around this surface as "s". The main portion of the Fermi surface is electron-like and has a coffee mug shape (light blue). This shape supports three possible extreme orbits as shown as "op" (which is at the opening of the surface), as "h" which is the orbit around the handle like shape, and "c" which is the orbit around the cup portion of the Coffee Mug-shaped Fermi surface. Interestingly, this cup-like shape is the most sensitive to the pressure and the frequency of this orbit increases with increasing pressure, in excellent agreement with our measurements. Finally, we have a small pocket of electron Fermi surface (dark blue), which has an ellipsoidal shape and is therefore denoted as "e". In this section, we will consider different systems to see if we can reproduce quantum oscillations with frequency near 1 kT as experimentally observed at the intermediate pressure mixed region.
It is of interest to see what kind Fermi surface and quantum oscillation we have from a finite layers of MoTe 2 as these could be indication of some surface states or maybe the decoupled few layers of MoTe 2 during the experiments. Hence, we have performed DFT calculations for single, double and triple layers of T d -MoTe 2 and look at the quantum oscillations from them. The calculations were done fully self consistently where both atomic positions and lattice parameters were optimized. We used the same cutoff and k-grid as discussed above. The spin-orbit interactions were also included. We used a super-cell with c-axis of 20 Ang, which is large enough to minimize the image interactions due to periodic structure. Figure 9 shows the supercell structures with one, two, and three layers of T d -MoTe 2 and the corresponding Fermi surfaces. In the case of single layer, we have two types of Fermi surfaces; two electron pockets with 0.14 kT orbits and the central hole pocket with 0.3 kT orbit. These electron and hole pockets are the foundation of the Weyl points present in these systems. However, as shown in Figure 9, the Fermi surface is very sensitive to the number of MoTe 2 layers present. In the case of double layer, the hole pocket is pushed down below the Fermi level and only two electron pockets remains. As we shall see below the shape of the Fermi surface from double layer is quite similar to the interface state but the orbits are about twice smaller than 1kT as experimentally observed. The Fermi surface of three layers of MoTe 2 is very different; it has cylindrical electron surfaces with oscillation orbits of 0.48-0.54 kT and small pocket with orbit of 0.04 kT. Based on these few layers of MoTe 2 , it is very unlikely that the observed quantum oscillations at the intermediate phase could be due to surface state or from somehow decoupled few layers of MoTe 2 .

F. Supplementary Note 5: Unsuccessful interface models
We first discuss hypothetical T d -like structures with different stacking orders of the MoTe 2 planes as a possible source of the observed quantum oscillations near 1 kT. As discussed in detail in our previous work 1 , the origin of T' and T d phases is based on how the MoTe 2 planes are stacked on each other. The phase transition is basically driven by the lowest energy phonon mode which is related to b-axis displacement of the MoTe 2 planes with respect to each other. As shown in Figure 10, sliding one plane along the b-axis in the T d phase indicates that there is another local minimum in the potential energy and this is related to the T' phase. Both the energy barrier and energy difference between these two stacking configurations is very small and because of that it is quite likely that we have many stacking faults during the T'-T d phase transition. Hence, one may wonder how the stacking faults would affect the Fermi surface and is it possible that the observed oscillation near 1 kT could be due to dislocated MoTe 2 planes?
In order to test this idea, we calculated the band structure and the Fermi surface of a MoTe 2 phase where the middle plane is shifted by about 0.32 Å along the b-axis which corresponds to the 2nd local minimum in the potential. This new structure is almost degenerate with the T d phase. However, the Fermi surface obtained in this phase is very similar to the perfect T d phase as shown in Fig. 5d with some additional holes on the fermi surface. The quantum oscillation frequencies from this hypothetical MoTe 2 phase are at 0.2 kT, 0.6 kT and near 1.8 kT (hole pocket). Hence, the dislocation of MoTe 2 plane can not explain the experimentally observed high frequency quantum oscillation near 1 kT.
We next consider another type of stacking fault as shown in Figure 11. We note that in the perfect T d phase, the MoTe 2 planes are stacked in a fashion that the Te-distortion is characterized by left (L) and right (R) displacements alternatively as shown in  Converting an "R" type MoTe 2 plane to "L" type cannot be accomplished by simply sliding the plane, because to go from "L" type to "R" type, the Te atoms have to move. Based on our initial reaction path calculations, there is no easy path to go from "L" to "R" plane. Hence, this sort of stacking fault is much more difficult to remove compared to a sliding dislocation. We construct a hypothetical MoTe 2 phase by rotating the middle plane to convert it from "R" to "L" orientation, labeling the new phase as T d -LL which is shown in Fig. 11. When we did full structure optimization in this stacking order of MoTe 2 planes, we obtained a distorted structure with cell angle of about 94.6 o . The Fermi-surface of this hypothetical structure is shown in the right panel of Fig. 11. The Fermi surface is very similar to that of ideal T d or T' phases where the available orbits are around 0.18 kT, 0.55 kT, and 2.2 kT. This structure does not yield any Fermi surface pocket with quantum oscillation near 1 kT. Hence, this stacking of MoTe 2 layers can not explain the observed quantum oscillations near 1 kT either. To summarize, several models based on MoTe 2 stacking faults failed to explain the observed 1 kT quantum oscillation. We also checked the Fermi surface of single, double or triple layers of MoTe 2 , none of which has a Fermi surface pocket that gives a 1 kT quantum oscillations.

G. Supplementary Note 6: Successful interface model
In order to show that we only get quantum oscillations near 1 kT when there is an interface between T d and T' phases, we now consider supercell structures that contain both T d and T' phases. Since the T' phase has a lattice angle of ≈ 93.4 o , we first confirmed that changing this angle to 90 o does not affect the Fermi surface significantly. Once the cell angles are all 90 o , creating a supercell that contains both T d and T' phase is straightforward. We used the lattice parameters of the T d phase at 10 kbar and created a supercell of six layers. We then took the top three layers out and put in three layers of T' phase. We then performed full structure optimization until the forces on all atoms were smaller than 0.01 eV/Ang, including those atoms at the interface. We used 64x32x3 k-point grids to calculate the Fermi surface.
While building a superlattice of T d and T' phases, there can exist two types of interfaces depending on how the MoTe 2 planes stack together. We note that in both T d and T' phases, the Mo atoms move towards the right ("R") or left "L" to lower the local symmetry and the planes are stacked to keep "R"-"L"-"R"-"L" sequence. Hence, when there is an interface between T d and T'-phases, we could keep this stacking order as shown on the left panel of Fig. 12. We call this the LR-type interface. This kind of interface should be easily converted to either T d or T' phase during the structural phase transition because sliding MoTe 2 planes from one minimum to the other does not cost too much energy and the energy barrier for sliding is very small. Hence we did not consider this type of interface further.
The other possibility for the interface is to match the T d and T' phase such that the R-L-R-L stacking ordering is not maintained. This creates a case where there is either L-L or R-R stacking at the interface as shown on the right panel of Figure 12. We note that eliminating this interface is not trivial. Converting a "L"-oriented MoTe 2 plane to a "R" oriented plane can not be achieved by a simple sliding of the plane as in the case of the LR-type interface. Mo atoms have to move to change the R to L orientation. DFT reaction-path calculations estimate that the potential barrier for the R-T transition is of the order of 0.5 eV, which is quite large. Hence, this sort of interface or domain wall will be more difficult to anneal out during the phase transition.
Probably both Mo and Te atoms move in a cooperative way to lower the potential barrier. In order to understand the energy of the two types of interface further, we calculated the interface interaction energy which is defined as The calculated interface energy for both the LR-type interface and the LL-type interface are given in Figure 12. Both interface formation energies are positive, meaning that converting the T d -T' mixed phase into a single phase of either T d or T' is energetically more favorable. To our surprise, the LL-type interface actually has a 7 meV/atom lower energy than the LR-type interface. Hence, between the two possible interfaces, the LL-type interface is energetically favored. This explains why we experimentally observe a mixed phase between the T d and T' phase transition, as eliminating this interface requires passing through a large energy barrier and is therefore not trivial. We next calculated the Fermi surface of the T d -T' supercell as shown in Figure12. Comparing the Fermi surfaces with a perfect 6L superlattice, we determine new states that are shown in the right panel of Figure 12. The larger one has a quantum oscillation orbit at 0.95-1.07 kT, which is in excellent agreement with the measurements peformed in the intermediate phase.
We also have a small Fermi pocket with quantum oscillation near 0.15 kT, which is close to several low frequency oscillations below 0.2 kT observed experimentally. These Fermi surfaces are mainly two dimensional but there is some dispersion along the c-axis as evidenced by the features on the Fermi surface. This small dispersion is due to the finite size of the unit cell, which is about 41.4 Å, almost too large for DFT calculations but not large enough to avoid small interactions between the images of the periodic cell. Even though the periodic superstructure calculation discussed above establishes that the observed quantum oscillation at 1 kT is due to the T d -T' interface, it does not demonstrate the interface states clearly, due to significant interactions along the c-axis and complications due to folding of the bands in the Brillion zone because of the supercell periodicity. To more clearly compare the interface Fermi surface to that of the bulk, we performed a similar calculation but this time we put together the three layers of T d and three layers of T'-phases as slabs rather than as a periodic superstructure as shown in Figure 13. In this way, we bring the T d and T'-slabs towards each other to form the T d and T' interface. By comparing the Fermi surface of the separated T d and T' slabs with that of joined T d -T' slab, we are able to directly identify the interface states.
Figure13a shows the separated (top) and joined (bottom) T d -T' slabs, respectively. We used the lattice parameters of the MoTe 2 T d phase at 10 kbar (i.e. 1GPa) pressure. The c-axis of the cell is taken about 4 times the size of c-axis of T d phase at 10 kbar. This gives about 9.5 Å vacuum when the three layer slabs are separated and 16.3 Å vacuum when they are joined. All the atomic positions were optimized until total forces on atoms are less than 0.01 eV/Å. Once we had the optimized positions, we carried out Fermi surface calculations using 64x34x1 k-point grids with and without spin orbit interactions. The resulting Fermi surfaces are plotted in Fig.13. The main effect of the spin-orbit interaction is similar to the case of the bulk; The small Fermi pockets near (0,k y ,0) are pushed below the Fermi level and other bands are split; As in the case of the bulk pure T d phase, we have two pairs of Weyl points near (±k x ,0,0). Comparing the Fermi surface of the separated and joined slabs clearly shows the new states appearing due to the T d -T' interface, which is shown in Fig.13d for the case of the Spin-Orbit calculation. We have two pockets near 1 kT and 0.1 kT. When we plot the quantum oscillation frequencies as Gaussians with equal intensities, the agreement between experiment and calculation is quite impressive as shown in Fig.13d (bottom panel). Hence our slabcalculations directly show that the interface state is responsible for the high frequency quantum oscillations near 1 kT that we observed in the measurements, consistent with the presence of a TIN.
It is possible that folding back of the bands at the Fermi level due to the increased thickness of the MoTe 2 plane induces some new states, which might coincide with those new states in our finite slab calculation. In order to rule out this possibility, we examined the 2D contour plots of the Fermi surface of bulk T d at 0.8 GPa with various k z values as shown in Fig.14. We T d -Phase P=.8 GPa Fermi Surface Cuts at k z =n/6 (n=0,1,2,3) considered k z = n/6 with some thickness (k z -0.02, k z + 0.02), here n = 0, 1, 2 and 3. The maximum of the extension of the Fermi surface in the k x -k y plane occurs at k z = 0. Overall, contours for other values of k z are smaller. In particular for k z = 1/3 (associated with 3-layers case), the circular contour becomes larger but it is smaller than the one at k z =0, conflicting with our interface calculations. Therefore, folding back of the Fermi surface bands with increased number of layers can not explain the additional orbits that we see in the joined-slab calculations.

H. Supplementary Note 7: Relation to Neutron Scattering Results
As a final note, we point out that we have performed very careful neutron diffraction measurements to characterise the crystal structure. Our neutron data already exclude all the hypothetical structures discussed here. However, it is important to show that even in the absence of neutron diffraction data, none of these potential structures can give rise to a quantum oscillations near 1 kT. Our supercell structure of T d -T' as shown in Fig. 12 is also ruled out by the neutron diffraction measurements. Hence we do not have a superlattice formation with triple c-axis, and the calculated structure of T d -T' shown in Fig. 12 should be considered as an approximate structure of the interface between two crystallites of T d and T'phases. Our slab calculations clearly show that the interface of T d -T' phase is needed to explain the observed high frequency oscillations. This is direct evidence of our picture of the TIN phase discussed in the main text. Here we show angular magnetoresistance measured at ambient pressure and 1.8 K when magnetic field rotates from c axis to b axis as the insert figure in Fig. 15(a). The beat frequency increases with the increasing of θ, the angle between magnetic field and c axis as shown in Fig. 15(b). The cross section areas of two electron bands, F α and F β slightly increase with increasing of θ at 1.8 K.
In order to understand this weak angle dependence of the SdH oscillations, we have calculated the extremal orbits as the magnetic field is turned as in our measurements. The results are summarized in Figure 15(d). The calculated angle dependence is also very weak up to 40 • . Due to spin-orbit splitting, near 40 • -60 • , the calculated orbit frequency is suddenly almost doubled with a resulting complicated orbit that involves both the cup surface and the interior of the mug-shape surface. After this sudden increase, a new orbit is obtained as the interior of the mug-shape surface (see Fig. 15(d) which has, interestingly almost the same magnitude as the original frequency. Hence, the overall angle dependence is very weak, as we found in our measurements. It is unlikely for the electrons to oscillate around such a complicated orbit coherently and therefore one may expect small intensity, which is consistent with decreasing intensity in our measurements.
The angle dependence of SdH oscillations at ambient pressure maps out two 3D topological Fermi surfaces of two electron pockets, analyzed using a multiband LK fitting in equation (11). From the magnetoresistance raw data at ambient pressure, we subtract the second order polynomial background. The FFT spectrum shows three bands, F γ = 32.5 T, F α = 240.5 T, F β = 258.0 T and their second and third order harmonic oscillations. The higher harmonic peaks of F α and F β indicate high quality and homogeneity of the measured sample. The 1.8 K SdH oscillation data were fitted to an equation describing multiple bands of a three dimensional LK formula, resulting in the corresponding Berry phases, (φ γ = π, φ α = 0.88π, and φ β = 0.88π), indicating that T d -MoTe 2 at ambient pressure is a possible three dimensional topological semimetal with 3D topological phase shift, δ = − 1 8 15,16 for electron pocket using equation (11).  The corresponding SdH oscillations were observed by calculating the oscillatory magnetoresistance ∆ρ = ρ − ρ background above 5.5 T to 14 T. (c) The FFT spectrum of SdH oscillations as a function of the frequency shows three pockets, whose oscillation frequencies slightly increase above θ > 50 • . (d) The extracted angle-dependence of the oscillation frequencies along with DFT-calculations. The angle dependence of the quantum oscillations for the cup-shape orbit as the magnetic field is turned from c-axis (left) to b-axis (right). The insets show the Fermi surface and the corresponding orbits; For field direction near 40-60 • , due to two closely related Fermi surfaces due to SO interactions and the complicated the mug-shape surface, the extremal orbit is rather complicated involving both the cup-portion of the surface and the interior of the mug-shape surface, which may explain the weak intensity of the oscillations for field directions near b-axis.

J. Supplementary Note 9: Analysis of Shubnikov-de Haas oscillations
The definitions of phase shift and Berry's phase are derived from the Lifshitz-Onsager quantization relation. By globally fitting the data to the Lifshitz-Kosevich formula using the program Bumps, we calculate uncertainty and correlation of the Dingle temperature and Berry's phase.
The general form of longitudinal conductivity is σ xx = ρxx (ρ 2 xx +ρ 2 xy ) , here, ρ xx and ρ xy are the longitudinal and transverse resisitivity. In MoTe 2 , the longitudinal resistivity is always much larger than transverse (Hall) resistivity ( ρ xx >> ρ xy , three to four orders larger below 2 GPa 20 ) and σ xx ≈ ρ −1 xx . The modified Lifshitz-Onsager quantization relation describes the closed trajectory of a charge carrier by an external magnetic field B as a function of Berry's phase and Zeeman splitting parameter S, Here, A n is the cross-sectional area of the Fermi surface related to the Landau level (LL) n, Berry phase Φ B , and splitting parameter is defined as S = 1 2 g m * m0 . m * is the effective mass of the charge carrier and g is the Landau g factor. To analyze the bulk SdH oscillations signal, we adopt the formula of ρ xx , where and θ = 2π(r( Here, ρ 0 is the nonoscillatory component of resistivity at zero field, and other parameters are Dingle temperature T d , oscillatory frequency B F , cyclotron frequency ω c = eB/m * , Boltzmann's constant k B and harmonic order r. The Landau level-broadening width Γ is caused by scattering and associated with Dingle temperature Γ = πk B T D . To extract the Zeeman splitting, we take the sum of the phase term as cos(θ + rSπ) + cos(θ − rSπ) = 2 cos θ cos(Srπ) = 2 cos(2π(r( sin(θ + rSπ) + sin(θ − rSπ) = 2 sin θ cos(Srπ) = 2 sin(2π(r( Here, 2 cos( 1 2 rπg m * m0 ) is a reduction factor due to the Zeeman splitting. Experimentally, oscillatory frequency (B F ) and effective mass (m*) are determined through the FFT of the quantum oscillations signal and the temperature dependence of the amplitude of the oscillations, typically determined at an extremum. δ is the topological phase shift, which arises from the dimensionality of the Fermi surface, δ = 0 ( or δ = ± 1 8 ) for a 2D (or 3D) system 15,16 . In 3D Weyl semimetals without lattice inversion symmetry, the topological phase shift could be either ± 1 8 or ± 5 8 18 . A nontrivial Berry's phase is defined as |γ − δ| = |1/2 − φ B /2π − δ| between 0 and 1/8. Berry's phase determination is very sensitive to the Zeeman effect and the Fermi surface might distort close to the quantum limit. The coefficients of R (Eqn. 4) are very small for the quantum oscillation frequencies and magnetic fields relevant to our measurements and do not affect our analysis.
Therefore, the longitudinal resistivity is These periodic SdH oscillations in magnetroresistance follow the Lifshitz-Kosevich (LK) formula for three-dimensional systems 17 as, The first order harmonic of the 3D LK formula dominates the oscillation signal 17 , giving If the Zeeman splitting term does not change within the experimental magnetic field (namely, each g factor for each band remains a constant over the experimental field range), this term is decoupled from the phase of oscillations, such that Thus, the g factor only modifies the oscillation amplitude (A 0 ) and will not change the phase of oscillations. Therefore, the Berry's phase and the phase shift δ are decoupled from the g factor and Zeeman parameter. Following convention, the amplitude A is kept positive. Typically, the g factor of these transition metal dichalcogenide are positive. Note that when 4n+1 < gm * m0 < 4n + 3, where n is an integer, Eq. 12 yields a negative value of A 0 . This condition is met in the conventional case when g = 2 and m * m0 = 1. In our fits, this results in a π phase shift, which is captured by the 1 2 term in the equation 1 2 -φ B 2π -δ. We expect that this condition is true also in our analysis. The values of φ B displayed in Supplementary Tables 5 and Supplementary Table  6 correspond to the measured phase shift, with a factor of 1 2 ×(2π) subtracted. An important caveat is that the value of g is not independently known, and it is therefore not known whether the extra factor of 1 2 ×(2π) is always appropriate. However, due to the large variation in m* (Supplementary Table 6), it would require a dramatic coincidence for every band to exhibit a value of gm * m0 that results in a positive amplitude. We demonstrate the LK fits under different field ranges for four pressure cases as shown in figure 16. The consistent π Berry's phases in Table 6 confirm that the g factor for each band is a constant and decoupled from Berry's phase. 22 P (GPa) field range (T) φγ (π) φ α (π) φ β (π)  Table 5: Berry's phase determined over different field ranges. Here, we show that the consistent Berry's phase is consistent for fits performed over different field ranges. The invariant of π Berry's phase suggests that g factor does not change within the field range.

K. Supplementary Note 10: Global fitting-Bumps
Finding the global minimum of fits to the Lifshitz-Kosevich formulam with multiple bands is nontrivial because of the large number of fitting parameters, which may be correlated. Bumps is a set of free and public routines for complicated curve fitting, uncertainty analysis and correlation analysis from a Bayesian perspective 19 . Bumps provides uncertainty analysis and explores all viable minima and finds confidence intervals on the parameters based on uncertainty from experimental errors. First, we applied gradient descent to quickly search local minimum and included the real experimental errors for global fitting of the LK formula, in other words, to the full magnetic field range of measured quantum oscillations. To see the distributions of uncertainty and correlation plots, we ran Bumps for the the rest of the variables: amplitude, Dingle temperature and Berry phase, and found convergence of these fitting parameters.
Bumps includes Markov chain Monte Carlo (MCMC) methods 19 to compute the joint distribution of parameter probabilities. The MCMC explores parameter space using a random walk and requires hundreds of thousands of function evolutions to explore the search space. The histogram range represents the 95 % credible interval, and the shaded region represents the 68 % credible interval. For full uncertainty analysis, Bumps uses a random walk to explore the viable parameter space near the minimum, showing pair-wise correlations between the different parameter values. The 2D correlation plots indicate the correlation relationship between multiple parameters in the fitting function. We checked the convergence of fits and compared different local minima to get the global minimum. Figure 16 shows the best fit curves of SdH oscillation signal from 1 atm to 1.8 GPa. We show the expectation value and uncertainties for fitting data at 1 atm (Fig. 17), 0.6 GPa (Fig. 18), 0.9 GPa (Fig. 19) and 1.8 GPa (Fig. 20). 23 P (GPa) label F (T) T d (K) m * (me) φ (π) µ(cm 2 /vS)   We obtained oscillation frequency F from FFT, effective mass m * and corresponding quantum mobility µ from the temperature dependence of FFT. Here, the sum of Berry's phase and phase shift, φ ( φ = 1 π φ B ± δ and phase shift determined by the dimensionality of Fermi surface, δ = 0 or δ = ± 1 8 for 2D or 3D ) could be directly determined from the LK formula.

24
Supplementary Figure 17: The uncertainty and correlation plot of fitting parameters in LK formula at ambient pressure. This is the (a) uncertainty of Lifshitz-Kosevich formula fitting to MoTe 2 data at 1.8 K and 1 atm. The Berry's phases of the γ(Gamma1), α(Gamma2), and β(Gamma3) oscillations are (0.97 ± 0.02)π, (0.89 ± 0.01)π and (0.88 ± 0.01)π. (b) The 2D correlation plots between each pair of parameters in the fits. There is positive correlation between maximum oscillation amplitude and Dingle temperature. The Berry's phase shows no correlation with Dingle temperature and oscillation amplitude.