Abstract
The interplay of Coulomb and electron–phonon interactions with thermal and quantum fluctuations facilitates rich phase diagrams in twodimensional electron systems. Layered transition metal dichalcogenides hosting charge, excitonic, spin and superconducting order form an epitomic material class in this respect. Theoretical studies of materials like NbS_{2} have focused on the electron–phonon coupling, whereas the Coulomb interaction, particularly strong in the monolayer limit, remained essentially untouched. Here, we analyze the interplay of short and longrange Coulomb as well as electron–phonon interactions in NbS_{2} monolayers. The combination of these interactions causes electronic correlations that are fundamentally different to what would be expected from the interaction terms separately. The fully interacting electronic spectral function resembles the noninteracting band structure but with appreciable broadening. An unexpected coexistence of strong charge and spin fluctuations puts NbS_{2} close to spin and charge order, suggesting monolayer NbS_{2} as a platform for atomic scale engineering of electronic quantum phases.
Introduction
Layered materials host in many cases pronounced electronic interaction phenomena ranging from eVscale excitonic binding energies in semiconductors to charge, spin, and superconducting order in metallic systems. Characteristic energy scales and transition temperatures associated with these interaction phenomena change often remarkably when approaching the limit of atomically thin materials.^{1,2,3} Two generic contributions to these materialthickness dependencies are quantumconfinement^{4,5,6} and enhanced local and longranged Coulomb interactions in monolayer thin materials.^{7,8,9,10,11,12} In addition, many layered materials feature sizable electron–phonon coupling.^{13,14,15,16} The resulting interplay of interactions, which are effective at different length and time scales (see Fig. 1), makes the phase diagrams of twodimensional materials and their response to external stimuli very rich.
The layered metallic transition metal dichalcogenides (TMDC),^{17,18} MX_{2}, where M denotes one of the transition metals V, Nb, or Ta and X stands for one of the chalcogens S or Se, presents a demonstrative case in this respect where the monolayer limit is becoming experimentally accessible.^{2,17,18} Within this material, class a competition of charge and spinordered, Mott insulating, as well as superconducting states can be found. Here, the Vbased compounds show tendencies toward magnetic^{19,20} as well as charge order^{21,22,23} in their monolayer and bulk phases, respectively, which might partially coexist in the fewlayer limit.^{20,24} In contrast, the subclass of Tabased compounds^{1,25,26,27,28,29,30} as well as NbSe_{2}^{2,3,31,32,33,34,35,36,37,38} show a competition between chargedensity waves, superconducting as well as Mottinsulating states. NbS_{2} appears to be a border case. It is superconducting in the bulk^{39,40,41} but does not display any chargedensity wave (CDW) formation there. In the case of fewlayer NbS_{2}, first experimental studies reported recently metallic transport properties down to three layers,^{42} whereas meanfield calculations reveal a tendency to form magnetic states.^{43,44} NbS_{2} is thus likely on the verge between different instabilities. Whether or how these instabilities are triggered by the interplay of the involved interactions is barely understood, up to now. Although a lot of focus was put on the investigation of electron–phonon coupling effects in the whole class of metallic TMDCs, the effects of the subtle interplay of the local and nonlocal Coulomb interaction terms have been mostly neglected, so far. It is thus necessary to draw our attention also to these short and longrange electron–electron interactions. Unfortunately, there is no theory that can handle, even qualitatively, the competition of these strong interactions beyond the perturbative regime, yet. To overcome this problem, we combine here the Dual Boson formalism^{45,46,47} with firstprinciples approaches and construct a stateoftheart materialrealistic theory of monolayer NbS_{2}, which properly treats electronic correlations as resulting from competing short and longrange Coulomb interactions. Thereby, we also account for the electron–phonon interactions to gain a universal understanding of all interaction effects.
Our calculations reveal a simultaneous enhancement of the charge and spin susceptibilities owing to the various interactions in monolayers of NbS_{2} and a sharp transition from tendencies of preferential spin ordering to charge ordering. Despite these strong interaction effects, the electronic spectral function as measured, e.g., in angularly resolved photoemission (ARPES) experiments largely resembles the noninteracting dispersion in accordance with the available experimental data. We trace this back to a compensation of the different interaction terms, which are partially effective on the singleparticle but not on the twoparticle level. From an experimental perspective, this means that finding a match between ARPES results and density functional theory (DFT) bands is not sufficient to rule out strong correlation/interaction effects, as the competition of the interactions masks correlation effects on the singleparticle level, whereas they are still visible on the twoparticle level, i.e., in the magnetic and charge susceptibility.
Results
Competing interactions in NbS_{2}
The noncorrelated band structure of NbS_{2} monolayers exhibits a halffilled metallic band surrounded by completely filled valence bands 1 eV below and completely empty conduction bands 3 eV above the Fermi level, as shown in Fig. 1a. This motivates a description of the competing interaction effects in terms of an extended HubbardHolstein model^{48} for the separated metallic band only
where \(c_{i\sigma }^\dagger\) and c_{ iσ } are the creation and annihilation operators of the electrons with spin σ on lattice site i, \(b^\dagger\) and b are the creation and annihilation operators of a local phonon mode, and \(n_i = c_{i\sigma }^\dagger c_{i\sigma }\) is the electron occupation number operator. This model includes on the singleparticle level the electron hopping t_{ ij } and a local phonon mode with energy ω_{ph}. We include an onsite Coulomb repulsion U and longrange Coulomb interactions V_{ ij }, as well as an electron–phonon coupling g which couples the local charge density to the given phonon mode. The latter can actually be integrated out which results in a purely electronic model with an effective dynamic local interaction
that is lowered and thereby effectively screened by the phonons.^{48,49,50} This treatment of the phonons as simple singlefrequency modes that are coupled locally to the electrons is an assumption necessary to keep the problem tractable. Otherwise, the nonlocal interaction V_{ ij } or V_{ q } in momentum space would also become frequency dependent. As we argue in the Methods section, there is a basis for this assumption, however, the simplification might change the exact position at which instabilities occur in the Brillouin zone. Furthermore, we would like to emphasize that, in our treatment, electronically generated phonon anharmonicities are automatically included, whereas bare phonon anharmonicities are not.
To realistically describe NbS_{2} monolayers, we derive the parameters entering Eq. (1) from first principles. Therefore, we generate a most accurate tightbinding model describing the metallic and the lowest two conduction bands in a first step, and use it afterwards to perform calculations within the constrained Random Phase Approximation (cRPA)^{51} to obtain the partially screened Coulomb interaction matrix elements within the same basis. The phonon frequency and the electron–phonon coupling are estimated based on densityfunctional perturbation theory calculations. The resulting threeband model is subsequently simplified in order to get the final singleband model describing the metallic band only as explained in the method section.
Our ab initio simulations yield an effective local Coulomb interaction U ≈ 1.8 eV, a nearestneighbor interaction V ≈ 1 eV as well as further longrange interaction terms. The typical bare phonon frequency ω_{ ph } and the electron–phonon coupling g for this material are estimated to be 20 and 70 meV, respectively (see methods for further details). Notably, both, the onsite Coulomb repulsion and the effective electron–electron attraction λ = 2g^{2}/ω_{ ph } = 0.5 eV, are on the order of the electronic band width ≈1.2 eV, as sketched in Fig. 1.
Spectral fingerprints of the interactions
Each interaction term on its own can thus trigger strong electronic correlations, which becomes evident from the electronic spectral functions shown in Fig. 2a. These have been calculated using the Dual Boson (DB) method taking into account each interaction term on its own and their combined effects. For local Coulomb interactions only (top left), the halffilled conduction band clearly splits into two Hubbard bands above and below the Fermi level. There is no spectral weight at the Fermi level and the system is insulating. Including the nonlocal Coulomb interaction terms (top right) markedly changes the spectral function. The lower Hubbard band still retains noticeable spectral weight. However, the upper Hubbard band overlaps now with a broad distribution of spectral weight reaching the Fermi level. That is, the nonlocal Coulomb interaction drives the system into the state of a correlated metal, similar to what has been shown for graphene.^{52} With only electron–phonon interaction (bottom left), the spectrum is also reminiscent of a correlated metal, with again strong spectral weight transfer away from the Fermi level toward polaronic bands at higher energies. Finally, simultaneous inclusion of all interactions yields the spectral function shown in the bottom right of Fig. 2a. We find a single band with a dispersion very similar to the DFT result of Fig. 1 (shown as a red line). Seemingly, the different interaction terms largely compensate each other despite the fact that they are effective at very different length and time scales. The major interaction effect visible in, e.g., ARPES experiments is that the band widens significantly compared to the thermal broadening inherent to any finite temperature measurement. ARPES experiments in other transition metal dichalcogenides monolayers (TaS_{2}^{53} and NbSe_{2}^{37}) are consistent with this picture: the dispersion follows roughly the DFT band structure with some broadening. A more detailed comparison with our results is, however, not possible, as in the experiments different materials have been used, lower temperatures were applied, and substrates were present.
Our materialspecific results can be compared with theoretical findings in model systems. In the HubbardHolstein model (U and λ in our language) on the triangular lattice, Mott and polaronic insulating states have been found,^{54,55} consistent with our results here. On the other hand, the combination U + V + λ presented here has so far not been studied since there were previously no methods that can deal with these competing interactions in the strongly correlated regime, where vertex corrections beyond GW are important.
Therefore, we need to scrutinize this behavior in more detail and examine the local selfenergy, which induces all correlation effects. In Fig. 2b, we show these selfenergies corresponding to the spectra in Fig. 2a. If we take only local Coulomb interactions into account (stars), we find a strongly enhanced selfenergy for small frequencies. By including also longrange Coulomb interactions (circles), the selfenergy is reduced around small frequencies. This trend is continued by including electron–phonon interactions (squares), which demonstrates how the longrange Coulomb and the electron–phonon interactions compensate the effects of the local Coulomb interaction. The full selfenergy including all interaction terms is thus strongly reduced around small frequencies, but has still sizeable contributions at all energies considered here, which results in the broadened spectral function without significant reshaping. It is interesting to note that when taking only the electron–phonon interactions into account (triangles), the selfenergy, and thus the degree of correlation increases by increasing the electron–phonon coupling g. However, in the presence of Coulomb interaction an enhanced electron–phonon coupling necessarily leads to a decrease of the selfenergy and hence to a decreased degree of correlation. Thus, the effect of electron–phonon coupling is the exact opposite depending on whether or not the Coulomb interaction is present in the model. It is therefore absolutely crucial to take all interactions simultaneously into account.
Competition of charge and spin fluctuations in NbS_{2} monolayers
The electronic correlations as resulting from the interplay of the electron–electron and electron–phonon interaction also manifest in the local twoparticle correlation functions of the system, which are shown as a function of the electron–phonon coupling g in Fig. 3. These local observables are calculated directly from the DB auxiliary singlesite system. The ratio of the static local charge and spin susceptibilities (lower panel; note the logarithmic scale) and the instantaneous double occupancy (upper panel) vary strongly as a function of g. Without electron–phonon coupling (g = 0) the system shows typical signs of strong MottHubbard correlation effects: the spin susceptibility is orders of magnitude larger than the charge susceptibility and the probability of finding two electrons at the same site is greatly reduced in comparison to the value of \(\left\langle {n_ \uparrow n_ \downarrow } \right\rangle\) = 0.25 found in noninteracting halffilled systems. Turning on the electron–phonon interaction screens the local Coulomb interaction according to Eq. (2), and makes the system less correlated. At sufficiently large g ≈ 70–80 meV, the susceptibility ratio and double occupancy even exceed their noninteracting values of 1 and 0.25 (gray lines), respectively. The numerical simulations get unstable close to a transition to the CDW phase. This is why we could perform simulations at 464 K only up to g = 70 meV (red circles). At a higher temperature of T = 2321 K (orange pluses), larger values of λ can be reached. For g = 40–70 meV, the two data sets agree reasonably well, at higher temperatures the double occupancy is less suppressed and the spin susceptibility is substantially smaller, see also the Methods section. Note that 464 K (0.04 eV) is well below the energy scales defined by the band width and the interactions.
Freestanding NbS_{2} monolayer, with g ≈ 70 meV as estimated in the methods, thus turn out to be on the verge to form a CDW ground state. The local properties presented in Fig. 3 also show what happens when both, the electron–phonon interaction and the nonlocal parts of the Coulomb interaction, are ignored. In that case (dashed green lines), the susceptibility ratio goes down another order of magnitude, and the double occupancy decreases to almost zero. These are all characteristics of a Mottinsulating phase. The local Hubbard interaction U is thus in principle strong enough to create an interactiondriven insulator, with a large spin susceptibility, local magnetic moments (small \(\left\langle {n_ \uparrow n_ \downarrow } \right\rangle\)) and strongly suppressed charge fluctuations, as we have already anticipated in the discussion of the spectral functions. Only through screening by the nonlocal Coulomb contributions, and by the electron–phonon coupling, can the system exhibit the large charge fluctuations (local “charge moments”, large \(\left\langle {n_ \uparrow n_ \downarrow } \right\rangle\)), that are necessary for a CDW. This shows that both, the Hubbard interaction U and the interactions that screen it, are nonperturbatively large in the freestanding monolayer, which casts doubt on approaches that do not explicitly include all interaction terms. Most importantly, the transition from the regime, which is dominated by spin fluctuations to the chargefluctuation dominated regime is very abrupt as the steep rise of the susceptibility ratio demonstrates. The strong fluctuations in different channels, around g ≈ 70 meV, signal the close proximity of competing charge and spin order and is indeed ubiquitous in correlated electron systems.^{56,57}
Next, we turn to the static momentumresolved susceptibilities. The noninteracting susceptibility of the singleband model, χ_{0}, shown in Fig. 4a, agrees with previously published data for NbS_{2} monolayers.^{44} In a noninteracting system the charge and spin susceptibility would be the same and coincide with χ_{0}. This is clearly not the case for the charge and magnetic susceptibilities resulting from our DB calculations shown in Fig. 4b–e.
Without electron–phonon coupling (g = 0) the spin susceptibility is enhanced indicating the presence of strong spin fluctuations. The charge susceptibility, on the other hand, is suppressed in the entire Brillouin zone owing to the Coulomb interaction, which is in line with the expectations for a correlated metal. Turning on the electron–phonon interaction (g = 70 meV) reduces the spin susceptibility, which is however still comparable to χ_{0}. At the same time, the charge susceptibility is strongly enhanced and is almost divergent at large momenta. At this point it is, however, important to note that the exact position of the ordering vector might change when the ordered phase is actually entered (here, we investigate just its onset based on the susceptibility in the normal phase) and when a more realistic phonon model is used. Nevertheless, these two observations show again one of our main findings that the interactions partially compete and screen each other, leading to a spin susceptibility that is only moderately enhanced. Most importantly, this competition does not lead to a complete cancellation, as is visible in the strong enhancement of the charge susceptibility. Owing to the interplay of these interactions a strong spin and charge response can thus coexist in this system.
Discussion
Using a combination of the Dual Boson approach and ab initio calculations, we investigated the interplay between the Coulomb and electron–phonon interactions in NbS_{2} monolayers and the resulting degree of electronic correlations. We found that both, the Coulomb and the phononmediated electron–electron interaction energies, are on the same order as the electronic band width allowing both of them to trigger strong electronic correlations. Both types of interactions on their own would drive NbS_{2} to the verge of an insulating state, as our analysis of electronic selfenergies shows. Remarkably, our simulations with Coulomb and electron–phonon interactions present yield a spectral function, which closely resembles the noninteracting band structure. Yet, in this situation electron correlations have not fully ceased but manifest themselves in a sizeable broadening of the spectral function. In this sense, NbS_{2} is very similar to socalled Hund’s metals,^{58,59,60,61,62} where the exchange coupling drives the electronic system of materials like Febased superconductors away from the Mott Hubbard insulating limit into a correlated metallic phase.
For NbS_{2}, the observed spectral broadening argues against simple nesting scenarios based on the bare bands for the CDW instabilities. Our findings rather show that the interplay between all interactions is responsible for driving the system in close proximity of a chargeordered state. The interactioninduced correlations result in strongly modified spin and charge susceptibilities compared with the noninteracting one. Specifically, we found that the competition between the longrange Coulomb and the electron–phonon interactions is responsible for NbS_{2} monolayers being on the edge between dominating spin and charge fluctuations. The transition from a preferential spin order to charge order is thereby abruptly driven by the electron–phonon interaction.
The resulting ground state is thus heavily dependent on the detailed balance between the internal interactions. To study and test this behavior experimentally, there are several points to be aware of. First, from matching ARPES and DFT data one cannot deduce that the meanfield calculation captures the main physics. Manybody effects lead to quasiparticle broadening as well as enhanced magnetic and charge susceptibilities, which can be measured directly in resonant Xray scattering^{63} or electron energy loss spectroscopy.^{64} Importantly, the predicted close vicinity of charge order and local magnetic moment formation can be experimentally tested. The electronic system of NbS_{2} can be manipulated via environmental screening or strain applied to the monolayer. Increasing the former will mostly reduce the longrange Coulomb interaction V, while the electron–phonon interaction remains largely untouched. Thereby, the effective screening of the local U due to the nonlocal V is reduced and the spin susceptibility should be enhanced. By applying strain the electron–phonon interaction can be varied without drastic changes to the Coulomb interaction. By increasing the electron–phonon interaction the system would be pushed into a chargeordered state.
Our calculations show that correlation effects are particularly prominent in the simultaneously enhanced spin and charge susceptibilities of NbS_{2}. Hence, we expect a strong response of the material to local perturbations, which can be experimentally realized through charged as well as magnetic adsorbates. Scanning tunneling microscopy experiments involving adsorbates or defects on NbS_{2}, similar to what has been done for NbSe_{2},^{65,66} allow the susceptibilities predicted in Fig. 4 to be probed in real space but could possibly also exploit NbS_{2} as a platform for quantum engineering, where one switches locally between spin and charge order. The combination of the lattice structure of NbS_{2}, the sizeable spinorbit coupling, and the enhancement of spin and charge susceptibilities clearly away from the Brillouin zone center suggests that the competing ordering tendencies is likely subject to frustration effects. For these reasons, monolayer NbS_{2} deserves future exploration not only in the light of fundamental interest but possibly also in relation to concepts of miniaturized neuromorphic computing.^{67}
Finally, our findings allow to speculate about possible superconducting properties. In this context, the enhanced spin and charge susceptibilities point toward interesting unconventional paring mechanisms. At the same time there are the before mentioned striking similarities between the selfenergy in NbS_{2} and the one found in Hund’s metals, yielding a similar scenario as in Febased superconductors. In addition, it needs to be pointed out that the appearance of a CDW phase is usually detrimental to superconductivity so that it likely needs to be suppressed to enhance T_{ c }. Given the complicated competition between magnetic and charge instabilities in NbS_{2}, an analysis including all of these aspects needs to be carried out to gain reliable insights into possible superconducting properties.
Methods
Parametrization of the extended HubbardHolstein model
All model parameters are derived from first principles based on DFT and cRPA calculations. To do so, we start with a DFT calculation in Fleur^{68} for NbS_{2} using a lattice constant of a = 3.37 Å, a k mesh of 18 × 18 × 1, a vacuum height of 32 Å, a relaxed sulfur–sulfur distance of Δ = 3.13 Å, and using FLAPW lexpansion cutoffs of 10 (Nb) and 8 (S) and muffin tin radii of 2.58 a_{0} (Nb) and 2.01 a_{0} (S) to calculate the band structure shown in Fig. 1a. As the spinorbit coupling leads to severe spin splittings at the K point only, but not around the Fermi level in NbS_{2} we neglect it in the following. Afterwards, we construct a threeband tightbinding model by projecting the original DFT wave functions onto the three dominant niobium orbitals (\(d_{z^2}\), d_{ xy }, \(d_{x^2  y^2}\)) using the Wannier90 code,^{69} whereby we ensure that the bands are properly disentangled. To preserve the orbital character of the Wannier functions, we do not perform maximal localization. The resulting threeband tightbinding model perfectly interpolates the original DFT band structure and can be used to evaluate the electronic dispersion at arbitrary k points.
The longrange Coulomb interaction is parametrized in a materialrealistic manner using the cRPA method.^{51} Therefore, we start with the fully screened dynamic Coulomb interaction W(q, ω) in reciprocal space that is defined by
where v(q) ∝ 1/q is the bare interaction in two dimensions and Π(q, ω) is the polarization function rendering all screening processes. According to the cRPA we can reformulate the latter Π(q, ω) ≈ Π_{mb}(q, ω) + Π_{rest}(q) by splitting it into a dynamic part arising from the halffilled metallic band (mb) and a static part resulting from the rest of the band structure. This is appropriate as we are interested in the lowfrequency properties of Π(q, ω) and W(q, ω) only, which are completely rendered by the metallic band and thus by Π_{mb}(q, ω). Using this formulation of the full polarization we can rewrite the fully screened interaction as follows
with U(q) being the partially screened Coulomb interaction defined by
As described in the Supplemental Methods, U(q) needs be evaluated within the same orbital basis as used for the tightbinding dispersions, using 3 × 3 matrices to represent the bare interaction v(q) and the dielectric function ε(q). Importantly, we can fit analytic expressions to all of the involved matrix elements U_{ αβ }(q), allowing us to evaluate U(q) at arbitrary q vectors.
In order to derive a singleband model, we neglect the orbital dependencies in a next step. In this case, the dynamic polarization matrix of the metallic band may be approximated via
where Π_{ sb }(q, ω) is the singleband polarization, which is going to be evaluated in the Dual Boson calculations. The factor \(\frac{1}{9}\) approximates the overlap matrix elements, which are in general orbital and momentum dependent. This is appropriate for small q and as long as all orbital weights are more or less the same. We found that this assumption is indeed valid in the halffilled situation discussed here. Using this, polarization corresponds to a singleband/orbital partially screened Coulomb interaction defined by
Thus, under the assumption of vanishing orbital dependencies we can define the partially screened Coulomb interaction of the singleband model as the arithmetic average of all matrix elements of the partially screened interaction matrix U(q) in the orbital basis. This U_{sb}(q) now represents the Fourier transform of the real space Coulomb interactions U and V as used in Eq. (1) and thus serves as the second important ingredient to our extended HubbardHolstein model.
Finally, we incorporate the phonon frequency and the electron–phonon coupling into our model to describe all important interactions at the same time. To this end, we employ DFPT^{70} calculations as implemented in the Quantum Espresso package^{71} using localdensity approximations potentials, a lattice constant of a = 3.24 Å, a vacuum height of 16 Å, a k mesh of 32 × 32 × 1 for the selfconsistent electronic calculation, and a q mesh of 8 × 8 × 1 for the phonons. Within the BZ, i.e., for increased q momenta, the most important electron–phonon couplings arise due to acoustic phonon modes in NbS_{2} (the optical modes couple via a Fröhlich interaction which is proportional to 1/q and is thus strongly decreased here). In more detail, the strongest coupling arises owing to the LA mode, which consequently softens and becomes unstable.
To estimate an average bare frequency for this mode in the monolayer, we make use of the other acoustic branches, which are not at all (ZA) or just slightly (TA) renormalized at the Brillouin zone’s M point. Thereby, we arrive at an estimation of ω_{ph} = 20 meV for the bare typical phonon frequency, which is comparable to the corresponding modes in bulk NbS_{2}.^{32} Using this bare frequency, we recalculate the renormalized phonon frequency \(\omega _{{\mathrm{ph}}}^{{\mathrm{re}}}(q)\) = \(\sqrt {\omega _{{\mathrm{ph}}}^2 + 2\omega _{{\mathrm{ph}}}g^2\chi _0(q)}\) using the random phase approximation susceptibility χ_{0}(q), where we approximate the phonon selfenergy as \(g^2\chi _0(q)\). From this, we find instabilities starting from g_{min} ≳ 50 meV and similar instabilities as in the full DFPT calculation for \(g_{{\mathrm{NbS}}_{\mathrm{2}}} \approx 60 \ldots 70{\kern 1pt}\) meV. This is comparable to the g_{max} = 0.13 eV found by Flicker and van Wezel^{35} for bulk NbSe_{2}. Accordingly, we use an interval of g = 0.0 … 0.1 eV in order to study the phononinduced effects, whereas g ≈ 70 meV is supposed to be our materialrealistic estimate. At this point it is important to note, that this is clearly just an approximate model. We neglect the phonon dispersions as well as the momentum dependency of the electron–phonon coupling and focus on a single phonon mode only. The latter is, however, well justified as the LA mode is the only mode becoming unstable in DFPT calculations. Furthermore, between the M and K points, the LA mode is rather flat allowing us to describe it as a local Einstein mode. Finally, the Froehlichlike coupling of the those optical modes, which have a finite coupling to the electrons is likely underestimated around Γ and overestimated around the K point in our model. This means, that a full phonon model might lead to changes in the exact position of the arising divergences in the susceptibilities. In more detail, it is likely that in a full model, the charge instability would emerge more within the Brillouin zone and less at its border.
Dual Boson approach
The resulting materialrealistic singleband HubbardHolstein model is solved using the Dual Boson (DB) method, which is based on the Dynamical MeanField Theory (DMFT)^{72} philosophy. That is, DB uses an auxiliary singlesite problem to take into account strong correlation effects selfconsistently. The DB method extends DMFT by also capturing nonlocal interactions via an effective, dynamic local interaction. Here, as in ref.^{73}, the impurity model is determined selfconsistently on the Extended DMFT level. Then, the DB method calculates the momentum and frequency resolved susceptibilities starting with a DMFTlike interacting Green’s function and then adding nonlocal vertex corrections (in the ladder approximation) to ensure charge conservation.^{45,46,47} The auxiliary impurity model was solved using a modified version of the open source CTHYB solver^{74,75} based on the ALPS libraries.^{76}
The Dual Boson calculations use the singleband dispersion E_{mb}(k) from the tightbinding model and the effective interaction U_{sb}(q) as their input. Both are evaluated on 144 × 144 × 1 k and q meshes. The electron–phonon coupling leads to an additional, retarded, local electron–electron interaction \(U_{\omega _n}^{{\mathrm{e}}  {\mathrm{ph}}}\) = \( 2g^2\frac{{\omega _{{\mathrm{ph}}}}}{{\omega _{{\mathrm{ph}}}^2 + \omega _n^2}}\), where g is the electron–phonon coupling, ω_{ph} is the phonon frequency and ω_{ n } is the nth Matsubara frequency.
Unless otherwise noted, all Dual Boson simulations were performed at β = 25 eV^{−1} (T = 464 K). Calculations without electron–phonon coupling were for temperatures down to β = 150 eV^{−1} (77 K). These showed few qualitative changes: the system remained in the strongly correlated phase. For the case of (strong) electron–phonon coupling, closer to the chargeordering transition, the temperature is important as ordering is more likely at low temperature, the same holds for spin ordering transitions. However, please note that the model parameters are derived for T = 0 K.
To learn more about the role of temperature, we show the local charge and spin susceptibility in Fig. 5, the ratio of which is plotted in 3, as a function of g and temperature. Near g ≈ 0.07 eV, the lowtemperature simulations approach the phase transition and the charge susceptibility sees a large change, whereas the spin susceptibility develops more smoothly. Comparing the susceptibility with that of the noninteracting system at the same temperature (dashed lines), both temperatures show the same trend, although the magnitude of deviations is generally larger for the lowtemperature system. At small g, U is the dominant interaction and the spin susceptibility is enhanced and the charge susceptibility reduced with respect to the noninteracting system. For both temperatures, we find a coupling strength g where both susceptibilities are enhanced compared to the noninteracting system (g ≈ 0.07 eV at the lower temperature and g ≈ 0.85 eV at the higher temperature). Thus, we find this simultaneous enhancement of both susceptibilities to be a general feature that does not require a specific temperature. On the other hand, the magnitude and location of the simultaneous enhancement depends on the temperature. In particular, the sharp rise in the charge susceptibility at lowtemperature signals the approach to the chargeorder transition, which depends on temperature.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank J. van Wezel for useful discussion. M.R. would like to thank the Alexander von Humboldt Foundation for support. E.G.C.P. v. L. and M.I.K. acknowledge support from ERC Advanced Grant 338957 FEMTO/NANO. T. W. and G. S. acknowledge support from DFG via RTG 2247 as well as the European Graphene Flagship.
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E.G.C.P. van Loon and M. Rösner contributed equally to this project. M.R. and G.S. performed the ab initio determination of the singleband Hamiltonian. E.G.C.P.v.L performed the Dual Boson calculations. All authors contributed to the manuscript.
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van Loon, E.G.C.P., Rösner, M., Schönhoff, G. et al. Competing Coulomb and electron–phonon interactions in NbS_{2}. npj Quant Mater 3, 32 (2018). https://doi.org/10.1038/s4153501801054
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