Saddle-point von Hove singularity and dual topological insulator state in Pt 2 HgSe 3
Barun Ghosh, Sougata Mardanya, Bahadur Singh, Xiaoting Zhou, Baokai Wang, Tay-Rong Chang, Chenliang Su, Hsin Lin, Amit Agarwal, Arun Bansil
SSaddle-point von Hove singularity and dual topological insulator state in Pt HgSe Barun Ghosh, Sougata Mardanya, Bahadur Singh † ,
2, 3
Xiaoting Zhou,
4, 5
Baokai Wang, Tay-Rong Chang,
5, 6
Chenliang Su, Hsin Lin ‡ , Amit Agarwal ‡‡ , and Arun Bansil ∗ Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India SZU-NUS Collaborative Center and International CollaborativeLaboratory of 2D Materials for Optoelectronic Science & Technology,Engineering Technology Research Center for 2D MaterialsInformation Functional Devices and Systems of Guangdong Province,College of Optoelectronic Engineering, Shenzhen University, ShenZhen 518060, China Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA Department of Physics and Astronomy, California State University, Northridge, CA 91330, USA Department of Physics, National Cheng Kung University, Tainan 701, Taiwan Center for Quantum Frontiers of Research and Technology (QFort), Tainan 701, Taiwan Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
Saddle-point van Hove singularities in the topological surface states are interesting because theycan provide a new pathway for accessing exotic correlated phenomena in topological materials. Here,based on first-principles calculations combined with a k · p model Hamiltonian analysis, we showthat the layered platinum mineral jacutingaite (Pt HgSe ) harbours saddle-like topological surfacestates with associated van Hove singularities. Pt HgSe is shown to host two distinct types of nodallines without spin-orbit coupling (SOC) which are protected by combined inversion ( I ) and time-reversal ( T ) symmetries. Switching on the SOC gaps out the nodal lines and drives the system intoa topological insulator state with nonzero weak topological invariant Z = (0; 001) and mirror Chernnumber n M = 2. Surface states on the naturally cleaved (001) surface are found to be nontrivialwith a unique saddle-like energy dispersion with type II van Hove singularities. We also discuss howmodulating the crystal structure can drive Pt HgSe into a Dirac semimetal state with a pair ofDirac points. Our results indicate that Pt HgSe is an ideal candidate material for exploring theproperties of topological insulators with saddle-like surface states. I. INTRODUCTION
Finding new topological materials with unique proper-ties is currently drawing intense interest as an open re-search frontier in condensed matter physics and relatedfields.
Initial ideas of time-reversal symmetry ( T ) pro-tected topological states have been generalized to incor-porate crystal symmetries, leading to the identification ofa variety of new topological states in insulators, semimet-als, and metals. Examples include mirror-symmetryprotected topological crystalline insulators (TCIs), weaktopological insulators (WTIs), Dirac/Weyl semimetals,nodal line semimetals, hourglass semimetals, triple-pointsemimetals, among others.
Theoretically predictedtopological properties of a number of materials havebeen demonstrated experimentally via spectroscopic andtransport measurements.
It has been recognized thata topological state can also be protected simultaneouslyby different crystal symmetries as is the case in Bi (Se,Te) where the protection involves both T and crystallinemirror symmetries. Such dual symmetry protectedtopological states can open up new possibilities for tuningtopological properties via controlled symmetry breaking.Topological surface states (TSSs) are the hallmarkand source of numerous useful properties in topologi- ∗ Corresponding authors’ emails: [email protected], [email protected], [email protected], [email protected] cal quantum materials. Depending on the symmetriesof their crystalline surfaces, the electronic dispersion( E k ) of TSSs can deviate substantially from the well-known Dirac-like form. Specifically, when a surfacelacks C ny with n > E k dispersion with saddle points is allowed by symme-try constraints. These saddle points in k -space lead toVan Hove singularities (VHSs) where densities of states(DOSs) diverge logarithmically in two-dimensions (2D).The importance of VHSs has been revived recently inthe theory of correlated twisted bilayer graphene and,in fact, the new concept of higher order VHSs has beenproposed. More generally, when VHSs lie close tothe Fermi level, the increased DOS amplifies electroncorrelation effects that can drive various quantum many-body instabilities involving the lattice, charge and spindegrees of freedoms.
When these VHSs lie at generic k points, they favor an odd-parity pairing, which canlead to unconventional superconductivity in the topolog-ical matrials. Despite theoretical prediction of TSSswith VHSs, experimental evidence of such states is stilllacking. The identification of new materials with saddle-like TSSs is thus of great importance.In this paper, we investigate the topological elec-tronic structure of layered platinum mineral jacutingaitePt HgSe and reveal a dual-symmetry-based protectionof its topological state and the existence of saddle-pointVHSs in the surface electronic spectrum. The monolayerPt HgSe has been predicted recently as a large band gapKane-Mele quantum spin Hall (QSH) insulator. A non- a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y trivial band gap of 0.53 eV was reported within the G W approximation: its Fermiology under electron and holedoping suggests the existence of VHSs and unconven-tional superconductivity. The QSH state in Pt HgSe monolayer has been experimentally demonstrated usingscanning tunneling microscopy (STM). Also, it is foundthat few nanometers thick as well as bulk jacutingaite isstable under ambient conditions on a timescale of monthsand even to a year. However, the bulk topological stateand the associated TSSs with VHSs remain unexplored.Our analysis reveals that Pt HgSe supports two dis-tinct types of nodal lines when spin-orbit coupling (SOC)effects are ignored. Including SOC effects in the com-putations gaps out the nodal lines and drives the sys-tem into a topological insulator state characterized bynonzero weak topological invariants Z = (0; 001), as wellas the mirror Chern number n M = 2. To highlight thenontrivial bulk band topology, we calculate the naturallycleavable (001) surface electronic structure and show theexistence of a unique symmetry allowed saddle-like E k dispersion of topological surface state with saddle-pointVHSs. Informed by our first-principles computations, wepresent a viable k . p model Hamiltonian for the topolog-ical surface states. We also investigate the effect of hy-drostatic pressure on bulk band topology and discuss atopological phase transition to a type-II Dirac semimetalstate with pressure. Our results reveal that Pt HgSe isan ideal material for exploring saddle-like surface stateswith VHSs in experiments.The remainder of the paper is organized as follows.In Sec. II, we discuss computational details along withthe crystal structure of Pt HgSe . The bulk topolog-ical properties are discussed in Sec. III. In section IV,we characterize the topological state and present sur-face electronic structure with and without SOC. The k . p model Hamiltonian for the topological surface states isdescribed in Sec. V. In Sec. VI, we present the evolu-tion of topological electronic structure under hydrostaticpressure. Finally, we summarize our findings in Sec. VII. II. COMPUTATIONAL DETAILS ANDCRYSTAL STRUCTURE
Electronic structure calculations were performedwithin the framework of density functional theory (DFT)with the projector-augmented-wave (PAW) pseudopoten-tials and a plane wave basis set using Quantum Espressopackage.
We used an energy cut-off of 50 Ry forthe plane wave basis set and a 9 × × k mesh for thebulk Brillouin zone integration. The generalized gradi-ent approximation (GGA) of Perdew, Burke, and Ernzer-hof (PBE) was used to include the exchange-correlationeffects. A tolerance of 10 − Ry was used for electronicenergy minimization. All the atomic positions were op-timized until the residual forces on each atom becomeless than 10 − Ry/au. We constructed a tight bindingmodel Hamiltonian by deploying atom-centered Wannier
FIG. 1: (a) Side and (b) top view of the layered crystal struc-ture of Pt HgSe . Pt(1) and Pt(2) denote two symmetryinequivalent Pt atoms in the unit cell. (c) Bulk and pro-jected (100) and (001) surface Brillouin zones. Various high-symmetry points are marked. functions and computed topological properties using theWannierTools package. We further calculated sur-face electronic spectrum by constructing a ten layer thickPt HgSe slab with a vacuum region of 16 ˚A to avoidinteractions between periodically repeated images usingthe VASP suite of codes. Jacutingaite Pt HgSe forms a bipartite lattice (con-taining two sublattices) with the trigonal space group P ¯3 m The crystal structure is layered whichcan be viewed as a 2 × × withadditional Hg atoms that are placed in the anti-cubo-octahedral voids of Se atoms [Figs. 1(a)-(b)]. Thereare two symmetry inequivalent Pt atoms in the primitiveunit cell that form two distinct hexagonal sublattices.The Pt(1) atom connects to six nearest Se atoms andforms Pt(1)Se local octahedral coordination while thePt(2) atom constitutes Pt(2)Se square structure. The Pt(1)-Se bond length is 2.55 ˚A which is slightlylarger than the Pt(2)-Se bond length of 2.47 ˚A. This crys-tal structure has three-fold rotation symmetry around z -axis { C z | } , inversion symmetry I , and mirror sym-metry { m | } . Additionally, it respects T symmetry. Γ Γ
Hg s Pt d xz Pt d yz Se p x Se p y Se p z Γ Γ Γ ΓΓ Γ (a) (b)(c) (d) A u B B g E g A B u A g A A M +3 , M +4 M , M M , M Γ Γ , Γ A A L ,L L ,L M , M FIG. 2: Bulk band structure of Pt HgSe (a) without and (b)with spin-orbit coupling (SOC). The irreducible representa-tions at time-reversal invariant momentum points are markedfor bands near the Fermi energy. The inset in (a) and (b) showthe closeup of the in-plane energy dispersion near K and H momentum points highlighted by gray boxes in the main fig-ure. A clear gap emerges at K and H points with SOC in(b). (c) and (d) show the orbital resolved band structure ofPt HgSe without and with SOC, respectively. III. BULK ELECTRONIC STRUCTURE ANDTOPOLOGICAL INVARIANTS
The bulk electronic spectrum of Pt HgSe withoutSOC is shown in Fig. 2(a). It is semimetallic in char-acter where the A u ( A g ) symmetry band is seen to crosswith the B g ( B u ) band at the K ( H ) point in the bulkBZ. These band crossings are linearly dispersed over asubstantial energy range along Γ M K
Γ (
ALHA ) planedirections. Similar Dirac-cone-like band features are alsothere in the band structure of graphite and their origin isattributed to the honeycomb lattice arrangements of theconstituents atoms. The orbital resolved band struc-ture in Fig. 2(c) shows that crossing bands are mainlycontributed by Hg s , Se p , and Pt d xz and d yz atomicorbitals. The band structure with SOC is illustrated inFig. 2(b) and Fig. 2(d). The Dirac-cone-like band cross-ings without SOC at K and H points are now gappedand a continuous band gap appears between the valenceand conduction bands.In order to characterize the nodal lines and their sym-metry protection, we systematically examine the bandcrossings in Fig. 3. A careful inspection of band cross-ings in full bulk BZ reveals that Pt HgSe hosts two dis-tinct types of nodal lines. The type I nodal lines (identi-fied by NL C ) are generated by accidental band crossingsand form an inversion symmetric pair of closed loops atgeneric k points around the Γ − A line inside the BZ.Importantly, these nodal lines are not hooked to a fixed momentum plane but trace an arbitrary path encirclingthe Γ A line [see red and blue curves in Fig. 3(a)]. Theyshow considerable energy spread in the momentum spaceas illustrated in Fig. 3(b) where the energies of the gapclosing points are plotted with color in the k x − k y − k z momentum space. We further demonstrate these nodalcrossing by plotting the band structure along the in-planedirections for a fixed k z = 0 . πc ) plane in Fig. 3(e).The type II nodal lines (NL KH ) stretch along the K − H high symmetry directions at the hinges of hexagonal BZ[green lines in Fig. 3(a)]. These nodal lines are es-sential and enforced by little group symmetries of the KH line. Notably, KH line is invariant under three-fold rotational symmetry C z and anti-unitary operator IT . For a spinless system, the eigenvalues of C z are1, e + i π , and e − i π . The conjugate symmetry operator IT however enforces a double degeneracy between stateswith e i π and e − i π eigenvalues. We have verified thesesymmetry states based on the first-principles wavefunc-tion analysis. We find that the symmetry adapted basisΨ = ( ψ + , ψ − ) T of the degenerate bands can be expressedas ψ ± = w | p x ± ip y (cid:105) + w | d xz ± id yz (cid:105) + w | d x − y ∓ id xy (cid:105) where w i =1 , , are normalized coefficients. We furtherexplore the nodal line energy dispersion in Figs. 3(c)and 3(g). We emphasize that similar type-II nodal lineshave also been reported in AA stacked graphite, the high-temperature superconductor MgB and it’s iso-structuralcounterparts like AlB . We present the Fermi surface of Pt HgSe in Fig. 3(d)with unique electron and hole pockets. These pocketsoriginate from both NL C and NL KH nodal lines and canlead to unique transport properties solely governed bynontrivial pockets.Figures 3(f) and 3(h) show the energy bands with SOCalong the selective k path of NL C and NL KH , respec-tively. Clearly, the SOC opens an energy gap at the nodalcrossing points, making valence and conduction bandseparated locally at each k point. This band gap openingfacilitates the calculation of symmetry-based indicator(SI) to determine the topological state. Following Ref.[7],the band insulators in space group P ¯3 m Z and a single Z indicator i.e. ( Z , Z , Z , Z ).By explicitly calculating the irreducible representationsof the occupied bands at different time-reversal invariantmomentum points, we find ( Z , Z , Z , Z ) = (0 , , , n M = 2, or a nonzero rotation invariant, n = 1. In both the cases, the inversion invariant has non-zerovalue ( n i = 1). In order to pin down the exact topologi-cal state, we further calculate the mirror Chern number, n M and find it to be 2. The calculated SI and topologi-cal invariants are listed in Table I. Thus, the topologicalphase of Pt HgSe is characterized by both (001) weaktopological invariants and a non-zero mirror Chern num-ber n M = 2. (c) (a) M K Γ KM H Γ K HA E n e r gy (b) K H A-0.8-0.400.40.8 E n e r gy ( e V ) K H A-0.8-0.400.40.8 E n e r gy ( e V ) ΓΓΓ Γ M K -0.8-0.400.40.8 E n e r gy ( e V ) k z =0.54 π /c (e) h + e - (g) (h) (f) k z ( Å - ) -0.4 0.0 0.4 0.0-0.4 0.4 k y ( Å - ) k x ( Å - ) E n e r gy ( e V ) M K -0.8-0.400.40.8 E n e r gy ( e V ) k z =0.54 π /c Γ Γ (d)(c) FIG. 3: (a) Configurations of nodal lines in the bulk BZ and their projection on (100) and (001) surface planes for Pt HgSe .Two distinct types of nodal lines are shown. Type-I nodal lines (NL C ) are located inside the bulk BZ and marked in red andblue colors. Type-II nodal lines (NL KH ) are located along the hinges and shown in solid green color. (b) Energy-momentumspread of NL C in the BZ. (c) Schematics of NL KH structure in the BZ. (d) Fermi surface with electron and hole pockets withoutSOC. Band structure at k z = 0 . πc ) plane (e) without and (f) with SOC. The nodal band crossings are clearly resolved alongΓ − M and Γ − K in (e). (g) and (h) show energy dispersion of NL KH nodal line without and with SOC, respectively.TABLE I: Calculated symmetry indicator and topological in-variants for Pt HgSe .( Z , Z , Z , Z ) ( ν ; ν , ν , ν ) n M n n i (0,0,1,2) (0;001) 2 0 1 IV. SURFACE ELECTRONIC STRUCTURE
We present the electronic spectrum of various surfacesof Pt HgSe in Fig. 4. The NL C nodal lines projection on(001) surface forms two closed loops whereas NL KH nodallines project at the corner points of (001) surface BZ.The topological surface states appearing due to NL C aretherefore more obvious over the (001) surface as found inFig. 4(a) without SOC. The two drumhead surface states(DSSs) nested outside the nodal line are clearly visible,consistent with the calculated nontrivial character of thenodal lines. Interestingly, the DSSs are more dispersiveand have opposite band curvature along M − Γ and M − K directions. They form unique saddle-like E k dispersionwith M being the saddle point. When SOC is included inthe computations, the DSSs split away from T symmetric M point, evolving into topological Dirac-cone-like stateswith a saddle-like energy dispersion [see Fig. 4(b)].Generally the tight-binding based methods for calcu-lating surface spectrum neglect possible surface poten-tial caused by charge redistribution near the surface area and provide only an approximate representation of thesurface states. To showcase the robustness of saddle-like topological states, we calculate 10 layers slab bandstructure by considering the self-consistent surface poten-tial effects within DFT computations and present resultsin Figs. 4(c) and 4(d). We find that these results arein reasonable agreement with the ones obtained usingsemi-infinite tight-binding slab calculations. More im-portantly, the saddle-like energy dispersion of the DSSsis preserved in DFT computations. Since (001) surface isa natural cleavable surface of Pt HgSe due to the pres-ence of weakly coupled pair of layers, these state can beeasily accessed in spectroscopic experiments.The (100) surface band structure is presented inFigs. 4(e) and 4(f) without and with SOC effects,respectively. Over the (100) surface, NL KH nodal linesprojection connects Γ − Y and Z − U symmetry linesas shown in Fig. 3(a). The topological DSSs connectthese projections which are seen clearly in Fig. 4(e).On considering SOC effects, the DSSs evolve into theDirac-cone like states with Dirac points at U and Y points, in accordance with the calculated topologicalinvariants. FIG. 4: Surface band structure of (001) semi-infinite slab cal-culated using Green’s function method (a) without and (b)with SOC. The sharp yellow lines represent surface states.Surface band structure of 10L thick Pt HgSe slab obtainedby DFT calculations (c) without and (d) with SOC. Theshaded green region highlights the projected bulk bands andsolid yellow lines mark the surface states. (e) and (f) are sameas (a) and (b) but for the (100) surface. The drumhead sur-face states connecting bulk NL KH nodal line projections areclearly visible. They evolve to topological Dirac cone stateswith SOC at Y and U point of (100) surface BZ. V. k · p MODEL HAMILTONIAN We now discuss a minimal low-energy k · p Hamilto-nian for the topological surface states that capture essen-tial features of these states. Based on our first-principlescalculations, the TSSs spread around M point on the(001) surface. Therefore, a k · p Hamiltonian around M = (0 , π ) point is sufficient to describe the TSSs. Onthe (001) surface at M , the little group C s contain a mir-ror plane symmetry. In the presence of SOC, the symme-try operators are given as M = − iτ σ and T = − iσ K ,where σ and τ are Pauli matrices in spin and sublatticespaces, respectively. The associated symmetry allowedbasis functions for TSSs Ψ = ( ψ Aα, ↑ , ψ Aα, ↓ , ψ Bα, ↑ , ψ Bα, ↓ ) T ,where A and B represent two sublattice of bipartite lat-tice, can be expressed as | ψ α,σ (cid:105) = λ s | s, σ (cid:105) + λ d yz | d yz , σ (cid:105) + λ d x − y | d x − y , σ (cid:105) + λ d z | d z , σ (cid:105) . Here, the subscript s = ↑ / ↓ denotes spin-up/spin-down,respectively, and λ s , λ d yz , λ d x − y , and λ d z describe nor-malization coefficients. Using the above basis, the min-imal four band Hamiltonian around the surface Diracpoint (up to second order in momentum) can be writ-ten as, H T SS ( p ) = 12 m ∗ ( p x + ηp y ) + v R ( p x σ − p y σ )+ v p x τ σ + λ p x p y τ σ + δ τ σ + δ τ σ . (1)where η , v R , v , λ and δ are real numbers and v R denotes Rashba parameter. The corresponding eigenen-ergies of H T SS ( p ) are E T SS ( p ) = 12 m ∗ ( p x + ηp y ) + ξ (cid:114) v R p + δ + p x ( v + λ p y ) + ξ (cid:48) δ (cid:113) v R p y + v p x (2)with p = p x + p y and ξ ( ξ (cid:48) ) = ±
1. Equation (2) showsthat the lower branch of conduction band cross with thetop branch of valence band at ( p x , p y ) = (0 , ± δ v R ) alongmirror invariant M − Γ line. This gives rise to the Diraccones states protected by mirror symmetry, as shown ex-plicitly in Figs. 5(a) and 5(b). In addition, for η <
0, wefind a pair of type II saddle-point VHSs, as illustratedin Fig. 5(c). VI. TOPOLOGICAL PHASE TRANSITION
We now demonstrate the possibility of tuning the topo-logical order of Pt HgSe and realizing a Dirac semimetalby modulating the unit cell volume in Fig. 6. For thispurpose, it is useful to define the SOC induced gap as∆ K = E Λ K − E Λ K at K and ∆ H = E Λ H − E Λ H at H between the Λ and Λ states that form a nodal linewithout SOC along KH direction (see Fig. 6 for details).The evolution of ∆ K and ∆ H with relative unit cell vol-ume V /V where V denotes equilibrium unit cell vol- (a) (b) 𝑫𝑫 𝑫𝑫 𝑫𝑫 𝑬𝑬 (C) -2.582 δ δ 𝒑𝒑 𝒙𝒙 𝒑𝒑 𝒚𝒚 E S S S S 𝒑𝒑 𝒙𝒙 𝒑𝒑 𝒚𝒚 FIG. 5: (a) Energy dispersion of the surface state Hamiltonian H TSS ( p x , p y ) for η = − MK direction ( S and S ),whereas the other two belong to the highest valence bandand are located along the M Γ direction. (c) Density of stateshowing VHSs associated with the topological surface states. ume is presented in Fig. 6(a). We find that ∆ K and∆ H are comparable in the gapped pristine state butshow opposite behavior on changing the unit cell volume( V ). On decreasing (increasing) V /V from its equilib-rium value, the Λ and Λ bands cross near H ( K ) pointand realize a tilted band crossing along KH direction[see Fig. 6(b)-(d)]. A detailed symmetry analysis showsthat the crossing bands have opposite C z rotation eigen-values and thus the Dirac point is symmetry protectedagainst band hybridization. Importantly, we find thatwhen ∆ K ∆ H >
0, the Λ and Λ are separated by acontinuous gap and realize a gapped topological insula-tor state. On the contrary, when ∆ K ∆ H <
0, the twostates cross along the KH line and the system realizessymmetry protected type II Dirac semimetal state.Our analysis suggests that V /V can act as a contin-uous knob for tuning the position as well as velocityof Dirac cones when ∆ K ∆ H <
0. Owing to the exis-tence of parabolic energy dispersion of Λ and Λ , theenergy location of Dirac points can be tuned to lie onthe Fermi level. We find that as we increase V /V froma value of 0.83, a pair of Dirac cones located on HKH move towards H point. The Dirac cones merge at H for V /V ∼ . V /V beyond ∼ . HgSe we find that a topologi-cal insulating phase exists as a critical region betweentwo gapless Dirac semimetal phases. Moreover, the Diracpoints in Pt HgSe are located on BZ hinges along KH (b) (c) (d)(e) DSM TI DSM VV E n e r gy ( e V ) G a p ( e V ) -0.100.1 0.85 0.90 0.95 1.05 1.15 1.201.00 1.10 K H K H K H (f) Λ Λ VV = = = Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ (a) (g) Δ K Δ H DSM DSMTI
FIG. 6: (a) Phase diagram showing the evolution of bandgap ∆ K at K and ∆ H at H point with relative unit cellvolume V /V . The product ∆ K ∆ H determines a topologicalstate. ∆ K ∆ H > K ∆ H < KH line for different relative cell volume (b) VV = 0 .
83, (c) VV = 1 .
00, and (d) VV = 1 .
19. Panels (e)-(g) il-lustrate the corresponding atomic displacements with respectto the equilibrium structure. The length of the arrows is pro-portional to the magnitude of displacement. line in contrast to the other well known Dirac semimet-als such as Na Bi, PtTe where they are located on Γ A line in hexagonal BZ. Such Dirac semimetal stateis unique to Pt HgSe and never been seen before. VII. CONCLUSION
In conclusion, based on our first-principles calculationscombined with a k . p model Hamiltonian analysis, weidentify and characterize the dual-symmetry-protectedtopological state of Pt HgSe . The material is shown toharbor two distinct types of nodal lines when SOC effectsare neglected in the computations. Inclusion of SOC gapsout the nodal lines and drives the system into a topolog-ical insulator state which is characterized by both theweak topological invariant Z = (0; 001) and the mirrorChern number n M = 2. The (001) surface band structurereveals the existence of unique saddle-like topological sur-face states with saddle-point VHSs. We further discussthe tenability of the topological state of Pt HgSe bymodulating its crystal structure. In this way, the systemis shown to undergo a unique topological phase transitionwhere a topological insulator state exists as an interme-diate phase between gapless Dirac semimetal states. Ouranalysis suggests that the naturally cleaved (001) sur-face of Pt HgSe presents an ideal testbed for exploringsaddle-like topological surface states with VHSs and theassociated physics in topological materials. ACKNOWLEDGEMENTS
Work at the ShenZhen University is finan-cially supported by the Shenzhen Peacock Plan(KQTD2016053112042971) and Science and Tech-nology Planning Project of Guangdong Province(2016B050501005). The work at Northeastern Univer-sity is supported by the U.S. Department of Energy(DOE), Office of Science, Basic Energy Sciences GrantNo. DE-FG02-07ER46352, and benefited from North-eastern Universitys Advanced Scientific ComputationCenter and the National Energy Research Scientific Computing Center through DOE Grant No. DE-AC02-05CH11231. T.-R.C. was supported from YoungScholar Fellowship Program by Ministry of Science andTechnology (MOST) in Taiwan, under MOST Grantfor the Columbus Program MOST107-2636-M-006-004,National Cheng Kung University, Taiwan, and NationalCenter for Theoretical Sciences (NCTS), Taiwan. Thiswork is supported partially by the MOST, Taiwan,Grants No. MOST 107-2627-E-006-001. H. 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