# Novel Anisotropy of Upper Critical Fields in Fe1+yTe0.6Se0.4

Yongqiang Pan  Department of Physics, Southeast University, Nanjing 211189,
China  Key Laboratory of Materials Physics, Institute of Solid State Physics,
Chinese Academy of Sciences, Hefei, 230031, China  Yue Sun  Corresponding
author<EMAIL_ADDRESS>Department of Physics, Southeast University, Nanjing
211189, China  Nan Zhou  Department of Physics, Southeast University, Nanjing
211189, China  Key Laboratory of Materials Physics, Institute of Solid State
Physics, Chinese Academy of Sciences, Hefei, 230031, China  Xiaolei Yi
Department of Physics, Southeast University, Nanjing 211189, China  Jinhua
Wang  School of Electrical and Electronic Engineering, Huazhong University of
Science and Technology, Wuhan 430074, China  Zengwei Zhu  School of Electrical
and Electronic Engineering, Huazhong University of Science and Technology,
Wuhan 430074, China  Hiroyuki Mitamura  Institute for Solid State Physics, The
University of Tokyo, Kashiwa 277-8581, Japan  Masashi Tokunaga  Institute for
Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan
Zhixiang Shi  Corresponding<EMAIL_ADDRESS>Department of Physics,
Southeast University, Nanjing 211189, China

###### Abstract

Studying the upper critical field ($\mu_{0}$$H$c2) and its anisotropy of
superconductors is of great importance because it can provide an unusual
insight into the pair-breaking mechanism. Since Fe1+yTe1-xSex exhibits the
high $\mu_{0}$$H$c2 and small anisotropic superconductivity, it has attracted
considerable attention. However, some issues related to $\mu_{0}$$H$c2 are
still unknown, including the effect of excess Fe content on $\mu_{0}$$H$c2
behavior and the origin of the crossover of the $\mu_{0}H_{\rm{c2}}^{c}$ – $T$
and $\mu_{0}H_{\rm{c2}}^{ab}$ – $T$ curves. In this work, the value of
$\mu_{0}$$H$c2 of Fe1+yTe0.6Se0.4 single crystals with controlled amounts of
excess Fe was obtained by resistivity measurements over a wide range of
temperatures down to $\sim$ 1.5 K, and magnetic fields up to $\sim$ 60 T. The
crossover of the $\mu_{0}H_{\rm{c2}}^{c}$ – $T$ and $\mu_{0}H_{\rm{c2}}^{ab}$
– $T$ curves was found to be independent of the excess Fe content. The angle
dependence of $\mu_{0}H_{\rm{c2}}$ was also checked. The
$\mu_{0}H_{\rm{c2}}(\theta)$ symmetry at higher temperature near $T_{c}$ could
be fitted by anisotropic G-L model, and novel fourfold symmetry of
$\mu_{0}H_{\rm{c2}}$ at lower temperature was found. Based on our spin-locking
pairing model, the crossover behavior originates from the anisotropic spin-
paramagnetic effect, and the novel fourfold symmetry of $\mu_{0}H_{\rm{c2}}$
could be understood by our extended anisotropic G-L model.

††preprint: FeTeSe-Hc2

##  I Introduction

The upper critical field $\mu_{0}$$H$c2 is sensitive to microscopic
superconducting (SC) parameters (e.g., the SC energy gap $\triangle_{\rm{SC}}$
and the mean free path $\ell$)[1, 2, 3] and is beneficial for understanding
the unconventional superconductivity, including the coherence length $\xi$,
electronic structure, and pair-breaking mechanism[4, 5]. According to the
Bardeen-Cooper-Schrieffer (BCS) theory, the superconductivity is based on
Cooper pairs. Cooper pairs are made up of two electrons with opposite spins
and momenta, which can be depaired by an external magnetic field via two
primary mechanisms[6]. One is the orbital depairing mechanism involving the
Lorentz force (orbital depairing effect, ODE), which is dominant in the high-
temperature region. The other is the Pauli spin-paramagnetic depairing
mechanism involving the Zeeman effect (spin-paramagnetic depairing effect,
SPDE)[7, 8], which is dominant in the low-temperature region. For single-band
superconductors, according to the Werthamer, Helfand, and Hohenberg (WHH)
theory[9], the two primary mechanisms are expressed using two dimensionless
parameters, the Maki parameter $\alpha$ and spin-orbit scattering
$\lambda_{\rm{so}}$[1, 10]. For multiband superconductors,
$\mu_{0}$$H_{\rm{c2}}$ can be described successfully using the two-band BCS
model[11, 12, 4, 13, 14, 15, 16, 17] and two-band Ginzburg-Landau (G-L)
theory[18, 19]. The two-band G-L theory yields a nonlinear temperature
dependence of $\mu_{0}$$H$c2($T$) when the temperature is close to the
critical temperature $T_{\rm{c}}$, which is different from the linear behavior
expected in the single-band theory. Additionally, the anisotropy $\gamma$ of
$\mu_{0}$$H$c2 ($\gamma_{H}$ = $H_{\rm{c2}}^{ab}$/$H$${}_{\rm{c2}}^{c}$, where
$H$${}_{\rm{c2}}^{ab}$ and $H$${}_{\rm{c2}}^{c}$ are the $H$c2 when
$H$$\parallel$$ab$ and $H$$\parallel$$c$, respectively) is related to the
dimensionality and topology of the electronic structure[20], which is crucial
for understanding multiband effects.

Iron-based superconductors (IBSs) exhibit rich distinctive features, such as
the two-band effect, ODE, and SPDE, which lead to a peculiar temperature
dependence of $\mu_{0}$$H_{\rm{c2}}$($T$)[8, 21, 22]. Among IBSs, the
11-system is unique in its structural simplicity which is favorable for
probing the SC pairing mechanism. Recent reports have shown that Fe1+yTe1-xSex
presents a strong-coupling superconductivity, a strong electron correlation,
and a crossover from BCS coupling to Bose-Einstein-condensation (BEC)
coupling[23, 24, 25, 26]. For the upper critical fields,
$\mu_{0}$$H$${}_{\rm{c2}}^{c}$ of Fe1+yTe0.6Se0.4 shows a multiband behavior
without the SPDE, which can be fitted by the two-band model[12]. By contrast,
$\mu_{0}$$H$${}_{\rm{c2}}^{ab}$ shows a single-band behavior due to the SPDE.
Besides, the impurity density, especially the excess Fe stoichiometry $y$[27],
greatly affects the value of $\mu_{0}H_{\rm{c2}}$(0 K) and behavior of
$\mu_{0}H_{\rm{c2}}(T)$. In the work by Matsuura $et$ $al$,
$\mu_{0}H_{\rm{c2}}$ (0 K) depends on the impurity concentration, as does the
initial slope of $\mu_{0}H_{\rm{c2}}(T)$, which will increase with increasing
impurity content[28, 29, 30, 31, 6, 32, 3, 33, 34]. However, Fe1+yTe0.6Se0.4
samples with fewer impurities present higher $\mu_{0}H_{\rm{c2}}(T)$, which is
abnormal and need to be studied.

Figure 1:  (a) Temperature dependence of the resistivity $\rho$ reduced by
$\rho$300K under zero field for three samples. The inset shows the enlarged
region near $T_{\rm{c}}$. (b)–(g)Magnetic field dependence of resistivity of
samples S1, S2, and S3, respectively. (h)–(i) Schematics of the applied
magnetic field directions for the cases of $H$$\parallel$$ab$ and
$H$$\parallel$$c$, respectively.

Interestingly, a crossover happens in $\mu_{0}$$H$c2 at low temperature
($T_{\rm{cr}}$), i.e., $H_{\rm{c2}}^{ab}$ $>$ $H_{\rm{c2}}^{c}$ in the high-
temperature region, whereas $H_{\rm{c2}}^{ab}$ $<$ $H_{\rm{c2}}^{c}$ in the
low-temperature region. This crossover on $\mu_{0}$$H$c2 is unusual and can be
found in Ba1-xKxFe2As2[21], $A$Cr3As3[35], $A_{2}$Cr3As3 [20, 36] ($A$
represent alkali metal). It is interpreted as the orbital limiting effect
persisted at all field angles, or Ising-like spin-singlet superconductivity.
Until now, the origin of this crossover has not been understood, which might
be due to a peculiar SC pairing mechanism. In addition, the angle dependence
of $\mu_{0}H_{\rm{c2}}(\theta)$ near the crossover point will be an
interesting issue, since the crossover in $\mu_{0}H_{\rm{c2}}$($T$) implies
the change of symmetry of $\mu_{0}H_{\rm{c2}}$($\theta$)[37]. In this work, a
series of Fe1+yTe0.6Se0.4 single crystals were synthesized with different
amount of excess Fe $y$ ranging from 0 to 0.14. The values of $\mu_{0}$$H$c2
were obtained under an external magnetic field up to $\sim$60 T. The
$\mu_{0}H_{\rm{c2}}$ anisotropy was investigated at the temperatures above and
below $T_{\rm{cr}}$ on the clean crystal free from excess Fe. A spin-locking
pairing model has been proposed to explain the novel $\mu_{0}H_{\rm{c2}}$
anisotropy.

##  II Experimental Details

Fe1+yTe0.6Se0.4 single crystals were synthesized via the standard self-flux
method as shown in our previous reports[38, 39, 40]. The crystals with
different amounts of excess Fe were prepared by annealing[27]. The
determination of the amount of excess Fe can be seen in our previous
report[41, 42] labeled as S1 (as-grown sample, Fe1.14Te0.6Se0.4), S2 (half-
annealed sample, Fe1.07Te0.6Se0.4), and S3 (fully-annealed sample,
Fe1.0Te0.6Se0.4)[41]. The electrical transport measurements were performed
using a commercial PPMS-9 (Quantum Design, $\sim$9 T) and a high magnetic
field generated from a 60-T magnet with a pulse width of $\sim$70 ms at the
Wuhan National High Magnetic Field Center (WHHMF) and with a pulse width of
$\sim$36 ms at the Institute for Solid State Physics, The University of
Tokyo[43].

##  III Results and discussions

Figure 1(a) shows the reduced zero-field resistivity $\rho$/$\rho$300K (where
$\rho$300K is the resistivity at $T$ = 300 K) for the synthesized single
crystals. $T$c determined by the 50$\%$ normal-state resistivity are
$\sim$13.3 K (sample S1), $\sim$14.6 K (sample S2), and $\sim$15.2 K (sample
S3), respectively. The residual resistivity ratio $RRR$, defined as
$\rho_{\rm{300K}}$/$\rho(T_{\rm{c}}^{\rm{onset}})$, are estimated as
$\sim$0.74, $\sim$0.92, and $\sim$2 for S1, S2, and S3, respectively. The
increase of $T_{\rm{c}}$ and decrease of residual resistivity manifest the
improvement of sample quality after removing excess Fe[41, 44]. Magnetic field
dependence of resistivity of S1, S2, and S3 are plotted at in Figs. 1(b)-(g).
Temperature dependence of $\rho$/$\rho$300K ($\rho$–$T$ curves) under
different magnetic fields is shown in Fig. S1 (Supplement Materials)[42]. The
temperature dependence of $\mu_{0}$$H$c2 can be obtained from the $\rho$–$T$
and $\rho$–$H$ curves using the criterion of the 50$\%$ normal-state
resistivity. With this criterion, the effects of the vortex motion expected
from the 10$\%$ criterion and the SC fluctuation expected from the 90$\%$
criterion can be minimized[17]. Additionally, $\mu$0$H$c2 obtained from both
field sweeps (performed at the WHHMF) and temperature sweeps (using the PPMS)
overlap with each other, demonstrating the consistence of the obtained
$\mu_{0}$$H$c2. Reduced temperature dependence of
$\mu$0$H$${}_{\rm{c2}}^{c}$($T$) and $\mu$0$H$${}_{\rm{c2}}^{ab}$($T$) for the
three samples are shown in Figs. 2(a)–(c) by points. The red curves and dark
cyan dash curves present the WHH fit on $\mu$0$H$${}_{\rm{c2}}^{ab}$($T$) and
two band fit on $\mu$0$H$${}_{\rm{c2}}^{c}$($T$), respectively.

For $\mu$0$H$${}_{\rm{c2}}^{c}$($T$), the fitting using the single-band WHH
model was not successful because of the linear $\mu$0$H$${}_{\rm{c2}}^{c}$
behavior at low temperatures [42], which is related to the multigap nature of
Fe1+yTe0.6Se0.4. Therefore, the two-band model was adopted, in the following
form in the dirty limit[12]:

$\displaystyle 0=a_{0}(\ln t+U(h)(\ln t+U(\eta h))$ $\displaystyle+a_{1}(\ln
t+U(h))+a_{2}(\ln t+U(\eta h)),$ (1)

where $a$0 = 2($\lambda$11$\lambda$22-$\lambda$12$\lambda$21)/$\lambda$0, $a$1
= 1+($\lambda$11-$\lambda$22)/$\lambda$0, $a$2 =
1-($\lambda$11-$\lambda$22)/($\lambda$0/2), $\lambda$0 =
(($\lambda$11-$\lambda$22)2+4$\lambda$12$\lambda$21)1/2, $t$ = $T$/$T$c, $h$ =
$\mu_{0}$$H$${}_{\rm{c2}}^{c}$$D$1/(2$\Phi$0/$T$), $\eta$ = $D$2/$D$1, and
$U$(x) = $\Psi$(1/2+x)-$\Psi$(1/2). $\Psi$(x) is the digamma function. $D$1
and $D$2 are the diffusivity of each band. $\lambda$11 and $\lambda$22 denote
the intraband coupling constants which can be derived from the $\mu$SR
experiment and can be adjusted[45]. $\lambda$12 and $\lambda$21 are the
interband coupling constants. Here, it is assumed that the intraband coupling
dominates the $\mu$0$H$${}_{\rm{c2}}^{c}$($T$), and the interband coupling
takes the value $\lambda$12 = $\lambda$21 to reduce the number of free
parameters. The fitting results of the $\mu$0$H$${}_{\rm{c2}}^{c}$($T$) data
obtained using the two-band model are shown in Figs. 2(a)-(c) by blue dash
curves. The fitting parameters can be seen in Table I. We can find
$\lambda$12$\lambda$21 $\ll$ $\lambda$11$\lambda$22, which is similar to
MgB2[16], indicating that the interband coupling is weak. The upper critical
fields obtained from the two-band model ($\mu$0$H$${}_{\rm{c2}}^{c,\rm{TB}}$(0
K)) are 45.50 (S1), 50.88 (S2), and 52.20 T (S3). The corresponding
$\xi_{ab}$(0 K), defined by the G-L equation $\xi$ab(0 K) =
($\Phi$0/2$\pi$$\mu_{0}$$H$${}^{c}_{c2}$(0 K))1/2, are 2.69 (S1), 2.54 (S2),
and 2.51 nm (S3), respectively. The slight increase of
$\mu$0$H$${}_{\rm{c2}}^{c,\rm{TB}}$(0 K) and decrease of $\xi$ab(0 K) with
reducing of excess Fe are related to the increase of $T_{\rm{c}}$. Meanwhile,
the values of -$\mu_{0}$d$H$${}_{\rm{c2}}^{c}$/d$T$ at $T$c are 3.64 (S1),
4.76 (S2), and 6.74 T/K (S3). The increase of
-$\mu_{0}$d$H$${}_{\rm{c2}}^{c}$/d$T$ at $T$c may lead to the change of
coupling strength from BCS coupling to BEC coupling[12, 46, 47, 42]

Table 1: Parameters for Fe1+yTe0.6Se0.4. samples | S1 | S2 | S3 |
---|---|---|---|---
$RRR$ | 0.74 | 0.92 | 2 |
$T$c (K) | 13.2 | 14.6 | 15.2 |
-$\mu_{0}$d$H_{\rm{c2}}^{c}$/d$T$$\mid$T=Tc (T/K) | 3.64 | 4.76 | 6.74 |
$\mu_{0}$$H$${}_{\rm{c2}}^{c,\rm{TB}}$(0 K) (T) | 45.50 | 50.88 | 52.20 |
$\xi$ab(0 K) (nm) | 2.69 | 2.54 | 2.51 |
$\lambda$11 | 0.250 | 0.278 | 0.307 |
$\lambda$12($\lambda$21) | 0.050 | 0.006 | 0.001 |
$\lambda$22 | 0.320 | 0.244 | 0.274 |
$D_{1}$ | 0.750 | 0.258 | 0.256 |
$\eta$ | 0.237 | 0.545 | 0.600 |
-$\mu_{0}$d$H$${}_{\rm{c2}}^{ab}$/d$T$$\mid$T=Tc (T/K) | 6.289 | 10.722 | 11.716 |
$\mu_{0}$$H$${}_{\rm{c2}}^{ab,\rm{orb}}$(0 K) (T) | 56.5 | 109.4 | 122.0 |
$\mu_{0}$$H$${}_{\rm{c2}}^{ab}$(0 K) (T) | 41.7 | 44.0 | 44.5 |
Maki parameter $\alpha$ | 1.30 | 3.45 | 3.90 |
Figure 2: Reduced temperature dependence of $\mu$0$H$${}_{\rm{c2}}^{c}$ and
$\mu$0$H$${}_{\rm{c2}}^{ab}$ of samples (a) S1, (b) S2, and (c) S3. Hollow
diamond and circle represent the $\mu_{0}H_{\rm{c2}}$ obtained from HMF.
Hollow triangle represent the $\mu_{0}H_{\rm{c2}}$ obtained from PPMS. (d)
Anisotropic $\gamma_{H}$ of samples S1, S2 and S3. The red arrow indicates the
crossover points of $\mu$0$H$c2. As the black dash line shown, all crossover
are located at same $T/T_{\rm{c}}$.

For $\mu_{0}$$H$${}_{\rm{c2}}^{ab}$($T$), these
$\mu_{0}$$H$${}_{\rm{c2}}^{ab}$($T$) curves show a similar convex shape. The
$\mu_{0}$$H$${}_{\rm{c2}}^{ab}$(0 K) of all samples exceed 40 T. The WHH model
considering both the Maki parameter $\alpha$ and the spin-orbital effect
parameter $\lambda$so is used to fit $\mu_{0}$$H$${}_{\rm{c2}}^{ab}$($T$)[48].
As shown in Figs. 2(a)–(c) and Table I, the orbital field
$\mu$0$H$${}_{\rm{c2}}^{ab,\rm{orb}}$(0 K) for three samples defined by
$-$0.693$T$c$\mu_{0}$d$H$c2/d$T$$\mid$T=Tc is 56.5 (S1), 109.4 (S2), and 122.0
T (S3), these values are larger than FeSe[49] but smaller than the typical
values of 122 system[50, 51] and 112 system[17]. In order to investigate the
influence of impurities on $\mu_{0}H_{\rm{c2}}^{ab}$, several important
parameters and their variation tendency are noticed. For sample S1, a small
$\alpha$ ($\sim$1.3) results in its $\mu$0$H$${}_{\rm{c2}}^{ab}$(0 K) ($\sim$
41.7 T) being not particularly small, meaning that the SPDE is comparatively
weak. Contrarily, for sample S3, a large $\alpha$ $\sim$3.9 results in a
limited $\mu$0$H$${}_{\rm{c2}}^{ab}$(0 K) ($\sim$44.5 T), which indicates that
the SPDE is enhanced by removing excess Fe. The $RRR$ dependence of
$\mu$0$H$${}_{\rm{c2}}^{ab}$(0 K) and the Maki parameter $\alpha$ are plotted
in Fig. 3 (orange and green areas). These parameters exhibit similar
increasing behavior with increasing $RRR$. We speculated that the disorder,
i.e., excess Fe, not only suppresses the $T_{c}$, but also weakens the SPDE
and reduces $\mu_{0}$$H$${}_{\rm{c2}}^{ab}$. Meanwhile, the coherence length
along $c$-axis of S3 is also calculated to be 9.5 Å by two-band BCS model
($\xi_{c}^{\rm{BCS}}$(0 K)) [42]. This value is larger than previous report
$\sim$4.4 Å[47]. Compared with the lattice parameter $c$ $\sim$ 6 Å, this
small $\xi_{c}^{\rm{BCS}}$(0 K) indicates that Fe1+yTe0.6Se0.4 may show some
quasi-two-dimensional behavior. Furthermore, $\mu$0$H$${}_{\rm{c2}}^{ab}$ is
smaller than $\mu$0$H$${}_{\rm{c2}}^{ab,\rm{orb}}$, and the difference becomes
larger at low temperatures, indicating that the SPDE is dominant at the low-
temperature region.

Figure 3: $RRR$ dependencies of $\mu$0$H$${}_{\rm{c2}}^{ab}$(0 K), the Maki
parameter $\alpha$, and $T$cr of samples S1, S2, and S3, respectively. Those
colored areas are represent the variation tendency of parameters, its are
corresponding to colored vertical coordinates.

The crossover can be observed in Figs. 2(a)-(d). Figure 2(d) shows the
temperature dependence of the $\mu$${}_{0}H_{\rm{c2}}$ anisotropy
$\gamma_{H}$($T$) defined as
$\mu_{0}$$H$${}_{\rm{c2}}^{ab}$/$\mu_{0}$$H$${}_{\rm{c2}}^{c}$. As the
temperature decreases, $\gamma_{H}$($T$) finally drops below 1 after the
crossover temperature $T_{\rm{cr}}$ (the red arrow in the Fig. 2(d)).
Interestingly, the $T_{\rm{cr}}/T_{c}$ are unchanged in different
Fe1+yTe0.6Se0.4 samples (shown as the black dashed line, $T_{\rm{cr}}$ =
0.22$T_{\rm{c}}$ $\sim$3.2 K). The value of $T_{\rm{cr}}$ is plotted in Fig. 3
(blue area), it indicates that the crossover is disorder-robust and intrinsic
property. The crossover of $\mu_{0}$$H$c2 in two directions is a novel
phenomenon observed in Fe(Te,Se) and (Ba,K)Fe2As2[11, 21, 8], and its origin
is still unclear. Previous works have tried to explain this crossover in terms
of the strong SPDE, that suppresses the $\mu$0$H$${}_{\rm{c2}}^{ab}$ at low
temperatures[8]. However, the origin of such a strongly anisotropic SPDE
remains unknown.

Figure 4: (a) Angle $\theta$ dependent $\mu_{0}H_{\rm{c2}}(\theta)$ at $T$ =
14 K, which are shown by blue solid points. The orange curve represents the
anisotropic G-L model fitting result. Inset is a schematic of the applied
magnetic field directions. (b) Angle $\theta$ dependent
$\mu_{0}H_{\rm{c2}}(\theta)$ in the range of -15∘ $\sim$ 200∘ at $T$ = 2.3 K,
which are shown by blue solid points. The hollow points represent the 180∘
rotated points of $\mu_{0}H_{\rm{c2}}(\theta)$ in the range of -15∘ $\sim$
200∘. The orange curve represents the extended anisotropic G-L model fitting
result. (c) Schematic diagram of the spin-locking phenomenon of Cooper pairs.
The spheres represent carriers and the arrows represent their spin direction.
The pair of arrows in different layers is either parallel or antiparallel. The
green ellipse signify the Cooper pair is constitutive of two carriers.

The appearance of crossover implies a dramatic change on $\mu_{0}H_{\rm{c2}}$
anisotropy. In order to investigate the symmetry of $\mu_{0}H_{\rm{c2}}$ above
and below $T_{\rm{cr}}$, we further measured the angle $\theta$ ($\theta$ is
the angle between $H$ and $c$-axis) dependence of magneto-resistance of S3
sample free from excess Fe at 14 K (near $T_{\rm{c}}$) and 2.3 K (below
$T_{\rm{cr}}\sim$ 3.2 K), respectively. The angle $\theta$ dependence of
$\mu_{0}H_{\rm{c2}}(\theta)$ was obtained according to the magneto-resistance
data (The raw data of $R(H)$ in different field directions are shown in Fig.
S4 in supplement materials). $\mu_{0}H_{\rm{c2}}(\theta)$ at 14 K is shown in
Fig. 4(a), exhibiting a ellipse shape with twofold symmetry. The maximum value
of $\mu_{0}H_{\rm{c2}}(\theta)$ appears at $H||ab$. The anisotropic G-L
model[52, 53, 32, 54] was used to fit the $\mu_{0}H_{\rm{c2}}(\theta)$ at 14
K,

$\displaystyle
H_{\rm{c2}}^{2}(\theta)=\frac{(H_{\rm{c2}}^{ab})^{2}}{\rm{sin}^{2}\theta+\gamma_{GL}^{2}\rm{cos}^{2}\theta}.$
(2)

The anisotropy parameter $\gamma_{\rm{GL}}$ estimated from the anisotropic G-L
model fitting is 2.24, consistent with the $\gamma_{H}$ $\sim$2.4 estimated
from $R$–$T$ curves under fields ($\mu_{0}H_{\rm{c2}}^{ab}$(14
K)$/\mu_{0}H_{\rm{c2}}^{c}$(14 K)).

The $\mu_{0}H_{\rm{c2}}(\theta)$ at 2.3 K are shown in Fig. 4(b). At this
temperature, $\mu_{0}H_{\rm{c2}}^{c}$ is slightly larger than
$\mu_{0}H_{\rm{c2}}^{ab}$. $\mu_{0}H_{\rm{c2}}(\theta)$ at 2.3 K shows a
butterfly-pattern with fourfold symmetry. Two minimum values appear in the
directions of $H||ab$ and $H\|c$, suggesting the SPDE get its largest value
when $H\|ab$, while the ODE get its largest value when $H\|c$ (shown as the
arrows in Fig. 4(b)). In theory, SPDE is interpreted as depairing mechanism
due to the Zeeman effect aligning the spins of the two electrons with the
applied field paralleled to $ab$-plane, while ODE is interpreted as depairing
mechanism due to the Lorentz force acting via the charge on the momenta of the
paired electrons with the applied field parallel to $c$-axis[6, 55]. Besides,
two maxima appear near 22∘ (338∘) and 158∘ (202∘), suggesting that both ODE
and SPDE are angle dependent, and ODE’s attenuation rate exceeds the
enhancement rate of in-plane SPDE when the $H$ rotate from 0∘ ($H\|c$) to 90∘
($H\|ab$).

To explain the observed strongly anisotropic SPDE and novel
$\mu_{0}H_{\rm{c2}}^{c}$($\theta$) anisotropy, a spin-locking pairing model in
quasi-two-dimensional superconductors is proposed[17] . In this model, the
half-itinerant carriers are proposed. In a normal state with $T$ slightly
above the pre-pairing critical temperature $T_{\rm{pf}}$, the majority of
carriers are half-itinerant, i.e., their spin orientation is locked in the
${ab}$-plane but their charge and spin are itinerant. These half-itinerant
carriers are mainly from Fe ions in Fe1+yTe1-xSex lattice, as shown in the
schematic in Fig. 4(c). When $T$ $\leq$ $T_{\rm{c}}$, these half-itinerant
carriers constitute Cooper pairs, and the long-range phase coherence of these
Cooper pairs and supercurrent are formed. The spin-locked Cooper pairs display
a peculiar anisotropy under an applied magnetic field. For $H$ $\parallel$
$ab$-plane, an angle exists between the spin direction of two carries in a
Cooper pair and $H$, their magnetization energy are
+$\mid$$\vec{M}$$\cdot$$\vec{H}$$\mid$ and
-$\mid$$\vec{M}$$\cdot$$\vec{H}$$\mid$, and the depairing energy reaches a
maximum value, which causes the paramagnetic effect to be dominant for
$\mu_{0}$$H_{\rm{c2}}^{ab}$. When $H$ $\parallel$ $c$-axis ($H$ is
perpendicular to the spin of all carriers of Cooper pairs), the magnetization
energy is always zero, and no SPDE occurs. In this case, the ODE is dominant
for $\mu_{0}$$H_{\rm{c2}}^{c}$, and the two-band effect could uncover in
Fe1+yTe0.6Se0.4. This model provides a good explanation of the origin of the
anisotropic SPDE.

Considering the spin-locking pairing model, an extended anisotropic G-L model
was used to fit the $\mu_{0}H_{\rm{c2}}(\theta)$ at 2.3 K. Due to spin-locked
Cooper pairs, the angle-dependent Zeeman splitting energy is assumed as a
separate term described by $\Delta
E_{Z}=2|\vec{M}\cdot\vec{H}|=|2g_{ab}\mu\sin\theta)|$ [56, 57, 58, 59]. The
ODE ($\Delta E_{orb}$) could be expressed with the formation of the Lorenz
force $F_{L}$, $\Delta
E_{orb}=|2F_{L}\cdot\xi_{GL}(\theta)/2|=|eHv_{ab}|\xi_{\rm{GL}}(\theta)$[6].
Here, the $g_{ab}$ and $v_{ab}$ are the Lande factor and Fermi velocity in
$ab$-plane, respectively (The derivation of the formula can be found in
supplement materials). Taking both Zeeman splitting effect and ODE in
anisotropic G-L model to get the extended anisotropic G-L model[42], the
$\mu_{0}H_{\rm{c2}}(\theta)$ can be written as:

$\displaystyle
H_{c2}(\theta)=\frac{\hbar^{2}}{2m^{*}\frac{(\xi_{GL}^{ab})^{2}}{\sin^{2}\theta+\gamma_{GL}^{2}\cos^{2}\theta}[\frac{\hbar
e}{m^{*}c}+|2g_{ab}\mu_{B}\sin\theta|+|ev_{ab}|(\frac{(\xi_{GL}^{ab})^{2}}{\sin^{2}\theta+\gamma_{GL}^{2}\cos^{2}\theta})^{0.5}]},$
(3)

where $\mu_{\rm{B}}$, $m^{*}$, and $\hbar$ are Bohr magneton, electron
effective mass, and reduced Planck constant, respectively. The fitting result
is shown in Fig. 4(b) by orange curve. Two minimum values appear when angle
$\theta=0^{\circ}$, $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$, indicates
that both SPDE and ODE have strong directionality, which is consistent with
the spin-locking pairing model. $\xi_{\rm{GL}}^{ab}$(2.3 K) and anisotropy
parameter $\gamma_{\rm{GL}}$ of $\mu_{0}H_{\rm{c2}}$(2.3 K) obtained from the
extended anisotropic G-L model are 2.73 nm and 1.24, respectively, which is
consistent with those shown in Table I. Theoretically, the SPDE in the
$ab$-plane is enhanced quickly with decreasing temperature, resulting in an
obvious suppression on $\mu_{0}H_{\rm{c2}}^{ab}$ at low temperatures. On the
other hand, the ODE along $c$-axis is enhanced slowly with decreasing
temperature, leading the value of $\mu_{0}H_{\rm{c2}}^{c}$ to linearly
increase and overshoot $\mu_{0}H_{\rm{c2}}^{ab}$ below $T_{\rm{cr}}$. When $H$
rotate away from 90∘ (0∘), SPDE (ODE) weaken quickly, resulting in a larger
$\mu_{0}H_{\rm{c2}}(\theta)$ than it at $ab$-plane ($c$-axis). Therefore, the
$\mu_{0}H_{\rm{c2}}(\theta)$ changes from a twofold symmetry near $T_{\rm{c}}$
to a fourfold symmetry at low temperatures.

##  IV Conclusion

$\mu_{0}$$H$c2 of Fe1+yTe0.6Se0.4 single crystals with selected amounts of
excess Fe were investigated by conducting resistivity measurements over a wide
range of temperatures and magnetic fields. $\mu$0$H$${}_{\rm{c2}}^{c}$ and
$\mu$0$H$${}_{\rm{c2}}^{ab}$ were fitted by the two-band model and the WHH
model, respectively. The crossover observed on $\mu$0$H$c2 is disorder-robust
and indicates the presence of a strong anisotropic SPDE. Furthermore, the
angle dependent $\mu$0$H$${}_{\rm{c2}}(\theta)$ exhibits a novel anisotropy
with a twofold symmetry near $T_{c}$, but a fourfold symmetry at low
temperatures. To understand the strong anisotropic SPDE and the novel
anisotropy of $\mu$0$H$c2, a spin-locking model was proposed and the novel
fourfold symmetry of $\mu$0$H$${}_{\rm{c2}}(\theta)$ could be fitted by our
extended anisotropic G-L model successfully.

## Acknowledgments

The present work was partly supported by the National Key R$\&$D Program of
China (Grant No. 2018YFA0704300), the Strategic Priority Research Program of
Chinese Academy of Sciences (Grant No. XDB25000000), and the National Natural
Science Foundation of China (Grant No. U1932217, No. 12204487).

Yongqiang Pan, Yue Sun, and Nan Zhou contributed equally to this paper.

###### Acknowledgements.

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Supplemental information

##  V Sample Characterizations

Temperature dependence and magnetic field dependence of resistivity of S1, S2,
and S3 samples have been plotted in Fig. S1. Removing excess Fe can improve
the superconductivity of Fe1+yTe0.6Se0.4[41, 27, 60, 61].

Figure S1: (a) and (h) Schematics of the applied magnetic field directions for
the cases of $H$$\parallel$$ab$ and $H$$\parallel$$c$, respectively. (b)–(d)
Temperature dependence of the resistivity of samples S1 (under $H$ = 0, 1, 2,
3, 4, 5, 6, and 7 T), S2 (under $H$ = 0, 0.5, 1, 2, 3, 5, 7, and 9 T), and S3
(under $H$ = 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 T), respectively, for
$H$$\parallel$$ab$. (e)–(g) Magnetic field dependence of resistivity of
samples S1 (for $T$ = 1.75, 2.45, 3.5, 4.2, 5.5, 6.5, 9, 11.5, 12.5, and 13.5
K), S2 (for $T$ = 1.6, 3.2, 4.2, 6, 8, 11, 12.5, 13.2, 14, and 16 K), and S3
(for $T$ = 2.45, 3.5, 5, 8, 10, 11, 12, 13, 14.5, 15, and 15.5 K),
respectively, for $H$$\parallel$$ab$. (i)–(k) Temperature dependence of
resistivity of samples S1, S2, and S3, respectively. The $H$ values are the
same as those of panels (b)–(d) but along the $c$-axis. (l)–(n) Magnetic field
dependence of the resistivity of samples S1 (for $T$ = 2, 2.6, 4.3, 6, 8, 10,
11, 12, 13, and 15 K), S2 (for $T$ = 2.1, 3.18, 4.2, 6, 7.5, 9, 10.5, 13.5,
14, and 16 K), and S3 (for $T$ = 2.6, 4.2, 6, 8, 10, 11, 12, 13, and 15 K),
respectively, for$H$$\parallel$$c$.

##  VI Fitting on $\mu_{0}$$H$c2

###  VI.1 $\mu_{0}$$H$${}_{\rm{c2}}^{c}$

Figure S2: Fit results on $\mu$0$H$${}_{\rm{c2}}^{c}$ by using BCS-BEC model.

The possibility of crossover from BCS to BEC has been discussed in main text.
Further investigation on reduced $\mu$0$H$${}_{\rm{c2}}^{c}$ fitted by
Disordered Attractive Hubbard Model model[58] have been shown in Fig. S2. Here
$D$ defines conduction band half-width, $U$ is the Hubbard attraction on the
lattice site, distribution width $\triangle$ serves as a measure of disorder,
$a$ is the lattice parameter and $b$ is reduced factor. $\triangle$/2$D$
reflects the concentration degree of disorder, $\triangle$/2$D$ $\geq$ 1 is
for the dirty limit, while $\triangle$/2$D$ $\leq$ 1 is for the clean limit.
$U$/2$D$ reflects the strength of coupling. $U$/2$D$ $\geq$ 1 represents the
BEC strong coupling limit, while $U$/2$D$ $\leq$ 1 represents the BCS weak
coupling limit. The BCS-BEC crossover happens at $U$/2$D$ $\sim$ 1\. As we can
see in Fig. S2, all the three samples exhibit the similar strength of SC
coupling with $U$/2$D$ $\sim$ 1, indicating that those samples are located in
crossover region from BCS to BEC.

Figure S3:  (a)-(c) $\mu$0$H$${}_{\rm{c2}}^{c}$($T$) of samples (a) S1, (b)
S2, and (c) S3 for $H$$\parallel$$c$. The points represent the obtained
experimental data, and the blue curves represent the results of the two-band
model fitting. Fit results on $\mu$0$H$${}_{\rm{c2}}^{c}$ of (a) S1, (b) S2,
and (c) S3 by using WHH model (black dashed curves) and two-band model (blue
dashed curves). $\mu$0$H$${}_{\rm{c2}}^{ab}$($T$) of samples (d) S1, (e) S2,
and (f) S3 for $H$$\parallel$$ab$. The blue dashed curves represent the WHH
fittings with finite $\alpha$ and $\lambda_{so}$. The brown dashed curves
represent the WHH fittings with $\alpha$ = 0 and $\lambda_{so}$ = 0. Figure
S4:  (a)-(c)The $R-H$ curves at $T=$ 14 K with different angle $\theta$ (0∘
represent $H\|c$ and 90∘ represent $H\|ab$), which was measured by PPMS.
(d)-(h) The $R-H$ curves at $T=$ 2.3 K with different angle $\theta$ (0∘
represent $H\|c$ and 90∘ represent $H\|ab$), which was measured in WHHMF. (i)
Angle dependent $\mu_{0}H_{c2}$ at $T=$ 2.3 K, the red points and the azure
curve represent the experiment data and the fitting result by equation.(12),
respectively.

$\mu$0$H$${}_{\rm{c2}}^{c}$ of S1, S2, and S3 have been plotted in Figs. S3
(a)-(c). Blue dashed curves represent the fitting results by using two-band
model as discussed in main text. The parameters also be listed in every
figures. Detail discussion can be found in main text. Black dashed curves
represent the fitting results by using single-band WHH model (solid curves),
which are failed.

###  VI.2 $\mu_{0}$$H$${}_{\rm{c2}}^{ab}$

Temperature dependence of $\mu_{0}H_{\rm{c2}}^{ab}$ of S1, S2, and S3 and the
WHH model fitting results are plotted in Figs. S3(d)-(f). The parameters also
be listed in every figures. Detail discussion can be found in main text.

##  VII Derivation of Extended Anisotropic G-L Model

According the Ginzburg-Landau (G-L) theory, the Gibbs free energy density of
superconductor can be written as following:

$\displaystyle{g_{s}}={g_{n}}+\alpha|\psi|^{2}+\frac{\beta}{2}|\psi|^{4}+.......$
(S1)

Here the $\psi$ is the order parameter and $|\psi|^{2}$=$n_{s}$, the $n_{s}$
is the superconducting electron density. The $g_{n}$ is the Gibbs free energy
density in normal state. Due to the $\psi$ should change with spatial
position, the equation (1) should be revised as following:

$\displaystyle{g_{s}(H_{a})}=f_{n}(0)+\alpha|\psi|^{2}+\frac{\beta}{2}|\psi|^{4}+\frac{1}{2m}|-i\hbar\bigtriangledown\psi-
eA\psi|^{2}+\frac{B^{2}}{2\mu_{0}}-B*H_{a}.$ (S2)

The angle-dependence of Zeeman splitting energy and orbital pair-breaking
energy should be considered (Fig. S5). As shown in Fig.S5, Zeeman splitting
energy can be written as $\bigtriangleup E_{z}$ = 2 $|M\cdot H|$ =
$|2g_{ab}\mu_{B}H\sin\theta|$. Since the Zeeman splitting energy does not
depend on the sign of $\sin\theta$, therefore, the absolute value has been
added to make the $\bigtriangleup E_{z}$ an even function with positive values
when $\theta$ is away from [$0^{\circ},180^{\circ}$]. Orbital pair-breaking
energy can be written as $\bigtriangleup
E_{\rm{orb}}=|2F_{L}\cdot\xi_{GL}(\theta)/2|=|eHv_{ab}|\xi_{GL}(\theta)$,
where the $F_{L}$ is the Lorentz force. The Lorentz force $F_{L}$ shows a
maximum value only when the orbital planes of Cooper pairs are perpendicular
to $H$. Corresponding, $v$ is always parallel to $ab$-plane ($v_{ab}$) and
perpendicular to both $H$ and $F_{L}$.

Figure S5:  Schematic diagrams of (a) Zeeman splitting energy and (b) orbital
pair-breaking energy.

Now we add the Zeeman splitting energy and orbital pair-breaking energy into
equation (2)[6]:

$\displaystyle{g_{s}(H_{a})}=f_{n}(0)+\alpha|\psi|^{2}+\frac{\beta}{2}|\psi|^{4}+\frac{1}{2m}|-i\hbar\bigtriangledown\psi-
eA\psi|^{2}+\frac{B^{2}}{2\mu_{0}}-B*H_{a}+|2g_{ab}\mu_{B}H\sin\theta|+|eHv_{ab}|\xi_{GL}(\theta).$
(S3)

$g_{ab}$ and $v_{ab}$ represent the Lande $g$-factor and Fermi velocity in
$ab$-plane, respectively. When the system is stable, $\psi$ adjusts itself to
minimize the overall free energy, this variational problem leads to the
celebrated G-L differential equation[32]:

$\displaystyle\frac{1}{2m}|-i\hbar\bigtriangledown-
eA|^{2}\psi+\alpha\psi+\beta|\psi|^{2}\psi=0.$ (S4)

This G-L differential equation is suitable for both equations (2) and (3).
Combining the definition of $\xi_{\rm{GL}}$:

$\displaystyle\xi_{GL}^{2}(T)=\frac{\hbar^{2}}{2m^{*}|\alpha(T)|}.$ (S5)

Equation (4) can be written as:

$\displaystyle{(\frac{\nabla}{i}-\frac{2\pi
A}{\Phi_{0}})}^{2}\psi=\frac{2m^{*}\alpha}{\hbar^{2}}\psi=\frac{\psi}{\xi_{GL}^{2}(T)}.$
(S6)

Firstly, when $H||c$-axis, $A_{y}=Hx$, Equation (6) can be written as:

$\displaystyle(-\nabla^{2}+\frac{4i\pi}{\Phi_{0}}Hx\frac{\partial}{\partial
y}+(\frac{2\pi H}{\Phi_{0}})^{2}x^{2})\psi=\frac{\psi}{\xi_{GL}^{2}}.$ (S7)

Since the effective potential depends only on $x$, it is reasonable to look
for a solution of the form, $\psi(x,y,z)=e^{ik_{y}y}e^{ik_{z}z}f(x)$,
substituting this into equation (7), we find:

$\displaystyle-f^{\prime\prime}(x)+(\frac{2\pi
H}{\Phi_{0}})^{2}(x-x_{0})^{2}f=(\frac{1}{\xi_{GL}^{2}}-k_{z}^{2})f.$ (S8)

where $x_{0}=(k_{y}\Phi_{0})/2\pi H$. According the equation (8), The total
de-pairing energy $\Delta E$ should be equated to
$[\hbar^{2}(\xi_{\rm{GL}}^{-2}-k_{z}^{2})]$/$2m^{*}$:

$\displaystyle\Delta E=\Delta E_{n}+\Delta E_{z}+\Delta E_{orb}$
$\displaystyle=(n+\frac{1}{2})\hbar\frac{2eH}{m^{*}c}+|2g_{ab}\mu_{B}H\sin\theta|+|eHv_{ab}|\xi_{GL}(\theta)=\frac{\hbar^{2}(\xi_{GL}^{-2}-k_{z}^{2})}{2m^{*}}.$
(S9)

The $H$ has its highest value ($H_{c2}$) if $n$ = 0 and $k_{z}$ = 0, so we
get[3, 33]:

$\displaystyle H_{c2}=\frac{\hbar^{2}}{2m^{*}\xi_{GL}^{2}(\theta)[\frac{\hbar
e}{m^{*}c}+|2g_{ab}\mu_{B}\sin\theta|+|ev_{ab}|\xi_{GL}(\theta)]}.$ (S10)

Here we take

$\displaystyle\xi_{GL}^{2}(\theta)=\frac{(\xi_{GL}^{ab})^{2}}{\sin^{2}\theta+\gamma_{GL}^{2}\cos^{2}\theta}.$
(S11)

into equation (10), we can get[27, 34, 62]:

$\displaystyle
H_{c2}(\theta)=\frac{\hbar^{2}}{2m^{*}\frac{(\xi_{GL}^{ab})^{2}}{\sin^{2}\theta+\gamma_{GL}^{2}\cos^{2}\theta}[\frac{\hbar
e}{m^{*}c}+|2g_{ab}\mu_{B}\sin\theta|+|eV_{ab}|(\frac{(\xi_{GL}^{ab})^{2}}{\sin^{2}\theta+\gamma_{GL}^{2}\cos^{2}\theta})^{0.5}]}.$
(S12)

When $\theta=0^{\circ}(180^{\circ})$, there only angle-dependent orbital pair-
breaking effect is active. When $\theta\neq 0^{\circ}(180^{\circ})$, both
angle-dependent orbital pair-breaking effect and Zeeman splitting effect are
active.

Angle dependence of $\mu_{0}H_{c2}$(2.3 K) and its fitting by equation (12)
are shown in Fig. S4(i). The polar diagram of $\mu_{0}H_{c2}$(14 K) and
$\mu_{0}H_{c2}$(2.3 K) can be found in Fig. 4(a) and Fig. 4(b) in main text.
Detail measurement data are shown in Figs. S4(a)-(c) with PPMS and Figs.
S4(d)-(h) with WHHMF.

##  VIII Anisotropy of $\xi$($T$)

The temperature dependence of $\xi$($T$) is discussed for sample S3 without
the influence from excess Fe. $\xi$($T$) for the single-band G-L model using
the following relationship:

$\displaystyle\xi(T)=\xi(0K)(1-T/T_{c})^{-1/2},$ (S13)

which has been applied to FeSe[49] and Ca-112 system[17]. On the other hand,
for the two-band model, it follows the relationship[63]:

$\displaystyle\xi(T)=\xi(0K)[f_{1}(1-T/T_{c})+f_{2}(1-T/T_{c})^{2}]^{-1/2},$
(S14)

where $f_{1}$ and $f_{2}$ are coefficients. This equation has been used for
MgB2 and LuNi2B2C[18, 63].

Figure S6: Temperature dependence of $\xi$($T$). The red and blue points
represent $\xi$$(T)$ in two directions. The orange- and blue dashed curves
represent single-band model fittings for the two directions, whereas the brown
and blue solid curves represent the two-band model fittings results for the
two directions. The black points and dashed curve represent
$\xi_{c}^{\rm{BCS}}$($T$) and its fitting obtained using the single-band
model, respectively.

When $H$$\parallel$$c$, since there is no spin-paramagnetic effect in this
direction, $\xi$ab($T$) (red diamonds in Fig. S6) with 0 K $\leq$ $T$ $\leq$
$T_{c}$ can be obtained from
($\Phi$0/2$\pi$$\mu$0$H$${}_{\rm{c2}}^{c}$($T$))1/2. As shown in Fig. S6, the
fitting for $\xi$ab($T$) obtained using the single-band model (orange dashed
curves) deviates from the experimental data when $T$ $<$ 0.7$T_{c}$.
Contrarily, the fitting obtained using the two-band model (brown dashed
curves) reproduces the experimental data very well over the whole temperature
region. It is consistent with the two-band model fitting for
$\mu$0$H$${}_{\rm{c2}}^{c}$ (Fig. S3(c)), proving that Fe1+yTe0.6Se0.4 is a
multigap system.

A peculiar situation occurs for $\xi_{c}$($T$) obtained from
$\Phi$0/2$\pi$$\xi$ab($T$)$\mu$0$H$${}_{\rm{c2}}^{ab}$($T$). $\xi_{c}$($T$)
and the fittings obtained using equations (13) and (14) are plotted in Fig. S6
with blue circles, blue dots, and an azure dashed curve, respectively.
Clearly, both fittings deviate strongly from the experimental data except for
the region close to $T_{c}$. This poor fitting is also typical of many other
superconductors[63, 64]. This demonstrates that both the single-band and two-
band theories fail to explain the $\xi$c($T$) trend, which is due to the
strong spin-paramagnetic effect, as discussed above.

$\xi_{c}$(0 K) should be obtained without the influence of the spin-
paramagnetic effect. According to the two-band BCS theory, when $T$
$\rightarrow$ $T_{c}$, $\xi$${}_{c}^{\rm{BCS}}$($T$) can be expressed as
follows[47]:

$\displaystyle\xi_{c}^{\rm{BCS}}(T)=\Phi_{0}/(2\pi
T_{c}(-\mu_{0}dH_{c2}^{ab}(0\rm{K})/dT)\xi_{\textit{ab}}(T)).$ (S15)

Due to the single-band behavior of $\mu$0$H$${}_{\rm{c2}}^{ab}$,
$\xi_{c}^{\rm{BCS}}$($T$) was fitted using equation (15). Both
$\xi_{c}^{\rm{BCS}}$($T$) near $T_{c}$ (black hollow blocks) and the fitting
result (black dashed curve) are plotted in Fig. S6. $\xi_{c}^{\rm{BCS}}$(0 K)
was roughly estimated as 9.5 Å, which is slightly larger than previous report
$\sim$ 4.4 Å[47]. The anisotropy of $\gamma_{\xi}$(0 K), defined as
$\xi_{ab}$(0 K)/$\xi_{c}^{\rm{BCS}}$(0 K), is 2.64 and is consistent with
($m_{c}$/$m_{ab}$)1/2 ($\sim$ 2.5 – 4)[41], indicating that the
superconductivity is anisotropic. Compared with the lattice parameter $c$
$\sim$ 6 Å, this small $\xi_{c}^{\rm{BCS}}$(0 K) indicates that
Fe1+yTe0.6Se0.4 may be a quasi-two-dimensional superconductor.


# Impact of electron spectra on morphology of pulsar halos at ultra-high
energies

Ying-Ying Guo  Qiang Yuan<EMAIL_ADDRESS>Key Laboratory of Dark Matter and
Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210023, China  School of Astronomy and Space Science, University of
Science and Technology of China, Hefei 230026, China

###### Abstract

The extended $\gamma$-ray halos around pulsars are unique probe of
transportation of high-energy electrons (and positrons) in vicinities of such
pulsars. Observations of morphologies of several such halos indicate that
particles diffuse very slowly around pulsars, compared with that in the Milky
Way halo. The energy-dependent morphologies are expected to be very important
in studying the energy-dependence of the diffusion coefficient. In this work
we point out that the spectrum of high-energy electrons takes effect in
shaping the $\gamma$-ray morphologies at the ultra-high-energy bands, and thus
results in a degeneracy between the electron spectrum and the energy-
dependence of the diffusion coefficient. The reasons for such a degeneracy
include both the Klein-Nishina effect of the inverse Compton scattering and
the curvature (if any) of the electron spectrum. It it thus necessary to take
into account the spectral shape of electrons when deriving the energy-
dependence of diffusion coefficient using ultra-high-energy $\gamma$-ray
measurements of extended pulsar halos.

###### keywords:

pulsar halos , diffusion propagation , energy losses

††journal: Journal of High Energy Astrophysics

##  1 Introduction

Charged cosmic rays (CRs) are expected to propagate diffusively in the
interstellar medium (ISM) due to scattering with random magnetized turbulence
[1, 2]. According to the resonant scattering theory between particles and the
interstellar turbulence, the rigidity-dependence of the diffusion coefficient
follows a power-law, $D(R)\propto R^{\delta}$, given a power-law form of the
spectral energy density of turbulence, $w(k)\propto k^{-2+\delta}$ where $k$
is the wave number of the magnetohydrodynamic waves. Fitting to the CR
secondary-to-primary ratios, the parameter $\delta$ is found to be about
$0.3-0.6$ for energies below TeV/n [3, 4]. Further precise measurements of the
secondary-to-primary ratios show that there are hardenings of such ratios at
high energies [5, 6], suggesting possible changes of the slope of rigidity-
dependence of the diffusion coefficient [7].

The secondary-to-primary ratios reflect an average effect of the propagation
of CRs in the Milky Way. It is very likely that the propagation is spatially
dependent. Such a scenario was proposed to explain some observational results
of CRs and diffuse $\gamma$ rays [8, 9, 10, 11]. Observations of extended
$\gamma$-ray halos surrounding middle-aged pulsars, which are expected to be
produced by the inverse Compton scattering (ICS) of high energy electrons (and
positrons) produced by the pulsars with the interstellar radiation field,
offer a more direct measurement of the particle diffusion in specific regions
of the ISM [12, 13]. The results show that electrons propagate much more
slowly around pulsars than that in the Milky Way halo, indicating an
inhomogeneous propagation of particles [14, 15]. Additional supports for an
inhomogeneous propagation were also given from the spectra of electrons and
positrons, nuclei, and the large-scale anisotropies [16, 17, 18].

Detailed morphological studies of the $\gamma$-ray pulsar halos are very
important in measuring the energy-dependence of diffusion coefficient, and
testing its generality in different places of the Milky Way. Due to the fast
energy losses of electrons in the ISM (synchrotron losses and the ICS losses),
higher energy electrons experience shorter lifetime which approximately scales
as $\tau(E_{e})\propto E_{e}^{-1}$. Depending on the slope $\delta$ of the
diffusion coefficient, the extensions of $\gamma$-ray halos may increase (for
$\delta>1$) or shrink (for $\delta<1$) when energy increases. This expectation
does not rely on the spectrum of electrons, and may hold for specific energy
range where the cooling dominates the diffusion, the ICS lies well in the
Thomson regime and the electron spectrum follows roughly a power-law. However,
in the ultra-high-energy (UHE) range (with $\gamma$-ray energy $E_{\gamma}$
higher than tens of TeV) many of the above assumptions are invalid, and this
simple expectation should not hold any more [19]. In this work, we study the
energy-dependent morphologies of pulsar halos in the UHE range assuming
different spectral shapes. We will show that the electron spectrum couples
with the diffusion process to shape the $\gamma$-ray morphologies and
complicates the measurement of the energy-dependence of the diffusion
coefficient. This effect cannot fe ignored since at UHE spectral cutoff is
common for source acceleration.

##  2 Injection and propagation of electrons from pulsars

The propagation of electrons (and positrons) in the ISM around a pulsar is
described by the diffusion-cooling equation

$\frac{\partial N}{\partial t}=\nabla\cdot(D\nabla N)+\frac{\partial}{\partial
E_{e}}(bN)+Q,$ (1)

where $E_{e}$ is the energy of electrons, $N=N(r,E_{e},t)$ is the differential
density of electrons, $D=D(E_{e})$ is the diffusion coefficient,
$b=b(E_{e})=-dE_{e}/dt$ is the energy loss rate, and $Q=Q(r,E_{e},t)$
represents the source term. In this work we focuses on electron propagation
near the pulsar, and neglect possible spatial variations of the diffusion
coefficient and the energy loss rate.

The diffusion coefficient is parameterized as a power-law of electron energy,
$D(E_{e})=D_{0}E_{e}^{\delta}$. The cooling of electrons includes mainly the
synchrotron cooling and ICS cooling. The energy loss rate of synchrotron
radiation is

$b_{\rm syn}(E_{e})=\frac{4}{3}\sigma_{T}c\gamma_{e}^{2}U_{B},$ (2)

where $\sigma_{T}$ is the Thomson cross section,
$\gamma_{e}=E_{e}/(m_{e}c^{2})$ is the Lorentz factor of electron, $m_{e}$ is
the electron mass, $c$ is the speed of light, and $U_{B}=B^{2}/(8\pi)$ is the
energy density of the magnetic field with strength $B$. In this work,
$B=3~{}\mu$G is assumed111The magnetic field affects the relative weights
between the synchrotron cooling rate and the ICS cooling rate. A weaker
magnetic field would result in more significant impacts on the halo
morphologies due to the Klein-Nishna effect and vise versa.. For the cooling
rate due to the ICS emission, we use the approximation given in Ref. [20]. We
assume three gray (black) body components of the interstellar radiation fields
(ISRF), the cosmic microwave background with temperature of 2.725 K and energy
density of 0.26 eV cm-3, an infrared background with temperature of 20 K and
energy density of 0.3 eV cm-3, and an optical component with temperature of
5000 K and energy density of 0.3 eV cm-3, respectively.

The injection function is assumed to be a point source in space, a power-law
with (super)-exponential cutoff in energy, and a profile following the pulsar
spindown in time:

$Q(\mathbf{r},E_{e},t)=Q_{0}\delta(\mathbf{r}-\mathbf{r}_{\rm psr})\times
E_{e}^{-\alpha}\exp[-(E_{e}/E_{c})^{\beta}]\times(1+t/\tau_{0})^{-2},$ (3)

where $Q_{0}$ is a normalization constant, $\alpha$ is the spectral index,
$E_{c}$ is the characteristic cutoff energy of electrons, $\beta$ describes
the sharpness of the cutoff, $\tau_{0}$ is the time scale that the spindown
luminosity of the pulsar starts to decay. The parameter $\tau_{0}$ is
difficult to be determined observationally, and we assume a nominal value of
$10^{4}$ yr in this work.

The propagation equation (1) can be solved analytically using the Green’s
function method with respect to variables $\mathbf{r}$ and $t$ (i.e.,
$\delta$-function type injection for space and time), for an arbitrary
injection spectrum [21]. Convolving the Green’s function with the injection
time profile, one can obtain the electron distribution at give time and
position.

The photon emissivity due to the ICS is then given by

$w_{\text{ICS}}(t,E_{\gamma},\mathbf{r})=\int_{0}^{\infty}d\epsilon\,n(\epsilon)\int_{E_{\text{min}}}^{\infty}dE_{e}\,N(t,E_{e},\mathbf{r})\,F(\epsilon,E_{e},E_{\gamma}),$
(4)

where $n(\epsilon)$ is the number density of background photons with energy
$\epsilon$, $N(t,E_{e},\mathbf{r})$ is the electron density obtained via
solving the propagation equation, and $F(\epsilon,E_{e},E_{\gamma})$ is the
$\gamma$-ray yield function of the ICS according to the Klein-Nishina cross
section. Integrating $w_{\text{ICS}}$ along the line of sight, one can obtain
the $\gamma$-ray surface brightness for specific energy $E_{\gamma}$.

##  3 Results

We take the Geminga pulsar halo as an example for the discussion. The distance
of Geminga pulsar is adopted as $d_{\text{psr}}=250$ pc, and the age is
$\tau_{\rm psr}=3.4\times 10^{5}$ yr [22]. The diffusion coefficient is
parameterized as a power-law of electron energy,
$D(E_{e})=D_{0}(E_{e}/100\,{\rm TeV})^{\delta}$. As a benchmark,
$D_{0}=10^{27}$ cm2 s-1 is assumed, which is several times smaller than that
inferred by HAWC observations [12]. A smaller value of the diffusion
coefficient is to avoid the complexity of the superluminal propagation at very
small scales [23, 24], and does not loses generality of this work.

Figure 1 illustrates the influence of $\delta$ on the evolution of the
$\gamma$-ray morphology with energy. The electron injection spectrum is
assumed to be a power-law function with index $\alpha=2.5$. Two values of
$\delta$, 0 and 1 are studied. Dashed lines in this plot correspond to
$\delta=0$, which show a clear contraction of the $\gamma$-ray extension with
increasing energy. In this case, the energy loss rate of electrons increases
with energy, but the diffusion coefficient keeps a constant, hence the
characteristic diffusion distance $r_{d}\sim 2\sqrt{D(E_{e})\tau(E_{e})}$
becomes smaller when energy is higher. For $\delta=1$, on the other hand, the
characteristic diffusion length keeps almost unchanged, and the source
extensions are similar at different energies as shown by solid lines. The
extension at 100 TeV is slightly more extended than those at lower energies,
due to that the ICS cooling rate becomes smaller and deviates from
$E_{e}^{-1}$ thanks to the Klein-Nishina effect.

Figure 1: Surface brightness distributions of $\gamma$ rays at different
energies for $\delta=1$ (solid lines) and $\delta=0$ (dashed lines). The
electron injection spectrum is a power-law form with spectral index
$\alpha=2.5$.  Figure 2: Impact on the $\gamma$-ray morphologies from
different electron injection spectra. The energy-dependent slope of the
diffusion coefficient is assumed to be $\delta=1$. Panel (a): comparison of
one-dimensional surface brightness distributions between power-law injection
spectra with indices $\alpha=1.5$ (solid lines) and $\alpha=2.5$ (dashed
lines). Panel (b): comparison of one-dimensional surface brightness
distributions between power-law injection spectrum with index $\alpha=2.5$
(dashed lines) and power-law with super-exponential cutoff spectrum
$E_{e}^{-1.5}\exp[-(E_{e}/100\,{\rm TeV})^{2}]$. Panel (c): variation of the
diffusion angle $\theta_{d}$ with $\gamma$-ray energy for different injection
spectra.

Figure 2 shows the impact on the $\gamma$-ray morphologies from different
injection spectra of electrons, for $\delta=1$. Panel (a) shows the one-
dimensional surface brightness distributions for power-law injection spectra
with indices $\alpha=1.5$ (solid lines) and $\alpha=2.5$ (dashed lines). Panel
(b) of Figure 2 illustrates the difference between the power-law injection
spectrum (with $\alpha=2.5$; dashed lines) and the power-law with super-
exponential cutoff spectrum ($\alpha=1.5$, $E_{c}=100$ TeV, $\beta=2$; solid
lines). To quantify the evolution of source extension, we fit the one-
dimensional surface brightness distribution with the following profile [12]

$f(\theta)\propto\frac{1}{\theta_{d}(\theta+0.06\theta_{d})}\cdot\exp\left[-(\theta/\theta_{d})^{2}\right],$
(5)

where $\theta_{d}=r_{d}/d_{\rm psr}$ is the characteristic diffusion angle,
and $\theta$ is the angular distance from the central pulsar. We assume $10\%$
relative errors of the surface brightness profiles for each angular bin and
adopt a minimum $\chi^{2}$ method to do the fit. The evolution of $\theta_{d}$
for the three injection spectra are shown in panel (c). We can see that
$\theta_{d}$ shows an increase tend when the energy increases for power-law
injection, but shows a clear decrease trend for the spectrum with cutoff. It
can also be noted that for $\alpha=1.5$, the $\gamma$-ray profiles are
typically more extended than the case of $\alpha=2.5$. All these results show
that the high-energy electron spectrum plays an important role in shaping the
$\gamma$-ray morpholigies.

To better understand these results, we scrutinize the distributions of
electrons at different energies in detail. Figure 3 show the radial profiles
of electron density as a function of distance from the pulsar, for different
post-propagation energies and injection spectra. To eliminate the Klein-
Nishina effect of the ICS, we test the case with synchrotron cooling only for
a power-law injection, as shown by the solid lines. The results show that the
spatial distributions of electrons with different energies are identical, just
as expected for a cooling rate of $E_{e}^{-1}$ and $\delta=1$. However, when
the ICS cooling is included (short-dashed lines), the cooling of higher energy
electrons is smaller than $E_{e}^{-1}$, leading to more extended distributions
of electrons with higher energies. As a result, the $\gamma$-ray morphology
would also become more extended at higher energies, as shown in Figure 2. This
also explains the morphology for harder spectrum ($\alpha=1.5$) is more
extended than that for softer spectrum ($\alpha=2.5$), since there are more
high-energy electrons for a harder spectrum. When there is cutoff of the
injection spectrum, we in turn observe that higher energy electrons are less
extended than low energy ones (long-dashed lines). In this case there are not
many electrons at the source with energy high enough to cool down to given
energy (e.g., $E_{e}=300$ TeV). This could also be understood as that, under
the continuous injection scenario, the current electron density is the time
integral of the previously injected electron with higher energies. The
decrease of high-energy electron flux for the cutoff spectrum is equivalent to
a shorter effective integration time. The increase of the diffusion
coefficient with energy ($\sim E_{e}^{1}$) does not compensate the decrease in
time, resulting in a decrease in $\theta_{d}$. Therefore, we would observe a
shrink of the $\gamma$-ray morphology with increasing energy if the source
spectrum has a cutoff.

Figure 3: Impact on the electron morphologies with respect energy loss rate
and injection spectra. The energy-dependent slope of the diffusion coefficient
is assumed to be $\delta=1$. All curves are normalized at 3 pc. In case of
only synchrotron cooling, the radial profiles of electrons of different
energies overlap with each other.

Finally, we show in Figure 4 the energy distribution of electrons which
produce $\gamma$-rays with fixed energy $E_{\gamma}$, for different injection
spectra. Due to the wide energy distributions of the ISRF and the ICS cross
section, the electron energy distribution to give fixed $E_{\gamma}$ is quite
wide. For power-law injections, a harder spectrum corresponds to a longer
high-energy tail of the electron energy distribution, which explains the
larger extension of the $\gamma$-ray morphology as we discussed in above. The
$\gamma$-ray morphologies from 1 TeV to 10 TeV evolve slowly with energy (see
panel (c) of Figure 2), because the energies of electrons producing those
$\gamma$ rays are mainly below 100 TeV, and the Klein-Nishina effect of the
cooling is not evident. However, from 10 TeV to 100 TeV, the Klein-Nishina
effect becomes important, and the $\gamma$-ray extensions become significantly
larger. For the injection spectrum with a curved shape, we can see that
electron energy distributions are more concentrated than the power-law
injection cases. Even if we observe the same morphology evolution of $\gamma$
rays, the inferred energy-dependent diffusion properties are different for
different injection spectra.

Figure 4: The fraction distribution of injection electron energies that
produce $\gamma$ rays with fixed energy $E_{\gamma}$.

##  4 Conclusion and Discussion

Pulsar halos are formed by high-energy electrons and positrons injected from
the associated pulsar wind nebulae via the ICS off the background photons [25,
26, 27, 28]. The $\gamma$-ray morphologies of pular halos are very important
in revealing the particle propagation in the ISM around pulsars. It has been
expected that the extension of the halo is proportional to the characteristic
diffusion length scale, $\theta_{d}\propto r_{d}\propto
2\sqrt{D(E_{e})\tau(E_{e})}$. However, this expectation is over-simplified. In
reality, the Klein-Nishina effect of the ICS cooling and the deviation of
source injection spectrum from a power-law distribution affect the electron
propagation and final distribution. As a consequence, the $\gamma$-ray
morphology evolution also depends on the injection electron spectrum.

This study investigates the impacts of the electron injection spectrum on the
$\gamma$-ray morphology evolution of pulsar halos. For simplicity, we take the
energy-dependent slope of the diffusion coefficient $\delta=1$ as a benchmark
for discussion. Given the $E_{e}^{-1}$ energy loss rate of electrons due to
synchrotron and ICS cooling, this diffusion coefficient predicts energy-
independent $\gamma$-ray morphologies. It has been shown that at relatively
low energies (e.g., $E_{\gamma}$ is smaller than tens of TeV), power-law
injection spectrum indeed gives energy-independent source morphologies as
expected. However, if $E_{\gamma}$ is as high as tens of TeV, even power-law
injection spectrum results in an evolution of the source extension. This is
because the Klein-Nishina effect of the ICS leads to a less efficient cooling
effect than the $E_{e}^{-1}$ form. For power-law with super-exponential cutoff
spectrum, a clear source contraction with increasing energy is given. We point
out that such a contraction is due to that there are not enough high-energy
electrons can contribute to those high-energy photons because of the spectral
cutoff. It is thus necessary to take into account the injection spectrum shape
when studying the energy-dependence of the diffusion coefficient using the UHE
$\gamma$-ray observations.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China
(No. 12220101003), the Project for Young Scientists in Basic Research of
Chinese Academy of Sciences (No. YSBR-061), and the Jiangsu Provincial
Excellent Postdoctoral Program with grant number 2022ZB472.

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# Invariant measures on finite rank subshifts

Nicolas Bédaride  ,  Arnaud Hilion  and  Martin Lustig  Aix Marseille
Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France
<EMAIL_ADDRESS><EMAIL_ADDRESS>Martin.Lustig@univ-
amu.fr

###### Abstract.

In this note we show that for any subshift $X$ of finite $S$-rank every
invariant measure $\mu$ is determined by its values on finitely many
cylinders. Under mild conditions these cylinders are given by the letters of
the alphabet in question.

###### Key words and phrases:

$S$-adic development, finite rank, cylinder measures

###### 2010 Mathematics Subject Classification:

Primary 37B10, Secondary 37A25, 37E25

For any finite discrete set $\cal A=\\{a_{1},\ldots,a_{d}\\}$ the infinite
product $\cal A^{\mathbb{Z}}$ is compact, with respect to the product
topology, and indeed it is a Cantor set. It is naturally equipped with a shift
map $T$, induced by subtracting 1 from the coordinate indices of $\cal
A^{\mathbb{Z}}$, and $T$ acts as homeomorphism on $\cal A^{\mathbb{Z}}$. A
subset $X\subseteq\cal A^{\mathbb{Z}}$ is called a subshift (over $\cal A$) if
$X$ is closed and if $T(X)=X$.

An invariant measure $\mu$ on $X$ is a finite $T$-invariant Borel measure on
$\cal A^{\mathbb{Z}}$ with support contained in $X$. Any such $\mu$ is
determined by its values $\mu([w])$ on the family of cylinders $[w]$, for all
words $w=y_{1}y_{2}\ldots y_{n}$ (with $y_{i}\in\cal A$) in the free monoid
$\cal A^{*}$ over the above alphabet $\cal A$. Here $[w]$ consists of all
biinfinite words $\ldots x_{j-1}x_{j}x_{j+1}\ldots\in\cal A^{\mathbb{Z}}$ with
$x_{i}=y_{i}$ for all $i=1,\ldots,n$. The integer $n\geq 0$ is called the
length of $w$ and is denoted by $|w|$.

For any subshift $X\subseteq\cal A^{\mathbb{Z}}$ we consider the linear
surjective map $\zeta$ from the cone $\cal M(X)$ of invariant measures on $X$
onto cone of letter measures $\cal C(X)\subseteq\mathbb{R}A$, given by
$\mu\mapsto(\mu([a_{1}]),\ldots,\mu([a_{d}]))$.

A very convenient tool to define a subshift $X$ is a directive sequence
${\overleftarrow{\sigma}}=(\sigma_{n})_{n\geq 0}$ of monoid morphisms
$\sigma_{n}:\cal A^{*}_{n+1}\to\cal A^{*}_{n}$, with $\cal
A_{n}=\\{a_{1,n},\ldots,a_{d(n),n}\\}$ for all $n\geq 1$ and $\cal A_{0}=\cal
A$. Here the subshift $X$ generated by $\overleftarrow{\sigma}$ consists of
all biinfinite words $x=\ldots x_{n-1}x_{n}x_{n+1}\ldots\in\cal
A^{\mathbb{Z}}$ such that every finite factor $x_{j}x_{j+1}\ldots x_{k}$ of
$x$ occurs also as factor in some
$\sigma_{0}\circ\ldots\circ\sigma_{n-1}(a_{k,n})$.

The directive sequence $\overleftarrow{\sigma}$ is also called an $S$-adic
development of $X$ if $S$ is a (possibly infinite) set of monoid morphisms
which contains all $\sigma_{n}$ that occur in $\overleftarrow{\sigma}$. The
set $S$ is of particular interest if the cardinality of all level alphabets
$\cal A_{n}$ is bounded; in this case we can identify all level alphabets
$\cal A_{n}$ with a single alphabet $\cal A$.

A directive sequence ${\overleftarrow{\sigma}}=(\sigma_{n})_{n\geq 0}$ is
called everywhere growing if the sequence of minimal lengths
$\beta_{-}(n)=\min\\{|\sigma_{0}\circ\ldots\circ\sigma_{n-1}(a_{k,n})|\mid
a_{k,n}\in\cal A_{n}\\}$ tends to $\infty$ for increasing $n$. Every subshift
$X$ possesses an everywhere growing $S$-adic development (see Monteil’s
Theorem 4.3 of [4] or Proposition 2.7 of [1]), and it makes sense to think of
$X$ as “small” if its possesses a directive sequence which is both, everywhere
growing and of bounded cardinalities $\mbox{card}(\cal A_{n})$. We say that
such a subshift $X$ has finite $S$-rank (since it seems closely related to
what has been called “finite (topological) rank” in the context of Bratteli-
Vershik systems, see for instance [4]).

To think of $X$ as “small” is justified for instance by the well known fact
that for any subshift $X$ of finite $S$-rank the cone $\cal M(X)$ is of finite
dimension. Indeed, this dimension is equal to the number $e(X)$ of invariant
ergodic probability measures on $X$, and (see for instance Remark 4.3 of [5])
$e(X)$ is bounded above by the cardinality $\mbox{card}(\cal A)$ of the
alphabet $\cal A$ which is used by any everywhere growing $S$-adic development
${\overleftarrow{\sigma}}=(\sigma_{n})_{n\geq 0}$ of $X$. Hence one has

$c(X)\leq e(X)\leq\mbox{card}(\cal A)\,,$

where we denote by $c(X)$ the dimension of the above cone $\cal C(X)$ of
letter measures.

For any subshift $X$, given as before by a directive sequence
$\overleftarrow{\sigma}=(\sigma_{m})_{m\geq 0}$, we will also consider, for
any $n\geq 0$, the dimension $c_{n}$ of the letter cone $\cal C(X_{n})$ of the
intermediate level-$n$ subshift $X_{n}$. The latter is defined by truncating
the given directive sequence $\overleftarrow{\sigma}$ at level $n$; in other
words, $X_{n}$ is the subshift generated by the directive sequence
$(\sigma_{m})_{m\geq n}$. In the special (but rather frequently occurring)
case where $c_{n}=c(X)$ for all $n\geq 0$, the directive sequence
$\overleftarrow{\sigma}$ is called thin. We say that a subshift $X$ is thin if
$X$ possesses an $S$-adic development which is everywhere growing, has bounded
level alphabet cardinalities, and is thin.

######  Theorem 1.

Let $X$ be a thin subshift over a finite alphabet $\cal A$. Then any two
invariant measures $\mu_{1}$ and $\mu_{2}$ on $X$ are equal if and only if one
has $\mu_{1}([a_{k}])=\mu_{2}([a_{k}])$ for the finitely many cylinders
$[a_{k}]$ given by all $a_{k}\in\cal A$.

Any monoid morphism $\sigma:\cal A^{*}\to\cal{A^{\prime}}^{*}$, for finite
alphabets $\cal A=\\{a_{1},\ldots,a_{d}\\}$ and
$\cal{A^{\prime}}=\\{a^{\prime}_{1},\ldots,a^{\prime}_{d^{\prime}}\\}$ defines
an incidence matrix $M(\sigma)=(|\sigma(a_{j})|_{a^{\prime}_{i}})_{i,j}$ of
seize $d^{\prime}\times d$, where $|w|_{x}$ denotes the number of occurrences
of the letter $x$ in the word $w$. Since $M(\sigma)$ describes also the linear
map $\mathbb{R}^{d}\to\mathbb{R}^{d^{\prime}}$ on the associated vector spaces
that is given by the abelianization of monoid morphism $\sigma$, one derives
directly from Theorem 1:

######  Corollary 2.

Let $\cal A$ be a finite alphabet, and for any $n\geq 0$ let $\sigma_{n}:\cal
A\to\cal A$ be a monoid morphism with invertible incidence matrix
$M(\sigma_{n})$. If the directive sequence $(\sigma_{n})_{n\geq 0}$ is
everywhere growing, then any invariant measure $\mu$ on the subshift $X$
generated by this sequence is determined by the evaluation of $\mu$ on the
letter cylinders, i.e. by the values $\mu([a_{k}])$ for all $a_{k}\in\cal A$.

The conclusion of Corollary 2 has recently been proved by Berthé et al. under
distinctly more restrictive hypotheses (see Corollary 4.2 of [3]). This
sparked among the authors the feeling that the results of our papers [1] and
[2] are yet not well enough divulged in the symbolic dynamics community. We
hence decided to write this note in order to show how the above statements
follow as directs consequences of our previous results.

###### Proof of Theorem 1.

For any $S$-adic development $\overleftarrow{\sigma}=(\sigma_{n}:\cal
A^{*}_{n+1}\to\cal A^{*}_{n})_{n\geq 0}$ of $X$ let $\zeta_{n}:\cal M(X)\to
C(X_{n})=:\cal C_{n}\subseteq\mathbb{R}^{\cal A_{n}}$ be the map for the level
$n$ subshift $X_{n}$ analogue to the above map $\zeta$ for $X$. The main
result of [1] (see also Theorem 3.2 of [2]) shows that for any everywhere
growing $\overleftarrow{\sigma}$ and any level $n\geq 0$ the linear map
$M(\sigma_{n})$ maps $\cal C_{n+1}$ onto $\cal C_{n}$, and that for the
resulting inverse limit cone $\cal V$ of the $C_{n}$ there is a surjective
linear map $\mathfrak{m}:\cal V\to\cal M(X)$ which splits over the maps
$\zeta_{n}$.

In Corollary 10.4 of [1] it is shown that, if all $\cal C_{n}$ have the same
finite dimension $c_{n}=c(X)$, then all of the maps $\zeta_{n}$ are bijective.
For $n=0$ this gives the claimed statement. $\sqcup$$\sqcap$

######  Remark 3.

The proof (and hence the statement of Theorem 1) is correct for the more
general case where the $S$-adic development is not of bounded level alphabet
cardinality, as long as it is everywhere growing and all $c_{n}$ are equal.

###### Proof of Corollary 2.

From the hypothesis that $M(\sigma_{n})$ is invertible it follows directly
that $c_{n+1}=c_{n}$, so that the hypotheses of Theorem 1 are satisfied.
$\sqcup$$\sqcap$

We finish this note with the observation that for any subshift $X$ of finite
$S$-rank and any everywhere growing $S$-adic development of $X$ over $\cal A$
it follows directly from the finiteness of $\cal A$ that there is a level
$n_{0}\geq 0$ with $c_{n}=c_{n_{0}}$ for all $n\geq n_{0}$. It follows from
Theorem 1 that any invariant measure $\mu_{n_{0}}$ on the intermediate
subshift $X_{n_{0}}$ is determined by the values $\mu_{n_{0}}([a_{k}])$, for
any $a_{k}\in\cal A$. Thus we can apply Proposition 4 below to conclude:

Any invariant measure $\mu$ on $X$ is determined by the values
$\mu([\sigma_{0}\circ\ldots\circ\sigma_{n_{0}-1}(a_{k})])$ for all
$a_{k}\in\cal A$.

Although we don’t know a precise reference, the following basic fact is
probably well known to the experts:

######  Proposition 4.

Let $\sigma:\cal A^{*}\to\cal B^{*}$ be a non-erasing monoid morphism (i.e.
$\sigma(a_{k})$ is non-trivial for any $a_{k}\in\cal A$), and let $X$ be any
subshift over $\cal A$. Let $Y$ be the subshift over $\cal B$ generated by the
set $\sigma(X)$. Then one has:

  1. (1)

The map $\sigma$ induces a well defined map $\sigma M:\cal M(X)\to\cal M(Y)$.

  2. (2)

The map $\sigma M$ is surjective.

  3. (3)

The map $\sigma M$ is functorial: For any second non-erasing monoid morphism
$\sigma^{\prime}$ one has $(\sigma\sigma^{\prime})M=\sigma M\sigma^{\prime}M$.
$\sqcup$$\sqcap$

######  Remark 5.

(1) The map $\sigma M$ isn’t quite as natural as one may wish it to be: If
$\mu$ is a probability measure (i.e. $\mu(X)=1$), then the image measure
$\sigma M(\mu)$ need not be a probability measure.

(2) A direct proof of Proposition 4 follows from the above cited surjectivity
of the map $\mathfrak{m}$, since any everywhere growing directive sequence for
$X$ is prolongated by $\sigma$ to an everywhere growing directive sequence for
$Y$, as long as $\sigma$ is non-erasing.

An alternative (but more elaborate) proof can be derived by elementary
methods, which does not depend on the particular directive sequence chosen to
generate the given subshift $X$.

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