Introduction

TiSe2, below 200K, undergoes a phase transition to a charge density wave (CDW), forming a commensurate 2 × 2 × 2 superlattice (\(P\bar{3}c1\)) of the original \(P\bar{3}m1\) structure. At the transition there is a softening of the zone boundary phonon, and changes are seen in the transport properties1,2. It has been observed that if the CDW is suppressed by pressure3 or intercalation of Cu atoms4, (unconventional) superconductivity appears. A quantum critical point has also been observed when pressure is applied5. The interplay of the CDW and superconductivity has been the subject of many studies, but even at the one-particle level whether the pristine (undoped) system is a semimetal or insulator is not well understood. A proper understanding of the origin of the one-particle spectrum is a prerequisite for understanding the superconductivity, and forms the subject of this work.

Bianco et al.6 discuss a controversy as to whether the instability originates either from the electrons or by the lattice. Such a distinction is not entirely well posed, because nuclear fluctuations are correlative to inverse of the static charge susceptibility7. A less ambiguous way to frame the question is one of a competition between a one-body effect, e.g. some analogue of a Jahn-Teller mechanism8,9,10, or a many-body effect, notably an excitonic insulator11,12,13,14,15,16. In their paper, Bianco et al.6 present a detailed review of theoretical and experimental understanding of the nature of the gap in the high temperature, \(P\bar{3}m1\) phase. It was established early on17 that the valence band maximum is of Se-4p character and located at the Γ point, while the conduction band minimum has Ti-3d character and located at L. Rossnagel carefully examined the available experimental information and noted that either a phonon-driven or an excitonic instability could account for the observed gap in the CDW. Not being able to discriminate, he took an agnostic view on which mechanism is predominant.

Experimental reports are voluminous and somewhat contradictory, especially for the high-temperature phase, in part because the distinction between a semimetal and a small-gap insulator at high temperatures is not sharp: if the gap is only a few times kBT there will be significant carrier populations in either case. The distinction is further obscured by extrinsic degenerate doping of the conduction band in typical samples18. A consensus has emerged, starting perhaps with the work of ref. 19 that the bandgap is small and positive. The gap is variously reported, in the range 0-0.2 eV, and especially in earlier studies was reported as a semimetallic phase with an inverted gap17,20,21 in the high temperature phase.

In an interesting theoretical work that argued for a band origin of the gap in the CDW phase22, the authors established that a hybrid functional (in contrast to LDA+U), could simultaneously determine the structure of the CDW instability and yield a positive gap. Their work anticipates some of the findings we present here, though it can only be suggestive since hybrid functionals have empirical parameters. Moreover neither hybrid functionals, nor LDA-based GW, account for the feedback between screening and band structure23,24. Reference 25 attempts to account for this interplay by adjusting admixture α in an exact-exchange (EXX) derived hybrid functional, so as to minimize the G0W0 correction to the eigenvalues. This ansatz markedly improves the G0W0-corrected EXX gap, bringing it close to (~0.1 eV larger than) the HSE06 gap they report. Still, it is ~0.25 eV larger than the QSGW gap reported in ref. 26. This work shows tuning density-functionals with assistance from G0W0 can improve their fidelity, but for the high fidelity needed here the theory must account for this feedback in a rigorous way, and crucially important, take into account whether excitonic effects are significant. The theory we present here does both.

It is very difficult to distinguish an excitonic condensate from a Peierls driven charge-density wave. One central argument for TiSe2 being an excitonic insulator stems from rather unusual shapes of the spectral function observed by ARPES. In one study13, a large spectral weight from backfolded bands was observed, which they interpreted as a signature of an excitonic insulator. In this work we indeed find backfolded bands, but they originate from Umklapp scattering processes, static in the CDW phase at 0 K and ephemeral owing to thermal nuclear fluctuations at higher temperatures. This will be discussed in more detail below. Another article27 reported a band of “Mexican hat” shape from a GW calculation, which they argued was indicative of an excitonic insulator. In an interesting study, the group of Abbamonte used momentum-resolved electron energy-loss spectroscopy to observe a softening of a plasmon at finite momentum near the transition temperature. As softening is a precursor to an excitonic condensate, it provides evidence that the transition is excitonic in nature, rather than Peierls14. This work however, examines instabilities around a frozen nuclear structure. As we show here, that assumption does not apply to TiSe2, and relaxing the assumption alters the conclusions both above and below Tc.

Early calculations of the energy band structure found it to be a semimetal with the conduction band at L falling below the valence band at Γ17, within the local-density approximation (LDA) to density-functional theory (DFT). The LDA predicts a negative gap in both the \(P\bar{3}m1\) and CDW phases, but as the LDA systematically underestimates bandgaps, it is not a reliable predictor of the gap. Bianco et al.6 managed to achieve a small opening of the gap in both phases with LDA+U by adding large enough U. This is obviously not predictive. Prior to this work, Cazzaniga et al.27 considered a GW calculation as a perturbation to the LDA, and found a gap of ~200 meV in the high-temperature \(P\bar{3}m1\) structure. However, both the positive gap and the Mexican hat structure turn out to be an artifact of the GLDAWLDA approximation, as we have reported previously26. Changes to the charge density, which single-shot GW as a first order perturbation to the LDA does not take into account, completely change the picture.

In this paper we use a self-consistent form of GW to demonstrate that excitonic effects are small and not responsible for the TiSe2 insulating state. We further demonstrate the crucial role of symmetry breaking in opening the band gap in the high-symmetry phase. We compute the band structure of TiSe2 for the high-symmetry (high temperature) and low-symmetry CDW (low temperature) crystal structures, as well as the band structure averaged over structures derived from ab initio molecular dynamics (AIMD) simulations at both high and low temperatures. The results show that breaking the \(P\bar{3}m1\) symmetry is responsible for the insulating state, whether by formation of a CDW (\(P\bar{3}c1\)) or by thermal fluctuations that displace atoms from the perfect-lattice locations and allow Umklapp scattering. Once the symmetry is broken, the insulating phase emerges without excitons, implying that TiSe2 is not an excitonic insulator. We begin with the CDW case as it is the simplest and provides a framework for the more complex high temperature case Fig. 1.

Fig. 1: Thermally driven disorder in TiSe2 high-symmetry (\(P\bar3m1\)) phase which leads to electron localisation.
figure 1

a AIMD snapshot of a 96-atom supercell showing the level of disorder and the overall structure, including as a visual aid an octahedron around a Ti atom with 6 Se atoms for vertices. b Brillouin zone of the ideal \(P\bar{3}m1\) phase with high symmetry points labelled. c, d rows of Se and Ti atoms respectively, taken from (a) emphasizing the disorder: idealised positions would lie in straight line while at any point in time the actual positions would be displaced as illustrated by this snapshot.

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Results and Discussion

The Charge-Density Wave Phase below 200 K

Below 200 K (TCDW), Di Salvo et al.1 found from neutron diffraction experiments that TiSe2 reconstructs into a “3qL” charge-density wave (CDW). In the CDW phase (\(P\bar{3}c1\) symmetry), the three-atom \(P\bar{3}m1\) unit cell undergoes a 2 × 2 × 2 reconstruction into a 24-atom superlattice. The 3qL superstructure is realised by superimposing displacements from a symmetric linear combination of three L point phonon modes. L points are located at a zone boundary: they are projected onto the Γ point under a 2 × 2 × 2 reconstruction. Thus the CDW is commensurate with a 2 × 2 × 2 superlattice of the \(P\bar{3}m1\) structure, with \(P\bar{3}c1\) symmetry.

The most important observation in the CDW phase is that QSGW28 finds an insulating solution (Fig. 2(a)) for TiSe2 with \(P\bar{3}c1\) symmetry. The bandgap opens adiabatically in a nontrivial manner, on a deformation path connecting \(P\bar{3}m1\) to \(P\bar{3}c1\) (see Supplemental Eq. 1 and Supplemental Fig. 1). No excitonic contributions are present at this level of theory. Spin-orbit coupling changes the band topology somewhat and reduces the gap to about 0.14 eV. The topology of the valence bands is remarkable with several distinct points all within kT of each other. The highest band along L-M (nominally the valence band maximum) almost appears like a flat, atomic-like state. The colours also show that a state at Γ below the VBM has Ti-3d character.

Fig. 2: Energy bands of the \(P\bar{3}c1\) superlattice in the QSGW approximation (left) and QS\(G\widehat{W}\) approximation (right), respectively, in the Brillouin zone corresponding to the \(P\bar{3}m1\) phase.
figure 2

Red indicates a projection onto Ti-3d character, blue onto Se character. The inset in the right figure compares QSGW (cyan) to QS\(G\widehat{W}\) (black) energy bands in a (−0.15, 0.25) eV energy window. The excitonic effects are seen to be very small.

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Excitonic effects

We can incorporate electron-hole effects in the GW approximation by adding ladder diagrams to the polarizability via a Bethe-Salpeter equation. We use QS\(G\widehat{W}\) to indicate an enhancement of QSGW when ladder diagrams are incorporated in W29. Such a capability was recently developed within Questaal, and ref. 29 examines in detail the change in spectral properties when W is augmented to \(\widehat{W}\) for a wide range of materials systems. It was shown to greatly reduce systematic errors inherent in QSGW theory, particularly when correlations are weak or moderate, as is the case with TiSe2. Thus QS\(G\widehat{W}\) provides a rigorous, ab initio framework to establish the role excitons play in TiSe2, and whether it might be an excitonic insulator. Figure 2 compares the QSGW and QS\(G\widehat{W}\) energy band structures in the CDW phase. Differences are very slight: the ladder diagrams reduce the fundamental gap from 0.17 eV to 0.15 eV (Fig. 2b). The weak influence of excitons on spectral functions is typical of narrow-gap systems: the nearly metallic electronic structure causes the RPA part of the polarizability to be large, and ladders make a small correction to it. Further, the excitonic binding energy is found to be less than 12 meV (see Supplemental Fig. 2), which is roughly one order of magnitude smaller than the band gap of the system. Similarly small band gap (~260 meV) and exciton binding energy (~20 meV) are observed in black Phosphorous as well30.

The three observations, that QSGW without excitonic effects yields a small gap in good agreement with available experiments, that the excitonic binding energy is much smaller than the gap, and that excitonic additions have almost negligible effect on the self-energy, lead us to conclude with some confidence that TiSe2 is not an excitonic insulator in the CDW phase. However, these observations do not address the high temperature phase. In the next section we show why this phase turns out to be insulating as well. Before proceeding, it is worth noting that the recent experimental observations31 predict the material to be insulating both above and below TCDW with respectively 74 and 15 meV band gaps. This is suggestive of an adiabatic continuity across TCDW and therefore the absence of an excitonic mechanism that would change any electronic properties of the system discontinuously across TCDW. However, experimental observations of insulating phase above the TCDW can not answer whether or not excitonic mechanisms are responsible. In the next section we will show the answer to this highly nontrivial question does not involve excitonic effects, but rather dynamical nuclear fluctuations. It also explains the adiabatic (or nearly adiabatic) continuity between the phases above and below TCDW.

The high symmetry phase, above 200 K: nuclear fluctuations at finite temperatures

Above the ordering temperature TCDW ~ 200 K, the CDW is lost and the system changes to the high-symmetry \(P\bar{3}m1\) phase. QSGW predicts the \(P\bar{3}m1\) structure to be semimetallic, with the conduction band minimum at L falling below the valence band maximum at Γ (Fig. 3a). This suggests that the earlier predictions of TiSe2 being an excitonic insulator may be correct, since QSGW does not include any excitonic correlations and can not open the gap either. However, we will establish in the following that this is not the case. (Note that without self-consistency, the GLDAWLDA gap comes out positive as shown in ref. 27. Self-consistency at the GW level, which takes into account feedback between screening and band structure has a marked effect in TiSe2, as explained in detail in ref. 26.)

When an ideal crystal structure corresponding to a given symmetry is used, the effect of nuclear fluctuations is usually ignored. In most band insulators such nuclear displacements only generate moderate quantitative corrections to the band gap. This effect has been computed by evaluating the electron-phonon self-energy (assuming harmonic phonons), and does not produce any qualitative changes. However, one extreme example of nuclear displacements is the Peierls transition in low dimensional systems. While Peierls envisioned a static distortion of the lattice, which also leads to electronic localization, it is possible that there is a dynamic counterpart to it: the atoms are located at the \(P\bar{3}m1\) crystal positions only when averaged over time. Nuclei are not fixed but fluctuate stochastically about the ideal lattice positions at any instant of time. At very low temperature excursions will be distributed as harmonic excursions around one of the several possible 3qL deformations to the CDW. Above the transition temperature the lattice assumes the \(P\bar{3}m1\) structure on average but will undergo fluctuations which will be weighted to a locale resembling one of the 3qL CDWs. What effect this has on the energy-band structure can only be determined from knowledge of the stochastic motion of the nuclei.

To this end we performed classical ab initio molecular dynamics AIMD simulations of a 96-atom simulation cell at the density-functional (DFT) level, and took selected snapshots separated far enough in time so as to be uncorrelated. This amounts to making an adiabatic approximation for the electronic structure, and follows a strategy similar to that used successfully to determine the dielectric response and temperature dependence of the bandgap renormalisation in Si 32. For context, \(P\bar{3}m1\) has 3 atoms in the unit cell: the 96-atom cell corresponds to a 4 × 4 × 2 replication of it, and a 2 × 2 × 1 superlattice of the CDW. The 24-atom CDW is the smallest system capable of revealing differences above and below the CDW phase transition; however, to allow for additional sampling of thermal fluctuations on length scales larger than supported with 24 atoms, we focused on the 96-atom cell. More complete details of how we performed the MD simulations are given in Methods section.

As we show below, while electronic structure of the ideal \(P\bar{3}m1\) is metallic at the QSGW level, it becomes insulating when QSGW calculations are performed for these MD snapshots (Compare panel (a) to panel (d) in Fig. 3 below). This is a rare instance when thermal nuclear fluctuations qualitatively change the electronic structure from metal to insulator. These are largely a result of Umklapp processes that couple k points near Γ to those near L. While the effect of nuclear fluctuations on the bandgap has been noted in other systems33,34, TiSe2 is unique in that the Umklapp processes give rise to a metal-insulator transition, and can be considered as a dynamic analogue of the classic static Peierls metal-insulator transition scenario. These Umklapp processes arise because of dynamical Peierls-like displacements that result from the “softness” of the system, which follow as a consequence of the internal energy being more favourable for lattices deformed relative to the \(P\bar{3}m1\) structure. It is noteworthy that ref. 22 suggested the electron-phonon interaction as a possible one-particle explanation for a positive gap in the high-symmetry phase.

Electronic structure from molecular dynamics trajectories

We computed the QSGW band structure from twelve snapshots calculated at nuclear configurations generated from the AIMD, at 120 K and 300 K (Fig. 3(c, d)). Details on how the AIMD were carried out are given in the Methods section below. The band edge states are quite fluid for both conduction and valence bands, and the valence band maximum fluctuates between the Γ and A points, but importantly the system is gapped, unlike the ideal \(P\bar{3}m1\) (Fig. 3(a)). The presence of a finite band gap at 300 K correlates with the experimental resistivity data1,35 where dR/dT is negative in this regime, typical of a semiconducting-like state.

Fig. 3: QSGW spectral functions folded to the Brillouin zone of the \(P\bar{3}m1\) structure.
figure 3

(a) the ideal \(P\bar{3}m1\) structure, (b) the CDW phase, (c) the statistically averaged snapshots from the 120 K AIMD simulations and (d) the same for the 300 K simulations. The colour scale ranges from no intensity (white) to full intensity (red) with blue in between.

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We find mean band gaps of 90 and 140 meV at 120 K and 300 K respectively, and at either temperature the gap varies between 0 and 250 meV, with RMS fluctuations of ~60 meV (Table 1). The lower bound of our estimated gap at 300 K is ~80 meV, which is close to the ~74 ± 15 meV gap estimated from ARPES measurements at the same temperature31. Moreover, an intriguing observation from our calculations is that the mean band gap is lower at 120 K (90 meV) than at 300 K (140 meV). The scale of this reduction—as thermal lattice fluctuation effects become quenched at low temperatures—is in remarkable agreement with the estimated experimental reduction in gap by ≈60 meV inside the CDW phase (since the experimentally estimated gap at 10 K is just 15 meV31), providing a natural explanation for this otherwise puzzling experimental result.

Table 1 Data extracted for the \(P\bar{3}c1\) in the experimental geometry1, and from molecular dynamics simulations in the the 96 atom cell at 120 K and 300 K
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How itinerant electrons localise is at the heart of microscopic principles of electronic correlations, as well as devices based on localisation. For a large class of semiconductors, electrons are Bloch states that open up a gap at the Fermi energy due to reflections at the Brillouin zone boundary, causing a transition from metal to band insulator. Most nonmagnetic semiconductors are such trivial band insulators whose electronic properties can be described by a single Slater determinant, and has a one-to-one correspondence with Landau Fermi liquid theory. By contrast, Mott insulators usually involving magnetic fluctuations and are multi-reference (multiple Slater determinants), forming a gap starting from the metallic phase and localising electrons. This phenomenon has no analogue in Landau Fermi liquid theory for electronic excitations. However, once the gap is formed, phonons, more often than not, play a secondary role of reducing the band gaps33,34 by enhancing the electronic screening. In solid state systems, it is rare to find an occasion where electronic localization in itself is mediated by lattice fluctuations, making TiSe2 a special case. It is in that sense that the electronic localization in TiSe2 is more akin to how localization happens in liquids. In TiSe2 the lattice remains significantly soft and labile over a large range of temperatures, so much so, that it can be a source of disorder in itself that is sufficient to localize electrons.

To compare spectra from AIMD snapshots to ARPES, we developed a band unfolding technique along the lines of refs. 36,37. Zone unfolding allows us to represent k-resolved spectral information of the 4 × 4 × 2 supercell as a spectral function in the \(P\bar{3}m1\) Brillouin zone. Moreover we can statistically average the AIMD snapshots to construct a time-averaged spectral function which corresponds to an ARPES measurement. The Brillouin zone of the primitive cell is folded into the supercell so that Nk-points in it k1, …, kN, become reciprocal lattice vectors G1, …, GN of the supercell (N = 24 in this case), with one of the G vectors being G = 0. While details of our implementation will be presented elsewhere, in brief it is a method to resolve an eigenstate ψnk of the supercell into linear combinations of eigenstates \({\widetilde{\psi }}^{{n}^{{\prime} }{{\bf{k}}}_{1}}\,\ldots\, {\widetilde{\psi }}^{{n}^{{\prime} }{{\bf{k}}}_{N}}\) of the primitive cell. Since ψnk is normalised, it will be distributed among partial contributions from the \({\widetilde{\psi }}^{{n}^{{\prime} }{{\bf{k}}}_{1}}\,\ldots\, {\widetilde{\psi }}^{{n}^{{\prime} }{{\bf{k}}}_{N}}\). The partial weights are the scattering amplitudes into these states from Umklapp processes. The G = 0 weight for eigenstate \({w}_{{\bf{k}}}^{n}\) is less than unity, approaching it when k is a good quantum number, i.e. if the supercell is merely a periodic replica of the primitive cell. Thus we can express the spectral function as

$$A({\bf{k}},\omega )=\sum _{i}{w}_{{\bf{k}}}^{i}\,\delta [\omega -({\varepsilon }_{i}-{E}_{F})]$$
(1)

where i runs over the eigenstates of the supercell at k. Eqn. (1) provides a reasonable description for A(k, ω) in the Brillouin zone of the primitive cell, for any snapshot, but averaging over snapshots gives a better representation. The result is shown in Fig. 3. The poles in Eq. (1) were broadened out by a Lorentzian broadening of 20 meV.

Figure 3a shows the energy bands of the ideal \(P\bar{3}m1\) structure. k is a good quantum number and bands are well defined bands without Umklapp processes. Figure 3b shows how Umklapp processes open a gap in the CDW. A replica of the highest valence band on the A–Γ line appears on the L–M line. This couples to the Ti-3d state on the L–M line, forming one bonding state and two antibonding states, thus opening a gap. Similarly, the Ti-d state at L picks up a replica at Γ; indeed the majority of the QP weight at L is transferred to Γ. Turning to the 120 K case, A(k, ω) falls between the \(P\bar{3}m1\) and CDW limiting cases. The Umklapp processes are weaker than in the CDW but nevertheless present, causing a gap to open very similar to the CDW case. The highest valence band on the A-Γ line is almost dispersionless, with significant incoherence, and very likely these factors are the driving force for many-body effects such as superconductivity in doped TiSe2. At 300 K, Umklapp processes are weaker but still present and the gap is preserved. This is a remarkable result, since only some kinds of deformations are sufficient to open a gap (see Fig. 3d).

The spectral weight at L points below the conduction band is missing in the ideal \(P\bar{3}m1\) structure and is only there when the lattice fluctuations are taken into account. Interestingly, such “anomalous” spectral weight can be observed in experiments too38. At low and intermediate temperatures, inside the CDW phase, ARPES experiments at the L points observe the backfolded hole bands, with high intensity. However at temperatures immediately above 200 K (i.e. outside the CDW phase, but close), one can still clearly observe some spectral weight with similar intensity as that of the CDW phase—although becoming more of a “blob" than a backfolded band—which gradually disappears with increasing temperature, until by 300 K it is mostly gone. The fact that this blob of spectral weight at and above 200 K persists and is temperature dependent are strong signatures that these are related to the fluctuations of the 2 × 2 × 2 CDW order. In our calculations at 300 K (Fig. 3d) the anomalous spectral weight at L is more structured than the blob seen in experiments at temperatures above the CDW phase, however the exact nature of the coherence of these features is sensitive to the de-phasing that comes from the finite size effects in the theory: the larger the supercell, the higher the de-phasing and the more quickly the blobs will be washed out with increasing temperature. With this caveat, our calculations taking into account thermal lattice fluctuations nicely explain the anomalous spectral weight seen experimentally that has previously been interpreted as precursor fluctuations of an excitonic order38.

An intriguing aspect of systems with an ordered phase is the fluctuation of the order parameter that persists at higher temperatures compared to the critical temperature for the ordered phase. Experimental methods can rarely measure the fluctuation itself above the ordered phase and often the information about the fluctuations are inferred from indirect microscopic signatures that reflect in various measurements. Indeed, the literature on TiSe2 has been confusing and contradictory because it is so difficult to distinguish competing mechanisms experimentally, while available theory could be suggestive but also not sufficient.

On the scale of the small bandgap, slight perturbations such as small nuclear deformations or approximations to the charge density, play an outsized role, as we have shown. It is only possible to disentangle the various effects with a description of the electronic structure that sufficiently describes the interplay between charge density, quasiparticles with and without excitonic contributions, and nuclear fluctuations, which QSGW (QS\(G\hat{W}\)) approximations provide, when combined with AIMD simulations. It shows explicitly that excitons play a negligible role, ruling out the common assertion that TiSe2 is an excitonic insulator. The gap forms in the low-temperature CDW phase because of level repulsion between Ti-3d states at L coupling to Se-4p states at Γ arising from matrix elements that appear by Umklapp processes from symmetry-breaking. A gap appears in the nominal \(P\bar{3}m1\) phase for similar reasons: but now the symmetry-breaking is dynamical, from instantaneous deformations about the \(P\bar{3}m1\) structure. TiSe2 offers a rare instance of electronic localization originating from lattice fluctuations.

A study of these indirect microscopic signatures in various materials is of extreme interest. Knowledge of the parameter space (temperatures and pressure) where these fluctuations survive is key to finding experimental parameters that can help in condensing the fluctuations into ordered phases at different temperatures and pressures. For example, the experimental signatures of preformed Cooper pairs in cuprates39,40,41,42, iron based superconductors43, organic superconductors,44 CDW fluctuations in various chalcogenides45 and cuprates46 and beginning of formation of Kondo cloud47 have been matters of intensive studies in the recent years. Often, multiple orders and their associated order-parameter fluctuations are present in these materials at a certain part of the phase diagram and these fluctuations can compete, co-operate or coexist. In such situations it becomes difficult to attribute certain microscopic signatures to the fluctuations associated with a particular order parameter. A similar situation occurs in TiSe2 where (since the spin fluctuations are absent) the charge fluctuations (particularly the excitonic correlations) and lattice fluctuations can leave their imprint on various measurements. Our work is a step change in this regard, where we unambiguously determine the signature of the CDW order parameter fluctuations that persist up to 300 K and show that the spectral blobs observed in the ARPES do not originate from excitonic correlations but from lattice fluctuations. The additional presence of the magnetic fluctuations in various cuprates, iron based superconductors make the analysis more involved, but a systematic diagrammatic improvement of our many-body perturbative approach is probably the desired way forward in addressing the outstanding problems in those classes of magnetic materials too.

Methods

Computational Methods

Band structure and QSGW

The Quasiparticle Self-Consistent GW form of GW28,48 dramatically improves its fidelity compared to the usual (DFT-based) GW. Details of the theory and its implementation in Questaal49 can be found in refs. 48,50. QSGW largely eliminates the starting-point dependence (a well known issue for GW implementations) and importantly, discrepancies with experiment become systematic, which clarifies which diagrams are missing. In the spectrum between Hartree Fock (which overestimates gaps) and DFT (which underestimates them) QSGW has a tendency to err slightly on the HF side. As we have shown recently29, adding ladder diagrams to the polarizability eliminates the tendency to underscreen, and mostly eliminates the systematic error in the RPA. As a consequence the fidelity of excited states and optical properties are much improved, particularly if correlations are modest as they are here. As this work shows, the fidelity achievable in QSGW and QS\(G\widehat{W}\) is essential to distinguish artifacts resulting in approximations of the theory from physical behaviour of TiSe2.

Ab initio molecular dynamics

All AIMD simulations were performed at the density functional theory (DFT) level using the Vienna Ab-Initio Simulation Package (VASP 5.4)51,52,53,54 with the Perdew–Burke–Ernzerhof (PBE) exchange correlation functional55 and the associated projector augmented wave (PAW) pseudopotentials56,57 with a plane-wave energy cutoff of 400 eV and periodic boundary conditions in all three directions. A 3 × 3 × 3 Γ-centred k-point grid was used, and Grimme’s D3 van der Waals dispersion-energy correction (DFT-D3)58 was applied with a timestep of 2 fs. Fermi smearing was applied with a smearing energy of 0.026 eV (consistent with a temperature of 300 K; for ease of comparison this was used at 120 K as well).

We determined the minimum size of k-point mesh and the choice of van der Waals correction for the AIMD by taking the 24 atom CDW cell and allowing both the atomic positions and the lattice dimensions to relax to minimize the energy, for several different combinations of k-space mesh and van der Waals dispersion correction. We chose a combination of mesh and dispersion correction that closely matches the experimentally determined cell volume of 521.47 Å. The results of these relaxations are shown in Table 2. Relaxation of the lattice with the Grimme DFT-D2 and DFT-D3 corrections and 2 × 2 × 2 and 3 × 3 × 3 meshes gives geometries close to the experimental value. The DFT-D3 van der Waals dispersion correction was selected because it gives volumes slightly smaller than the experimental volume, which is consistent with there being some contraction from relaxation (essentially a zero temperature result). The 3 × 3 × 3 k-space mesh (bolded) was selected because it is the smallest tested mesh that contains the Γ point and gives a volume within 0.5% of the experimental volume.

Table 2 24 atom cell volume with different combinations of van der Waals correction (VDW) and Γ-centred k space mesh, where Γ denotes that only the gamma point is used
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Configurations for the 96 atom system were obtained below and above the CDW phase transition from AIMD simulations at 120 K and 300 K, respectively. The system was first equilibrated at constant temperature and pressure (the so-called NpT ensemble) using VASP’s implementation of the Parinello-Rahman algorithm,59,60 which allows the cell volume and shape to fluctuate, and an applied pressure of 1 atmosphere. The NpT runs were used to determine average lattice parameters which were then used in subsequent runs at constant temperature and volume (NVT ensemble). Configurations were sampled at intervals of 2 ps from the NVT runs and used to compute the thermally-averaged electronic structure of the TiSe2 system above and below the CDW transition temperature.

Determining lattice parameters from NpT runs: At both 120 K and 300 K, we found that the hexagonal lattice structure was preserved, with fluctuations in the 120 degree angle between the two lattice vectors in the xy plane (<0.3°), and a smaller (<0.2°) fluctuation between the two in-plane lattice vectors and the out-of-plane lattice vector. At 300 K, we ran NpT dynamics for 32 ps and observed that the lattice parameters fluctuated but did not drift over this timescale, with average lattice vector lengths of approximately 14.19 Å, 14.14 Å, and 12.07 Å, with root-mean-squared fluctuations of 0.03, 0.06 and 0.09 Å, respectively. These lattice dimensions can be related to the dimensions of the minimum sized CDW lattice by halving the first two (~14 Å) values. The two xy lattice vectors can be taken, within the fluctuations, to be the same so we average their values to produce the average 300 K lattice vectors given in Table 3. At 120 K, we ran NpT dynamics for 22 ps and observed a gradual increase in the z-oriented lattice vector over the first 12 ps. We therefore averaged only over the final 10 ps to find average lattice vector lengths of 14.13 Å, 14.12 Å, and 12.14 Å, all with root-mean-squared fluctuations of 0.02 Å. We again averaged the lengths of the two vectors in the xy plane to produce the average 120 K a lattice constant given in Table 3.

Table 3 Lattice parameters (a and c) for the different TiSe2 structures studied here
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Crystal structure and fluctuations at finite temperature: Table 1 shows that average bond lengths and angles are similar in the \(P\bar{3}c1\) and \(P\bar{3}m1\) phases, so it would be difficult to distinguish them based on pair distribution functions. Indeed, the thermally broadened radial distribution functions look similar at both 120 K and 300 K, albeit with slightly greater broadening of peaks at 300 K. Figure 4 shows species-projected radial distribution functions from trajectories below (120 K) and above (300 K) the CDW phase transition, and we see that the high temperature and low temperature radial distribution functions are very similar. Fluctuations that broaden the peaks wash out much of the difference in the structures from the subtle CDW symmetry breaking, with greater broadening at 300 K than at 120 K. Hence, we might expect the high and low temperature spectral functions to resemble each other at both DFT and QSGW levels, notwithstanding the differences between them for the perfect crystal forms.

Fig. 4: Species-projected radial distribution functions from molecular dynamics trajectories at 120 K (upper panel) and 300 K (lower panel).
figure 4

Red curves show radial distribution functions for Ti–Ti separations, blue curves for Se–Se separations, and purple curves for Ti–Se separations.

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