Local orthorhombic lattice distortions in the paramagnetic tetragonal phase of superconducting NaFe{}_{1-x}Ni{}_{x}As

# Local orthorhombic lattice distortions in the paramagnetic tetragonal phase of superconducting NaFe$_{1-x}$Ni$_x$As

## Abstract

Understanding the interplay between nematicity, magnetism and superconductivity is pivotal for elucidating the physics of iron-based superconductors. Here we use neutron scattering to probe magnetic and nematic orders throughout the phase diagram of NaFeNiAs, finding that while both static antiferromagnetic and nematic orders compete with superconductivity, the onset temperatures for these two orders remain well-separated approaching the putative quantum critical points. We uncover local orthorhombic distortions that persist well above the tetragonal-to-orthorhombic structural transition temperature in underdoped samples and extend well into the overdoped regime that exhibits neither magnetic nor structural phase transitions. These unexpected local orthorhombic distortions display Curie-Weiss temperature dependence and become suppressed below the superconducting transition temperature , suggesting they result from a large nematic susceptibility near optimal superconductivity. Our results account for observations of rotational symmetry-breaking above , and attest to the presence of significant nematic fluctuations near optimal superconductivity.

12

## I introduction

Iron pnictide superconductors are a large class of materials hosting unconventional superconductivity that emerges from antiferromagnetically ordered parent compounds [Fig. 1(a)]. Unique to the iron pnictides is a tetragonal-to-orthorhombic structural transition at , where the underlying lattice changes from exhibiting four-fold () above to two-fold () rotational symmetry below , that occurs either simultaneously with or above the antiferromagnetic (AF) phase transition temperature [Fig. 1(b)] stewart (); dai (). The large electronic anisotropy present in the paramagnetic orthorhombic phase has been ascribed to an electronic nematic state JHChu2010 (); JHChu2012 (); RMFernandes2014 () that couples to shear strain of the lattice; the orthorhombicity [, where and are in-plane orthorhombic lattice parameters] therefore acts as a proxy for the nematic order parameter. In the paramagnetic tetragonal state, the nematic susceptibility can be measured via determining the resistivity anisotropy induced by anisotropic in-plane strain HHKuo2016 () or by measuring the elastic shear modulus Yoshizawa (); AEBohmerCRP (). By fitting temperature dependence of the nematic susceptibility with a Curie-Weiss form, a nematic quantum critical point (QCP) with the Weiss temperature has been identified near optimal superconductivity for different iron-based superconductors HHKuo2016 (); AEBohmerCRP (). Theoretically, the proliferation of nematic fluctuations near the nematic QCP can act to enhance Cooper pairing Tmaier (); Metlitski (); SLederer (); SLederer17 ().

Although symmetry-breaking is typically associated with the structural transition at , there are numerous reports of its observation well above and in overdoped compounds Kasahara (); TSonobe (); RZhou2016 (); EThewalt (); XRen (); EPRosenthal (); SLiu (). These observations are either reflective of an intrinsic rotational-symmetry-broken phase above , which can occur in the bulk Kasahara (); TSonobe (); RZhou2016 () or on the surface of the sample EThewalt (), or simply result from a large nematic susceptibility XRen (); EPRosenthal (); SLiu (); HHKuo2013 (). In the first case, there is a small but non-zero nematic order parameter throughout the material above , although no additional symmetry-breaking occurs below despite the sharp increase of the nematic order parameter. For the latter scenario, only local orthorhombic distortions can be present and the system remains tetragonal on average. One way to differentiate the two scenarios is to directly and quantitatively probe the distribution of the inter-planar atomic spacings (-spacings) and its temperature dependence.

Ideally, when the system becomes orthorhombic, two different in-plane -spacings corresponding to different in-plane lattice parameters can be resolved; on the other hand, when there are only local orthorhombic distortions, the -spacing distribution only broadens while the average structure remains tetragonal [Fig. 1(c)]. However, experimentally it can be very difficult to distinguish the two scenarios when is too small for a splitting to be resolved, then a broadening is also seen even when the system goes through a tetragonal-to-orthorhombic phase transition. In such cases, it is more instructive to examine the temperature dependence of the experimentally obtained broadening, characterized either by or by the width of the -spacing distribution, [Fig. 1(c)]. For a phase transition, the broadening should exhibit a clear order-parameter-like onset; for local orthorhombic distortions in an average tetragonal structure, the broadening instead tracks the nematic susceptibility, which exhibits a Curie-Weiss temperature dependence JHChu2012 () [Fig. 1(c)]. An additional complication is that AF order typically becomes spin-glass-like and sometimes incommensurate near the nematic QCP DKPratt2011 (); APDioguardi2013 (); XYLu2013 (); DHu2015 (); GTan2016 (), and given the strong magnetoelastic coupling in the iron pnictides RMFernandes2014 (); AEBohmerCRP (), it is unclear how such changes in AF order affect the nematic order.

In this work, we use high-resolution neutron diffraction and neutron Larmor diffraction to map out the phase diagram of NaFeNiAs DRParker2010 (), focusing on the interplay between magnetic order, nematic order, and superconductivity near optimal superconductivity. Unlike most other iron pnictide systems, we find in NaFeNiAs to be continuously suppressed towards near optimal doping, while the order remains long-range and commensurate. This allows us to demonstrate that and in NaFeNiAs remain well-separated near optimal superconductivity, indicating distinct quantum critical points associated with nematic and AF orders similar to quantum criticality in electron-doped BaFeNiAs Rzhou (). Utilizing the high resolution provided by neutron Larmor diffraction Martin11 (); xylu16 (), we probed the nematic order parameter in underdoped NaFeNiAs below and surprisingly, uncovered local orthorhombic distortions well above and in overdoped samples without a structural phase transition. Although the average structure is tetragonal in these regimes, broadening of the -spacing distribution is unambiguously observed. Such local orthorhombic distortions were hinted at in previous high-resolution neutron powder diffraction measurements on electron-overdoped NaFeCoAs, where a small broadening of Bragg peaks at low temperature was observed DRParker2010 (). Regardless of whether orthorhombic distortions are long-range due to a structural phase transition or local in nature, resulting from a large nematic susceptibility, we find that they become suppressed inside the superconducting state similar to AF order. Our results therefore elucidate the interplay between AF order, nematicity, and superconductivity in NaFeNiAs; at the same time, our observation of local orthorhombic distortions with a Curie-Weiss temperature dependence across the phase diagram accounts for rotational symmetry-breaking seen in nominally tetragonal iron pnictides. In addition, our measurements demonstrate that neutron Larmor diffraction can be used to determine the nematic susceptibility of free-standing iron pnictides without the need to apply external stress or strain. These results should stimulate future high-resolution neutron/X-ray diffraction work to study orthorhombic lattice distortion and its temperature dependence in the nominally tetragonal phase of iron-based superconductors.

## Ii Results

### ii.1 Overall phase diagram of NaFe1−xNixAs

Our results are reported using the orthorhombic structural unit cell with lattice parameters and for NaFeAs DRParker2009 (); sli (). The momentum transfer is denoted as in reciprocal lattice units (r.l.u.) with , and . In this notation, magnetic Bragg peaks are at with . Samples were mostly aligned in the scattering plane, which allows scans of magnetic peaks along and ; the sample was also studied in the plane. We have carried out neutron diffraction, neutron Larmor diffraction, and inelastic neutron scattering experiments on NaFeNiAs (see Methods section for experimental details).

Figure 1(d) shows the overall phase diagram determined from our experiments, with , , and marked. Although for optimal-doped and overdoped regimes the samples on average exhibit a tetragonal structure at all temperatures, there are local orthorhombic distortions resulting in broadening of -spacing distribution that can be characterized by or . The orthorhombic distortion is plotted in a pseudo-color scheme as a function of temperature and doping near optimal doping in Fig. 1(d). Figure 1(e) shows the Ni-doping dependence of the ordered magnetic moment and at K, and for superconducting samples. With increasing Ni-doping , the AF ordered moment and decrease monotonically, and no magnetic order is detected in the sample [Fig. 1(e)]. While the magnetic order parameter for the sample resembles that of NaFeAs [Figs. 2(e) and 2(f)], magnetic order becomes strongly suppressed upon entering the superconducting state for [Fig. 2(g)], similar to other iron pnictides DKPratt2009 (); ADChristianson2009 ().

### ii.2 Re-entry into the paramagnetic state in NaFe1−xNixAs with x=0.012

For the sample, magnetic order onsets at K and becomes strongly suppressed upon entering the superconducting state below and re-enters into the paramagnetic state without any long-range order below K [Fig. 2(h)]. Given the sharp superconducting transition at (Supplementary Fig. 1 and Methods section), is well inside the superconducting state. This is similar to the behavior of nearly-optimal-doped Ba(FeCo)As RMFernandes2010 (), although AF order in Ba(FeCo)As is short-range and incommensurate DKPratt2011 (). To confirm that the magnetic order in our sample is long-range and commensurate, we carried out scans along the , and directions in and scattering planes [Fig. 2(a)], with results summarized in Figs. 2(b)-2(d). As can be seen, magnetic order remains long-range along all three high-symmetry directions (with spin-spin correlation lengths exceeding ) for the sample near optimal superconductivity, before disappearing near . These wave-vector scans also confirm the complete disappearance of long-range magnetic order below . For comparison, we note that magnetism in electron-doped Ba(FeCo)As (6%) DKPratt2011 (), BaFeNiAs (5%) XYLu2013 (), and NaFeCoAs (2.3%) GTan2016 () exhibits cluster spin glass and incommensurate magnetic order near optimal superconductivity likely related to impurity effects APDioguardi2013 (); MGKim2012 (). The absence of such behavior in NaFeNiAs is likely a result of significantly lower dopant concentration in NaFeNiAs (1.3%) near optimal doping. Our inelastic neutron scattering measurements on the sample confirm the presence of a neutron spin resonance, which can act as a proxy for the superconducting order parameter, is unaffected when cooled below (Supplementary Fig. 2 and Methods section).

### ii.3 Nematic order and local orthorhombic distortions in NaFe1−xNixAs

Having established the evolution of AF order and its interplay with superconductivity in NaFeNiAs, we examined the Ni-doping evolution of the nematic order in NaFeNiAs. To precisely determine the evolution of the orthorhombic distortion, we used high-resolution neutron diffraction and neutron Larmor diffraction to investigate the temperature evolution of the orthorhombic lattice distortion (Supplementary Figs. 3 and 4 and Methods section). For NaFeNiAs with , we can see clear orthorhombic lattice distortion below , also confirmed by anomalies in the temperature dependence of electrical resistivity measurements (Supplementary Fig. 5 and Methods section). Figures 3(a), 3(b), and 3(c) show temperature and Ni-doping dependence of the orthorhombic distortion . For NaFeNiAs with at temperatures above , and for at all temperatures, the system is on average tetragonal and should in principle have . Surprisingly, we see clear temperature-dependent . Moreover, while below behaves as expected for an order parameter associated with a phase transition, in temperature regimes with an average tetragonal structure exhibits a Curie-Weiss temperature dependence, suggesting it arises from local orthorhombic distortions. In all cases, we find that decreases dramatically below , indicating that orthorhombic distortion, whether long-range or local, competes with superconductivity. The competition between superconductivity and long-range nematic order is similar to Ba(FeCo)As SNandi () and can be captured by a phenomenological Landau theory based on an effective action in terms of the corresponding order parameters (see Methods Section):

 F[Δ,δ]=C2δ2+D4δ4−α2|Δ|2+β4|Δ|4+γ|Δ|2δ2, (1)

where the last term described the competition between nematicity and superconductivity. As a result, the nematic order parameter is noticeably suppressed inside the superconducting phase compared to its value () in the normal phase, so that (see Methods section for the derivation)

 δ2≊δ20−(2γD)|Δ|2, (2)

whereas the superconducting order parameter itself remains essentially unchanged due to tiny values of (see eq. (8) in Methods section). In the tetragonal phase (), the competition between local orthorhombic distortions and superconductivity is reflective of the suppression of nematic susceptibility below RMFernandes2010_2 ().

We emphasize that the local orthorhombic distortions we uncovered in the tetragonal phase of NaFeNiAs are distinct from the phase separation into superconducting tetragonal and antiferromagnetic orthorhombic regions found in Ca(FeCo)As under biaxial strain Ca122-Boehmer1 (); Ca122-Boehmer2 (). In the latter compound, the quantum phase transition between the superconducting tetragonal and AF orthorhombic phases is first-order, and the resulting phase separation into these two phases with different in-plane lattice parameters allows the material to respond to biaxial strain in a continuous fashion; this would occur even if there were no quenched disorder. In NaFeNiAs, the quantum phase transition is second order and therefore an analogous phase separation does not occur. Instead, the local orthorhombic distortions we observe in NaFeNiAs likely result from the large nematic susceptibility near optimal superconductivity, pinned by quenched disorder.

Given that the orthorhombic distortion with Curie-Weiss temperature dependence arises from local orthorhombic distortions, an alternative way to characterize such distortion is broadening of -spacing distribution width (see Methods section). In Figs. 4(a)-(d), we show obtained from our neutron Larmor diffraction results for NaFeNiAs. Given the local orthorhombic distortions arise from quenched disorder coupled with a large nematic susceptibility near a nematic QCP, it should track temperature dependence of the nematic susceptibility, since the quenched disorder should depend weakly on temperature. Therefore, we have fitted in Figs. 4(a)-(d) with the Curie-Weiss form and extracted the Weiss temperature as a function of doping, as shown in Fig. 4(e). Our results are well described by the Curie-Weiss form with changing from positive in underdoped to negative in overdoped regime [Fig. 4(e)], suggesting a nematic QCP near optimal superconductivity. These results are reminiscent of temperature and doping dependence of the nematic susceptibility from elastoresistance HHKuo2016 () and shear modulus measurements AEBohmerCRP (), thus suggesting that temperature dependence of is a direct measure of the nematic susceptibility without the need to apply external stress.

## Iii Discussion

In NaFeNiAs, the orthorhombic distortion and the structural phase transition temperature are and K for sli (); GTan2016 (); for they become and K. We find no evidence for a structural phase transition for samples with , thus suggesting the presence of a putative nematic QCP at , where . These results are consistent with recent Muon spin rotation and relaxation study of magnetic phase diagram of NaFeNiAs sky18 (). They are also consistent the with Ni-doping dependence of determined from Curie-Weiss fits to temperature dependence of the , which changes from positive to negative near [Fig. 4(e)]. Since our neutron Larmor diffraction measurements were carried out using polarized neutron beam produced by an Heusler monochromator, which has an energy resolution of about meV Martin11 (); xylu16 (), the local orthorhombic distortions captured in our measurements are either static or fluctuating slower than a time scale of ps, where is the reduced Planck constant Doloc99 (); Dai00 (). One possible origin of such slow fluctuations may be in-plane transverse acoustic phonons that exhibit significant softening in the paramagnetic tetragonal phase when approaching a nematic instability Yli18 (). Future neutron scattering experiments with energy resolutions much better than meV are desirable to separate the static and slowly fluctuating contributions. Our results also indicate that the nematic QCP would occur at a value that is distinctively larger than that of the magnetic QCP, in the absence of superconductivity. In the phase diagram of iron pnictides with decoupled and , due to the competition between superconductivity with both nematic and magnetic orders, magnetic order forms a hump peaked at near optimal doping [Fig. 1(d)], and the structural phase transition disappears in a similar fashion at a larger .

Theoretically, a determinantal quantum Monte Carlo study of a two-dimensional sign-problem-free lattice model reveals an Ising nematic QCP in a metal at finite fermion density Schattner (). In the nematic phase, the discrete lattice rotational symmetry is spontaneously broken from four fold to two fold, and there are also nematic quantum critical fluctuations above the nematic ordering temperature. Within the numerical accuracy of the determinantal quantum Monte Carlo study, the uniform nematic susceptibility above the nematic ordering temperature has a Curie-Weiss temperature dependence, signaling an asymptotic quantum critical scaling regime consistent with our observation Schattner (). Alternatively, the observed Curie-Weiss temperature dependent behavior of the nematic susceptibility can be understood from spin-driven nematic order theory, where magnetic fluctuations associated with the static AF order induce formation of the nematic state Karahasanovic (). In this picture, the effect of lattice strain coupled to the nematic order parameter produces a mean-field Curie-Weiss like behavior, arising from the nemato-elastic coupling which has direction-dependent terms in the propagator for nematic fluctuations. The Curie-Weiss temperature dependent nematic susceptibility should occur in the entire phase diagram where there is a significant softening of the elastic modulus Karahasanovic (). This means that Curie-Weiss temperature dependence of the local orthorhombic distortions we observe is a signature of nemato-elastic coupling, which does not suppress the magnetic fluctuations that cause the nematic order, but transforms the Ising-nematic transition into a mean-field transition Karahasanovic ().

Our discovery of local orthorhombic distortions exhibiting with Curie-Weiss temperature dependence across the phase diagram of NaFeNiAs results from the proliferation of nematic fluctuations and large nematic susceptibility near the nematic QCP. Quenched disorder that are always present in such doped materials act to pin the otherwise fluctuating local nematic domains, resulting in static (or quasi-static) local orthorhombic distortions that can lead to rotational-symmetry-breaking observation seen with multiple probes Kasahara (); EThewalt (); TSonobe (); XRen (); EPRosenthal (); RZhou2016 (); SLiu (). We have definitively observed local nematic distortions in NaFeNiAs that are static or quasi-static, in contrast to local distortions seen in SrNaFeAs using pair distribution function analysis that contain significantly more dynamic contributions BFrandsen2017 (), and which would not cause rotational-symmetry-breaking seen by static probes. Our observation of local nematic distortions highlights the presence of nematic fluctuations near the nematic QCP, which can play an important role in enhancing superconductivity of iron pnictides Tmaier (); Metlitski (); SLederer (); SLederer17 (), while the intense Ising-nematic spin correlations near the nematic QCP may be the dominant pairing interaction DJScalapino (); QMSi_NRM (); YSong2015 ().

## Iv Methods

Elastic neutron scattering experimental details. Elastic neutron experiments were carried out on the Spin Polarized Inelastic Neutron Spectrometer (SPINS) at the NIST Center for Neutron Research (NCNR), United States and the HB-1A triple-axis spectrometer at the High-Flux-Isotope Reactor (HFIR), Oak Ridge National Laboratory (ORNL), United States. We used pyrolytic graphite [PG(002)] monochromators and analyzers in these measurements. At HB-1A, the monochromator is vertically focused with fixed incident neutron energy meV and the analyzer is flat. At SPINS, the monochromator is vertically focused and the analyzer is flat with fixed scattered neutron energy meV. A PG filter was used at HB-1A and a Be filter was used at SPINS to avoid contamination from higher-order neutrons. Collimations of 40- 40-sample-40-80 and guide-40-sample-40-open were used on HB-1A and SPINS, respectively.

To measure the structural distortion in NaFeNiAs () at SPINS, we changed the collimation to guide-20-sample-20-open to improve the resolution and removed the Be filter. Our measurement was carried out nominally around a weak nuclear Bragg peak , but the measured intensity at this position mostly come from higher-order neutrons [ for neutrons and for neutrons]. While we do not resolve two split peaks in the orthorhombic state, clear broadening can be observed. Typical scans along the direction centered at are shown in Supplementary Fig. 3. in Fig. 3(b) is obtained by assuming at K and fitting broadening at lower temperatures as two split peaks with fixed widths of the peak at K.

Inelastic neutron scattering experimental details. Our inelastic neutron scattering experiment was carried out on the HB-3 triple-axis spectrometer at HFIR, ORNL, United States. Vertically-focused pyrolytic graphite [PG(002)] monochromator and analyzer with fixed scattered neutron energy meV were used. A PG filter was used to avoid higher-order neutron contaminations. The collimation used was 48-40-sample-40-120.

Using inelastic neutron scattering we studied the neutron spin resonance mode dai (); meng () in NaFeNiAs with . Energy scans at above ( K) and below ( and 9 K) are compared in Supplementary Fig. 2(a). The scans below after subtracting the K scan are compared in Supplementary Fig. 2(b). A clear resonance mode at meV similar to optimal-doped NaFeCoAs clzhang2016 () is observed, with almost identical intensities at and 9 K. Constant-energy scans along at different temperatures are compared in Supplementary Fig. 2(c), confirming the results in Supplementary Fig. 2(a) and 2(b). Temperature dependence of the resonance mode is shown in Supplementary Fig. 2(d), over-plotted with temperature dependence of the orthorhombicity and the AF order parameter. Intensity of the resonance mode increases smoothly below and , displaying no response when AF order is completely suppressed below . These results demonstrate the coexistence of robust superconductivity and nematic order without AF order in NaFeNiAs () below .

Larmor diffraction experimental details. Our neutron Larmor diffraction measurements were carried out at the three axes spin-echo spectrometer (TRISP) at Forschungs-Neutronenquelle Heinz Maier-Leibnitz (MLZ), Garching, Germany. Neutrons are polarized by a super-mirror bender, and higher-order neutrons are eliminated using a velocity selector. We used double-focused PG(002) monochromator and horizontal-focused Heusler (CuMnAl) analyzer in these measurements. Incident and scattered neutron energies are fixed at meV ().

The detailed principles of neutron Larmor diffraction has been described in detail elsewhere tkeller (); xylu16 (). In such experiments, polarization of the scattered neutrons is measured as a function of the total Larmor precession phase . By analyzing measured , information about the sample’s -spacing distribution can be obtained.

For an ideal crystal with -spacing described by a -function, is independent of with . accounts for the non-ideal polarization of the neutrons and can be corrected for by Ge crystal calibration measurements. In real materials due to internal strain, sample inhomogeneity or in the case of iron pnictides, a twinned orthorhombic phase, the -spacing should instead be described by a distribution , with . is the deviation from the average -spacing . is then described by

 P(ϕtot)=P0∫∞−∞f(ϵ)cos(ϕtotϵ)dϵ. (3)

Thus, can be regarded as the Fourier transform of the lattice -spacing distribution . By measuring , it is possible to resolve features with a resolution better than in terms of , limited by the range of accessible .

The distribution of -spacing is commonly described as a Gaussian function with full-width-at-half-maximum (FWHM) , also denoted as in the rest of the paper. Eq. 3 then becomes

 P(ϕtot)=P0exp(−ϵ2FWHM16ln2ϕ2tot). (4)

In iron pnictides with a non-zero nematic order parameter, due to twinning becomes the sum of two Gaussian functions. Assuming the two Gaussian peaks have identical FWHM , Eq. 4 becomes

 P(ϕtot)=P0exp(−ϵ2FWHM16ln2ϕ2tot)×√r2+(1−r)2+2r(1−r)cos(ϕtotΔϵ), (5)

where and denotes the relative populations of the two lattice -spacings and , and mgkim (). Therefore, the nematic order parameter can be extracted by fitting using Eq. 5.

When is too small to be directly resolved by Larmor diffraction, can be well described by either Eq. 4 or Eq. 5. In such cases, we either extract from Eq. 4 (Fig. 4) or extract by assuming at K, and extract , then fit to Eq. 5 by fixing to this value (Fig. 1(e) and Fig. 3). Measurements of at several different temperatures for NaFeNiAs () are shown in Supplementary Fig. 4, and fit to Eq. 5 as described.

A key feature of Eq. 5 is an oscillation in , which can be seen in raw data in Supplementary Figs. 4(d)-(i) (open symbols in Fig. 3(c)), in these cases the measurement provides definitive evidence of an orthorhombic state. For other panels in Supplementary Fig. 4, due to limited range of , can be equally well-described by Eq. 4 (solid symbols in Fig. 3(c)), for such data we cannot differentiate between a true splitting and a broadening from measurement done at a single temperature.

Magnetic susceptibility and electrical resistivity measurements. To ensure that for NaFeNiAs () is well inside the superconducting state, we show in Supplementary Fig. 1 its magnetic susceptibility as a function of temperature. As can be seen, the sample displays a sharp superconducting transition at K, with a width K. is well inside the superconducting state, unaffected by the width of the superconducting transition.

The temperature and doping dependence of the in-plane electrical resistivity were measured using the standard four-probe method, the results are normalized to and summarized in Supplementary Fig. 5. The superconducting transitions for all measured samples are sharp. The kinks associated with the structural transition at can be clearly identified in underdoped samples (Supplementary Figs. 5(a)-(d)), similar to NaFeCuAs awang (). These kinks are progressively suppressed with increasing Ni concentration, and disappear in overdoped samples. determined from electrical resistivity measurements are in good agreement with those obtained from Larmor diffraction.

Coexistence of superconductivity with lattice nematicity. Here we first consider the case without any long-range magnetic order, as is realized in NaFeNiAs for . In that case, the effective Landau free energy can be written in terms of only the superconducting order parameter and the orthorhombicity :

 F[Δ,δ]=C2δ2+D4δ4−α2|Δ|2+β4|Δ|4+γ|Δ|2δ2 (6)

Here we assume that the superconducting order parameter transforms under the tetragonal point symmetry, i.e. it does not break the rotational symmetry of the lattice. Since the lattice-nematic order parameter breaks this symmetry, the coupling to superconductivity is quadratic in . Above, the coefficient is in fact the elastic shear modulus , which is the inverse of the nematic susceptibility. The latter has a Curie-Weiss behavior (see Fig. 4 in the main text):

 χnem=1C66=1C(0)66T∗T−T∗ (7)

Here is the “bare” value of the shear modulus in the absence of the nematic transition. Note that the above formula can been derived rigorously from an effective model of the lattice orthorombicity coupled to an electronic nematic order parameter xylu16 (). Here we simply take to be the phenomenological Curie–Weiss temperature extracted from fitting the -spacing in Fig. 4(e). Note that if is positive (for ), we identify it with the nematic transition temperature , such that below .

Minimizing this effective action with respect to the two order parameters , we obtain in the mixed state with non-zero values of both parameters:

 Δ2 = αD−2γ|C|βD−4γ2=|Δ0|2−(2γβ)δ201−4γ2βD (8) δ2 = β|C|−2γαβD−4γ2=|δ0|2−(2γD)Δ201−4γ2βD, (9)

where and are the values of the order parameters in the absence of coupling between them. In the coexistence phase, the free energy becomes:

 F=F(0)SC−14(|C|−2γαβ)δ2=F(0)SC−D4δ2(1−4γ2βD), (10)

where . Note that for the coexistence phase to be stable, the last term in the above expression must be positive, which is only possible if , or equivalently, .

There is no perceptible change in the superconducting transition temperature below , implying . Substituting this into Eq. (8), we obtain:

 2γβδ20≪|Δ0|2 (11)

By contrast, the suppression of the orthorhombicity below is substantial: (see Figs. 3(b)-(c)), meaning that from Eq. (9). Substituting this into Eq. (11), we obtain:

 4γ2βD≪1, (12)

in other words, we can approximate the denominator in Eqs. (8) and (9) to be 1. This is also consistent with the requirement from Eq. (10) for the coexistence phase to be stable.

In summary, the phenomenological Landau free energy explains qualitatively the experimental data in the coexistence phase of superconductivity and nematicity. Furthermore, comparison with the experiment allows us to impose the strong condition on the smallness of the coupling constant in terms of the inequality (12).

Coexistence of three phases. Below , NaFeNiAs has a long-range antiferromagnetic (AF) order, and the free energy in Eq. (10) has to be modified to include the magnetic order parameter :

 F3[M,Δ,δ]=F[Δ,δ]−a2M2+b4M4−μ|δ|⋅M2+w|Δ|2M2, (13)

where we have included phenomenological coupling constants and . The sign of is positive, in accord with our experimental observation that AF order and superconductivity compete with each other (see Figs. 2(g)-(h) in the main text). The sign in front of on the other hand is negative, indicating magneto-elastic coupling that favors the coexistence of magnetism and orthorhombic distortion. Because of this coupling, it is clear that will acquire an additional component proportional to inside the AF phase:

 δ=δ(M=0)+κM2 (14)

Because and repel each other via the last term in Eq. (13), this implies, in view of Eq. (14), that a new term proportional to will be generated in the action, coupling the square of the superconducting order parameter linearly to the lattice orthorhombicity.

Working with the full free energy in Eq. (13) is impractical because of the large number of phenomenological parameters that are difficult to determine experimentally. Nevertheless, it offers a qualitative insight into the coexistence between AF, lattice nematicity, and superconductivity, as the above discussion shows.

As a parenthetical remark, we note that the term in the free energy may appear surprising at first sight, as one might have expected that lattice distortion and magnetization should couple biquadratically. The reason for the linear coupling is because the stripe AF order in the iron pnictides breaks the lattice symmetry, as does the shear strain  xylu16 (); fang08 (); fernandes11 (); wang15 (). Note that this conclusion holds independently of whether the microscopic origin of nematicity is purely magnetic fang08 (); fernandes11 () or due to orbital ordering of Fe orbitals wang15 (); lee (); kruger (); lv ().

Data availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

Acknowledgments The single crystal growth and neutron scattering work at Rice is supported by the U.S. DOE, BES under contract no. DE-SC0012311 (P.D.). A part of the material’s synthesis and characterization work at Rice is supported by the Robert A. Welch Foundation Grant Nos. C-1839 (P. D.) and C-1818 (A.H.N.). A.H.N. also acknowledges the support of the US National Science Foundation Grant No. DMR-1350237. C.D.C. acknowledges financial support by the NSFC (51471135), the National Key Research and Development Program of China (2016YFB1100101), Shenzhen Science and Technology Program (JCYJ20170815162201821), and Shaanxi International Cooperation Program (2017KW-ZD-07). We acknowledge the support of the High Flux Isotope Reactor, a DOE Office of Science User Facility operated by ORNL in providing the neutron research facilities used in this work.

Author contributions: Single cyrstal growth and neutron scattering experiments were carried out by W.W., Y.S., C.C., Y.L. with assistance from K.F.T., T.K.,L.W.H., W.T., S.C. and P.D. Theoretical understandings were performed by R.Y. and A.H.N. The entire project is overseen by P.D and C.C. The paper was written by P.D., W.W., Y.S., A.H.N., and all authors made comments.

The authors declare no competing financial and non-financial interests.

Correspondence and requests for materials should be addressed to C.C. (caocd@nwpu.edu.cn) or P.D. (e-mail: pdai@rice.edu)

### Footnotes

1. thanks: These authors made equal contributions to this work.
2. thanks: These authors made equal contributions to this work.

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