Abstract
The Coulomb interaction among massless Dirac fermions in graphene is unscreened around the isotropic Dirac points, causing a logarithmic velocity renormalization and a cone reshaping. In less symmetric Dirac materials possessing anisotropic cones with tilted axes, the Coulomb interaction can provide still more exotic phenomena, which have not been experimentally unveiled yet. Here, using siteselective nuclear magnetic resonance, we find a nonuniform cone reshaping accompanied by a bandwidth reduction and an emergent ferrimagnetism in tilted Dirac cones that appear on the verge of charge ordering in an organic compound. Our theoretical analyses based on the renormalizationgroup approach and the Hubbard model show that these observations are the direct consequences of the longrange and shortrange parts of the Coulomb interaction, respectively. The cone reshaping and the bandwidth renormalization, as well as the magnetic behaviour revealed here, can be ubiquitous and vital for many Dirac materials.
Introduction
Dirac materials^{1} are a novel class of solidstate systems, in which the lowenergy electronic excitations are described by pseudorelativistic massless Dirac fermions (DFs) with linear energy dispersion around the Fermi energy E_{F}. Triggered by the studies in twodimensional (2D) graphene^{2} and the surface of threedimensional topological insulators^{3}, extended now to threedimensional Weyl and Dirac semimetals with strong spin–orbit coupling^{4,5,6}, many intriguing properties of DFs have been revealed and have constituted active topics in modern condensed matter physics. The role of Coulomb interaction is one of such issues of particular interest^{7,8,9,10,11,12,13,14,15,16,17}. For instance, in chargeneutral 2D massless DF systems, composed of two gapless points at E_{F}, the longrange (LR) part of the Coulomb potential V(q) (q: wave vector) is unscreened owing to the vanishing density of states (DOS) at E_{F}. Consequently, the LR character of the interaction (V(q)∝1/q) is preserved at low energy, which couples to the fermionic excitations and induces a logarithmic correction to the Fermi velocity v_{F} and associated physical quantities^{7,8,9}. The logarithmic velocity renormalization induces a nonlinear reshaping of the cones around each of Dirac points (DPs), as observed in graphene near the charge neutrality point^{11,12,13,14}.
However, graphene is a special case of 2D massless DF systems, in which isotropic Dirac cones with vertical axes have the DPs at particularly symmetric points on the Brillouin zone boundary^{2}. Indeed, theoretical studies have revealed that massless DFs possessing anisotropic cones and the DPs at arbitrary kpoints emerge more generally in a broad class of materials^{1,18,19,20}. A typical example in 2D is the organiclayered compound α(BEDTTTF)_{2}I_{3} (αI_{3}) (BEDTTTF=bis(ethylenedithio)tetrathiafulvalene) (Fig. 1a), which has a pair of Dirac cones occurring at two distinct points (±k_{D}) in the 2D first Brillouin zone (Fig. 1c)^{20,21,22,23,24,25,26,27,28,29,30}. The electronic structure of αI_{3}, described on the base of molecular orbitals as usual in this type of compounds^{31}, is rather involved compared with graphene due to the presence of four sites per unit cell (Fig. 1b) with anisotropic hopping amplitudes^{32,33}. The system has only the inversion symmetry^{32,33,34}, which, in conjunction with the anisotropic hopping, brings about a tilt of Dirac cones and drives the 2D DPs away from high crystallographic symmetry positions^{20,21,23,25,35,36} (Fig. 1c). A remarkable feature is that, because of the 3/4filled nature of the electronic bands^{23,24,25,26,33}, the two gapless points are anchored at E_{F} by this band filling in αI_{3}.
Another issue of great physical interest in αI_{3} in terms of the Coulomb interaction problems is that, within the pressure–temperature (P–T) phase diagram (Fig. 1g)^{32,37,38,39,40,41}, the 2D massless DF phase appears in the vicinity of an insulating phase with charge order, as first pointed out by transport measurements^{22}. This contrasts with the case of graphene, in which no phase transitions have been observed at least in the absence of a quantizing magnetic field^{10}. The chargeordered phase in αI_{3}, which is induced by the strong shortrange (SR) electron correlations in this 3/4filled system^{40,41}, is suppressed when applying a P above a critical value of P_{C}≈1.2 GPa (Fig. 1g) and turns into the 2D massless DF phase^{22,39}. Once the highP phase is reached, the Dirac cones become stable against further pressurization; in fact, the gapless point is fixed at E_{F} on varying the hopping integrals in a finite range by virtue of the 3/4filled nature of the electronic bands, as revealed by bandstructure calculations^{23,25,26,33,42,43,44,45}. The presence of such a phase transition in this system potentially offers the possibility to test the impact of the SR electron correlations on the behaviours of 2D massless DFs. Moreover, the tilt of anisotropic Dirac cones^{36} coupled with the SR and LR parts of the Coulomb interaction opens new possibilities in the physics of 2D massless DFs. For instance, it is predicted to bring about a nonuniform reshaping of titled cones^{46}, novel nonFermi liquid behaviours near the quantum critical point^{16,17}, where two DPs merge^{47,48}, enhanced shot noise for quantum transport^{49} and anomalous charge/spin textures inside the unit cell^{48,50}. Studying the electronic structures and the role of the Coulomb interaction in pressurized αI_{3}, which remains unclear up to date, is thus of primary importance to understand the various effects of the Coulomb interaction in 2D massless DFs.
In this article, we focus on the 2D massless DF phase in αI_{3} emerging under a hydrostatic pressure (P>P_{C}) and present experimental evidence for interaction effects of massless DFs. Employing siteselective nuclear magnetic resonance (NMR), we uncover three distinct interaction phenomena induced by the electron–electron Coulomb interaction. First, NMRshift measurements in conjunction with renormalizationgroup (RG) analyses reveal a Tdriven cone reshaping around each of the DPs due to the LR part of the Coulomb interaction. Because of this reshaping, tilted cones become effectively isotropic at low energies. Second, quantitative RG analyses establish that the best fit to the data inevitably requires a strong bandwidth reduction inherent to the SR electron correlation, as often discussed in strongly correlated materials. Finally, an anomalous ferrimagnetic spin polarization is observed, which is accounted for by the onsite Coulomb repulsion, as revealed by a simulation based on the Hubbard model presented here. These experimental and theoretical investigations demonstrate that αI_{3} under P is an intrinsically interacting 2D massless DF system, in which both the LR and SR parts of the Coulomb interaction strongly influence the electronic behaviours.
Results
Basic principles to probe tilted Dirac cones
Our strategy to investigate tilted Dirac cones in αI_{3} is as follows. The crystal structure of αI_{3} has a 2D unit cell with four molecular sites (dubbed sites A, A’ (=A), B and C), each of which constitutes a sublattice in the crystalline abplane (Fig. 1a,b). The four molecular orbitals on these sites form a pair of tilted Dirac cones near E_{F} (Fig. 1c). Around the gapless point at E_{F}, a very unique situation is realized where the Bloch state has different weights in the amplitudes of the four molecular orbitals. The bandstructure calculation^{25} revealed that these weights, dubbed sitespectral weights hereafter, show anisotropic k dependence around each of 2D DPs with a clear contrast between nonequivalent sites. The corresponding sitespectral weight for the sublattice j=A (=A’), B and C around the DP at k_{D}, (equation (11)), is shown in Fig. 1d–f, where q=(q_{x}, q_{y}) is defined as q=k−k_{D} and ζ=± is the band index (Fig. 1c). (For details, see Methods.) Notably, the anisotropy of the sitespectral weights makes a particular distinction between the site B and C. Namely, the Bloch electrons with large v_{F} (in the steep slope of the tilted cones) have a large weight on the Bsite wavefunction, whereas the Bloch states with small v_{F} (on the opposite side of the cones in the gentle slope) have a large weight on the Csite wavefunction (Fig. 1e,f and Supplementary Fig. 1a,c). Thus, if one probes the local electronic states on sites B and C separately by means of a siteselective measurement, it is possible to reveal the electronic nature of the Bloch states in the steep part and the gentle part of the tilted Dirac cones individually. Taking advantage of this feature, we performed a siteselective NMR in this compound to separately elucidate the electronic states in the two slopes of the Dirac cones. Specifically, the Knight shift, derived from the NMR line shift measured at a temperature T, is converted into the local electronspin susceptibility on the site j, , which is given by a thermal average of the sitespectral weight around E_{F} summed over all q for both electrons (ζ=+) and holes (ζ=−). Hence, the siteselective NMR in αI_{3} works as an effectively qresolved probe of 2D DFs thermally excited around each of DPs. Indeed, the electronic excitations in the steep and gentle parts of the Dirac cones can be almost independently probed by and , respectively, as we will demonstrate below.
NMR observation of tilted Dirac cones
To address the electron interaction issues of 2D massless DFs, we have carried out ^{13}CNMR measurements in αI_{3} at P=2.3 GPa (>P_{C}; see Fig. 1g) on the four molecular sites in the unit cell, j=A, A’ (=A), B and C. The two ^{13}C nuclei (spin I=1/2) introduced at the centre of BEDTTTF (ET) molecules (inset of Fig. 1a) are used for ^{13}C NMR, which are known as a sensitive probe of electronic states at E_{F} in this class of compounds^{51}. Figure 2a shows the typical NMR spectra observed at a magnetic field of H=6 T, applied in the crystalline abplane. Eight lines are observed in the spectra, associated with the four molecular sites in the unit cell. By changing the field orientation in the crystalline abplane and examining the angular dependence of the line positions, the eight lines are to be assigned to two doublets from sites B and C and one quartet from the site A (=A’)^{37}. (The site A and A’ are not distinguished hereafter). Figure 2b shows the typical angular dependence of the NMR total shift for each molecular site, defined as the centreofmass position of the doublet or the quartet. The jsite total shift for a given T and a fieldangle (measured from the crystalline a axis; Fig. 2c) is expressed as , where the first term is the conductionelectron term (Knight shift) and the second term stands for the coreelectron contribution (chemical shift). Note that the socalled hyperfine coupling constant, , and σ_{j}() are strongly dependent (with little T and P dependence), while the susceptibility is isotropic in this compound^{52}. Because the electronic excitations around the gapless point (at E_{F}) are vanishingly small at low temperatures, the total shift S_{j} at the lowest T (=3 K in the present experiment) is expected to provide the chemical shift term σ_{j}. Thus, subtracting σ_{j}() from and employing the value of the hyperfine coupling constant reported at ambient pressure^{37}, the total shift is converted into the local electronspin susceptibility (for details, see Methods).
Figure 2d shows the temperature dependence of at the site j=A, B and C, which arises from the interband thermal (electron–hole) excitations across the gapless point at E_{F} and is proportional to the k_{B}T average of the jsite DOS around E_{F}: . Decreasing from T=300 K, exhibits Tindependent features down to T≈200 K, followed by a rapid decrease with a clear difference in the size of the susceptibility between nonequivalent sites, >> (see also Supplementary Fig. 2), and finally becomes vanishingly small at all sites. The observation, in particular the crossover from ∼const. to ∼0 on cooling, indicates that there is an energyindependent large D_{j}(E) at high energies (inset I of Fig. 2e) and a vanishingly small D_{j}(E) at low energies around E_{F} (inset II of Fig. 2e). This is consistent with the bandstructure calculations^{24,25,50}, where a flat DOS is predicted at high energies above the vanHove singularity (locating at about 12 meV off E_{F}) and linear energy dependence is suggested around the bandcrossing point at E_{F}. Note that D_{j}(E) around the gapless point (E=E_{F}=0) is given by a q summation of the sitespectral weight (Fig. 1d–f) at a given energy E for the band ζ (equation (14)). Then, the anisotropic qdependence of around the DPs of tilted Dirac cones (Supplementary Fig. 1) is expected to bring about D_{C}(E)>D_{A}(E)>D_{B}(E) at low energy. This is in excellent agreement with the observed relation >> below T≈200 K (Fig. 2d) and is the direct consequence of the fact that the site B and C selectively probes the steep and gentle slopes of titled cones, respectively, consistent with the prediction of the effective tightbinding (TB) model given in ref. 25. All these observations demonstrate the existence of tilted Dirac cones with E_{F} located around the gapless point.
Fermi velocity renormalization
However, the strong nonlinear temperature dependence in below T≈150 K does not comply with the expectation of TB calculations, which leads to ∝T at low temperatures^{25,50}. To better visualize this point, we plot the first derivative of the susceptibility (d/dT), as shown in Fig. 2e. With decreasing T, d/dT exhibits a peak at ≈120 K for the site B, and at ≈50–60 K for the site A and C, and then drops continuously to zero at all sites towards low temperatures. These features are in striking contrast to the TB calculation^{25}, where d/dT increases on cooling but saturates at low T (Supplementary Fig. 3d–f). Indeed, varies almost quadratic in T at all sites below the inflection point (), which suggests a nonlinear suppression of D_{j}(E) around E_{F} (=0) in an energy range of . As the total DOS, D(E), is proportional to the inverse square of v_{F} in 2D massless DF systems for the noninteracting case (refs 2, 20), a suppression of DOS in turn corresponds to an enhancement of v_{F}. Thus, the observed peak structure in d/dT strongly indicates that a Tdriven renormalization of v_{F} grows below T≈. The most probable origin of this effect is the LR part of the Coulomb interaction between electrons, which is unscreened at E_{F} in chargeneutral massless DF systems and is known to cause a logarithmic correction to v_{F} either driven by tuning carrier densities^{1,7,8,9,10,11,12,13} or temperatures^{9,46}. We recall, however, the value of is twice higher for the site B (≈120 K) than for the site A and C (≈50–60 K). At first glance, this may add an extra complication to the data interpretation but in fact turns out to be a direct consequence of the anisotropy of the sitespectral weight (Fig. 1d–f) and the tilt of Dirac cones, as we shall see below.
Renormalizationgroup analyses
To further understand the nonlinear temperature dependence of at each site, we have examined the selfenergy correction effect due to the LR Coulomb interaction. For this, we employed a RG approach based on an effective Hamiltonian near the gapless point^{20,25}, whose energymomentum dispersion is given by
where w_{0}=(w_{0x}, w_{0y}) and v=(v_{x}, v_{y}) are velocities reflecting the tilt and anisotropy of the cone, respectively (for details, see Methods). At the oneloop level, the selfenergy correction leads to a renormalization of v but does not affect w_{0} (Supplementary Fig. 4a)^{46}. The RG flow of v is expressed as
where N=4 is the number of fermion species, corresponding to two DPs and two spin projections, q=q(cos ϕ, sin ϕ) is measured from k_{D}, l=ln(Λ/q) is the momentum scale, Λ (=0.667 Å^{−1}) is a momentum cutoff of the size of the inverse lattice constant^{33} and is circular around the DP, is the coupling, F(g_{ϕ}) has the form and ɛ is the dielectric constant. (Note that equation (2) is obtained in the leading order in 1/N assuming N>>1, which is valid both for the weak and strong Coulomb interaction.)
Assuming the four velocities given by the effective TB calculation ( and v^{TB})^{25} as initial velocities at q=Λ, we have calculated the RG correction effects on (Fig. 3a–c) D_{j}(E) (Fig. 4a–c) and the energy spectrum (Fig. 4d–f). (For the justifications of employing this TB model as well as the velocities and v^{TB} in performing RG calculations, see Methods.) Here, we note that the temperature is used as an explicit scale parameter in the calculation of that determines the RG flow. To get a reasonable agreement between the calculation and experiment, a phenomenological parameter u is introduced to adjust the velocities at q=Λ such that and v′=u v^{TB}. The two parameters in the calculation, (u, ɛ), are then optimized from a leastsquare fit to the experimental susceptibilities. Good agreements are obtained in the fit especially at the site A and C (Fig. 3a–c), which lead (u, ɛ)≈(0.35, 1). (Note that the fitting results are sensitive to the choice of u while they are little dependent on ɛ in the range ɛ≈1–30; for details, see Methods and Supplementary Figs 4–7). The calculation demonstrates that the nonlinear T dependence of below T≈ can be properly ascribed to the logarithmic renormalization of v_{F}. In Fig. 4a–c, the calculated shape of D_{j}(E) is shown around the gapless point at E_{F}. A strong suppression from the Elinear DOS is seen at low energies due to the renormalization. Figure 4d–f, depicts the corresponding energy spectrum around the DP (at k_{D}), where the colours indicate the magnitude of the sitespectral weight, , in Fig. 1d–f, respectively. A nonlinear reshaping of the tilted cone induced by the renormalization is clearly visible around the gapless point. It should be stressed that a good agreement is accomplished only when a small value of u (≈0.35) is used. The fact that we have u<1 indicates a reduction of the initial velocities or of the hopping amplitudes between the adjacent lattice sites. In conventional strongly correlated materials, the SR part of the Coulomb interaction is well known to induce this sort of hopping (or bandwidth) renormalization due to the frequency dependence of the selfenergy^{53}. We believe that the observed ureduction effect in the initial velocities occurs because of this selfenergy correction due to the SR Coulomb interaction, which is not considered implicitly in the original RG calculation.
Remarkably, the calculation well reproduces the observed difference in the thermal energy scale of the Tdriven renormalization in , ΔE^{j} (≈k_{B}), at the site B and (A, C) as indicated by thin arrows in Fig. 3a–c. This distinction stems from the tilt of the Dirac cones and the resultant momentum dependence of the energy cutoff δ around the DP, , where is the ϕ dependent Fermi velocity in the band ζ (Supplementary Fig. 1a,b). Namely, the energy cutoff is large for the largev_{F} DFs, dominantly probed by the site B, while it is small for the smallv_{F} DFs, having a large weight on the site C (and A). Hence, the renormalization starts from a higher T in (T) than in (T) (and (T)), producing the observed energy scale difference ΔE^{B}>E^{A,C}, consistent with the previous RG calculation of Isobe et al.^{46}. It is also worth mentioning that tilted cones become more isotropic at lower energies near E_{F} because of the nonuniform velocity renormalization around each DP, as reflected in Fig. 4d–f. This is because the anisotropic term in the Hamiltonian is small (v_{x}/v_{y}≈1) in αI_{3} (ref. 25), and we have w_{0}<<v around the DP due to the RG flow (Supplementary Fig. 4a), leading the tilting term (w_{0}) to be effectively negligible near E_{F}.
From all these, it is concluded that our RG analyses appropriately capture many of the essential parts of the experimental results. They constitute experimental evidence for the bandwidth renormalization (the ureduction effect) due to the SR repulsion between electrons as well as the Tdriven logarithmic renormalization of v_{F} by the LR part of the Coulomb interaction. Nevertheless, we note that, at low temperatures, the agreement is less satisfactory for the site B compared with the other sites (Fig. 3c), suggesting the presence of another correlation effect. Indeed, we will clarify this point by a latticemodel simulation, as described below.
Emergent ferrimagnetic spin polarization
The temperature dependence of (T) in the experiment is appreciably stronger and more complex than what is predicted by the RG calculation (Fig. 3c). Indeed, the experimental (T) exhibits an anomalous sign change at T≈60 K and an upturn with a negative slope below T≈40 K (inset of Fig. 3c), while the RG calculation shows monotonic temperature dependence. The observation of <0 is in sharp contrast to >0 and in the experiment (Fig. 5a), indicating an emergent ferrimagnetic spin polarization in which the local magnetic field points antiparallel to the applied field at the site B while it is parallel to the field at all other sites (Fig. 6). To further understand this sublatticescale magnetism, we have investigated the Hubbard model with an onsite repulsive (Hubbard) interaction, U, at a meanfield level within the random phase approximation (RPA). For the RPA calculation, we have considered both the interband and intraband contributions to the spin susceptibility with a wave vector Q=0 (for details, see Methods). Figure 5b presents the calculated temperature dependence of the RPA spin susceptibility at the site B. Using U=0.14 eV, the RPA calculation (in particular the interband term) clearly reproduces the observed negative spin susceptibility for site B (<0) at similar temperatures. Moreover, the negative susceptibility appears only at site B in the calculation (Supplementary Fig. 3a–c) in good agreement with the experiment (Fig. 5a and Supplementary Fig. 2). These facts show that the ferrimagnetic polarization is induced by the onsite Hubbard interaction. The fact that the negative polarization emerges solely on the site B might be relevant to a superexchangelike interaction between the site A and A’ (via B) (see Fig. 6). Density functional calculation^{24} suggests a largest hopping amplitude on this path (b2 in Supplementary Fig. 8), and Xray and Raman scattering measurements^{32,38} point to the largest hole density (despite the small spin density) at the intermediate site B in the unit cell. Then, if there is an antiferromagnetic (ferrimagnetic) coupling between sites A and B (A’ and B), a large energy gain is expected due to the kinetic energy of electrons, which favours the observed pattern of the ferrimagnetic polarization.
Generally speaking, RPA tends to overestimate the effect of correlations, as it does not consider the selfenergy correction to the energy bands^{54}. In fact, the onsite interaction we use, U=0.14 eV, is chosen smaller than what would be typically used (U≈0.4 eV) (refs 25, 26). This discrepancy brings another support that the selfenergy correction due to the SR interaction is of great importance in this system, consistent with the bandwidth reduction effect we discussed in the RG calculation. Finally, we note that RPA is unable to reproduce the observed nonlinear T dependence of at low temperatures. This is because the selfenergy correction due to the LR Coulomb interaction, which is the main origin of the nonlinear , is not taken into account in RPA (for details, see Methods and Supplementary Fig. 3d–f).
Discussion
So far, we have demonstrated three distinct Coulombinteraction phenomena in the 2D massless DF phase of αI_{3}, which develop systematically at different temperature scales (or energy scales of thermal excitations), as summarised in Table 1. A bandwidth renormalization (or the ureduction of v_{F}) occurs due to the SR Coulomb interaction which appears to exist from room temperature down to lowest T. At temperatures T≤ (or in the corresponding energy range around the gapless point at E_{F}), a Tdriven logarithmic renormalization of v_{F} and the resultant nonuniform reshaping of the Dirac cones appear, due to the LR part of the Coulomb interaction. With further decreasing T, a ferrimagnetic spin polarization shows up in the unit cell because of the onsite Coulomb repulsion between electrons.
First, we mention that the observed nonuniform reshaping of titled Dirac cones in αI_{3} should affect other physical observables at low temperature or at low magnetic field. As the cones become effectively isotropic around each of DPs (Fig. 4d–f), the shape of the crosssection of the cone is different at high energy and close to the gapless point, which should cause a Tdependent anisotropy of the inplane electrical conductivity. Another experiment that would be able to see the reshaping is infrared spectroscopic measurements, known as a powerful tool to reveal the Landau level (LL) structure in graphene^{55}. In a perpendicular magnetic field (H) normal to the 2D plane, the massless DFs in αI_{3} exhibit the LL spectrum, (refs 56, 57), where ζ=± distinguishes the electron and hole bands crossing at the gapless point, n (=0, ±1, ±2, ⋯) is the LL index, l_{B}=(ℏ/eH)^{1/2} is the magnetic length and is a tilting parameter. As E_{F} locates at the gapless point in αI_{3} and the n=0 LL is halffilled, one may be able to detect, for instance, the dipole transition n=0→n=1 in the absorption line at the energy . The velocities (v_{x}, v_{y}) with a logarithmic correction may appear in ΔE_{10} at low field; v_{x} (≈v_{y}) will increase by a factor of four by changing ln(Λ/q) from 1 to 5 (Supplementary Fig. 4a) and (1−Ω^{2})^{3/4} increases by a factor of two (from ∼0.5 to ∼1.0). One would expect to see this change in ΔE_{10} at low temperatures where the LL broadening, mainly caused by the thermal scattering of carriers in the n=0 LL (refs 27, 28), becomes sufficiently small.
Theoretically, the renormalization of the coupling constant of the Coulomb interaction (the RG flow of v_{F}) makes the system flow to a weak coupling regime at low energies (or at low T in the case of a Tdriven RG flow as in our experiment)^{10}. As organic compounds are very clean and are little influenced by impurities, this consequence of the RG flow implies that lowT electron correlation effects, if any, would be induced by the SR part of the Coulomb interaction, as usually the case in conventional strongly correlated materials. In fact, the insulating phase possessing charge order^{22,26,31,32,33,34,35,36,37,38,39,40,41} emerges next to the 2D massless DF phase on the P–T phase diagram in αI_{3} (Fig. 1g), suggesting the vital role of the SR Coulomb interactions in the massless DF phase of this 3/4filled system, in line with recent meanfield calculations^{48}. In the halffilled system graphene, strong electron correlations are predicted to stabilize Mott insulators and charge density waves^{10}. However, the typical experimental conditions seem to locate away from these situations^{1,10,58}, and no phase transition has been reported yet. Under a strong magnetic field, on the other hand, a gapped liquid phase is observed with a quantized Hall conductivity at fractional filling factors, stabilized by the SR part of the Coulomb interaction^{10}. Similar physics may occur in thin films of αI_{3} as well, where the integer quantum Hall effects have recently been observed^{59}.
To conclude, our NMR measurements combined with theoretical calculations have demonstrated three Tdependent Coulomb interaction effects of 2D massless DFs (Table 1) in pressurized αI_{3} (P>P_{C}), having tilted Dirac cones and gapless points fixed at E_{F}. We found that the LR part of the Coulomb interaction, which is unscreened around the gapless DPs, causes a Tdriven renormalization of the velocity and induces a nonuniform reshaping of tilted Dirac cones. Quantitative analyses of the cone reshaping based on the RG approach further necessitates a large bandwidth reduction due to the SR electron correlation. Moreover, we showed that the onsite Coulomb repulsion gives rise to a ferrimagnetic spin polarization as unveiled by the numerical calculations using the Hubbard model. These findings can be distinguished from the case of weakly interacting 2D massless DFs in graphene with vertical Dirac cones and are consistent with the emergent correlated phase on the verge of the massless DF phase in the P–T phase diagram. Continuing this study to the vicinity of the phase transition at P_{C} (≈1.2 GPa) would be of particular interest, which may connect the physics of the massless DFs and conventional strongly correlated materials.
Methods
Sample preparation
Single crystals of α(BEDTTTF)_{2}I_{3} (αI_{3}) (ref. 60) with the dimensions of 0.1 × 0.5 × 2.0 mm^{3} were synthesized from ^{13}Cenriched BEDTTTF (ET) molecules using the conventional electrochemical method. To perform ^{13}CNMR measurements, the central carbon atoms of BEDTTTF molecules connected by a double bond were 99% enriched by carbon13 (^{13}C) isotopes (inset of Fig. 1a) with a nuclear spin I=1/2.
Pressurization scheme
A hydrostatic pressure of P=2.3 GPa was applied to the sample using a BeCu/NiCrAl clamptype pressure cell, with the Daphne 7373 oil as a pressure medium. At this pressure, the oil locates close to the liquid–solid phase transition point at room temperature^{61,62}. To avoid applying uniaxial strains to the sample, the cell was kept at a sufficiently high temperature (T≈50 °C) during the pressurization. With decreasing temperature from T=300–3 K, a pressure reduction of ΔP∼0.1 GPa occurs inside the cell^{61,62}. The inner pressure at the lowest T we measured (3 K) is, however, substantially higher than the transition pressure to the chargeordered phase at T≈0 (P_{C}≈1.2 GPa; see Fig. 1g)^{22,30,39,40,41,63,64}, indicating that the Preduction will not change the physics. Hence, we neglect this effect throughout the paper.
^{13}CNMR measurements
^{13}CNMR measurements were performed in αI_{3} in a magnetic field H of 6 T applied parallel to the crystalline abplane (Fig. 1a). To get NMR signals, the standard spinecho techniques were used with a commercially available homodyne spectrometer. Spinecho signals were recorded at a fixed radio frequency after the conventional spinecho pulse sequence of t_{π/2}−τ−t_{π} (with τ=5–25 μs and t_{π/2}−t_{π}=0.6–1.2 μs) and were converted into the NMR spectra via Fourier transformation. The resonance frequency of the natural abundance ^{13}C nuclei in TMS (tetramethylsilane (CH_{3})_{4}Si) was used as the origin of the NMR shift. We note that the present work is targeted at lower T and higher P compared with the earlier NMR studies^{65,66}, which is more suitable for exploring the nature the lowT 2D massless DF phase emerging at P>P_{C} on the phase diagram (Fig. 1g)^{22,27,28,29,30,39}.
Line assignments of the NMR spectra
The details of line assignments for the ^{13}CNMR spectra in this compound are given elsewhere^{37,65}. Here, we shall describe only the essence. The ^{13}CNMR spectra of αI_{3} show large temperature dependence (Fig. 2a and Supplementary Fig. 9) and fieldangle () dependence (Supplementary Fig. 10). They have eight ^{13}C lines that consist of two doublets (from the B and the C molecules) and one quartet (from the A (=A’) molecule) (Fig. 2a). The doublet and the quartet are caused by the nuclear dipole–dipole interaction in a ET molecule (between the two ^{13}C nuclear magnetic moments around the molecular centre; see the inset of Fig. 1a)^{37,51,65}. The NMR total shift S_{j} for a particular site j is determined from the centreofmass position of the corresponding ^{13}C lines, which is expressed by a sum of the Knight shift K_{j} () and the chemical shift σ_{j} terms, , as mentioned in the text. (The orbital vanVleck contribution is negligible in ETbased salts because of the low lattice symmetry^{51}.) Both and σ_{j}() are highly anisotropic in this system^{37,65,66,67}, causing clear dependence of the spectra. The anisotropy of the shift is, however, largely different between the molecule A(A’) and molecules B and C, reflecting the fishbonelike arrangement of ET molecules in the crystalline abplane (Fig. 1b). Thus, by rotating the magnetic field H in this plane, one can assign the ^{13}C lines to the different sublattices^{37}, with the aid of the Xray diffraction data under pressure^{33} (Supplementary Fig. 10).
Conversion of the NMR shift to the spin susceptibility
In αI_{3}, the Tdependence curve of the jsite total shift shows a largely distinct feature for different field orientations. For instance, the T dependence of is large at =60^{°} showing a prominent decrease with decreasing T (Supplementary Fig. 11a), whereas it is small at =120^{o} with an increase on cooling (Supplementary Fig. 11b). This difference can be accounted for by the anisotropy of the Tindependent hyperfine coupling constant , where can be either positive or negative depending on the value of , as we will describe below. First, we note that the chemical shift σ_{j}() is Tindependent^{51} as well as little affected by P (see Supplementary Discussion). Thus, the observed T dependence of the total shift S_{j} can be ascribed to the Tvariation of the Knight shift K_{j}. Here, is the hyperfinecoupling constant averaged for the two central ^{13}C nuclei in the molecule j (Fig. 1a), which is dependent reflecting the anisotropy of the coupling tensor^{33,37,66,67}. The principal values of the tensor are, however, weakly affected by the change of T and P (Supplementary Figs 12 and 13). Moreover, the spin susceptibility is expected to be isotropic in this compound^{52}. These points clearly indicate that K_{j} can practically be expressed as K_{j}(T, ) (for details, see Supplementary Discussion).
Notice that there are no excitations around the gapless DP in the ground state of massless DF systems. This means, at the lowest T, the Knight shift K_{j}(T, ) is expected to become vanishingly small^{68}, and the total shift resumes to the Tindependent chemical shift σ_{j}() (Supplementary Fig. 12). For the site B, there is a negative slope in the T dependence of the total shift below T≈40 K, showing a small increase of the shift (of ∼5 p.p.m.) towards lower T (inset of Supplementary Fig. 11b). This is to be associated to the ferrimagnetic spin polarization, which causes a local magnetic field that points opposite to the external field only at the site B (Fig. 6)^{42}. The effect is, however, negligible at the lowest T since the thermally excited polarization vanishes as T→0 (Fig. 5b and Supplementary Fig. 3c). Hence, we fitted to the angular dependence of for all sites at the lowest measured temperature (T=3 K) and assumed this fitted curve to be the chemical shift value at each angle . The total shift for a particular site is then converted to K_{j}(T, ) by subtracting this σ_{j}(). The subtraction is done at a field orientation where the total shift becomes close to the maximum in order to minimize the ambiguity of the chemical shift, namely, =60° for the site A(A’) and =120° for the site B and C (Supplementary Figs 11 and 12). The value of the hyperfine coupling constant is calculated for these angles by employing the coupling tensors given at ambient pressure^{37} by means of Xray diffraction data reported at high pressure^{33}. This yields =9.0, =6.4 and =9.9 in kOe μ_{B}^{−1}. In terms of these coupling constants, the Knight shift K_{j}(T,) is eventually converted to the local electronspin susceptibility (Fig. 2d).
Effective tightbinding model for the tilted Dirac cone
In order to construct reasonable arguments for the fitting analyses to the observed (Fig. 2d), we have introduced a fourband bandstructure calculation of ref. 25. The lowenergy effective model based on this calculation shall be used as a noninteracting reference to the data analyses in particular in the RG calculation. (For details, see Methods: RG calculations). The rationales behind this choice are described here.
In αI_{3}, it has been shown that the gapless DPs at ±k_{D} (Fig. 1c) are fixed at E_{F} for a certain parameter region in the fourband TB parameter space with and without finite site potentials^{23,33,42,43,44,45}. However, it is difficult to use bare TB parameters as adjustable variables in the fitting analyses of the lowT part of (in Fig. 2d). This is because the TB calculations lead to linear dispersion around the DP^{21,23}, which causes ∝T, owing to the excitations around the gapless point at E_{F} (ref. 68). Our experiment, on the other hand, exhibits nonlinear Tdependence in at all sites below T≈ (Fig. 2d,e) and a negative (T) at low T (≤/2≈60 K; see the inset of Figs 3c and 5a). A model calculation based on simple linear dispersion can hardly account for these features. Thus, instead of fitting the data with TB parameters, we will use them as a minimal noninteracting reference and perform more sophisticated RG calculations based on a continuum model derived from that reference.
For the minimal model in our study, we use the bandstructure calculation of ref. 25, which is practically based on a noninteracting TB model with adjustable sitedependent potentials, associated to the four molecular sites in the unit cell, j=A(1), A’(2), B(3) and C(4) (Fig. 1b). (This model shall be dubbed the effective TB model throughout this study.) Strictly speaking, this model takes into account the electron–electron Coulomb interaction up to the nearestneighbour terms and, at first glance, appears not to be a noninteracting model. However, as one works within a meanfield framework, the interaction merely ends up in additional site potentials in the expression of the Hamiltonian^{40,41}. In this sense, ref. 25 can be also considered as a TB model with adjustable sitedependent potentials from a practical point of view. Importantly, it is well known that the presence of the gapless point at E_{F} is unaffected by modulations of this kind of site potentials within a range in this compound^{19,33,42,43}. The chosen values of the site potentials in ref. 25, which are given by
are within this range and are thus acceptable. Using these potentials, the Hamiltonian of the effective TB model in ref. 25 can be eventually expressed by a 4 × 4 matrix, ɛ_{ij} [i,j=A(1), A’(2), B(3) and C(4)], whose Fourier transformed matrix elements are given by
with the kinetic terms
The hopping integrals are better determined by ab initio calculations such that the resultant electronic bands become compatible with experimental observations in this system. For this, we employed the hopping integrals reported by the firstprinciple densityfunctional calculation at T=8 K (ref. 24), as in ref. 25, which are given for the nearest neighbours by (in the unit of eV)
and for the next nearest neighbours as
(For the definition of the integrals, see Supplementary Fig. 8.) The largest integrals, and , are known to vary about 15% by raising T from 8 to 300 K (ref. 24), though the variation is less than a few per cent below 100 K. As our fitting analyses primarily focus on this lowT region, it is reasonable to omit the Tdependence and keep using the hopping integrals estimated at T=8 K, {; p=a1−a4’}, at all T as is done in ref. 25. By diagonalizing the Hamiltonian (equation (4)) in conjunction with the hopping integrals (equations (5)–(7), , ) and the site potentials (equation (2)), one obtains the four energy bands with tilted Dirac cones at E_{F}, as shown in Fig. 1c.
We note that it is very important to use these hopping integrals in equations (3) and (4) to reproduce the observed sign change of the Hall coefficient R_{H} from R_{H}>0 to R_{H}<0 with decreasing T^{69}. (For details, see refs 26, 70.)
From all these, we used the effective TB model in ref. 25 as our noninteracting reference to the RG analyses. It should be stressed that we do not mean to incorporate interaction effects at this level, and indeed the model we assumed is a purely noninteracting TB model with acceptable sitedependent potentials. Note that these values of site potentials are realistic because they lead to a sitedependent charge differentiation which is compatible with the observed Xray and Raman scattering results in the conducting phase; see refs 25, 32, 34.
Generalized Weyl Hamiltonian and sitespectral weight
Around the bandcrossing DPs, where the Fermi energy E_{F} (put as E=0 hereafter) is fixed in αI_{3} due to the stoichiometry, the lowenergy continuum model is shown to be given by the generalized Weyl Hamiltonian^{20,23,25,35}
which describes the electronic states in the vicinity of one of the DPs. Here, w_{0}=(w_{0x,}w_{0y}) and v=(v_{x}, v_{y}) are effective velocities describing the tilt and the anisotropy of the Dirac cone, respectively; σ^{0} is the 2 × 2 unit matrix; (σ^{x}, σ^{y}) are the Pauli matrices; and q=(q_{x}, q_{y}) is the 2D wave vector measured from the DP at k_{D}. Note that the twofold valley degeneracy associated to the two DPs at ±k_{D} (Fig. 1c) will be omitted for simplicity, and we will hereafter focus on a single valley (at k_{D}). (When the valley degeneracy is to be included in some of the expressions, we will specifically mention it.) The Hamiltonian (equation (8)) is defined in a space spanned by the Luttinger–Kohn bases^{71}: and . These bases are the two degenerate Bloch states at k_{D}, which are given by a (normalized) superposition of the highest occupied molecular orbital on each of the four different BEDTTTFs (ETs) in the unit cell^{36}
where j=A(1), A’(2), B(3) and C(4) represents the different sites (see Fig. 1b). Diagonalization of equation (8) yields the eigenvalue E_{ζ}(q) (equation (1)) in terms of the two bands (ζ=±) and the eigenstates (Goerbig, M.O., private communication.)
where tan ϕ_{q}=v_{y}q_{y}/v_{x}q_{x}. We note that the two states in equation (10) have an equal weight for any value of q, as in the twoband model of the graphene Dirac cone, whereas the four states in equssation 10 have not necessarily the same weight^{25,72}. To see this, we define a (normalized) qdependent sitespectral weight by taking a projection of onto , (ref. 36) (Goerbig, M.O., private communication.), which reads
where is the relative phase between and .
Taking the lowenergy limit of the effective TB model in ref. 1 (see equations (3)–(7), , , , ), one can derive the four effective velocities in the generalized Weyl Hamiltonian (equation (8)), which are given by
Using these velocities, the phase ϕ_{q} in equation (11) can be approximated as , where ϕ is the angle between q and the k_{x}axis. It is shown from equation (11) that the sitespectral weight acquires an antiphase relation between the j=B and the other sites, namely, and . Moreover, and have an equal size for the bases λ=LK1 and λ=LK2, while one obtains [or ]. This causes an oscillation of equation (11) as a function of ϕ around the DP, with a large amplitude and an opposite phase on the j=Bsite and the j=Csite, whereas the ϕ dependence is small on the j=A(A’) site (Supplementary Fig. 1c). As the cone is tilted in the k_{x}direction (inset of Supplementary Fig. 1a), the Fermi velocity becomes highly anisotropic around the cone^{20,25,50} (Supplementary Fig. 1a), and can be expressed as
The most striking consequence of this anisotropy, in conjunction with equation (11), is that there is a large asymmetry in the sitespectral weight for the site A(A’), B and C around the DP. Namely, the site B predominantly reflects the largev_{F} electrons in the steep slope of the cone (ϕ≈π); the site C mostly probes the smallv_{F} electrons in the gentle slope (ϕ≈0); and the site A(A’) probes the entire electronic states on average around the DP (see Fig. 1d–f and Supplementary Fig. 1a,c). This asymmetry of the sitespectral weight results in a clear difference in the size of the jsite DOS, D_{j}(E, ζ), for the different sites—D_{C}(E, ζ)>D_{A(A′)}(E, ζ)>D_{B}(E, ζ) (ref. 25)—where D_{j}(E, ζ) (per valley and ET molecule) is defined as the q summation of at a given energy in the band ζ, which is expressed as
Here, V_{C} is the 2D unitcell volume in the conducting abplane. The contrasting features of the B and C sitespectral weights around the DP provide unique opportunities to probe the excitations of largev_{F} Dirac electrons (in the steep slope) and smallv_{F} Dirac electrons (in the gentle slope) separately in terms of a siteselective local measurement such as NMR.
We note that in the original effective TB calculation by Katayama et al.^{25}, the velocities are given in the unit of energy (meV), w_{0}=(w_{0x}, w_{0y})=(−38.9, 4.8) and v=(v_{x}, v_{y})=(51.5, 43.9), because both the primitive vectors and the reciprocal lattice vectors are set to have a unit length in their notation. To recover the ordinary physical unit (length/time), one has to multiply the velocities either by a/ℏ or b/ℏ, using the values of the lattice constants a and b at the current pressure (2.3 GPa), where ℏ is the Planck constant divided by 2π. By linearly extrapolating the Xray diffraction data obtained at 1.76 GPa (ref. 33) to 2.3 GPa, the lattice constants are estimated as a=8.567 Å and b=10.282 Å. The velocities in equation (12) are obtained in terms of these lattice constants.
Renormalizationgroup calculations
The observed nonlinear temperature dependence of the spin susceptibility below the inflection point in Fig. 2d,e, cannot be understood within the noninteracting Diracfermion picture, as we mentioned above (see Methods: effective TB model). In this temperature range, the screening effect should be weak reflecting the vanishing thermal excitations around the gapless point at E_{F}. For this kind of situation, it is well known from the RG study of graphene Dirac cone that the LR part of the unscreened Coulomb interaction among electrons causes a logarithmic divergence of the Fermi velocity v_{F} around the DP^{7,8,9,10,11,12,13,14}. A similar argument has been recently proposed for the tilted Dirac cone in αI_{3} (ref. 46), in which a RG flow of the Fermi velocity is suggested as a function of T. Hence, the most straightforward and reasonable way to understand the lowT nonlinear feature of Fig. 2d would be to attribute it to the Tdriven renormalization of v_{F} due to the LR Coulomb interaction.
To check this hypothesis, we have performed a RG calculation based on the generalized Weyl Hamiltonian (equation (8)) and tried to fit the data. A circular momentum cutoff of Λ=0.667 Å^{−1} around the DP (q=0) is introduced in the RG theory, which is of the size of the averaged inverse lattice constant, Λ=2π/L with L=(a+b)/2=9.425 Å at 2.3 GPa. For the initial values of the velocities at the cutoff momentum q=Λ (that is, v and w_{0} in equation (8)), we employ velocities derived from the effective TB model of ref. 25, and v^{TB} (equation (12)), as discussed in the previous subsection (Methods: Generalized Weyl Hamiltonian and sitespectral weight). In the oneloop order largeN expansion of the RG theory, v=(v_{x},v_{y}) are renormalized following equation (2) (given in the main text) and grows logarithmically as functions of Λ/q (where q is measured from the DP), whereas w_{0}=(w_{0x},w_{0y}) are not renormalized (Supplementary Fig. 4a). We note that equation (2) takes into account the screening effect of the Coulomb interaction including the polarization bubbles in the selfenergy. It is applicable to any size of the coupling , where ɛ is the dielectric constant and N=4 is the number of fermion species corresponding to the two DPs in the Brillouin zone and two spin projections (that means the twofold valley degeneracy is considered).
Reflecting the renormalization of v, the eigenenergy with the RG correction becomes (with the band index ζ=±)
Correspondingly, the local DOS and the electronspin susceptibility for the site j are respectively given by the expressions
where V_{C}=88.086 Å^{2} is the 2D unitcell volume in the conducting abplane estimated at 2.3 GPa, N=4 is the number of fermion species (again, the twofold valley degeneracy is included), E_{Z}=gμ_{B}H is the electron Zeeman energy and f(E) is the Fermi distribution. The integration with respect to q in equations (16) and (17) is done up to the momentum cutoff (q=Λ). Note that T plays the role of the flow parameter in equation (17) (that is, a Tdriven RG flow).
In Supplementary Figs 4–6, we present the calculated profiles of the velocities (Supplementary Fig. 4), the jsite DOS (Supplementary Fig. 5) and the jsite electronspin susceptibility (Supplementary Fig. 6) based on the RG equation (equation (2)) and equations (15)–(17), , . Here, to get a reasonable agreement with the experiment, we introduced a phenomenological parameter, u (≤1), in the calculations which is defined by the expressions
This parameter reflects a suppression of the velocity or a reduction of the hopping amplitude t_{ij} between the lattice site i and j due to the SR part of the Coulomb interaction^{53}, as mentioned in the main text. Then, the RG flow, which is determined by equation (2), is controlled by two parameters—the dielectric constant ɛ and the phenomenological parameter u.
Supplementary Fig. 4b,c, presents the parameter dependence of the flow of the velocities v_{x} and v_{y}. The dielectric constant ɛ affects the power of the flow function v=v(Λ/q) (Supplementary Fig. 4b), while the parameter u determines both the initial values of the velocities and the power of the flow (Supplementary Fig. 4c). Supplementary Fig. 5b,c, depicts the ɛ and u dependence of the DOS for the site j=A(A’). It is clearly seen that a prominent suppression of the DOS develops around E=E_{F} (=0) by reducing the dielectric constant ɛ (which corresponds to an increase in the coupling g_{ϕ} at the cutoff momentum q=Λ). The reduction of the parameter u, on the other hand, leads to an enhancement of the DOS because the DOS is linked to the inverse square of the velocities^{20}. In Supplementary Fig. 6b,c, we show the parameter dependence of the calculated electronspin susceptibility. As the susceptibility probes the k_{B}Taverage of the DOS near E_{F} (=0), the suppression of the DOS for small ɛ causes a reduction of the susceptibility at low temperatures (Supplementary Fig. 6b). The enhancement of the susceptibility for a small value of u can be also understood in the same fashion (Supplementary Fig. 6c).
Supplementary Fig. 7a shows the result of the variance analyses of the Tdriven RGflow fit to the experimental spin susceptibilities for the site j=A(A’) and C (Fig. 2d) based on equation (17). Here, the variance, plotted in the u−−Var() space, is defined by
where T_{i} is the experimental temperature points (i=1,2, ⋯, n), and (T_{i}) and are the measured and calculated susceptibilities at T=T_{i}, respectively. Note that the variance is defined only for the results on the site A(A’) and C, since the results on the site B has less good agreement with the RGcalculation (see Supplementary Fig. 7b–e). This is because of the emergent ferrimagnetic spin polarization—the negative susceptibility on the site B at low temperatures (inset of Fig. 3c)—which is due to the SR part of the Coulomb interaction, not considered in this calculation. (For details, see Methods: Simulations with the Hubbard model.) The calculated susceptibilities at selected values of u and are shown for three different sites j=A(A’), B and C together with the experimental data in Supplementary Fig. 7b–e. Except for the site B, the agreement with the calculations and the experiments is pretty good for small values of u and ɛ. The variance Var() has a minimum around (u, ɛ)=(0.35, 1) with little ɛ dependence up to ɛ≈10^{1.5} (Supplementary Fig. 7b,c) and increases rapidly as one moves away from this minimum especially when u is increased. Thus, we take these values of (u, ɛ)=(0.35, 1) as the optimal parameters hereafter. (Here, the result for ɛ=1 is chosen because the ɛdependence is small and affects the result little.) The small value of u=0.35(<1) is in agreement with the aforementioned reduction of the hopping amplitude due to the SR repulsion^{53} and indicates the presence of moderate electron correlations. (This point naturally supports our discussion based on the Hubbard model in the next subsection to deal with the observed negative spin susceptibility on the site B.)
Lastly, it may be worthwhile to mention that the flow of the velocities v=(v_{x}, v_{y}) with w_{0}=(w_{0x}, w_{0y}) staying constant in Supplementary Fig. 4a leads to a situation where v_{x}, v_{y}w_{0x}, w_{0y} is realized for a large value of the momentum scale Λ/q (1) (that is, in the vicinity of the DP). This means that the tilting term (w_{0}) becomes effectively negligible with respect to the anisotropy term (v) at low energy in the generalized Weyl Hamiltonian (equation (8)). Moreover, the values of the two velocities, v_{x} and v_{y}, remain very close to each other down to the vicinity of the DP (for instance, ln(Λ/q)=5 in Supplementary Fig. 4a corresponds to =0.0045 Å^{−1} in terms of Λ=0.667 Å^{−1}). These points together suggest that the anisotropy of the Dirac cone becomes very small near the DP and the cone is practically isotropic at low energy, as one can see in the calculated cone in Fig. 4d–f. This is can well account for the observed site dependence of the susceptibility, the root of which is linked to the tilt and the anisotropy of the cone. The site dependence becomes vanishingly small at low temperatures (Supplementary Fig. 2), which reflect the renormalization of the velocity that makes the cone to be more and more isotropic at low energy.
Simulations with the Hubbard model
In the previous subsection, we have shown that the RG calculation based on the lowenergy continuum model (equation (8)) well reproduces the observed nonlinear Tdependence of at the site A(A’) and C (Supplementary Fig. 7b–e) when the velocity suppression due to the SR Coulomb interaction is phenomenologically taken into account. However, the agreement is less good at the site B; in particular, the negative susceptibility <0 below T≈60 K cannot be reproduced at all, suggesting the presence of other interaction effects.
To understand the origin of the observed <0, we have performed another simulation of the susceptibility based on the RPA. For this, we start from the standard Hubbard model with the onsite Hubbard interaction U, by extending earlier study^{50}. The model Hamiltonian is given for the present system by the expression
where is the creation operator on the site j=A(1), A’(2), B(3) and C(4) in the unit cell α(=1, …, N_{u.c}.) with the spin σ(=, ), N_{u.c.} is the total number of the unit cell, and t_{i}α_{:jβ} is the nearestneighbour and next nearestneighbour hopping amplitude between the lattice site (i, α) and (j, β) (see Supplementary Fig. 8). (It should be noticed that this Hamiltonian (equation (20)) lacks the sitedependent potential term (equations (3) and (4)) we employed above. From qualitative points of view, the inclusion of these potentials does not alter the results much, and, for the sake of simplicity, we shall omit this term in this subsection.)
The amplitude t_{i α:jβ} at finite temperatures are estimated from the hopping integrals given by the firstprinciple calculations^{24} at 8 K, {; p=a1−a4’} (equations (6) and (7)), and at 300 K, {; p=a1−a4’}, as given in the following (in eV)
in combination with the interpolation formula given in ref. 50
Within the meanfield approximation, the diagonalization of the Hubbard Hamiltonian (equation (20)) yields
where we define , δ_{ij} is a vector connecting the nearestneighbour lattice sites i and j, is the eigenvalue (E_{1σ}>E_{2σ}>E_{3σ}>E_{4σ}), d_{iησ}(k) is the corresponding eigenvector, f(E) is the Fermi distribution and μ is the chemical potential. Note that the average electron number is determined selfconsistently from the condition , reflecting the ¾filling of the band. In the normal state, one has ; thus, the spin σ is omitted hereafter.
We introduce the bare sitespin susceptibility matrix, , whose (ij)element is given by
in terms of a form factor
where iδ (δ>0) is an infinitesimally small imaginary part. Within the RPA approach, the spin fluctuations are estimated using the expression^{50}
(where is the 4 × 4 unit matrix), and the total RPA jsitespin susceptibility (for Q=0 and ω=0) is given by
Now, we decompose the bare susceptibility into two parts (for the reason that will become clear below): , where and correspond to the intraband and interband contributions to the bare susceptibility (for Q=0), respectively. The intraband RPA susceptibility is then defined by
and the interband contribution to the total RPA susceptibility is expressed as
In Supplementary Fig. 3a, the calculated temperature dependence of the intraband RPA susceptibility is shown for U=0.14 eV in comparison to the noninteracting case (U=0) for the site j=A(A’), B and C. It is seen that the intraband susceptibility for a finite value of U becomes always larger than the case for U=0 at all temperature and all site j. The interband contribution, on the other hand, is found to provide a sitedependent correction to the susceptibility (Supplementary Fig. 3b). Namely, the interband RPA susceptibility gives a positive contribution on the site A(A’) and C, whereas the contribution is negative on the site B. The negative interband contribution on the site B () develops with increasing U and in turn leads to a negative total RPA susceptibility () above a critical U value of U_{C}≈0.12 eV. The position of the minimum shifts towards higher energies with increasing U (Supplementary Fig. 3c). By taking a value of U=0.14 eV, the minimum of the total RPA susceptibility appears at around T≈50 K, which agrees well with the experiment (inset of Fig. 3c). These results demonstrate that the SR part of the Coulomb interaction between electrons causes a ferrimagnetic spin polarization at low temperature. This leads to a situation where the site B is subjected to a negative local magnetic field, pointing opposite to the external field, while the other sites (A, A’ and C) feel a positive field, as schematically illustrated in Fig. 6.
To have an overall comparison of the RPA calculations with the experiment, the first derivative of the susceptibility is analysed for the case U=0, the total RPA susceptibility (equation (29)) for U=0.14 eV, and the observed spin susceptibility (Fig. 2e), as depicted for the nonequivalent sites in Supplementary Fig. 3d–f. The calculations capture the experimental features relatively well on the site A(A’) and C above the peak temperature (T≈50 K), whereas the calculation does not agree with the experiment at all on the site B. At low temperatures, on the other hand, the agreement between the calculation and experiment becomes worse even for the site A(A’) and C. That is, in the low temperature limit, the calculations show a saturation both for U=0 and the finite U, while the experiment exhibits a monotonic decrease towards zero (Supplementary Fig. 3d and f). We believe that this disagreement arises because the present RPA calculation does not incorporate the Tdriven v_{F}renormalization effect due to the LR part of the Coulomb interaction, as discussed in the previous subsection (see Methods: RG calculations). The v_{F}renormalization results in a superlinear temperature dependence in the spin susceptibility (Supplementary Fig. 6), which causes a decrease of the first derivative of with decreasing temperature as reflected in the experiment.
Finally, we comment on the comparison of the experiment with an orthodox RPA fitting. This is done by assuming a simplified RPA (sRPA) expression for the spin susceptibility, which is defined by
where is the bare spin susceptibility and U_{j} is an adjustable parameter reflecting the onsite Hubbard interaction. (Note that and correspond to the diagonal term in equations (26) and (28), respectively, for Q=0 and ω=0 with the Hubbard interaction U in equation (20) replaced by the sitedependent parameter U_{j}.) Supplementary Fig. 14 presents the leastsquare fitting results to the experiment using the sRPA expression, which yields U_{A(A′)}=0.6, U_{B}=1.3 and U_{C}=0.4 (in eV). It is clearly seen that the observed nonlinear temperature dependence of the susceptibility cannot be reproduced by the sRPA fit at all sites. In particular, there is an unphysical divergence in the calculation, which is linked to the large U_{j} values used in the calculation that are too large compared with the typical values applicable to this system^{24,26,31,35,40,41,48,50}. It is thus concluded that one has to consider the fullmatrix elements of equation (28) in order to obtain a reasonable agreement with the experiment.
Data availability
The data that support the findings of this study are available from M.H. upon requests.
Additional information
How to cite this article: Hirata, M. et al. Observation of an anisotropic Dirac cone reshaping and ferrimagnetic spin polarization in an organic conductor. Nat. Commun. 7:12666 doi: 10.1038/ncomms12666 (2016).
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Acknowledgements
We gratefully acknowledge valuable discussions with Y. Suzumura, N. Nagaosa, H. Isobe, M. Imada, M. Ogata, H. Matsuura, H. Fukuyama, C. Hotta, S. Sugawara, T. Osada, T. Taniguchi, M. Potemski, M.H. Julien and H. Mayaffre. In particular, we thank M. Horvatić and M. O. Goerbig for their thoughtful advices on the analyses and dedicated discussions. We also thank D. Liu for providing us unpublished results and for fruitful discussions. This work was supported by MEXT/JSPJ KAKENHI under Grant Noes 20110002, 21110519, 24654101, 25220709, 15K05166, 15H02108, JSPS Postdoctoral Fellowship for Research Abroad (Grant No. 66, 2013) and MEXT Global COE Program at University of Tokyo (Global Center of Excellence for the Physical Sciences Frontier; Grant No. G04).
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The samples were prepared by M.T. The data were taken, analysed and interpreted by M.H. and K.I. with the help of K.M., C.B., D.B., G.M., A.K. and K.K. The renormalizationgroup calculation was done by D.B. and analysed by M.H. with the help of D.B. and C.B. The simulations using the Hubbard model were carried out by G.M. and A.K. The project was supervised by K.K, and the manuscript was written by M.H. with K.M., D.B., A.K., C.B. and K.K.
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Hirata, M., Ishikawa, K., Miyagawa, K. et al. Observation of an anisotropic Dirac cone reshaping and ferrimagnetic spin polarization in an organic conductor. Nat Commun 7, 12666 (2016). https://doi.org/10.1038/ncomms12666
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DOI: https://doi.org/10.1038/ncomms12666
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