## Abstract

We use neutron scattering to study Vanadium (hole)-doped LiFe_{1−x}V_{x}As. In the undoped state, LiFeAs exhibits superconductivity at *T*_{c} = 18 K and transverse incommensurate spin excitations similar to electron overdoped iron pnictides. Upon Vanadium doping to form LiFe_{0.955}V_{0.045}, the transverse incommensurate spin excitations in LiFeAs transform into longitudinally elongated ones in a similar fashion to that of potassium (hole)-doped Ba_{0.7}K_{0.3}Fe_{2}As_{2} but with dramatically enhanced magnetic scattering and elimination of superconductivity. This is different from the suppression of the overall magnetic excitations in hole-doped BaFe_{2}As_{2} and the enhancement of superconductivity near optimal hole doping. These results are consistent with density function theory plus dynamic mean field theory calculations, suggesting that Vanadium doping in LiFeAs may induce an enlarged effective magnetic moment *S*_{eff} with a spin crossover ground state arising from the inter-orbital scattering of itinerant electrons.

## Introduction

The flexible spin states of Fe^{2+} ions are interesting due to its extensive distribution, from the lowermost mantle in the earth^{1} to the innermost hemoglobin in our human body^{2} and from the most common ferrite magnet to the most recent iron-based superconductors.^{3,4,5} Ignoring the orbital angular momentum, each Fe^{2+} ion with six electrons distributed in five *d* orbitals has three possible spin states, *S* = 0, 1, or 2 [Fig. 1a]. In iron-based superconductors, where Fe^{2+} ion is surrounded by crystal electric field of As atoms, the fivefold degeneracy of the *d* orbitals is split into a twofold degenerate *e*_{g} and a threefold degenerate *t*_{2g} states nearby the Fermi level [Fig. 1a].^{3} Although it is widely believed that Hund’s rule plays a critical role in determining the electronic configuration of iron,^{6} the *S* = 2 spin state has never been observed in metallic iron pnictides.^{5,7}

The total moment sum rule within a Heisenberg model for system with spin *S* requires \(\left\langle {m}^{2}\right\rangle ={(g{\mu }_{{\mathrm{B}}})}^{2}S(S+1)\), where *g* ≈ 2 is the Landé *g*-factor, when magnetic scattering is integrated over all energy and momentum space within the Brillouin zone (BZ).^{8} Since neutron scattering directly measures the energy (*E*) and momentum (**Q**) dependence of the dynamic structure factor *S*(**Q**, *E*), it is therefore possible to estimate the effective spin-fluctuating moment *S*_{eff} by integrating the magnetic spectral weight over all energies and wave vectors within the BZ. In previous inelastic neutron scattering on electron-doped BaFe_{2}As_{2} compounds, it is established that electron doping via Ni substitution in BaFe_{2−x}Ni_{x}As_{2} reduces slightly the total spin-fluctuating moment, changing from \(\left\langle {m}^{2}\right\rangle \approx 3.5\)\({\mu }_{{\mathrm{B}}}^{2}\)/Fe at *x* = 0 to \(\left\langle {m}^{2}\right\rangle \approx 2.7\)\({\mu }_{{\mathrm{B}}}^{2}\)/Fe at *x* = 0.3.^{9,10,12} These values approximately correspond to *S*_{eff} ≈ 1/2.^{9,11} For hole-doped Ba_{1−x}K_{x}Fe_{2}As_{2}, the total spin-fluctuating moment decreases rapidly with increasing *x*,^{9} and *S*_{eff} reduces to 0.14 at *x* = 1.^{13} For electron-doped NaFe_{1−x}Co_{x}As,^{5} the total spin-fluctuating moment also decreases slightly with increasing *x* but has almost identical value as that of BaFe_{2−x}Ni_{x}As_{2},^{14} thus suggesting universal nature of the *S*_{eff} ≈ 1/2 for two different families of iron pnictides. This is different from the superconducting LiFeAs, where the low-energy spin fluctuations coupled to superconductivity are about a factor 4 smaller than that of the Co-underdoped superconducting BaFe_{2}As_{2}.^{15,16} The total fluctuating moment of LiFeAs is \(\left\langle {m}^{2}\right\rangle \approx 1.5\pm 0.3\)\({\mu }_{{\mathrm{B}}}^{2}\)/Fe, more than a factor of 2 smaller than that of BaFe_{2}As_{2}.^{17,18} On the other hand, the total spin-fluctuating moment for iron chalcogenide compounds FeSe and FeTe are significantly larger. In the case of FeSe, the total spin-fluctuating moment of \(\left\langle {m}^{2}\right\rangle \approx 5.2\)\({\mu }_{{\mathrm{B}}}^{2}\)/Fe arises entirely from dynamic magnetic excitations as the system has no static ordered moment.^{19} For FeTe, the effective moment of Fe may change from *S*_{eff} ≈ 1 at 10 K to *S*_{eff} ≈ 3/2 at 300 K, suggesting entangled local magnetic moments with the itinerant electrons on warming.^{20}

The total moment sum rule strictly speaking is only valid for insulating local moment systems, where quantum effect and itinerant electron-induced magnetism are not important.^{8} In iron-based superconductors, itinerant electrons and hole–electron Fermi surface nesting are known to be important for magnetism and superconductivity [Fig. 1b].^{21} Theoretically, many physical properties of iron-based superconductors can be understood within the orbital-selective Mott phase picture, where electrons in Fe^{2+} ion with different orbitals can behave differently.^{22,23,24} In this model, hole-doped compounds are believed to be more correlated and therefore should have large effective local moments.^{22,23,24} From combined density functional theory (DFT) and dynamic mean field theory (DMFT) calculations,^{6} the total fluctuating moments in the paramagnetic state are nearly invariant around 2.2 *μ*_{B}/Fe in a wide variety of iron pnictide/chalcogenide families. In X-ray emission spectroscopy (XES) experiments, values of magnetic spin moments in the paramagnetic phase are about 1.3 *μ*_{B}/Fe for iron pnictides, including BaFe_{2}As_{2} and Ba(Fe,Co)_{2}As_{2}.^{7} These values are considerably smaller than the predictions from DFT+DMFT calculation but larger than those from neutron-scattering experiments,^{9,10,12} possibly due to the fast time scale of the XES measurements.^{7} Therefore, to fully understand the key ingredients of magnetism in iron-based superconductors, it is important to study the evolution of *S*_{eff} in different families of hole-doped iron-based superconductors in a controlled and systematic way.

While it is easy to dope holes in BaFe_{2}As_{2} family of iron pnictides by substituting Ba^{2+} cation with K^{1+}, replacing Fe with Cr/Mn introduces impurities and/or disorders instead of holes.^{25,26,27} For NaFeAs family, systematic neutron scattering and angle-resolved photoemission spectroscopy (ARPES) experiments reveal that replacing Fe with Cu dopes holes into the system and induces large ordered magnetic moment for *x* > 0.1 in NaFe_{1−x}Cu_{x}As.^{28,29} However, there are no inelastic neutron-scattering experiments to determine the total fluctuating moment for this family of compounds. In the case of LiFeAs family, ARPES experiments indicate that doping V into Fe sites actually introduces hole carriers with a rate of 0.3 hole per V dopant and selectively enlarges the inner hole Fermi surface [Fig. 1b],^{30} even though the doping-induced impurity and disorder effects might still play a role in the transport and susceptibility measurements with higher V doping ratio. Although hole doping via K substitution in Ba_{1−x}K_{x}Fe_{2}As_{2} induces superconductivity, <2% V doping in LiFe_{1−x}V_{x}As quickly suppresses the superconductivity in pure LiFeAs, even with a perfect nesting condition established between the inner hole and electron Fermi pockets at *x* = 0.084 [see arrows in Fig. 1b].^{30} The rapid suppression of superconductivity in LiFe_{1−x}V_{x}As has been suggested as due to magnetic impurity effect of V dopant, but there are no neutron-scattering experiments to establish the effect of V doping to spin excitations of pure LiFeAs .^{15,16,17}

## Results

We report the magnetic excitations in hole-doped nonsuperconducting LiFe_{0.955}V_{0.045}As, which has nearly nested hole–electron Fermi surfaces with *d*_{xz/yz} and *d*_{xy} orbital characters, respectively, and exhibits non-Fermi-liquid behavior.^{30} We observe enhanced commensurate magnetic fluctuations at antiferromagnetic (AF) wave vectors, suggesting that the non-Fermi-liquid behavior is intimately associated with magnetic fluctuations. The broadness and diffuseness of the spin excitations are also distinct from the sharp commensurate spin fluctuations in electron-doped LiFe_{0.88}Co_{0.12}As,^{18} probably reflecting the local nature of quantum spin fluctuations.^{31} More importantly, we find that the intensities of spin fluctuations are strongly enhanced in the energy range of our measurements with a larger *S*_{eff}, in sharp contrast to the reduced effective moment in hole-doped Ba_{1−x}K_{x}Fe_{2}As_{2}.^{9,13} Considering the different superconducting behavior in Ba_{1−x}K_{x}Fe_{2}As_{2} and LiFe_{1−x}V_{x}As, we argue that the inter-orbital scattering processes between *d*_{xz/yz} and *d*_{xy} orbitals increase the fluctuating moment but are detrimental to superconductivity.^{32} These results are consistent with the notion that intra-orbital scattering processes between hole and electron pockets, particularly those involving *d*_{yz}–*d*_{yz} orbital characters, are good for superconductivity in different classes of iron-based superconductors.^{33,34,35,36,37}

We first plot in Fig. 2 the two-dimensional images of spin excitations in LiFe_{0.955}V_{0.045}As at different energies and their comparison with the DFT+DMFT calculations for 10% hole doping (about LiFe_{0.967}V_{0.033}As). At energy transfer *E* = 3 ± 1 meV, spin excitations occur at commensurate *Q*_{AF} = (1,0) and (0,1) positions [Figs. 1c and 2a], different from the transverse incommensurate spin fluctuations in LiFeAs.^{15,16,17} Instead, spin excitations display a small elongation along the longitudinal direction, mimicking the low-energy spin excitations of hole-doped Ba_{1−x}K_{x}Fe_{2}As_{2}.^{38} As the energy transfer increases from *E* = 3 to *E* = 18 meV, the elongation of the elliptical spin fluctuations changes from the longitudinal to transverse direction, similar to that in Ba_{0.67}K_{0.33}Fe_{2}As_{2},^{9} reflecting the hole–electron Fermi surface nesting contribution from itinerant electrons.^{32,38} The wave vector-dependent spectra at the corresponding energies are confirmed by the DFT+DMFT calculations in the 10% hole-doped LiFeAs [Fig. 2e–h]. These results are significantly different from Mn-substituted BaFe_{2}As_{2}, where Mn doping induces diffusive magnetic scattering at the checkerboard wave vector (1,1).^{27} The broad and diffusive magnetic scattering at (1,0)/(0,1) may be attributed to the short-range spin fluctuations from localized moment and/or not perfect nesting condition between hole and electron Fermi surfaces.

To compare the magnitude of spin fluctuations in LiFeAs, LiFe_{0.88}Co_{0.12}As, and LiFe_{0.955}V_{0.045}As, we calculate in Fig. 1d the energy dependence of the local dynamic spin susceptibility *χ″*(*E*) for these materials obtained by integrating wave vector-dependent dynamic spin susceptibility *χ*″(**Q**, *E*) over the entire BZ.^{5} Compared with those of LiFeAs and LiFe_{0.88}Co_{0.12}As,^{18} *χ*″(*E*) in LiFe_{0.955}V_{0.045}As is clearly enhanced about <80 meV, although we do not have data >80 meV. Such an enhancement is also captured in the DFT+DMFT calculations as shown in red and blue solid lines of Fig. 1d. The local dynamic spin susceptibility of LiFe_{0.955}V_{0.045}As in the measured energy range is somewhat larger than that of BaFe_{2}As_{2}.^{39,40} For comparison, magnetic scattering of LiFeAs is considerably smaller than that of the Co-doped superconducting Ba(Fe_{0.92}Co_{0.08})_{2}As_{2}.^{15,16}

The differences between LiFeAs, LiFe_{0.88}Co_{0.12}As, and LiFe_{0.955}V_{0.045}As can be further confirmed in Fig. 3a in which we plot constant-energy cuts of spin excitations along the transverse and longitudinal directions. In all cases, we obtain the magnitude of magnetic scattering by normalizing it to phonon intensities,^{14} making it possible to compare magnetic scattering of different materials measured on different instruments. It is clear that the magnetic scattering in LiFe_{0.955}V_{0.045}As is both wider and higher than those in LiFeAs and LiFe_{0.88}Co_{0.12}As at *E* = 5 meV, resulting in a large increase in *χ″*(*E*) [Fig. 1d]. This cannot be explained by the extra Vanadium magnetic impurities since the 3*d*^{3} electronic configuration of V^{2+} ions only has *S* = 3/2, which is smaller than the expected *S* = 2 for 3*d*^{6} of Fe^{2+} in iron-based superconductors. Therefore, the enhanced magnetic scattering has to be associated with the introduction of hole carriers, which effectively modifies the spin configuration of Fe^{2+} ions.

In Fig. 3b, we show the line width of spin excitations and the corresponding dynamic spin-spin correlation length for LiFeAs, LiFe_{0.88}Co_{0.12}As, and LiFe_{0.955}V_{0.045}As. The sudden drop of spin correlation length in LiFe_{0.955}V_{0.045}As suggests that short-range spin correlations of local moments may become important. Figure 3c shows the line widths of spin excitations along both the transverse and longitudinal directions at different energies in LiFe_{0.955}V_{0.045}As. The crossing around 10 meV is consistent with the isotropic spin excitations in Fig. 2b. Therefore, LiFe_{1−x}V_{x}As is distinct from Cr/Mn-doped BaFe_{2}As_{2} in which magnetic impurities induce Néel-type diffusive AF spin fluctuations at low concentrations.^{25,26,27}

Figure 4a, b compare the DFT+DMFT calculated wave vector and energy dependence of *χ*″(**Q**, *E*) for LiFeAs and LiFe_{0.967}V_{0.033}As, respectively. Consistent with Fig. 1d, we find that LiFe_{0.967}V_{0.033}As has larger magnetic spectral weight, especially at low energies. In Fig. 4c, we show the estimated effective spin *S*_{eff} for different iron-based superconductors. It is clear that all values are distributed between *S* = 0 and *S* = 1 with most of them around *S*_{eff} = 1/2. The discrepancy of hole-doping dependence of the magnetic excitations between BaFe_{2}As_{2} and LiFeAs families may arise mostly from the fact that we have only integrated the energy range of *χ*^{″}(*E*) from 0 to about 60 meV where we have solid data for V-doped and pure LiFeAs. Although these numbers cannot be compared with absolute values of *S*_{eff} determined for BaFe_{2}As_{2} and FeSe, the trend of increasing total magnetic susceptibility with hole doping in V-doped LiFeAs is unmistakable. Therefore, it is interesting to understand the microscopic origin of magnetic excitations in V-doped LiFeAs.

## Discussion

Theoretically, the reduction of effective spins in iron pnictides/chalcogenides can be understood in either an itinerant or a localized moment picture. The former considers Fe^{2+} ions embedded in a Fermi sea of conduction electrons,^{41,42} and the lower spin states can be achieved through a multiple-stage Kondo screening process,^{31,43} where the system is in a quantum critical incoherent metallic state down to 0 K. In the local moment scenario, due to the sensitivity of spin states of Fe^{2+} to the ionic radius, the effective magnetic moment in iron-based material can be tuned by controlling the strength of crystal electric field in the local FeAs_{4} tetrahedral structure.^{44,45} This picture can successfully explain some phenomena such as temperature-dependent magnetic moments^{46,47} and coherent–incoherent crossover.^{31,48,49,50,51} However, it is still difficult to understand the large variety of the fluctuated moments in iron-based superconductor by assuming a reduced spin state with *S* = 1 or 0. To achieve an intermediate crossover between spin state *S* = 1 and *S* = 0, a dynamic “spin mixing” process was proposed^{44,45} as schematically shown in Fig. 4d. When two Fe^{2+} ions exchange their electrons, their spin states would effectively change from *S* = 0 to *S* = 1 and vice versa. This results in a continuous evolution of the effective spin state with 0 < *S*_{eff} < 1. In this scenario, the probability of each *S* = 0 or 1 state is closely related to both the local structure and conduction electrons.^{44} We note that this spin crossover scenario is a localized model and may not completely reflect the itinerant nature of iron-based superconductors. Further theoretical and experimental evidences are needed to clarify the situation.

In insulating copper oxides, local moments are locked into the *S* = 1/2 state by the elimination of doubly occupied states with a large charge gap.^{52} Doping charge carriers introduce doubly occupied states and lead to the destruction of local moments, resulting in the suppression of the neutron-scattering spectral weight in the energy range of typical neutron-scattering experiments.^{8} Similarly, in iron pnictides, the effective moment on each Fe^{2+} ion is also affected by the population of doubly occupied states or spin singlets due to Kondo screening. However, this cannot be simply attributed to the introduction of charge carriers since the doping dependence of *S*_{eff} in Ba_{1−x}K_{x}Fe_{2}As_{2} and LiFe_{1−x}V_{x}As is rather different. Such a difference may be rooted in the different sites of the dopants. Since Vanadium is directly introduced into the FeAs layer while K^{+} is not, the Fe–As bond length is affected by V doping^{30} and a high spin state is favored in V-doped LiFeAs. With increasing V- doping, the nesting condition improves between the inner hole Fermi surface with *d*_{xz∕yz} orbital character and the inner electron pocket mainly derived from *d*_{xy} orbital character.^{30} Therefore, the enhanced inter-orbital scattering between *d*_{xz∕yz} and *d*_{xy} orbitals, which suppresses superconductivity,^{32} favors the “spin mixing” process [Fig. 4d] and enhances fluctuated moments. This is different from the intra-orbital *d*_{yz}–*d*_{yz} hole–electron Fermi surface nesting scattering, which favors superconductivity and induces a neutron spin resonance only at the AF wave vector in detwinned FeSe^{33} and underdoped iron pnictides.^{37,53}

To conclude, we performed time-of-flight neutron-scattering measurements on LiFe_{0.955}V_{0.045}As and did DFT+DMFT calculation on the corresponding 10% hole-doped LiFeAs system. We found that the low-energy spin fluctuations are influenced by itinerant electrons following the same doping dependence of spin excitations in BaFe_{2}As_{2}. More importantly, both the experimental and theoretical results suggest that magnetic excitations are dramatically enhanced by doping holes with Vanadium. We discuss the possible origin of such enhancement and argue that the inter-orbital scattering between *d*_{xz/yz} and *d*_{xy} orbitals are bad for superconductivity and may dynamically mix the spin states of Fe^{2+} ions.

## Methods

### Sample preparation

We grew single crystals of LiFe_{0.955}V_{0.045}As by self-flux method.^{55} In all, 7.2 g of single crystals were used in our time-of-flight neutron-scattering experiments, of which 1.8 g were synthesized by using isotope ^{7}Li to reduce neutron absorption. Samples were wrapped by Aluminum foil with Hydrogen-free glue in the glove box filled with Argon gas since the samples are air and humidity sensitive. All samples were co-aligned and glued on five 40 × 40 mm^{2} Aluminum plates as shown in Fig. S1b. The [1, 0, 0] direction is along the horizontal direction and [1, 1, 0] along the diagonal [Fig. S1a].

### Neutron-scattering experiments

Our inelastic neutron-scattering measurements on LiFe_{0.955}V_{0.045}As were carried out on the time-of-flight 4D-Space Access Neutron Spectrometer (4SEASONS) at Materials and Life Science Experimental Facility (MLF), Japan Proton Accelerator Research Complex (J-PARC).^{56} We define the momentum transfer *Q* in three-dimensional reciprocal space in Å^{−1} as **Q** = *H***a**^{*} + *K***b**^{*} + *L***c**^{*}, where *H*, *K*, and *L* are Miller indices and \({{\bf{a}}}^{* }=\hat{{\bf{a}}}2\pi /a\), \({{\bf{b}}}^{* }=\hat{{\bf{b}}}2\pi /b\), and \({{\bf{c}}}^{* }=\hat{{\bf{c}}}2\pi /c\) with *a* = *b* ≈ 5.316 Å, and *c* = 6.315 Å.^{17,18} Samples are co-aligned in the [*H*, 0, *L*] scattering plane with mosaic <5° and incident beam (*E*_{i} = 16, 39, 75, and 200 meV) parallel to the *c*-axis of the crystals. In principle, absolute magnetic neutron-scattering intensity from LiFe_{0.955}V_{0.045}As can be normalized by comparing the scattering with a Vanadium standard.^{5} However, LiFe_{0.955}V_{0.045}As has considerable unknown amount of Vanadium-containing flux from the growth, rendering this method unreliable. To obtain the absolute intensity, we therefore use phonon normalization method as discussed below.

### Background subtraction and data analysis

The background subtraction processes are shown in Fig. S2.^{14} The raw magnetic scattering appears in a fourfold symmetry at wave vectors [1, 0], [0, 1], [−1, 0], and [0, −1] [Fig. S2a] due to the tetragonal symmetry of the lattice structure. We symmetrized the scattering spectra in order to enhance the magnetic signal and then remove the background [Fig. S2b]. Background was obtained by masking the signal within the white squares [Fig. S2c] and assumed to be radially symmetric. A polynomial function of \(\left|{\bf{Q}}\right|\) was used to fit the background intensity [Fig. S2e] and then was subtracted from the raw data. The results were shown in Fig. S2d. Figure S2f, g are cuts along the [*H*, 0] and [1, *K*] directions, respectively. We note that there are still residual background intensities, which likely come from phonon scattering and affect the calculation of local dynamical spin susceptibility. Different treatments to these intensities might result in the differences of the final estimated local dynamical spin susceptibility. Therefore, we recalculate the local dynamical spin susceptibility of pure LiFeAs and LiFe_{0.88}Co_{0.12}As from the previous data with the same method we used for V-doped LiFeAs. In this way, we can have a better comparison between different compounds. The recalculated local dynamical spin susceptibilities are shown in Fig. 1d.

### Absolute neutron-scattering intensity normalization

We performed our inelastic neutron scattering on LiFeAs and LiFe_{0.88}Co_{0.12}As at the wide Angular-Range Chopper Spectrometer and Cold Neutron Chopper Spectrometer at Spallation Neutron Source, Oak Ridge National Laboratory and on LiFe_{0.955}V_{0.045}As at 4SEASONS, MLF, J-PARC. The neutron-scattering intensity of the LiFeAs and LiFe_{0.88}Co_{0.12}As have already been normalized by standard Vanadium sample. The neutron-scattering intensity of LiFe_{0.955}V_{0.045}As is normalized by phonon intensities since a standard Vanadium is unavailable during the experiment. As discussed in the main text, V normalization is actually less reliable in V-doped LeFeAs because of the unknown amount of V-containing flux in the sample. Therefore, phonon normalization using the same phonon for LiFeAs, which we re-analyzed from using previous data, and V-doped LiFeAs should be more reliable than the standard Vanadium scan when comparing the results from different spectrometers.^{14}

We identified two phonons, one is around (1, 1) and the other originates from (2, 0). The incident neutron energy is almost the same, about 39 meV for LiFe_{0.955}V_{0.045}As and 35 meV for LiFeAs and LiFe_{0.88}Co_{0.12}As. The one-dimensional cuts of these phonons are shown in Fig. S3, and the integrated intensity of each phonon can then be calculated. By comparing the phonon intensities of different materials, we estimated the scale factor, which is defined as the ratio of phonon intensity between LiFeAs/LiFe_{0.88}Co_{0.12}As and LiFe_{0.955}V_{0.045}As. Since the intensity of LiFeAs and LiFe_{0.88}Co_{0.12}As has been normalized already by the standard Vanadium scan, we could calculate the absolute value of the intensities in LiFe_{0.955}V_{0.045}As. The data of LiFeAs and LiFe_{0.88}Co_{0.12}As can be used as a cross-check and the absolute value we obtained for LiFe_{0.955}V_{0.045}As is consistent with these two references.

Figures S3a, b show the acoustic phonon near (1,1) at 4.5 and 5.5 meV [Fig. S3e], and Figs. S3c, d display another phonon around (2,0) [Fig. S3f] at 12.5 and 13.5 meV. In Fig. S3g, we plot the estimated factor from Fig. S3a–d. An average scale factor about 200 is obtained.

Alternatively, the incoherent nuclear scattering can be used as another reference to normalize the data. In Fig. S4, we show the incoherent peak for LiFe_{0.955}V_{0.045}As, LiFeAs, and LiFe_{0.88}Co_{0.12}As. The incident energy *E*_{i} is 39 and 75 meV for LiFe_{0.955}V_{0.045}As and 35 and 80 meV for LiFeAs and LiFe_{0.88}Co_{0.12}As. The estimated scale factor is about 70, significantly smaller than the value from phonon normalization. We note here that the residual flux trapped in the single crystal samples is an issue that may affect the estimation of the absolute value. This has been observed in our previous paper on Co-doped NaFeAs in which a difference of 30% was observed.^{14} This situation would be even more serious since the LiFe_{0.955}V_{0.045}As actually was grown out of a nominal composition of LiFe_{0.9}V_{0.1}As. In particular, the neutron incoherent scattering cross-section of V is huge compared with that of Fe, Li, and As. The trapped flux with rich V having a large neutron incoherent cross-section would lead to a serious underestimation of the scale factor. Therefore, we argue that the phonon normalization is more accurate than the normalization by incoherent peaks.

More importantly, we emphasize that, even with this smaller scale ratio, we can still observe the enhancement of magnetic scattering in LiFe_{0.955}V_{0.045}As [Fig. S5], demonstrating the robustness of our conclusion that the low-energy magnetic excitations in V-doped LiFeAs is greatly enhanced. In the main text, we used the scale factor 200 to normalize all the data. In Fig. S5, we display the local dynamical spin susceptibility of LiFe_{0.955}V_{0.045}As from both normalization methods, as well as others for comparison.

### Transport results

Figure S6 shows the magnetic susceptibility and electric resistivity in a series of V-doped LiFeAs. The effective magnetic moments were obtained from the fits with a Curie–Weiss form \(\frac{1}{\chi -{\chi }_{0}}=(T-\theta )/C\), in which *θ* is the Curie temperature and *χ*_{0} is the temperature-independent constant. It is clear that the effective magnetic moment increases with the concentration of Vanadium, as shown in inset of Fig. S6a. Figure S6b shows that the temperature dependence of resistivity of LiFe_{1−x}V_{x}As varies from *x* = 0 to *x* = 0.15, showing the evolution from metallic to insulating behavior. A non-Fermi liquid region was identified with *x* between 0.05 and 0.15, suggesting enhanced electronic correlation [Fig. S6b inset]. We note that, although the susceptibility and resistivity probe the energy scales different from neutron scattering, the increase of moments and resistivity is qualitatively consistent with the enhancement of low-energy magnetic excitations.

### DFT+DMFT calculations

We use DFT+DMFT^{57} to compute the electronic structure and spin dynamics of V-doped LiFeAs in the paramagnetic state. The DFT part is based on the full-potential linear augmented plane wave method implemented in Wien2K^{58} in conjunction with Perdew–Burke–Ernzerhof generalized gradient approximation^{59} of the exchange correlation functional. DFT+DMFT is implemented on top of Wien2K and documented in ref. ^{60}. In the DFT+DMFT calculations, the electronic charge was computed self-consistently on DFT+DMFT density matrix. The quantum impurity problem was solved by the continuous time quantum Monte Carlo (CTQMC) method^{61,62} with a Hubbard *U* = 5.0 eV and Hund’s rule coupling *J* = 0.8 eV in the paramagnetic state.^{6,63,64} Bethe–Salpeter equation is used to compute the dynamic spin susceptibility where the bare susceptibility is computed using the converged DFT+DMFT Green’s function while the two-particle vertex is directly sampled using CTQMC method after achieving full self-consistency of DFT+DMFT density matrix. The detailed method of computing the dynamic spin susceptibility is documented in ref. ^{6} and was shown to be able to compute accurately the spin dynamics of many iron pnictide superconductors. The experimental crystal structure (space group I4/mmm, #139) of V-doped LiFeAs with lattice constants *a* = *b* = 3.7914 Å and *c* = 6.3639 Å^{65} is used in the calculations. Virtual crystal approximation is employed to approximate the V-doping effect. In Fig. S7, we plot the calculated dynamic spin susceptibility from two inter-orbital scattering channels, which are *d*_{xy}–*d*_{xz} and *d*_{xy}–*d*_{yz} in LiFeAs and LiFe_{0.967}V_{0.033}As. The enhancement of electron scattering in both channels in LiFe_{0.967}V_{0.033}As is closely associated with the Fermi surface nesting between the inner hole pockets (*d*_{xz}/*d*_{yz}) and the inner electron pocket (*d*_{xy}). Such an enhanced electron scattering between different orbitals is consistent with the “spin mixing” process as shown in Fig. 4c in the manuscript and drives the system into the *S* = 1/2 state. Furthermore, such spin-state fluctuations might be responsible for the biquadratic exchange coupling, which is essential for the nematic instability existing extensively in iron pnictide/chalcogenide superconductors.

## Data availability

The neutron-scattering and numerical data used in this work, and in the figures of this manuscript, are available upon request from the corresponding authors.

## Code availability

The computer codes used to carry out the DFT+DMFT calculations used in this work are available upon request from the corresponding author.

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## Acknowledgements

The neutron-scattering work at Beijing Normal University is supported by the Fundamental Research Funds for the Central Universities (Grant Nos. 310432101 and 2014JJCB27) and the National Natural Science Foundation of China (Grant No. 11734002) (to X.L.). Z.Y. was supported by the NSFC (Grant No. 11674030), the Fundamental Research Funds for the Central Universities (Grant No. 310421113), and the National Key Research and Development Program of China grant 2016YFA0302300. The calculations used high-performance computing clusters at BNU in Zhuhai and the National Supercomputer Center in Guangzhou. The neutron-scattering work at Rice is supported by the U.S. DOE BES under contract no. DE-SC0012311 (to P.D.). The neutron experiment at the Materials and Life Science Experimental Facility of the J-PARC was performed under a user program 2017B0216.

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P.D., G.T., X.L., and C.J. designed the research. Single crystals were grown by G.D. and L.X. under the instructions of X.W. and C.J. Neutron-scattering experiments were carried out by Z.X., G.T., Y.R., L.T., Y.L., P.L., H.W., Y.W., R.K., K.I., and D.L.A. Theoretical calculations were carried out by Z.Y. The paper was written by P.D., Y.L., Z.X., and Z.Y. All authors made comments on the paper.

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Xu, Z., Dai, G., Li, Y. *et al.* Strong local moment antiferromagnetic spin fluctuations in V-doped LiFeAs.
*npj Quantum Mater.* **5, **11 (2020). https://doi.org/10.1038/s41535-020-0212-x

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