# Interface suppressed nematicity and enhanced superconductivity of
FeSe/NdFeO3 in the low doping regime

Chihao Li  Laboratory of Advanced Materials, State Key Laboratory of Surface
Physics,
and Department of Physics, Fudan University, Shanghai 200438, China  Yuanhe
Song  Laboratory of Advanced Materials, State Key Laboratory of Surface
Physics,
and Department of Physics, Fudan University, Shanghai 200438, China  Xiaoxiao
Wang  Laboratory of Advanced Materials, State Key Laboratory of Surface
Physics,
and Department of Physics, Fudan University, Shanghai 200438, China  Minyinan
Lei  Laboratory of Advanced Materials, State Key Laboratory of Surface
Physics,
and Department of Physics, Fudan University, Shanghai 200438, China  Xiaoyang
Chen  Laboratory of Advanced Materials, State Key Laboratory of Surface
Physics,
and Department of Physics, Fudan University, Shanghai 200438, China  Haichao
Xu<EMAIL_ADDRESS>Laboratory of Advanced Materials, State Key
Laboratory of Surface Physics,
and Department of Physics, Fudan University, Shanghai 200438, China  Shanghai
Research Center for Quantum Sciences, Shanghai 201315, China  Rui Peng
<EMAIL_ADDRESS>Laboratory of Advanced Materials, State Key Laboratory of
Surface Physics,
and Department of Physics, Fudan University, Shanghai 200438, China  Shanghai
Research Center for Quantum Sciences, Shanghai 201315, China  Donglai Feng
<EMAIL_ADDRESS>Laboratory of Advanced Materials, State Key Laboratory of
Surface Physics,
and Department of Physics, Fudan University, Shanghai 200438, China  National
Synchrotron Radiation Laboratory and School of Nuclear Science and Technology,
University of Science and Technology of China, Hefei, 230026, China  New
Cornerstone Science Laboratory, University of Science and Technology of China,
Hefei, 230026, China  Collaborative Innovation Center of Advanced
Microstructures, Nanjing 210093, China

###### Abstract

The discovery of interface-enhanced superconductivity in single-layer
FeSe/oxides has generated intensive research interests. Beyond the family of
FeSe interfaced with various TiO2 terminated oxides, high pairing temperature
up to 80 K has been recently observed in FeSe interfaced with FeOx-terminated
LaFeO3. Here we successfully extend the FeSe/FeOx superconducting interface to
FeSe/NdFeO3, by constructing 1uc-FeSe/6uc-NdFeO3/Nb:SrTiO3 heterostructures.
Intriguingly, well-annealed FeSe/NdFeO3 exhibits a low doping level of
0.038$\sim$0.046 e${}^{-}/$Fe which deviates universally magic doping level
(0.10$\sim$0.12 e${}^{-}/\rm{Fe}$) and provides a new playground for studying
the FeSe/oxide interface in the low electron-doped regime. Comparing it with
thick FeSe films at the comparable electron doping level induced by surface
potassium dosing, FeSe/NdFeO3 shows a larger superconducting gap and the
absence of a nematic gap, indicating an enhancement of the superconductivity
and suppression of nematicity by the FeSe/FeOx interface. These results not
only expand the FeSe/FeOx superconducting family but also enrich the current
understanding on the roles of the oxide interface.

††preprint: APS/123-QED

##  I Introduction

The discovery of high-temperature interfacial superconductivity in one unit
cell (1uc) FeSe/SrTiO3 [42] has ignited intensive research interests. The
superconducting pairing temperature in 1uc FeSe/SrTiO3 (Tg $\sim$ 65K) [10,
39] exhibits a substantial enhancement compared to that of bulk FeSe crystals
(superconducting critical temperature Tc $\sim$ 8K) [13]. Interface
engineering on the expitaxial strain, charge transfer, and interfacial
electron-phonon coupling have been conducted [34, 29, 28, 44, 36, 7, 28, 29,
46, 47, 18, 32, 40], revealing the significant contributions of interface
charge transfer and interfacial electron-phonon coupling to the enhancement of
superconducting pairing [7, 46, 36, 47, 18, 32, 40].

To examine the generality of the mechanism underlying the enhancement of
interfacial superconductivity and to explore new interfacial superconductors,
it is crucial to investigate FeSe films interfaced with other oxides.
Extending the superconducting interfaces, a range of FeSe interfaces with
TiOx-terminated oxides has been explored. These include 1uc FeSe on BaTiO3
[29], on anatase TiO2 (001) [6], on rutile TiO2 (100) [32], SrTiO3 (110) [46,
52, 48], on EuTiO3 [22] and on LaTiO3 [14]. Angle-resolved photoemission
spectroscopy (ARPES) measurements have revealed that well-annealed 1uc FeSe on
different TiOx-terminated interfaces generally exhibit substantial electron
doping (0.10$\sim$0.12 e${}^{-}/\rm{Fe}$), referred as “magic doping” [19,
14]). It puts 1 uc FeSe/titanate at the heavy electron doping regime of the
phase diagram, where nematicity is absent [39, 28, 29, 44, 50]. Moreover,
replica bands reflecting the interfacial electron-phonon coupling are commonly
observed in these FeSe-oxide interfaces [18, 32, 20, 7, 46]. However, it
remains challenging to construct new types of FeSe/oxide interfaces beyond the
FeSe/TiOx structure in order to test the generality of this mechanism.

Figure 1: (a) Crystal structure of NdFeO3. (b) Schematic illustration of the
heterostructure. (c) Reflection high-energy electron diffraction (RHEED)
pattern of the annealed Nb:SrTiO3 substrates, showing sharp Kikuchi lines. (d)
RHEED pattern of the epitaxial 6 uc NdFeO3 films, showing two-dimensional
RHEED streaks. (e) The intensity oscillation of the diffracted and specular
RHEED spots during the growth of a calibration sample. (f) The intensity
oscillation of the diffracted and specular RHEED spots during the growth of
the 6 uc NdFeO3. (g) Valence band comparison on the 6 uc NdFeO3-δ/Nb:SrTiO3
film after each annealing experiment. The valence bands shift to higher
binding energy with longer annealing times, suggesting increasing electron
doping due to oxygen vacancies. (h) Valence band comparison between Se flux-
annealed 6-uc NdFeO3-δ/Nb:SrTiO3 and Se flux-annealed 6-uc LaFeO3/Nb:SrTiO3
[37]. They exhibit comparable oxygen vacancy density as indicated by the
valence band position. (i) Scanning Tunneling Microscopy (STM) image of the
annealed 6uc-NdFeO3/Nb:SrTiO3.

Very recently, Song et. al reported the successful fabrication of high-quality
monolayer FeSe films on the FeOx-terminated LaFeO3 [37], in which the
superconducting pairing temperature reaches 80 K, surpassing the boiling
temperature of nitrogen. This work suggests that a similar pairing enhancement
mechanism operates at the FeSe/LaFeO3 interface as observed in the FeSe/TiOx-
terminated interfaces [37]. Yet, the FeSe/LaFeO3 interface remains the only
reported instance of the FeSe/FeOx superconducting interface. Could it be
possible to achieve epitaxial growth of 1uc-FeSe on other XFeO3 (X=Nd, Pr, Bi,
and etc.) perovskites? By expanding the FeSe/FeOx family, we can potentially
establish new platforms to explore the characters of FeSe-based interfacial
superconductivity. In particular, it is worth noting that the reported
FeSe/LaFeO3 exhibits a lower electron doping (0.087 e${}^{-}/\rm{Fe}$)
compared to the magic doping observed in FeSe/TiOx-terminated interfaces. By
studying FeSe on various XFeO3 substrates, valuable insights can be gained
into the FeSe/oxide interface under doping conditions that differ from
FeSe/titanates.

Here we have successfully fabricated a new high-quality FeSe/FeOx interface by
growing a 1uc FeSe film on a 6uc epitaxially grown NdFeO3 layers on Nb:SrTiO3
substrates. Through in-situ ARPES measurements, we have determined that the
electron doping level of the well-annealed 1uc FeSe/NdFeO3/Nb:SrTiO3 interface
is only 0.038$\sim$0.046 e${}^{-}/\rm{Fe}$, which is lower than the doping
levels observed in the various 1uc FeSe on TiOx-terminated oxides (0.10$\sim$
0.12e${}^{-}/\rm{Fe}$) [39, 28, 29, 6, 32, 22, 14] and in FeSe/LaFeO3 (0.087
e${}^{-}/\rm{Fe}$) interfaces [37], placing the superconducting interface of
FeSe/NdFeO3 in the low doping regime of electron-doped FeSe phase diagram.
Compared to FeSe thick films with similar levels of electron doping [44, 50,
35], the FeSe/NdFeO3 system exhibit an enhancement of superconductivity and
the suppression of nematic splitting. This observation suggests that the
FeSe/FeOx interface plays an additional role beyond the electron doping. Our
findings not only extend the FeSe/FeOx superconducting family, but also
provide evidence for the tailored band structure achieved through the
interface, shedding light on the intricate interplay between interfacial
superconductivity and nematicity.

##  II MBE growth of 1uc FeSe/6uc NdFeO3/Nb:SrTiO3

NdFeO3 (NFO) bulk crystals exhibit a structurally distorted perovskite
structure with lattice constants of $a$=5.450 $\AA$, $b$=5.587 $\AA$,
$c$=7.761 $\AA$ [Fig. 1(a)] [17]. The pseudo-cubic lattice constant of NFO is
$a^{\prime}$=3.895 $\AA$ and the lattice mismatch with SrTiO3 substrate is
only -0.25$\%$. Commercial TiOx terminated 0.5$\%$wt Nb-doped SrTiO3(001)
substrates (NSTO) with 0.1∘ $\sim$ 0.15 ∘ miscut angle (CrysTec) were used as
substrates. The NSTO substrates were annealed at 500∘C in the molecular beam
epitaxy (MBE) chamber (base vacuum $7\times 10^{-10}$ mbar) for 1 hour, using
an pressure of $2\times 10^{-7}$ mbar under ozone atmosphere. The RHEED
pattern of the annealed NSTO substrates shows sharp Kikuchi lines [Fig. 1(c)].
Subsequently, NFO films were grown on the annealed substrates by ozone-
assisted molecular beam epitaxy at 650∘C with pressure of $2\times 10^{-7}$
mbar under ozone atmosphere. By alternately opening the pneumatic shutters of
the Nd source and the Fe source, the oxidized Nd and Fe layers were
alternately grown on the substrate in an atomic layer-by-layer mode. The
relative stoichiometry and the absolute amount of each element (La and Fe) was
first calibrated by quartz crystal microbalance (QCM) measurements and then by
real-time intensity oscillations of reflection high-energy electron
diffraction (RHEED) during the growth [9, 28, 29]. After calibration, stable
oscillations of RHEED intensity can be obtained, indicating a stoichiometric
Nd-Fe atomic flux ratio [Fig.1 (e)]. Subsequently, 6 uc NdFeO3 films were
grown on freshly-annealed Nb:SrTiO3 substrates following the calibrated
stoichiometry [Fig. 1(f)]. The RHEED streaks suggest two-dimensional growth
character of the grown NdFeO3 films [Fig. 1(d)]. Using this atomic layer-by-
layer epitaxial growth method, we control NdFeO3 to be FeOx-terminated.

Oxygen vacancies in the substrates are vital for epitaxially growing high-
quality FeSe films [42, 39, 36, 8, 27, 14, 6, 32, 46, 52, 48, 22]. To induce
sufficient oxygen vacancies, the NdFeO3 films were annealed four times under
vacuum, and the valence bands were examined using ARPES after each annealing
step. As shown in Fig. 1(g), the valence bands (VB) of NdFeO3-δ/Nb:SrTiO3
shift towards higher binding energy upon vacuum annealing, suggesting
increasing electron doping with increasing oxygen vacancies. In contrast to
the wide insulating band gap observed in NdFeO3, an increase in the density of
states near the Fermi level is observed in NdFeO3-δ films with higher oxygen
vacancy densities resulting from stronger annealing conditions. This
observation suggests the presence of low-energy states induced by oxygen
vacancies [4, 39, 12, 1, 23]. Subsequently, the NdFeO3-δ/Nb:SrTiO3 films were
transferred to a Chalcogenide-MBE chamber and annealed under Se flux at 950 ∘C
for 45 minutes. After the annealing, the valence band peaks of the NdFeO3
films at 4 eV below the Fermi energy are similar to those of the annealed
LaFeO3 films under Se flux, suggesting comparable oxygen vacancy densities
[see Fig. 1(h)]. Note that the annealed NdFeO3-δ films exhibit a higher
spectral weight near the Fermi level [see Fig. 1(g-h)] , when compared to
LaFeO3-δ [37]. This observation indicates that the concentration of oxygen
vacancies in the surface layers of NdFeO3 is no less than that in LaFeO3.

The topography measurements were performed by a scanning tunneling
microscope[STM, (RHK Technology)] that in-situ connected with the ARPES
chamber and MBE chambers in the combined ultra-high vacuum (UHV) system. The
samples were measured at 17 K under pressure of $\sim$ 1 $\times$ 10 -10 mbar.
The annealed NdFeO3/Nb:SrTiO3 films exhibit atomically flat surfaces, as
observed by in-situ scanning tunneling microscopy (STM) in Fig. 1(i).
Subsequently, 1uc FeSe films were grown on the NdFeO3/Nb:SrTiO3
heterostructures at a temperature of 490 ∘C, using a Fe/Se flux rate ratio of
1:10, followed by annealing at 520 ∘C for 3 hours. The STM morphology image of
the annealed 1uc FeSe/NdFeO3/Nb:SrTiO3 [Fig. 2(b)] exhibits well-defined
terrace edges, predominantly 1uc in thickness with a few 2 uc areas
[Supplementary Fig. S2].

Figure 2: (a) Structure of 1uc-FeSe/6uc-NdFeO3/Nb:SrTiO3. (b) STM morphology
of annealed 1uc-FeSe/6uc-NdFeO3/Nb:SrTiO3. (c) Fermi surface of 1uc-FeSe/6uc-
NdFeO3/Nb:SrTiO3 heterostructure. Open circles refer to fitted $k_{F}$ from
in-plane mappings (cut by cut) and the grey elliptical circle refer to 2 non-
degenerate electron pockets at the M point, suggesing relatively low electron
doping level. (d) Band dispersion near $\Gamma$ point along the cut $\\#$1
shown in Fig (c). The hole band of FeSe is labeled as $\alpha$. (e) Band
dispersion near M point along the cut $\\#$2 shown in Fig (c), labeling the 2
electron bands as $\gamma_{1}$ and $\gamma_{2}$. (f) MDC near the M point. The
FeSe/ NdFeO3 exhibits non-degenerate electron pockets at the Fermi level. (g)
Fitted electron bands plotted on the band dispersion along the cut $\\#$1
shown in Fig (c). All the ARPES measurements were conducted at 9K.

##  III Low electron doping in 1uc FeSe/6uc NdFeO3/Nb:SrTiO3

In-situ angle-resolved photoemission spectroscopy (ARPES) studies were
performed on the grown 1uc-FeSe/6uc-NdFeO3/Nb:SrTiO3 (hereafter referred as
FeSe/NFO) with the 21.2eV(He-$\rm I\alpha$) helium discharge lamp from Fermi
Instrument and the VG-Scienta DA30 electron analyzer, under the ultra-high
vacuum better than 2.5 ${\times}$ 10 -11 mbar. All ARPES measurements were
conducted at 9K. The energy resolution is 6 meV and the angular resolution is
0.3∘. Around the $\Gamma$ point, a parabolic band denoted as $\alpha$ band was
observed, with its band top locate at 45 meV below the Fermi energy [Fig.
2(d)], which is higher than that of FeSe/SrTiO3 of 80 meV below the Fermi
energy and FeSe/LaFeO3 of 70 meV below the Fermi energy, suggesting that the
electron doping level of the FeSe/NdFeO3 interface is lower than that in
FeSe/SrTiO3 interface and FeSe/LaFeO3 interface.

Figure 2(e) shows the measured band structure near the M point, where two
electron bands, labeled as $\gamma_{1}$ and $\gamma_{2}$. The electron bands
$\gamma_{1}$ and $\gamma_{2}$ can be further identified by extracted momentum
distribution curves (MDCs) and fitted bands according to peak positions of
MDCS [Fig. 2(f-g)]. The sizes of the semi-minor axis ($k_{1}$) and the semi-
major axis ($k_{2}$) of the elliptical Fermi pocket were determined to be
0.096 $\AA^{-1}$ and 0.172 $\AA^{-1}$, respectively. According to Luttinger’s
theorem, the electron doping level of the FeSe/NFO interface, derived from the
Fermi surface volume, is 0.040 e${}^{-}/\rm{Fe}$, which is lower than that of
1uc FeSe/SrTiO3 interface(0.10$\sim$0.12 e${}^{-}/\rm{Fe}$) and 1uc
FeSe/LaFeO3 interface (0.087 e${}^{-}/\rm{Fe}$). To confirm this, experiments
were repeated on a series of 1uc FeSe/NdFeO3/Nb:SrTiO3 samples, and the
electron doping level ranged from 0.038 e${}^{-}/\rm{Fe}$ to 0.046
e${}^{-}/\rm{Fe}$ (refer to supplementary Fig. S3), indicating a consistently
low doping level at the FeSe/NFO interface. The band structure of the FeSe/NFO
is similar to that of the FeSe/SrTiO3 and FeSe/LaFeO3 interfaces, except for a
downward shift in the chemical potential [39, 37].

Such a low electron doping level of 1uc FeSe/NdFeO3 falls within the
underdoped region of electron-doped FeSe [44], suggesting that the charge
transfer effect at the FeSe/NdFeO3 interface is weaker compared to that of
FeSe/SrTiO3 and FeSe/LaFeO3, despite the presence of an similar quantity of
oxygen vacancies as suggested by the valence band structure in Fig. 1(h).

##  IV Suppressed nematicity in 1uc FeSe/6uc NdFeO3/Nb:SrTiO3

Figure 3: Band dispersion and fitting at M point of different samples: (a)
K-doped FeSe with the doping level of 0.046e${}^{-}/\rm{Fe}$; (b) FeSe/NFO
with doping level of 0.040 e${}^{-}/\rm{Fe}$. (c) Fitting comparison between
K-doped FeSe and FeSe/NFO. (d) Band dispersion of FeSe/NFO at the $\Gamma$
point; (e) Band dispersion of K-doped FeSe films at the $\Gamma$ point. The
band tops which do not cross the Fermi level are marked by red arrows.

Figure 3 shows the band structure of a surface potassium(K)-doped FeSe sample
and the FeSe/NdFeO3 samples in comparable doping levels. Around M, for K-doped
FeSe, the band bottom of $\gamma_{2}$ is at $E_{\rm{F}}$-65 meV and the band
bottom of $\gamma_{1}$ is at $E_{\rm{F}}$-35 meV, showing a 30 meV splitting
of the band bottoms at M [Fig. 3(a,c)]. In contrast, by fitting the peak
positions of the MDCs to a parabolic dispersion, we find that the band
$\gamma_{1}$ and the band $\gamma_{2}$ of FeSe/NdFeO3 samples exhibit the same
band bottom at $E_{\rm{F}}$-34 meV, without splitting [Fig. 3(b,d)]. Around
$\Gamma$, for K-dosed FeSe films [Fig.3(d)], there is one hole band crossing
the Fermi level from the undoped FeSe underlayer, while the topmost doped FeSe
shows two hole bands with the band top located at $E_{\rm{F}}$-10 meV for the
band most close to the Fermi level [44]. However, for FeSe/NdFeO3, the top of
hole band at the $\Gamma$ point is located further below the Fermi level
($E_{\rm{F}}$-45 meV), which is the character of 1uc FeSe/oxides [Fig. 3(e)].
Therefore, there is a stark contrast in the band structure between 1 uc
FeSe/NdFeO3 and K-doped FeSe thick films with comparable electron doping
levels (0.040 e${}^{-}/\rm{Fe}$ for FeSe/NdFeO3 and 0.046 e${}^{-}/\rm{Fe}$
for K-doped FeSe thick films [44]).

The splitting of bands $\gamma_{1}$ and $\gamma_{2}$ is commonly observed in
FeSe bulk crystals [25, 45, 49], FeSe thick films [44, 24, 39] and underdoped
K-dosed FeSe [44] below the nematic transition temperature, representing
different occupation between the $d_{xz}$ and $d_{yz}$ orbitals due to
nematicity [3, 33]. The larger energy separation of the bands reflects the
stronger strength of the nematic order [39]. Here the negligible energy
separation in 1 uc FeSe/NdFeO3 suggests an absence of nematicity. It should be
noted that such doping corresponding to a nematic regime in thick FeSe films
[44], and thus the interface plays intriguing roles in varying the band
structure and suppressing nematicity beyond simple electron doping.

Figure 4: (a) Symmetrized band dispersion of # 2 shown in Fig. 2(e), showing
the superconducting pairing gap $\Delta$. (b) Energy distribution curve at the
momentum of $k_{1}$ shown in Fig (a) with the fitted superconducting pairing
gap $\Delta_{k_{1}}$=8.5 meV and the single-particle scattering rate
$\Gamma_{1}$ of 18.4 meV. (c) Energy distribution curve at the momentum of
$k_{2}$ shown in Fig (a) with the fitted superconducting pairing gap
$\Delta_{k_{2}}$=10.3 meV and the single-particle scattering rate $\Gamma_{1}$
=11.3 meV. All the ARPES measurements were taken at 9K. (d) Phase diagram of
nematic order induced energy splitting and superconducting pairing gap for
FeSe-FeOx, FeSe/SrTiO3 and K-doped FeSe against electron doping level. The
solid-circle-shaped data refer to FeSe-FeOx. The open-circle-shaped data refer
to K-doped FeSe data adapted from ref. [44] (Wen et. al), the open-triangle-
shaped data and the open-square-shaped data refer to K-doped FeSe data adapted
from ref. [50] (Zhang et. al), the open-hexagon-shaped data refer to K-doped
FeSe data adapted from ref. [35] (Song et. al), the solid-square-shaped data
refer to well-annealed 1uc FeSe/SrTiO3 data adapted from ref. [39] (Tan et.
al), the solid-pentagon-shaped data refer to well-annealed 1uc FeSe/SrTiO3
data adapted from ref. [36] (Song et. al) and the solid-triangle-shaped data
refer to well-annealed 1uc FeSe/SrTiO3 data adapted from ref. [53] (Hu et.
al). FeSe-FeOx interfaces exhibit superconducting pairing strength enhancement
and nematicity suppression compared to K-doped FeSe.

##  V Superconducting gap and phase diagram

Figure 4(a) presents the symmetrized photoemission spectrum of 1uc-FeSe/6uc-
NdFeO3/Nb:SrTiO3 measured at 9 K at the M point. The observed back-bending
behavior is a hallmark of Bogliubov quasiparticle dispersion, and a
superconducting gap opening is observed. Figure 4(b) and Fig. 4(c) show the
energy distribution curve (EDC) and corresponding fits for the superconducting
gap at two momentum positions. According to spectral function fitting, the
superconducting gap sizes $\Delta$ at $k_{1}$ (the minor axis endpoint of the
Fermi pocket at M point) and $k_{2}$ (the major axis endpoint of the Fermi
pocket at M point) are 8.5 meV and 10.3 meV respectively. Furthermore,
superconducting spectral function fitting provides insights into the single-
particle scattering rate $\Gamma_{2}$, which reflects the level of impurity
scattering and the quality of the films. The single-particle scattering rate
coefficient $\Gamma_{2}$ is 11.3 meV at $k_{2}$ (the outer elliptical electron
pocket), which is similar to that of the high-quality 1uc FeSe/SrTiO3
heterostructures ($\Gamma_{2}$ = 12 meV) [36], indicating that the 1uc
FeSe/6uc NdFeO3/Nb:SrTiO3 samples are of good quality without strong impurity
scattering. Note that the superconducting gap is absent in K-doped FeSe films
with a comparable doping level [44, 35], suggesting that the origin of the
enhanced superconductivity in FeSe/NdFeO3 is beyond simple electron doping.

In Fig.4(d), we plot the energy scales of superconducting gap and nematic
splitting in 1uc FeSe films interfaced with various FeOx terminated oxides as
a function of electron doping [37], together with those of the surface doped
thick FeSe films [50, 44, 35] and well annealed 1uc FeSe/SrTiO3[39, 7, 14, 22,
29, 32]. For electron-doped FeSe without an oxide interface, the comparable
electron doping level of FeSe/NdFeO3 is close to the boundary of
superconductivity and in a partially suppressed nematic order state [44, 35].
For 1uc FeSe/NdFeO3 heterostructure, a superconducting pairing gap $\Delta$ of
10.3 meV is observed with no energy splitting at M point, suggesting that
nematicity is suppressed while superconducting pairing strength is enhanced by
the FeSe/NdFeO3 interface.

##  VI Discussion

Previous ARPES studies have revealed that for the 1uc FeSe/TiOx interface, the
doping level of 1uc FeSe films is pined to 0.10$\sim$0.12 e${}^{-}/\rm{Fe}$
(i.e. “magic doping”) [19, 14]. However, the well-annealed 1uc-FeSe/6uc-
NdFeO3/Nb:SrTiO3 samples have lower electron doping levels of 0.038$\sim$0.046
e${}^{-}/\rm{Fe}$, deviating from the “magic doping”. Based on the VB shift
and density of states (DOS) at EF, it is evident that the density of oxygen
vacancies is equal to or exceeds that in FeSe/LaFeO3. Therefore, the NdFeO3
acts as “infinite” charge reservoir, and the doping level of FeSe/NdFeO3 is
not constrained by oxygen vacancies but could potentially be limited by a
smaller work function mismatch. This calls for future studies into the
variations of work functions across different FeSe-FeOx and FeSe-TiOx
interfaces.

FeSe/NdFeO3 is the first well-annealed FeSe/oxide interface that exhibits low
electron doping in 1uc FeSe films, which provides new insights into the role
of the interface. It is shown that nematicity can be enhanced by tensile
strain [30, 39, 21, 51]. Consequently, it is expected nematicity would be
strongest in 1 uc FeSe/STO with the strongest tensile strain if without
interfacial electron transfer [39]. However, our results provide direct
evidence that the nematicity is even weaker in 1 uc FeSe/NdFeO3 than the
similarly doped thick FeSe with less tensile strain. Further theoretical
development is required to elucidate the origin of the suppressed nematicity
by the interface, while it is likely that the altered band structure of FeSe
by the oxide interface plays some role. In iron-based superconductors, nested
electron and hole pockets at the Fermi surface are suggested to induce strong
spin fluctuation [11] and drive nematicity [43]. When the nesting conditions
deteriorate, the nematic order is suppressed [26]. In comparison to K-dosed
thick FeSe with similar electron doping, the hole-like band of 1-uc
FeSe/NdFeO3 is far from the Fermi surface, and the significantly poorer
nesting conditions may contribute to the suppression of nematicity.

The interaction between the electrons in FeSe and interfacial optical photons
has been suggested to boost superconductivity in FeSe/STO and FeSe/LaFeO3 [37,
36, 7]. There are two optical phonon modes in NdFeO3 corresponding to the
relative motion of Fe-O atoms within the oxygen octahedron of 55.8 meV and
79.8 meV, respectively [5]. The 79.8 meV optical phonon mode is similar in
energy scale to the previously reported phonon modes in the FeSe/LaFeO3
interface[37]. Although the data statistics of the EDC at M point are not high
enough for replica band fitting, the integrated EDC shows a peak-dip-hump
structure M point, and the energy separation between the peak and hump is
around 80 meV, supporting the coupling between electrons and the FK phonons of
NdFeO3 [36, 37] (Details can be seen in supplementary Fig.S5). Furthermore,
given that FeSe/NdFeO3 is closer to the nematic phase but lacks nematic order,
it is intriguing to investigate whether nematic fluctuations play an
additional role in enhancing superconductivity in the low electron-doped
region, in addition to the cooperative effect of interfacial electron-phonon
coupling (EPC).

To summarize, we have successfully grown FeSe/NdFeO3 interface, and
demonstrate it as a new member of the FeSe/FeOx interfacial superconductor
family. Note that there are multiple magnetic ordered states in NdFeO3[41, 31,
38, 2, 15, 16], and the extending of FeSe/FeOx interfacial superconductor
family provides a new platform to engineer and study the interaction between
superconductivity and magnetism. Moreover, the charge transfer of the
interface can be controlled at the underdoped region (0.038$\sim$0.046
e${}^{-}/\rm{Fe}$), which is the first underdoped case in the well-annealed
1uc FeSe/oxide family. The observed nematicity suppression provides new
information in understanding the interaction between oxide interfaces and
FeSe.

###### Acknowledgements.

This work is supported in part by the National Science Foundation of China
under the grant Nos. 11922403, 12274085, 12074074 and 92365302, the National
Key R&D Program of the MOST of China (2023YFA1406300), the New Cornerstone
Science Foundation, the Innovation Program for Quantum Science and Technology
(Grant No. 2021ZD0302803), and Shanghai Municipal Science and Technology Major
Project (Grant No.2019SHZDZX01).

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# Nematic spin correlations pervading the phase diagram of FeSe1-xSx

Ruixian Liu  Center for Advanced Quantum Studies and Department of Physics,
Beijing Normal University, Beijing, 100875 P. R. China  Wenliang Zhang  Yuan
Wei  Photon Science Division, Swiss Light Source, Paul Scherrer Institut,
CH-5232 Villigen PSI, Switzerland  Zhen Tao  Center for Advanced Quantum
Studies and Department of Physics, Beijing Normal University, Beijing, 100875
P. R. China  Photon Science Division, Swiss Light Source, Paul Scherrer
Institut, CH-5232 Villigen PSI, Switzerland  Teguh C. Asmara  Photon Science
Division, Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI,
Switzerland  European X-Ray Free-Electron Laser Facility GmbH, 22869
Schenefeld, Germany  Yi Li  Center for Advanced Quantum Studies and Department
of Physics, Beijing Normal University, Beijing, 100875 P. R. China  Vladimir
N. Strocov  Photon Science Division, Swiss Light Source, Paul Scherrer
Institut, CH-5232 Villigen PSI, Switzerland  Rong Yu  Department of Physics,
Renmin University of China, Beijing 100872, China  Qimiao Si  Department of
Physics and Astronomy, Rice Center for Quantum Materials, Rice University,
Houston, TX 77005, USA  Thorsten Schmitt<EMAIL_ADDRESS>Photon
Science Division, Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen
PSI, Switzerland  Xingye Lu<EMAIL_ADDRESS>Center for Advanced Quantum
Studies and Department of Physics, Beijing Normal University, Beijing, 100875
P. R. China

###### Abstract

We use resonant inelastic X-ray scattering (RIXS) at the Fe-L3 edge to study
the spin excitations of uniaxial-strained and unstrained FeSe1-xSx ($0\leq
x\leq 0.21$) samples. The measurements on unstrained samples reveal dispersive
spin excitations in all doping levels, which show only minor doping dependence
in energy dispersion, lifetime, and intensity, indicating that high-energy
spin excitations are only marginally affected by sulfur doping. RIXS
measurements on uniaxial-strained samples reveal that the high-energy spin-
excitation anisotropy observed previously in FeSe is also present in the
doping range $0<x\leq 0.21$ of FeSe1-xSx. The spin-excitation anisotropy
persists to a high temperature up to $T>200$ K in $x=0.18$ and reaches a
maximum around the nematic quantum critical doping ($x_{c}\approx 0.17$).
Since the spin-excitation anisotropy directly reflects the existence of
nematic spin correlations, our results indicate that high-energy nematic spin
correlations pervade the regime of nematicity in the phase diagram and are
enhanced by the nematic quantum criticality. These results emphasize the
essential role of spin fluctuations in driving electronic nematicity and open
the door for uniaxial strain tuning of spin excitations in quantum materials
hosting strong magnetoelastic coupling and electronic nematicity.

Nematic order in quantum materials refers to an electronic state characterized
by broken rotational symmetry and retained orientational order (or
translational symmetry) Fradkin2015 . In iron-based superconductors (FeSCs),
nematic order manifests as strong $C_{2}$ symmetric electronic anisotropies in
the paramagnetic orthorhombic state with a small orthorhombic lattice
distortion $\delta=(a-b)/(a+b)$ Fernandes2014 ; Si_NRM2016 ; Si_Hussey2023 ;
Fernandes2022 ; Coldea2021 ; Chu2010 ; Chu2012 ; Yi2011 ; Lu2014 ; Kuo2016 ;
Lu2018 . Under uniaxial strain ($\varepsilon$) along the $a/b$ axis, such
anisotropy can persist to a temperature range beyond the nematic transition,
revealing the fluctuating regime in the tetragonal state Chu2010 ; Chu2012 ;
Yi2011 ; Lu2014 ; Kuo2016 ; Lu2018 . Nematic order/fluctuation has been
demonstrated to be a universal feature of FeSCs Si_Hussey2023 ; Fernandes2022
; Kuo2016 . Since electronic nematicity is essential for determining the
exotic electronic properties of FeSCs and driving other emergent orders
therein, it has attracted tremendous research interest in the past decade
Fernandes2014 ; Si_NRM2016 ; Si_Hussey2023 ; Fernandes2022 ; Coldea2021 .

FeSe1-xSx, hosting exotic electronic properties and a pristine nematic
transition without subsequent magnetic transition, has been a focus for
studying electronic nematicity and its correlation with other intertwined
order/fluctuations Si_Hussey2023 ; Coldea2018 ; Shibauchi2020 ; Coldea2021 .
Figure 1(a) illustrates the compositional phase diagram of FeSe1-xSx
Coldea2021 . The parent compound FeSe exhibits a tetragonal-to-orthorhombic
structural (nematic) transition at $T_{s}\approx 90$ K and a superconducting
transition at $T_{c}\approx 8$ K Hsu2008 ; McQueen2009 . The nematic state at
$T<T_{s}$ is a quantum-disordered magnetic state with intense
antiferromagnetic (AF) spin excitations as revealed by neutron scattering and
RIXS studies of twinned FeSe Wang2016_1 ; Wang2016_2 ; Rahn2019 ; Johnny21 ;
FWang15 ; YuSi15 . Furthermore, our previous RIXS study on uniaxial-strain
detwinned FeSe revealed underdamped spin excitations with strong anisotropy
between the excitations along $[H,0]$ and $[0,K]$ directions, which persists
to a temperature slightly above $T_{s}$ under moderate uniaxial strain and has
been suggested to be driven by spin nematicity Lu2022 . With increasing sulfur
doping, the nematic transition is gradually suppressed to $T_{s}=0$ at the
putative nematic quantum critical point (NQCP) ($x_{c}\approx 0.17$), while
$T_{c}$ increases slightly from $T_{c}\approx 8$ K for $x=0$ to $T_{c}\approx
10$ K at $x\approx 0.08$ and decrease to $T_{c}\approx 5$ K across the NQCP
Coldea2021 . Beyond the nematic ordering region at $T\lesssim T_{s}$ and
$x\lesssim x_{c}$, the nematic susceptibility of FeSe1-xSx derived from
elastoresistance measurements revealed a widely spread nematic fluctuating
regime in the tetragonal phase ( $T\gtrsim T_{s}$ and $x\gtrsim x_{c}$)
Hosoi2016 . The NQCP leads to maximized nematic susceptibility Hosoi2016 , and
induces dramatic changes in various electronic properties Licciardello2019 ;
Culo2021 ; Sato2018 ; Si_Hussey2023 . However, it is unclear how the AF spin
excitations evolve across the nematic ordering phase boundary and the NQCP. In
particular, on sulfur doping, as the nematic order weakens and disappears at
the NQCP, it remains to be explored experimentally to what extent the spin-
excitation anisotropy (nematic spin correlations) will pervade the phase
diagram and be affected by the NQCP.

Figure 1:  (a) Electronic phase diagram of FeSe1-xSx. The red vertical bars
mark the doping levels measured in the present study. (b) Schematics of
scattering geometry with a crystal glued on a uniaxial strain device. The
scattering angle was set to $2\theta_{s}=130^{\circ}$. (c) Total fluorescence
yield (TFY) X-ray absorption spectroscopy (XAS) data of FeSe1-xSx ($x=0.17$).
(d) RIXS spectra of uniaxial-strained $x=0.17$ sample measured at
$\mathbf{q}_{\shortparallel}=(0.375,0)$ and $(0,0.375),T=20$ K. The red and
green solid curves are the overall fittings of the spectra. The red and green
dashed curves are fittings of the spin excitations. The magenta dashed curves
and the blue solid curves describe the fluorescence tail and the elastic
peaks, respectively.

To address these issues, we use Fe-L3 RIXS to measure the spin excitations of
unstrained and uniaxial-strained FeSe1-xSx ($x=0-0.21$) across the NQCP (Fig.
1(a)) rixs_rmp ; chunjing_prx ; kejin13 ; johnny19 . To explore the nematic
spin correlations, one needs to apply uniaxial strain along $a/b$ axis of
FeSe1-xSx and measure the possible spin-excitation anisotropy Lu2022 . In this
work, we use a uniaxial strain device based on differential thermal expansions
of invar alloy and aluminum to apply uniaxial strain on FeSe1-xSx (Fig. 1(b))
Sunko2019 ; SI . While the measurements on unstrained samples reveal
persistent spin excitations with minor doping dependence (Figs. 2 and 3), the
measurements on uniaxial-strained samples reveal an enhancement of the spin-
excitation anisotropy near the NQCP. Moreover, the prominent spin-excitation
anisotropy persists in high doping levels ($x=0.21$) and temperature ($T>100$
K in $x=0.11$ and $T>200$ K in $x=0.18$) far beyond the nematic ordering
region, demonstrating that the high-energy nematic spin correlations pervade a
wide doping and temperature range in the phase diagram of FeSe1-xSx. Our
results demonstrate that the electronic instability underlying the nematic
fluctuating regime dominates a wide temperature and doping range in the phase
diagram and significantly affects high-energy spin excitations. This discovery
corroborates the spin-nematic picture and provides new insight for
understanding the interplay between the intertwined orders/fluctuations in
FeSe1-xSx.

The doping levels studied in this work are marked by vertical red bars in the
phase diagram of FeSe1-xSx (Fig. 1(a)). Figure 1(b) illustrates the scattering
geometry and the reciprocal space of the RIXS measurements, which were
performed near the Fe-L3 edge (incident energy $E_{i}\approx 708$ eV) as shown
in the total fluorescence yield (TFY) X-ray absorption spectroscopy (XAS) data
in Fig. 1(c). The RIXS and XAS measurements were carried out with the RIXS
experimental station at the ADRESS beamline of the Swiss Light Source at the
Paul Scherrer Institut ADRESS ; SAXES ; SI .

To apply uniaxial strain on the sample, a thin FeSe1-xSx crystal was glued
onto the titanium bridge in the center of the uniaxial-strain device using
epoxy (Fig. 1(b)). Upon cooling, the differential thermal expansion
coefficients between the aluminum frame ($\alpha\approx-24\times 10^{-6}$/K)
and the invar-alloy blocks ($\alpha\approx-2\times 10^{-6}$/K) can generate a
uniaxial strain up to
$\varepsilon=\varepsilon_{xx}-\varepsilon_{yy}\approx-0.8\%$ on the neck of
the titanium bridge at low temperature, which can be transferred to the thin
crystal glued on it Sunko2019 ; SI . The uniaxial strain on the sample’s
surface can be accurately measured using a microscope SI .

Figure 2: (a), (c) $q_{\shortparallel}$-dependent RIXS spectra along $H$
direction of unstrained $x=0.06$ and $0.19$ samples. (b) RIXS spectra along
$H$ (solid curves) and $K$ (dashed curves) directions of uniaxial-strained
$x=0.11$ sample. (d), (e) RIXS spectra measured along (d) $H/K$ and (e)
$[H,H]$ directions of uniaxial-strained $x=0.17$ sample #1. (f) RIXS spectra
along $H/K$ directions of uniaxial-strained $x=0.17$ (sample #2) and $0.18$.
The color areas mark the difference between $I_{h}(q_{\shortparallel})$ and
$I_{k}(q_{\shortparallel})$.

To facilitate the discussion of the results presented below, we denote the
RIXS spectra and the spin excitations along $H,K$ and $[H,H]$ directions as
$I_{h/k/hh}(q_{\shortparallel},E)$ and $S_{h/k/hh}(q_{\shortparallel},E)$,
respectively. The RIXS spectrum collected at $\mathbf{q}_{\shortparallel}$ can
be expressed as

$I(E)=I_{\rm fluo}(E)+S(E)+I_{\rm el}(E),$ (1)

where $I_{\rm fluo}(E)$, $I_{\rm el}(E)$, and $S(E)$ are the intensity vs.
energy loss of fluorescence, elastic peak, and spin excitation, respectively.
Figure 1(d) shows two representative RIXS spectra
$I_{h}(q_{\shortparallel},E)$ (red circles) and $I_{k}(q_{\shortparallel},E)$
(green circles) ($q_{\shortparallel}=0.375$) of an $x=0.17$ sample measured
under a uniaxial strain of $\varepsilon\approx-0.6\%$ (at $T=20$ K) along the
$b$ axis. Because the spin excitations of twined/unstrained FeSC crystals
would be $C_{4}$ symmetric
($I_{h}(q_{\shortparallel},E)=I_{k}(q_{\shortparallel},E))$ in the $[H,K]$
reciprocal space, the substantial difference between
$I_{h}(q_{\shortparallel},E)$ and $I_{k}(q_{\shortparallel},E)$ clearly
reveals an excitation anisotropy with $C_{2}$ symmetry induced by the uniaxial
strain. The fitting analysis of the spectra following Eq. (1) agrees well with
the raw data (red and green circles) SI  and reveals a strong anisotropy
between $S_{h}(q_{\shortparallel},E)$ (red dashed curve) and
$S_{k}(q_{\shortparallel},E)$ (green dashed curve), unveiling the existence of
nematic spin correlations in FeSe1-xSx.

Figure 3: (a) $E_{0}$ and $\gamma/2$ of the spin excitations along $H$
($S_{h}(q_{\shortparallel},E))$ and $[H,H]$ ($S_{hh}(q_{\shortparallel},E))$
for unstrained $x=0.06,0.11,0.17,0.18$, and $0.19$. (b) $E_{0}$ and $\gamma/2$
of the spin excitation at $H=0.375$ of unstrained samples as a function of the
doping level $x$. (c) Doping dependence of the spin-excitation intensity
$S_{h}(q_{\shortparallel})$ for $q_{\shortparallel}=0.375$ (red diamonds) and
the momentum integrated $S_{h}(q_{\shortparallel})$ in the momentum range
$q_{\shortparallel}=[0.25,0.425]$ (black circles). (d), (e) Comparison of
$E_{0}(q_{\shortparallel})$ between $S_{h}(q_{\shortparallel},E)$ and
$S_{k}(q_{\shortparallel},E)$ for (d) $x=0$ (black lines), $0.11$ (red
diamonds) and (e) $x=0.17$. (f) Energy integrated spin-excitation intensity
$S_{h}(q_{\shortparallel})$ (open symbols) and $S_{k}(q_{\shortparallel})$
(filled symbols) for $x=0.11$ (red diamonds), $0.17$ #1 (black circles), and
$0.17$ #2 (green triangles). (g) Doping-dependent spin-excitation anisotropy
$S_{h}(q_{\shortparallel})/S_{k}(q_{\shortparallel})$ for $x=0$ (blue dashed
curve), $0.11$ (red diamonds), $0.17$ (black and illuminated orange circles),
and $0.18$ (green squares). The red and black dashed curves are guides to the
eyes.

Having established the experimental strategy, we have carried out systematic
RIXS measurements on unstrained and uniaxial-strained FeSe1-xSx samples in the
doping range $x=0.06-0.21$ to sort out the development of the spin excitations
and their anisotropy (Figs. 2-4) across the electronic phase diagram. Figure 2
displays the momentum-dependent RIXS spectra measured at $T=15$ and $20$ K on
unstrained $x=0.06$ (Fig. 2(a)), strained $x=0.11$ ($\varepsilon\approx-0.8\%$
Fig. 2(b)), unstrained $x=0.19$ (Fig. 2(c)), strained $x=0.17$ #1
($\varepsilon\approx-0.6\%$, Figs. 2(d), 2(e)), and strained $x=0.18$
($\varepsilon\approx-0.4\%$, Fig. 2(f)) and the $x=0.17$ #2
($\varepsilon\approx-0.6\%$, Fig. 2(f)) samples, in which $x=0.06$ and $0.11$
are in the nematic ordering regime, $x=0.17$ at the NQCP, and $x=0.19$ in the
tetragonal phase SI . All the spectra were normalized according to the doping-
independent fluorescence in the energy-loss range of [1, 10] eV. The waterfall
plots in Fig. 2 show that dispersive spin excitations along $H/K$ and $[H,H]$
directions are observed in a wide doping range covering the nematic regime.
For the unstrained samples, the momentum-dependent RIXS spectra of $x=0.19$
are similar to those of the $x=0.06$ ($T_{s}\approx 74$ K) sample, implying
only subtle change of the spin excitations across the phase boundary of the
nematic order. For the strained samples, the RIXS spectra collected both in
the nematic ordering regime ($x=0.11$, $T_{s}\approx 65$ K), and close to the
NQCP ($x=0.17$ and $0.18$ samples), exhibit prominent excitation anisotropy
($I_{h}(q_{\shortparallel},E)$>$I_{k}(q_{\shortparallel},E)$ in Fig. 2.

To extract the information about the spin excitations and understand the spin-
excitation anisotropy quantitatively, we use a damped harmonic oscillator
function Rahn2019 ; Lu2022  to describe the spin excitations collected with
RIXS:

$S(q,E)=A\,\frac{2\,\gamma\,E}{\left(E^{2}-E_{0}^{2}\right)^{2}+(\gamma\,E)^{2}},$
(2)

where $E_{0}(\mathbf{q}_{\shortparallel})$ is the undamped energy and
$\gamma(\mathbf{q}_{\shortparallel})$ is the damping factor. Generally, the
undamped energy $E_{0}(\mathbf{q}_{\shortparallel})$ and the excitation
lifetime $\gamma(\mathbf{q}_{\shortparallel})$ are essential information to
characterize line-shape of the excitations. If in the fitting with Eq (2)
$E_{0}>\gamma/2$ ($E_{0}<\gamma/2$), the spin excitations are defined as
underdamped (overdamped).

Figure 3 summarizes the fitting results of the RIXS spectra collected in the
doping range covering the nematic phase and the NQCP ($x_{c}\approx 0.17$).
Figure 3(a) shows the energy dispersion $E_{0}(\mathbf{q_{\shortparallel}})$
(colored symbols) and the damping rate $\gamma/2$ (colored lines) for the spin
excitations of unstrained samples, for which the unstrained
$I_{h}(q_{\shortparallel},E)$ of $x=0.11$ and $0.17$ were obtained by
averaging the $I_{h}(q_{\shortparallel},E)$ and $I_{k}(q_{\shortparallel},E)$
of uniaxial-strained samples. The energy dispersion
$E_{0}(\mathbf{q_{\shortparallel}})$ exhibits negligible doping dependence
along both $[H,0]$ and $[H,H]$ directions. Moreover, the damping rate
$\gamma/2$ also shows a minor doping dependence. As $\gamma/2$ is overall
lower than $E_{0}$ in the whole momentum range studied, all the spin
excitations observed here are underdamped paramagnons. The doping dependence
of $E_{0}$ and $\gamma/2$ indicates that the high-energy spin excitations are
not affected by sulfur doping. This is further evidenced by the doping-
independent $E_{0}$ and $\gamma/2$ (Fig.3(b)), and the largely unchanged
energy-integrated intensity $S_{h}(q_{\shortparallel})$ (red diamonds in Fig.
3(c)) for $\mathbf{q_{\shortparallel}}=(0.375,0)$ in more dopings. For
$S_{h}(q_{\shortparallel}=0.375)$, $E_{0}$ is constantly larger than
$\gamma/2$, consistent with underdamped spin excitation modes. Figure 3(c)
shows the energy- and $q_{\shortparallel}$-integrated (within the momentum
integral interval ${q_{\shortparallel}}=[0.25,0.425]$) intensity of the spin
excitations along the $H$ axis of unstrained samples (black filled circles).
Again, the integrated spectral weight does not show a clear increasing or
decreasing tendency, i.e., it remains unchanged from $x=0$ to $0.19$.

Figure 4: Temperature and doping dependence of the spin-excitation anisotropy
of uniaxial-strained FeSe1-xSx. (a) Temperature dependence of the RIXS spectra
for $\mathbf{q}_{\shortparallel}=(0.375,0)$ and $(0,0.375)$ for $x=0.18$
measured at $T=20,200$ and $320$ K with $\varepsilon\approx-0.6\%$ at $T=20$
K. The inset shows the temperature-dependent intensity ratio
$S_{h}(q_{\shortparallel})/S_{k}(q_{\shortparallel})$ with
$q_{\shortparallel}=0.375$ for $x=0.11$ and $0.18$.
$S_{h}(q_{\shortparallel})/S_{k}(q_{\shortparallel})$ are obtained from the
fitting of the spin excitations. (b) Doping-dependent RIXS spectra
($x=0.15,0.17,0.18$ and $0.21$) with $\mathbf{q}_{\shortparallel}=(0.375,0)$
and $(0,0.375)$. The colored areas mark the difference between the spectra
$I_{h}(q_{\shortparallel}$) and $I_{k}(q_{\shortparallel}$). The inset shows
the ratio between $I_{h}(q_{\shortparallel}$) and $I_{k}(q_{\shortparallel})$
for $x=0.15,0.17(\\#1),0.18$ and $0.21$. (c) Doping dependence of the
$S_{h}(q_{\shortparallel})/S_{k}(q_{\shortparallel})$ with
$q_{\shortparallel}=0.375$.

Figure 3(d) exhibits the undamped energy dispersion
$E_{0}(\mathbf{q_{\shortparallel}})$ of $S_{h}(q_{\shortparallel},E)$ and
$S_{k}(q_{\shortparallel},E)$ for uniaxial-strain detwinned $x=0$ and $0.11$
($T_{s}\approx 65$ K). The $E_{0}(\mathbf{q_{\shortparallel}})$ along $H$ and
$K$ directions overlap in these two dopings. In comparison,
$S_{h}(q_{\shortparallel},E)$ and $S_{k}(q_{\shortparallel},E)$ in $x=0.17$
exhibit a clear difference in $E_{0}(\mathbf{q_{\shortparallel}})$ (Fig.
3(e)), indicating that the uniaxial strain can induce a pronounced difference
in the energy dispersion of spin excitations near the NQCP. Figure 3(f)
displays the energy integrated intensity of the spin excitations
$S_{h}(q_{\shortparallel})$ and $S_{k}(q_{\shortparallel})$ for $x=0.11$, and
$0.17$, from which we determine the ratio
$S_{h}(q_{\shortparallel})/S_{k}(q_{\shortparallel})$ (Fig. 3(g)), which
reflects the magnitude of spin-excitation anisotropy. The spin-excitation
anisotropy in $x=0.11$ is overall slightly lower than that of FeSe (blue
dashed curve), indicating that the intrinsic nematic spin correlations in the
nematic state weaken as the sulfur doping increases, which suppresses the
nematic transition.

With further increase of doping, the nematic order is completely suppressed at
$x=0.17$; i.e., this doping creates a tetragonal system at finite temperature.
Surprisingly, RIXS measurements on $x=0.17$ with $\varepsilon\approx-0.6\%$
reveal a pronounced spin-excitation anisotropy that is even stronger than
those in detwinned FeSe ($\varepsilon\approx-2\delta\approx-0.54\%$ at $T=20$
K) and FeSe0.89S0.11 ($\varepsilon\approx-0.8\%$ at $T=15$ K) (Fig. 3(g)).
Since the $x=0.17$ sample falls in the regime of nematic quantum criticality,
the enhancement of the nematic spin correlations could be driven by strong
nematic fluctuations near the NQCP fnote .

To further assess the enhancement of nematic spin correlations near the NQCP
and illustrate the nematic spin correlations in the fluctuating regime, we
show in Fig. 4 the temperature and doping dependence of the spin-excitation
anisotropy $S_{h}(q_{\shortparallel})/S_{k}(q_{\shortparallel})$ at
$q_{\shortparallel}=0.375$. We find that the spin-excitation anisotropy
$S_{h}(q_{\shortparallel})/S_{k}(q_{\shortparallel})\approx 1.65$ at $T=15$ K
in $x=0.11$ decreases only by $\sim 15\%$ when warmed to $T=100$ K
($>T_{s}=65$ K) (inset of Fig. 4(a)), indicating that the nematic spin
correlations can persist to higher temperatures well above the $T_{s}$ in the
nematic ordering regime ($x<x_{c}$). Furthermore, we show in Fig. 4(a)
$I_{h}(q_{\shortparallel},E)$ and $I_{k}(q_{\shortparallel},E)$ of $x=0.18$
(slightly higher than $x_{c}$) measured under a moderate uniaxial strain
$\varepsilon\approx-0.4\%$ ($T=20$ K). The spectra show clear difference at
$T=20$ K, generating
$S_{h}(q_{\shortparallel})/S_{k}(q_{\shortparallel})\approx 1.8$ (inset of
Fig. 4(a)). Upon warming, while the uniaxial strain was relaxed gradually, the
anisotropy, however, persists at $T=200$ K (decreased by $\sim 40\%$ ), and
finally vanishes at $T=320$ K (Fig. 4(a)).

Figure 4(b) shows the spectra $I_{h}(q_{\shortparallel},E)$ and
$I_{k}(q_{\shortparallel},E)$ measured at the same $q_{\shortparallel}=0.375$
near the regime of NQC. Similar spectral difference is observed in $x=0.15$
($T_{s}=50$ K, $\varepsilon\approx-0.6\%$) and $x=0.21$
$(\varepsilon\approx-0.4\%)$ at $T=20$ K (color areas in Fig. 4(b)). The inset
of Fig. 4(b) depicts the doping dependence of the spin excitation anisotropy
$I_{h}(q_{\shortparallel},E)/I_{k}(q_{\shortparallel},E)$ for $x=0.15$
($\varepsilon\approx-0.6\%$, blue), $x=0.17$ ($\varepsilon\approx-0.6\%$,
red), $x=0.18$ ($\varepsilon\approx-0.4\%$, green), and $x=0.21$
($\varepsilon\approx-0.4\%$, black), revealing a decreasing tendency for
higher doping. We note that the anisotropy of the RIXS spectra persists to an
energy scale $\sim 1$ eV of the electron-hole pair tail, which could be
attributed to an anisotropy in incoherent charge scattering Degiorgi2017 .

Figure 4(c) summarizes the doping-dependent spin excitation anisotropy
$S_{h}(q_{\shortparallel})/S_{k}(q_{\shortparallel})$ with
$q_{\shortparallel}=0.375$ extracted from the fitting of the spin excitations
SI . Consistent with the enhancement at $x=0.17$ (Fig. 3(g)), the anisotropy
in Fig. 4(c) reaches a maximum in the doping region $x\approx 0.15-0.18$. Such
a doping evolution of the nematic spin fluctuations is in line with the exotic
electronic properties at the NQCP ($x_{c}\approx 0.17$) Hosoi2016 ;
Licciardello2019 ; Culo2021 ; Sato2018 . It is well known that quantum
critical fluctuations usually dominate static electronic transport properties
and low-energy charge/spin dynamics Sachdev_book . The $E\sim 200$ meV nematic
spin correlations exhibiting a maximum near the NQCP strongly suggest that the
nematic fluctuations can also dominate the physics at a much higher energy
scale (under a uniaxial-strain field).

The nematic fluctuating regime in FeSCs has been well established in various
studies of (quasi-)static properties, such as the softening of the shear
modulus $C_{66}$ Anna2014 ; Anna2015 ; Fujii2018 , the divergent nematic
susceptibility obtained from elastoresistance $-2m_{66}$ Kuo2016 ; Hosoi2016 ,
and the persistence of local orthorhombicity (short-range orthorhombic
structure) in the tetragonal state of both iron pnictides and FeSe Lu2016 ;
Ben2017 ; Weiyi18 ; Ben2019 . As such local orthorhombicity persists in both
iron pnictides and iron chalcogenides, it should be a common feature of the
nematic fluctuating regime of FeSCs. Our identification of spin-excitation
anisotropy at high energies is consistent with the general notion that high-
energy fluctuations are concomitant with short-spatial-range correlations. Our
results, thus, uncover a common feature that high-energy nematic spin
fluctuations permeate across FeSCs.

In summary, we find the high-energy spin-excitations in FeSe1-xSx ($x\lesssim
0.21$) are only marginally affected by sulfur doping, and illustrate that the
nematic spin correlations pervade wide doping ($0\lesssim x\lesssim 0.21$) and
temperature region of the phase diagram, corroborating the spin-nematic
picture. The enhancement of the nematic spin correlations near the NQCP
suggests that the nematic fluctuations could affect the physics at a higher
energy scale. The strain-induced spin-excitation anisotropy in the tetragonal
state of FeSe1-xSx establishes uniaxial strain as an effective way to tune
spin and/or charge fluctuations in similar quantum materials hosting electron-
lattice coupling.

###### Acknowledgements.

The work at Beijing Normal University is supported by the National Key
Projects for Research and Development of China (Grant No. 2021YFA1400400) and
the National Natural Science Foundation of China (Grant Nos. 12174029, and
11922402) (X.L.). The RIXS experiments were carried out at the ADRESS beamline
of the Swiss Light Source at the Paul Scherrer Institut (PSI). The work at PSI
is supported by the Swiss National Science Foundation through project no.
200021_207904, 200021_178867, and the Sinergia network Mott Physics Beyond the
Heisenberg Model (MPBH) (project numbers CRSII2 160765/1 and CRSII2 141962).
Y.W. and T.C.A. acknowledge financial support from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No. 884104 (Y.W.) and No. 701647 (T.C.A.) (PSI-FELLOW-
III-3i) Work at Renmin has in part been supported by the National Science
Foundation of China Grant No. 12174441. Work at Rice was primarily supported
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
under Award No. DE-SC0018197, and by the Robert A. Welch Foundation Grant No.
C-1411. Q.S. acknowledges the hospitality of the Kavli Institute for
Theoretical Physics, supported in part by the National Science Foundation
under Grant No. NSF PHY1748958, during the program “A Quantum Universe in a
Crystal: Symmetry and Topology across the Correlation Spectrum", as well as
the hospitality of the Aspen Center for Physics, which is supported by NSF
grant No. PHY-1607611.

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where we recall that $Q_{0}$ has been set to $0$, but we keep using $Q_{0}$
for notational convenience. The curve $\Sigma$ admits a normalization
$\tilde{\Sigma}$ of genus zero which has the following explicit
parameterization (with uniformizing coordinate $z\in\mathbb{P}^{1}$)

(42)
$x(z)=-\Lambda\frac{\prod_{i=1}^{r}(P_{i}+z)}{\prod_{i=0}^{r-2}(Q_{i}-z)},\qquad
y(z)=-\frac{z\prod_{i=0}^{r-2}(Q_{i}-z)}{\Lambda\prod_{i=1}^{r}(P_{i}+z)}.$

We also define $\omega_{0,1}=ydx$ and
$\omega_{0,2}(z_{1},z_{2})=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}$ to obtain
the spectral curve $(\tilde{\Sigma},x,\omega_{0,1},\omega_{0,2})$.

We will often need to view $(\tilde{\Sigma},x,\omega_{0,1},\omega_{0,2})$ as a
family of spectral curves over a base parameterized by $\mathbf{P},\mathbf{Q}$
(see [BBC+23, Section 5.1] for a discussion on families of spectral curves).
Then, we will repeatedly use the result that the correlators are analytic in
globally admissible families as proved in [BBC+23, Theorem 5.8]. Global
admissibility of a family in our situation boils down to the verification of
the three conditions (gA1)-(gA3) given in Definition 4.9 of loc.cit.

Notice that the fiber $x^{-1}(\infty)$ is of rank $r$ and consists of the
points $z=Q_{0},\ldots,Q_{r-2},\infty$. Then, we have the following simplified
version of one of the main theorems of [BCU]:

######  Theorem 5.1.

Consider the spectral curve $(\tilde{\Sigma},x,\omega_{0,1},\omega_{0,2})$
defined from (42), and the corresponding correlators
$\left(\omega_{g,n}(z_{1},\ldots,z_{n})\right)_{g\in\mathbb{Z}_{\geq
0}\,,\,n\in\mathbb{Z}_{\geq 1}}$ constructed using the topological recursion
formula. The expansion coefficients of the correlators $\omega_{g,n}$ at
$x=\infty$ coincide with the $F_{g,n}$ of the partition function $\mathcal{Z}$
satisfying the $\mathcal{W}$-constraints (30). More precisely, for $i\in[n]$,
fix $1\leq a_{i}\leq r$. Assuming that $z_{i}$ is near $Q_{a_{i}-1}$ if
$a_{i}\in[r-1]$ or that $z_{i}$ is near $\infty$ if $a_{i}=r$, we have

$\frac{\omega_{g,n}(z_{1},\dots,z_{n})}{\mathsf{d}x(z_{1})\cdots\mathsf{d}x(z_{n})}-\frac{\delta_{g,0}\delta_{n,2}}{(x(z_{1})-x(z_{2}))^{2}}=\sum_{k_{1},\ldots,k_{n}\geq
1}F_{g,n}\big{[}\begin{smallmatrix}a_{1}&\cdots&a_{n}\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}\prod_{i=1}^{n}x(z_{i})^{-k_{i}-1}\,,$

under the assumption that the $Q_{0},\ldots,Q_{r-1}$ are pairwise distinct.

Putting together the above theorem and the $b$-Hurwitz interpretation of
$\mathcal{Z}$ from Section 4 proves that the $\omega_{g,n}$ are generating
functions for certain weighted Hurwitz numbers.

######  Proposition 5.2.

Consider the spectral curve $(\tilde{\Sigma},x,\omega_{0,1},\omega_{0,2})$
defined from (42), and corresponding correlators
$\left(\omega_{g,n}(z_{1},\ldots,z_{n})\right)_{g\in\mathbb{Z}_{\geq
0}\,,\,n\in\mathbb{Z}_{\geq 1}}$ constructed using the topological recursion
formula. Assume that for all $i\in[n]$, $z_{i}$ is near $Q_{0}=0$. Under the
identification $\Lambda=t$ we have

$\frac{\omega_{g,n}(z_{1},\dots,z_{n})}{\mathsf{d}x(z_{1})\cdots\mathsf{d}x(z_{n})}-\frac{\delta_{g,0}\delta_{n,2}}{(x(z_{1})-x(z_{2}))^{2}}=\sum_{\mu_{1},\dots,\mu_{n}\in\mathbb{Z}_{\geq
1}}H^{(0)}_{G;g}(\mu_{1},\dots,\mu_{n})|\operatorname{Aut}(\mu)|\prod_{i=1}^{n}\mu_{i}x(z_{i})^{-\mu_{i}-1}\,,$

for the weight
$G(z)=\frac{\prod_{i=1}^{r}(P_{i}+z)}{\prod_{i=1}^{r-2}(Q_{i}-z)}$, where
$\mathbf{P},\mathbf{Q}\in\mathbb{C}^{*}$.

###### Proof.

Theorem 5.1 shows that expanding the correlators $\omega_{g,n}$ such that
$z_{1},\dots,z_{n}$ are near $Q_{0}=0$, gives the coefficients
$F_{g,n}\big{[}\begin{smallmatrix}1&\cdots&1\\\
k_{1}&\cdots&k_{n}\end{smallmatrix}\big{]}$. These coefficients are precisely
the ones that appear in the reduced partition function
$\mathcal{Z}^{\operatorname{red}}$. Then, as long as
$Q_{1},\ldots,Q_{r-2}\in\mathbb{C}^{*}$ are pairwise distinct, the
specialization of Theorem 4.7 to $\mathfrak{b}=0$ proves that $\tau^{(0)}_{G}$
coincides with $\mathcal{Z}^{\operatorname{red}}$ for the weight
$G(z)=\frac{\prod_{i=1}^{r}(P_{i}+z)}{\prod_{i=1}^{r-2}(Q_{i}-z)}$, up to the
identification $\Lambda=t$. More precisely, extracting the expansion
coefficients of $\tau^{(0)}_{G}$ and $\mathcal{Z}^{\operatorname{red}}$ yields
the following equality

$F_{g,n}\big{[}\begin{smallmatrix}1&\cdots&1\\\
\mu_{1}&\cdots&\mu_{n}\end{smallmatrix}\big{]}=H^{(0)}_{G;g}(\mu_{1},\dots,\mu_{n})|\operatorname{Aut}(\mu)|\prod_{i=1}^{n}\mu_{i}\,$

which proves the statement of the proposition when $Q_{1},\ldots,Q_{r-2}$ are
pairwise disjoint.

In order to remove the restriction on $Q_{1},\ldots,Q_{r-2}$, we view our
spectral curve defined using (41) as a family over the base
$\left(\mathbb{C}^{*}\right)^{r-2}$ parameterized by $Q_{1},\dots,Q_{r-2}$.
Then, we will show that this family is globally admissible to conclude that
the correlators $\omega_{g,n}$ are analytic [BBC+23, Theorem 5.8]. If
$Q_{1},\ldots,Q_{r-2}$ are pairwise disjoint, the curve $\Sigma$ from (41) is
easily seen to be smooth, and hence automatically globally admissible. On the
other hand, if the $Q_{i}$ are not pairwise disjoint, the curve acquires a
single singular point at $x=\infty,y=0$ and hence satisfies (gA1). At any
preimage in $\tilde{\Sigma}$ of this singular point, the local parameter is
$\bar{s}=0$, and thus satisfies the local admissibility condition (gA2).
Finally for (gA3), condition C-ii is always satisfied when $\bar{s}=0$, and
checking the non-resonance condition for points with $\bar{s}=0$ is a
straightforward computation. ∎

###  5.2. TR for arbitrary rational weights

Now we extend the results of the previous subsection to prove topological
recursion for weighted Hurwitz numbers with arbitrary rational weights. We
will take appropriate limits of the statement proved in Proposition 5.2, using
the results of [BBC+23]. Consider the weight

$G(z)=\frac{\prod_{i=1}^{p}(P_{i}+z)}{\prod_{i=1}^{q}(Q_{i}-z)}$

where $p,q\geq 0$. Consider the curve $S$ defined by

(43) $x(z)=t\frac{\prod_{i=1}^{p}(P_{i}+z)}{z\prod_{i=1}^{q}(Q_{i}-z)},\qquad
y(z)=\frac{z^{2}\prod_{i=1}^{q}(Q_{i}-z)}{t\prod_{i=1}^{p}(P_{i}+z)}.$

As before, define $\omega_{0,1}=ydx$ and
$\omega_{0,2}(z_{1},z_{2})=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}$ to obtain
the spectral curve $(S,x,\omega_{0,1},\omega_{0,2})$.

######  Theorem 5.3.

Consider the correlators
$\left(\omega_{g,n}(z_{1},\ldots,z_{n})\right)_{g\in\mathbb{Z}_{\geq
0}\,,\,n\in\mathbb{Z}_{\geq 1}}$ constructed using topological recursion on
the spectral curve $(S,x,\omega_{0,1},\omega_{0,2})$. When
$z_{1},\ldots,z_{n}$ are near $0$, we have

$\frac{\omega_{g,n}(z_{1},\dots,z_{n})}{\mathsf{d}x(z_{1})\cdots\mathsf{d}x(z_{n})}-\frac{\delta_{g,0}\delta_{n,2}}{(x(z_{1})-x(z_{2}))^{2}}=\sum_{\mu_{1},\dots,\mu_{n}\in\mathbb{Z}_{\geq
1}}H^{(0)}_{G;g}(\mu_{1},\dots,\mu_{n})|\operatorname{Aut}(\mu)|\prod_{i=1}^{n}\mu_{i}x(z_{i})^{-\mu_{i}-1}\,,$

for the weight
$G(z)=\frac{\prod_{i=1}^{p}(P_{i}+z)}{\prod_{i=1}^{q}(Q_{i}-z)}$.

###### Proof.

The case of $q=p-2$ is precisely the content of Proposition 5.2 for $r=p$. We
need to address the remaining cases $q>p-2$ and $q<p-2$ separately.

The case $q<p-2$:  Consider the curve (41) with the parameter $r=p$, set
$\Lambda=tQ_{q+1}\cdots Q_{p-2}$, and denote the resulting curve by
$\Sigma_{1}$. The normalization of the curve obtained in the limit
$Q^{-1}_{q+1},\ldots,Q^{-1}_{p-2}\to 0$ of $\Sigma_{1}$ is precisely the curve
$S$. Thus, if the correlators $\omega_{g,n}$ were analytic, applying the above
limit to Proposition 5.2 for the curve
$(\tilde{\Sigma_{1}},x,\omega_{0,1},\omega_{0,2})$ would yield the statement.
The analyticity for Hurwitz numbers is immediate from the definition, as
explained in the proof of Theorem 4.8.

Let us prove the analyticity of the correlators $\omega_{g,n}$ using [BBC+23,
Theorem 5.8]. To apply this theorem, we need to check that the spectral curve
$(\tilde{\Sigma}_{1},x,\omega_{0,1},\omega_{0,2})$ viewed as a family over the
base $\mathbb{C}^{p-q-2}$ parameterized by $Q^{-1}_{q+1},\ldots,Q^{-1}_{p-2}$
is globally admissible. If all the $Q^{-1}_{q+1},\ldots,Q^{-1}_{p-2}\neq 0$,
the curve is globally admissible as proved in the last paragraph of the proof
of Proposition 5.2. If some of the $Q^{-1}_{q+1},\ldots,Q^{-1}_{p-2}$ go to
$0$, the only possible singularity of $\Sigma_{1}$ is still $(x,y)=(\infty,0)$
and thus condition (gA1) is satisfied. However, the structure of the
singularity changes – the point $z=0$ in $\tilde{\Sigma}_{1}$ is now a
preimage of this singular point. At $z=0,$ the local parameter $\bar{s}=-1$,
and the local admissibility condition (gA2) is still satisfied. Finally for
condition (gA3), condition $C-ii$ is satisfied automatically when $\bar{s}\leq
0$ (and the non-resonance for points with $\bar{s}=0$ is again an easy
calculation).

The case $q>p-2$:  Consider the curve (41) with the parameter $r=q+2$, set
$\Lambda=t/(P_{p+1}\cdots P_{q+2})$, and denote the resulting curve by
$\Sigma_{2}$. First, let us take the limit $P^{-1}_{q+2}\to 0$. In this limit,
the curve becomes reducible as is evident using the homogeneous coordinates
$[X_{0}:X_{1}]=x$ and $[Y_{0}:Y_{1}]=y$,

$Y_{1}\left(\sum_{i=1}^{q+2}Y_{1}^{i-1}(-Y_{0})^{q+2-i}X_{1}^{i-1}X_{0}^{q+2-i}\left(X_{1}\frac{e_{i-1}(P_{1},\ldots,P_{q+1})}{P_{p+1}\cdots
P_{q+1}}+(-1)^{q}X_{0}\frac{e_{i-1}(Q_{0},Q_{1},\dots,Q_{q})}{t}\right)\right)=0\,.$

However, the extra component $Y_{1}=0$ (or $y=\infty$) does not contribute to
topological recursion and can be discarded as explained in [BBC+23, Section
6.2.2]. Finally, if we send the remaining parameters
$P^{-1}_{p+1},\ldots,P^{-1}_{q+1}\to 0$ we recover the spectral curve $S$ of
interest, and we can apply the analyticity result of [BBC+23, Theorem 5.8] to
finish the proof. As the verification of global admissibility is analogous to
the previous case (the only difference being that $(x,y)=(0,\infty)$ is the
singular point), we omit this. ∎

The above theorem first appeared in [BDBKS20] and generalizes the results of
[ACEH20]. Here, we give a completely new proof using $\mathcal{W}$-algebra
representations.

######  Remark 5.4.

Typically, in papers relating topological recursion and weighted Hurwitz
theory such as [BDBKS20, ACEH20], Theorem 5.3 is formulated for the following
slightly different spectral curve:

$\tilde{x}(z)=\frac{z\prod_{i=1}^{q}(Q_{i}-z)}{t\prod_{i=1}^{p}(P_{i}+z)},\qquad\tilde{y}(z)=\frac{t\prod_{i=1}^{p}(P_{i}+z)}{\prod_{i=1}^{q}(Q_{i}-z)}.$

Then, Hurwitz numbers are obtained by expanding the corresponding correlators
$\tilde{\omega}_{g,n}$ at $z=0$ (thus, $\tilde{x}=0$) in powers of $\tilde{x}$
(as we have presented in the introduction Theorem 1.2). However, notice that
$\tilde{x}=1/x$ and $\tilde{y}d\tilde{x}=-ydx$ where $x,y$ define the curve
$S$ (43). The correlators $\tilde{\omega}_{g,n}$ and $\omega_{g,n}$ only
differ by a sign $(-1)^{n}$ which is canceled by the signs coming from
$\frac{d\tilde{x}}{\tilde{x}}=-\frac{dx}{x}$ in the expansion.

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