1T-TaS2 as a quantum K. T. Lawa and Patrick A. Leeb,1

aDepartment of Physics, Hong Kong University of Science and Technology, Hong Kong, China; and bDepartment of Physics, Massachusetts Institute of Technology, Cambridge MA 02139

Contributed by Patrick A. Lee, May 26, 2017 (sent for review April 24, 2017; reviewed by Steven A. Kivelson and N. Phuan Ong)

1T-TaS2 is unique among transition metal dichalcogenides in that Mott . This fact was pointed out by Fazekas and Tosatti it is understood to be a correlation-driven insulator, where the (8) in 1976. Band calculations show that band folding creates a unpaired electron in a 13-site cluster experiences enough correla- cluster of bands near the Fermi surface. Rossnagel and Smith tion to form a Mott insulator. We argue, based on existing data, (9) found that, due to spin–orbit interaction, a very narrow band that this well-known material should be considered as a quantum is split off, which crosses the Fermi level with a 0.1- to 0.2-eV spin liquid, either a fully gapped Z2 spin liquid or a Dirac spin liq- gap to the other subbands. The narrow bandwidth means that uid. We discuss the exotic states that emerge upon doping and a weak residual repulsion is sufficient to form a Mott insulator, propose further experimental probes. thus supporting the proposal of Fazekas and Tosatti, and dis- tinguishes 1T-TaS2 from other TMD. Apparently, the formation spin liquid | Mott insulator | transition metal dichalcogenide of the commensurate clusters is essential for the strong correla- tion behavior in these 4d and 5d systems. Currently, the assign- ment of cluster Mott insulator to the undoped 1T-TaS2 ground he transition metal dichalcogenide (TMD) is an old subject state is widely accepted. A band about 0.2 eV below the Fermi Tthat has enjoyed a revival recently due to the interests in its energy has been interpreted as the lower Hubbard band in angle- topological properties and unusual . The layer resolved photo-emission spectroscopy (ARPES) (10, 11), and the structure is easy to cleave or intercalate and can exist in single- electronic-driven nature of the 200 K transition has been con- layer form (1, 2). This material was studied intensively in the firmed by time-dependent ARPES on the basis of the fast relax- 1970s and 1980s and was considered the prototypical example ation time of the spectra (12). of a charge density wave (CDW) system (3). Due to imperfect Surprisingly, with a few exceptions, the issue of magnetism nesting in two dimensions, in most of these materials, the onset associated with the Mott insulator has not been discussed in the of CDW gaps out only part of the Fermi surface, leaving behind a literature. In the standard picture of a Mott insulator, the spins metallic state that often becomes superconducting. Conventional form local moments, which then form an antiferromagnetic (AF) band theory and electron–phonon coupling appear to account ground state due to exchange coupling. No such AF ordering has for the qualitative behavior (3). There is, however, one notable been reported in 1T-TaS2. There is not even any sign of the local exception, namely 1T-TaS2. The Ta forms a triangular lattice, moment formation, which usually appears as a rise in the spin sandwiched between two triangular layers of S, forming an ABC- susceptibility with decreasing , following a Curie type stacking. As a result, the Ta is surrounded by S, forming Weiss law. As seen in Fig. 2, the drops an approximate octahedron. In contrast, the 2H-TaS2 forms an at the CDW transitions, but remains almost flat below 200 K (3). ABA-type stacking, and the Ta is surrounded by S, forming a (The small rise below 50 K can be attributed to 5 × 10−4 impu- trigonal prism. In a single layer, inversion symmetry is broken in rity spin per Ta.) This serious discrepancy from the Mott pic- 2H structure. The system is a good metal below the CDW onset ture did not escape the attention of Fazekas and Tosatti (8); they around 90 K, and, eventually, the spin–orbit coupling gives rise attempted to explain this by arguing that the g factor is very small to a special kind of superconductivity called Ising superconduc- due to spin–orbit coupling. However, their argument depends tivity (4–6). In 1T-TaS2, inversion symmetry is preserved. The sensitively on the assumption of a cubic environment for the Ta system undergoes a CDW transition at about 350 K with a jump atoms, which is now known not to be true. In fact, Rossnagel and in the resistivity. It is known that this transition is driven by an Smith (9) claim that a single band crosses the Fermi level that is incommensurate triple-Q CDW (ICDW). A similar transition mainly made up of xy and x 2 − y 2 orbitals. This picture suggests is seen in 1T-TaSe2 at 470 K. However, whereas TaSe2 stays metallic below this transition, 1T-TaS2 exhibits a further resis- tivity jump around 200 K that is hysteretic, indicative of the first- Significance order nature of this transition. These transitions are also visible in the spin susceptibility data shown in Fig. 2. In early samples, In with an odd number of electrons per unit cell, band the resistivity rises only by about a factor of 10 as the temperature theory requires that they are metals, but strong interaction is lowered from 200 K to 2 K and, below that, obeys Mott hop- can turn them into insulators, called Mott insulators. In this ping law (log resistivity goes as T −1/3) (7). More recent samples case, the electrons form local moments that, in turn, form an show better insulating behavior, and it is generally agreed that antiferromagnetic ground state. In 1973, P. W. Anderson pro- the ground state is insulating. The 200 K transition is accom- posed that, in certain cases, quantum fluctuations may pre- vent magnetic order and result in a paramagnetic ground state panied√ by a lock-in√ to a commensurate CDW (CCDW), form- ing a 13 × 13 structure. As shown in Fig. 1, this structure called a quantum spin liquid. After many years of searching, a is described as clusters of stars of David where the sites of the few examples have been discovered in the past several years. stars move inward toward the site in the middle. The stars of We point out that a well-studied material, TaS2, may be a spin David are packed in such a way that they form a triangular lat- liquid candidate. We propose further experiments that probe tice. Thus, the unit cell is enlarged to have 13 Ta sites. The for- the exotic properties of this new state of . mal valence of Ta is 4+, and each Ta site has a single 5d elec- Author contributions: K.T.L. and P.A.L. performed research; and P.A.L. wrote the paper. tron. We have an odd number of electrons per unit cell. (We first restrict ourselves to a single layer. Interlayer effects will be Reviewers: S.A.K., Stanford University; and N.P.O., Princeton University. discussed later in this section.) According to band theory, the The authors declare no conflict of interest. ground state must be metallic. The only option for an insulat- Freely available online through the PNAS open access option. ing ground state in the pure material is a correlation driven 1To whom correspondence should be addressed. Email: [email protected].

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PHYSICS possible if the spin liquid on each layer is fully gapped. In this case, NMR relaxation rate 1/T1. In the former case, gauge fluctua- a sufficiently weak interlayer hopping will not be able to destroy tions may lead to some modifications of these power laws. As the topological protection of the spin liquid state, as the transition noted earlier, the NMR data seem to point to the Dirac spin liq- to a topologically trivial state necessarily involves a gap closing. uid over a finite temperature range. Thus, it is possible for a gapped spin liquid to survive as the true Next we come to the gapped spin liquids. In mean field the- ground state even in the presence of some amount of interlayer ory, the most common description is one with gapped bosonic coupling. On the other hand, even if the spin liquid is not the true (27) together with gapped visons (28); these are called ground state, for small interlayer coupling, there may exist a finite Z 2 spin liquids, and the visons are the “vortices” of the Z2 gauge temperature range where the physics is dominated by that of the fields. We note that, in two dimensions, U (1) gapped spin liquid single-layer spin liquid. There is evidence against the formation is not allowed, because the compact U (1) gauge field will lead of dominant interlayer singlet from the 1/T1 data of ref. 17. They to confinement. A fermionic mean field theory can also lead to do not find an exponential decay with temperature as the domi- a Z 2 state with fermionic spinons (29, 30). However, this state is nant interlayer singlet model would imply. Instead, they find a T 2 smoothly connected to the one with bosonic spinons, because a behavior that may support a Dirac-type spin liquid in the tem- bosonic can bind with a vison to form a fermionic spinon, perature range between 50 K and 200 K. Below 50 K, the system and it is a question of energetics as to whether the low-energy becomes inhomogeneous and hard to interpret. spinons are fermions or bosons. There are more exotic possibili- The purpose of this paper is to bring the exciting possibility ties, but our expectation is that, without further breaking of sym- of finding spin liquid physics in 1T-TaS2 to the attention of the metry, the Z 2 gapped spin liquid is likely the only common pos- community and discuss what kind of spin liquids are consistent sibility. This state is characterized by low-energy excitations that with existing data. We also discuss further experiments that can are gapped spinons (fermions or bosons) and visons. To explain be done to probe this new . To this end, it will be the spin susceptibility data, we will have to argue that the back- interesting to look at ultrathin√ √ crystals in addition to bulk sam- ground subtraction in Fig. 2 is such that the susceptibility has ples. For example, the 13 × 13 structure in free-standing tri- dropped to a small value below the first-order transition at 200 K; layer crystals has been reported to be more robust than in the this means that the spin gap and the exchange scale J are above bulk crystal, being stable even at room temperature (22). It will 200 K. This argument will allow us to get around the issue of the also be interesting to grow atomically thin samples such as mono- absence of the Curie–Weiss law due to local moments. layers by molecular beam epitaxy (MBE). There is one more possibility of a gapped spin liquid that has been widely discussed for triangular lattice, and that is the chi- Possible Spin Liquid States in TaS2 ral spin liquid (31). This state spontaneously breaks time-reversal A key feature of spin liquid is the appearance of “fractionalized” symmetry. There should be an easily detected Ising-type transition excitations, such as particles that carry spin 1/2 but not charge. at a finite temperature, but one could argue that it is not accessi- These particles have been called spinons (15). The spinons can ble because its temperature scale is above the first-order transi- be fermions or bosons. Spin liquids can be divided into two tion at 200 K. The chiral spin liquid has many observable conse- broad classes, gapped or gapless. The gapless spin liquids generi- quences that have been widely discussed, including Kerr rotation, cally have fermionic spinons, which may form a Fermi surface or chiral spin edge modes, and spontaneous quantized thermal Hall Dirac nodes (23). The spinon Fermi surface is characterized by a conductivity. The absence of signatures of time-reversal symmetry mass corresponding to a hopping matrix element of order J , the breaking in the µSR data makes this possibility unlikely. exchange energy. A linear term in the with coef- We note that recent numerical work using density matrix ficient γ given by this mass has been seen experimentally (24). renormalization group (DMRG) and variational Monte Carlo Theoretically, gauge fluctuation is predicted to convert the lin- methods supports the existence of a region of spin liquid in a ear term to T 2/3 power, but it has proven difficult to distinguish J1 − J2 Heisenberg model on a triangular lattice where the next between the two over the limited temperature range available in nearest neighbor (NNN) J2 is in the range 0.08 < J2/J1 < 0.15 the experiment so far. A second key signature is a linear T term (32–36). Currently, many groups find it difficult to distinguish in the thermal conductivity, which is usually observed only in a between the Z 2 spin liquid, the chiral spin liquid, and the U (1) metal (25). It is widely believed that the spin liquid observed in Dirac spin liquid. These states seem to have very similar ener- the organics belongs to this type (13, 14). Interestingly, the heat gies. It is also important to remember that our system has strong SU (2) capacity of 1T-TaS2 shows a linear intercept with small upturn spin–orbit coupling. Consequently, there is no symme- at low (26). Nevertheless, we believe this observa- try for the pseudospin, and the Heisenberg model is not a good tion does not constitute evidence for a spinon Fermi surface for starting model. We expect anisotropic exchange terms. We also the following reason. The coefficient of this linear term is about expect ring exchange terms, because the system appears to be 2 mJ/mol·K2, about 4 times that of copper. This coefficient cor- quite close to the Mott transition. Thus, the phase space may be responds to a Fermi energy of about 0.16 eV; we note that the quite large to support some form of spin liquid. bandwidth of the band at the Fermi level, from band theory, is considerably narrower than this (9). Because we expect J to be Effect of Doping or and Further Experimental at least several times smaller than the bandwidth, the observed Consequences γ is much too small to be due to a spinon Fermi surface. It is Next we discuss what is known experimentally when the system probably due to the impurity moments seen in the spin suscepti- is doped or when pressure is applied. It has been reported that bility, forming a random singlet-type state. (In the organics, the 1% Fe doping destroys the CCDW state (37). A Fermi surface γ is several tens in units of mJ/mol·K2 (24) and corresponds to appears near the Γ point above this doping level (11), and super- J of 250 K.) Thus, the heat capacity data effectively rule out a conductivity with about 3 K Tc appears. Further doping creates spinon Fermi surface. Anderson localization and kills the superconductivity. With a A second kind of gapless spin liquid is the Dirac spin liquid. small amount of pressure of 1 GPa, the CCDW is destroyed and Here the spinons are gapless at certain nodal points and follow replaced by the ICDW, which is, in turn, destroyed with a pres- a linear spectrum around these points. There are two versions sure of 7 GPa (38). The state is metallic and superconducting, of Dirac spin liquid, U(1) or Z2, depending on whether there with Tc about 5 K everywhere above 4 GPa. We summarize the exists gapless gauge fields or not. In the latter case, the density low-temperature state in the pressure and doping concentration of states is linear in energy, giving rise to T 2 heat capacity and plane in Fig. 3.

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(45, in magnets placed or conductors when phenomena undoped dramatic the show that proposed been leading holes, that doped (20). possible the superconductivity of is to pairing it interlayer and induce cuprates, lad- will of the doping context in the that in to compounds analogous is der situation this singlet, interlayer an has possibility. a coupling also spin–orbit is of FL* The effect explored. the been not but has discussed, This much symmetry. been time-reversal breaks which superconductor, Liquid. Spin Chiral surface a Doping Fermi surface. a Fermi spinon form own their and form spinons may spinons the holons of volume the some of Alternatively, with superconductor. bind nodal condense, can holons a bosonic the in that is resulting liquid of spin consequence Dirac natural the A doping photons. gauge gapless dissipative has Liquid. or Spin Dirac the Doping oscilla- (44). imaging Friedel (STM) via microscope or tunneling anomaly scanning by Koln dis- tions via be measured can momentum, be states Fermi the principle, two of FL* the size the earlier, the whereas by discussed tinguished planes), As (because between metallic. layer hop is can the is state hole to metal physical perpendicular holon a direction stacked only the the quantum system, in exhibit will multilayer insulating Nevertheless, metal a gapped. holon In the as sample, oscillations. appear enough up will clean show sur- metal a Fermi will in holon a distinction show the will This but FL* The FL*. face, tunneling. in and ARPES gapless spinon in the is clearly (because but metal vol- holon gapped) surface the is violates in Fermi gapped the is dramatically excitation that tron which fact the state, is to ume ground due theorem metallic Luttinger are they (40). Anderson super- by to route envisioned are (RVB) as bond supercon- conductivity holons valence conventional resonating the a the is form case, this and ductor; this condense In will fermions. which the bosons, as when described clearly most are emerges spinons possibility this theory, field mean it conventional because the FL* obey called not been does with has but theorem state surface Luttinger Fermi This metal a 2. holon like of the behave Because factor than surface. smaller a is Fermi by volume a case the forms spin, turn, carries hole in the which, hole, physical cor- a surface Fermi the of volume metal. the holon and case, to lattice, the responds triangular not the for is (42); this holons the by predominantly pied facutrMt nuao.Telgtydpdsaei ieyan likely is state doped lightly The insulator. 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PHYSICS exotic metal with unusual experimental consequences, some of discussions and review of theoretical ideas. After the initial submission of which we discussed. Ultrathin samples are particularly attractive, this paper, we became aware of unpublished data shown in refs. 17 and 18 that have been incorporated in the final version. We thank these authors because they help avoid the complications of interlayer coupling for sharing their data and insight with us. P.A.L. acknowledges the National and can be doped by gating to minimize disorder. Science Foundation for support under Grant DMR-1522575. K.T.L. acknowl- edges the support of the Research Grant Council and the Croucher Founda- ACKNOWLEDGMENTS. We thank Debanjan Chowdhury for bringing sev- tion through HKUST3/CRF/13G, C6026-16W, 16324216, and a Croucher Inno- eral key experimental papers to our attention, and he and T. Senthil for vation Grant.

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