# Interrelationships between nematicity, antiferromagnetic spin fluctuations
and superconductivity: Role of hotspots in FeSe1-xSx revealed by high pressure
77Se NMR study

K. Rana  Ames National Laboratory, U.S. DOE, Ames, Iowa 50011, USA  Department
of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  D. V.
Ambika  Ames National Laboratory, U.S. DOE, Ames, Iowa 50011, USA  Department
of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  S. L.
Bud’ko  Ames National Laboratory, U.S. DOE, Ames, Iowa 50011, USA  Department
of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA  A. E.
Böhmer  Ames National Laboratory, U.S. DOE, Ames, Iowa 50011, USA  Department
of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
Institut für Experimentalphysik IV, Ruhr-Universität Bochum, 44801 Bochum,
Germany  P. C. Canfield  Ames National Laboratory, U.S. DOE, Ames, Iowa 50011,
USA  Department of Physics and Astronomy, Iowa State University, Ames, Iowa
50011, USA  Y. Furukawa  Ames National Laboratory, U.S. DOE, Ames, Iowa 50011,
USA  Department of Physics and Astronomy, Iowa State University, Ames, Iowa
50011, USA

###### Abstract

The sulfur-substituted FeSe, FeSe1-xSx, is one of the unique systems that
provides an independent tunability of nematicity, antiferromagnetism and
superconductivity under pressure ($p$). Recently Rana et al. [Phys. Rev. B
101, 180503(R) (2020)] reported, from 77Se nuclear magnetic resonance (NMR)
measurements on FeSe0.91S0.09 under pressure, that there exists a clear role
of nematicity on the relationship between antiferromagnetic (AFM) spin
fluctuations and superconducting transition temperature ($T_{\rm c}$) where
the AFM spin fluctuations are more effective in enhancing $T_{\rm c}$ in the
absence of nematicity than with nematicity. Motivated by the work, we carried
out 77Se NMR measurements on FeSe1-xSx with $x$ = 0.15 and 0.29 under pressure
up to 2.10 GPa to investigate the relationship in a wide range of $x$ in the
FeSe1-xSx system. Based on the new results together with the previously
reported data for $x$ = 0 [P. Wiecki et al., Phys. Rev. B 96, 180502(R)
(2017)] and 0.09 [K. Rana et al. Phys. Rev. B 101, 180503(R) (2020)], we
established a $p$ \- $x$ \- temperature ($T$) phase diagram exhibiting the
evolution of AFM spin fluctuations. From the systematic analysis of the NMR
data, we found that the superconducting (SC) state in nematic state arises
from a non Fermi liquid state with strong stripe-type AFM spin fluctuations
while the SC state without nematicity comes from a Fermi liquid state with
mild stripe-type AFM spin fluctuations. Furthermore, we show that the
previously reported impact of nematicity on the relationship between AFM
fluctuations and superconductivity holds throughout the wide range of $x$ from
$x$ = 0 to 0.29 in FeSe1-xSx under pressure. We discuss the origin of the role
of nematicity in terms of the different numbers of hotspots on Fermi surfaces
with and without nematicity.

##  I Introduction

The interplay between magnetic fluctuations, electronic nematicity and the
unconventional nature of superconductivity has received wide interest after
the discovery of high $T_{\rm c}$ superconductivity in iron pnictides
Kamihara2008 . In most Fe-based superconductors (FeSCs), superconductivity
appears close to the quantum phase transitions of two long-range orders: the
nematic order, which is an electronically driven structural transition from
high-temperature tetragonal (HTT) with C4 symmetry to low-temperature
orthorhombic (LTO) having C2 symmetry, and the antiferromagnetic (AFM) order
with spontaneously oriented Fe-3$d$ spins characterized by stripe-type spin
structure Scalipino2012 ; Johnston2010 ; Canfield2010 ; Stewart2011 . In those
systems, the nematic transition temperature ($T_{\rm s}$) is usually at or
just above the transition temperature for the AFM state, ($T_{\rm N}$), and
superconductivity emerges upon the simultaneous suppression of both the
nematic and the AFM transitions by carrier doping and/or the application of
pressure ($p$). These results clearly indicate a close relationship between
AFM and nematic phases, however, the individual contribution to
superconductivity from these two phases is difficult to separate due to the
close proximity of the two phases.

In this respect, the S-substituted FeSe system, FeSe1-xSx, provides a suitable
platform to investigate the individual contribution of nematic and AFM
fluctuations to superconductivity. FeSe1-xSx has the simplest of crystal
structures among the Fe-based superconductors, with quasi-two dimensional
Fe(Se,S) layers in the $ab$ plane, stacked along the $c$ axis. At $x$ = 0,
FeSe undergoes a nematic phase transition at $T_{\rm s}$ $\sim$ 90 K followed
by a superconducting (SC) transition at $T_{\rm c}\sim$ 8.5 K without AFM
ordering at ambient pressure Bohmer2018 ; Hsu2008 ; McQueen2009 ; Baek2015 .
This allows the study of magnetic fluctuations inside the nematic order and
its relationship with superconductivity Imai2009 . The nematic phase in FeSe
can be suppressed by pressure application, with $T_{\rm s}$ decreasing down to
32 K at $p$ = 1.5 GPa Wiecki2018 ; Wang2017 . $T_{\rm c}$ shows a complex
multi-domed structure with $p$, reaching a maximum $T_{\rm c}$ $\sim$ 37 K at
$p\sim$ 6 GPa Mizuguchi2008 ; Margadonna2009 ; Medvedev2009 . At the same
time, an AFM ordered state appears above $p$ = 0.8 GPa Terashima2015 ;
Bendele2010 , and $T_{\rm s}$ merges with $T_{\rm N}$ above $p$ = 1.7 GPa
Kothapalli2016 ; Bohmer2019 ; Gati2019 , limiting the range for studying the
effects of nematicity on superconductivity without AFM state.

The S-substitutions for Se in FeSe1-xSx is also a well known way to control
the nematic phase where $T_{\rm s}$ is suppressed to zero at the critical $x$
value, $x_{\rm c}\sim$ 0.17, with increasing $x$. $T_{\rm c}$ first increases
up to 10 K around $x$ = 0.09 making a maximum and then decreases gradually at
higher $x$ close to $x_{\rm c}$ Watson2015 ; Wiecki2018 ; Reiss2017 ;
Abdel2015 . Although AFM order is not seen at ambient pressure in the
FeSe1-xSx system, an enhancement of AFM fluctuations was found at $x$ $\sim$
0.09 by nuclear magnetic resonance (NMR) measurements Wiecki2018 . The NMR
study also suggested that AFM fluctuations are important in determining the
superconductivity in FeSe1-xSx even in the vicinity of a nematic quantum phase
transition (QPT) at $x_{\rm c}$ $\sim$ 0.17. On the other hand, recent
spectroscopic-imaging scanning tunneling microscopy Hanaguri2018 , thermal
conductivity and specific heat Sato2018  measurements show an abrupt change in
$T_{\rm c}$ around $x_{\rm c}$ Mizukami2021  and the considerable change in
the size and anisotropy of the SC gap is observed at the nematic QPT, implying
different SC states inside and outside nematic states Coldea2021 . These
results may indicate that the nematicity also plays an important role in the
SC states of FeSe1-xSx.

In fact, nematicity has been reported to affect superconductivity by changing
the symmetry Fernandes2012 ; Fernandes2014  and magnitude Wiecki2018 ;
Tanatar2016  of AFM spin fluctuations. AFM spin fluctuations in FeSCs are
characterized by the in-plane stripe-type wave vectors: $\bf{Q_{\rm
1}}=(\pi,0)$ and/or $\bf{Q_{\rm 2}}=(0,\pi)$ (using the single-iron Brillouin
zone notation) Paglione2010 ; Dai2015 . In the HTT C4 symmetric state where
nematicity is absent, the AFM spin fluctuations may originate from both
$\pm\bf{Q_{\rm 1}}$ and $\pm\bf{Q_{\rm 2}}$ wave vectors. In contrast, in the
C2 symmetric state when nematicity is present, only one of the two wave
vectors, either $\pm\bf{Q_{\rm 1}}$ or $\pm\bf{Q_{\rm 2}}$, will contribute to
AFM spin fluctuations, due to the breaking of symmetry equivalency
Fernandes2012 ; Fernandes2014 . Recent 77Se NMR studies on FeSe1-xSx under
pressure actually showed the large change in the magnitude of AFM spin
fluctuations with and without nematicity Rana2020 ; Kuwayama2021 ;
Kuwayama2019 ; Kuwayama2020 . In addition, from 77Se NMR studies on FeSe1-xSx
with $x$ = 0.09 under pressure, the impact of nematicity on the relationship
between AFM spin fluctuations and $T_{\rm c}$ has been proposed, where the AFM
spin fluctuations are more effective in enhancing superconductivity in the
absence of nematicity as compared to when it is present Rana2020 ; Rana2022 .
On the other hand, as described above, the importance of nematicity for the
appearance in superconductivity has been discussed in FeSCs Kontani2010 ;
Shibauchi2020 ; Yang2015 ; Kuo2016 ; Lederer2015 ; Lederer2017 . Therefore it
is important to investigate the relationships between AFM spin fluctuations,
nematicity, and superconductivity in a wide range of $x$ in FeSe1-xSx.

In this paper, we carried out 77Se NMR studies under pressure up to 2.1 GPa
for $x=0.15$ and $0.29$ compounds to investigate the universality of the role
of nematicity on the relationship for a wide range of $x$ in FeSe1-xSx. From
the analysis of the NMR data for the two compounds together with the previous
data for $x=0$ Wiecki2017  and $x=0.09$ Rana2020  systems under $p$, we report
that the SC state in nematic state arises from a non-Fermi liquid (nFL) with
strong AFM spin fluctuations while the SC state without nematicity arises from
a Fermi liquid (FL) with moderate AFM spin fluctuations in these systems.
Furthermore, we confirm that nematicity has a clear impact on the relationship
between AFM spin fluctuations and superconductivity throughout the FeSe1-xSx
system under $p$ as previously suggested Rana2020 . The AFM spin fluctuations
in the C4 phase were found to be more effective in enhancing $T_{\rm c}$
compared to AFM spin fluctuations in the C2 phase by a factor of $\sim 7\pm
2$. We provide a possible explanation for this observation in terms of the
higher total number of hotspots in the Fermi surface that meet the nesting
condition in the C4 symmetric state, in comparison with the case of the C2
symmetric state by a factor of 8. Finally it suggested that AFM correlation
length ($\xi_{\rm AFM}$) in the C2 states is generally longer than in the C4
phase which could be related to the different nature of SC states with and
without nematicity.

##  II Experimental Details

NMR measurements of 77Se nuclei ($I$=1/2, $\gamma_{\rm N}/2\pi=8.118$MHz/T)
were carried out using a laboratory-built phase-coherent spin-echo pulse
spectrometer. The measurements were carried out under a fixed magnetic
external field of $H$ = 7.4089 T applied either along the $c$ axis or the
tetragonal [110] direction in the $ab$ plane. The single crystals with sulfur
contents $x=0.15$ and 0.29 in FeSe1-xSx were grown using chemical vapor
transport as outlined in Refs. Wiecki2018 ; Bohmer2016 . Multiple small single
crystals were oriented and stacked with spacers inside the NMR coil. A
NiCrAl/CuBe piston-cylinder was used for application of $p$ up till $\sim$ 2
GPa. Daphne 7373 was used as the $p$ mediating fluid, and the nuclear
quadrupolar resonance (NQR) of Cu in Cu2O at 77 K was used for $p$ calibration
Fukazawa2007 ; Reyes1992 . Spectra were obtained from spin-echo signals by
using fast Fourier transform. $T_{1}$ was measured with a saturation-recovery
method and $1/T_{1}$ at each $T$ was determined by fitting the nuclear
magnetization $M$ versus time $t$ using the single exponential function
$1-M(t)/M(\infty)=e^{-t/T_{1}}$, where $M(t)$ and $M(\infty)$ are the nuclear
magnetization at $t$ after saturation and the equilibrium nuclear
magnetization at $t$ $\rightarrow$ $\infty$, respectively.

Figure 1: Pressure ($p$) and $x$ dependence of 77Se NMR spectra of FeSe1-xSx
at a temperature of 20 K under the magnetic field of 7.4089 T applied along
the tetragonal [110] direction in the $ab$ plane. The spectra for (a) $x=0$
was taken from Ref. Wiecki2017  and (b) $x=0.09$ from Ref. Rana2020  whereas
the spectra for FeSe1-xSx at ambient pressure were taken from Ref. Wiecki2018
. Red points represent the spectra in the nematic state which are fitted as
the sum (red lines) of two Lorentzian peaks (blue lines). Green line in (a) is
the spectrum in antiferromagnetic state. Single peak spectra in the HTT phase
are shown with black lines.

##  III Results

###  III.1 77Se NMR Spectrum and Knight shift

Figures 1(a)-1(d) show the 77Se NMR spectra measured at $T$ = 20 K under
various pressures ($p$ = 0 - 2.10 GPa) for $x$ = 0.15 and 0.29, together with
those for $x$ = 0 and 0.09 reported previously Wiecki2017 ; Rana2020 . Here we
applied magnetic field along [110] direction in the HTT phase ($H\parallel
ab$). As in the case of $x$ = 0 ($T_{\rm s}$ = 90 K) and 0.09 ($T_{\rm s}$ =
65 K) at ambient $p$, the two split lines of the NMR spectrum (shown in red)
are observed in nematic state for $x$ = 0.15 ($T_{\rm s}=35$ K at ambient $p$)
as shown at the bottom of Fig. 1(c). Those two peak structures are due to the
twinned nematic domains Baek2015 ; Bohmer2015 ; Wiecki2017 . As can be seen in
Fig. 1(c) for $x$ = 0.15, the splitting in the NMR spectra is not seen at $p$
= 0.2 GPa and higher (even down to 1.7 K at 0.2 GPa, as shown in the
supplementary Sup ). This indicates a very small critical pressure of $p_{\rm
c}$ $\sim$ 0.2 GPa for nematic QPT at $x$ = 0.15. In the case of $x=0.29$ ($>$
$x_{c}$ $\sim$ 0.17), no split of the NMR spectra is observed from ambient
pressure up to 1.95 GPa as shown in Fig. 1(d), showing no-nematic state in the
compound. NMR spectra for nonzero $x$ values are broader than those for $x=0$
due to the introduction of inhomogeneity with sulfur substitution of selenium.

Figure 2: Temperature and pressure dependencies of 77Se NMR Knight shift ($K$)
at various measured pressures under a magnetic field ($H$) of 7.4089 T for (a)
$K_{\rm{ab}}$ and (b) $K_{\rm{c}}$ for $x=0.15$. (c) $K_{\rm ab}$ and (d)
$K_{\rm c}$ for $x$ = 0.29.

Figures 2(a) and 2(b) show the $T$ dependences of $K$ for $H\parallel ab$
($K_{\rm{ab}}$) and for $H\parallel c$ ($K_{\rm c}$) of $x=0.15$ at various
pressures from ambient up to 2.0 GPa. With decreasing $T$, both $K_{\rm ab}$
and $K_{\rm c}$ decrease and become nearly constant below $\sim$50 K. The
values of $K_{\rm ab}$ decrease slightly with increasing $p$ while $K_{\rm c}$
shows less $p$ dependence. In particular, no clear $p$ dependence is observed
in $K_{\rm c}$ at low $T$ below $\sim$ 150 K within our experimental
uncertainty. Similar $T$ and $p$ dependences of $K_{\rm ab}$ and $K_{\rm c}$
are observed for $x=0.29$ as shown in Figs. 2(c) and 2(d).

To see the $x$ and $p$ dependences of $K$ at high and low temperatures more
clearly, we plot the $K_{\rm{ab}}$ and $K_{\rm{c}}$ measured at 295 K (top)
and at 20 K (bottom) in Fig. 3(a) and 3(b), respectively. We also plot the
average values of Knight shift ($K_{\rm{avg}}\equiv\frac{2K_{\rm{ab}}+K_{\rm
c}}{3}$) in Fig. 3(c). Here the values of $K$ for $x=0$ and 0.09 were taken
from Refs. Wiecki2017  and Rana2020 , respectively. The average values of
$K_{\rm{ab}}$ for the two peaks were used when nematic state was present. In
general, $K$ consists of the $T$-independent orbital component ($K_{0}$) and
the $T$-dependent spin component ($K_{\rm{spin}}$) which is proportional to
static uniform magnetic susceptibility and thus to the density of states at
the Fermi energy $N(E_{\rm F})$. As we show later in Sec. IV, $K_{0}$ is
nearly independent of $x$ and $p$. Therefore the $x$ and $p$ dependences of
$K_{\rm ab}$, $K_{\rm c}$ and $K_{\rm ave}$ can be attributed to the spin part
of $K$. Given those dependences, the $N(E_{\rm F})$ in FeSe1-xSx system has a
slight suppression at 295 K with increasing $x$ and/or $p$ up to $\sim$ 2.0
GPa. In contrast, the $N(E_{\rm F})$ is nearly independent of $x$ at low
temperatures with a tiny decrease with $p$. We note that the change in
$N(E_{\rm F})$ due to a Lifshitz transition under pressure suggested in the
quantum oscillations measurements for $x$ = 0.11 Reiss2019  and inferred from
NMR measurements for $x$ = 0.12 Kuwayama2021  was not detected in the $p$
dependence of $K$ values within our experimental uncertainty in the measured
FeSe1-xSx systems. It is interesting to point out that $T_{\rm c}$ varies
significantly with $x$ and $p$ even though the $N(E_{\rm F})$ is nearly
independent of those. This is in contrast to conventional BCS superconductors,
in which $N(E_{\rm F})$ generally correlates with $T_{\rm c}$. These results
strongly indicate that AFM spin fluctuations play an important role in the
appearance of SC in FeSe1-xSx, as will be discussed below.

Figure 3: Pressure and $x$ dependences of the Knight shift ($K$) under a
magnetic field ($H$) of 7.4089 T for (a) $H\parallel ab$, (b) $H\parallel c$,
and (c) the average value of Knight shift
($K_{\rm{avg}}\equiv\frac{2K_{\rm{ab}}+K_{\rm c}}{3}$) at different
temperatures at 295 K (top) and 20 K (bottom). The data for $x$ = 0 and 0.09
are from Ref. Wiecki2017  and Ref. Rana2020  respectively and the $K_{c}$
values under $p$ for $x$ = 0 are missing due to the unavailability of the
data.

###  III.2 77Se Spin-lattice Relaxation Rates ($1/T_{1}$)

Figure 4: Temperature dependence of 77Se 1/$T_{1}T$ (left axes) for $x=0.15$
at various pressures under a magnetic field ($H$) = 7.4089 T, with $H\parallel
ab$ (red circles) and $H\parallel c$ (gray circles). The green line for each
panel (right axes) shows the temperature dependence of $CK_{\rm spin,c}^{2}$
where $K_{\rm spin,c}$ is the spin part of $K$ estimated by subtracting the
$T$ independent $K_{0}$ and $C$ is a scaling factor. For all pressures
measured, $K_{0}$ and $C$ are found to be constant, 0.175 % and 8.5,
respectively. Downward black arrows show $T_{\rm c}$ under $H\parallel ab$ =
7.4089 T determined by the ${\it in~{}situ}$ ac susceptibility measurements
Sup  and downward blue arrows show the $T$ below which $1/T_{1}T$ = constant
(black dashed line) behavior is observed, defined as $T_{\rm FL}$. The inset
in each panel shows the $T$ dependence of the ratio $R$ $\equiv$
$(1/T_{1}T)_{ab}$/$(1/T_{1}T)_{c}$. The black and red horizontal lines
represent the expected values for stripe-type ($R$ = 1.5) and Néel-type ($R$ =
0.5) antiferromagnetic fluctuations, respectively.

Figures 4(a)-4(h) show the $T$ dependence of 77Se $1/T_{1}T$ for $H\parallel
ab$ [$(1/T_{1}T)_{\rm ab}$] by red circles for the various pressures at
$x=0.15$. In general, 1/$T_{1}T$ is related to the dynamic magnetic
susceptibility as $1/T_{1}T\sim\gamma^{2}_{\rm N}k_{\rm
B}\sum_{\mathbf{q}}|A(\mathbf{q})|^{2}\chi^{\prime\prime}(\mathbf{q},\omega_{\rm
N})/\omega_{\rm N}$, where $A(\mathbf{q})$ is the wave-vector $\mathbf{q}$
dependent form factor and $\chi^{\prime\prime}(\mathbf{q},\omega_{\rm N})$ is
the imaginary part of $\chi(\mathbf{q},\omega_{\rm N})$ at the Larmor
frequency $\omega_{\rm N}$ Moriya1963 . Since $K$ measures the $\mathbf{q}=0$
uniform magnetic susceptibility, by comparing the $T$ dependencies between
1/$T_{1}T$ and $K$, one can obtain information on the $T$ evolution of
$\sum_{\mathbf{q}}\chi^{\prime\prime}(\mathbf{q},\omega_{\rm N})$ with respect
to that of $\chi^{\prime}(0,0)$. Above $\sim$ 100 K, $(1/T_{1}T)_{\rm ab}$
decreases monotonically with decreasing $T$ similar to $K_{\rm{ab}}$ and
$K_{\rm{c}}$. In contrast, different $T$ dependences below $\sim$ 100 K were
observed: $K_{\rm{ab}}$ and $K_{\rm{c}}$ continue to decrease on lowering $T$
and show a nearly constant behavior below 50 K for $K_{ab}$ and 25 K for
$K_{c}$ respectively, whereas $(1/T_{1}T)_{\rm ab}$ starts increasing around
$\sim$ 100 K when decreasing $T$. This deviation is attributed to the presence
of AFM spin fluctuations as in $x=0$ and 0.09 in FeSe1-xSx Wiecki2018 ;
Rana2020 . The decrease in $1/T_{1}T$ observed around or below $T_{\rm c}$
marked by black arrows is due to SC transition. The suppression of 1/$T_{1}T$
above $T_{\rm c}$ is also observed [see Fig. 4(a)] whose origin has been
discussed in terms of a possible SC fluctuation effect Kasahara2016  or
pseudogap behavior Shi2018 .

The temperature dependence of $(1/T_{1}T)_{\rm ab}$ at low temperatures
changes when $p$ is passed $p_{\rm c}\sim$ 0.2 GPa for $x$ = 0.15. As shown in
Fig. 4(a), at ambient $p$, the $(1/T_{1}T)_{\rm ab}$ at low $T$ shows a Curie-
Weiss like enhancement which is expected for two-dimensional AFM spin
fluctuations Moriya1963 . On the other hand, for $p>p_{\rm c}$ shown in Figs.
4(c)-4(h), $(1/T_{1}T)_{\rm ab}$ moderately increases below $\sim$ 100 K
before exhibiting a constant behavior below a characteristic temperature
$T_{\rm{FL}}$ marked by blue arrows. Here we determined $T_{\rm{FL}}$ from the
$1/T_{1}T$ data for $H||ab$ since $1/T_{1}T$ are more sensitive to AFM
fluctuations for $H||ab$ than $H||c$ Wiecki2018 . A similar temperature
dependence of 1/$T_{1}T$ has been reported for $x=0.09$ Rana2020  and the
state below $T_{\rm{FL}}$ was regarded as a Fermi liquid (FL) state because it
follows the Korringa behavior where both $1/T_{1}T$ and $K^{2}$ are constant,
similar to the FL state in exchange enhanced metals Moriya1963 ; Narath1968 .
It is important to note that while the SC state with C4 symmetry arises from a
FL state with AFM spin fluctuations, the SC state with C2 symmetry arises from
the nematic state with AFM spin fluctuations that show a Curie-Weiss-like
behavior.

Figure 5: Temperature dependence of 77Se 1/$T_{1}T$ (left axes) for $x=0.29$
at various pressures under a magnetic field ($H$) of 7.4089 T, with
$H\parallel ab$ (red circles) and $H\parallel c$ (gray circles). The green
line for each panel (right axes) shows the temperature dependence of $CK_{\rm
spin,c}^{2}$ where $K_{\rm spin,c}$ is the spin part of $K$ estimated by
subtracting the $T$ independent $K_{0}$ and $C$ is a scaling factor. For all
pressures measured, $K_{0}$ and $C$ are found to be constant, 0.175 % and
7.75, respectively. Downward blue arrows show the $T_{\rm FL}$ below which
$1/T_{1}T$ = constant (black dashed line) behavior is observed. The insets in
(a), (c), (e), and (g) show the $T$ dependence of the ratio $R$ $\equiv$
$(1/T_{1}T)_{ab}$/$(1/T_{1}T)_{c}$. The black and red horizontal lines in the
insets represent the expected values for stripe-type ($R$ = 1.5) and Néel-type
($R$ = 0.5) AFM spin fluctuations, respectively.

Similarly, Figs. 5(a)-5(g) show the $T$ dependence of $(1/T_{1}T)_{\rm ab}$ by
red circles for the measured pressures at $x=0.29$. At high temperatures above
$\sim$ 100 K, $(1/T_{1}T)_{\rm ab}$ increases monotonically with increasing
$T$, similar to $K_{ab}$ and $K_{c}$ for all the measured $p$ region up till
1.95 GPa. As in the case of $x$ = 0.15, the deviation between $(1/T_{1}T)_{\rm
ab}$ and $K$ is seen below $\sim$ 100 K. The characteristic $T_{\rm{FL}}$ is
also observed at $\sim$ 20–30 K for all $p$ values, where a constancy in
$1/T_{1}T$ and $K$ (both $K_{\rm ab}$ and $K_{\rm c}$) is observed.

Since the $T$ dependence of 1/$T_{1}T$ at low temperatures seems to be
different below and above $x_{\rm c}$ and/or $p_{\rm c}$, it is important to
see whether the wave vectors associated with the AFM spin fluctuations change
or not. According to previous NMR studies on Fe pnictides KitagawaSrFe2As2 ;
Kitagawa2010 ; FukazawaKBaFe2As2  and related materials Furukawa2014 ;
Pandey2013 ; Ding2016 , the ratio, $R\equiv\frac{T_{1,c}}{T_{1,ab}}$ provides
valuable information on the $\bf{Q}$ dependence of the AFM spin fluctuations,
where $T_{1,c}$ and $T_{1,ab}$ represent the $T_{1}$ measured under
$H\parallel c$ and $H\parallel ab$, respectively. If AFM spin fluctuations are
characterized by stripe-type with $\bf{Q_{\rm 1}}$ and/or $\bf{Q_{\rm 2}}$, an
$R$ = 1.5 is expected for isotropic fluctuations while, in the case of Néel
type AFM spin fluctuations with the wave vector $\bf{Q}$ = ($\pi$, $\pi$), $R$
is expected to be 0.5. Therefore we measured $1/T_{1}T$ for $H\parallel c$ for
both $x=0.15$ and 0.29. The results are shown by gray circles in Figs. 4 and
Figs. 5, respectively where the $T$ dependence of $R$ at different pressures
is shown in the corresponding inset. At high temperatures above $\sim$ 100 K,
$1/T_{1}T$ values are nearly same for $H\parallel ab$ and $H\parallel c$
directions and $R$ $\sim$ 1 for both $x=0.15$ and 0.29. When AFM spin
fluctuations start developing as $T$ is lowered below 100 K, a difference is
seen for the two measured directions, and $R$ becomes $\sim$ 1.5 or more. This
is very similar to what was observed for $x=0.09$ Rana2020  and suggests that
AFM spin fluctuations are characterized by the stripe-type ones with
$\bf{Q_{\rm 1}}$ and/or $\bf{Q_{\rm 2}}$ in FeSe1-xSx throughout the measured
$p$ region up to $\sim$ 2 GPa.

##  IV Discussion

Figure 6: (a) $x$-pressure($p$)-temperature($T$) phase diagram of FeSe1-xSx
determined from NMR and resistivity measurements. The nematic transition
temperatures $T_{\rm s}$ was determined by the splitting of the 77Se NMR
spectra with $H\parallel ab$ in the tetragonal [110] direction which includes
data from Refs. Wiecki2017 ; Rana2020 . The SC transition temperature $T_{\rm
c}$ was determined by ${\it in~{}situ}$ ac susceptibility (closed magneta
circles) measurements at $H=0$ Sup ; Rana2020  and resistivity measurements
from Ref. Udhara2016  (magenta $\times$) and Ref. Reiss2017  (magenta +).
$T_{\rm N}$ for AFM transition temperature was determined by $\chi_{\rm AC}$
(closed green triangles) Wiecki2017  and resistivitiy measurements (green
$\times$) Udhara2016 . (b)-(e) $p-T$ Contour plots showing the magnitude of
antiferromagnetic spin fluctuations for various values of $x$ along with the
transition temperatures as described in (a). The 1/$T_{1}T$ data for $x=0$ and
0.09 are from Ref. Wiecki2017  and Ref. Rana2020 , respectively. The Fermi
liquid (FL) temperature ($T_{\rm FL}$) is the temperature at which Korringa
relation was observed where both 77Se nuclear relaxation rate divided by
temperature ($1/T_{1}T$) and 77Se Knight shift values are constant. The
temperature region where Korringa relation is not satisfied, is indicated as
non Fermi Liquid (nFL) state. The solid and dotted lines are guides for the
eyes.

In this section, we show the relationship between AFM spin fluctuations,
nematicity and superconductivity in FeSe1-xSx. To discuss the magnitude of AFM
spin fluctuations, we estimate the contribution of AFM spin fluctuations from
the observed $1/T_{1}T$ assuming that the observed $1/T_{1}T$ can be
decomposed into two components of the AFM [$(1/T_{1}T)_{\rm AFM}$] and the $q$
independent [$(1/T_{1}T)_{\rm 0}$] components: 1/$T_{1}T$ = $(1/T_{1}T)_{\rm
AFM}$ \+ $(1/T_{1}T)_{0}$. The second term $(1/T_{1}T)_{0}$ is expected to be
proportional to $K_{\rm spin}^{2}$. Here $K_{\rm spin}$ is given by $K-K_{0}$
where $K_{0}$ is the $T$-independent orbital shift. Therefore, by comparing
1/$T_{1}T$ to $K_{\rm spin}^{2}$, one can estimate $(1/T_{1}T)_{\rm AFM}$. The
green curves in Figs. 4 and 5 are the estimated temperature dependence of
$CK_{\rm spin}^{2}$; the values are shown on right axes. $C$ is a proportional
constant introduced to fit the temperature dependence of $(1/T_{1}T)_{\rm ab}$
at high temperatures above $\sim$100 K. From the fittings, we found that
$K_{0}$ = 0.175 % is independent of $p$ and $x$ and the $p$ independent values
of $C$ = 8.5 and 7.75 for for $x$ = 0.15 and 0.29, respectively. Here we used
$K_{c}$ data to compare the temperature dependence of $(1/T_{1}T)_{\rm ab}$
since the enhancements in $1/T_{1}T$ due to the stripe-type AFM spin
fluctuations are prominent along the $c$ axis Kitagawa2010 . For the cases
where $K_{c}$ values were not available as in Figs. 4(b), 4(e), 4(g) and Figs.
5 (b), 5(e), 5(f), smooth extrapolations were made to estimate $K_{c}$ based
on the data at nearest pressures for each compound. Thus, the differences
between $(1/T_{1}T)_{\rm ab}$ (red circles) and green curves are attributed to
the contributions of AFM spin fluctuations, $(1/T_{1}T)_{\rm AFM}$.

The estimated AFM contributions to $1/T_{1}T$ are shown with the phase diagram
in Figs. 6(b)-6(d) where contour plots of $(1/T_{1}T)_{\rm AFM}$ are shown for
$x$ = 0.15 and 0.29 as a function of $p$ together with the data for $x$ = 0
and 0.09 estimated from our previous papers Wiecki2017 ; Rana2020 . The
$p-x-T$ phase diagram of FeSe1-xSx in Fig. 6(a) was constructed from NMR
measurements from ambient to $p$ $\sim$ 2 GPa as well as the data available
from literature. $T_{\rm c}$ (closed magenta circles) Sup ; Rana2020  and
$T_{\rm N}$ (closed green triangles) Wiecki2017  were deduced using ${\it
in~{}situ}$ ac susceptibility measurements at zero field, and $T_{\rm s}$
(closed orange squares) Wiecki2017 ; Rana2020  were determined from the NMR
spectra. For FeSe under pressure, transition temperatures for the nematic
(open orange triangles), AFM (green crosses) and superconducting states
(magenta crosses) determined from resistivity curves were taken from Ref.
Udhara2016 . Similarly, for $p$ = 0 in FeSe1-xSx, the transition temperatures
for the nematic (orange pluses) and superconducting (magenta pluses) were
taken from Ref. Reiss2017  which were also based on resistivity measurements.

In the contour plots [Figs. 6(b)-6(e)], we also show $T_{\rm s}$ and $T_{\rm
c}$ by symbols in orange and magenta, respectively. The white squares
represent the characteristic temperature ($T_{\rm{FL}}$) below which FL states
are observed. It is clearly seen in the contour plots that AFM spin
fluctuations are more enhanced inside the nematic state than outside the
nematic state. One of the interesting phenomena in FeSe1-xSx is that a FL
state is observed only in the C4 state after suppressing the nematic state
with $p$ or $x$ throughout the presented $p-x-T$ phase diagram. A FL behavior
was also seen through resistivity measurements for $x=0.11$ under pressure
Reiss2019 . This was recently attributed as the signatures of quantum
Griffiths phase close to the nematic quantum phase transition through
magnetoresistivity measurements Reiss2021 . It is interesting to note that the
smooth extrapolation of $T_{\rm FL}$ to $T_{\rm FL}$ = 0 seems to cross the
nematic QPT pressure of $p_{\rm c}$ $\sim$ 0.5 and 0.2 GPa for $x$ = 0.09 and
0.15, respectively. Therefore our results also suggest the strong connection
between nematic QPT and the appearance of FL states.

Figure 7: Plot of $T_{\rm c}$ at zero field versus $(1/T_{1}T)_{\rm AFM}$ at
$T$ = 15 K. The values for $x=0$ are taken from Refs Wiecki2017 ; Bohmer2015 ;
Udhara2016  and for $x=0.09$ from Ref. Rana2020 . Black and blue lines show
linear fits for C4 (closed symbols) and C2 (open symbols) phases,
respectively. The three points inside the dashed black oval denote the values
taken close to the critical pressure for nematic quantum phase transition and
were not included in the linear fits. Figure 8: Schematic Fermi surfaces and
nesting with stripe-type wave vector at $q_{z}=0$ in the FeSe1-xSx system. (a)
Schematic Fermi surface for the tetragonal state based on Refs. Reiss2017 ;
Watson2017  where $q_{x}$ and $q_{y}$ are in units of $\pi/a$ where a is
related to the tetragonal lattice constant $d$, by $a\equiv d/\sqrt{2}$. (b)
Schematic Fermi surface for the detwinned nematic state based on Ref.
Watson2017  where $q_{x}$ and $q_{y}$ are in units of $\pi/a$ and $\pi/b$
where $a$ and $b$ are the orthorhombic lattice constants. Here red lines
represent the hole pockets at the $\Gamma$ point and blue lines represent the
electron pockets at the Z and A points for the tetragonal and orthorhombic
states respectively. (c-d) The number of hotspots ($n_{\rm{hs}}$) due to the
intersection of single hole and single electron pockets when either is
translated by the wave vectors $\bf{Q_{\rm 1}}$ or $\bf{Q_{\rm 2}}$ in (c)
tetragonal and (d) nematic states.

Now we discuss the relationship between $T_{\rm c}$ and the magnitude of AFM
spin fluctuations. In Fig. 7, we plot $T_{\rm c}$ at zero magnetic field
against the magnitude of AFM spin fluctuations. Here we used the values of
(1/$T_{1}T)_{\rm AFM}$ at $T$ = 15 K to represent the magnitude of AFM spin
fluctuations. As has been pointed out in Ref. Rana2020 , the data points are
mainly divided into two groups with the open (C2 state) and closed (C4 state)
squares, and $T_{\rm c}$ seems to be proportional to AFM spin fluctuations in
both C2 and C4 states as shown in the blue and black lines, respectively.
However, clearly the slope for the C4 states is higher that that for the C2
state. This indicates that the AFM spin fluctuations without nematicity
enhance $T_{\rm c}$ much more than those with nematicity. Here, the black line
has a $7\pm 2$ times higher slope than the blue line. It is interesting to
point out that the three points inside the black dashed oval between the two
lines correspond to the data close to $p_{\rm c}$, suggesting a smooth
crossover between the two linear behaviors at the nematic QPT.

One of the reasons why AFM spin fluctuations in the C4 state lead to higher
$T_{\rm c}$ compared to those in the C2 state in FeSe1-xSx could be due to the
different Fermi surfaces in the paramagnetic tetragonal and nematic
orthorhombic phases. Figures 8(a) and 8(b) show the schematic Fermi surfaces
at $q_{z}$ = 0 for the C4 and C2 states based on Refs. Reiss2017 ; Watson2017
. The main differences are seen at $\bf{q}=(q_{x},q_{y})=(0,0)$ ($\Gamma$
point) and at $\bf{q}=\pm\bf{Q_{\rm 1}}$ and $\bf{q}=\pm\bf{Q_{\rm 2}}$ (Z
points in the C4 state and A points in the C2 state). For simplification, the
orbital contribution to the Fermi surface are not indicated.

In the C4 state of FeSe, two circular hole pockets (red lines) at $\Gamma$
point and two oval shaped electron pockets (blue lines) at the Z points are
found in ARPES measurements Watson2017 . This structure of the Fermi surface
is also observed in ARPES measurements with $x=0.18$ at the low $T$ of 10 K
where nematicity is absent Reiss2017 . When the hole pocket at $\Gamma$ point
is translated by either $\bf{Q_{\rm 1}}$ or $\bf{Q_{\rm 2}}$, 16 intersections
with the electron pockets at the Z points can exist as shown in Fig. 8(c).
Here $n_{\rm hs}=16$ denotes the number of intersections when the hole pockets
are translated by one wave vector. Since the four wave vectors ($\pm\bf{Q_{\rm
1}}$ and $\pm\bf{Q_{\rm 1}}$) are allowed for C4 symmetry, a total of
$16\times 4=64$ possible intersections can be found.

In the C2 nematic state, on the other hand, an oval hole pocket (red line) at
the $\Gamma$ point and a single peanut shaped electron pocket (blue line) at
the A points are present as shown in Fig. 8(b) Watson2017 . When the hole
pocket at $\Gamma$ is translated by either $\bf{Q_{\rm 1}}$ or $\bf{Q_{\rm
2}}$ to the electron pocket at A points in the nematic state, there are 4
possible intersections (that is, $n_{\rm hs}$ = 4) for one wave vector as
shown in Fig. 8(d). Since two wave vectors (either $\pm\bf{Q_{\rm 1}}$ or
$\pm\bf{Q_{\rm 1}}$) are prominent in the nematic state, a total of $4\times
2=8$ intersections can be present.

These intersections that meet the nesting conditions are called hotspots and
are deemed important in determining $T_{\rm c}$ in high-$T_{\rm c}$ cuprates
where AFM spin fluctuations are relevant Wang2017 . The total number of the
hotspots in the C4 symmetric tetragonal state increases by a factor of 64/8=8
with respect to the C2 orthorhomibc nematic state. This number seems to be
consistent with the experimentally determined factor of $\sim$ $7\pm 2$ in the
slopes of the lines of Fig. 7. Although this is quite qualitative, our results
suggest that “hotspots” are one of the key parameters to determine $T_{\rm c}$
in Fe-based superconductors. More detailed theoretical studies for
quantitative analysis about the dependency of $T_{\rm c}$ on $n_{\rm hs}$ are
highly called for.

Figure 9: Pressure dependence of $(1/T_{1}T)_{\rm AFM}$ in FeSe1-xSx. Here the
values for $x=0$ and $x=0.09$ are taken from Refs. Wiecki2017 ; Bohmer2015
and Ref. Rana2020 , respectively.

Finally we comment on the $x$ and $p$ dependences of AFM correlation length
$\xi_{\rm AFM}$. Figure 9 shows the $x$ and $p$ dependent of
$(1/T_{1}T)_{\rm{AFM}}$ at $T$ = 15 K in FeSe1-xSx where the open and closed
symbols represent the C2 and C4 states, respectively. It is clearly seen that
the values of $(1/T_{1}T)_{\rm{AFM}}$ in the C2 phase are higher than those in
the C4 phase. According to Millis-Monien-Pines (MMP) model, $(1/T_{1}T)_{\rm
AFM}$ is proportional to the square of $\xi_{\rm AFM}$ in real space
($1/T_{1}T\propto\xi_{\rm AFM}^{2}$) Millis1990 ; Baek2020 . Therefore our
results may suggest the $\xi_{\rm AFM}$ in the C4 state is shorter than in the
nematic C2 state. It is interesting to point out that SC states with
nematicity (C2 state) and without nematicity (C4 state) are induced by AFM
spin fluctuations with different values of $\xi_{\rm AFM}$. It would also be
interesting if the change in $\xi_{\rm AFM}$ between the C4 and C2 states
relate to the BCS-BEC transition at $x_{\rm c}$ $\sim$ 0.17 observed by ARPES
measurements at ambient pressure Hashimoto2021  where longer $\xi_{\rm AFM}$
is suggested when BCS superconductivity is present and a shorter $\xi_{\rm
AFM}$ is suggested in the BEC case. If so, the shortening of $\xi_{\rm AFM}$
under $p$ in FeSe1-xSx would be compatible to the BCS-BEC transition at the
respective critical pressure for nematic QPT around 0.5 GPa for $x~{}=~{}0.09$
and 0.2 GPa for $x~{}=~{}0.15$. Further studies are called for searching BCS-
BEC transition under pressure in FeSe1-xSx.

##  V Summary

In summary, 77Se-NMR measurements were carried out in the FeSe1-xSx system
under pressure for $x$ = 0.15 and 0.29, and were analyzed together with the
previously reported measurements for $x$ = 0 Wiecki2017  and 0.09 Rana2020 ,
to provide a comprehensive study of the interrelationships between nematicity,
magnetism and superconductivity in this system. We established the $p-x-T$
phase diagram featuring the presence of nematic quantum phase transitions, the
appearance of Fermi liquid phases, superconductivity with C4 and C2 rotational
symmetries and the contour plots of the magnitude of AFM spin fluctuations.
When analyzing the dependence of superconducting transition temperature on the
magnitude of AFM spin fluctuations, the different linear relationships on the
basis of symmetry are apparent. The C4 symmetric AFM spin fluctuations are
more effective in enhancing $T_{\rm c}$ by a factor of 7 $\pm$ 2 higher than
compared to their C2 symmetric counterparts. This was explained in terms of
the presence of higher total number of hotspots in the C4 symmetric Fermi
surface compared to the Fermi surface with C2 symmetry in the FeSe1-xSx system
by a factor of 8. These results indicate that the hotspots may play an
important role in determining $T_{\rm c}$ in FeSe1-xSx. Future theoretical
investigations will be interesting to clarify the relationship between the
total number of hotspots and $T_{\rm c}$, leading a step closer towards
understanding the details of Cooper-pairing mechanism in unconventional
superconductors.

##  VI Acknowledgments

We appreciate Rafael Fernandes for helpful discussions. The research was
supported by the U.S. Department of Energy (DOE), Office of Basic Energy
Sciences, Division of Materials Sciences and Engineering. Ames Laboratory is
operated for the U.S. DOE by Iowa State University under Contract No. DE-
AC02-07CH11358.

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# Strong Local Nondeterminism of Spherical Fractional Brownian Motion

Xiaohong Lan
School of Mathematical Sciences
University of Science and Technology of China Corresponding author. Research
of X. Lan is supported by NSFC grants 11501538\. E-mail<EMAIL_ADDRESS>Yimin Xiao
Department of Statistics and Probability
Michigan State University Research of Y. Xiao is partially supported by NSF
grants DMS-1612885 and DMS-1607089. E-mail<EMAIL_ADDRESS>
###### Abstract

Let $B=\left\\{B\left(x\right),\,x\in\mathbb{S}^{2}\right\\}$ be the
fractional Brownian motion indexed by the unit sphere $\mathbb{S}^{2}$ with
index $0<H\leq\frac{1}{2}$, introduced by Istas [12]. We establish optimal
upper and lower bounds for its angular power spectrum
$\\{d_{\ell},\ell=0,1,2,\ldots\\}$, and then exploit its high-frequency
behavior to establish the property of its strong local nondeterminism of $B$.

Key words: Angular power spectrum, Karhunen-Loève expansion, Spherical
fractional Brownian motion, Strong local nondeterminism.

2010 Mathematics Subject Classification: 60G60, 60G22, 60G17.

##  1 Introduction

The spherical fractional Brownian motion (SFBM, for brevity) was introduced by
Istas in 2005 [12], as an extension of the spherical Brownian motion of Lévy
[18] as well as a spherical analogue of fractional Brownian motion indexed by
the Euclidean spaces. Later Istas [13, 14] established the Karhunen-Loève
expansion and studied quadratic variations of the spherical fractional
Brownian motion.

The purpose of this paper is to investigate the property of strong local
nondeterminism (SLND) for the spherical fractional Brownian motion. This is
motivated by studies of sample path properties of Gaussian random fields
indexed by the Euclidean space $\mathbb{R}^{N}$ and by the currently
increasing interest in stochastic modeling of spherical data in statistics,
cosmology and other applied areas (see below).

The concept of local nondeterminism (LND) of a Gaussian process was first
introduced by Berman [3] to unify and extend his methods for studying the
existence and joint continuity of local times of real-valued Gaussian
processes. Roughly speaking, a Gaussian process is said to have the LND
property if has locally approximately independent increments, see [3, Lemma
2.3] for precise description. This property allowed Berman to overcome some
difficulties caused by the complex dependence structure of a non-Markovian
Gaussian process for studying its local times. Pitt [24] and Cuzick [6]
extended Berman’s LND to Gaussian random fields. However, the property of LND
is not enough for establishing fine regularity properties such as the law of
the iterated logarithm and the uniform modulus of continuity for the local
times or self-intersection local times of Gaussian random fields. For studying
these and many other problems on Gaussian random fields, the appropriate
properties of strong local nondeterminism (SLND) have proven to be more
powerful. Instead of recalling definitions of various forms of strong local
nondeterminism for (isotropic or anisotropic) Gaussian random fields indexed
by $\mathbb{R}^{N}$ and their applications, we refer to Xiao [26, 27, 28] for
more information.

Recently, Lan, Marinucci and Xiao [16] have studied the SLND property of a
class of Gaussian random fields indexed by the unit sphere $\mathbb{S}^{2}$,
which are also called spherical Gaussian random fields. The main difference
between [16] and the aforementioned work for Gaussian fields indexed by the
Euclidean space is that [16] takes the spherical geometry of $\mathbb{S}^{2}$
into full consideration and its method relies on harmonic analysis on the
sphere. More specifically, Lan, Marinucci and Xiao have considered a centered
isotropic Gaussian random field $T=\\{T(x),x\in\mathbb{S}^{2}\\}$. That is,
$T$ satisfies

$\mathbb{E}\big{(}T(x)T(y)\big{)}={\mathbb{E}}\big{(}T(gx)T(gy)\big{)}$ (1)

for all $g\in SO(3)$ which is the group of rotations in $\mathbb{R}^{3}$. See
[20] for a systematic account on random fields on $\mathbb{S}^{2}$. By
applying harmonic analytic tools on the sphere, Lan, Marinucci and Xiao [16]
have proved that the SLND property of an isotropic Gaussian field $T$ on
$\mathbb{S}^{2}$ is determined by the high-frequency behavior of its angular
power spectrum. Moreover, by applying SLND, they have established exact
uniform modulus of continuity for a class of isotropic Gaussian fields on
$\mathbb{S}^{2}$.

Since SFBM $B=\left\\{B\left(x\right),\,x\in\mathbb{S}^{2}\right\\}$ is not
isotropic in the sense of (1), the results on SLND in [16] are not directly
applicable. In our approach we will make use of the Karhunen-Loève expansion
of the spherical fractional Brownian motion obtained by Istas [13] and derive
optimal upper and lower bounds for the coefficients
$\\{d_{\ell},\ell=0,1,\ldots\\}$ (see (7) below for the definition). These
bounds for $\\{d_{\ell}\\}$ correct the last part of Theorem 1 in [13] and
will be useful for studying the dependence structures and sample path
properties of SFBM. This paper provides an important step towards this
direction. More specifically, we demonstrate that the coefficients
$\\{d_{\ell}\\}$ play the same role as the angular power spectrum of an
isotropic Gaussian field on $\mathbb{S}^{2}$ in [16] and their high frequency
behavior determines the property of strong local nondeterminism of SFBM. For
this reason, we will also call the sequence $\\{d_{\ell},\ell=0,1,\ldots\\}$
the angular power spectrum of SFBM.

Similarly to the cases of Gaussian random fields indexed by the Euclidean
space $\mathbb{R}^{N}$ (cf. [26, 27, 28]), we expect that the SLND property in
Theorem 3.2 will be useful for studying regularity (e.g., the exact modulus of
continuity, exact modulus of nondifferentiability, etc) and fractal properties
of SFBM. This will be carried out in a subsequent paper [17].

Our analysis on SFBM and other spherical Gaussian random fields is strongly
motivated by applications in a number of scientific areas, such as geophysics,
astrophysics, cosmology, and atmospheric sciences (see e.g. [2, 4, 7, 8]).
Huge data sets from satellite missions such as the Wilkinson Microwave
Anisotropy Probe (_WMAP_) of NASA (see http://map.gsfc.nasa.gov/) and the
_Planck_ mission of the European Space Agency (see
http://sci.esa.int/planck/53103-planck-cosmology/) have been collected and
made publicly available. Spherical random fields (usually assumed to be
Gaussian) have been proposed for modeling such data sets.

Related to aforementioned aspects, we also mention that in probability and
statistics literature various isotropic or anisotropic Gaussian random fields
on $\mathbb{S}^{2}$ have been constructed and studied (see e.g. [9, 10, 11,
15, 23]). Excursion probabilities and topological properties of excursion sets
of isotropic Gaussian random fields on $\mathbb{S}^{2}$ have been studied in
[5, 21]. Many interesting questions on probabilistic and statistical
properties of anisotropic Gaussian random fields on the sphere can be raised.
In order to study these problems. it would be interesting to establish
appropriate properties of strong local nondeterminism for anisotropic Gaussian
random fields on $\mathbb{S}^{2}$.

The rest of the paper is organized as follows. In Section 2, we recall briefly
some background material on SFBM, including its Karhunen-Loève expansion from
[13], and an analysis of its random coefficients. Our main result in this
section is Theorem 2.2, which provides optimal upper and lower bounds for the
angular power spectrum $\\{d_{\ell},\ell=0,1,\ldots\\}$ of SFBM. In Section 3,
we combine the high frequency behavior of $\\{d_{\ell}\\}$ and Proposition 7
in [16] to establish the property of strong local nondeterminism for SFBM.

##  2 SFBM and asymptotic behavior of its angular power spectrum

Let $N$ be the North pole on $\mathbb{S}^{2},$ and
$d_{\mathbb{S}^{2}}\left(x,y\right)$ the geodesic distance between $x$ and $y$
on $\mathbb{S}^{2}$. Recall from Istas [12] the definition of SFBM.

###### Definition 2.1

The SFBM $B=\left\\{B\left(x\right),\,x\in\mathbb{S}^{2}\right\\}$ is a
centered Gaussian random field with

$B\left(N\right)=0,\ a.s.$ (2)

and

$\mathbb{E}\left[B\left(x\right)-B\left(y\right)\right]^{2}=d_{\mathbb{S}^{2}}\left(x,y\right)^{2H},$
(3)

for any $x,y\in\mathbb{S}^{2}$ and $0<H\leq\frac{1}{2}.$

It is well known that fractional Brownian motion indexed by $\mathbb{R}^{d}$
can be defined for every $0<H\leq 1$. This is different from the case when the
index set is $\mathbb{S}^{2}$. Istas [12] proved that SFBM exists if and only
if the Hurst index $H\in(0,\frac{1}{2}].$ When $H=\frac{1}{2}$, it is the
classical Lévy spherical Brownian motion, see [18, 22].

Our work is based on the following Karhunen-Loève expansion of
$\big{\\{}B\left(x\right),$ $x\in\mathbb{S}^{2}\big{\\}}$ proved by Istas [13,
Theorem 1]:

$B\left(x\right)=\mathop{\displaystyle\sum}\limits_{\ell=0}^{\infty}\mathop{\displaystyle\sum}\limits_{m=-\ell}^{\ell}i\sqrt{\pi
d_{\ell}}\,\varepsilon_{\ell m}\left(Y_{\ell m}\left(x\right)-Y_{\ell
m}\left(N\right)\right).$ (4)

Here $\left\\{Y_{\ell m}:\ell=0,1,...,m=-\ell,...,\ell\right\\}$ are the
spherical harmonic functions on $\mathbb{S}^{2}$; that is, they are
eigenfunctions of the spherical Laplacian:

$\Delta_{{\mathbb{S}}^{2}}=\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left\\{\sin\theta\frac{\partial}{\partial\theta}\right\\}+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}},$

where $(\theta,\phi)\in\left[0,\pi\right]\times\left[0,2\pi\right)$ denotes
the spherical coordinates of $x\in\mathbb{S}^{2}$. More precisely,
$\left\\{Y_{\ell m}:\ell=0,1,...,m=-\ell,...,\ell\right\\}$ satisfy

$\Delta_{\mathbb{S}^{2}}Y_{\ell m}(\theta,\phi)=-\ell(\ell+1)Y_{\ell
m}(\theta,\phi).$

It is known (cf. [20]) that $\left\\{Y_{\ell
m}:\ell=0,1,...,m=-\ell,...,\ell\right\\}$ form an orthonormal basis for the
space $L^{2}\left(\mathbb{S}^{2},d\sigma\right)$, where $d\sigma=\sin\theta
d\theta d\phi$ is the Lebesgue measure on $\mathbb{S}^{2}$. An explicit form
for spherical harmonics is given by

$\begin{split}Y_{\ell
m}\left(\theta,\phi\right)&=\sqrt{\frac{2\ell+1}{4\pi}\frac{\left(\ell-m\right)!}{\left(\ell+m\right)!}}P_{\ell
m}\left(\cos\theta\right)e^{im\phi},\ \ \hbox{ for }m\geq 0,\\\ Y_{\ell
m}\left(\theta,\phi\right)&=\left(-1\right)^{m}\overline{Y}_{\ell,-m}\left(\theta,\phi\right),\qquad\qquad\qquad\quad\hbox{
for }m<0,\end{split}$ (5)

where $\overline{z}$ denotes the complex conjugate of $z\in\mathbb{C},$ and
$P_{\ell m}\left(\cos\vartheta\right)$ are the associated Legendre functions
(cf. [20, pp.315–316]) defined in terms of the Legendre polynomials
$\\{P_{\ell},\ell=0,1,\ldots\\}$ as

$P_{\ell
m}\left(x\right)=(-1)^{m}(1-x^{2})^{m/2}\frac{d^{m}}{dx^{m}}P_{\ell}\left(x\right)$

for $m=0,1,\ldots,\ell$, and

$P_{\ell
m}\left(x\right)=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!}\,P_{\ell,-m}\left(x\right)$

for $m$ negative. Moreover, the following orthonormality property holds:

$\int_{\mathbb{S}^{2}}Y_{lm}\left(x\right)\overline{Y}_{l^{\prime}m^{\prime}}\left(x\right)d\sigma\left(x\right)=\delta_{l}^{l^{\prime}}\delta_{m}^{m^{\prime}}.$
(6)

Now recall from [13] that, for all $\ell\geq 0$, the coefficients $d_{\ell}$
in (4) are defined by

$d_{\ell}=\frac{1}{2\pi}\int_{\mathbb{S}^{2}}d_{\mathbb{S}^{2}}\left(x,N\right)^{2H}P_{\ell}\left(\left\langle
x,N\right\rangle\right)d\sigma\left(x\right),$ (7)

where $\left\langle\cdot,\cdot\right\rangle$ is the usual inner product in
$\mathbb{R}^{3}$. Similarly to the angular power spectrum of an isotropic
Gaussian random field on $\mathbb{S}^{2}$, the sequence
$\left\\{d_{\ell},\ell=0,1,2,...\right\\}$ plays an important role in
determining the dependence structure and other probabilistic properties of the
SFBM $\\{B\left(x\right),x\in\mathbb{S}^{2}\\}.$ Hence we will also refer to
$\left\\{d_{\ell},\ell=0,1,2,...\right\\}$ as the _angular power spectrum_ of
SFBM.

Moreover, it follows from Theorem 1 of Istas [13] that the Karhunen-Loève
expansion (4) holds in
$L^{2}\left(\mathbb{S}^{2}\times\Omega,d\sigma\otimes\mathbb{P}\right)$ sense
and in $L^{2}\left(\mathbb{P}\right)$ sense for every fixed
$x\in\mathbb{S}^{2}$. Hence one can represent the random coefficients
$\varepsilon_{\ell m},\ell=0,1,...,m=-\ell,...,\ell$ in (4) by

$\varepsilon_{\ell m}=-i\left(\pi
d_{\ell}\right)^{-1/2}\int_{\mathbb{S}^{2}}B(x)\overline{Y_{\ell
m}}(x)d\sigma(x),$

where the equality holds in $L^{2}(\mathbb{P})$ sense. Notice that
$\\{\varepsilon_{\ell m}\\}_{lm}$ is a set of complex-valued Gaussian random
variables. Obviously, $\mathbb{E}\left(\varepsilon_{\ell m}\right)=0$ for all
$\ell,m,$ due to the zero-mean property of $B(x).$ Moreover, recall (5) and
(6), we can verify that

$\varepsilon_{\ell m}=\left(-i\right)^{m}\varepsilon_{\ell,-m}\ \hbox{ and }\
\mathbb{E}\left(\varepsilon_{\ell_{1}m_{2}}\overline{\varepsilon}_{\ell_{2}m_{2}}\right)=\delta_{\ell_{2}}^{\ell_{1}}\delta_{m_{2}}^{m_{1}}.$
(8)

Therefore, $\left\\{\varepsilon_{\ell m},\ell\geq 0,m=0,1,...,\ell\right\\}$
is a set of $i.i.d.$ standard complex Gaussian random variables.

The following is the main result of this section. The bounds in (9) correct
the last part of Theorem 1 in [13]. The high-frequency behavior of $d_{\ell}$
in (9) is essential for proving the SLND property of $B$ in Theorem 3.2 below.
Together, they will allow us to study precise analytic and geometric
properties of the sample functions of SFBM. See [17] for further information.

###### Theorem 2.2

Let $\left\\{B(x),x\in\mathbb{S}^{2}\right\\}$ be the spherical fractional
Brownian motion of index $H\in(0,1/2]$. There exists a uniform constant
$K_{1}$ such that

$\begin{split}&K_{1}^{-1}\leq d_{0}\leq K_{1}\\\ K_{1}^{-1}\ell^{-(2H+2)}&\leq
d_{\ell}\leq K_{1}\ell^{-(2H+2)}\quad\hbox{ for all
}\,\ell=1,2,....\end{split}$ (9)

Proof. We work in spherical coordinates $\left(\theta,\phi\right)$ with
$\theta\in\left[0,\pi\right]$ and $\phi\in\left[0,2\pi\right].$ By the
definition of $d_{\ell}$ in (7) and a change of variable $x=\cos\theta$ we
obtain

$d_{\ell}=\int_{0}^{\pi}\theta^{2H}P_{\ell}\left(\cos\theta\right)\sin\theta
d\theta.$ (10)

Recall from [13] the Dirichlet-Mehler representation for
$P_{\ell}\left(\cos\theta\right)(0<\theta<\pi)$,

$P_{\ell}\left(\cos\theta\right)=\frac{\sqrt{2}}{\pi}\int_{\theta}^{\pi}\frac{\sin\left((\ell+\frac{1}{2})\varphi\right)}{\sqrt{\cos\theta-\cos\varphi}}d\varphi,$

we see that (10) becomes

$\begin{split}d_{\ell}&=\frac{\sqrt{2}}{\pi}\int_{0}^{\pi}\theta^{2H}\sin\theta\int_{\theta}^{\pi}\frac{\sin\left(\left(\ell+\frac{1}{2}\right)\varphi\right)}{\sqrt{\cos\theta-\cos\varphi}}d\varphi
d\theta\\\
&=\frac{\sqrt{2}}{\pi}\int_{0}^{\pi}\sin\left(\Big{(}\ell+\frac{1}{2}\Big{)}\varphi\right)\left[\int_{0}^{\varphi}\frac{\theta^{2H}\sin\theta}{\sqrt{\cos\theta-\cos\varphi}}\,d\theta\right]\,d\varphi.\end{split}$

From the elementary fact that
$\sin\frac{\theta}{2}\leq\frac{\theta}{2}\leq\theta\left[1-\frac{1}{6}\left(\frac{\theta}{2}\right)^{2}\right]\leq
2\sin\frac{\theta}{2}$ for all $0\leq\theta\leq\pi,$ it follow that

$\left(\sin\frac{\theta}{2}\right)^{2H}\leq\left(\frac{\theta}{2}\right)^{2H}\leq
2^{2H}\cdot\left(\sin\frac{\theta}{2}\right)^{2H},\quad 0\leq\theta\leq\pi.$

Hence, we have

$\widetilde{d}_{\ell}\leq d_{\ell}\leq 2^{2H}\widetilde{d}_{\ell},$ (11)

where

$\widetilde{d}_{\ell}=\frac{2^{2H+1/2}}{\pi}\int_{0}^{\pi}\sin\left(\left(\ell+\frac{1}{2}\right)\varphi\right)\int_{0}^{\varphi}\frac{\left(\sin\frac{\theta}{2}\right)^{2H}\sin\theta}{\sqrt{\cos\theta-\cos\varphi}}\,d\theta
d\varphi.$

Meanwhile, by a change of variable $u=\sin\frac{\theta}{2}$ in the inside
integral, we have

$\begin{split}\int_{0}^{\varphi}\frac{\left(\sin\frac{\theta}{2}\right)^{2H}\sin\theta}{\sqrt{\cos\theta-\cos\varphi}}d\theta&=2^{\frac{3}{2}}\int_{0}^{\sin\frac{\varphi}{2}}\frac{u^{2H+1}}{\sqrt{\sin^{2}\frac{\varphi}{2}-u^{2}}}du\\\
&=2^{\frac{1}{2}}\left(\sin\frac{\varphi}{2}\right)^{2H+1}\int_{0}^{1}\frac{v^{H}}{\sqrt{1-v}}dv\\\
&=2^{\frac{1}{2}}B\left(H+1,\frac{1}{2}\right)\left(\sin\frac{\varphi}{2}\right)^{2H+1},\end{split}$

where $B\left(\cdot,\cdot\right)$ is the Beta function defined on
$\mathbb{R}^{+}\times\mathbb{R}^{+}.$ Consequently,

$\widetilde{d}_{\ell}=\frac{2^{2H+1}}{\pi}B\left(H+1,\frac{1}{2}\right)\int_{0}^{\pi}\sin\left(\left(\ell+\frac{1}{2}\right)\varphi\right)\left(\sin\frac{\varphi}{2}\right)^{2H+1}d\varphi.$
(12)

For $\ell=0,$

$\widetilde{d}_{0}=\frac{2^{2H+1}}{\pi}B\left(H+1,\frac{1}{2}\right)\int_{0}^{\pi}\left(\sin\frac{\varphi}{2}\right)^{2H+2}d\varphi,$

which is readily seen that $\left|\widetilde{d}_{0}\right|\leq 8.$ For
$\ell\geq 1,$ we will make use of some techniques from complex analysis. Let
$z=e^{i\varphi/2},$ then the integral in (12) can be written as

$\int_{0}^{\pi}\sin\left(\left(\ell+\frac{1}{2}\right)\varphi\right)\left(\sin\frac{\varphi}{2}\right)^{2H+1}d\varphi=2\mathop{\mathrm{I}m}\int_{\mathcal{C}}f_{\ell}\left(z\right)dz,$
(13)

where $\mathcal{C}=\left\\{z:z=e^{it},\ 0\leq t\leq\pi/2\right\\}$ is the unit
circle in the first quadrant with the direction counterclockwise and
$f_{\ell}\left(z\right)$ is the principal branch of the complex function

$F_{\ell}\left(z\right)=\frac{z^{2\ell}}{2^{2H}i^{2H+1}}\left(z-\frac{1}{z}\right)^{2H+1},\quad
z\in\mathbb{C},$

Obviously, $f_{\ell}(z)$ is analytic in $\mathbb{C}\backslash\left\\{0,\pm
1\right\\}$. Moreover, let $\epsilon>0$ be an arbitary small value,
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ the two line segments defined by

$\mathcal{L}_{1}=\left\\{z:z=i\left(1-t\right),\ \epsilon\leq t\leq
1\right\\}$

and

$\mathcal{L}_{2}=\left\\{z:z=t,\epsilon\leq t\leq 1-\epsilon\right\\};$

Meantime, let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be the circles with
radius $\epsilon$ in the first and second quadrants, defined respectively by

$\mathcal{C}_{1}=\left\\{z:z=\epsilon e^{it},0\leq t\leq\pi/2\right\\}$

and

$\mathcal{C}_{2}=\left\\{z:z=\epsilon e^{i\left(\pi-t\right)}+1,0\leq
t\leq\pi/2\right\\}.$

Now consider the following integrals

$\displaystyle\int_{\mathcal{L}_{1}}f_{\ell}\left(z\right)dz$ $\displaystyle=$
$\displaystyle\frac{i^{2\ell-2H-1}}{2^{2H}i^{2H+1}}\int_{\epsilon}^{1}\left(1-t\right)^{2\ell-2H-1}\left[-\left(1-t\right)^{2}-1\right]^{2H+1}dt$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{\ell}}{2^{2H}}\int_{\epsilon}^{1}u^{2\ell-2H-1}\left[u^{2}+1\right]^{2H+1}dt,$
$\displaystyle\int_{\mathcal{L}_{2}}f_{\ell}\left(z\right)dz$ $\displaystyle=$
$\displaystyle\frac{1}{2^{2H}i^{2H+1}}\int_{\epsilon}^{1}t^{2\ell-2H-1}\left(t^{2}-1\right)^{2H+1}dt$
$\displaystyle=$
$\displaystyle\frac{i^{2H+1}}{2^{2H}}\int_{\epsilon}^{1}\left(1-v\right)^{\ell-H-1/2}v^{2H+1}dt,$

and

$\int_{\mathcal{C}_{1}}f_{\ell}\left(z\right)dz=\frac{1}{2^{2H}i^{2H+1}}\int_{0}^{\frac{\pi}{2}}\left(\epsilon
e^{i\left(\pi-t\right)}\right)^{2\ell-2H-1}\left(\epsilon^{2}e^{2i\left(\pi-t\right)}-1\right)^{2H+1}dt,$
$\int_{\mathcal{C}_{2}}f_{\ell}\left(z\right)dz=\frac{1}{2^{2H}i^{2H+1}}\int_{0}^{\frac{\pi}{2}}\left(1+\epsilon
e^{i\left(\pi-t\right)}\right)^{2\ell-2H-1}\left(\epsilon^{2}e^{2i\left(\pi-t\right)}+2\epsilon
e^{i\left(\pi-t\right)}\right)^{2H+1}dt.$

Careful calculations show that

$\mathop{\mathrm{I}m}\int_{\mathcal{L}_{1}}f_{\ell}\left(z\right)dz=0,\
\lim_{\epsilon\rightarrow
0}\left|\int_{\mathcal{C}_{j}}f_{\ell}\left(z\right)dz\right|=0,\ j=1,2$ (14)

and

$\lim_{\epsilon\rightarrow
0}\int_{\mathcal{L}_{2}}f_{\ell}\left(z\right)dz=\frac{i^{2H+1}}{2^{2H}}B\left(2H+2,\ell-H+1/2\right)$
(15)

Thus, by the Cauchy integral theorem in complex analysis, we have

$\begin{split}\int_{\mathcal{C}}f_{\ell}\left(z\right)dz&=-\lim_{\epsilon\rightarrow
0}\int_{\mathcal{L}_{1}+\mathcal{C}_{1}+\mathcal{L}_{2}+\mathcal{C}_{2}}f_{\ell}\left(z\right)dz\\\
&=-\lim_{\epsilon\rightarrow
0}\int_{\mathcal{L}_{1}}f_{\ell}\left(z\right)dz-\frac{i^{2H+1}}{2^{2H}}B\left(2H+2,\ell-H+\frac{1}{2}\right)\end{split}$

which leads to that

$\mathop{\mathrm{I}m}\int_{\mathcal{C}}f_{\ell}\left(z\right)dz=2^{-2H-1}B\left(2H+2,\ell-H+\frac{1}{2}\right)\sin\left[\Big{(}H+\frac{1}{2}\Big{)}\pi\right].$
(16)

in view of the equalities (14) and (15). Therefore, by combining the
equalities (16) and (12), (13) above, we obtain that

$\widetilde{d}_{\ell}=\frac{2}{\pi}B\left(H+1,\frac{1}{2}\right)B\left(2H+2,\ell-H+\frac{1}{2}\right)\sin\left[\Big{(}H+\frac{1}{2}\Big{)}\pi\right].$

Here, recall the formula

$B\left(a,b\right)=\frac{\Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)},a>0,b>0$

with Gamma function
$\Gamma\left(y\right)=\left(y-1\right)\Gamma\left(y-1\right)$ for any $y>1$
and the stirling’s approximation

$\Gamma\left(y\right)=\sqrt{2\pi
y}\left(\frac{y}{e}\right)^{y}\left(1+O\left(\frac{1}{y}\right)\right),\ \
\hbox{ as }\ y\rightarrow\infty,$

where the error term $O\left(\frac{1}{y}\right)$ being the same order as
$\frac{1}{y},$ we derive the following estimates: there exists a uniform
constant $C_{1}$ such that for any $\ell\geq 1,$

$C_{1}^{-1}\ell^{-\left(2H+2\right)}\leq\widetilde{d}_{\ell}\leq
C_{1}\ell^{-\left(2H+2\right)}.$

Let $K_{1}=2C_{1},$ then the inequalities in (9) are derived in view of (11).

##  3 Strong local nondeterminism

In order to prove the strong local nondeterminism property of
$\left\\{B\left(x\right),x\in\mathbb{S}^{2}\right\\},$ we first recall the
following lemma, which is a consequence of Proposition 7 in [16].

###### Lemma 3.1

Assume a sequence $\left\\{d_{\ell},\ell=0,1,...\right\\}$ satisfies the
condition (9). Then there exists a constant $C_{2}>0$ depending on $H$ only,
such that for all choices of $n\in\mathbb{N},$ all
$x,x_{1},...,x_{n}\in\mathbb{S}^{2},$ and $\gamma_{j}\in\mathbb{R},$
$j=1,2,...,n$, we have

$\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}d_{\ell}\bigg{[}Y_{\ell
m}(x)-\sum_{j=1}^{n}\gamma_{j}Y_{\ell m}(x_{j})\bigg{]}^{2}\geq
C_{2}\varepsilon^{2H},$ (17)

where $\varepsilon=\min\left\\{d_{\mathbb{S}^{2}}\left(x,x_{k}\right),\
k=1,...,n\right\\}$.

Now we are ready to state and prove the following theorem. Its conclusion is
referred to as the property of strong local nondeterminism of SFBM
$\left\\{B\left(x\right),x\in\mathbb{S}^{2}\right\\}$.

###### Theorem 3.2

For a SFBM $\left\\{B\left(x\right),x\in\mathbb{S}^{2}\right\\}$, there exists
a constant $K_{2}>0$ depending only on the Hurst index $H\in(0,1/2],$ such
that for all integers $n\geq 1$ and all $x,x_{1},...,x_{n}\in\mathbb{S}^{2},$
we have

$\mathrm{Var}\left(B\left(x\right)|B\left(x_{1}\right),...,B\left(x_{n}\right)\right)\geq
K_{2}\min_{0\leq k\leq n}d_{\mathbb{S}^{2}}\left(x,x_{k}\right)^{2H},$ (18)

where
$\mathrm{Var}\left(B\left(x\right)|B\left(x_{1}\right),...,B\left(x_{n}\right)\right)$
denotes the conditional variance of $B(x)$ given
$B\left(x_{1}\right),...,B(x_{n})$, and $x_{0}=N$.

###### Remark 3.3

Note that the strong local nondeterminism (18) here is slightly different from
the SLND property proved in [16] for isotropic Gaussian random fields: the
minimum on the right hand side of (18) is not only taken over
$x_{1},...,x_{n}$ but also over $x_{0}=N$. This is because of the assumption
$B\left(N\right)=0$ in the definition of SFBM. From statistics viewpoint,
$\mathrm{Var}\left(B\left(x\right)|B\left(x_{1}\right),...,B\left(x_{n}\right)\right)$
is the squared error of predicting the value $B(x)$, given observations of $B$
at locations $x_{1},\ldots,x_{n}$. Since we already know that
$B\left(N\right)=0$, this information may reduce the prediction error.

Proof of Theorem 3.2. It is known that for a Gaussian random field $B$,

$\mathrm{Var}\left(B(x)|B\left(x_{1}\right),...,B\left(x_{n}\right)\right)=\inf\left\\{\mathbb{E}\bigg{[}\Big{(}B(x)-\mathop{\displaystyle\sum}\limits_{j=1}^{n}\gamma_{j}B(x_{j})\Big{)}^{2}\bigg{]}\right\\},$

where the infimum is taken over all
$\left(\gamma_{1},...,\gamma_{n}\right)\in\mathbb{R}^{n}.$ Hence, in order to
establish (18), it is sufficient to prove that there exists a positive
constant $C_{3}$ such that

$\mathbb{E}\bigg{[}\Big{(}B(x)-\mathop{\displaystyle\sum}\limits_{j=1}^{n}\gamma_{j}B(x_{j})\Big{)}^{2}\bigg{]}\geq
C_{3}\,\varepsilon^{2H}$ (19)

holds for all $\gamma_{1},...,\gamma_{n}\in\mathbb{R}$, where
$\varepsilon=\min\left\\{d_{\mathbb{S}^{2}}\left(x,x_{k}\right),\
k=0,...,n\right\\}.$ Let
$\gamma_{0}=1-\mathop{\displaystyle\sum}\limits_{j=1}^{n}\gamma_{j},$ then it
follows from the Karhunen-Loève expansion (4) of $B\left(x\right)$ and the
properties of random coefficients $\\{\varepsilon_{\ell m}\\}_{lm}$ in (8)
that the left hand side of (19) is equal to

$\begin{split}&\mathbb{E}\Bigg{\\{}\bigg{[}\mathop{\displaystyle\sum}\limits_{\ell=0}^{\infty}\mathop{\displaystyle\sum}\limits_{m=-\ell}^{\ell}i\sqrt{\pi
d_{\ell}}\,\varepsilon_{\ell m}\Big{(}Y_{\ell
m}\left(x\right)-\mathop{\displaystyle\sum}\limits_{j=0}^{n}\gamma_{j}Y_{\ell
m}\left(x_{j}\right)\Big{)}\bigg{]}^{2}\Bigg{\\}}\\\
&=\mathop{\displaystyle\sum}\limits_{\ell=0}^{\infty}\mathop{\displaystyle\sum}\limits_{m=-\ell}^{\ell}\pi
d_{\ell}\mathbb{E}\big{(}\left|\varepsilon_{\ell
m}\right|^{2}\big{)}\bigg{|}Y_{\ell
m}\left(x\right)-\mathop{\displaystyle\sum}\limits_{j=0}^{n}\gamma_{j}Y_{\ell
m}\left(x_{j}\right)\bigg{|}^{2}\\\
&=\pi\mathop{\displaystyle\sum}\limits_{\ell=0}^{\infty}\mathop{\displaystyle\sum}\limits_{m=-\ell}^{\ell}d_{\ell}\bigg{|}Y_{\ell
m}\left(x\right)-\mathop{\displaystyle\sum}\limits_{j=0}^{n}\gamma_{j}Y_{\ell
m}\left(x_{j}\right)\bigg{|}^{2},\end{split}$ (20)

where $x_{0}=N.$ Therefore (19) is an immediate consequence of Lemma 3.1 in
view of the equalities (20). This completes the proof of (18).

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