Lattice uctuation induced pseudogap in quasi-one-dimensional Ta2NiSe5

: In conventional solid-state systems, the development of an energy gap is often associated with a broken symmetry. However, strongly correlated materials can exhibit energy gaps without any global symmetry breaking -- the so-called pseudogap, most notably in the Mott insulating state 1 and the fluctuating superconducting or charge density wave states 2-3 . To date, lattice induced pseudogap remains elusive. With angle-resolved photoemission spectroscopy (ARPES) and single crystal x-ray diffraction, we identify a pseudogap in the quasi-1D excitonic insulator candidate Ta 2 NiSe 5 . Strong lattice contribution is revealed by the pervasive diffuse scattering well above the transition temperature and the negative electronic compressibility in the pseudogap state. Combining first-principles and microscopic model calculations, we show that inter-band electron-phonon coupling can create fluctuating phonon-mediated electron-hole pairing or hybridization. This suppresses the spectral weight on the Fermi surface, causing a metal-to-insulator-like transition without breaking the global symmetry. Our work establishes the precedence of a pseudogap with a lattice origin, highlighting Ta 2 NiSe 5 as a room-temperature platform to study lattice-induced charge localization and low dimensional fluctuations.


Main:
In mean-field theory, a second-order phase transition is accompanied by spontaneous symmetry breaking. Informed by systems' symmetry, the phase transition of a macroscopic system with ~10 23 particles can be described by only a few parameters, namely the order parameters 4 . In many electronic transitions, an energy gap can be used as a liaison to the order parameter ( Fig.1(a), grey line) [5][6] . However, energy gaps can also occur without breaking any global symmetry 2-3,7-8 , most notably the enigmatic "pseudogap" in the high-Tc cuprate superconductors due to strong Coulomb interactions or superconducting fluctuations [9][10][11] . These non-symmetry-breaking gaps are believed to originate from strong electronic correlations and the lattice contribution is considered at most indirect [12][13][14] . Non-symmetry-breaking gaps are also observed in low-dimensional charge density wave (CDW) systems, where fluctuating electron density modulations arise at a given spatial frequency [15][16][17][18] . However, it remains unclear if lattice fluctuation itself is sufficient to form localized electron-hole pairs and drive a metal-to-insulator transition (MIT) without breaking the global symmetry.
In the pursuit of a potential room-temperature Bose-Einstein condensate, recent studies revealed evidence for exciton formation, strong lattice instability and electron-phonon coupling in quasi-1D ternary chalcogenide Ta2NiSe5 [19][20][21][22] . Upon warming, a q = 0 monoclinic-to-orthorhombic structural transition happens at Ts~329K, above which a semi-metallic electronic structure is supposedly restored and protected by the mirror symmetry of Ta chains about the Ni chain 23 .
However, an insulating behaviour persists up to 550K 19 . This is in striking contrast to archetypal MIT systems such as the perovskite nickelates 24 and chain compound TTF-TCNQ 25 , where the structural normal state often occurs in sync with the metallic electronic normal state. To date, the nature of the high-temperature electronic state in Ta2NiSe5 remains controversial [26][27][28][29][30][31] . In addition, contentions remain regarding whether the system's ground state hosts an exciton condensate [32][33][34][35][36][37] .
To examine possible lattice fluctuation induced insulating states and resolve the controversy, we systematically investigate the system's electronic and lattice degrees of freedom, tuned across the transition via both thermal and nonthermal methods.

High-temperature lattice instability:
Ta2NiSe5 crystallizes in a layered structure stacked with van der Waals interactions. Within each layer, the Ta and Ni atoms form a chain structure along the a-axis of the crystal (Fig. 1(c), inset). The system undergoes a structural phase transition from a high-temperature orthorhombic Cmcm phase to a low-temperature monoclinic C2/c phase at Ts~329K 19 , indicated in both resistivity and specific heat measurements ( Fig. 1(b)), consistent with previous reports 34 . Crossing the transition, the Ta atoms slightly shear along the chain direction, resulting in an increase of the β angle from 90° to 90.53° (as exaggerated in Fig 1(c)) 19 . The second order nature of the transition is indicated by the gradual separation of |Q(1 3 4)| and |Q(-1 3 4)| peaks measured by the highresolution synchrotron X-ray diffraction ( Fig.1(c) and Extended Data Fig.1(a)). In contrast, a dramatically enhanced thermal diffuse scattering signal is observed at and above Ts ( Fig. 1(d)), indicating the presence of extensive phonon softening. Strong lattice fluctuation is also reflected by a pronounced decrease of the (2 0 0) peak intensity (Extended Data Fig. 1(b)) in a broad temperature range above Ts, which is mostly affected by the squared average of the atomic displacement along the chain direction. The observed strong lattice fluctuation above Ts aligns with the presence of soft transverse acoustic 38 and anharmonic optical modes [20][21][22] .

High-temperature pseudogap state:
Given the persistent insulating behaviour above Ts, we investigate the electronic structure with high-resolution ARPES along the -X high symmetry direction (geometry in Extended Data Fig. 2) across the transition temperature. Here, the high statistics and energy resolution enable spectra restoration up to ~150 meV above the Fermi level (EF) after resolution-convolved Fermifunction division 11 . As illustrated in Fig. 2(b), the low temperature spectra (T<Ts) show a pronounced single-particle gap, where the dispersion and orbital composition are consistent with density functional theory (DFT) calculations ( Fig. 2(a)). In contrast, the high-temperature spectra (T>Ts) deviate from the calculated band structure in that a pronounced spectral weight depletion is observed around EF (±100meV) despite the disappearance of the low temperature flat valence band top ( Fig. 2(b) and Extended Data Fig. 3(a-b)). Such a strong intensity depletion cannot be addressed in the generic band theory: drastic orbital character change alone cannot account for the missing spectral weight in both Ta and Ni states, evidenced in both photon polarization channels. The optical matrix-element effect is also unlikely given the abruptness of the intensity drop in momentum space, as well as robust spectral weight depletion seen across a broad range of photon energies and Brillouin zones (Extended Data Fig. 3(c-d)). Such persistent spectral depletion at EF naturally accounts for the insulating behaviour in resistivity and optical conductivity above Ts. the structure-related band reconstruction at non-zero binding energy tracks the order parameter within a relatively small temperature range below Ts (Fig. 2(d)) 19 . On the other hand, the zeroenergy spectral weight evolves continuously from 120K to 380K with little influence from the structural transition. Clearly, a much higher temperature is needed to fully restore the expected spectral weight of a gapless electronic normal state, consistent with resistivity data where metallicity is only recovered above 550K 19 . This single-particle "gapped" state does not require symmetry reduction of the orthorhombic lattice structure, and is analogous to the pseudogap found in cuprates. Its occurrence correlates with the pronounced lattice fluctuations observed in the same temperature range.

Low-temperature carrier doping dependence:
While the link between lattice fluctuations and the high-temperature pseudogap is established above Ts, the role of the lattice in the broken-symmetry ground state requires nonthermal tuning to clarify. Pump-probe experiments have achieved higher electronic or lattice temperatures, but contentions remain as to which channel determines the eventual restoration of metallicity [26][27][28][29] . Recently, surface potassium dosing is shown to be an effective nonthermal method to restore the gapless state at low temperatures 39-40. Through the combined use of a micron-spot synchrotron ARPES and in-situ potassium dosing in the low temperature broken-symmetry state, we achieved an unprecedented level of quantification that is necessary to examine the thermodynamic stability of the ground-state electronic subsystem. Here, the relative potassium dosage is accessed through the core-level X-ray photoelectron spectroscopy (XPS) of K 3p orbital, which corresponds to the electronic normal state at sufficiently high temperatures achieved in the high-fluence pump-probe study 28 .
Furthermore, the low-energy electronic structure shows a striking correspondence to an anomalous chemical potential evolution. Here, the chemical potential change is evaluated from the energy shift of the fully filled Ta 4f core-level ( Fig. 3(b-c)), and cross referenced with the Se valence band shift (dashed line in Fig. 3(a)) to rule out external contributions 41 . Negative electronic compressibility (NEC), where the energy of the electronic subsystem decreases despite the addition of electrons, appears when the pseudogap is being replenished in the second stage ( Fig. 3(d), green region). Considering that purely electronic systems only exhibit repulsive interactions to added electrons, such an NEC behaviour requires the involvement of additional degrees of freedom, such as the fluctuating lattice, in order to maintain the thermodynamic stability of the full system.

Microscopic explanation via inter-band electron-phonon coupling:
Exciton condensate formation was proposed as a viable route toward MIT soon after the success of the BCS theory of superconductivity 7 . In Ta2NiSe5, one leading contention is whether it is electron-phonon coupling or direct electron-hole Coulomb attraction that dominates the energy gap in the insulating state. We found that both the thermal and nonthermal evolution of the spectral function can be captured using a minimal model (Method and Extended Data Fig. 6) with symmetry-informed electron-phonon coupling. Following the symmetry of the monoclinic instability of the high-temperature orthorhombic lattice, a long-wavelength optical phonon, whose displacement breaks the local orthorhombic symmetry, couples the conduction band bottom (blue) to the valence band top (red) (Fig. 4(a), left) 42 . Enhanced by the low dimensionality, strong lattice fluctuation above Ts readily enables the band hybridization, and results in the electronic pseudogap without causing a global symmetry breaking ( Fig.4(a), middle). This reproduces the pseudogap spectra above Ts (Fig. 2(b)). Further reducing temperature results in a divergence of the phonon number (Extended Data Fig. 7), which signifies the phonon condensation and drives the transition to the monoclinic phase. Subsequently, a hard hybridization gap forms below Ts (Fig.4(a), right).
On the other hand, increasing electron carrier density lifts the Fermi level from the charge-neutral point and reduces the inter-band phonon dressing. Thus, doping ultimately drives the monoclinic phase into the orthorhombic phase even at low temperatures, which is quantitatively confirmed by both the many-body lattice model and first-principles calculation (Extended Data Fig. 8-9). The observed NEC also reflects strong electron-phonon coupling [43][44] (Extended Data Fig. 9), since the additional electronic energy from the added carriers is absorbed into the lattice degree of freedom.
Therefore, the phase diagram of Ta2NiSe5 along both the temperature and electron-doping axes can be sketched in Fig. 4(b), where the region shaded in green represents the lattice fluctuation induced pseudogap state above Ts. Clearly, both thermal and quantum fluctuations effects are strong.
On the other hand, without the phonon breaking the lattice mirror symmetry, direct prototypical inter-band Coulomb interaction alone can only repel the conduction and valence bands in their entirety, and is unable to create a hybridization gap or excite electrons from Ta to Ni orbitals 23 (Extended Data Fig.10(a)). Even in the presence of lattice-induced gap, Coulomb interaction up to 200 meV can only marginally increase the gap size, and severely weakens the conduction band bottom spectral intensity due to much reduced band overlap (Extended Data Fig.   10(b)).

Discussion and outlook:
While electron-phonon coupling is a symmetry prerequisite and a dominant component in the low temperature energy gap, the strength of direct electron-hole Coulomb attraction V may also be estimated from the valence-conduction band overlap in the semi-metallic state. Such band overlap ( Fig. 4(a)) will be linearly reduced with increasing V and total electron-hole population ne + np (Extended Data Fig. 10(a)). By comparing the high temperature spectra ( ≅ 240 meV, Extended Fig.3(b)), the ultrafast photodoped spectra ( ≅ 260 meV, photodoping enhanced ne + np) 28 , and heavily K-dosed spectra ( ≅ 260 meV, heavily screened V, Extended Data Fig. 4, cycle 19), an upper bound of ~70meV can be placed for V (see Method). This electronic Coulomb interaction would incur less than 5% enhancement to the low temperature gap (Extended Data Fig.   11), consistent with the earlier identification of a mostly lattice-induced insulating state in Ta2NiSe5.
In the pseudogap phase without any global structural symmetry lowering over an extensive temperature range above Ts, phonons cannot be treated under the Born-Oppenheimer approximation, where the lattice's impact on the electronic structure is only reflected by a nonzero lattice distortion 〈 〉. Instead, phonons act on the electronic structure through a fluctuating state where 〈 〉 = 0 but 〈 2 〉 ≠ 0 ( Fig.1(a) middle inset and Fig. 4(b)). In this regard, the pseudogap state may be conceptually likened to the 'preformed excitons' proposed in recent studies 31  Excess iodine was removed from the surfaces of the crystals with ethanol.

Physical properties measurement
Electric resistivity and heat capacity measurements were carried out by using a commercial PPMS (Quantum Design). The electric resistivity was measured by the four-probe method with the current applied in the ac-plane of a Ta2NiSe5 single crystal. The specific heat measurement was performed in the temperature range from 200K to 400K where the background signal was recorded in the same temperature range.

Angle-resolved photoemission spectroscopy (ARPES)
Synchrotron-based ARPES measurements were performed at beamline BL5-2 of Stanford Synchrotron Radiation Laboratory (SSRL), SLAC, USA, and BL 7.0.2 of Advanced Light Source (ALS), USA. The samples were cleaved in situ and measured under ultra-high vacuum below 3×10 -11 Torr. Data was collected by R4000 and DA30L analyzers. The total energy and angle resolutions were 10 meV and 0.2°, respectively.

Thermal diffuse scattering
Hard X-ray single-crystal diffraction is carried out at the energy of 44 keV at beamline QM2 of the Cornell High Energy Synchrotron Source (CHESS). The needle-like sample is chosen with a typical lateral dimension of 100 microns, then mounted with GE Varnish on a rotating pin before being placed in the beam. A Pilatus 300M 2D detector is used to collect the diffraction pattern with the sample rotated 360 degrees around three different axes at 0.1° step and 0.1s/frame data rate at each temperature. The full 3D intensity cube is stacked and indexed with beamline software.

First-principles DFT calculation
Ab initio calculations are performed using the Quantum ESPRESSO package 46 . The structural relaxation is calculated using the r 2 SCAN 47 functional with a semiempirical Grimme's DFT-D2 Van-der-Waals correction 48 . A 30×30×15 k-mesh was used with a 100 Ry wavefunction energy cutoff.

Full many-body model simulations allowing fluctuations
We perform many-body model simulations to address the impact of electron-phonon coupling and lattice fluctuations. According to the fitting with experimental results, we map the electronic structure Ta2NiSe5 into a two-band model (Extended data Fig. 6 The direct hybridization between the conduction and valence bands is forbidden in the orthorhombic phase by the inversion symmetry 23 . However, this hybridization is enabled by an anti-symmetric lattice distortion, parametrized as a momentum-dependent displacement and quantized as the phonon mode = + − † . As the Fermi momentum kF is much smaller than 2 / 0 , we further restrict the e-ph coupling to the zone centre = . Therefore, other phonon modes are decoupled from electrons and can be ignored. The phonon frequency is chosen to be 2THz [20][21][22] . To generate the simulations in Fig.4 of the main text, we set the coupling strength = 70 meV and Coulomb interaction = 0. We perform an exact diagonalization (ED) simulation on an 8-site cluster (effectively 16 sites due to the two bands). The diagonalization of the Hamiltonian leads to the ground-state wavefunction | ⟩ and all excited states (denoted as | ⟩ , with | = 0⟩ = | ⟩ ). The finitetemperature spectral function, for each band, is calculated through a canonical ensemble average where is the partition function. The ensemble-averaged equal-time observables, e.g., the average phonon occupation, are defined in a similar manner To capture the high momentum resolution comparable to experiments, we employ the twisted average boundary condition (TABC) in the simulation, with 50 equal-spacing phases for the spectral simulation and 30 phases for the equal-time observables. In addition to achieving the momentum resolution, the TABC is known to reduce the finite-size effects, particularly for the model with = 0 interactions 49 . The ED simulation provides the solution of the full many-body state in 1D and preserves reflection symmetry, i.e. ⟨ ⟩ ≡ 0 ; however, the fluctuation of displacements ⟨ 2 ⟩ ≠ 0, reflecting the phonon squeezed state.

Mean-field simulations for the symmetry breaking state at T < Ts
To reflect the spectral properties with the presence of symmetry breaking, we simulate the spectral function by projecting the phonon fluctuations to a classical displacement. Specifically, we assume that the ground state collapses to a symmetry-breaking state characterized by a finite . Treating Using the ⟨ 2 ⟩ obtained by ED simulations for various temperatures, we estimate the mean-field by = √⟨ 2 ⟩ and evaluate the spectral functions for the corresponding symmetry-broken states.

Estimation of the upper bound of the Coulomb interaction
The strength of direct electron-hole Coulomb attraction V is estimated from the valence-conduction band overlap in the semi-metallic state.
If the system has a finite inter-band Coulomb interaction in the form of the Hamiltonian According to the Luttinger volume, the doped electrons for the 15 th -dosing (Fig. 3(c)) reaches ~0.4 per unit cell, indicating (Δ + Δ )/ = 0.46. As the Fermi velocity is comparable in two bands, it leads to (Δ − Δ )/ ≈ 0.152. Therefore, with an upper bound of Δ ∼ 20meV, we obtain V < 67meV. Such a Coulomb interaction is too small to address the spectral gap (over 200meV) at the low temperature gapped state. Therefore, from a quantitative perspective independent from the symmetry analysis, we conclude that the Coulomb interaction does not play a major role in the observed insulating state.

Data availability
The data will be deposited to an open-access Yale repository and released upon final publication.