Switching of band inversion and topological surface states by charge density wave

Topologically nontrivial materials host protected edge states associated with the bulk band inversion through the bulk-edge correspondence. Manipulating such edge states is highly desired for developing new functions and devices practically using their dissipation-less nature and spin-momentum locking. Here we introduce a transition-metal dichalcogenide VTe2, that hosts a charge density wave (CDW) coupled with the band inversion involving V3d and Te5p orbitals. Spin- and angle-resolved photoemission spectroscopy with first-principles calculations reveal the huge anisotropic modification of the bulk electronic structure by the CDW formation, accompanying the selective disappearance of Dirac-type spin-polarized topological surface states that exist in the normal state. Thorough three dimensional investigation of bulk states indicates that the corresponding band inversion at the Brillouin zone boundary dissolves upon the CDW formation, by transforming into anomalous flat bands. Our finding provides a new insight to the topological manipulation of matters by utilizing CDWs’ flexible characters to external stimuli.

S ince the discovery of topological insulators, a wide variety of topological phases have been intensively developed and established in realistic materials 1,2 . The upcoming target is to explore the guiding principles for the manipulation of these topological states. A key parameter to characterize the topological nature of materials is the band inversion realized by the crossing and anti-crossing of energy bands with opposite parities. In bulk materials, the control of band inversion has been mostly done through the tuning of spin-orbit coupling (SOC) by element substitution 3,4 , with some exceptions related to the topological crystalline insulators 5 . In the present work, we focus on charge density wave (CDW), i.e., the spontaneous modulation of charge density and lattice that modifies the periodicity and symmetry of the host crystal. In TaS 2 , for example, the Star-of-David type CDW superstructure induces the effective narrowing of valence bands thus causing the Mott transition [6][7][8] . From this viewpoint, CDW could also modify the band structures that involve the band inversion, and induce topological change. Moreover, it is worth noting that CDW can be flexibly controlled by external stimuli.
The transition metal dichalcogenides (TMDCs) are a wellknown family of layered materials that host a variety of CDWs reflecting their quasi-two dimensionality 9 . Manipulations of CDW states have been intensively investigated and realized, especially in the aforementioned archetypal material TaS 2 , by various stimuli, such as pressure 7 , electric field 10 , and optical pulse 11,12 . The feasibility of CDW thus paves the way toward triggering the exotic phase transitions and generating new functions. More recently, there has been increasing interest in SOC effect in TMDCs [13][14][15][16][17][18] . Especially in the tellurides, (Mo,W) Te 2 [14][15][16] and (Ta,Nb)IrTe 4 17,18 are reported to be topological Weyl semimetals. Here, the peculiar quasi-one-dimensional chain-like structures inherent to tellurides, as well as their stronger SOC compared with selenides and sulfides, are essential in realizing the topologically non-trivial states. In this stream, we focus on the telluride material VTe 2 , to investigate the interplay of band topology and CDW instability. VTe 2 has CdI 2 structure in the high-temperature normal 1T phase, consisting of trigonal layers formed by edge-sharing VTe 6 octahedra (Fig. 1a). With cooling, it undergoes a phase transition to the CDW phase at around 475 K, appearing as a jump in the temperature-dependent resistivity 19 . The resulting CDW state exhibits a (3 × 1 × 3) superstructure characterized by double zigzag chains of vanadium atoms ( Fig. 1d; hereafter we refer it as 1T″ phase, see also Supplementary Note 1) [19][20][21] . This superstructure commonly appears in group-V transition metal ditellurides MTe 2 (M = V, Nb, Ta) 20,22 . Its relation with CDW has been discussed to originate from the peculiar chemical σ-bonding among t 2g d-orbitals connecting three adjacent M sites, in, for example, other TMDCs including TaS 2 23,24 and purple bronze AMo 6 O 17 (A = Na, K) [25][26][27] . Previous band calculation on NbTe 2 , on the other hand, implies the importance of Fermi surface nesting and electron-phonon coupling as the origin of CDW with 1T″ lattice distortion 28 . However, the modifications of electronic structure by this CDW transition have not been systematically investigated so far. It is worth noting that the contraction of metal-metal bond length (Δa max /a) via the CDW transition in VTe 2 is fairly large (~9.1%) 20 and comparable with that in TaS 2 (~7.0%) 29 . We also note that the CDW in VTe 2 can be optically manipulated, as reported by recent time-resolved diffraction studies 30,31 .
In this Article, we investigate the electronic structures of VTe 2 , by employing spin-and angle-resolved photoemission spectroscopy (ARPES) and first-principles calculations. We focus on the modifications of the bulk bands through the CDW formation, and its relevance to the Dirac surface states stemming from the topological band inversion. Experimentally it is not easy to perform highly precise ARPES measurement on the normal 1T phase of pristine VTe 2 (>475 K), since such a high temperature may cause the sample degradation. Therefore we start from TiTe 2 showing a simple 1T structure down to the lowest temperature 32 , and develop the single crystalline V 1−x Ti x Te 2 to access both 1T and 1T″ phases at appropriate temperatures. Figure 1f displays the schematic electronic phase diagram of V x Ti 1−x Te 2 , based on the temperature-dependent ARPES measurements. With increasing Ti, the CDW phase transition becomes gradually suppressed to the lower temperature region (see Supplementary Note 2). For investigating both the normal and CDW phases within a single sample, we synthesized the minimally Ti-doped single crystals (0.10 ≤ x ≤ 0.13) showing the transition just below room temperature (280-250 K). To discuss the electronic structures in momentum space, we introduce the following notation of the Brillouin zones (BZ). In 1T, the 2-dimensional (3-dimensional) BZ is represented by the hexagonal plane (prism) reflecting the trigonal symmetry (Fig. 1b, c). In 1T″, the BZ changes into a smaller one of lower symmetry as indicated in Fig. 1e. In this paper, we use the 1T BZ notation to present the band structures in 1T and 1T″ in a common fashion. To account for the in-plane anisotropy of vanadium chains that is essential in 1T″, we define K 1 , K 2 and M 1 , M 2 as depicted in Fig. 1e (see Supplementary Note 1 for the details of BZ).

Results
Overviewing the electronic structures and modification by CDW. Let us start by briefly overviewing the anisotropic modification of electronic structures from 1T to 1T″ by presenting the ARPES data successfully focused on a CDW single-domain region. Figure 2a shows the ARPES image for the normal-state 1T-V 0.87 Ti 0.13 Te 2 taken at 300 K with a He-discharge lamp (photon energy hν = 21.2 eV). We find V-shaped band dispersions along K À M À K that clearly cross the Fermi level (E F ). Looking at the higher binding energy (E B ) region, these V-shaped bands are connected to the Dirac-cone-like bands reminiscent of surface states in topological insulators, with the band crossing (Dirac points) at M. The topological character of these Dirac bands will be discussed later. On the other hand, Fig. 2b displays the ARPES results on a single-domain CDW state in 1T″-VTe 2 (200 K, synchrotron light hν = 90 eV). Because of the zigzag type CDW formation, the system now loses the threefold rotational symmetry, and the three equivalent M points in 1T turn into one M 1 and two M 2 . Here, the V-shaped band and Dirac-like bands remain at M 1 , whereas at M 2 side the unusual flat band is observed and the Dirac-like state is vanished. Thus, the 1T-1T″ CDW transition induces the huge directional change of electronic structure accompanying the selective disappearance of Dirac-like states. In the following, we discuss these band structures in detail, by comparing with band calculations. Figure 3a, b respectively displays the bulk band calculation for the normal-state V 0.87 Ti 0.13 Te 2 along high symmetry lines (Γ-K-M-Γ) and the Fermi surface at k z = 0 (see Supplementary Note 3). They are characterized by the circular and triangular hole Fermi surfaces respectively around Γ and K. The ARPES results on the normal-state V 0.90 Ti 0.10 Te 2 (350 K, marked as "#1" in Fig. 1f) are shown in Fig. 3c, d. Here a He-discharge lamp (hν = 21.2 eV) is used as the light source. The black broken curves in Fig. 3c are the guides for the bands plotted by tracking the peaks of the energy/momentum distribution curves (EDCs/MDCs) (see Supplementary Note 4). We find hole-like bands centered at around Γ and K points, in a qualitative agreement with the calculation in Fig. 3a (We note that the tiny electron pocket at Γ is not clearly detected, probably reflecting the finite energy mismatch of the bands compared with the calculation.). In the EDCs along the M À K line (Fig. 3d), we can clearly trace a dispersive band that crosses E F .
To grasp the essential electronic modification via the CDW formation, we survey the temperature-and doping-dependent ARPES results (He-discharge lamp, hν = 21.2 eV). Note that the 1T″ phase inevitably contains the in-plane 120-degree CDW domains reflecting the threefold symmetry of 1T, and usually ARPES measurements include the signals of multiple domains (see Supplementary Note 5). Figure 3e, f respectively shows the ARPES image and corresponding EDCs of V 0.90 Ti 0.10 Te 2 in the CDW 1T″ phase (multi-domain, 20 K, #2). Comparing with the hightemperature 1T phase (#1), we notice a very unusual flat band appearing at E B~0 .1-0.25 eV as denoted by the red curve in Fig. 3e, spreading over the measured momentum region. This is most well recognized in Fig. 3f as the series of EDC peaks at E B~0 .2 eV around the K 1 = K 2 point. The blue broken curve in Fig. 3e, on the other hand, indicates the dispersive band crossing E F resembling the high-temperature normal phase (see Supplementary Note 6 for the detailed temperature-dependence). Such coexistence of flat/dispersive bands results from the mixing of CDW domains. In the pristine 1T″-VTe 2 (multi-domain, 15 K, #3) as shown in Fig. 3g, h, the similar flat band with slightly different energy and dispersion is more clearly observed (the red broken curve), together with the 1Tlike dispersive band (the blue broken curve). The appearance of this anomalous flat band is thus the common signature of the CDW 1T″ phase, which however is seemingly beyond the simple band folding and gap opening in the Fermi surface nesting scenario. The localized nature of this electronic structure will be discussed later.
Bulk and surface band structures in 1T normal phase. Here we introduce the topological aspect that can be relevant to the 1T-1T″ CDW transition. In the normal 1T phase, the band calculation suggests the band inversion involving V3d and Te5p orbitals at around the M and L points. The calculations of 1T-V 0.87 Ti 0.13 Te 2 at several k z , plotted along the direction parallel to Γ-M (k ΓM , see (~0.76 π/c). In the corresponding (0 0 1) slab calculation (Fig. 4c), a new band dispersing around E − E F = −0.6 eV near M can be recognized, that does not exist in the bulk calculations. It roughly follows the trajectory of the virtual crossing points of bulk bands A and B for no SOC. This is a surface state topologically protected by the band inversion at (a*/2, 0, k z ) occurring due to the moderate k z dispersions of V3d and Te5p bands. Indeed, in the K À M À K direction (Fig. 4d), this surface state shows a Dirac cone-like dispersion connecting bands A and B. Figure 4e shows the ARPES image of normal-state V 0.90 Ti 0.10 Te 2 along K À M À K (350 K, hν = 21.2 eV). By carefully analyzing the EDC/MDC (see Supplementary Note 7), we can indeed quantify the bottom of bulk band A (E B~0 .30 eV), the top of bulk band B (~0.90 eV), and the crossing point of the surface Dirac cone (DP,~0.66 eV). We note that this bulk band A forms the triangular Fermi surfaces centered at K points in the k z = 0 plane, as shown in the Fermi surface image in Fig. 4f (V 0.87 Ti 0.13 Te 2 , 300 K, hν = 83 eV).
We further perform the hν-dependent ARPES to confirm the two-dimensionality of the Dirac surface states and to clarify the k z -dependent bulk electronic structure that is relevant to the band inversion. Figure 4h-j displays the ARPES images of 1T-V 0.90 Ti 0.10 Te 2 near the M point (see the red arrow in Fig. 4g), recorded at 320 K with different photon energies, hν = 63, 69, and 78 eV, respectively corresponding to k z~0 , π/2c, and π/c. Circle markers with vertical (horizontal) error bars represent the peak positions of EDCs (MDCs) (see Supplementary Note 7). Figure 4k displays the schematic band dispersions overlaid with the experimental peak plots extracted from Fig. 4h-j. Here we find that the bulk bands A and B clearly show the finite k z -dispersions (respectively larger than 0.1 and 0.4 eV at M). As can be seen in Fig. 4b, this k z -dispersion is essential for the band inversion at (a*/2, 0, k z ), the origin of the topological surface state appearing around the M point. On the other hand, the Dirac surface state, that is highlighted by the overlaid orange curves in Fig. 4k, is almost independent of hν, indicating the two-dimensional nature of the topological surface state.
Bulk and surface band structures in 1T″ CDW phase. To unambiguously elucidate the anisotropic electronic structures in the CDW 1T″ phase, here we utilize the small spot size (typically 300 μm) of the synchrotron light beam and separately measure the in-plane CDW domains of VTe 2 (see Supplementary Note 5). Figure 5a shows the Fermi surface image of the single-domain 1T″-VTe 2 (200 K, hν = 90 eV). We find that the two sides of the triangular Fermi surface around K observed in the normal 1T phase (Fig. 4f) are completely absent in the CDW state, and the remaining one forms the quasione-dimensional Fermi surface marked by the red broken curves. Figure 5b, d displays the ARPES images along representative two cuts (cut #1 and #2 as denoted in Fig. 5a), nearly along K 1 À M 1 À K 1 and K 1 À M 2 À K 2 , respectively. Though they are originally the equivalent momentum cuts in the normal 1T phase, distinctive features are clearly observed. Along K 1 À M 1 À K 1 , there are several bands crossing E F including the 1T-like dispersion (the black broken curve), together with the Dirac-cone-like band (the white broken curve) as clearly seen in the MDCs (Fig. 5c). On the other hand, along K 1 À M 2 À K 2 , the bands crossing E F as well as the Dirac band completely disappear (see the MDCs in Fig. 5e), and the peculiar flat band (the black broken curve in Fig. 5d) appears around E B~0 .2-0.3 eV. These indicate that the CDW induces the drastic directional modification of electronic structure.
Here, to confirm the topological nature of the Dirac surface state in VTe 2 , we performed spin-resolved ARPES on a multidomain sample (see Supplementary Note 8 for the experimental setup). Figure 5f depicts the schematic 2D BZ with CDW multidomains, together with the measurement region (the red arrow). Figure 5g shows the spin-integrated ARPES image with the peak plots obtained from spin-resolved spectra (15 K, hν = 21 eV, s polarization). By comparing with the results from the singledomain measurement (Fig. 5b-e), these peaks can be easily assigned to either M 1 or M 2 side. The red (blue) and purple (cyan) triangle markers respectively represent the peak positions of spin-up (-down) spectra at M 1 and M 2 sides. As shown in the spin-resolved spectra in Fig. 5h, the lower branch of the Dirac cone band clearly shows the spin polarization with sign reversal at Similarly to the normal phase, hν-dependent ARPES measurement is performed on the CDW state VTe 2 (15 K, multi-domain) to clarify the k z -dependence of electronic structures. which respectively detects at k z = 0, π/2c, and π/c plane. The circles with vertical (horizontal) error bars represent the EDC's (MDC's) peak positions. k Schematic band dispersion along K À M À K with experimentally obtained peak plots. The red, green, and blue markers are the peak plots for hν = 63, 69, and 78 eV, respectively from h-j. The orange curve represents the schematic of the Dirac surface state (DSS), whereas the blurred gray curves are the bulk bands (Bulk A and B). NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-16290-w ARTICLE NATURE COMMUNICATIONS | (2020) 11:2466 | https://doi.org/10.1038/s41467-020-16290-w | www.nature.com/naturecommunications the schematic band dispersions along K 1 À M 1 À K 1 and K 1 À M 2 À K 2 , overlaid with the experimental peak plots extracted from Fig. 5j-l. At the M 1 side, the V-shaped band with high intensity clearly shows the finite k z -dispersion of >0.2 eV, together with the hν-independent Dirac surface state. These are similar to the case of normal 1T phase along K À M À K (Fig. 4k). On the other hand, at the M 2 side, the emergent flat band shows negligible k z -dependence, in contrast to the V-shaped band observed at the M 1 side. With this we can conclude that the electronic state at the M 2 side has unusually localized nature, which should give rise to the dissolution of the band inversion along k z direction.
Picture of CDW state based on orbital bonding. To grasp the essential feature of the CDW state, here we introduce the local orbital picture. For simplicity, we adopt an orthogonal octahedral i Spinresolved spectra near E F at θ = ±4 • . j-l hν-dependent ARPES images of a multi-domain sample recorded with 54 eV (j), 61.5 eV (k), and 69 eV (l) photons (s polarization, 15 K). Circles markers with vertical (horizontal) error bars represent the peak positions of EDCs (MDCs). The black and red markers are respectively assigned to the dispersive bulk band and the Dirac surface state at M 1 side, while the white markers correspond to the flat bulk state at M 2 side. m Schematic band dispersion along K 1 À M 1 À K 1 with experimental band plots. The orange curve represents the schematic of the Dirac surface state, whereas the blurred gray curves are those for the bulk bands. n Same as m, but along XYZ coordination by considering the VTe 6 octahedron as shown in Fig. 6a, and focus on V3d t 2g (d XY , d YZ , d ZX ) orbitals that dominate the density of states near E F (see Supplementary Note 10). Note that this is different from the global xyz setting adopted in Fig. 4b, where z corresponds to the stacking direction.
Here we choose Z as the V-Te bond direction that is perpendicular to the vanadium's chain direction b m . Figure 6b shows the calculated partial density of states (PDOS) for vanadium d XY , d YZ , and d ZX orbitals. In 1T (Fig. 6b left), they are naturally degenerate reflecting the trigonal symmetry. In 1T″ (Fig. 6b right), on the other hand, d YZ /d ZX and d XY orbitals have strikingly different distributions. We can classify the PDOS for 1T″ into three major parts, lower, middle, and upper, respectively lying around E B~0 .4, −0.5, and −1.3 eV. They are indicative of bonding (B), nonbonding (NB), and antibonding (AB) bands, arising from the trimerization-like displacements of three adjacent vanadium atoms formed by the σ-bonding of d YZ / d ZX orbitals 23,33 , as marked by the pink oval in Fig. 6a for d YZ . Through the d YZ /d ZX trimerization, the remaining d XY stays relatively intact, thus its PDOS mostly contributes to the middle band. Here, the d YZ /d ZX bonding band makes a peak in PDOS at around E B~0 .3 eV, corresponding to the flat bands lying around Γ À K 2 À M 2 (see Supplementary Note 10 for the band dispersions). The experimentally observed flat band in Fig. 5d should be thus reflecting the localized nature of d YZ /d ZX trimers.
Such orbital bonding picture is also useful for the intuitive understanding of Fermi surfaces at BZ boundary. Figure 6c depicts the schematics of the Fermi surfaces with the locations of the Dirac crossing points (yellow circle makers at M points). According to the calculation in 1T, the three sides of the triangular Fermi surface mainly consist of d YZ , d ZX , and d XY orbitals, respectively (see Supplementary Note 10). This can be regarded as the virtual combination of one-dimensional Fermi surfaces formed by the σ-bonding of d YZ , d ZX and d XY orbitals, raised as the basic concept of the hidden nesting scenario in CdI 2type TMDCs 23 and purple bronzes AMo 6 O 17 (A = Na, K) [25][26][27] . In the CDW 1T″ phase, the two sides of the Fermi surfaces composed of d YZ /d ZX orbitals ( M 2 sides) turn into the localized flat bands as a consequence of the vanadium trimerization, while the one composed of d XY remains intact. This explains the quasi-2D to quasi-1D change of the triangular Fermi surface at BZ boundary as obtained in the present result. Figure 6d summarizes the observed anisotropic reconstruction of the bulk and topological surface states occurring around the BZ boundaries by the CDW formation. As discussed, the band structures observed in the normal 1T phase are basically in a good agreement with our calculations, indicating the triangular hole Fermi surfaces around the K points. At M points, the Dirac-type topological surface states exist in addition to the V-shaped bulk bands. We attribute this to the band inversion of V3d and Te5p orbitals occurring at (a*/2, 0, k z ), which can be traced a V 3d PDOS (arb.units)  (Fig. 4k). In the CDW state, on the other hand, drastic changes show up in the electronic structure. The originally triangular hole Fermi surface around the K point loses its two sides, and the remaining one forms the one-dimensional-like Fermi surface at the M 1 side. We note that this corresponds to the vanadium zigzag chain direction. At the M 2 side, the V-shaped bulk band transforms into the flat band at E B~0 .25 eV, reflecting the vanadium trimerization. The original Dirac surface state survives at M 1 , but can be no longer seen at M 2 . The disappearance of the Dirac surface state can be ascribed to the absence of the band inversion at M 2 , where the k z -dependence of the bulk state is lost due to the flat band formation (Fig. 5n), in contrast to the M 1 side (Fig. 5m). Thus, in this system, the CDW accompanying the metal trimerization plays a crucial role in selectively switching off the band inversion and corresponding topological surface state.

Discussion
Such a drastic modification of band inversion by CDW can be further explained based on the change in crystal symmetry. In the high-temperature 1T phase P 3m1 ð Þ, as mentioned earlier, the Dirac cones appear at the M points, due to the band inversions involving the Te5p and V3d orbitals occurring along M-L lines. Here, the M-L line resides in a mirror plane, which prohibits the hybridization between the band consisting of Te5p antibonding orbitals (odd with respect to the mirror plane) and that derived from the predominantly V3d with finite Te5p bonding orbitals (even with respect to the mirror plane), without the help of SOC. Eventually, the non-relativistic bands cross at k z~0 .76 π/c along M-L depicted as the broken curves in Fig. 4b, thus causing the band inversion, due to the k z dependent energy eigenvalues. Only by the SOC, they get mixed and make a gap (~200 meV) around the crossing point. In the low-temperature 1T" phase (C2/m), on the other hand, while the M 1 point keeps the similar situation with the M point in high-temperature phase, M 2 no longer resides in the mirror plane. At M 2 , the V3d and Te5p (bonding, antibonding) can now significantly mix and form the well hybridized flat band with no k z dependence, leading to the disappearance of the band inversion. This can be also confirmed by evaluating the orbital components in our bulk calculation (see Supplementary note 11). It indicates the significant d YZ /d ZX -p Z hybridization (here X, Y, Z represent the octahedral setting, see Fig. 6a) forming the flat band at E B~0 .3-0.4 eV. In contrast, the mirror plane remains for Γ À M 1 , thus the band inversion and the related Dirac surface state can sustain.
Finally, we would like to mention that the topological character of materials should be determined by considering the detailed band structures for the whole system and quantifying the Berry curvature and topological invariants (Z 2 index, (spin) Chern number, etc). The difficulty in the present case, however, is that the accuracy of the band calculation on 1T″ phase is still insufficient. Though the qualitative features (such as the quasi-onedimensional Fermi surface and the flat band formation) are quite well reproduced (see Supplementary Note 12 for the band unfolding calculation), the detailed band structures including their relative energy positions are not in good enough agreement. This makes it difficult to fully discuss the topology which sharply depends on the crossings of band energy levels, particularly on its modification across the CDW transition. In this viewpoint, we would rather like to stress that our present results and microscopic analysis are offering one way to pursue the topological character of a complex material that is still difficult to predict.
In conclusion, we systematically clarified the bulk and surface electronic structures of (V,Ti)Te 2 by utilizing ARPES and firstprinciples calculations. We revealed that the strongly orbitaldependent reconstruction of the electronic structures occurs through the CDW formation, giving rise to the topological change in particular bands. Considering that CDWs are often flexible to external stimuli 7,10-12 , we can also expect to manipulate the CDW-coupled topological state. The effect of thinning the crystals down to atomic thickness is also worth investigating, which may give rise to hidden CDW states 10 and electronic phase transitions 34,35 . The combination of CDW ordering and topological aspects may lead to the new stage of manipulating the quantum materials.

Methods
Sample preparation. High-quality single crystals of V 1−x Ti x Te 2 were grown by the chemical vapor transport method with TeCl 4 as a transport agent. Temperatures for source and growth zones are respectively set up as 600°C and 550°C. The Ti concentrations (x) were characterized by energy dispersive x-ray spectrometry (EDX) measurements. To access both 1T and 1T″ phases, we used V 0.90 Ti 0.10 Te 2 , V 0.87 Ti 0.13 Te 2 , and VTe 2 samples with the transition temperature of~280 K, 250 K and~475 K, respectively.
ARPES measurements. ARPES measurements were made at the Department of Applied Physics, The University of Tokyo, using a VUV5000 He-discharge lamp and a VG-Scienta R4000 electron analyzer. The photon energy and the energy resolution are set to 21.2 eV (HeIα) and 15 meV. The spot size of measurements was~2 × 2 mm 2 . The data in Figs. 2a, 3c-h and 4e were obtained with this condition. Photon-energy-dependent and domain-selective ARPES measurements were conducted at BL28A in Photon Factory (KEK) by using a system equipped with a Scienta SES2002 electron analyzer. Photon-energy-dependent measurements on 1T-V 0.90 Ti 0.10 Te 2 and 1T″-VTe 2 were performed respectively using circular and spolarized light (hν = 45-90 eV). The energy resolution was set to 30 meV. The relation between the incident light energy and the detected k z value is estimated from the experimentally obtained work function (~3.7 eV) and inner potential (~8.8 eV). The data in Figs. 4h-j and 5j-l were obtained in this condition. Domain-selective measurements on 1T″-VTe 2 were performed at 200 K using a 90 eV circularly polarized light with a spot size of~100 × 300 μm 2 . The energy resolution was set to 50 meV. The data in Fig. 5b, d were obtained from the different in-plane domains of a single sample, by translating the sample without changing its orientation. The Fermi surface images in Fig. 4f, 5a were also obtained with this instrument.
Spin-resolved ARPES measurements were performed at Efficient SPin Resolved SpectroScOpy end station attached to the APPLE-II-type variable polarization undulator beamline (BL-9B) at the Hiroshima Synchrotron Radiation Center (HSRC) 36 . The analyzer of this system consists of two sets of very-low-energy electron diffraction spin detectors, combined with a hemispherical electron analyzer (VG-Scienta R4000). Measurements of 1T″-VTe 2 were performed using a s-polarized 21 eV light at 15 K. In the spin-resolved measurements, we set the energy and angle resolutions to 120 meV and ±1.5 deg, respectively. We adopted the Sherman function S eff = 0.25 for analyzing the obtained data. The data in Fig. 5g-i were obtained with this condition.
In all the ARPES experiments, samples were cleaved at room temperature insitu, and the vacuum level was better than 5 × 10 −10 Torr through the measurements. The Fermi level of the samples were referenced to that of polycrystalline golds electrically connected to the samples.
Band calculations. The relativistic electronic structure of 1T-V 0.87 Ti 0.13 Te 2 was calculated within the density functional theory (DFT) using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional corrected by the semilocal Tran-Blaha-modified Becke-Johnson potential, as implemented in the WIEN2k package 37 . The effect of Ti doping was treated within the virtual crystal approximation 38 . The BZ was sampled by a 20 × 20 × 20 k-mesh and the muffin-tin radius R MT for all atoms was chosen such that its product with the maximum modulus of reciprocal vectors K max becomes R MT K max = 7.0. To describe the surface electronic structure, the bulk DFT calculations were downfolded using maximally localized Wannier functions 39,40 composed of V3d and Te5p orbitals, and the resulting 22band tight-binding transfer integrals implemented within a 200-unit supercell. For the pristine 1T-and 1T″-VTe 2 , the DFT electronic structure calculations were carried out by OpenMX code (http://www.openmx-square.org/), using the PBE exchange-correlation functional and a fully relativistic j-dependent pseudopotentials. We adopted and fixed the crystalline structures of 1T-and 1T″-VTe 2 reported in ref. 20,41 and sampled the corresponding BZ by a 8 × 8 × 8 k-mesh.

Data availability
The datasets that support the findings of the current study are available from the corresponding author on reasonable request.