Abstract
Understanding the organizing principles of interacting electrons and the emergence of novel electronic phases is a central endeavor of condensed matter physics. Electronic nematicity, in which the discrete rotational symmetry in the electron fluid is broken while the translational one remains unaffected, is a prominent example of such a phase. It has proven ubiquitous in correlated electron systems, and is of prime importance to understand Febased superconductors. Here, we find that fluctuations of such broken symmetry are exceptionally strong over an extended temperature range above phase transitions in \({{{{{\rm{Ba}}}}}}{{{{{{\rm{Ni}}}}}}}_{2}{({{{{{{\rm{As}}}}}}}_{1x}{{{{{{\rm{P}}}}}}}_{x})}_{2}\), the nickel homologue to the Febased systems. This lends support to a type of electronic nematicity, dynamical in nature, which exhibits a particularly strong coupling to the underlying crystal lattice. Fluctuations between degenerate nematic configurations cause splitting of phonon lines, without lifting degeneracies nor breaking symmetries, akin to spin liquids in magnetic systems.
Introduction
The normal state of unconventional superconductors generally exhibits a variety of exotic electronic states emerging out of the interplay between intertwined orders. It is at least as intriguing as the superconducting state itself. The electronic nematic state is one such exotic state, which has proven particularly insightful in unveiling the properties of Febased superconductors. In these materials, nematicity is a canonical example of vestigial order which grows out of the magnetic fluctuations of two degenerate magnetic ground states^{1}. It has been proposed that the superconducting pairing is enhanced—even possibly mediated  by quantum critical nematic fluctuations^{2}, but their most prominent effect is rooted in their coupling to the crystal lattice. By softening the C_{66} shear modulus in e.g. Ba(Fe_{1−x}Co_{x})_{2}As_{2}, this coupling ultimately yields a lattice distortion at a structural tetragonaltoorthorhombic phase transition above the superconducting dome^{3,4}.
This coupling to the lattice changes the phonon spectra and dispersion, which in turn provides new routes to probe electronic nematicity. In the fluctuating regime, it was recently shown that the spatial dependence of the nematic fluctuations can directly be inferred from the softening of acoustical phonons^{5,6,7} at small but finite momentum (q ≠ 0). At the Brillouin zone center (q = 0), the largest effects are observed in the ordered phase, through the lifting of the degeneracy of the a and baxis polarized inplane vibrations of the square FeAs lattice with E_{g} symmetry. The resulting relative splitting Δω/ω of the modes in the orthorhombic phase can be as large as 8% in the Febased superconductors’ parent compounds such as BaFe_{2}As_{2}^{8,9,10} or EuFe_{2}As_{2}^{11}, exceeding by far the expectation based on the small orthorhombicity \(\delta=\frac{ab}{a+b} \sim 1{0}^{3}\). The much weaker effects reported in nonmagnetic FeSe^{12} suggest that the coupling of nematic degrees of freedom to the lattice in Febased superconductors primarily occurs through the spin channel rather than through the orbital one^{12}.
At room temperature, BaNi_{2}As_{2} has a tetragonal crystal structure (space group I4/mmm) similar to BaFe_{2}As_{2}, but unlike its Fecounterpart, it is superconducting, albeit below a modest critical temperature T_{c} ~ 0.6 K^{13}. While earlier electronic structure studies concluded low electronic correlations in this system, pointing at conventional phononmediated BCS superconductivity^{14,15}, more recent investigations advocate for an exotic normal state, which exhibits a manifold of charge density waves (CDW) instabilities and structural phase transitions interesting in their own right^{16,17,18,19,20,21,22,23}, and possible nematicdriven superconducting pairing^{24}. No longrange magnetic order has been reported so far, and it has been argued that the CDW plays a role similar to that of magnetism in the Febased superconductors^{19}, suggesting that BaNi_{2}As_{2} could be seen as a charge analogue of BaFe_{2}As_{2}.
Here we investigate the lattice and electron dynamics of \({{{{{\rm{Ba}}}}}}{{{{{{\rm{Ni}}}}}}}_{2}{({{{{{{\rm{As}}}}}}}_{1x}{{{{{{\rm{P}}}}}}}_{x})}_{2}\), and report on an exceptionally large splitting of the doubly degenerate Raman active planar vibrations of the NiAs tetraedra. In sharp contrast to the behavior in the ironbased systems, where this splitting was taken as evidence for nematic symmetry breaking, in BaNi_{2}(As_{1−x})P_{x})_{2} it occurs well above any reported structural phase transition temperatures. This calls for a distinction between the lifting of a degeneracy and a dynamical spectral splitting. We show that our observation can be accounted for by a particularly strong coupling of electronic B_{1g} nematic fluctuations, likely of orbital nature^{20}, to the lattice degrees of freedom in this material. This indicates that the tetragonal phase of BaNi_{2}As_{2} hosts an electronic nematic phase, dynamical in nature. We show that the broadening and splitting of the planar phonons can be described in terms of an entangled superposition of the two degenerate Isingnematic states that are coupled to a cloud of vibrational quanta. This bears analogies with the phenomenology of spin liquids—dynamical states without longrange magnetic order but longrange entanglement—and suggests in turn that similarly rich physics could be expected in such nematic liquids.
Results
From a point group analysis follows that the tetragonal phase of BaNi_{2}As_{2}hosts four Ramanactive optical phonons of A_{1g}, B_{1g}, and E_{g} symmetry at the Brillouin zone center. The corresponding eigendisplacements are shown in Fig. 1a. In this figure, we further report on room temperature Raman scattering measurements performed on BaNi_{2}As_{2} single crystals. The experiments were carried out in backscattering geometry with XZ, ZZ, XX and XY configurations, where the first (respectively second) letter refers to the orientation of the incident (resp. scattered) light polarization with respect to the axis of the tetragonal unit cell (Supplementary Note 3). All four Raman active optical phonon modes were detected. The A_{1g} mode is seen in the ZZ configuration at 172.9 cm^{−1}, as well as in the XX channel, where it partially overlaps with the B_{1g} mode, at 158.6 cm^{−1}. The two modes observed in the XZ channel are the doubly degenerate E_{g} modes referred to as E_{g,1} (41.4 cm^{−1}) and E_{g,2} (235.2 cm^{−1}). With the notable exception of the lowest E_{g,1} mode, these energies are in good agreement with the predictions of ab initio calculations (see Supplementary Note 3). These calculations also allowed us to estimate the strength of the electron–phonon coupling for the different modes and revealed that the phonon exhibiting the largest coupling is the A_{1g} mode, which consistently displays a weak Fano asymmetry. On the other hand, despite the rather modest calculated electron–phonon coupling, the E_{g,1} mode is very broad (fullwidthathalfmaximum (FWHM) ~ 22 cm^{−1}) at room temperature, indicating additional decay channels.
The singular behavior of the E_{g,1} phonon is confirmed upon cooling. The conventional behavior of phonon is exhibited by the A_{1g}(Fig. 1b) mode which harden and narrow at low temperatures. In contrast, the E_{g,1} mode initially softens upon cooling, starts broadening around T^{*} ~ 200 K before splitting at lower temperatures, where two peaks can be resolved; see Fig. 1c and Supplementary Note 3. Just above the firstorder transition^{13} to a triclinic phase at T_{Tri} = 133 K (on cooling), within which the phonon spectra qualitatively change (Supplementary Note 3 and Fig. 2), the splitting is as large as 22 cm^{−1}, that is, more than 50% of the mode’s original frequency.
The degeneracy of an E_{g} phonon can only be lifted if the fourfold symmetry of the Ni planes is broken. This occurs across the tetragonaltoorthorhombic structural transition in the Febased compounds Ba(Fe_{1−x}Co_{x})_{2}As_{2}^{8,9,10}, EuFe_{2}As_{2}^{11} or FeSe^{12}, where E_{g} modes split into B_{2g} and B_{3g} modes. The largest reported splitting in AFe_{2}As_{2} (A = Ba or Eu) is ~10 cm^{−1} (~8% of the mode frequency)^{8,9,10,11}, significantly larger than in FeSe (~2.6 cm^{−1})^{12}. In both cases, this splitting is already considered unusually large, in the sense that it exceeds the expectation based on the lattice distortion. The splitting in our measurements is quantitatively much larger and moreover onsets (with a broadening of the mode) at a temperature significantly higher than that at which the fourfold symmetry breaking takes place.
Before discussing the doping dependence of the effect in BaNi_{2}As_{2}, we briefly review the potential sources of symmetry breaking that could yield a splitting of the E_{g,1} phonon. As FeSe, BaNi_{2}As_{2}does not exhibit any magnetic order but a unidirectional, biaxial, incommensurate CDW (ICDW) above T_{Tri} has recently been reported^{16,19,20}. We performed a detailed temperaturedependent xray diffraction (XRD) investigation of the intensity of the CDW satellite at q_{ICDW} = (±0.28, 0, 0) (Fig. 2a, note that throughout the manuscript, we will only refer to reciprocal lattice vectors in the tetragonal unit cell). A very weak diffuse scattering signal can be tracked up to room temperature, but a strong increase of the peak intensity is only observed below ~155K. A secondorder phase transition consisting of an orthorhombic distortion of the lattice can be detected through highresolution dilatometry^{20} at ~142 K. In the parent compound, it also manifests itself as a minimum in the derivative of the resistance against temperature dR/dT, labeled T_{ρ} (Supplementary Note 2) in Fig. 2a. In contrast to more pronounced distortions, the identification of the twin structure associated with the structural change was limited in our XRD measurements to a broadening of high order Bragg reflections (e.g. (8,0,0)). This turns to our advantage as it allows us to put an upper bound on the corresponding lattice distortion \(\delta=\frac{ab}{a+b} \sim 1{0}^{4}\), an order of magnitude weaker than that reported in BaFe_{2}As_{2} and in good agreement with thermodynamic measurements^{21}.
Cooling further, the ICDW superstructure peak is suppressed in the triclinic phase in which a commensurate CDW (CCDW) signal develops at q_{CCDW} = (±1/3, 0, ±1/3). We did not detect any additional phase transition in the temperature range at which the E_{g,1} mode splitting onsets, and can already conclude at this stage that this splitting is occurring in the tetragonal I4/mmm phase. Our firstprinciple calculations confirmed that the amplitude of the orthorhombic structural distortion is in all cases much too small to account for the gigantic energy splitting of the E_{g,1}phonons reported here (Supplementary Note 3). Furthermore, in stark contrast to Febased materials^{8,12}, the E_{g,1} mode splitting increases linearly and does not show any sign of saturation down to T_{Tri}. In the same temperature range, a subtle broadening of the E_{g,2} mode (of much lower intensity) occurs.
Next, we confirmed the behavior of the E_{g} modes by studying the impact of arsenic substitution with phosphorus. This has previously been reported to suppress the triclinic transition^{21,25}, and to enhance the orthorhombic distortion^{20,21}. In Fig. 2, we show the results for \({{{{{\rm{Ba}}}}}}{{{{{{\rm{Ni}}}}}}}_{2}{({{{{{{\rm{As}}}}}}}_{1x}{{{{{{\rm{P}}}}}}}_{x})}_{2}\) with x = 3.5% (T_{Tri} = 95 K) and x = 7.6% (T_{Tri} = 55 K), for which we observe a similar splitting of the E_{g,1} mode, which increases linearly as temperature decreases reaching almost 30 cm^{−1} at T_{Tri} (~65% of the mode frequency). Upon further increase of the Pconcentration (x = 10%), for which the triclinic (and therefore the CCDW) transition is completely suppressed, the maximal amplitude of the splitting is reduced and appears to saturate at the lowest temperatures. In all doped samples, the strong increase of the ICDW satellite intensity at q_{ICDW} is smoother than in the parent compound and occurs at a temperature very close to T_{ρ}. The observation of the E_{g} phonon broadening and splitting above any lowering of the symmetry of the compound, strongly suggests a coupling of the mode to fluctuations. Given the lack of magnetism in BaNi_{2}As_{2}, orbital degrees of freedom are the most likely candidates. This can directly be tested using electronic Raman scattering, which is a particularly sensitive probe of the charge fluctuations. In Fig. 3, we show the temperature dependence of the electronic Raman response in the B_{1g} and B_{2g} channels for the x = 6.5% sample (for clarity, the Raman active phonon has been subtracted from the B_{1g} spectrum—Supplementary Note 3). In both channels, the electronic response consists of a broad continuum, extending up to 1500 cm^{−1}, akin to the particlehole excitations seen in many correlated metals^{26,27,28,29,30}. It apparently displays a conventional metallic behavior, with a smooth increase of the low frequency (≤250 cm^{−1}) response upon cooling, reflecting the decrease of the quasiparticle scattering rate Γ (which is inversely proportional to the slope of the Raman response, χ″(ω)/ω∣_{ω→0} in the static limit). The main difference between the two channels is quantitative: the low energy B_{1g} intensity gain spans over a broader energy range (≥500 cm^{−1}) than the B_{2g}, it is overall larger and accelerates significantly at temperatures where the E_{g,1} phonon splitting becomes evident, below ~140 K). This observation is in line with recent elastoresistivity measurements^{16,17} and can be interpreted as a signature of B_{1g} nematic fluctuations in BaNi_{2}As_{2}. In contrast to the electronic nematicity of the Febased superconductors, observed in the B_{2g} channel, the respective B_{1g} response in BaNi_{2}As_{2} appears overdamped, suggesting a strong coupling of the lattice to the nematic fluctuations.
Next, we develop a qualitative understanding of the ’splitting’ of the E_{g} phonon spectrum, in the absence of a structural phase transition. While the latter would lift the degeneracy and naturally cause such a splitting, we show how when it does not occur, the coupling of nematic fluctuations to the lattice can yield a spectral splitting of a doubly degenerated mode that bears striking similarities to our experimental observation.
We first focus on three phonon branches, a transverse acoustic (TA) mode that couples to nematic order and the two degenerate optical E_{g} modes with lattice Hamiltonian \({H}_{{{{{{\rm{latt}}}}}}}=\frac{1}{2}{\sum }_{{{{{{\boldsymbol{q}}}}}}\kappa }{\omega }_{{{{{{\boldsymbol{q}}}}}}\kappa }{a}_{{{{{{\boldsymbol{q}}}}}}\kappa }^{{{\dagger}} }{a}_{{{{{{\boldsymbol{q}}}}}}\kappa }\). Here, κ refers to the phonon branch of frequency ω_{qκ}. Since nematic fluctuations take place at small momenta, we ignore the momentum dependence of the optical modes \({\omega }_{{{{{{\boldsymbol{q}}}}}},{E}_{g}}\approx {\omega }_{0}\). The TA mode frequency \({\omega }_{{{{{{\boldsymbol{q}}}}}},{{{{{\rm{TA}}}}}}}={c}_{{{{\hat{{{\boldsymbol{q}}}}}}}}q\) possesses a directiondependent sound velocity \({c}_{{{{\hat{{{\boldsymbol{q}}}}}}}}\). For momenta \(\hat{q}\) along [100], [010], \({c}_{{{{\hat{{{\boldsymbol{q}}}}}}}}^{2}\) is proportional to the B_{1g}elastic constant \({C}_{{{{{{{\rm{B}}}}}}}_{1g}}={C}_{11}{C}_{12}\). In our consideration, we include electronic degrees of freedom with nematic character. Having in mind the absence of magnetic order in this system as well as recent observation of orbital fluctuations above the triclinic transition^{20}, suggest that orbital degrees of freedom are the most relevant here. A natural electronic object of proper symmetry is the orbital polarization \({O}_{i}={\sum }_{\sigma=\{\uparrow,\downarrow \}}\left({d}_{i\sigma,xz}^{{{\dagger}} }{d}_{i\sigma,xz}{d}_{i\sigma,yz}^{{{\dagger}} }{d}_{i\sigma,yz}\right)\) at lattice site i; or its Fourier transform in momentum space O_{q}. The Ni 3d_{xz} and 3d_{yz} orbitals are symmetry related in the tetragonal phase and broken nematic symmetry implies \(\left\langle {O}_{{{{{{\boldsymbol{q}}}}}}}\right\rangle=\left\langle O\right\rangle {\delta }_{{{{{{\boldsymbol{q}}}}}},{{{{{\boldsymbol{0}}}}}}}\,\ne\, 0\). In the following, we draw general conclusions that do not rely on the microscopic origin of such ordering (these go beyond the scope of this paper and will be developed in a subsequent work), nor of its microscopic nature providing that its symmetry allows a coupling to the relevant phonons.
The most direct coupling of the orbital polarization O_{q} to elastic modes occurs through TA phonons with displacements u_{x,y}
This interaction imposes that orbital ordering and the tetragonaltoorthorhombic transition occur simultaneously. Such coupling between a degenerate electronic state to TA phonons can in principle cause nematic order, akin to a cooperative Jahn–Teller effect^{31}. Even when the nematicity is primarily of electronic origin, the nematoelastic coupling of Eq. (1) will always increase the tendency towards nematic order^{32}, consistent with the JahnTeller argument. It leads in any event to a lattice softening near the transition and allows probing the nematic susceptibility via measurements of the elastic constant \({C}_{{B}_{1g}}\)^{3,32}, the elastoresistivity^{17} or the B_{1g}Raman response^{33}.
By symmetry, the coupling of the orbital polarization to optical E_{g} phonon modes is given by
In the case of a finite nematic order parameter \(\left\langle O\right\rangle \ne 0\), the degeneracy of the E_{g} phonons is lifted \({\omega }_{0}\to \sqrt{{\omega }_{0}^{2}\pm {g}_{{E}_{g}}\left\langle O\right\rangle }\), as the protecting symmetry is broken. The splitting of the squared frequencies is directly proportional to the order parameter.
While symmetry breaking is necessary to lift the degeneracy and to induce a nematic order parameter, it is not the only approach to achieve spectral splitting. Next, we show that this can indeed arise from a purely dynamical effect, which relates to the dynamic JahnTeller effect^{34}, albeit rather originating from fluctuations of electronic degrees of freedom than from a large zeropoint energy of vibronic modes. Similar phenomena have been reported in the halffield Holstein model^{35}.
In order to get a qualitative understanding of such a dynamic splitting we consider a simple toy model that directly follows from our above description. We assume that O_{i} behaves as an effective Ising variable \({O}_{i}={A}_{0}{\tau }_{i}^{z}\), with typical amplitude A_{0}. The Ising pseudospin states \(\left\Uparrow \right\rangle \) and \(\left\Downarrow \right\rangle \)—defining the basis for the Pauli matrix τ^{z}— describe the two orbital polarizations. Then the Hamiltonian takes the form
Here, \(\lambda={g}_{{E}_{g}}{A}_{0}/{\omega }_{0}^{2}\) is a dimensionless nematoelastic coupling constant. In the ordered state, the two modes become \({\omega }_{0}\to {\omega }_{0}\sqrt{1\pm \lambda \left\langle {\tau }^{z}\right\rangle }\) in agreement with the discussion above. Note that the model can be easily adapted to the coupling of B_{2g} nematicity, relevant for Febased superconductors, using u_{i,x}. u_{i,y} instead of \({u}_{i,x}^{2}{u}_{i,y}^{2}\) in the previous equation. This yields exactly the same results, albeit with a different strain dependence. From the splitting of the E_{g} phonons^{8} we can estimate λ ~ 0.08 for BaFe_{2}As_{2}, much weaker than λ ~ 0.7 of the Nisystem. To model quantum fluctuations, we introduced the last term that gives rise to tunneling processes between the two degenerate states^{36}. Using a variational approach^{37,38}, one can show that the tunneling rate can be thought of as a renormalized quantity \({{{\Omega }}}_{0}\to {{\Omega }}={{{\Omega }}}_{0}{e}^{\int\nolimits_{0}^{{\omega }_{c}}\frac{{{{{{\rm{Im}}}}}}{{\Gamma }}(\omega )}{{(\omega+{{{\Omega }}}_{0})}^{2}}{{{\rm{d}}}}\omega }\) rooted in a more complex dynamic nematic susceptibility of the form \({\chi }_{{{{{{\rm{nem}}}}}}}({{\Omega }})={[{{{\Omega }}}_{0}^{2}{\omega }^{2}+{{\Gamma }}(\omega )]}^{1}\), similar to other pseudospin problems^{39}.
The electronic degrees of freedom will now give rise to some coupling of the \({\tau }_{i}^{z}\) at different lattice sites, responsible for actual nematic order. If we assume a mean field description of the tetragonal phase, different lattice sites decouple. Still \({H}_{{{{{{\rm{latt,}}}}}}{E}_{g}}+{H}_{c,{E}_{g}}\), that describes the E_{g} phonons, is a manybody model of interacting pseudospin and lattice vibrations. It is possible to obtain an exact solution of the model with manybody eigenenergies:
where \(2{\epsilon }_{\pm }={\omega }_{0}(\sqrt{1+\lambda }\pm \sqrt{1\lambda })\), the indices m_{1,2} = 0, 1, 2, ⋯ refer to phonon occupations of the E_{g,x} and E_{g,y} modes, and s = ± 1 corresponds to the pseudospin states \(\left\pm \right\rangle=\frac{1}{\sqrt{2}}(\left\Uparrow \right\rangle \pm \left\Downarrow \right\rangle )\). From an analysis of the spectral functions follows that both phonon modes are degenerate and at T = 0 have two dominant peaks at ω_{0±} = E_{1,0,±} − E_{0,0,±} = E_{0,1,±} − E_{0,0,±} with splitting Δω = ω_{+} − ω_{−}; see Fig. 4a, b. For λ = 0, ω_{0+} = ω_{0−} = ω_{0}, i.e. there is no splitting. However, since the pseudospin splitting \({E}_{{m}_{1},{m}_{2},+}{E}_{{m}_{1},{m}_{2},}\) of the manybody eigenstates depends on the phononpopulation via \({\epsilon }_{}^{2}{({m}_{1}{m}_{2})}^{2}\), both E_{g}modes split in two main satellites by \({{\Delta }}\omega=\sqrt{{{{\Omega }}}^{2}+2{\omega }_{0}^{2}(1\sqrt{1{\lambda }^{2}})}{{\Omega }}\). For weak coupling λ ≪ Ω/ω_{0} (relevant for Febased systems) holds \({{\Delta }}\omega={\lambda }^{2}{\omega }_{0}^{2}/{{\Omega }}\) such that the splitting is small Δω/ω_{0} ≪ 1. However, as soon as λ is larger than Ω/ω_{0} we have Δω ≈ λω_{0} and the splitting of both E_{g}modes is of order unity (note that the system becomes unstable as λ → 1, which signals a structural instability).
To obtain a qualitative understanding of the origin of this behavior we use our earlier result for the static lifting \( \sim \lambda \left\langle {\tau }^{z}\right\rangle {\omega }_{0}\) of the degeneracy and replace it by \(\sqrt{{\lambda }^{2}\left\langle {({\tau }^{z})}^{2}\right\rangle }{\omega }_{0} \sim \lambda {\omega }_{0}\). This simply indicates that when the fluctuations between degenerate nematic configurations are strongly coupled and slow compared to the timescale of phonons (Ω ≲ ω_{0}), a split spectral structure for the phonons similar to that induced by static ordering can be obtained. In Fig. 4a, b we also show the behavior at finite T where several additional satellites enter the analysis, but the overall behavior remains unchanged. Finally, we can include externally applied stress σ_{ext} that explicitly breaks the fourfold symmetry via \({H}_{\sigma }=\!{\sum }_{i}{\sigma }_{{{{{{\rm{ext}}}}}}}{\tau }_{i}^{z}\). Now, the degeneracy of the two E_{g} phonons is lifted. We then observe merely a gradual transfer of weight between the split peaks, see Fig. 4c.
The precise nature of the orbital fluctuations and of the electronic state coupling to the phonon in BaNi_{2}As_{2} remains to be clarified experimentally. Nevertheless, it is possible to experimentally investigate the symmetrylifting for BaNi_{2}As_{2}, by performing measurements under strain, in a comparative study with FeSe. Quite generally, Febased superconductors are particularly soft and can be detwinned with very modest stress. This can be seen in a Raman experiment through the suppression in the intensity of one of the two degenerate B_{2g} or B_{3g} modes^{9,10,11,12}. Here, we used the approach proposed in ref. 40, gluing a BaNi_{2}As_{2} sample onto a glassfiber reinforced plastic substrate with the edges of the tetragonal unit cell aligned with the fibers. The resulting symmetry breaking strain is estimated to 0.4% at 150 K, which decreases the triclinic transition temperature by about 5K and yields a small but measurable shift of the A_{1g} phonon of ~0.5 cm^{−1} (Supplemmentary Information). In sharp contrast to the Febased compounds (Fig. 4f), the intensity ratio between the two E_{g} features barely changes in BaNi_{2}As_{2} (Fig. 4e), as we only a observe a small spectral weight transfer between the two satellites, in line with the above prediction, hereby confirming the dynamical nature of the nematic electronic phase of BaNi_{2}As_{2}.
Discussion
We summarize our findings on the phase diagram shown in Fig. 5 in which we report the doping dependence of the characteristic temperatures of the BaNi_{2}(As_{1−x})P_{x})_{2} system determined from a combination of XRD, Raman, resistivity and specific heat experiments. The main result of this study is the pronounced broadening and splitting of the E_{g} modes which occurs at temperatures significantly larger than that of static structural distortions and/or of the apparition of CDW orders. The effect is qualitatively and quantitatively very different from that associated with nematicity in Febased compounds. Indeed, rather than manifesting itself at a symmetry breaking phase transition or via longwavelength fluctuations (probed e.g. through the softening of elastic constants or Raman scattering in the symmetry channel of the nematic order parameter), the behavior of the E_{g} modes can be explained by a strong symmetryallowed coupling between the lattice and dynamic B_{1g} local nematic fluctuations visible even without phase transition. We argue that the E_{g} phonons follow the dynamics of an Ising variable, likely related to resonant transitions between distinct electronic orbital states, which causes a splitting in the phonon spectrum even in the absence of broken symmetry. These transitions can in principle be driven both by quantum and thermal fluctuations, and disentangling their effects is generally not trivial at finite temperature. We note however that the splitting of the E_{g} modes can be as large as ~30 cm^{−1}. This effectively corresponds to a temperature scale of ~45 K, above which fluctuations between the two configurations of the system can in principle be thermally driven. This is most likely the case for most of the investigated samples, with the notable exception of the one containing 10% of phosphorus, in which triclinic transition is completely suppressed and T_{c} is enhanced. We note that the splitting amplitude remains essentially temperature independent below ~50 K (Supplementary Note 3), where quantum fluctuations can in principle be expected to take the lead, calling for further investigation on this interesting regime.
We end our discussion by drawing an analogy with generalized liquids states, strongly correlated states that do not break a symmetry, such as the Fermi^{41}, orbital^{42} or the spin^{43} liquids. The regime of strong fluctuations between degenerate nematic states that splits the E_{g} phonons without breaking the rotational symmetry in BaNi_{2}(As_{1−x})P_{x})_{2} discussed here can therefore be understood as a nematic liquid. The case of spin liquids in which strong zeropoint fluctuations between degenerate configurations prevent longrange magnetic ordering^{43} might bear the strongest conceptual similarity with the present case. One can therefore expect that some of the phenomenology of the quantum spin liquid might carry over to nematic liquids and give rise to related phenomena, such as unconventional superconductivity or exotic quantum states with longrange entanglement. It strongly supports the view that the muchenhanced superconducting transition temperature of the studied materials upon doping is closely tied to the emergence of dynamic nematic fluctuations uncovered in our measurements.
Methods
Single crystal growth
Single crystals of \({{{{{\rm{Ba}}}}}}{{{{{{\rm{Ni}}}}}}}_{2}{({{{{{{\rm{As}}}}}}}_{1x}{{{{{{\rm{P}}}}}}}_{x})}_{2}\) were grown using a selfflux method. NiAs binary was synthesized by mixing the pure elements Ni (powder, Alfa Aesar 99.999%) and As (lumps, Alfa Aesar 99.9999%) that were ground and sealed in a fused silica tube and annealed for 20 h at 730 °C. All sample handlings were performed in an argon glove box (O_{2} content <0.7 ppm). For the growth of \({{{{{\rm{Ba}}}}}}{{{{{{\rm{Ni}}}}}}}_{2}{({{{{{{\rm{As}}}}}}}_{1x}{{{{{{\rm{P}}}}}}}_{x})}_{2}\), a ratio of Ba:NiAs:Ni:P = 1: 4(1 − x): 4x: 4x was placed in an alumina tube, which was sealed in an evacuated quartz ampule (i.e. 10^{−5} mbar). The mixtures were heated to 500 °C − 700 °C for 10 h, followed by heating slowly to a temperature of 1100 °C − 1150 °C, soaked for 5 h, and subsequently cooled to 995–950 °C at the rate of 0. 5 °C/h to 1 °C/h, depending on the phosphorus content used for the growth. At 950–995 °C, the furnace was canted to remove the excess flux, followed by furnace cooling. Platelike single crystals with typical sizes 3 × 2 × 0.5 mm^{3} were easily removed from the remaining ingot. The crystals were brittle having shiny brassyellow metallic lustre. Electron microscope analysis of the \({{{{{\rm{Ba}}}}}}{{{{{{\rm{Ni}}}}}}}_{2}{({{{{{{\rm{As}}}}}}}_{1x}{{{{{{\rm{P}}}}}}}_{x})}_{2}\) crystals was performed using a benchtop scanning electron microscope (SEM) (Supplementary Note 1). The energy dispersive xray (EDX) analysis on the \({{{{{\rm{Ba}}}}}}{{{{{{\rm{Ni}}}}}}}_{2}{({{{{{{\rm{As}}}}}}}_{1x}{{{{{{\rm{P}}}}}}}_{x})}_{2}\) crystals revealed phosphorus content x = 0.035 ± 0.005, 0.076 ± 0.005, and 0.10 ± 0.005.
Single crystal Xray diffraction
The phosphorus concentrations of the investigated samples were further confirmed by structural refinement from xray diffraction at room temperature using STOE imaging plate diffraction system (IPDS2T) equipped with Mo Kα radiation^{20}. Detailed temperature dependencies of the I and CCDW superstructure peaks were obtained using a four circle diffractometer. The samples were cooled under vacuum in a DE202SG/700K closedcycle cryostat from ARS, surrounded by a Beryllium dome. The incoming beam was generated from a Molybdenum Xray tube with a voltage of 50 kV and a current of 40 mA. The beam was collimated and cleaned up by a 0.8 mm pinhole before hitting the samples. We specifically followed the superstructure reflections close to the (4, 1, 1) and (1, 0, 3) Bragg peaks for the I and CCDW, respectively.
Polarizationresolved confocal Raman scattering
Confocal Raman scattering experiments were performed with a JobinYvon LabRAM HR Evolution spectrometer in backscattering geometry, with a laser power of ≤0.8 mW that was focused on the sample with a ×50 magnification longworkingdistance (10.6 mm) objective. The laser spot size was ≈2 μm in diameter. Lowresolution mode (1.54 cm^{−1}) of the spectrometer with 600 grooves/mm was used to maximize the signal output. For the phonon measurements, a He–Ne laser (λ = 632.8 nm) was used as the incident source, whereas the electronic background was best observed using the 532 nm line of a Nd:YAG solid state laser. Direct comparison of the structural phase transition temperature upon cooling as measured in specific heat and Raman indicated a laserinduced heating limited to less than 2K. The Raman spectra were Bose corrected and the phonons analyzed using a damped harmonic oscillator profile (with the exception of the A_{1g} mode that displayed a Fano asymmetry and was treated accordingly).
Additional details on the experiment, phonon calculations or on the theoretical model presented here are given in Supplementary Notes 3 and 4.
Data availability
The Raman data reported in this study have been deposited at the KIT Open, under the following identification number KITopenID: 1000148276. The data that support the findings of this study are available from the corresponding author, Matthieu Le Tacon, upon reasonable request.
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Acknowledgements
The authors are grateful to P. Abbamonte, A. E. Boehmer, J. Demsar, F. Hardy, C. W. Hicks, and J. Paglione and L. Wang for valuable discussions. We further thank Y. Li for allowing us to reproduce his data in Fig. 4f. The contribution from M.M. was supported by the Karlsruhe Nano Micro Facility for Information (KNMFi). We further acknowledge the support of the KNMFi and Dr. Torsten Scherer for the EDX measurements. We acknowledge the funding by the Deutsche Forschungsgemeinschaft (DFG; German Research Foundation) ProjectID 422213477TRR 288 (Projects B01, B03, and A02) and support by the state of BadenWürttemberg through bwHPC. K.W. acknowledges funding by the SNSF through a postdoc mobility fellowship. S.M.S. acknowledges funding by the Deutsche ForschungsgemeinschaftProjektnummer 441231589.
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M.L.T. conceived and supervised the project. Y.Y. and T.L. acquired and analyzed the Raman scattering data. A.A.H. and T.L. grew the single crystals. T.L., S.M.S., F.W., and M.M. carried out XRD experiments and analysis. M.F. performed transport experiments. C.M. performed dilatometry experiments. K.W. performed specific heat experiments. R.H. performed firstprinciple calculations and R.W. and J.S. developed the theoretical model. M.L.T., R.W., and J.S. wrote the manuscript with inputs from all the coauthors.
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Yao, Y., Willa, R., Lacmann, T. et al. An electronic nematic liquid in BaNi_{2}As_{2}. Nat Commun 13, 4535 (2022). https://doi.org/10.1038/s41467022321127
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DOI: https://doi.org/10.1038/s41467022321127
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