## Finding correlations in a Dirac-cone material

Researchers have long been on the lookout for signatures of electron-electron interactions in materials whose electrons have linear energy dispersions represented by Dirac cones, such as graphene. However, these effects have remained frustratingly small. Hirata *et al.* used nuclear magnetic resonance to study the layered organic material α-(BEDT-TTF)_{2}I_{3}, in which a phase featuring Dirac cones is known to be adjacent to one with enhanced electronic correlations. The unusual temperature dependence of spin-related properties in this material indicated strong correlations among the linearly dispersing electrons.

*Science*, this issue p. 1403

## Abstract

The Coulomb interaction in systems of quasi-relativistic massless electrons has an unscreened long-range component at variance with conventional correlated metals. We used nuclear magnetic resonance (NMR) measurements to reveal unusual spin correlations of two-dimensional Weyl fermions in an organic material, causing a divergent increase of the Korringa ratio by a factor of 1000 upon cooling, in marked contrast to conventional metallic behavior. Combined with model calculations, we show that this divergence stems from an interaction-driven velocity renormalization that almost exclusively suppresses zero-momentum spin fluctuations. At low temperatures, the NMR relaxation rate shows an unexpected increase; numerical analyses show that this increase corresponds to internode excitonic fluctuations, a precursor to a transition from massless to massive quasiparticles.

Weyl fermions in solids are massless quasiparticles described by a linear energy-momentum dispersion relation that mimics the relativistic Weyl-Dirac theory (*1*). Their effective model is characterized by a pseudospin-½ degree of freedom whose projection onto the momentum, known as the chirality, produces unconventional charge (*2*, *3*) and spin (*4*–*7*) responses. Here, we focus on a two-dimensional (2D) material and use the term Weyl fermions (WFs) to describe massless quasiparticles that obey the generalized Weyl Hamiltonian in 2D materials (*4*, *5*, *7*–*10*). In contrast to the short-range electronic correlations in ordinary metals, the Coulomb interaction among WFs has a long-range component that is unscreened at the band-crossing points, owing to the vanishing density of states at the Fermi energy *E*_{F} (*2*). As a result, anomalous phenomena can occur, such as an upward renormalization of the electron velocity in graphene (*1*, *2*, *11*), in marked contrast to its suppression in conventional correlated materials. For strong coupling, characterized by a large value of α [the dimensionless coupling constant given by the ratio of the Coulomb potential to the electronic kinetic energy (*1*, *2*)], theoretical studies have predicted an excitonic mass gap opening (or chiral symmetry breaking), which originates from the incipient instability of massless fermions as first discussed in high-energy physics (*12*) and more recently in the context of condensed matter (*13*, *14*). However, experimental characterization of interacting WFs under strong coupling has remained limited because α has been found to be rather small in a range of materials (*1*, *2*).

Here, by combining nuclear magnetic resonance (NMR) experiments and model calculations, we demonstrate the realization of a strongly coupled 2D WF system in the organic charge-transfer salt α-I_{3} [α-(BEDT-TTF)_{2}I_{3}, where BEDT-TTF is bis(ethylenedithio)tetrathiafulvalene]. α-I_{3} is a layered material comprising BEDT-TTF conducting layers and I_{3} insulating layers (Fig. 1A). In the bulk of conducting layers, a ¾-filled system is realized in which a charge-ordered insulating phase caused by short-range electron correlations appears below 135 K (*15*–*19*). An application of hydrostatic pressure (*P*) increases the bandwidth and reduces the correlations, in turn suppressing charge order (Fig. 1B) (*20*). Above a threshold *P* = 1.2 GPa, transport measurements found a semiconducting behavior with a vanishingly small gap (*15*), which led to the predictions of a 2D WF phase with a pair of spin-degenerate tilted Dirac cones (*8*–*10*, *21*) resulting from the presence of space and time inversion symmetries; the cones originate from BEDT-TTF molecular orbitals, and their nodes are fixed at *E*_{F} because of the ¾-filling of the electronic band. Experimentally, direct evidence for the cones has been found by recent NMR measurements (*4*) at a pressure of 2.3 GPa. Reflecting the low-symmetric crystal lattice that only possesses inversion symmetry, the band-crossing nodes are located away from high-symmetry points within the first Brillouin zone; in an extended region of the hopping parameter space, these nodes are stable to perturbations, such as pressure, that do not break space and time inversion symmetries (*8*–*10*).

The presence of WFs competing with charge order suggests a strong influence of electron-electron interactions on the nature of WFs (*5*, *7*, *16*, *19*, *22*). Indeed, the recent measurement of the Knight shift (*4*), probing the real part of the longitudinal static uniform susceptibility Re χ_{||}(**Q** → **0**, ω = 0) (where **Q** is a wave number vector and ω is a frequency), found a logarithmic velocity enhancement caused by the long-range component of the interaction as well as a ferrimagnetic spin polarization by short-range correlations (*7*, *22*), where opposing magnetic moments having unequal signs appear. These findings make α-I_{3} an ideal playground for investigating the effects of electron-electron interactions in WF materials. In addition, at high temperatures, α-I_{3} moves into a conventional 2D metal-like regime (*4*), reflecting the flatness of the density of states (DOS) above |*E*_{W} – *E*_{F}| ~ 12 meV (~150 K) (where *E*_{W} is a threshold up to which the DOS is linear in energy; see insets of Fig. 2A) (*8*, *21*). By varying the experimental energy scale (i.e., temperature), one can thus explore the electronic properties of 2D WFs at low *T* as compared to the metal-like state at high *T.*

To gain a deeper knowledge of the electron-electron interactions, we measured the ^{13}C-NMR spin-lattice relaxation rate divided by temperature (*T*), 1/*T*_{1}*T*, which probes the **Q** average of the imaginary part of the transverse dynamic spin susceptibility Im χ_{⊥}(**Q**, ω) (where ω lies in the MHz region) (*23*–*25*). Figure 2A presents the temperature dependence of 1/*T*_{1}*T* and the squared Knight shift *K*^{2} (*4*) in the WF state of α-I_{3} (at 2.3 GPa), measured on the ^{13}C nuclei at the center of BEDT-TTF molecules. [A magnetic field of 6 T was applied parallel to the 2D conducting layers for NMR measurements (*26*).] Both quantities are approximately flat above 150 K and decrease notably upon cooling below this temperature. In standard metals with a DOS constant in energy, the quantity 1/(*T*_{1}*TK*^{2}) is constant upon cooling; this is known as the Korringa law (*23*–*25*). In contrast, in a Dirac cone system, both 1/*T*_{1}*T* and *K*^{2} rapidly drop upon cooling, reflecting the vanishingly small DOS around *E*_{F} (*27*). The overall behavior in Fig. 2A points to a crossover from a Korringa-like metal to a gapless state below 150 K, in line with the predicted DOS profile (insets of Fig. 2A).

To see the nature of the interaction in a standard metal, it is useful to study the behavior of the so-called Korringa ratio, K = 1/(*T*_{1}*TS*_{0}β*K*^{2}), which measures the strength of the short-range electron correlations (*24*, *25*). Here, *S*_{0} = (4π*k*_{B}/*ħ*)(γ_{n}/γ_{e})^{2} (where γ_{n} is the nuclear gyromagnetic ratio, γ_{e} is the electron gyromagnetic ratio, *k*_{B} is the Boltzmann constant, and* ħ* is the reduced Planck constant) and β is a form factor representing the anisotropy of the nuclear hyperfine interaction. The value of K is on the order of unity and does not considerably vary with *T* in weakly correlated systems, whereas a sizable deviation from unity appears in strongly correlated metals, with K > 1 signifying enhanced antiferromagnetic spin fluctuations and K < 1 signifying enhanced ferromagnetic spin fluctuations (*24*, *25*). Figure 2B shows the temperature dependence of K obtained from the data in Fig. 2A using the value of β previously reported at ambient pressure in α-I_{3} (*28*), which is approximately independent of temperature and pressure (*4*, *26*). Above 150 K, a *T*-independent behavior with a size of K ~ 3 is seen, such as has been observed in the typical 2D organic metals θ-(BEDT-TTF)_{2}I_{3} (*29*) and κ-(BEDT-TTF)_{2}Cu(NCS)_{2} (*30*). This corroborates the 2D metallic picture at high *T* with moderate short-range correlations.

In contrast, a breakdown of the Korringa law sets in below 150 K, and an increase of K by several orders of magnitude appears with decreasing temperature, leading to K ~ 10^{3} at 10 K. The notable increase of K directly originates from the distinct *T* dependence of *K*^{2} and 1/*T*_{1}*T* (Fig. 2A). For a system with 2D Dirac cones, the DOS varies linearly with energy around the gapless point at *E*_{F}, which, however, does not violate the Korringa law and leads to K ≈ 1.71 (*26*). Moreover, even when the DOS is nonlinear, as in the reshaped 2D cone systems (*2*), this law does not necessarily break down; indeed, when the DOS approximately varies as a power of energy, the Korringa law holds exactly (*26*). Thus, the observed *T* dependence in Fig. 2B indicates that the finite-**Q** excitations that appear in 1/*T*_{1}*T* play a pivotal role at low *T*.

For WFs in 2D systems, the conduction and valence bands touch at *E*_{F}, giving rise to a pair of spin-degenerate nodes at ±**k**_{0} that are not topologically protected, and lack of conventional screening causes the Coulomb interaction to remain long-ranged (*1*, *2*). An in-plane magnetic field lifts the spin degeneracy and causes the Zeeman splitting of the up- and down-spin nodes. The electronic excitations at low temperature appear exclusively around these four nodes; the excitations can be categorized into two processes that are characterized by contrasting momentum transfers (*ħ***Q**), C_{Q}_{~0} and (Fig. 3C). Note that 1/*T*_{1}*T* probes the sum of C_{Q}_{~0} and , whereas *K* examines only C_{Q}_{~0} (in particular **Q** = **0**). Therefore, the different profiles of the traces in Fig. 2A signify a very distinct influence of the interactions on C_{Q}_{~0} and .

To evaluate the impact of long-range interactions, we performed a renormalization group (RG) calculation at one-loop level for dealing with the self-energy correction, based on the leading-order large-*N* approximation (*4*) and the tilted Weyl Hamiltonian describing the 2D WFs in α-I_{3} (*8*–*10*, *26*). The bare Coulomb coupling constant, α ≈ *e*^{2}/ε*ħv*, is estimated to be ~8.4 by using ε ≈ 30 and *v* = 2.4 × 10^{4} m s^{−1} as determined from fitting *K* (*4*), where *e* is the elementary charge, ε is the permittivity, and *v* is the electron velocity (*26*). The RG calculation can properly trace the observed excess suppression of *K*^{2} with respect to 1/*T*_{1}*T* and hence the divergent increase of K upon cooling (Fig. 3, A and B). The contrasting temperature dependence of *K*^{2} and 1/*T*_{1}*T* can be accounted for by the *T*-driven RG flow of the coupling constant (fig. S1) and the resultant upward velocity renormalization, which only suppresses the **Q** = **0** response. Thus, the uniform part of the static susceptibility Re χ_{||}(**Q** → **0**, 0) (∝ *K*), probing C_{Q}_{~0}, is directly affected (Fig. 3E), leading to a continuous drop of *K*^{2} upon cooling (*4*, *7*). Conversely, the **Q**-summed Im χ_{⊥}(**Q**, ω) (∝ 1/*T*_{1}*T*), probing the sum of C_{Q}_{~0} and , is affected less; the process C_{Q}_{~0} dies off upon cooling by renormalization, whereas the process , in particular in the interband electron-hole channel (inset of Fig. 3A), diminishes less and becomes prevailing at low *T* (Fig. 3D), causing moderate *T* dependence. Note that the effect of the Zeeman splitting, which would induce a saturation of 1/*T*_{1}*T* and *K* by field-induced pockets (fig. S2), is negligible, as the renormalization drastically suppresses the DOS at *E*_{F} (fig. S3) (*26*). Furthermore, the tilt of the cone has a very small effect (Fig. 3B). These considerations demonstrate that the divergent increase of K is a hallmark of general 2D WFs either tilted or vertical, which is directly promoted by the renormalization.

A similar violation of the Korringa law also appears in conventional half-filled correlated metals near a Mott transition, where an increase of K up to at most 10 upon cooling is reported (*23*). Its origin is, however, the growing finite-**Q** components of χ(**Q**, ω) that push up 1/*T*_{1}*T* but keep *K* intact, which can be clearly differentiated from the present case where the enhancement of *K* is driven by the suppressed **Q** = **0** term.

Surprisingly, 1/*T*_{1}*T* shows an additional upturn below 3 K and increases by a factor of 2 toward 1.7 K (Fig. 4A). [Note that *K* is vanishingly small in this *T* range (*4*) and is difficult to use for quantitative analyses (*26*).] Given the sharp nature of this upturn, it is likely that slow spin dynamics emerge at low energies. Although the *T*-driven RG flow (fig. S1) reduces the Coulomb coupling α toward lower temperatures, its value remains rather large in the present temperature range: α ≈ 2.0 at 5 K. Considering this sizable value, the upturn suggests emergent spin fluctuations associated with the incipient instability of gapless fermions that is driven by the Coulomb interaction.

In the presence of the Coulomb force that preserves its long-range nature, theoretical studies in 2D WF systems have revealed an electronic instability favoring excitonic pair formations (*31*). Above a critical value for the Coulomb coupling α (> α_{C} ~ 1), this leads to an excitonic transition accompanied by mass acquisition (*2*, *13*), akin to the chiral symmetry breaking that has been intensively studied in the relativistic high-energy theory (*12*). To examine the influence of this instability, we calculated χ(**Q**, ω) according to the ladder approximation (fig. S4) with the tilted Weyl Hamiltonian and evaluated the excitonic gap function at mean-field level, taking into account the velocity renormalization (*26*). We confirmed that the spin-triplet even-parity pairing can produce the upturn of 1/*T*_{1}*T* as a manifestation of the growing excitonic spin fluctuations that develop in advance of the condensation (Fig. 4B and its inset). The fluctuations set in at a temperature similar to that in Fig. 4A when one assumes a coupling of α = 2, which agrees with our RG flow results (fig. S1).

Furthermore, the calculation shows that the upturn is mainly induced by the spin transverse fluctuations (inset of Fig. 4B and fig. S5), reflecting the fact that the interband electron-hole channel has a perfect nesting for but not for C_{Q}_{~0}, as two cones are tilted in opposite directions (Fig. 3C). This is in good agreement with the RG simulation (inset of Fig. 3A) finding that the electron-hole channel of the process dominates the low-energy excitations at low temperature. Therefore, the upturn of 1/*T*_{1}*T* can be perceived as a precursor to the internode () excitonic condensation. The tilt is essentially not important for this phenomenon, although it affects details such as the size of the coupling α and the favored process (i.e., C_{Q}_{~0} or ) (fig. S6) (*26*).

Recent transport measurements in pressurized α-I_{3} have observed an in-plane insulating behavior below 5 to 10 K (*16*, *32*). The temperature dependence of the resistivity is not of Arrhenius type but is rather close to a power-law or logarithmic form, pointing to emergent fluctuations that strongly affect the transport properties above the excitonic transition temperature (*33*). This lends further support to the NMR indication that the preexisting excitonic fluctuations in close proximity to the order govern the electronic nature in this low-*T* region.

Apart from the 2D WF picture discussed so far, let us briefly examine other possible mechanisms that can contribute to 1/*T*_{1}*T* and *K*. Relaxation of the nuclear magnetization into paramagnetic impurities is one such example, which is, however, unlikely in the clean organic salt α-I_{3} where extremely high mobilities exceeding 10^{5} cm^{2} V^{−1} s^{−1} have been reported at low temperature (*15*). Indeed, even below 3 K, 1/*T*_{1} increases with decreasing temperature (fig. S7); this contradicts the impurity scenario, which would cause a leveling off of 1/*T*_{1} or an activation behavior (*34*, *35*). Relaxation caused by other (massive) electrons is another possibility, but this is also ruled out because it leads to a saturation of both 1/*T*_{1}*T* and *K*^{2} as in standard 2D metals, in clear contrast to the observations. Lastly, the orbital hyperfine interaction, which is predicted to give a divergent contribution in 3D topological Weyl semimetals (*36*), can be neglected in the weak spin-orbit 2D system α-I_{3}, as it plays an important role only in the strong spin-orbit 3D systems.

The anomalous spin dynamics whose signatures we found are a direct consequence of the chiral nature of the Weyl Hamiltonian (*1*, *10*), which is common to various Weyl-Dirac systems, regardless of dimensionality, pseudospin, or topology (*1*). Therefore, this physics would also apply to a range of 3D topological Weyl and Dirac semimetals if the contribution from ordinary electrons is sufficiently small. This work paves the way for the exploration of the rich physics of “strongly interacting WFs” in solids (*2*), which has until now mainly been discussed on a theoretical basis.

## Supplementary Materials

www.sciencemag.org/content/358/6369/1403/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S7

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

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**Acknowledgments:**We thank K. Nomura for critical discussions and reading of the manuscript; D. Basko for technical help with RG calculations; M. O. Goerbig for support and discussions; J. S. Kinyon for fruitful discussions; and H. Fukuyama, N. Nagaosa, H. Isobe, H. Kohno, Y. Suzumura, M. Ogata, T. Osada, C. Hotta, H. Yasuoka, D. Liu, H. Nojiri, T. Kihara, H. Mukuda, M. Tokunaga, Y. Maeno, W. Li, M. Potemski, M.-H. Julien, H. Mayaffre, and M. Horvatić for helpful discussions and comments. Supported by MEXT/JSPS KAKENHI grants 20110002, 21110519, 24654101, 25220709, 15K05166, 15H02108, 17K05532 and 17K14330; JSPS Postdoctoral Fellowship for Research Abroad grant 66, 2013; the MEXT Global Center of Excellence Program at the University of Tokyo (Physical Sciences Frontier grant G04); the Kurata Memorial Hitachi Science and Technology Foundation (G.M. and A.K.); and the Motizuki Fund of the Yukawa Memorial Foundation (M.H.). The data presented in this paper are available upon request to the first author.