††thanks: Present address: Department of Chemistry, Princeton University,
Princeton, New Jersey 08544, USA

# Topological Hall effect in the antiferromagnetic Dirac semimetal EuAgAs

Antu Laha<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology, Kanpur 208016, India  Ratnadwip Singha  Saha Institute of Nuclear
Physics, HBNI, 1/AF Bidhannagar, Calcutta 700 064, India  Sougata Mardanya
Department of Physics, Indian Institute of Technology, Kanpur 208016, India
Bahadur Singh<EMAIL_ADDRESS>Department of Condensed Matter
Physics and Materials Science, Tata Institute of Fundamental Research, Mumbai
400005, India  Amit Agarwal  Department of Physics, Indian Institute of
Technology, Kanpur 208016, India  Prabhat Mandal  Saha Institute of Nuclear
Physics, HBNI, 1/AF Bidhannagar, Calcutta 700 064, India  Z. Hossain
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Kanpur
208016, India

###### Abstract

The non-trivial magnetic texture in real space gives rise to the intriguing
phenomenon of topological Hall effect (THE), which is relatively less explored
in topological semimetals. Here, we report large THE in the antiferromagnetic
(AFM) state in single crystals of EuAgAs, an AFM Dirac semimetal. EuAgAs hosts
AFM ground state below $T_{N}$ = 12 K with a weak ferromagnetic component. The
in-plane isothermal magnetization below $T_{N}$ exhibits a weak metamagnetic
transition. We also observe chiral anomaly induced positive longitudinal
magnetoconductivity which indicates a Weyl fermion state under applied
magnetic field. The first-principles calculations reveal that EuAgAs is an AFM
Dirac semimetal with a pair of Dirac cones, and therefore, a Weyl semimetailic
state can be realized under time-reversal symmetry breaking via an applied
magnetic field. Our study establishes that EuAgAs is a novel system for
exploiting the interplay of band topology and the topology of the magnetic
texture.

The rich interplay between the electronic structure and topology in materials
ascertain a variety of topological states. These include topological states as
diverse as Dirac or Weyl semimetals (DSMs or WSMs), topological insulators,
topological magnets, and topological superconductors Armitage  _et al._
(2018); Yan and Felser (2017); Hasan and Kane (2010); Qi and Zhang (2011);
Neupane  _et al._ (2013); Laha  _et al._ (2019, 2020a, 2020b); Morali  _et
al._ (2019); Belopolski  _et al._ (2019); Liu  _et al._ (2018); Wang  _et al._
(2018); Zhu  _et al._ (2020); Mook  _et al._ (2016); Li  _et al._ (2017,
2016a); Yao  _et al._ (2018). The topological states of matter are robust and
cannot be destroyed by local disorder and perturbations. Among these, the
topological magnets are particularly appealing since they support tunable
phases and electromagnetic responses which are exciting for fundamental
science as well as for next-generation technological applications. Notable
examples are magnetic WSMs where magnetization can enhance the net Berry flux
through Weyl nodes tuning and as a consequence, the anomalous Hall effect
(AHE) is enhanced Liu  _et al._ (2018); Wang  _et al._ (2018); Liang  _et al._
(2018); Suzuki  _et al._ (2016). Despite their recent experimental
realization, the current choice of the available materials is quite limited.
Therefore, search for new topological magnets with novel electromagnetic
responses is of immense importance.

Furthermore, topologically non-trivial spin textures in magnetic systems give
rise to another type of Hall effect, termed as topological Hall effect (THE).
Such nontrivial spin textures in real space are generated either by
geometrical frustration or antisymmetric Dzylashinsky-Moriya (DM) interaction
Bruno  _et al._ (2004). When an electron moves through these spin textures, it
acquires a real space Berry phase, which acts as an effective magnetic field.
This effective magnetic field is proportional to the scalar spin chirality of
the magnetic structure and leads to THE Wang  _et al._ (2019). The nontrivial
spin textures can be described by a finite topological number
$Q=(1/4\pi)\int\hat{n}\cdot(\partial\hat{n}/\partial
x\times\partial\hat{n}/\partial y)~{}dxdy$, where $\hat{n}$ is the unit vector
in the direction of local magnetization. An integer value of $Q$, with
$|Q|\geq 1$, results in THE Neubauer  _et al._ (2009); Kanazawa  _et al._
(2011); Schulz  _et al._ (2012); Wang  _et al._ (2016); Xiao  _et al._ (2019).
Recent studies have shown that achiral polar crystals with $C_{nv}$ symmetry
allow DM interaction which can stabilize the Neel-type skyrmion phase with
large THE Puphal  _et al._ (2020); Kurumaji  _et al._ (2017); Kézsmárki  _et
al._ (2015). While the THE has been observed in various families of materials
as diverse as B20 compounds, antiferromagnets, pyrochlore lattices, correlated
oxide thin films Sürgers  _et al._ (2014); Ueland  _et al._ (2012); Taguchi
_et al._ (2001); Machida  _et al._ (2007); Ohuchi  _et al._ (2015); Vistoli
_et al._ (2019); Neubauer  _et al._ (2009); Kanazawa  _et al._ (2011); Huang
and Chien (2012); Taniguchi  _et al._ (2004); Fabris  _et al._ (2006), it
remains largely unexplored in topological WSMs with a few exceptions such as
CeAlGe Puphal  _et al._ (2020).

In this Letter, we report large THE in single crystalline EuAgAs, a new
antiferromagnetic (AFM) DSM. EuAgAs crystallizes in a hexagonal symmetry with
$P6_{3}/mmc$ ($D_{6h}$) space group. This symmetry group can generate finite
scalar magnetization-chirality as well as skyrmion or bi-skyrmion phases in
magnetic materials, suitable for realizing the THE Wang  _et al._ (2016); Xiao
_et al._ (2019). We explore the interplay between the topology and magnetism
in EuAgAs below $T_{N}=$12 K. We also observe chiral anomaly induced positive
longitudinal magnetoconductivity associated with the Weyl fermion state under
applied magnetic field. From the first-principles calculations, we conclude
that EuAgAs realizes an AFM DSM state with a pair of Dirac cones lying on the
$C_{3z}$ axis. Thus, our study establishes EuAgAs as a new AFM DSM with
anomalous transport properties.

Figure 1: (a) Temperature-dependent magnetic susceptibility in zero-field-
cooled (ZFC) and field-cooled (FC) conditions for $B$$\parallel$$c$ and
$B$$\parallel$$ab$. (b) The ZFC magnetic susceptibility (cyan markers) and the
Curie-Weiss fitting (solid red line) to the $1/\chi$ curve in the temperature
range $20-300$ K for $B$$\parallel$$ab$. (c) Isothermal magnetization ($M$)
and the first derivative of $M$ ($dM/dB$) at 10 K for $B$$\parallel$$ab$. The
inset shows $M$ vs. $B$ at 4 K for $B$$\parallel$$c$ and $B$$\parallel$$ab$.
(d) Temperature dependence of the specific heat at various magnetic field
values up to 9T. The solid blue line in the inset fits the formula
$C_{P}/T=\gamma+\beta T^{2}$.

EuAgAs single crystals were grown by flux method with bismuth as an external
flux Nayak  _et al._ (2018). Magnetotransport, magnetization, and specific
heat measurements were carried out in physical property measurement system and
SQUID-VSM (Quantum Design). The experimental and computational details are
given in supplementary material (SM) SM ; Rodríguez-Carvajal (1993); Hohenberg
and Kohn (1964); Kohn and Sham (1965); Blöchl (1994); Kresse and Furthmüller
(1996); Kresse and Joubert (1999); Perdew  _et al._ (1996); Anisimov  _et al._
(1997, 1991). To resolve the magnetic ordering in EuAgAs, we have measured the
temperature-dependent zero-field-cooled (ZFC) and field-cooled (FC)
susceptibility ($\chi$) along the crystallographic $c$-axis
($B$$\parallel$$c$) and $ab$-plane ($B$$\parallel$$ab$) as shown in Fig. 1(a).
The $\chi$ exhibits a peak at 12 K due to AFM ordering of the Eu2+ moments
which is suppressed by the application of a magnetic field of 3 T. It is
evident from Fig. 1 that the nature of $\chi$($T$) curve for
$B$$\parallel$$ab$ is quite different from typical AFM system Laha and Hossain
(2018). The $\chi$($T$) shows nonmonotonic $T$ dependence below $T_{N}$. After
a sharp drop below $T_{N}$, $\chi$($T$) starts to increase and saturates below
8 K. A small bifurcation between ZFC and FC curves is also observed below 8 K.
This behavior of $\chi$ suggests the presence of a weak FM component and
competition between AFM and ferromagnetic (FM) interactions Cho  _et al._
(1996). Above 20 K, we have fitted the 1/$\chi$ for $B$$\parallel$$ab$ with
the modified Curie-Weiss law, $\chi(T)=\chi_{0}+C/(T-\theta_{p})$ [Fig. 1(b)].
Here, $\chi_{0}$, $C$, and $\theta_{P}$ are the temperature-independent
susceptibility, Curie constant, and Curie-Weiss temperature respectively. The
estimated effective magnetic moment of Eu2+ is 8.03 $\mu_{B}$ for
$B$$\parallel$$ab$ and 7.68 $\mu_{B}$ for $B$$\parallel$$c$, which are close
to the theoretical value of $g\sqrt{S(S+1)}\mu_{B}=7.94\mu_{B}$ for $S=7/2$.
The fit yields $\theta_{P}\approx$ 10.4 K and 8.7 K for $B$$\parallel$$ab$ and
$B$$\parallel$$c$, respectively [see Supplementary Materials (SM)] SM .

To obtain further insight into the magnetic ground state, we have measured
isothermal magnetization $M(B)$ for both orientations $B$$\parallel$$ab$ and
$B$$\parallel$$c$. For $T$$<$$T_{N}$, the $M$ increases faster with increase
in $B$ and saturation-like behaviour appears at lower field for
$B$$\parallel$$ab$ ($\sim$ 2 T) as compared to $B\parallel c$ ($\sim$ 3 T)
[see the inset of Fig.1(c)]. The $M(B)$ curve at 10 K shows a weak anomaly
below 1 T for $B$$\parallel$$ab$ as shown in Fig.1(c). To analyze this
behavior, we plot $dM/dB$ as a function of $B$, in which a peak is observed
around $\sim$ 0.3 T. At low temperature, the $M(B)$ curve for
$B$$\parallel$$ab$ exhibits an upward curvature and a weak hysteresis at low
field (see SM for details) SM . These behaviors indicate a weak metamagnetic
transition for $B$$\parallel$$ab$. However, no anomaly is observed for
$B\parallel c$ at low field. It is also clear from Figs. 1(a) and (c) that the
easy axis of magnetization is on the $ab$-plane. The observed behavior is
qualitatively similar to that reported for isostructural EuCuAs compound Tong
_et al._ (2014). Both the value and $T$ dependence of $\chi$ below $T_{N}$ in
EuCuAs are also sensitive to the direction of the applied magnetic field and
$\theta_{P}$ is positive. To reconcile the observed behavior of magnetic
properties, A-type AFM ordering was proposed for EuCuAs where Eu2+ spins align
ferromagnetically within $ab$ plane and antiferromagnetically between the
planes. Our theoretical results discussed below also support such a spin state
for EuAgAs. We have also measured field dependence of magnetization at a high
temperature above $T_{N}$ as shown in SM SM . Even well above $T_{N}$, the $M$
is large and shows nonlinear B dependence (e.g. at 20 K).

The temperature dependence of specific heat ($C_{P}$) at various fields up to
9 T is shown in Fig. 1(d). $C_{P}$($T$) shows a sharp $\lambda$-like peak at
$T_{N}$=12 K which suppresses with increase in magnetic field. The fit to
zero-field $C_{P}$($T$) curve in the range 16-30 K with the expression
$C_{P}=\gamma T+\beta T^{3}$ yields $\gamma\sim 334$ mJ mol-1 K-2 [inset of
Fig. 1(d)]. Large value of $\gamma$ has also been reported for EuFe2P2 and
EuCr2As2 and was attributed to magnon contributions from the ordering of Eu
momentsFeng  _et al._ (2010); Paramanik  _et al._ (2014).

Figure 2: (a) Temperature-dependent resistivity at zero magnetic field. (b)
Transverse magnetoresistance ($B\perp I$) as a function of $B$ at different
temperatures up to 100K. (c) Longitudinal magnetoresistance ($B\parallel I$)
as a function of $B$ at various temperatures. (d) The magnetic field
dependence of longitudinal conductivity ($\sigma=1/\rho$) at various $T$
($\theta=0^{\circ}$). The solid red lines show the fit to Eq. (1), which
includes antilocalization and chiral anomaly induced corrections to
conductivity. (e) $1/C_{0}$ vs. $T^{2}$, the red solid line indicates a linear
fitting. (f) Transverse conductivity ($\sigma=\rho/(\rho^{2}+\rho_{yx}^{2})$)
for 50 K ($\theta=90^{\circ}$) fits to the expression
$\sigma(B)=\sigma_{WAL}+\sigma_{N}=(\sigma_{0}+a\sqrt{|B|})+(\rho_{0}+AB^{2})^{-1}$.
(g) Resistivity as a function of $B$ at various angle $\theta$ at 50 K. (h)
Schematic diagram of the experimental set up.

We now discuss the electrical resistivity ($\rho$) and magnetoresistance (MR)
of EuAgAs. Figure. 2(a) shows that $\rho$ decreases almost linearly with
decreasing $T$ down to 40 K. With a further decrease in $T$ below 40 K, $\rho$
starts increasing due to the influence of magnetic ordering and reaches a
maximum at $T_{N}$=12 K. The transverse magnetoresistance (TMR) shows a cusp-
like feature at very low magnetic field which is attributed to the weak
antilocalization (Fig.2(b)). The transition from the AFM state to a spin-
polarized state is marked by a sharp maximum around 3 T. In the spin-polarized
state, the TMR becomes negative and saturates at high fields. The TMR is
negative up to 20 K. However, TMR becomes positive well above $T_{N}$ ($T>$ 20
K). In contrast, the longitudinal magnetoresistance (LMR) is negative both
below and above $T_{N}$ and increases monotonically with the field, as shown
in Fig. 2(c). The negative LMR increases continuously with a decrease in $T$
from 2.7% at 100 K to 66% at 4 K. The negative MR at low-temperature may
partly arise due to the suppression of spin-disorder scattering as in the case
of several Eu-based compounds where MR can be scaled with magnetization as
discussed in SM SM ; Chan  _et al._ (1998); Majumdar  _et al._ (1999).
However, the negative LMR in the paramagnetic region well above $T_{N}$ is
surprising. Also, the significantly larger value of LMR and the nature of its
field dependence suggest that LMR at low temperature is dominated by a
nonmagnetic component. Regarding the physical mechanism behind the observation
of the negative LMR, the possibility of the current jetting effect can be
ruled out as discussed in the SM SM ; Hu  _et al._ (2005). We attribute the
negative LMR to the chiral anomaly associated with the presence of opposite
chirality Weyl Fermions. This is supported by the successful fitting of the
magnetoconductivity (MC) data to the semiclassical formula [Fig. 2(d)] Kim
_et al._ (2013),

$\sigma(B)=\sigma_{\rm WAL}~{}(1+C_{0}B^{2})+\sigma_{N}~{}.$ (1)

Here, $\sigma_{\rm WAL}=(\sigma_{0}+a\sqrt{|B|})$ is the conductivity due to
weak anti-localization effect, and $\sigma_{N}=(\rho_{0}+AB^{2})^{-1}$
represents the normal conductivity originating from the conventional Fermi
surface (other than Weyl points). $\rho_{0}$ denotes the zero-field
resistivity, and $a$ and $A$ are two constants. The positive MC (or negative
MR) depends on $C_{0}B^{2}$, where the positive value of $C_{0}$ originates
from the finite ${\bf E}\cdot{\bf B}$ term which produces chiral charge
current in WSM Das and Agarwal (2020); Nishihaya  _et al._ (2018); Ishizuka
and Sato (2019). Remarkably, we observe chiral anomaly and weak
antilocalization up to relatively high temperature [Fig 2(d)] as in the case
of YbPtBi Guo  _et al._ (2018). Notably, we do not find any $B-$linear
resistivity terms in our fitting which indicates there are no electron pockets
in the quantum limit Abrikosov (2000). The temperature dependence of $C_{0}$
can be described by the relation $C_{0}\propto
v_{F}^{3}\tau_{v}/(T^{2}+\mu^{2}/\pi^{2})$, where $v_{F}$, $\tau_{v}$ and
$\mu$ are the Fermi velocity, chirality-changing scattering time and chemical
potential respectively Li  _et al._ (2016b). The linear fitting to the curve
$1/C_{0}$ vs. $T^{2}$ as shown in Fig. 2(e) further supports the presence of
chiral anomaly in the temperature range 20-100 K. The estimated value of
$\tau_{v}$ is $2.8\times 10^{-12}$ s. Similar range of scattering timescales
have also been seen in other Weyl semimetals Jadidi  _et al._ (2020); Guo  _et
al._ (2018); Zhang  _et al._ (2016). The chiral anomaly induces positive MC
only for the longitudinal case when $I\parallel B$, and this is not present in
the transverse case (${\bf E}\cdot{\bf B}=0$). Notably, the transverse MC can
be expressed as
$\sigma^{T}(B)=(\sigma_{0}+a\sqrt{|B|})+(\rho_{0}+AB^{2})^{-1}$, without
chiral anomaly. The successful fitting of MC data at 50 K to this expression
further confirms that the origin of LMR and TMR are quite different and the
cusp-like feature is due to the weak antilocalization [Fig. 2(f)]. We have
further fitted the MC data with the Hikami-Larkin-Nagaoka formula to establish
the weak antilocalization effect up to 100 K and shown the variation of phase
coherence length with temperature in SM SM ; Shrestha  _et al._ (2017); Xu
_et al._ (2014); Hou  _et al._ (2015).

Figure 2(g) shows magnetic field dependent resistivity at various angle
$\theta$ between $B$ and $I$ for 50 K (The direction of $B$ and $I$ are
schematically shown in Fig. 2(h)). The MR shows a maximum negative value at
$\theta=0^{\circ}$ (=-10$\%$) and decreases with increasing $\theta$. MR
becomes positive for $\theta>30^{\circ}$ and reaches maximum at
$\theta=90^{\circ}$ (=29$\%$). Similar angle-dependent behavior of the MR has
also been reported in other WSMs Li  _et al._ (2016c); Guo  _et al._ (2018).
We have also measured the directional dependence of MR by fixing the angle
between $B$ and $I$ and observed a twofold symmetric anisotropy pattern at 4 K
(details in SM) SM .

The experimental Hall resistivity after removing the MR contribution using the
expression $\rho_{yx}=[\rho_{yx}(B)-\rho_{yx}(-B)]$ is shown in Fig. 3(a). The
negative and linear $\rho_{yx}$ for 20 K $\leq T\leq$ 300 K indicates that
electrons are the majority carriers. We estimate the carrier density ($n$) and
Hall mobility ($\mu$) using the relations $n$=1/($eR_{\textrm{0}}$) and
$\mu$=$R_{\textrm{0}}$/$\rho_{(B=0)}$, where $R_{\textrm{0}}$ is the slope of
$\rho_{yx}(B)$ curve. The temperature dependence of $n$ and $\mu$ are shown in
Fig. 3(b). We find that the carrier density is of the order of $10^{20}$cm-3
which remains almost temperature independent.

The Hall resistivities for $T\leq$ 10 K show an anomaly below 3 T, which is
quite prominent at low temperature. This behavior may be the reminiscent of
the anomalous Hall effect (AHE) observed in AFM Weyl semimetals. AHE is
empirically modelled as $\rho_{yx}^{A}=S_{0}\rho^{2}M$, $S_{0}$ being a field
independent parameter. However, we find that below 3 T, the observed
$\rho_{yx}$ deviates from the fitting of the empirical Hall resistivity
$\rho_{yx}=\rho_{yx}^{N}+\rho_{yx}^{A}$, as shown in Fig. 3(c). We also tried
fitting the $\rho_{yx}$ with the conventional two-carriers model but it fails
to reproduce the anomaly (details in SM) SM . This clearly suggests the
presence of an extra contribution in the Hall resistivity, which we attribute
to the topological Hall effect Kanazawa  _et al._ (2011); Puphal  _et al._
(2020).

Figure 3: (a) The Hall resistivity $\rho_{yx}$ as a function of $B$
($B\parallel c$) at various temperatures. (b) Charge carrier density ($n$) and
mobility ($\mu$) vs. temperature. (c) Hall resistivity (green markers) fitted
to the conventional Hall contribution
$\rho_{yx}$=$\rho^{N}_{yx}+\rho_{yx}^{A}$ at 4 K (blue solid lines). The clear
deviation from the conventional Hall resistivity indicates the presence of an
additional term - the topological Hall resistivity (orange markers). The inset
shows Anomalous Hall angle as a function of $B$ at 4 K. (d) Topological Hall
resistivity at various temperatures.

By combining both AHE and THE in the AFM state, the total Hall resistivity can
be expressed as Kanazawa  _et al._ (2011)

$\rho_{yx}=\rho_{yx}^{N}+\rho_{yx}^{A}+\rho_{yx}^{T}=R_{0}B+S_{0}\rho^{2}M+\rho_{yx}^{T}~{}$
(2)

Here, $\rho_{yx}^{T}$ denotes the novel topological Hall resistivity arising
from the chiral magnetization-texture. Since $\rho_{yx}^{T}$ vanishes in the
fully spin polarized state due to the absence of spin-chirality, we execute a
linear fit to the curve $\rho_{yx}/B$ vs $\rho^{2}M/B$ in the field range 4 T
$\leq B\leq$ 9 T. From the fitting, we estimate, $R_{0}=-2.51~{}\mu\Omega$ cm
T-1 and $S_{0}=-1.72\times 10^{-3}$ ($\mu\Omega$ cm)-1 $\mu_{B}^{-1}$. Using
these parameters, $\rho_{yx}^{T}$ can be obtained by subtracting
$\rho_{yx}^{N}+\rho_{yx}^{A}$ from $\rho_{yx}$, which is shown in Fig.3(c)
Puphal  _et al._ (2020); Kanazawa  _et al._ (2011); Li  _et al._ (2013). We
estimate the maximum amplitude of the $\rho_{yx}^{T}$ at 2 K to be
$\sim-7~{}\mu\Omega$ cm [Fig.3(d)]. This value of $\rho_{yx}^{T}$ is quite
large. In order to compare the THE, we estimate the topological Hall angle,
$\Theta_{xy}^{T}=\sigma_{xy}^{T}/\sigma$, where $\sigma_{xy}^{T}$ and $\sigma$
are the topological Hall conductivity and magnetoconductivity respectively.
The maximum amplitude of $\Theta_{xy}^{T}$ ($\sim 0.17$) is comparable with
that obtained in the magnetic Weyl semimetals GdPtBi ($\sim 0.17$) and YbPtBi
($\sim 0.18$) Suzuki  _et al._ (2016); Guo  _et al._ (2018). We have also
estimated the anomalous Hall angle ($\theta_{xy}^{A}=\sigma_{xy}^{A}/\sigma$)
(inset of Fig.3(c)). The maximum amplitude of $\theta_{xy}^{A}$ is 0.11, which
is compared with some other known AFM Weyl semimetals in Table-I of SM SM ;
Nakatsuji  _et al._ (2015); Nayak  _et al._ (2016); Suzuki  _et al._ (2016);
Singha  _et al._ (2019); Zhu  _et al._ (2020).

Figure 4:  (a) Orientation of Eu spins in the antiferromagnetic EuAgAs. (b)
The density of states (DOS). The partial DOS of spin-up and spin-down states
are shown in red and blue, respectively. The DOS associated with the localized
Eu $f$ states is shown in gray. (c) Band structure along the high-symmetry
directions without spin-orbit coupling (SOC). The red and blue identify spin-
up and spin-down states. (d) Band structure with SOC with spins along the
$ab$-plane. The Dirac band crossings are seen along the $\Gamma$-$A$
direction.

We delineate the AFM DSM state of EuAgAs in Fig. 4. The unit cell consists of
six atoms, in which Eu atoms occupy the Wyckoff position 2a and form hexagonal
stacking along the $z-$direction. The Ag and As atoms are placed at Wyckoff
positions 2d and 2c respectively to form hexagonal layers which sandwiched
between the Eu-layers, satisfying the overall inversion symmetry. To consider
the possible magnetic configuration of EuAgAs, we investigated three AFM spin
configurations (see SM for details) SM . The lowest energy is obtained for the
in-plane AFM [110] configuration as shown in Fig. 4(a). This is consistent
with our measured magnetic properties. The AFM ordering of the Eu $4f$ spins
is further demonstrated in Fig. 4(b) where the density of states (DOS) in the
valence band region is dominated by the Eu $f$ states which is equal but
opposite for spin-up and spin-down states. Moreover, EuAgAs has a vanishingly
small DOS at the Fermi level which confirms its semi-metallic electronic
state. This is further showcased in the band structure without spin-orbit
coupling (SOC) in Fig. 4(c). The valence and conduction bands derived for both
the spin states are seen to cross along the $\Gamma-A$ direction with an
inverted band ordering at the $\Gamma$-point. The band structure with SOC is
shown in Fig. 4(d). We find that the EuAgAs crystals have an inversion
symmetry $I$, along with an effective time-reversal symmetry,
$S=\Theta\tau_{1/2}$, where $\Theta$ is the time-reversal operator and
$\tau_{1/2}$ is translation vector connecting spin-up and spin-down atoms.
Owing to these two symmetries, each of the band in AFM EuAgAs remains two-fold
spin degenerate. We find that while the nodal band crossings without SOC are
lifted, there are two symmetry-related four-fold degenerate band crossing
points along the $\Gamma-A$ directions as shown in Fig. 4(d). EuAgAs thus
realizes a robust AFM DSM ground state with a pair of Dirac cones in vicinity
of the Fermi energy, on the $C_{3z}$ rotation axis.

We next turn to discuss the possible reason behind the AHE and THE observed in
EuAgAs. To confirm if the Dirac nodes in EuAgAs can split into a pair of Weyl
nodes on applying a magnetic field, we calculated the electronic structure of
the fully polarized state of EuAgAs (see SM) SM . We find that each Dirac node
indeed splits into a pair of Weyl nodes of opposite chiral charge, realizing a
ferromagnetic Weyl semimetal. The distance between two pair of Weyl nodes is
$k_{w1}$ = 0.1096 Å-1 and $k_{w2}$ = 0.227 Å-1. The calculated anomalous Hall
conductivity using the expression
$|\sigma^{A}_{yx}|=[(k_{w1}+k_{w2})/2\pi]\times(e^{2}/h)$ Burkov and Balents
(2011); Uchida  _et al._ (2019); Šmejkal  _et al._ (2018); Hirschberger  _et
al._ (2016) is found to be 208 $\Omega^{-1}$cm-1, which is lower than our
experimentally observed maximum amplitude of $\sim 2700$ $\Omega^{-1}$cm-1 at
2 K. The fact that EuAgAs belongs to a symmetry group which supports real
space Berry curvature associated with nontrivial spin texture similar to
MnNiGa and MnPdGa Wang  _et al._ (2016); Xiao  _et al._ (2019) makes THE a
natural choice for the additional contribution.

In summary, using first-principle calculations we show that EuAgAs hosts an
antiferromagnetic DSM ground state with a pair of Dirac cones. The observed
magnetic properties confirm that EuAgAs shows an AFM transition at $T_{N}=$12
K along with a weak metamagnetic transition below $T_{N}$. Chiral anomaly
induced positive longitudinal MC confirms the presence of a Weyl fermion state
in EuAgAs under an applied magnetic field. It further exhibits anomalous and
topological Hall effects. We provide compelling evidence of topological Hall
effect in a spin only divalent europium compound that is likely to inspire
further studies on topological Hall effect in DSMs.

Acknowledgements. Research support from IIT Kanpur is gratefully acknowledged.
Z. H. also acknowledges support from SERB India (Grant No. CRG/2018/000220).
A. A. acknowledges Science and Engineering Research Board (SERB) and
Department of Science and Technology (DST) of the government of India for
financial support. The work at TIFR Mumbai is supported by the Department of
Atomic Energy of the Government of India under project number
12-R&D-TFR-5.10-0100. We thank Arup Ghosh for Hall resistivity measurements.

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# Correlations of vorticity inside a coherent vortex

I.V.Kolokolov${}^{1}\,{}^{2}$, V.V.Lebedev${}^{1}\,{}^{2}$,
M.M.Tumakova${}^{1}\,{}^{3}$ 1 Landau Institute for Theoretical Physics, RAS,
142432, Chernogolovka, Moscow region, Russia;
2 National Research University Higher School of Economics,
101000, Myasnitskaya ul. 20, Moscow, Russia.
3 National Research University Higher School of Economics, 190008,
St.Petersburg, Russia

###### Abstract

We examine fluctuations of vorticity inside the coherent vortex, appearing as
a consequence of the inverse energy cascade in two-dimensional turbulence.
Temporal and spacial correlations can be characterized by the pair correlation
function. The interaction between the fluctuations leads to non-zero value of
the third moment of vorticity. We examine the pair correlation function and
the third moment for the model where the pumping is short correlated in time.
We find explicit expressions for the Gaussian spacial correlation function of
the pumping force. They confirm the general predictions obtained earlier.

##  I Introduction

Two-dimensional turbulence is a subject of numerous investigations
boffetta2012two . The state can be observed in fluid films on scales larger
than the film thickness. From the practical point of view, the most
interesting such fluid “film” is atmosphere. Of course, atmosphere is very
complex object and its detailed description is extremely difficult problem.
However, some general features of two-dimensional turbulence could be useful
for understanding atmospheric phenomena. Note in this respect the trend of the
production of large-scale eddies from small-scale ones thanks to the non-
linear hydrodynamic interaction in two-dimensional fluids 67Kra ; 68Lei ;
69Bat . In presence of the external forcing (pumping) the trend leads to
formation of the inverse energy cascade at scales larger than the pumping
length KraM .

In a finite box, the transfer of the energy to large scales leads to formation
of big vortices with the diameter of the order of the size of the box. Such
vortices were observed both in laboratory experiments ShaF ; FilLevchOrlov17
and in numerical simulations MCH04 ; chertkov2007dynamics . At some conditions
the big vortex has an infinite life time, that is it possesses long time
correlations. We call such big vortices coherent. In the work
laurie2014universal  the flat profile of the mean velocity of the coherent
vortex was observed in numerical simulations and some arguments explaining the
profile were presented. In the works KL15 ; kolokolov2016structure ;
kolokolov2016velocity ; frishman17 ; kolokolov2017static  the quasilinear
regime of the flow fluctuations inside the coherent vortex was ground, the
flat velocity profile was derived and some velocity correlation functions were
discussed. In the work KL20  the criterion of the formation of the coherent
vortex was established, the criterion was confirmed in direct numerical
simulations DFKL22 .

We consider the case where the two-dimensional turbulence is excited by a
permanent small-scale pumping (external force). If the force is static or has
statistical properties homogeneous in time then finally the statistically
homogeneous in time turbulent state is achieved. Just in this situation the
coherent vortices could appear. The laboratory experiments and the numerical
simulations show that the coherent vortex is isotropic in average. The
isotropy is explained by fast rotation suppressing the anisotropic
perturbations on the background of the isotropic rotation. Let us stress that
the mean flow of the coherent vortex is highly non-solid differential
rotation. Such motion essentially deforms small-scale fluctuations,
suppressing their nonlinear interaction. The property explains the
applicability of the quasilinear approximation for the flow fluctuations
inside the coherent vortex. The deformation makes correlation functions of the
flow fluctuations highly anisotropic. The structure of the correlation
functions is of crucial importance for such processes as transport of the
passive scalar (the temperature field or the concentration of pollutants)
inside the coherent vortex.

We are interested in the correlation functions of the fluctuating vorticity.
In the quasilinear approximation the correlation functions are reduced to the
pair correlation function. To find the correlation function, one has to
examine the evolution of the small-scale flow fluctuations, produced by
pumping, inside the coherent vortex. Some universal scaling properties of the
correlation function were formulated in our work KLT23 . However, it is
important to check the results by explicit calculations by using some concrete
model. We chose the model where the pumping is assumed to be random processes
short correlated in time. The spacial pair correlation function of the
external force is assumed to be a Gaussian function of the distance between
the points. The analytical calculations in the model enabled us to find
explicit analytic expressions for the correlation function of vorticity,
confirming the predictions of the work KLT23 .

Though the nonlinear interaction of the flow fluctuations inside the coherent
vortex is weak, it plays an essential role in some processes. As an example,
we note the energy transport inside the coherent vortex. In addition, the
strength of the interaction grows as one goes away from the center of the
vortex. Thus, evaluation of the strength is needed to establish the vortex
boundary and the character of the fluctuations near the boundary. The strength
of the interaction can be characterized by the third moment of the vorticity,
that is zero in the quasilinear approximation. Some general properties of the
moments of the vorticity were formulated in the work KL23 . However, again, it
is important to check the results by explicit calculations by using some
concrete model. We calculated the third moment in the framework of the above
model and found the explicit analytic expression, confirming the predictions
of the work KL23 .

The structure of our text is as follows. In the section II we give some
general relations concerning the fluctuations inside the coherent vortex. In
the section III we evolve the perturbation theory needed for calculation of
the correlation function of vorticity. In the section IV we examine the pair
correlation function of vorticity. In the section V we calculate the third
moment of vorticity. In the section VI we sum up the results of our research.

##  II General relations

We consider two-dimensional turbulence excited by an external random force
with a stationary statistics. We analyze the statistically stationary state
realized after some transient process. In the state the characteristics of the
coherent vortices are independent of time. Let us give two conditions needed
for appearing the coherent vortices in the system. The first condition is

$\epsilon\gg L^{2}\alpha^{3},$ (1)

where $\epsilon$ is the energy production rate per unit mass (supplied by the
external forcing), $L$ is the size of the box and $\alpha$ is the bottom
friction coefficient. The condition (1) means that the energy carried by the
inverse cascade accumulates at the scale $L$. The second condition is

$\alpha\lesssim\nu k_{f}^{2},$ (2)

where $\nu$ is the kinematic viscosity coefficient and $k_{f}$ is
characteristic wave vector of the external force. The condition (2) was
derived in the work KL20  and confirmed by direct numerical simulations, see
Ref. DFKL22 .

It is convenient to examine the coherent vortex in the reference frame
attached to the vortex center. In the reference frame the mean (coherent) flow
is differential rotation, which can be described by the polar (tangential)
velocity $U$ depending on the distance $r$ from the observation point to the
vortex center. Locally, the differential rotation is the shear flow with the
shear rate

$\Sigma=r\partial_{r}(U/r),$ (3)

depending on $r$ as well. Note that for the solid rotation, where $U\propto
r$, the shear rate (3) is zero. For the flat velocity profile, derived in the
works KL15 ; kolokolov2016structure ; kolokolov2017static , $U$ is independent
of $r$. Then the shear rate behaves as $\Sigma\propto r^{-1}$, that us
$\Sigma$ diminishes as $r$ grows. Thus, the suppression of the interaction of
the flow fluctuations, associated with the coherent vortex, slackens as $r$
increases.

The main object of our investigation are correlation functions of the
vorticity $\varpi$, which fluctuates on the background of the average
vorticity $\Omega$ in the coherent vortex. We assume that the fluctuations are
produced by pumping, which has relatively small correlation length
$k_{f}^{-1}$. At examining the fluctuations near a circle of radius $R$ much
larger than the pumping correlation length, $k_{f}R\gg 1$, we can limit
ourselves to considering a narrow neighborhood of the circle. Then after
passing into the reference system rotating with the angular velocity
$\Omega(R)$ we obtain the following linear equation for $\varpi$

$(\partial_{t}+\hat{\mathcal{L}})\varpi=\phi,\quad\hat{\mathcal{L}}=\Sigma
x_{2}\frac{\partial}{\partial x_{1}}+\alpha-\nu\nabla^{2},$ (4)

where $\Sigma$ is the shear rate (3), taken at $r=R$. The tangential variable
$x_{1}=R\varphi$ ($\varphi$ is polar angle) and the radial variable
$x_{2}=r-R$ are local Cartesian coordinates in the reference system. In the
equation (4) $\phi$ is curl of the external force $\bm{f}$ per unit mass,
$\phi=\mathrm{curl}\,\bm{f}$.

The criterion justifying the quasilinear approximation is

$\Sigma\gg(\epsilon k_{f}^{2})^{1/3},$ (5)

where the right part of the relation is Kolmogorov non-linear rate at the
pumping scale. In turn, the quantity should be much larger than the viscous
damping on the same scale:

$(\epsilon k_{f}^{2})^{1/3}\gg\nu k_{f}^{2}.$ (6)

The criterion (6) is no other than the condition of large Reynolds number at
the pumping scale. The condition is necessary for exciting turbulence.
Combining the inequalities (5,6), one finds that

$\Sigma\gg\nu k_{f}^{2},$ (7)

inside the coherent vortex.

The equation (4) demonstrates that the problem is reduced to the hydrodynamic
motion on the background of a stationary shear flow. The presence of the shear
flow breaks the spatial homogeneity. More precisely, the homogeneity is broken
along the second axis. The fact makes complicated investigations of the
solutions of the equation (4). The equation (4) without pumping $\phi$ was
analyzed in the work souzy . We are interested in the correlation functions of
$\varpi$. To find the correlation functions one should solve the equation (4)
for arbitrary $\phi$ and then average the corresponding product over the
statistics of $\phi$.

In the work we use the model of pumping short correlated in time. Then
statistical properties of pumping are determined by its pair correlation
function

$\langle\phi(t_{1},\bm{x})\phi(t_{2},\bm{y})\rangle=-2\epsilon\delta(t_{1}-t_{2})\nabla^{2}\Xi(\bm{x}-\bm{y}).$
(8)

The angular brackets in Eq. (8) and below designate mean values obtained by
averaging over time in experiment. In the theoretical framework the angular
brackets mean averaging over the system statistics. Laplacian $\nabla^{2}$ in
Eq. (8) is explained by the relation $\varpi=\mathrm{curl}\,\bm{v}$.

Further we exploit the particular shape of the pair correlation function of
pumping

$\Xi(\bm{x})=\exp(-k_{f}^{2}|\bm{x}|^{2}/2).$ (9)

Then $\epsilon$ in Eq. (8) is just the pumping rate (energy production rate
per unit mass). The pumping correlation function (9) enables us to find
explicitly the pair correlation function $\langle\varpi\varpi\rangle$ and the
third moment $\langle\varpi^{3}\rangle$. However, their qualitative properties
are universal and are independent of the particular shape of the pair
correlation function of pumping.

In the work KLT23  the time $\tau_{\star}$ is introduced, determining the
characteristic time, at which the second moment of $\varpi$ is formed. The
time is defined as

$\tau_{\star}=\left(\Sigma^{2}\nu
k_{f}^{2}\right)^{-1/3},\quad\Sigma\gg\tau_{\star}^{-1}\gg\nu k_{f}^{2}.$ (10)

The inequalities in Eq. (10) are explained by the condition (7). Note also
that $\alpha\tau_{\star}\ll 1$. The inequality is explained by the same
condition (7) and the condition (2) needed for realizing the coherent vortex.

If one is interested in effects related to the interaction of the fluctuations
then one should introduce the nonlinear term into the linear equation (4). It
is determined by the nonlinear term in Navier-Stokes equation, responsible for
the interaction of the fluctuations. Adding the nonlinear term to the linear
equation (4), one finds

$\partial_{t}\varpi+\hat{\mathcal{L}}\varpi+\nabla\left(\bm{v}\varpi-\langle\bm{v}\varpi\rangle\right)=\phi,$
(11)

where $\bm{v}$ is the velocity of the fluctuating flow.

An introduction of the average $\langle\bm{v}\varpi\rangle$ into Eq. (11) is
related to the fact that we assume $\langle\varpi\rangle=0$,
$\langle\phi\rangle=0$. The first condition is related to the definition of
the fluctuating flow. The second condition means that the external forcing
does not produce vorticity in average. The term $\langle\bm{v}\varpi\rangle$
depends on coordinates inside the coherent vortex and, consequently, its
divergence is non-zero. It is the reason to include the term into Eq. (11).

Due to the incompressibility condition $\nabla\bm{v}=0$ it is possible to
introduce the stream function $\psi$, related to the velocity components and
to the vorticity as

$v_{1}=\frac{\partial\psi}{\partial x_{2}},\quad
v_{2}=-\frac{\partial\psi}{\partial x_{1}},\quad\varpi=-\nabla^{2}\psi.$ (12)

To restore the velocity $\bm{v}$ from the vorticity field $\varpi$, one has to
solve the Laplace equation $\nabla^{2}\psi=-\varpi$ and then to calculate the
velocity components in accordance with Eq. (12). Let us stress that the stream
function $\psi$, introduced by Eq. (12), is related to the fluctuating part of
the flow.

In the quasilinear approximation the third order moment
$\langle\varpi^{3}\rangle$ is zero. To find a non-zero contribution to the
moment $\langle\varpi^{3}\rangle$, one has to take into account the
interaction between the flow fluctuations related to the non-linear term in
Navier-Stokes equation. In our setup one has to use the non-linear equation
(11) where the interaction is described by the second-order term
$\bm{v}\nabla\varpi$.

All our calculations are performed at the condition $\Sigma>0$. However, the
results can be easily extended to negative $\Sigma$. Say, the simple symmetry
reasoning show that the third moment $\langle\varpi^{3}\rangle$ changes its
sign at $\Sigma\to-\Sigma$. As to the pair correlation function
$\langle\varpi(t,\bm{x})\varpi(0,\bm{y})\rangle$, the transformation
$\Sigma\to-\Sigma$ means that one should change the sign of the second
component $x_{2}\to-x_{2}$, $y_{2}\to-y_{2}$ without touching $x_{1},y_{1}$.

##  III Perturbation theory

The interaction of the flow fluctuations can be consistently examined in terms
of Wyld diagrammatic technique Wyld61 , based on the dynamic equation (11).
The diagram technique can be derived from the representation of the
correlation functions of the vorticity $\varpi$ as functional integrals over
$\varpi$ and an auxiliary field $\mu$ with the weight $\exp(-{\mathcal{I}})$
MSR73 , where

$\displaystyle{\mathcal{I}}={\mathcal{I}}_{2}+{\mathcal{I}}_{int},$ (13)
$\displaystyle{\mathcal{I}}_{2}=\int
dt\,d^{2}x\,\mu(\partial_{t}+\hat{\mathcal{L}})\varpi$
$\displaystyle+\epsilon\int
dt\,d^{2}x\,d^{2}r\,\nabla^{2}\Xi(\bm{x}-\bm{r})\mu(t,\bm{x})\mu(t,\bm{r}),$
(14) $\displaystyle{\mathcal{I}}_{int}=\int
dt\,d^{2}x\,\mu\bm{v}\nabla\varpi.$ (15)

Here the velocity $\bm{v}$ is implied to be expressed via the vorticity
$\varpi$. Details of the technic can be found in the review HRS .

The pair correlation function of the vorticity is written as the following
functional integral

$\langle\varpi(t,\bm{x})\varpi(0,\bm{y})\rangle=\int
D\varpi\,D\mu\,e^{-{\mathcal{I}}}\varpi(t,\bm{x})\varpi(0,\bm{y}),$ (16)

We introduce also the following pair average

$\langle\varpi(t,\bm{x})\mu(0,\bm{y})\rangle=\int
D\varpi\,D\mu\,e^{-{\mathcal{I}}}\varpi(t,\bm{x})\mu(0,\bm{y}).$ (17)

This correlator (17) is the Green function, since it determines the response
of the system to an additional external force. Note that the average
$\langle\mu\mu\rangle$ is zero.

One can evolve the perturbation theory for any correlation function expanding
the weight $\exp(-{\mathcal{I}})$ in the third order term
${\mathcal{I}}_{int}$ (15) and calculating the resulting Gaussian functional
integrals. The integrals are expressed in terms of the “bare” correlation
functions determined by the quadratic term (14):

$\displaystyle\langle\varpi(t,\bm{x})\mu(0,\bm{y})\rangle_{0}=\int{\mathcal{D}}\varpi{\mathcal{D}}\mu\,e^{-{\mathcal{I}}_{2}}\varpi(t,\bm{x})\mu(0,\bm{y}),$
(18)
$\displaystyle\langle\varpi(t,\bm{x})\varpi(0,\bm{y})\rangle_{0}=\int{\mathcal{D}}\varpi{\mathcal{D}}\mu\,e^{-{\mathcal{I}}_{2}}\varpi(t,\bm{x})\varpi(0,\bm{y}).$
(19)

The average of the type $\langle\varpi\dots\mu\dots\rangle_{0}$ is determined
by Wick theorem Wick50 )and it is equal to the sum of products of the pair
averages (18,19) organized by all possible pairings. Each term of the
perturbation series corresponds to a Feynman diagram.

The expressions (14,18) lead to the following equation for the bare Green
function

$(\partial_{t}+\hat{\mathcal{L}})\langle\varpi(t,\bm{x})\mu(0,\bm{y})\rangle_{0}=\delta(t)\delta(\bm{x}-\bm{y}).$
(20)

Remind that any Green function is zero at negative times due to causality.
Therefore $\langle\varpi(t,\bm{x})\mu(0,\bm{y})\rangle$ is zero at $t<0$. The
expression for the “bare” pair correlation function can be derived using Eq.
(19):

$\displaystyle\langle\varpi(t,\bm{x})\varpi(0,\bm{y})\rangle_{0}=-2\epsilon\int
d\tau\int d^{2}rd^{2}z\nabla^{2}\Xi(\bm{r}-\bm{z})$
$\displaystyle\langle\varpi(t,\bm{x})\mu(\tau,\bm{r})\rangle_{0}\langle\varpi(0,\bm{y})\mu(\tau,\bm{z})\rangle_{0}.\quad$
(21)

The correlation function is proportional to the energy production rate
$\epsilon$, as it should be.

Due to the presence of the shear flow the space homogeneity is broken.
Therefore the pair correlation functions depend generally on both coordinates.
In the case Fourier transform should be performed over both coordinates. We
define Fourier transform for the Green function and the pair correlation
function as follows

$\displaystyle\langle\varpi(t,\bm{x})\mu(0,\bm{y})\rangle_{0}=\int\frac{d^{2}k\,d^{2}q}{(2\pi)^{4}}e^{i\bm{k}\bm{x}-i\bm{q}\bm{y}}{\mathcal{G}}(t,\bm{k},\bm{q}),$
(22)
$\displaystyle\langle\varpi(t,\bm{x})\varpi(0,\bm{y})\rangle_{0}=\int\frac{d^{2}k\,d^{2}q}{(2\pi)^{4}}e^{i\bm{k}\bm{x}+i\bm{q}\bm{y}}{\mathcal{F}}(t,\bm{k},\bm{q}).$
(23)

The definition leads to the rules convenient for us.

For the Fourier transform ${\mathcal{G}}(t,\bm{k},\bm{q})$ (22) we derive from
Eq. (20) the following differential equation

$\displaystyle\left(\partial_{t}-\Sigma k_{1}\frac{\partial}{\partial
k_{2}}+\alpha+\nu k_{1}^{2}+\nu
k_{2}^{2}\right){\mathcal{G}}(t,\bm{k},\bm{q})$
$\displaystyle(2\pi)^{2}\delta(t)\delta(\bm{k}-\bm{q}).$ (24)

Since the equation (24) is of the first order, it can be easily solved by the
method of characteristics to obtain

$\displaystyle{\mathcal{G}}(t,\bm{k},\bm{q})=(2\pi)^{2}\theta(t)\delta(k_{1}-q_{1})\delta\left(k_{2}-q_{2}+\Sigma
k_{1}t\right)$ $\displaystyle\exp\left(-\alpha t-\nu q_{1}^{2}t-\nu
q_{2}^{2}t+\nu\Sigma
q_{2}q_{1}t^{2}-\frac{1}{3}\nu\Sigma^{2}q_{1}^{2}t^{3}\right),$ (25)

where $\theta(t)$ is the Heaviside step function.

We find from Eq. (9)

$\displaystyle\int
d^{2}x\,\exp(-i\bm{k}\bm{x})\Xi(\bm{x})=\frac{2\pi}{k_{f}^{2}}\exp\left(-\frac{\bm{k}^{2}}{2k_{f}^{2}}\right).$
(26)

Therefore we obtain for the pair correlation function (21)

$\displaystyle{\mathcal{F}}(t,\bm{k},\bm{q})=2\epsilon\int
dt_{1}\int\frac{d^{2}p}{2\pi
k_{f}^{2}}p^{2}\exp\left(-\frac{p^{2}}{2k_{f}^{2}}\right)$
$\displaystyle{\mathcal{G}}(t+t_{1},\bm{k},\bm{p}){\mathcal{G}}(t_{1},\bm{q},-\bm{p}).$
(27)

The expression (27) can be represented as the Feynman diagram depicted in Fig.
1. Here the combined solid-dashed line represents the Green function (22), the
bullet represents the pumping, and the integration over the wave vector
$\bm{p}$ is implied.

Figure 1: Feynman diagram representing the pair correlation function (27).

To find a non-zero contribution to the moment $\langle\varpi^{3}\rangle$, one
has to take into account the interaction between the flow fluctuations
described by the third order term ${\mathcal{I}}_{int}$ (15). The leading
contribution to the third order moment is of the first order in the term:

$\langle\varpi^{3}\rangle=-\int{\mathcal{D}}\varpi\,{\mathcal{D}}\mu\,\exp(-{\mathcal{I}}_{2})\varpi^{3}{\mathcal{I}}_{int}.$

Using Wick theorem, one obtains

$\displaystyle\langle\varpi^{3}\rangle=-6\int
dt\,d^{2}x\,\langle\varpi(0,\bm{0})\mu(t,\bm{x})\rangle_{0}$
$\displaystyle\frac{\partial}{\partial\bm{x}}\langle\varpi(0,\bm{0})\varpi(t,\bm{x})\rangle_{0}\langle\varpi(0,\bm{0})\bm{v}(t,\bm{x})\rangle_{0}.$
(28)

In terms of the stream function $\psi$, see Eq. (12), one can rewrite Eq. (28)
as

$\displaystyle\langle\varpi^{3}\rangle=6\int_{0}^{\infty}dt\,\int
dx_{1}dx_{2}\langle\varpi(t,\bm{0})\mu(0,\bm{x})\rangle_{0}$
$\displaystyle\left[-\frac{\partial}{\partial
x_{1}}\langle\varpi(t,\bm{0})\varpi(0,\bm{x})\rangle_{0}\frac{\partial}{\partial
x_{2}}\langle\varpi(t,\bm{0})\psi(0,\bm{x})\rangle_{0}\right.$
$\displaystyle\left.+\frac{\partial}{\partial
x_{2}}\langle\varpi(t,\bm{0})\varpi(0,\bm{x})\rangle_{0}\frac{\partial}{\partial
x_{1}}\langle\varpi(t,\bm{0})\psi(0,\bm{x})\rangle_{0}\right].$ (29)

Here we have taken into account that the correlation functions depend on the
time differences and that the Green function is zero at negative times $t$ due
to causality.

##  IV Pair correlation function

The pair correlation function (23) can be examined using the obtained
expressions. Substituting the expression (27) into Eq. (23) we find

$\displaystyle\langle\varpi(t,\bm{x})\varpi(0,\bm{y})\rangle_{0}=2\epsilon\int_{0}^{\infty}dt_{1}\int\frac{d^{2}p\,p^{2}}{2\pi
k_{f}^{2}}$ $\displaystyle\exp\left(-\frac{p^{2}}{2k_{f}^{2}}+W\right),$
$\displaystyle W=-\alpha(t_{1}+t_{3})-\nu p_{1}^{2}(t_{1}+t_{3})-\nu
p_{2}^{2}(t_{1}+t_{3})$ $\displaystyle+\nu\Sigma
p_{2}p_{1}(t_{1}^{2}+t_{3}^{2})-\frac{1}{3}\nu\Sigma^{2}p_{1}^{2}(t_{1}^{3}+t_{3}^{3})+ip_{1}x_{1}$
$\displaystyle+i(p_{2}-\Sigma p_{1}t_{3})x_{2}-ip_{1}y_{1}-i(p_{2}-\Sigma
p_{1}t_{1})y_{2}.$

where $t>0$, $t_{3}=t+t_{1}$. The expression can be rewritten as

$\displaystyle\langle\varpi(t,\bm{x})\varpi(0,\bm{y})\rangle_{0}=2\epsilon
k_{f}\frac{\partial}{\partial k_{f}}\int\frac{dp_{1}dp_{2}}{2\pi}$
$\displaystyle\int_{0}^{\infty}dt_{1}\exp\left(-\frac{p_{1}^{2}+p_{2}^{2}}{2k_{f}^{2}}+W\right).$

The exponent here is represented as

$\displaystyle\exp\left[-\frac{1}{2}(\begin{array}[]{cc}p_{1}&p_{2}\end{array})\hat{L}\left(\begin{array}[]{c}p_{1}\\\
p_{2}\end{array}\right)\right],$ (33)

where $\hat{L}$ is the matrix with the components

$\displaystyle
L_{11}=k_{f}^{-2}+2\nu(t_{1}+t_{3})+(2/3)\nu\Sigma^{2}(t_{1}^{3}+t_{3}^{3}),$
$\displaystyle L_{12}=L_{21}=-\nu\Sigma(t_{1}^{2}+t_{3}^{2}),$ $\displaystyle
L_{22}=k_{f}^{-2}+2\nu(t_{1}+t_{3}).$

Determinant of $\hat{L}$ is

$\displaystyle\det\hat{L}=k_{f}^{-4}+k_{f}^{-2}4\nu(t_{1}+t_{3})$
$\displaystyle+k_{f}^{-2}(2/3)\nu\Sigma^{2}(t_{1}^{3}+t_{3}^{3})+4\nu^{2}(t_{1}+t_{3})^{2}$
$\displaystyle+(1/3)\nu^{2}\Sigma^{2}t_{1}^{4}+(4/3)\nu^{2}\Sigma^{2}t_{1}^{3}t_{3}$
$\displaystyle-2\nu^{2}\Sigma^{2}t_{1}^{2}t_{3}^{2}+(4/3)\nu^{2}\Sigma^{2}t_{1}t_{3}^{3}+(1/3)\nu^{2}\Sigma^{2}t_{3}^{4}.$
(34)

Taking the integral over $p_{1},p_{2}$, one finds

$\displaystyle\langle\varpi(t,\bm{x})\varpi(0,\bm{y})\rangle_{0}={2\epsilon}k_{f}\frac{\partial}{\partial
k_{f}}$
$\displaystyle\int_{0}^{\infty}dt_{1}\exp[-\alpha(t_{1}+t_{3})](\det\hat{L})^{-1/2}$
$\displaystyle\exp\left[-\frac{1}{2}(x_{1}-\Sigma t_{3}x_{2}-y_{1}+\Sigma
t_{1}y_{2})^{2}\lfloor p_{1}^{2}\rfloor\right.$
$\displaystyle-\frac{1}{2}(x_{2}-y_{2})^{2}\lfloor
p_{2}^{2}\rfloor-(x_{1}-\Sigma t_{3}x_{2}-y_{1}$ $\displaystyle\left.+\Sigma
t_{1}y_{2})(x_{2}-y_{2})\lfloor
p_{1}p_{2}\rfloor\vphantom{\frac{1}{2}}\right],$ (35)

where

$\displaystyle\lfloor
p_{1}^{2}\rfloor=(\det\hat{L})^{-1}[k_{f}^{-2}+2\nu(t_{1}+t_{3})]$
$\displaystyle\lfloor
p_{2}^{2}\rfloor=\frac{1}{\det\hat{L}}\left[k_{f}^{-2}+2\nu(t_{1}+t_{3})+\frac{2}{3}\nu\Sigma^{2}(t_{1}^{3}+t_{3}^{3})\right],$
$\displaystyle\lfloor
p_{1}p_{2}\rfloor=(\det\hat{L})^{-1}\nu\Sigma(t_{1}^{2}+t_{3}^{2}).$

As it should be, the correlation function (35) depends solely on
$x_{1}-y_{1}$.

First of all, we calculate the second moment $\langle\varpi^{2}\rangle$.
Taking $t=0,\bm{x}=0,\bm{y}=0$, we find from Eq. (35)

$\displaystyle\langle\varpi^{2}\rangle={2\epsilon}k_{f}\frac{\partial}{\partial
k_{f}}\int_{0}^{\infty}dt_{1}\exp(-2\alpha t_{1})(\det\hat{L})^{-1/2},$ (36)

where one should substitute $t_{3}=t_{1}$. As we will see,
$t_{1}\sim\tau_{\star}$. Neglecting the terms of the order of $\nu
k_{f}^{2}t_{1}\ll 1$ we find in the main approximation from Eq. (34)

$\displaystyle\det\hat{L}\approx
k_{f}^{-4}+(4/3)k_{f}^{-2}\nu\Sigma^{2}t_{1}^{3}.$

Substituting the expression into Eq. (36), one finds after integrating over
$t_{1}$

$\langle\varpi^{2}\rangle=2\epsilon
k_{f}^{2}\tau_{\star}\frac{2}{\sqrt{\pi}}\left(\frac{4}{3}\right)^{2/3}\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{7}{6}\right).$
(37)

At the calculation one substituted $\exp(-2\alpha t_{1})$ by unity since
$\alpha\tau_{\star}\ll 1$.

Consider the case, where $\bm{x}=\bm{y}=0$ and $t\gg\tau_{\star}$. Then

$\displaystyle\langle\varpi(t,\bm{0})\varpi(0,\bm{0})\rangle_{0}$
$\displaystyle={2\epsilon}k_{f}\frac{\partial}{\partial
k_{f}}\int_{0}^{\infty}dt_{1}\frac{\exp[-\alpha(t_{1}+t_{3})]}{(\det\hat{L})^{1/2}},$
(38)

where $t_{3}=t+t_{1}$. Then $t_{1}\sim t$ and

$\det\hat{L}\approx k_{f}^{-2}(2/3)\nu\Sigma^{2}[t_{1}^{3}+(t+t_{1})^{3}].$

The approximation is valid provided $t\ll(\nu k_{f}^{2})^{-1}$. Since
$\alpha\lesssim\nu k_{f}^{2}$, we, again, can substitute $\exp(-2\alpha
t_{1})$ by unity in Eq. (38). Then one obtains

$\displaystyle\langle\varpi(t,\bm{0})\varpi(0,\bm{0})\rangle=\sqrt{6}\,c_{1}\frac{\epsilon
k_{f}^{2}\tau_{\star}^{3/2}}{t^{1/2}},$ (39)

where $c_{1}$ is the following numerical factor

$c_{1}=\int_{0}^{\infty}\frac{ds}{\sqrt{(1+s)^{3}+s^{3}}}\approx 1.6969.$

The expression (39) is correct provided $\tau_{\star}\ll t\ll(\nu
k_{f}^{2})^{-1}$, at $t\sim\tau_{\star}$ it turns to Eq. (37).

If $t=0$, then the pair correlation function depends on $\bm{x}-\bm{y}$.
Therefore one can take $\bm{y}=0$ to obtain

$\displaystyle\langle\varpi(0,\bm{x})\varpi(0,\bm{0})\rangle_{0}={2\epsilon}k_{f}\frac{\partial}{\partial
k_{f}}\int_{0}^{\infty}dt_{1}\frac{\exp(-B)}{(\det\hat{L})^{1/2}},$ (40)

where

$\displaystyle\det\hat{L}=k_{f}^{-4}+8k_{f}^{-2}\nu
t_{1}+(4/3)k_{f}^{-2}\nu\Sigma^{2}t_{1}^{3}$
$\displaystyle+8\nu^{2}t_{1}^{2}+(4/3)\nu^{2}\Sigma^{2}t_{1}^{4},$ (41)
$\displaystyle B=\frac{1}{2}(x_{1}-\Sigma t_{1}x_{2})^{2}\lfloor
p_{1}^{2}\rfloor+\frac{1}{2}x_{2}^{2}\lfloor p_{2}^{2}\rfloor$
$\displaystyle+(x_{1}-\Sigma t_{1}x_{2})x_{2}\lfloor p_{1}p_{2}\rfloor,$

that is

$\displaystyle B=(2\det\hat{L})^{-1}\left\\{[k_{f}^{-2}+4\nu
t_{1}]x_{1}^{2}\right.$ $\displaystyle-2\Sigma t_{1}[k_{f}^{-2}+2\nu
t_{1}]x_{1}x_{2}+[k_{f}^{-2}+4\nu t_{1}$
$\displaystyle\left.+\Sigma^{2}t_{1}^{2}k_{f}^{-2}+(4/3)\nu\Sigma^{2}t_{1}^{3}]x_{2}^{2}\right\\}.$
(42)

We will consider different regions corresponding to different characteristic
values of $t_{1}$. In any case, we assume $\alpha t_{1}\ll 1$, therefore the
omitted in Eq. (40) the factor $\exp(-2\alpha t_{1})$.

Let us consider the case $x_{1}=0$. If $k_{f}|x_{2}|\gg 1$ then the expression
(40) is exponentially small. That is why we analyze the opposite limit. If
$k_{f}|x_{2}|\ll(\Sigma\tau_{\star})^{-1}$ then the dependence of
$\langle\varpi(0,\bm{x})\varpi(0,\bm{0})\rangle$ on $x_{2}$ is negligible and
we return to the second moment (37). Therefore we concentrate on the region

$1\gg k_{f}|x_{2}|\gg(\Sigma\tau_{\star})^{-1},$ (43)

where a power behavior is realized. There are two different regions of
integration over $t_{1}$ in the expression (40): small $t_{1}$ and large
$t_{1}$, supplying the leading contributions to
$\langle\varpi(0,\bm{x})\varpi(0,\bm{0})\rangle$.

In the first region (small $t_{1}$) the dominant term in the curly brackets in
Eq. (42) is $\Sigma^{2}t_{1}^{2}k_{f}^{-2}x_{2}^{2}$ and we find from Eqs
(41,42) in the main approximation

$\det\hat{L}=k_{f}^{-4},\quad
B=\frac{1}{2}\Sigma^{2}t_{1}^{2}k_{f}^{2}x_{2}^{2}.$ (44)

Therefore we find the characteristic $t_{1}$, $t_{1}\sim(\Sigma
k_{f}x_{2})^{-1}$. Then we obtain from Eq. (43)

$\Sigma^{-1}\ll t_{1}\ll\tau_{\star}.$ (45)

The inequalities (45) justify the expressions (44). Substituting the
expressions (44) into Eq. (40), one finds the small $t_{1}$ contribution:

$\langle\varpi(0,\bm{x})\varpi(0,\bm{0})\rangle_{s}=\sqrt{2\pi}\frac{2\epsilon
k_{f}}{\Sigma|x_{2}|},$ (46)

The second contribution stems from the region of large $t_{1}$, where the
leading contributions to the determinant (41) and to the quantity (42) are

$\displaystyle\det\hat{L}=\frac{4}{3k_{f}^{2}}\nu\Sigma^{2}t_{1}^{3},\quad
B=\frac{3}{8}(\nu t_{1})^{-1}x_{2}^{2}.$ (47)

Therefore the characteristic $t_{1}$ is estimated as
$t_{1}\sim\nu^{-1}x_{2}^{2}$ and we find from Eq. (43)

$(\nu k_{f}^{2})^{-1}\gg t_{1}\gg\tau_{\star}.$ (48)

The inequalities (48) justify the expressions (47). Substituting the
expressions (47) into Eq. (40), one finds the same expression (49). Thus, at
$x_{1}=0$ the answer (49) should be doubled:

$\langle\varpi(0,\bm{x})\varpi(0,\bm{0})\rangle=\sqrt{2\pi}\frac{4\epsilon
k_{f}}{\Sigma|x_{2}|},$ (49)

At the boundary $k_{f}|x_{2}|\sim(\Sigma\tau_{\star})^{-1}$ the expression
(49) matches the expression for the second moment (37).

Now we consider the case $x_{1}\neq 0$, $x_{2}=0$. If $k_{f}|x_{1}|\ll 1$ then
we return to the second moment (37) for the correlation function (40). We will
analyze the region

$1\ll k_{f}|x_{1}|\ll\Sigma(\nu k_{f}^{2})^{-1}$ (50)

where the correlation function (40) possesses a power behavior. By analogy
with the above analysis, one can consider contributions from small and large
$t_{1}$. Since $k_{f}|x_{1}|\gg 1$, the integrand in (40) becomes
exponentially small at small $t_{1}$. Therefore the contribution is
negligible.

Now we pass to the contribution of large $t_{1}$, where the leading
contributions to the determinant (41) and to the quantity (42) are

$\displaystyle\det\hat{L}=\frac{4}{3k_{f}^{2}}\nu\Sigma^{2}t_{1}^{3},\quad
B=\frac{3}{8}\frac{x_{1}^{2}}{\nu\Sigma^{2}t_{1}^{3}}.$ (51)

We conclude that the characteristic value of $t_{1}$ is $t_{1}\sim
x_{1}^{2/3}(\nu\Sigma^{2})^{-1/3}$. Then we obtain from the inequalities (50)
the same region (48) for $t_{1}$. The inequalities (48) justify the
expressions (51). Substituting the expressions (51) into Eq. (40) and taking
the resulting integral over $t_{1}$, one finds

$\langle\varpi(0,\bm{x})\varpi(0,\bm{0})\rangle=\Gamma\left(\frac{7}{6}\right)\frac{2^{3/2}3^{1/3}\epsilon
k_{f}}{\Sigma^{2/3}\nu^{1/3}|x_{1}|^{1/3}}.$ (52)

At the boundary value $x_{1}\sim k_{f}^{-1}$ the expression (52) matches the
expression for the second moment (37).

Figure 2: Plot of the function $f(u)$ (55).

Now we generalize the above analysis, assuming that both, $x_{1}$ and $x_{2}$,
are nonzero and are determined by the inequalities (43,50). Then the
contribution to $\langle\varpi(0,\bm{x})\varpi(0,\bm{0})\rangle$ from small
$t_{1}$ is absent (exponentially small) and we concentrate on the region of
large $t_{1}$, determined by the same conditions (48). Then the leading
contributions to the determinant (41) and to the quantity (42) are

$\det\hat{L}=\frac{4}{3k_{f}^{2}}\nu\Sigma^{2}t_{1}^{3},\quad
B=\frac{3}{8}\frac{(x_{1}-\Sigma t_{1}x_{2})^{2}}{\nu\Sigma^{2}t_{1}^{3}}.$
(53)

Eq. (53) is a generalization of Eqs. (47,51).

Substituting the expressions (53) into Eq. (40) and passing to the
dimensionless variable $\vartheta$, $\vartheta=\nu t_{1}/x_{2}^{2}$, one
obtains

$\displaystyle\langle\varpi(0,\bm{x})\varpi(0,\bm{0})\rangle=\sqrt{2\pi}\frac{2\epsilon
k_{f}}{\Sigma|x_{2}|}f\left(\frac{\nu x_{1}}{\Sigma x_{2}^{3}}\right),$ (54)
$\displaystyle
f(u)=\frac{\sqrt{3}}{2\sqrt{2\pi}}\int_{0}^{\infty}\frac{d\vartheta}{\vartheta^{3/2}}\exp\left[-\frac{3}{8\vartheta}\left(1-\frac{u}{\vartheta}\right)^{2}\right].$
(55)

Here $f(-0)=1$ and we reproduce Eq. (46). If $|u|\gg 1$, then

$f\approx\frac{3^{1/3}\Gamma(7/6)}{\sqrt{\pi}|u|^{1/3}},$

reproducing Eq. (52). The function $f(u)$ is plotted in Fig. 2. One should be
careful with small $u$: the function $f(u)$ experiences the jump at $u=0$,
that is observed in Fig. 2. The jump is explained by a finite contribution to
the integral in Eq. (55) at small positive $u$ from the region $\vartheta\sim
u$, at negative $u$ such contribution is absent. However, in accordance with
Eqs. (43,50) $|u|$ cannot be smaller than $\nu k_{f}^{2}/\Sigma$. Thus the
region $|u|\lesssim\nu k_{f}^{2}/\Sigma$ is unachievable in (54),(55).

##  V Third moment of vorticity

The leading contribution to the third moment of vorticity is determined by the
expression (29). It is instructive to introduce Fourier transformations for
the objects figuring in the expression

$\displaystyle{F}(t,\bm{q})=\int
d^{2}x\exp(-i\bm{q}\bm{x})\langle\varpi(t,\bm{0})\varpi(0,\bm{x})\rangle_{0},$
(56) $\displaystyle\Phi(t,\bm{k})=\int
d^{2}x\exp(-i\bm{k}\bm{x})\langle\varpi(t,\bm{0})\psi(0,\bm{x})\rangle_{0},$
(57) $\displaystyle{G}(t,\bm{p})=\int
d^{2}x\exp(i\bm{p}\bm{x})\langle\varpi(t,\bm{0})\mu(0,\bm{x})\rangle_{0}.$
(58)

In terms of the quantities (56,57,58), the expression (29) is rewritten as

$\displaystyle\langle\varpi^{3}\rangle=6\int_{0}^{\infty}dt\,\int\frac{d^{2}q\,d^{2}k}{(2\pi)^{4}}(q_{1}k_{2}-q_{2}k_{1})$
$\displaystyle{F}(t,\bm{q}){\Phi}(t,\bm{k}){G}(t,\bm{q}+\bm{k}).$ (59)

Note the wave vector conservation law.

One can express the quantities (56,57,58) via the Green function and the pair
correlation function. We find

$\displaystyle{G}(t,\bm{p})=\int\frac{d^{2}k}{(2\pi)^{2}}{\mathcal{G}}(t,\bm{k},\bm{p})$
$\displaystyle=\theta(t)\exp\left(-\alpha t-\nu\bm{p}^{2}t+\nu\Sigma
p_{2}p_{1}t^{2}-\frac{1}{3}\nu\Sigma^{2}p_{1}^{2}t^{3}\right).$ (60)

Note, that one can rewrite Eq. (25) as

$\displaystyle{\mathcal{G}}(t,\bm{k},\bm{q})=(2\pi)^{2}\delta(k_{1}-q_{1})$
$\displaystyle\delta\left(k_{2}-q_{2}+\Sigma k_{1}t\right){G}(t,\bm{q}).$ (61)

We find for the object (57)

$\displaystyle{F}(t,\bm{q})=\int\frac{d^{2}k}{(2\pi)^{2}}{\mathcal{F}}(t,\bm{k},\bm{q})$
$\displaystyle=2\pi\frac{2\epsilon}{k_{f}^{2}}\int_{0}^{\infty}dt_{1}(q_{1}^{2}+q_{3}^{2})\exp\left(-\frac{q_{1}^{2}+q_{3}^{2}}{2k_{f}^{2}}\right)$
$\displaystyle{G}(t+t_{1},q_{1},q_{3}){G}(t_{1},-q_{1},-q_{3})$

where $q_{3}=q_{2}+\Sigma q_{1}t_{1}$. Assuming $t>0$, we find

$\displaystyle{F}(t,\bm{q})=2\pi\frac{2\epsilon}{k_{f}^{2}}\int_{0}^{\infty}dt_{1}(q_{1}^{2}+q_{3}^{2})\exp\left(-\frac{q_{1}^{2}+q_{3}^{2}}{2k_{f}^{2}}\right)$
$\displaystyle\exp\left[-(\alpha+\nu q_{1}^{2}+\nu
q_{3}^{2})(t_{1}+t_{3})\vphantom{\frac{1}{2}}\right.$
$\displaystyle\left.+\nu\Sigma
q_{1}q_{3}(t_{1}^{2}+t_{3}^{2})-\frac{1}{3}\nu\Sigma^{2}q_{1}^{2}(t_{1}^{3}+t_{3}^{3})\right],$
(62)

where $t_{3}=t+t_{1}$, $q_{3}=q_{2}+\Sigma q_{1}t_{1}$. Analogously for $t>0$
we find

$\displaystyle\Phi(t,\bm{k})=2\pi\frac{2\epsilon}{k_{f}^{2}}\int_{0}^{\infty}dt_{2}\frac{k_{1}^{2}+k_{3}^{2}}{k_{1}^{2}+k_{2}^{2}}\exp\left(-\frac{k_{1}^{2}+k_{3}^{2}}{2k_{f}^{2}}\right)$
$\displaystyle\exp\left[-(\alpha+\nu k_{1}^{2}+\nu
k_{3}^{2})(t_{2}+t_{4})\vphantom{\frac{1}{2}}\right.$
$\displaystyle\left.+\nu\Sigma
k_{3}k_{1}(t_{2}^{2}+t_{4}^{2})-\frac{1}{3}\nu\Sigma^{2}k_{1}^{2}(t_{2}^{3}+t_{4}^{3})\right],$
(63)

where $t_{4}=t+t_{2}$, $k_{3}=k_{2}+\Sigma k_{1}t_{2}$.

To find the leading contribution to the third moment, one has to substitute
the expressions (60,62,63) into Eq. (59) and to calculate the resulting
integral. Note that the contribution can be represented by Feynman diagram
depicted in Fig. 3. The designations here are the same as in Fig. 3: the
combined solid-dashed line represents the Green function (22), the bullet
represents the pumping. Further we pass to integration over the components
$k_{1},k_{3},q_{1},q_{3}$ in Eq. (59). Then one should substitute

$q_{2}=q_{3}-\Sigma q_{1}t_{1},\quad k_{2}=k_{3}-\Sigma k_{1}t_{2},$ (64)

into the expressions (62,63) for $F,\Phi$. Note that the components
$k_{1},k_{3},q_{1},q_{3}$ are restricted by pumping, and therefore they cannot
be much larger than $k_{f}$.

Figure 3: Feynman diagram representing first contribution to the third moment.

Further we examine the main contribution to the integral (59). As we will see,
the contribution originates from the region of integration where
$t,t_{1}\gg\tau_{*},t_{2}$, $k_{1},q_{1}\ll k_{f}$ whereas $k_{3},q_{3}\sim
k_{f}$. Therefore one can neglect $k_{1},q_{1}$ in comparison with
$k_{3},q_{3}$. Next, we neglect $t_{2}$ in comparison with $t,t_{1}$ and keep
solely the third order terms in $t,t_{1}$ in the exponents in Eqs. (60,62,63)
in terms of the components $(k_{1},k_{3},q_{1},q_{3})$, see Eq. (64). We
neglect the terms with $\alpha$ as well. Then

$\displaystyle{G}(t,\bm{k}+\bm{q})\to\theta(t)\exp\left[-\nu\Sigma^{2}t_{1}^{2}tq_{1}^{2}\vphantom{\frac{1}{2}}\right.$
$\displaystyle\left.-\nu\Sigma^{2}t_{1}t^{2}q_{1}(q_{1}+k_{1})-\frac{1}{3}\nu\Sigma^{2}(q_{1}+k_{1})^{2}t^{3}\right],$
(65) $\displaystyle{F}(t,\bm{q})\to
2\pi\frac{2\epsilon}{k_{f}^{2}}\int_{0}^{\infty}dt_{1}q_{3}^{2}\exp\left(-\frac{q_{3}^{2}}{2k_{f}^{2}}\right)$
$\displaystyle\exp\left[-\frac{1}{3}\nu\Sigma^{2}q_{1}^{2}(t_{1}^{3}+t_{3}^{3})\right],$
(66)
$\displaystyle\Phi(t,\bm{k})=2\pi\frac{2\epsilon}{k_{f}^{2}}\int_{0}^{\infty}dt_{2}\frac{k_{3}^{2}}{k_{1}^{2}+k_{2}^{2}}$
$\displaystyle\exp\left(-\frac{k_{3}^{2}}{2k_{f}^{2}}\right)\exp\left[-\frac{1}{3}\nu\Sigma^{2}k_{1}^{2}t^{3}\right].$
(67)

In the situation one can integrate over $t_{2}$ in Eq. (67) to obtain

$\int_{0}^{\infty}dt_{2}\frac{1}{k_{1}^{2}+(\Sigma
t_{2}k_{1}-k_{3})^{2}}\to\pi\Sigma^{-1}k_{1}^{-2}\theta(k_{1}k_{3}).$ (68)

The Heaviside function in Eq. (68) means that the integral over $t_{2}$ for
identical signs of $k_{1},k_{3}$ is much larger than for different signs of
$k_{1},k_{3}$. Therefore we find from Eq. (67)

$\displaystyle\Phi(t,\bm{k})\to\frac{(2\pi)^{2}\epsilon}{\Sigma
k_{f}^{2}}\theta(k_{1}k_{3})\frac{k_{3}^{2}}{k_{1}^{2}}\exp\left(-\frac{k_{3}^{2}}{2k_{f}^{2}}\right)$
$\displaystyle\exp\left(-\frac{1}{3}\nu\Sigma^{2}k_{1}^{2}t^{3}\right).$ (69)

Substituting the expressions (65,66,69) into Eq. (59)and taking the integrals
over $k_{3},q_{3}$, we find

$\displaystyle\langle\varpi^{3}\rangle=6\epsilon^{2}k_{f}^{2}\iint_{0}^{\infty}dt\,dt_{1}\,t_{1}\int
dk_{1}\,dq_{1}\,\frac{q_{1}}{k_{1}}$
$\displaystyle\exp\left\\{-\nu\Sigma^{2}\left[\frac{2}{3}(t_{1}+t)^{2}q_{1}^{2}\right.\right.$
$\displaystyle\left.\left.+\left(t_{1}t^{2}+\frac{2}{3}t^{3}\right)q_{1}k_{1}+\frac{2}{3}t^{3}k_{1}^{2}\right]\right\\}.$
(70)

Taking then the integral over $q_{1}$, one obtains

$\displaystyle\langle\varpi^{3}\rangle=-\frac{6\epsilon^{2}k_{f}^{2}}{\Sigma\nu^{1/2}}\iint_{0}^{\infty}dt\,dt_{1}\,t_{1}\int
dk_{1}\,\sqrt{2\pi}\frac{b}{a^{3/2}}$
$\displaystyle\exp\left\\{\nu\Sigma^{2}\left[\frac{b^{2}}{2a}-\frac{2}{3}t^{3}\right]k_{1}^{2}\right\\},$
(71)

where $a=(4/3)(t_{1}+t)^{3}$, $b=t_{1}t^{2}+(2/3)t^{3}$. Taking then the
integral over $k_{1}$, one finds

$\displaystyle\langle\varpi^{3}\rangle=-\frac{12\pi\epsilon^{2}k_{f}^{2}}{\Sigma^{2}\nu}\iint_{0}^{\infty}dt\,dt_{1}\,t_{1}\frac{(b/a)}{\sqrt{(4/3)t^{3}a-b^{2}}}.$
(72)

Passing to the variable $\eta=t_{1}/t$, one obtains the logarithmic integral

$\displaystyle\langle\varpi^{3}\rangle=-\frac{9\pi\epsilon^{2}k_{f}^{2}}{\Sigma^{2}\nu}c_{2}\int\frac{dt}{t},$
(73)

where $c_{2}$ is the numerical factor

$\displaystyle
c_{2}=\int_{0}^{\infty}d\eta\frac{\eta(3\eta+2)}{(1+\eta)^{3}}(12+36\eta+39\eta^{2}+16\eta^{3})^{-1/2},$
$\displaystyle c_{2}\approx 0.187.$

Since $\eta\sim 1$, then $t_{1}\sim t$. Note also that

$k_{1},q_{1}\sim(\nu\Sigma^{2}t^{3})^{-1/2},$ (74)

as follows from the corresponding integrals.

The integration over $t$ in the integral (73) is restricted from above by the
viscous time $(\nu k_{f}^{2})^{-1}$ since at larger times the neglected in the
exponents in Eqs. (60,62,63) terms of the type $\nu k_{3}^{2}t$ start to work.
We neglected $k_{1},q_{1}$ in comparison with $k_{f}$, therefore
$t\gg\tau_{\star}$, as it follows from Eq. (74). Thus, we conclude that the
time $t$ lies in the interval

$(\nu k_{f}^{2})^{-1}\gtrsim t\gtrsim\tau_{\star}.$ (75)

In terms of $q_{1},k_{1}$ the interval (75) is written as

$k_{f}\gtrsim q_{1},k_{1}\gtrsim\frac{\nu k_{f}^{2}}{\Sigma}k_{f},$ (76)

as it follows from Eq. (74). Note that the integral (68) is gained near

$t_{2}\approx k_{3}(\Sigma k_{1})^{-1}\sim k_{f}(\Sigma k_{1})^{-1}.$ (77)

As it follows from Eqs. (74,77), the ratio

$t_{2}/t\sim(\nu k_{f}^{2}t)^{1/2}$

is small in the interval (75). As it is demonstrated in the work KL20 , the
coherent vortex is formed if the condition $\alpha\lesssim\nu k_{f}^{2}$ is
satisfied. Since the terms of the type $\nu k_{f}^{2}t$ can be neglected, the
terms with $\alpha$ can be neglected as well. Thus, our approximations are
justified in the interval (75).

The logarithmic integral (73) is gained in the wide interval of $t$ determined
by the conditions (75). To find the integral it is enough to substitute the
lower and the higher values of $t$ from the conditions (75) as the limits in
the logarithmic integral (73) and we find

$\displaystyle\langle\varpi^{3}\rangle=-\frac{6\pi\epsilon^{2}k_{f}^{2}}{\Sigma^{2}\nu}c_{2}\ln\frac{\Sigma}{\nu
k_{f}^{2}},$ (78)

with the logarithmic accuracy. Remind that we assumed $\Sigma>0$ in our
calculations. Therefore the sign minus in Eq. (78) means, that the third
moment has the sign, opposite to the sign of the shear rate $\Sigma$.

##  VI Conclusion

We analytically investigated the pair correlation function of the fluctuating
vorticity and its third moment inside the coherent vortex appearing as a
consequence of the inverse cascade in two-dimensional turbulence. The
particular choice of the pumping (external forcing) enabled us to obtain
explicit analytic expressions for the objects. The expressions
(37,39,49,52,54) are in accordance with the general analysis given in the work
KLT23 . The expression (78) for the third moment is in accordance with the
general predictions given in the work KL23 .

One new ingredient is the logarithm in the expression (78) derived with the
logarithmic accuracy. It is related to the fact that the third moment is
gained from the wide region of times $\tau_{\star}<t<(\nu k_{f}^{2})^{-1}$,
whereas the second moment is gained from the region $t\sim\tau_{\star}$. The
existence of the wide region of times is, obviously, universal, that is
independent of the concrete shape of pumping. We are going to investigate the
role of the fact for higher moments of $\varpi$.

It is worth noting that there are also fluctuations with the scale of the
order of the system size $L$. It was shown in Frishman1  that such
fluctuations are important in analysis of velocity statistics. However for
$k_{f}L\gg 1$ their contribution to vorticity moments is negligible.

I.V. Kolokolov’s work was carried out within the framework of the scientific
program of the National Center for Physics and Mathematics (the project
”Physics of High Energy Densities. Stage 2023-2025”), M.M.Tumakova thanks the
Basis Foundation for its support, V.V.Lebedev thanks the support of the grant
of the Ministry of Higher Education and Science of the Russian Federation,
project No. 075-15-2022-1099.

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