Abstract
Excitonic insulator is a coherent electronic phase that results from the formation of a macroscopic population of bound particlehole pairs—excitons. With only a few candidate materials known, the collective excitonic behavior is challenging to observe, being obscured by crystalline lattice effects. Here we use polarizationresolved Raman spectroscopy to reveal the quadrupolar excitonic mode in the candidate zerogap semiconductor Ta_{2}NiSe_{5} disentangling it from the lattice phonons. The excitonic mode pronouncedly softens close to the phase transition, showing its electronic character, while its coupling to noncritical lattice modes is shown to enhance the transition temperature. On cooling, we observe the gradual emergence of coherent superpositions of band states at the correlated insulator gap edge, with strong departures from meanfield theory predictions. Our results demonstrate the realization of a strongly correlated excitonic state in an equilibrium bulk material.
Introduction
Attractive interactions between fermions are known to lead to a proliferation of bound pairs of particles at low temperatures causing a transition into superconducting or superfluid phases. In a semiconductor or a compensated semimetal, the Coulomb attraction between electrons and holes may induce a similar transition where electronhole pairs, the excitons, develop macroscopic coherence^{1,2,3,4}. The resulting state, characterized by an interactioninduced gap, has been dubbed as the excitonic insulator^{5}. However, so far only a few materials have been identified as excitonic insulator candidates^{6,7,8,9,10,11}—possibly because its formation requires strong attraction or matching energy dispersions of the electron and holelike carriers^{1}. These restrictions can be overcome by creating a nonequilibrium exciton population and cooling below their degeneracy temperature, in which case the coherent state may be observable only as a transient due to the finite lifetime of the excited state^{12,13,14}. An equilibrium excitonic phase in a bulk material, on the other hand, would allow for a far wider range of questions to be asked and answered regarding the excitonic states of matter and their formation.
As an example, controlling the bare band gap of an excitonic insulator allows one to explore a range of correlated regimes: from a weakly correlated electonhole condensate analogous to the Bardeen–Cooper–Schrieffer (BCS) condensate in fermionic superfluids in the negativegap (semimetallic) regime to a weakly interacting gas of tightly bound excitons in the opposite limit of a gapped (semiconductor) band structure^{3,5,15}. In the former case, weakly bound excitons are characterized by size ξ_{ex} larger than the interparticle distance l_{eh} and the exciton wavefunctions overlap strongly, while in the latter one, a dilute gas of tightly bound excitons with ξ_{ex} ≪ l_{eh} exists also above the transition temperature, with their chemical potential going to zero at the transition, in analogy with the BoseEinstein condensation (BEC). The “BCS” and “BEC” regimes are also characterized by a different dynamics of the excitons. In particular, for a semimetallic normal state (as in Fig. 1d), the exciton, regardless of its energy, may decay into unbound particlehole pairs (Landau damping), leading to overdamped dynamics. On the other hand, for a semiconductorlike normal state, the exciton energy is within the direct gap at T > T_{c}, such that energy conservation ensures the undamped dynamics of the excitons. Most interesting is the strongly correlated crossover regime^{16,17}, where neither of the weakly coupled bosonic or fermionic descriptions applies and the signatures of excitons above T_{c} may coexist with the strongly coupled electronhole plasma.
However, proving the excitonic origin of the phase transition is challenging as it is expected to be obscured by an accompanying structural transition of the crystalline lattice. Formation of a macroscopic excitonic population in a real material may break some of the lattice symmetries: translational one if the excitons are indirect (i.e., have a nonzero centerofmass momentum)^{9} or pointgroup ones otherwise^{3}. In the latter case, the nontrivial effect on the lattice comes about due to the transformation properties of the particle and hole wavefunctions involved in forming an exciton: if those are distinct, e.g., an slike electron is paired with a dlike hole, the exciton wavefunction would have symmetry lower than the lattice one, even if the relative motion of the particle and hole is in a fully symmetric slike state. The discrete nature of the pointgroup symmetry broken in the excitonic insulator at the transition temperature T_{c} bears important consequences for its properties: in particular, its excitations are expected to have a finite energy gap, in contrast to the nonequilibrium excitonic condensates, where a continuous U(1) symmetry associated with approximate exciton number conservation is broken resulting in superfluidity and a gapless BogoliubovGoldstone mode at T < T_{c}^{12,13,14}.
The above mentioned difficulties are pertinent to the case of Ta_{2}NiSe_{5}, a material showing a semiconducting behavior at low temperatures with a phase transition from hightemperature orthorhombic phase to a monoclinic one at T_{c} = 328 K^{18}, breaking two of the mirror symmetries (Fig. 1, a and b). Symmetrywise, this transition corresponds to the reduction of pointgroup symmetry from D_{2h} to C_{2h} with the order parameter transforming as the B_{2g} irreducible representation of D_{2h} (xzlike quadrupole). The electronic structure of Ta_{2}NiSe_{5} has been predicted to have a small or negative (as in Fig. 1d) direct gap at the Brillouin zone center^{19,20}, in agreement with experiments above T_{c}^{15,21}. The two bands closest to Fermi energy have quasionedimensional (1D) character and are derived from superpositions of Ta 5d and Ni 3d orbitals at the multiple sites of the unit cell. Hybridization between them in the k_{x} = 0 plane, forbidden by x → −x symmetry above T_{c}, can serve as the order parameter that is microscopically induced by a condensation of the resulting interband excitons^{22,23}. This points to the quadrupolar character of these excitons.
Experimentally, below T_{c} formation of a gap has been observed in transport^{15} and optical^{21} measurements, and an anomalous dispersion of the holelike band^{24} has been taken as an indication for the excitonic character of the transition. Above T_{c}, a gaplike spectral weight suppression has been observed in ARPES studies^{25}, suggesting a BEClike picture for the excitons, while the low values of transport gap suggest otherwise^{15}. On the other hand, the changes in spectral and transport properties could also be due to the change of the lattice structure below T_{c}. While Ta_{2}NiSe_{5} has been actively investigated since then^{15,21,23,25,26,27}, a structural origin of the transition has not been excluded^{28,29,30}. Additionally, the origin of the lowtemperature spectral gap has not been directly probed: while the valence band dispersion at low temperatures is consistent with a symmetrybreaking hybridization forming below T_{c}^{24}, a direct proof of hybridization (regardless of the driving force behind the transition) requires showing the states of two bands being mixed into coherent superpositions of the states at the gap edge as in Fig. 1e.
The questions above can be addressed directly by Raman spectroscopy that probes the excitations of the system by an inelastic twophoton process. Polarization analysis of the incoming and outgoing photons further enables one to select excitations with a specific symmetry^{31,32}. Applied to Ta_{2}NiSe_{5} above T_{c}, ac polarization geometry (Fig. 1c, bottom) probes excitations with the symmetry of actype quadrupole (i.e., B_{2g}, see Methods), the same as that of the order parameter, allowing direct observation of the soft mode expected at a secondorder phase transition (Fig. 1d). Being even in parity, these excitations are invisible in a conventional light absorption experiment due to the dipole selection rules. The character of the soft mode reveals the origin of the transition. In an excitonic transition in a semimetal, critical fluctuations would have a broad relaxational lineshape due to the Landau damping and are enhanced at low frequencies close to T_{c} (Fig. 1f, right). On the other hand, in a structural transition driven by an optical phonon, a sharp spectral peak would soften to zero energy at T_{c} (Fig. 1f, left). In addition, the structural transition can also be driven by an instability of the acoustic modes (ferroelastic instability^{33}), which would lead to absence of signatures in Raman scattering, as coupling to light vanishes at q = 0 for acoustic modes^{34}.
Hybridization between bands can be further revealed by studying the contribution of electronhole excitations to Raman scattering. Above T_{c}, the ac scattering geometry probes the interband transitions (Fig. 1d) between the valence (red) and conduction (blue) bands that have an actype quadrupole (B_{2g}) character. The aa scattering geometry, on the other hand, probes fully symmetric excitations (Fig. 1c, top), and is limited to intraband transitions only. Below T_{c}, hybridization mixes the states of two bands into coherent superpositions (Fig. 1e). This results in interference effects at the gap edge, in analogy with the effect of coherence factors in a supercondutor^{35}. In particular, for ac geometry a destructive interference occurs between transitions from “red” to “blue” states and vice versa resulting in an exact cancellation (Fig. 1e, on the right). In contrast, for aa geometry, the destructive interference is between two types of interband transitions, which do not cancel exactly, as “red” and “blue” bands couple to light differently (Fig. 1e). This results in the intensity close to the gap edge being strongly suppressed in ac geometry with respect to aa. In Fig. 1g we present the spectra expected in two polarization channels based on a meanfield model of an excitonic insulator (see Methods and Supplementary Note 2 for details). In contrast to the above description, if hybridization is absent and the lowtemperature gap is between the conduction and valence bands, the gap edge corresponds to a purely interband transition. The intensity in ac geometry is then expected to be dominant, clearly distinct from the hybridization gap case.
Here, we employ polarizationresolved Raman spectroscopy to prove the excitonic origin of the transition in Ta_{2}NiSe_{5}, as opposed to structural one, and the hybridization nature of the lowtemperature gap. We further determine that the resulting excitonic insulator is in the strongly correlated BCSBEC crossover regime. In particular, close to T_{c} we observe critical softening of overdamped quadrupolar excitations Fig. 2b, f) that are consistent with excitonic fluctuations in a semimetal. In contrast, we find no softening of the optical phonon modes (Figs. 3 and 4b). At low temperatures, by comparing the intensities in aa and ac polarization geometries (Fig. 2e) we find direct evidence for hybridizationinduced gap and coherent mixing of the two semimetallic bands driven by the excitonic order (as in Fig. 1d), i.e., spontaneously formed symmetrybreaking hybridization between the bands. With heating, the gap observed in fully symmetric channel fills in, rather than closes, characteristic of strong correlations beyond the meanfield regime. By estimating the exciton coherence length, we find that for Ta_{2}NiSe_{5} the excitonic condensate lies within the strongly correlated BCSBEC crossover regime, and argue that the whole body of experimental data for Ta_{2}NiSe_{5} is consistent with this identification.
Results
Overview
In Fig. 2a–d we present an overview of the Raman spectra. For ac geometry, probing excitations with the symmetry of actype quadrupole above T_{c}, same as that of the order parameter (Fig. 1c, bottom), a pronounced enhancement of the lowenergy response is observed around T_{c}. This is characteristic of a soft mode development near a secondorder phase transition. For aa geometry, the most prominent feature is the redistribution of intensity towards higher energies below T_{c} with a pronounced gaplike suppression below 380 meV; at low energies conventional phonon peaks are observed (Fig. 2c and Supplementary Fig. 8).
The presence of a symmetrybreaking phase transition at T_{c} is evident from the appearance of new sharp optical phonon modes in both geometries. Their appearance is related to the change in selection rules below T_{c} (see Methods); in what follows, we will call this intensity admixture from an another polarization as “leakage”.
This reflects that below T_{c} the two polarization geometries are no longer orthogonal, thus all excitations may appear in both geometries. In Fig. 4a we further quantify this effect by showing the temperature dependence of the integrated intensity of the lowestenergy fully symmetric (above T_{c}) phonon mode in ac scattering geometry, which grows substantially below 328 K. The parameters of the fully symmetric phonon modes are also consistent with a recent study of Raman scattering in aa geometry^{36}.
Identification of the excitonic soft mode
We now focus on the lowenergy lineshapes in the ac geometry spectra close to the transition temperature, detailed in Figs. 2f and 3. Three distinct peaks are observed, which correspond to three optical phonons of B_{2g} symmetry expected from the space group of Ta_{2}NiSe_{5} (Cmcm), ruling the presence of other optical modes out. The striking feature of the raw data is the notably asymmetric shape of the two lowestenergy modes, that cannot be described as conventional Lorentzian oscillators. Instead, the data above T_{c} is well described by a generalized Fano model (see Methods and Supplementary Note 1.3 for details), including three phononic oscillators interacting with a continuum of overdamped excitonic excitations described by a purely relaxational response
where its imaginary part χ″(ω) exhibits a maximum at Ω_{e}(T). This continuum is clearly distinct from the structural phonons and suggests the presence of an overdamped bosonic mode emerging from the electronic system consistent with a Landaudamped exciton in a semimetal (Fig. 1d). The parameter Ω_{e}(T) can then be represented by an analogy with an overdamped oscillator as \({{{\Omega }}}_{{\rm{e}}}(T)\equiv {\omega }_{0}^{2}/{{\Gamma }}\), ω_{0} being the exciton frequency (which is a collective mode of the semimetal, similar to the collective Cooper pair mode above T_{c} in a superconductor^{37}) and Γ ≫ ω_{0} —the damping rate. This form can be alternatively derived using the timedependent GinzburgLandau equation (see Supplementary Note 1.3.1). Upon cooling towards T_{c}, Ω_{e}(T) linearly decreases as \({{{\Omega }}}_{{\rm{e}}}(T)\propto T{T}_{{\rm{c}}}^{el}\) (see Supplementary Fig. 5), where \({T}_{{\rm{c}}}^{el}=137(16)\) K, consistent with a critical softening of this excitonic collective mode. Just below T_{c}, the ac Raman response is additionally enhanced at the lowest frequencies (Fig. 3e, f). We associate this enhancement with coupling to acoustic B_{2g} modes with finite momenta, that is mediated by a quasiperiodic pattern of structural domains (which form below T_{c}) that takes the quasimomentum recoil (see Methods for the details).
Most importantly, we do not observe phonon softening: instead, the two bare lowest B_{2g} mode frequencies harden on cooling, Fig. 4b. Consequently, an instability of a zonecenter optical phonon^{29} can be ruled out by our results. Away from T_{c}, the hardening on cooling is consistent with the expectations due to lowestorder phonon anharmonicity (lines in Fig. 4b); however, a more pronounced increase is observed close to T_{c}. We attribute this increase to a nonlinear coupling between the excitonic fluctuations and the phonons. In particular, introducing the electronic order parameter φ, the phonon frequency can be expressed as ω_{ph}(T, φ) ≈ ω_{ph}(T, 0) + α_{ph}φ^{2}. For α_{ph} > 0 (which points to a competition between the electronic and lattice orders), below T_{c} an additional increase of the phonon frequency is expected. In our results, the increase starts above T_{c}, where 〈φ〉 = 0, pointing to the influence of the fluctuations beyond the meanfield; in particular, the increase starts at 350 K, suggesting a fluctuation region of around 20 K for the phonon properties.
To study the effects of the excitonphonon coupling on the transition, we analyze the static order parameter susceptibilities (Fig. 5). From the Fano model, one can deduce the individual susceptibilities of the excitonic continuum, optical phonons as well as the combined one, that includes the effects of the coupling between them. The purely electronic contribution to the inverse susceptibility (Fig. 5b) \({\chi }_{{\rm{cont}}}^{1}(0,T)={{{\Omega }}}_{{\rm{e}}}(T)/{t}_{{\rm{e}}}^{2}\) (from Eq. (1)) follows the CurieWeiss form \(\propto (T{T}_{{\rm{c}}}^{el})\) above T_{c}, indicating the softening of Ω_{e}(T). Extrapolating this trend to T < T_{c} suggests that a purely electronic transition would have taken place at \({T}_{{\rm{c}}}^{el}\,<\,{T}_{{\rm{c}}}\) marked by the divergence of \({\chi }_{{\rm{cont}}}(0,{T}_{{\rm{c}}}^{{\rm{el}}})\). On the other hand, the bare optical phonon susceptibility remains almost constant and is even reduced around T_{c}.
Nevertheless, the coupling between the normal modes of the lattice and the excitonic continuum can increase the transition temperature. Extrapolating the susceptibility of the full Fano model, which includes the excitonphonon interaction, from above T_{c} (red line in Fig. 5) we obtained a transition temperature \({T}_{{\rm{c}}}^{{\mathrm{comb}}}=238(18)\) K, larger than \({T}_{{\rm{c}}}^{{\mathrm{el}}}\) by about 100 K. However, \({T}_{{\rm{c}}}^{{\mathrm{comb}}}\) is still smaller then the actual T_{c}, because apart from the optical phonons that we have observed, the coupling to the acoustic B_{2g} strain modes has also to be considered. Such a coupling has been demonstrated to increase the temperature of an electronic nematic ordering to the actually observed one in ironbased superconductors^{38,39}. These effects can be understood within the Landau theory, where the electronic order parameter φ couples linearly to (optical) phononic and strain order parameters (denoted as η_{i} and ε_{ac}, respectively) of the same symmetry resulting in the free energy expansion
where κ is the B_{2g} shear modulus. Minimizing the quadratic term one obtains the condition for the transition temperature \({\chi }_{{\rm{cont}}}^{1}(0,{T}_{{\rm{c}}}){\lambda }^{2}/\kappa \mathop{\sum }\nolimits_{i = 1}^{3}{\lambda }_{{\rm{opt}},i}^{2}{\chi }_{{\rm{opt}},i}(0,{T}_{{\rm{c}}})=0\), as opposed to \({\chi }_{{\rm{cont}}}^{1}(0,{T}_{{\rm{c}}}^{el})=0\) in the purely electronic case, leading to an enhanced transition temperature T_{c}. In Fig. 5 we illustrate this effect (dashed black line) by choosing the value of the interaction with strain such that the total susceptibility diverges at T_{c}. We find the value of the excitonstrain coupling obtained this way to be consistent with a recent study of the sound velocity renormalization above T_{c}^{40} (see Supplementary Note 1.3.5), and the Raman response from finitemomentum acoustic phonons below T_{c} in Fig. 3e, f.
The analysis above leads us to the following conclusions: (a) the excitonic mode exhibits a strong tendency to soften, suggesting a purely electronic transition temperature of \({T}_{{\rm{c}}}^{el}=137(16)\) K; and (b) the transition temperature is boosted to the observed T_{c} = 328 K due to the coupling to noncritical optical phonon modes and acoustic strain.
Coherent factors in the excitonic insulator
Having established the excitonic origin of the transition in Ta_{2}NiSe_{5} we now demonstrate that an interband hybridization emerges in the excitonic insulator phase. At low temperatures, an intense peak at about 380 meV in the aa geometry emerges with a much subtler feature in the ac geometry (Fig. 2e). This stark contrast cannot be attributed to the difference of the Raman coupling in two geometries: while the intensity in the ac geometry is indeed weaker than in the aa one at all temperatures, the evolution of the intensity in aa geometry at T < T_{c} is much more pronounced, allowing to relate this effect to the symmetry breaking (see also Supplementary Note 1.5). Moreover, a substantial part of the signal observed in ac geometry at low temperatures can be attributed to the change in selection rules below T_{c}, making the ac intensity actually related to the interband transition even smaller (see Supplementary Fig. 11). Both band structure calculations^{22} and experiments^{41,42,43} suggest that only two bands are mostly affected by this symmetry breaking, allowing us to limit ourselves to the simplest excitonic insulator model of Fig. 1. The strong temperature dependence of this feature below T_{c} (also discussed below) also precludes its attribution to transitions away from k_{x} = 0 plane, where hybridization can be present already above T_{c}. For a hybridizationinduced gap in a semimetal, a suppression of interband transitions at the gap edge in the ac geometry is indeed expected (Fig. 1e). Within a simplified twoband model of the excitonic insulator (Fig. 1g), a divergence of the intensity at the gap edge occurs in aa geometry, but not in the ac one, where interference effects lead to an exact cancellation of the intensity at the gap edge. These predictions are in agreement with the pronounced peak being observed in the aa geometry only (which is further exacerbated if the leakage from aa to ac geometry is accounted for, see Supplementary Fig. 11). On the other hand, in the case of a semiconductor with a large direct gap, such that hybridization effects can be ignored, the opposite is true: the interband transition would have predominantly ac quadrupolar character allowing its observation only in ac geometry (see Methods and Supplementary Note 2 for details), in stark contrast to the data in Fig. 2e. Thus, the observation of an intense peak emerging at low temperatures in aa geometry clearly favors a semimetallic band structure with a hybridizationinduced gap.
Now we discuss the temperature dependence of the highenergy Raman response in aa geometry (Fig. 2e). Within the meanfield theory, the energy of the peak is related to the amplitude of the order parameter, which is expected to diminish upon heating towards T_{c} (Fig. 1g), and zero intensity is expected within the hybridization gap. We have observed a pronounced redistribution of intensity in a broad energy range, with intensity appearing at the lowest energies at temperatures significantly smaller than T_{c}. Such behavior appears inconsistent with the meanfield expectation, pointing to a role of correlations beyond the meanfield picture. This is further emphasized by all the curves crossing at 280 meV—the socalled “isosbestic point”^{32,44,45}, characteristic of strongly correlated systems (see also the discussion of BCSBEC crossover effects and pseudogap below). It was previously observed in materials where spectral gap has a manybody origin, such as SmB_{6}^{46} or cuprates^{47}. While band gap renormalization in semiconductors can be also related to electronphonon interactions^{48}, the broadening of the gap edge features is expected to be rather small in that case; on the contrary, the spectral features we have observed quickly become broader than their energy on heating (Fig. 2e). Furthermore, the renormalization effects are expected to diminish with phonon energy^{49} and in our case all phonon energies are below 50 meV, suggesting the scale of the electronphonon effects to remain below 100 meV.
Discussion
Our results unambiguously point to the excitonic physics playing a crucial role in Ta_{2}NiSe_{5}; we further found evidence for strong correlation effects beyond the meanfield expectations. The important role of correlations in the formation of excitonic order in Ta_{2}NiSe_{5} is further corroborated by the large ratio \(\frac{2{{\Delta }}}{{k}_{{\rm{B}}}{T}_{{\rm{c}}}}\approx 13\), k_{B} being the Boltzmann constant (2Δ ≈ 380 meV taking the energy of the spectral peak in Fig. 2e as an estimate for the order parameter value), well beyond the BCS prediction, but consistent with the suppression of T_{c} occurring in BCSBEC crossover^{16}. Moreover, we can estimate the coherence length of the excitonic order using the bare electronic transition temperature with the BCS expression \({\xi }_{{\rm{ex}}}=\frac{\hslash {v}_{{\rm{F}}}}{1.76\pi {k}_{{\rm{B}}}{T}_{{\rm{c}}}^{el}}\). As the system is highly anisotropic, we use the value of v_{F} along the most dispersive a direction, where the exciton size is expected to be largest. This yields ξ_{ex} = 32 Å, while the distance between particles can be estimated in a quasi1D system from \({l}_{{\rm{eh}}}=\frac{\pi }{2{k}_{{\rm{F}}}}\approx 16\) Å (k_{F} ~ 0.1 Å^{−1}, m^{*} ≈ 0.37m_{0}^{41}). The estimate places the excitons in Ta_{2}NiSe_{5} in the correlated BCSBEC crossover regime l_{eh} ~ ξ_{ex}.
Identification of Ta_{2}NiSe_{5} as a correlated excitonic insulator finally allows to clarify the character of the electronic structure above T_{c}. Semimetallic character is suggested by the absence of a discernible gap in spectroscopy (Fig. 2e and refs. ^{21,41}) and the overdamped lineshape of the excitonic fluctuations (Fig. 3). A direct semiconducting gap at T > T_{c} would further contradict the absence of a noticeable peak in ac Raman intensity (Fig. 2e) at low temperatures that requires strong mixing of the bands as in Fig. 1e. On the other hand, an apparent gap feature has been observed in ARPES studies above T_{c}^{25}. It is known however, that strong fluctuations of a particlehole order, such as a charge density wave, result in a pseudogap^{50,51,52} in a metallic system—a suppression of the density of states already above T_{c}, which has been experimentally observed close to charge or spindensity wave transitions^{53,54}. In all of these cases, the pseudogap opens above the ordering transition due to the thermal order parameter fluctuations, regardless of the microscopic mechanism of the transition. Strong excitonic fluctuations in the BCSBEC crossover regime may be expected to lead to similar effects, which explains the spectral weight suppression observed in ARPES^{25}. Moreover, as the density of states in the pseudogap is finite^{50}, it can be reconciled with both the zerogap behavior of the hightemperature transport^{15}, as well as the overdamped character of the excitonic mode (Fig. 3).
In conclusion, by using polarizationresolved Raman spectroscopy, we have directly revealed the excitonic fluctuations driving a phase transition in Ta_{2}NiSe_{5}, shown the coherence of the lowtemperature insulator by revealing the coherent band superpositions, and identified the fingerprints of strong electronic correlations, similar to the ones occuring in the BCSBEC crossover. The strongly correlated excitonic insulator nature of Ta_{2}NiSe_{5} accommodates for its anomalous features and provides a unified view of its electronic structure, consistent with the previous experiments^{21,24,25}. Furthermore, as the band structure of this material can be tuned by chemical substitution or pressure^{15}, this opens perspectives for the exploration of different correlation regimes in excitonic insulators.
Methods
Sample preparation
Single crystals of Ta_{2}NiSe_{5} were grown using the chemical vapor transport (CVT) method. Elemental powders of tantalum, nickel and selenium were mixed with a stoichiometric ratio and then sealed in an evacuated quartz ampoule with a small amount of iodine as the transport agent. The mixture was placed in the hot end of the ampoule (~950 ^{∘}C) under a temperature gradient of about 10 ^{∘}C cm^{−1}. After about a week mmsized needlelike single crystals were found at the cold end of the ampoule. These crystals are shiny and cleave easily. We have used Xray diffraction and Electron Dispersive Xray Spectroscopy (EDS) to verify the exact composition of the crystals and their uniformity.
Resistance measurement
The resistance along the aaxis (direction along the Ta/Ni chains) of a very thin needlelike single crystal was measured in a fourprobe configuration using a Quantum Design DynaCool PPMS system (see Supplementary Note 1.1 for details).
Polarizationresolved Raman measurements
The samples used for Raman measurement were cleaved to expose (010) crystallographic plane (ac plane). The cleaved surface was then examined under a Nomarski microscope to find a strainfree area.
Raman measurements were performed in a quasibackscattering geometry from the samples placed in a continuous heliumgasflow cryostat. The 647 nm line from a Kr^{+} ion laser was used for excitation. Incident light was focused into a 50 × 100 μm^{2} spot of freshly cleaved crystal surface. For data taken below 310 K, laser power of 8 mW was used. To reach temperature above 310 K, we kept the environmental temperature at 295 K and increased laser power to reach higher sample temperature in the excitation spot. The reported temperatures were corrected for laser heating, which was determined to be 1.29 ± 0.17 K mW^{−1} from the Stokes/antiStokes intensity measurement consistent with detailed balance (see Supplementary Note 1.2 for details). We employed a custom triplegrating spectrometer with a liquidnitrogencooled chargecoupled device detector for analysis and acquisition of the Raman signal. All the data were corrected for the spectral response of the spectrometer and the CCD detector. For lowfrequency Raman shift data below 50 meV, the spectral resolution was set to 0.06 meV for data taken below 100 K; to 0.10 meV for data taken from 100 to 295 K; and to 0.19 meV for data taken above 295 K. For highfrequency spectral features above 50 meV about 2.5 meV resolution was used.
Two incident/scattered light polarization configurations were employed to probe excitations in different symmetry channels. The relationship between the scattering geometries and the symmetry channels is given in Table 1.
The Raman response function χ″(ω, T) was calculated from the measured Raman intensity I(ω, T) by the relation I(ω, T) = [1 + n(ω, T)]χ″(ω, T), where n(ω, T) is the Bose factor, ω is Raman shift energy, and T is temperature.
Fitting model for the Raman susceptibility
We adapt an extended Fano model^{55} comprising three phonon modes interacting with an electronic continuum to fit the acpolarization Raman susceptibility data above T_{c}, see Figs. 2f and 3 of the main text. It was found that the electronic continuum is best described by a simple relaxational response \({\chi }_{{\rm{cont}}}^{^{\prime\prime} }(\omega )=\frac{{t}_{{\rm{e}}}^{2}\omega }{{\omega }^{2}+{{{\Omega }}}_{{\rm{e}}}^{2}}\), corresponding to the overdamped limit of the DrudeLorentz model (see Eq. (1) in the main text). Here t_{e} is the Raman coupling vertex to electronic response and \({{{\Omega }}}_{{\rm{e}}}\equiv \frac{{\omega }_{0}^{2}}{{{\Gamma }}}\) is a single fit parameter representing ratio of the square of electronic excitation frequency to its relaxation rate.
The resulting purely phononic and electronic parts of the Raman susceptibility are shown in Fig. 3 in green and blue lines correspondingly. The temperature dependencies of the frequencies of the lowest phonons are shown in Fig. 4b. The electronic part of the inverse susceptibility (Fig. 5b, red points) is, on the other hand, proportional to the electronic oscillator parameter Ω_{e}(T) (see Eq. (1)). The static susceptibilities shown in Fig. 5 are obtained by taking the Fano model susceptibility at ω = 0. As the model susceptibility is an analytic function and matches the raw data well, the result is equivalent to that of a KramersKronig transformation.
By analyzing the data just below T_{c} (Fig. 3 e, f), we have found an additional enhancement at low frequencies can not be described with the same model as the characteristic continuum energy Ω_{e}(T) only grows below T_{c} (see Supplementary Fig. 5) and the resulting continuum intensity at low energies only weakens (Fig. 3d–g). This suggests the presence of an additional source of lowfrequency response. The appearance of a lowenergy mode can be appreciated from the formation of domains observed in transmission electron microscopy (see Supplementary Note 1.3.2 and ref. ^{18}). In particular, the domains have an elongated needlelike shape along the a axis, with their size along the c axis on the order of \(\bar{d}=200\) Å. Acoustic phonon, scattering on this quasiperiodic structure, can take a recoil momentum of the order \(2\pi /\bar{d}\); this way a finite \({q}_{{\rm{d}}}=\pm \!2\pi /\bar{d}\) phonon can contribute to the Raman intensity (where the momentum transfer is approximately zero). At finite q, introducing the acoustic phonon coordinate \({x}_{q}=\sqrt{q}{u}_{{\rm{q}}}\), where u_{q}  is the Fourier transform of the atomic displacement amplitude, one obtains that the Raman vertex \({t}_{{\rm{s}}}={\tau }_{{\rm{s}}}\sqrt{q}\) and the coupling to the excitonic mode \({v}_{{\rm{s}}}={\beta }_{{\rm{s}}}\sqrt{q}\) of the acoustic phonon are both proportional to \(\sqrt{q}\). Both vanish at q = 0 and thus the acoustic modes can not be observed in the absence of domains. The energy of the relevant B_{2g} acoustic mode is ω_{s}(q) = c_{s}q, where c_{s} is the sound velocity, that can be deduced from the Xray measurements^{40} to be c_{s} ≈ 30 meV Å away from T_{c}. Note that the coupling to the soft excitonic mode results in the renormalization of the observed c_{s} close to T_{c}; in Supplementary Note 1.3.5 it is shown that the softening of c_{s} close to T_{c} can be fully attributed to this effect. To fit the data we further fix the ratio \({\beta }_{{\rm{s}}}^{2}/{c}_{{\rm{s}}}\) such that the total static susceptibility (Eq. 1) diverges at T_{c} and assume that the domain size i in units of the crystal unit cell constant c follows the Poisson distribution. In addition to that, we allow for an intrinsic linewidth of the phonons at finite q: ω_{s} → ω_{s} − ir_{s}q. Thus, the parameters τ_{s} and r_{s} remain the only fit parameters. Additional details can be found in Supplementary Note 1.
Effect of coherence factors on lowtemperature Raman scattering
To describe an excitonic insulator at low temperature we use a twoband meanfield model (see Supplementary Note 2 for details): \({\hat{H}}_{{\rm{MF}}}={\sum }_{{\bf{p}}}{\varepsilon }_{{\rm{c}}}({\bf{p}}){\hat{c}}_{{\rm{c}}}^{\dagger }({\bf{p}}){\hat{c}}_{{\rm{c}}}({\bf{p}})+{\varepsilon }_{{\rm{v}}}({\bf{p}}){\hat{c}}_{{\rm{v}}}^{\dagger }({\bf{p}}){\hat{c}}_{{\rm{v}}}({\bf{p}})+W({\hat{c}}_{{\rm{c}}}^{\dagger }({\bf{p}}){\hat{c}}_{{\rm{v}}}({\bf{p}})+{\hat{c}}_{{\rm{v}}}^{\dagger }({\bf{p}}){\hat{c}}_{{\rm{c}}}({\bf{p}}))\), where ε_{c(v)}(p) is the dispersion of the conduction (valence) band and W—is the excitonic order parameter (spin index is omitted for brevity). The order parameter mixes two bands such that the eigenstates are superpositions of band states with coherent factors, analogous to the BCS ones. To calculate the Raman intensity, we use the following form of the electronlight vertex, consistent with the system’s symmetry above T_{c}: \({\hat{R}}_{{\rm{aa}}}={\sum }_{p}{g}_{{\rm{c}}}{\hat{c}}_{p}^{\dagger }{\hat{c}}_{p}+{g}_{{\rm{v}}}{\hat{v}}_{p}^{\dagger }{\hat{v}}_{p};\ {\hat{R}}_{{\rm{ac}}}={g}_{{\rm{ac}}}{\sum }_{p}{\hat{c}}_{p}^{\dagger }{\hat{v}}_{p}+{\hat{v}}_{p}^{\dagger }{\hat{c}}_{p},\) where \({\hat{R}}_{{\rm{aa}}}\) is the vertex for aa polarization geometry and \({\hat{R}}_{{\rm{ac}}}\)— for ac geometry, and g_{c/v}, g_{ac} are coupling constants.
For the semimetallic case one can linearize the dispersion ε_{c,v}(p) around the Fermi surface. The resulting Raman intensity obtained using Fermi’s Golden Rule is:
for ω > 2W and zero otherwise, where ν_{0} is the density of states. One can see that at the gap edge, the ac intensity vanishes due to the vanishing matrix element of the transition, as is shown in Fig. 1e, while the aa diverges as a squareroot, similar to the case of a BCS superconductor^{35}, as is shown in Fig. 1g.
For the semiconductorlike 1D band structure with a direct gap 2E_{0}, on the other hand, we find that the interference effects are weakened such that a singularity at the gap edge is present in aa and ac scattering (See Supplementary Note 2 for details).
Data availability
The data that support the plots within this paper and other findings of this study are available from the authors upon reasonable request.
Code availability
Wolfram Mathematica codes for data analyses are available from one of the contributing authors (M.Y.) upon reasonable request.
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Acknowledgements
We thank K. Haule for discussions. The spectroscopic work conducted at Rutgers was supported by NSF Grant No. DMR1709161 (M.Y. and G.B). P.A.V. acknowledges the Postdoctoral Fellowship support from the Rutgers University Center for Materials Theory. The sample growth and characterization work conducted at the Technion was supported by the Israel Science Foundation Grant No. 320/17 (H.L., I.F. and A.K.). H.L. was supported in part by a PBC fellowship of the Israel Council for Higher Education. The work at NICPB was supported by the European Research Council (ERC) under Grant Agreement No. 885413.
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P.A.V. and M.Y. contributed equally to this project. G.B. conceived and supervised the project. G.B. and M.Y. designed the experiment. I.F., H.L. and A.K. grew and characterized the crystals. M.Y. performed the spectroscopic experiments. M.Y and P.A.V. performed the data analysis and prepared the figures. All the authors contributed to discussions and writing the manuscript.
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Volkov, P.A., Ye, M., Lohani, H. et al. Critical charge fluctuations and emergent coherence in a strongly correlated excitonic insulator. npj Quantum Mater. 6, 52 (2021). https://doi.org/10.1038/s41535021003514
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DOI: https://doi.org/10.1038/s41535021003514
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