Abstract
The interplay of electronic correlations, spinâorbit coupling and topology holds promise for the realization of exotic states of quantum matter. Models of strongly interacting electrons on honeycomb lattices have revealed rich phase diagrams featuring unconventional quantum states including chiral superconductivity and correlated quantum spin Hall insulators intertwining with complex magnetic order. Material realizations of these electronic states are, however, scarce or inexistent. In this work, we propose and show that stacking 1T-TaSe2 into bilayers can deconfine electrons from a deep Mott insulating state in the monolayer to a system of correlated Dirac fermions subject to sizable spinâorbit coupling in the bilayer. 1T-TaSe2 develops a Star-of-David charge density wave pattern in each layer. When the Star-of-David centers belonging to two adyacent layers are stacked in a honeycomb pattern, the system realizes a generalized KaneâMeleâHubbard model in a regime where Dirac semimetallic states are subject to significant MottâHubbard interactions and spinâorbit coupling. At charge neutrality, the system is close to a quantum phase transition between a quantum spin Hall and an antiferromagnetic insulator. We identify a perpendicular electric field and the twisting angle as two knobs to control topology and spinâorbit coupling in the system. Their combination can drive it across hitherto unexplored grounds of correlated electron physics, including a quantum tricritical point and an exotic first-order topological phase transition.
Similar content being viewed by others
Introduction
Prospects of quantum information technologies have motivated an intense search for systems, which intertwine topology and electronic correlations1,2,3,4. Strongly interacting and spinâorbit-coupled electrons on the honeycomb lattice, as theoretically described by the KaneâMeleâHubbard model5, feature quantum spin Hall (QSH), MottâHubbard, collective magnetic and chiral superconducting states. While several weakly interacting QSH systems are now well-studied experimentally2,6, material realizations of honeycomb KaneâMeleâHubbard fermions with strong correlations and spinâorbit coupling are rare7,8. Here, we introduce a new âvan der Waals engineeringâ platform to serve this purpose.
We show that stacking of 1T-TaSe2 into bilayers can deconfine electrons from a deep Mott insulating state realized in the monolayer to a system of correlated Dirac fermions subject to sizable spinâorbit coupling. Central to this transition is the possibility of van der Waals materials to stack in different configurations. For a specific honeycomb arrangement (Fig. 1) the kinetic energy associated with the electronic hopping t turns out to be of the same order of magnitude as the effective local Coulomb repulsion U. The system features, therefore, electronic correlations, which turn out to put the system right on the verge between QSH and correlated antiferromagnetic insulating states at charge neutrality and support chiral superconductivity under doping. We finally demonstrate that tuning the system via electric fields and twisting of the layers relative to each other sensitively affects the low-energy electronic structure in terms of emerging Dirac mass and spinâorbit coupling terms and leads to completely unexplored regimes of correlated electrons.
Results and discussion
From a correlated insulator to emergent Dirac fermions
Layered group-V transition metal dichalcogenides (TMDCs) such as 1T-TaSe2 or 1T-TaS2 feature a low-temperature commensurate charge density wave (CCDW) where Ta-atoms are displaced into Star-of-David (SoD) patterns (Fig. 1a)9,10. In this phase, the SoDs form a triangular \(\sqrt{13}\times \sqrt{13}\) superlattice in every layer and host correlated electrons: 1T-TaS2 shows a metal-to-insulator transition when entering the CCDW phase10. In 1T-TaSe2, the bulk remains conductive till lowest temperatures, while the surface exhibits a Mott transition around 250âK11,12. Recently, 1T-TaSe2 has been fabricated down to monolayer thickness13,14,15 and a pronounced thickness dependence of the electronic structure has been reported15. As bonds between the layers are mainly of van der Waals type, different stacking configurations are observed in experiments16,17 and have a strong impact on the electronic structure18,19,20.
We compare the CCDW state in the monolayer to a bilayer with honeycomb stacking, where the SoD centers form a buckled honeycomb lattice (Fig. 1b). The bottom layer SoD centers form sublattice A and the top layer sublattice B. This stacking is one of many possible configurations, which can generally differ, both, by the local atomic stacking and by the stacking of the SoD centers. The local stacking considered here, has the Ta atoms of the bottom layer approximately beneath the lower Se atoms of the top layer (Fig. 1c). While this kind of local stacking is not the most commonly observed one of the T-phase TMDCs15,17, a corresponding stacking sequence has been found in transmission electron microscopy studies of 1T-TaS217 and turns out to be metastable in density functional theory (DFT) simulations of 1T-TaSe221. Regarding the stacking of the SoD centers further van der Waals DFT total energy calculations (see Supplementary Information) show that the particular honeycomb arrangement studied here, is metastable and energetically on the order of 10âmeV per formula unit above the lowest energy configuration. This is similar to the configuration observed experimentally in ref. 15 and it is thus plausible that also the honeycomb configuration considered here, is within reach of experiments. Experimental approaches including tear-and-stack22 and Scanning Tunneling Microscopy voltage pulse-based manipulation schemes23 can present possible avenues to control and switch metastable stacking configurations and to reach the honeycomb configurations discussed in our paper. At small twist angles all different kinds of configurations can be realized locally in the moiré, including likely those honeycomb cases discussed here.
To study the electronic structure of such engineered stackings, we combine ab-initio calculations in the frameworks of DFT and the random phase approximation (RPA) with effective low-energy models, which we investigate with dynamical mean-field theory (DMFT) and two-particle self-consistent (TPSC) many-body approaches, see âMethodsâ section.
In the CCDW phase, the band structure of monolayer 1T-TaSe2 obtained with non-spin-polarized DFT is characterized by a single (Ta) flat band at the Fermi level10,18,19,20, which has a bandwidth of less than 20âmeV (Fig. 2a, left). Hence, the CCDW formation largely quenches in-plane hopping of the electrons. In the honeycomb-stacked bilayer (with no twist), two dispersive bands with a bandwidth of the order of 200âmeV emerge from the low-energy flat band of the monolayer (Fig. 2a, right). Comparison of the mono- and bilayer bandwidths shows that interlayer hopping effects must dominate over intralayer hoppings by approximately an order of magnitude. In this sense, CCDW TaSe2 bilayers are the exact opposite of graphene bilayer systems, since in the latter out-of-plane coupling is an order of magnitude weaker than in-plane hopping24,25. For the 1T-TaSe2 honeycomb bilayer, the upper and lower low-energy bands touch as Dirac points at the Brillouin zone corners K and K'. In the undoped system, these Dirac points are exactly at the Fermi level.
We next construct a Wannier Hamiltonian to describe the Dirac bands with one Wannier function for each SoD center, i.e., two Wannier orbitals per bilayer CCDW superlattice unit cell (Supplemental Fig. 2). The resulting nearest-neighbor hopping between a sublattice A site in the bottom and a neighboring sublattice B site in the top layer amounts to tâ=ââ34âmeV and is the leading term of the Wannier Hamiltonian. There are further terms in the Wannier Hamiltonian, which are, however, at least an order of magnitude smaller than t.
The effective Hubbard interaction U for the SoD Wannier orbitals of CCDW TaSe2 calculated in RPA is Uâââ130âmeV, which is also in line with the experimental estimates in ref. 15 and calculations for TaS219. The ratio of hopping to Coulomb interaction is decisive in determining the strength and kind of electronic correlation phenomena taking place. Our calculations yield U/â£tâ£âââ3.8.
To study the resulting electronic correlations, we performed simulations of the Hubbard model for the non-twisted CCDW 1T-TaSe2 bilayer in the framework of DMFT and the TPSC approach26,27. The quasi-particle weight Z shown in Fig. 2c is a measure of the electronic correlation strength. In the temperature range Tâ=â60â230âK, both DMFT and TPSC consistently yield essentially constant Zâââ0.75. Our system is thus at intermediate local correlation strength and far away from the paramagnetic MottâHubbard transition taking place above UMott/tâââ8.2. The DMFT and TPSC studies of the full Hubbard model for the non-twisted CCDW 1T-TaSe2 bilayer are, thus, in line with studies of the idealized Hubbard model on the honeycomb lattice28.
The TPSC calculations give insight to spin-fluctuations taking place in the system (Supplemental Fig. 4). The inverse intra- and inter-sublattice terms of the static magnetic susceptibility at wave number qâ=â0, as well as the antiferromagnetic correlation length ξAFM are shown in Fig. 2d. We observe that antiferromagnetic fluctuations with alternating spin orientation between the two sublattices (A, B) are dominant and strongly enhanced at temperatures Tââ²â100âK. These fluctuations indicate that the non-twisted CCDW 1T-TaSe2 bilayer is close to a quantum phase transition from a Dirac semimetal to an antiferromagnetic insulator, which occurs for ideal Hubbard honeycomb systems exactly in the range of Uc/tâââ3.6â3.86,29,30.
Spinâorbit coupling
The aforementioned magnetic correlation phenomena are sensitive to details of the low-energy electronic structure and spinâorbit coupling (SOC), which we discuss in the following based on a symmetry analysis. The space group of non-twisted CCDW 1T-TaSe2 bilayer in the honeycomb structure is \(P\bar{3}\) (#147), comprising inversion symmetry and three-fold rotations C3 around an axis perpendicular to the bilayer. Imposing also time-reversal symmetry, every band must be two-fold degenerate. Therefore, SOC-induced qualitative changes of the band structures can occur near the Dirac points at K and Kâ². A corresponding k â p expansion (see Supplementary Information) reads
where the pseudospin S describes the sublattice degree of freedom, Ï acts on the electron spin, and Ïâ=â±1 labels the valley (K, Kâ²) degree of freedom. This Hamiltonian comprises three contributions; the first contribution is a two-dimensional massless Dirac term, with the sublattice-pseudospin playing the role of the spin inherent to the Dirac equation. This term is analogous to the massless Dirac term in graphene6,24,25. SOC is responsible for the second and third contributions to H0: a valleyâspinâsublattice coupling λSOCâ=â0.74âmeV, which is often called KaneâMele spin-orbit term31,32, and a sublattice-staggered Rashba term αR2, which belongs to the R2 class according to the classification form ref. 33. A finite KaneâMele term λSOC opens a gap and turns the system described by H0 into a QSH insulator31,32. Importantly, the KaneâMele term here is enhanced in comparison to its counterpart in graphene by two orders of magnitude24,25,34, and corresponds to a temperature Tâââ10âK, which is well accessible in experiments. Given that U/â£tâ£âââ3.8, the non-twisted CCDW 1T-TaSe2 bilayer implements a material realization of the KaneâMeleâHubbard model in a regime very close to the topological quantum phase transition from a QSH insulator to an antiferromagnetic insulator (Fig. 2e).
Dirac fermions and their topology are affected by different kinds of mass fields. An energy difference between electrons localized in sublattice A and B leads to a so-called Semenoff mass term M, which would enter the Hamiltonian H0 of equation (1) in the form MSz. This term breaks sublattice invariance and thereby inversion symmetry and leads to a transition from a QSH to a band insulator at â£Mâ£â=ââ£Î»SOCâ£31,35. In the non-twisted CCDW 1T-TaSe2 bilayer, the intrinsic Semenoff mass M0â=â0 is required to vanish by symmetry. Our Wannier construction yields M0â=â0.14âmeV, which is small as compared to all other relevant terms in the system. Possible origins of this small symmetry breaking can be the Wannier constructions and also faint asymmetries accumulated during self-consistency iterations in the DFT calculations. Vertical electric fields Ez, as realizable in field effect transistor geometries (Fig. 1c), break inversion symmetry, translate into staggered sublattice potentials, and therefore corresponding extrinsic Semenoff mass contributions ÎM (Fig. 2b). DFT calculations (Supplementary Fig. 3) yield the approximate relation ÎMâââeEzd/ϵâ¥, where dâ=â6.4âà is the interlayer distance, e is the elementary charge, and ϵâ¥â=â3.72 plays the role of an effective dielectric constant. The QSH to band insulator transition is reached for Ezâââ0.5âmVâà â1â=â50âkVâcmâ1 (Fig. 2b), which is well within reach of experiments36.
Taken together, our calculations show that the honeycomb-stacked non-twisted CCDW 1T-TaSe2 bilayer is located in a region of the phase diagram (Fig. 2e) with three different phases (QSH insulator, band insulator and antiferromagnetic insulator) coming together. Electron correlation is known to change the order of the QSH-band insulator transition from second to first-order37. Contrary to the standard non-interacting QSH to band insulator transition, where the gap closes and reopens continuously with vanishing gap at the transition point, the QSH gap remains finite and the system changes discontinuously to a band insulating state at the transition point. Application of vertical electric fields in the system at hand represents hence a possibility to realize this exotic interaction-induced first-order transition.
Twisted CCDW 1T-TaSe2 bilayers
Layered van der Waals systems allow to realize different stacking configurations via twisting, i.e., relative rotations between the layers. General twist angles θ lead to incommensurate moiré patterns superimposed to the CCDW lattice. Arguably the simplest case of twisting is a rotation angle of θâ=â180â, which leads to a system with identical Bravais lattice but different symmetry of the supercell basis (Fig. 1c): in case of honeycomb stacking of the CCDW, the space group of the 180â twisted structure of CCDW TaSe2 bilayer is reduced to P3, meaning that inversion symmetry is lost with respect to the non-twisted case. The resulting band structure (Fig. 3a) is qualitatively similar to the non-twisted honeycomb case regarding the overall shape and width of the low-energy bands. Thus, also the 180â twisted case will be far away from the paramagnetic Mott transition. However, the low-energy band structure is markedly different in the 180â twisted case. First, the conduction band is almost flat between K and M. Second, inversion symmetry breaking lifts band degeneracies: our DFT calculations reveal a staggered potential and an associated intrinsic Semenoff mass term of M0â=â8.55âmeV, which opens a gap at the K and K' points already without SOC and without external electric fields. Furthermore, additional SOC terms are now allowed by symmetry (see Supplementary Information) and completely lift the remaining band degeneracies except for the time-reversal symmetric points Î and M.
Based on a symmetry analysis, we obtain the following low-energy model in the vicinity of K and Kâ²:
The additional terms with respect to the non-twisted honeycomb structure equation (1) are the intrinsic Semenoff mass (M0), the spinâvalley coupling term (B), the KaneâMeleâRashba interaction (λR), and a Rashba interaction belonging to the R1 class33 giving rise to pure Rashba spin-polarization patterns (αR1). The last term (with coupling constant λD) can also be seen as an effective k-dependent magnetic field parallel to the z-axis. Our DFT calculations yield M0â=â8.55âmeV, Bâ=ââ1.85âmeV, â£Î»SOC⣠â²â0.05âmeV and λRâ=â3.21âmeV. These terms affect the dispersion and imprint an intricate sublattice and spin structure to the low-energy bands, which can be manipulated by vertical electric fields as shown in Fig. 3b.
Except for Ezâ=ââ4.2âmeVâà â1 (ÎMâ=ââ6.68âmeV and M =âM0â+âÎMâ=â1.87âmeV), the system is always gapped. We calculated the Z2 topological invariant for the non-interacting 180â twisted bilayer in comparison to the non-twisted bilayer case,, as well as for two cases in between where the ratio B/λSOC is varied (Fig. 3c, and Supplemental Information). In the non-twisted bilayer, the system is in a QSH state unless an extrinsic sufficiently large Semenoff mass term or an additional Rashba SOC term λR are added. In the 180â twisted case, the situation is very different regardless whether or not the intrinsic Semenoff mass term is compensated by an external electric field and regardless of λSOC. Indeed, many changes in the SOC terms suppress the QSH state in the 180â twisted bilayer: The comparably large Rashba λR and the spinâvalley coupling terms B and a strong reduction in λSOC. Each of these alone is sufficient to suppress the QSH state. At vertical electric field Ezâ=ââ4.2âmVâà â1 the gaps at K and K' close, and a band touching point emerges.
While it is clear that there will be tendencies towards interaction-induced (quasi)ordered phases as well, here, the kind of ordering is likely different from the non-twisted case but largely unexplored. The band touching at Ezâ=ââ4.2âmVâà â1 implements a situation similar to saddle points in a two-dimensional dispersion, where already arbitrarily weak interactions would trigger different kinds of magnetic or excitonic instabilities38. How these instabilities translate into the intermediately correlated and strongly spinâorbit-coupled case of 180â twisted TaSe2 is a completely open question.
Conclusions and outlook
The field of twistronics with materials like bilayer graphene is based on the idea that weak interlayer coupling can flatten highly dispersive bands and thereby boost electronic correlations39,40,41. The system introduced here takes the opposite route of deconfining formerly Mott localized electrons. This approach should be generally applicable to interfaces of Mott localized electrons under two conditions: the interlayer coupling should substantially exceed the in-plane one and at the same time define a connected graph linking all sites of the system. Possible example systems range from stacking faults in the bulk of CCDW layered Mott materials17 to molecular systems42.
Especially the twisting degree of freedom opens new directions to experiments. Since interlayer hopping is the dominant kinetic term in deconfined Mott systems like bilayers of CCDW 1T-TaSe2, we expect incommensurability effects to be much more pronounced than in twisted graphene systems39. θâ=â30â twisted CCDW 1T-TaSe2 bilayer should realize a quasicrystal with twelvefold rotation symmetry and provide an experimental route to correlated electrons and emerging collective states in a quasicrystalline environment.
The prototypical case of CCDW 1T-TaSe2 bilayer demonstrates how deconfinement of Mott electrons leads to exotic states of quantum matter: the non-twisted bilayer approaches a quantum tricritical region of competing QSH, trivial band insulating, and antiferromagnetic insulating states. At 180â twist angle different kinds of electrically controllable band degeneracies with associated many-body instabilities, hypothetically of excitonic type, emerge. Clearly, the phase space for manipulating deconfined Mott electrons is high dimensional. We here identified the combination of twist angle and perpendicular electric field as decisive for TaSe2 bilayers. Further means to control emerging electronic states include dielectric engineering43 and charge doping. Our calculations showed that the non-twisted CCDW 1T-TaSe2 bilayer in honeycomb stacking approximates the (KaneâMele) Hubbard model on the honeycomb lattice with U/â£tâ£âââ3.8 very well. In this regime, doping the system away from the Dirac point towards the van Hove singularity is expected to lead to chiral superconductivity44,45,46,47, most likely of dâ+âid-type.
Methods
DFT calculations
We perform DFT48,49 calculations by using the Vienna ab-initio simulation package (VASP)50,51 with the generalized gradient approximation of Perdew, Burke, and Ernzerhof (GGA-PBE) for the exchange-correlation functional52,53. We obtain the total energies, relaxed structures and electronic structure of mono- and bilayer 1T-TaSe2 systems. We calculate the total energies for various possible stackings in undistorted and CCDW bilayer 1T-TaSe2 using Î-centered k-meshes of 15âÃâ15 Ãâ1 and 9âÃâ9âÃâ1, respectively, and taking into account van der Waals (vdW) corrections within DFT-D2 and cross-checking with DFT-D354,55, see Supplemental Fig. 1. The ionic relaxation is done using the conjugate gradient algorithm until all force components are smaller than 0.02âeVâà â1.
Since the DFT-D2 corrections yield correct interlayer distances but do not correctly capture the CCDW distortions, we adopted the following relaxation procedure to calculate the commensurate \(\sqrt{13}\times \sqrt{13}\) CCDW bilayer structures:
-
1.
We perform relaxations for a \(\sqrt{13}\times \sqrt{13}\) supercell of the monolayer (without vdW corrections), using a superlattice constant of aâ=â12.63âà according to ref. 15. We fix the vertical positions of the Ta atoms, while allowing for Ta in-plane displacements. The Se atoms are allowed to freely relax in all three directions.
-
2.
We include then a second layer and optimize the interlayer distance, d, while keeping all relative intralayer distances fixed. We find dâ=â6.4âà for the ideal honeycomb stacking in CCDW bilayer 1T-TaSe2.
-
3.
We relax the CCDW bilayer following the same procedure described for the monolayer, i.e., without vdW corrections and vertical positions of the Ta atoms according to the optimized interlayer distance d fixed, while allowing for in-plane displacements. The Se atoms are allowed to freely relax in all three directions.
We cross check the results obtained by this procedure against calculations with vdW corrections according to the DFT-D3 method55. In contrast to DFT-D2, the CCDW distortions are well described in the DFT-D3. Thus, full relaxations of all atomic positions have been performed in the DFT-D3 framework for the mono- and bilayer. Both our step-by-step procedure using DFT-D2 method, and the full relaxation using DFT-D3 give equivalent results for the total energies (see Supplemental Fig. 1b), for the crystal and band structures.
For the non-collinear magnetic calculation, i.e., when SOC is included, we set the net magnetic moment to zero in all atoms of the unit cell, and use a Î-centered k-mesh of 6âÃâ6âÃâ1.
Estimation of the screened Hubbard interaction U via RPA
We estimate the local Hubbard interaction U for the flat bands around the Fermi level in the CCDW 1T-TaSe2 bilayer from the ab-initio calculation of the screened Coulomb interaction, using RPA for the undistorted bilayer. We follow a similar procedure as in ref. 56, which we summarize below:
-
We initially calculate the WANNIER90 tight-binding model for the three low-energy Ta bands \({\mathcal{C}}\), whose orbital character is mostly \(\{{d}_{{z}^{2}},{d}_{{x}^{2}-{y}^{2}},{d}_{xy}\}\) (see Fig. 2(a) in ref. 56) in the undistorted monolayer 1T-TaSe2.
-
The static RPA-screened Coulomb interaction tensor Wαβγδ(q, Ï â 0) is calculated for undistorted monolayer 1T-TaSe2, where q is a reciprocal wave vector on a Î-centered mesh of 18âÃâ18âÃâ1, and \(\alpha ,\beta ,\gamma ,\delta \in {\mathcal{C}}\). We neglect qâ=â0 terms in our RPA analysis in order to avoid unphysical effects.
-
In the CCDW 1T-TaSe2 bilayer, the \({d}_{{z}^{2}}\) orbitals from Ta atoms in the SoD centers have the largest contribution for the bands around the Fermi level. Thus, for each q, we consider only the tensor element W(q)ââ¡âWαααα(q) with \(\alpha ={d}_{{z}^{2}}\).
-
Then, the local Hubbard interaction U in a single star of David is calculated by averaging over the \({d}_{{z}^{2}}\) orbital weight from each Ta atom (labeled by \({w}_{{d}_{{z}^{2}}}({\bf{R}})\)) in the star of David:
$$U=\mathop {\sum}\limits_{{\bf{R}},{\bf{R}}^{\prime} \in {\davidsstar} }{w}_{{d}_{{z}^{2}}}({\bf{R}})U({\bf{R}}-{\bf{R}}^{\prime} ){w}_{{d}_{{z}^{2}}}({\bf{R}}^{\prime} )$$(3)where U(R) is the discrete Fourier transform of W(q).
DMFT and TPSC many-body calculations
For non-twisted CCDW 1T-TaSe2 bilayer, we construct tight-binding-Hubbard Hamiltonians of the type
from a Wannierization of the ab-initio DFT band structure (Supplementary Information), and estimate the on-site repulsion U to be about 130âmeV by means of RPA calculations. We study these effective low-energy models from a many-body perspective to judge the type of correlations in the system. In DMFT the lattice Hamiltonian is mapped onto a self-consistently determined single impurity problem, solvedâin our caseâwithin numerical exact quantum Monte Carlo in the hybridization expansion flavor (CT-HYB) (for a review, see57). The resulting sublattice-resolved self-energy, Σ, is local (k-independent) but it contains frequency-dependent non-perturbative corrections beyond Hartree-Fock to all orders and can account for MottâHubbard metal-to-insulator transitions. All DMFT calculations are performed using w2dynamics58. The double counting is accounted for using the fully localized limit. For two-dimensional systems it is important to estimate non-local effects at the level of the self-energy, not included in DMFT. To this goal, we apply the TPSC method59, which produces accurate results in the weak-to-intermediate coupling regime, if compared to lattice quantum Monte Carlo calculations in the single band Hubbard model. For non-twisted CCDW 1T-TaSe2 bilayer, which is modeled by a multi-band system we use the multi-site formulation of TPSC26 while neglecting the Hartree term to avoid double counting of correlation effects already accounted for in DFT. Moreover, to be able to apply TPSC to this system we project out spin off-diagonal terms and take only the diagonal spin-up contributions from DFT while still assuming a paramagnetic state. The combination of TPSC accounting for the k-dependence of Σ and DMFT, in which we can include all off-diagonal terms between spin-orbitals and access antiferromagnetic ordering at strong coupling consitutes a powerful tool to determine the many-body nature of 1T-TaSe2.
Data availability
The data that support the findings of this study is available from the corresponding author upon reasonable request.
References
Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559â1562 (1982).
Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057â1110 (2011).
Sato, M. & Ando, Y. Topological superconductors: a review. Rep. Prog. Phys. 80, 076501 (2017).
Rachel, S. Interacting topological insulators: a review. Rep. Prog. Phys. 81, 116501 (2018).
Hohenadler, M. & Assaad, F. F. Correlation effects in two-dimensional topological insulators. J. Phys.: Condens. Matter 25, 143201 (2013).
Wehling, T. O., Black-Schaffer, A. M. & Balatsky, A. V. Dirac materials. Adv. Phys. 63, 1â76 (2014).
Marrazzo, A., Gibertini, M., Campi, D., Mounet, N. & Marzari, N. Prediction of a large-gap and switchable Kane-Mele Quantum Spin Hall insulator. Phys. Rev. Lett. 120, 117701 (2018).
Wu, X., Fink, M., Hanke, W., Thomale, R. & Di Sante, D. Unconventional superconductivity in a doped quantum spin Hall insulator. Phys. Rev. B 100, 041117 (2019).
Wilson, J. A., Salvo, F. J. D. & Mahajan, S. Charge-density waves and superlattices in the metallic layered transition metal dichalcogenides. Adv. Phys. 24, 117â201 (1975).
Rossnagel, K. On the origin of charge-density waves in select layered transition-metal dichalcogenides. J. Phys.: Condens. Matter 23, 213001 (2011).
Perfetti, L. et al. Spectroscopic signatures of a bandwidth-controlled Mott transition at the surface of 1T-TaSe2. Phys. Rev. Lett. 90, 166401 (2003).
Colonna, S. et al. Mott phase at the surface of 1T-TaSe2 observed by Scanning Tunneling Microscopy. Phys. Rev. Lett. 94, 036405 (2005).
Nakata, Y. et al. Selective fabrication of Mott-insulating and metallic monolayer TaSe2. ACS Appl. Nano Mater. 1, 1456â1460 (2018).
Börner, P. C. et al. Observation of charge density waves in free-standing 1T-TaSe2 monolayers by transmission electron microscopy. Appl. Phys. Lett. 113, 173103 (2018).
Chen, Y. et al. Strong correlations and orbital texture in single-layer 1T-TaSe2. Nat. Phys. 16, 218â224 (2020).
Müller-Caspary, K. et al. Atomic-scale quantification of charge densities in two-dimensional materials. Phys. Rev. B 98, 121408 (2018).
Hovden, R. et al. Atomic lattice disorder in charge-density-wave phases of exfoliated dichalcogenides (1T-TaS2). Proc. Natl Acad. Sci. USA 113, 11420â11424 (2016).
Freericks, J. K., Krishnamurthy, H. R., Ge, Y., Liu, A. Y. & Pruschke, T. Theoretical description of time-resolved pump/probe photoemission in TaS2 : a single-band DFT.DMFT(NRG) study within the quasiequilibrium approximation. Phys. Status Solidi (B) 246, 948â954 (2009).
Darancet, P., Millis, A. J. & Marianetti, C. A. Three-dimensional metallic and two-dimensional insulating behavior in octahedral tantalum dichalcogenides. Phys. Rev. B 90, 045134 (2014).
Ritschel, T., Berger, H. & Geck, J. Stacking-driven gap formation in layered 1T-TaS2. Phys. Rev. B 98, 195134 (2018).
Lazar, P., Martincová, J. & Otyepka, M. Structure, dynamical stability, and electronic properties of phases in TaS2 from a high-level quantum mechanical calculation. Phys. Rev. B 92, 224104 (2015).
Weston, A. et al. Atomic reconstruction in twisted bilayers of transition metal dichalcogenides. Nat. Nanotechnol. 15, 592â597 (2020).
Ma, L. et al. A metallic mosaic phase and the origin of Mott-insulating state in 1T-TaS 2. Nat. Commun. 7, 10956 (2016).
Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109â162 (2009).
Katsnelson, M. I. Graphene: Carbon in Two Dimensions (Cambridge University Press, 2012).
Zantout, K., Altmeyer, M., Backes, S. & ValentÃ, R. Superconductivity in correlated BEDT-TTF molecular conductors: critical temperatures and gap symmetries. Phys. Rev. B 97, 014530 (2018).
Zantout, K., Backes, S. & ValentÃ, R. Effect of nonlocal correlations on the electronic structure of LiFeAs. Phys. Rev. Lett. 123, 256401 (2019).
Tran, M.-T. & Kuroki, K. Finite-temperature semimetal-insulator transition on the honeycomb lattice. Phys. Rev. B 79, 125125 (2009).
Arya, S., Sriluckshmy, P. V., Hassan, S. R. & Tremblay, A.-M. S. Antiferromagnetism in the Hubbard model on the honeycomb lattice: a two-particle self-consistent study. Phys. Rev. B 92, 045111 (2015).
Raczkowski, M. et al. The Hubbard model on the honeycomb lattice: from static and dynamical mean-field theories to lattice quantum Monte Carlo simulations. Phys. Rev. B 101, 125103 (2020).
Kane, C. L. & Mele, E. J. Z2 Topological order and the Quantum Spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).
Ezawa, M. Valley-polarized metals and quantum anomalous hall effect in silicene. Phys. Rev. Lett. 109, 055502 (2012).
Zhang, X., Liu, Q., Luo, J.-W., Freeman, A. J. & Zunger, A. Hidden spin polarization in inversion-symmetric bulk crystals. Nat. Phys. 10, 387â393 (2014).
Kochan, D., Irmer, S. & Fabian, J. Model spin-orbit Hamiltonians for graphene systems. Phys. Rev. B 95, 165415 (2017).
Di Sante, D. et al. Towards topological quasifreestanding stanene via substrate engineering. Phys. Rev. B 99, 035145 (2019).
Klein, J. et al. Electric-field switchable second-harmonic generation in bilayer MoS2 by inversion symmetry breaking. Nano Lett. 17, 392â398 (2017).
Amaricci, A., Budich, J., Capone, M., Trauzettel, B. & Sangiovanni, G. First-order character and observable signatures of topological quantum phase transitions. Phys. Rev. Lett. 114, 185701 (2015).
Kotov, V. N., Uchoa, B., Pereira, V. M., Guinea, F. & Castro Neto, A. H. Electron-electron interactions in graphene: current status and perspectives. Rev. Mod. Phys. 84, 1067â1125 (2012).
Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene. Proc. Natl Acad. Sci. USA 108, 12233 (2011).
Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43â50 (2018).
Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80â84 (2018).
Tsukahara, N. et al. Evolution of Kondo resonance from a single impurity molecule to the two-dimensional lattice. Phys. Rev. Lett. 106, 187201 (2011).
Pizarro, J. M., Rösner, M., Thomale, R., ValentÃ, R. & Wehling, T. O. Internal screening and dielectric engineering in magic-angle twisted bilayer graphene. Phys. Rev. B 100, 161102 (2019).
Black-Schaffer, A. M. & Doniach, S. Resonating valence bonds and mean-field d -wave superconductivity in graphite. Phys. Rev. B 75, 134512 (2007).
Nandkishore, R., Levitov, L. S. & Chubukov, A. V. Chiral superconductivity from repulsive interactions in doped graphene. Nat. Phys. 8, 158â163 (2012).
Kiesel, M. L., Platt, C., Hanke, W., Abanin, D. A. & Thomale, R. Competing many-body instabilities and unconventional superconductivity in graphene. Phys. Rev. B 86, 020507 (2012).
Black-Schaffer, A. M. & Honerkamp, C. Chiral d -wave superconductivity in doped graphene. J. Phys.: Condens. Matter 26, 423201 (2014).
Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136, B864âB871 (1964).
Kohn, W. & Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133âA1138 (1965).
Kresse, G. & Hafner, J. Norm-conserving and ultrasoft pseudopotentials for first-row and transition elements. J. Phys.: Condens. Matter 6, 8245â8257 (1994).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758â1775 (1999).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865â3868 (1996).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple [Phys. Rev. Lett. 77, 3865 (1996)]. Phys. Rev. Lett. 78, 1396â1396 (1997).
Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem. 27, 1787â1799 (2006).
Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 132, 154104 (2010).
Kamil, E. et al. Electronic structure of single layer 1T-NbSe2 : interplay of lattice distortions, non-local exchange, and MottâHubbard correlations. J. Phys.: Condens. Matter 30, 325601 (2018).
Gull, E. et al. Continuous-time Monte Carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349â404 (2011).
Wallerberger, M. et al. w2dynamics: Local one- and two-particle quantities from dynamical mean field theory. Comput. Phys. Commun. 235, 388â399 (2019).
Vilk, Y. M. & Tremblay, A.-M. Non-perturbative many-body approach to the Hubbard model and single-particle pseudogap. J. Phys. 7, 1309â1368 (1997).
Acknowledgements
We thank D. Di Sante, P. Eck, E. van Loon, M. Schüler, and C. Steinke for useful conversations. J.M.P. and T.W. acknowledge funding from DFG-RTG 2247 (QM3) and the European Graphene Flagship. S.A. and G.S. are supported by DFG-SFB 1170 Tocotronics, and further acknowledges financial support from the DFG through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matterâct.qmat (EXC 2147, project-id 390858490). T.M., K.Z., and R.V. acknowledge funding from the DFG through grant VA117/15-1. P.B. acknowledges financial support from the Italian Ministry for Research and Education through PRIN-2017 project âTuning and understanding Quantum phases in 2D materials-Quantum 2Dâ (IT-MIUR Grant No. 2017Z8TS5B). This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. We gratefully acknowledge the Gauss Center for Supercomputing e.V. (www.gauss-center.eu) for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Center (www.lrz.de). Computing time at HLRN (Berlin and Göttingen) is acknowledged.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Contributions
J.M.P. performed the DFT and RPA calculations. S.A. calculated the topological invariant diagrams. S.A., K.Z., and T.M. performed the DMFT and TPSC calculations. P.B. derived the k â p model. R.V., G.S., and T.W. supervised the project. J.M.P., S.A., G.S., and T.W. analyzed the results. All authors contributed to write the paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisherâs note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
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 articleâs Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the articleâs 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.org/licenses/by/4.0/.
About this article
Cite this article
Pizarro, J.M., Adler, S., Zantout, K. et al. Deconfinement of Mott localized electrons into topological and spinâorbit-coupled Dirac fermions. npj Quantum Mater. 5, 79 (2020). https://doi.org/10.1038/s41535-020-00277-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41535-020-00277-3
This article is cited by
-
Heavy fermions vs doped Mott physics in heterogeneous Ta-dichalcogenide bilayers
Nature Communications (2024)
-
Hall anomalies of the doped Mott insulator
npj Quantum Materials (2023)