### 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)^{8,9,10}, 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 Figs. 15–23, 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 (Supplementary Figs. 3–7, 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 first-principles 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 *μ*SR 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 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 high-frequency 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 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 Fermi-surface 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 by ~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.