text:
\section{Introduction}

Vortices are characteristic quantized topological
excitations of a superfluid. In the case of a Bose
superfluid a vortex represents a line singularity
in the phase of the condensate wave function, the
phase having an increment of $2\pi$ as one turns around
this line. Vortices in superfluid $^4$He \cite{donnelly,barenghi}
and in cold bosonic
atoms \cite{fetter,Pit16} have been extensively studied over the years both
as an example of the ubiquitous vorticity
phenomena in many-body systems as well as a
signature of the presence of phase coherence in the system.
Vortices are expected to be also present in a supersolid.

The supersolid phase (SSP) of matter has attracted considerable interest
because the two symmetries that are spontaneously broken at the
same time seem incompatible at first sight \cite{bal10}. On the one hand the
translational symmetry is broken so that particles localize
with crystalline order. On the other hand gauge symmetry is broken and this
leads to a condensate and to the appearance of superfluid properties.

Supersolidity was
proposed long ago for solid $^4$He \cite{and69,che70,leg70}. However, experiments have shown
that low temperature ($T$) solid $^4$He, in spite of displaying a number of anomalous properties, does
not conform to the bulk supersolid paradigm \cite{cha13,bon12}. Cold bosons have turned out
to be a more fruitful platform to address supersolidity.  As discussed in the following,
a number of properties expected for a supersolid have recently been verified for systems made of
bosonic atoms with a permanent magnetic moment (dipolar bosons).

One should expect differences between the appearance and dynamics of a vortex in a superfluid
and in a supersolid due to its completely different environment.
In a superfluid, density can be non-uniform, e.g. close to
an interface or in the whole volume as for cold atoms in a trap. These are large scale
inhomogeneities compared to the vortex
core radius, and the vortex is free to move inside the superfluid which it sees as locally uniform.
This behavior is similar to that of vortices in
classical hydrodynamics: vortices are free to move inside the fluid
and each vortex moves with the local velocity due to the presence of other vortices
\cite{lamb}.
The density in a supersolid has a strong spatial modulation
on the microscopic scale. Since, in order to minimize its kinetic energy, a vortex is a low density
``seeker'', it is no longer free to move as in the superfluid phase (SFP).
Rather, it behaves as if it were moving in a
periodic potential with a spatial scale comparable to its core size.
In addition, this density inhomogeneity is not simply due to
an external potential but the interparticle
interactions play a key role in establishing it.
This is quite a new regime and the main purpose of
the present paper is to uncover and discuss some of its properties.

Experiments, in agreement with the predictions of theory,
show that dipolar bosons can be found in different
phases.
When the dipole interaction strength is small,
the system is in the superfluid phase.
As the interaction strength increases, the dipolar boson
dispersion relation
(excitation energy as a function
of the momentum of the excitation, ${\cal E}(k)$) displays
a roton minimum  whose gap amplitude depends
on the relative strengths of short-range and dipolar interactions \cite{lewe,odell}.
Such dispersion relation has some similarities with that
of superfluid $^4$He \cite{donnelly,Pit16}.
When the ratio of the dipolar to the short-range interaction
increases beyond a critical value,
a spontaneous density modulation occurs
which is driven by the softening of the
roton mode at finite momentum $k_R$,
and the resulting system has supersolid character
\cite{wenzel,blackie_baillie3,Chomaz2018,lewe,kora,ancilotto}.
The density modulation, with wavelength $\sim 2\pi/k_R$,
results in an ordered array of
``droplets'',  made of many atoms each, elongated in the direction of the polarization axis.
Global phase coherence is maintained between adjacent droplets
due to a low-density superfluid background that
allows atom tunneling from one superfluid droplet to another.
As the ratio between interactions is further increased,
the superfluid background almost disappears
so that phase coherence is no longer present between droplets,
and the phase becomes a ``normal solid phase'' (NSP in the following),
even if each droplet can be superfluid.

Further increase of that ratio
eventually results
in a transition to a totally different regime, characterized by
the formation of
{\it self-bound} droplets \cite{Kad16,Fer16,schmitt,santos_wachtler,blackie_baillie1,ferlaino},
with order-of-magnitude higher densities, the binding
arising from the interplay between the
two-body dipolar interactions and the effect of quantum
fluctuations \cite{lima_pelster,santos_wachtler}.

In dipolar bosons supersolid behavior occurs in the intermediate
regime between superfluid and self-bound droplets regimes.
We remark that the term droplet will
be used here to indicate the individual clusters making
up the ordered structure of a modulated phase in such intermediate regime,
and will not refer to the self-bound droplet regime.

A number of theoretical studies predicted supersolid behavior in dipolar
Bose-Einstein condensates (BEC) in different geometries
\cite{bombin,cinti1,wenzel},
dipolar gases confined
in a quasi-2D pancake shaped trap \cite{blackie_baillie3,blackie_baillie1},
or in a tube \cite{ancilotto}.
The order of the superfluid-supersolid transition has been studied
in Ref. \cite{pohl}.

Experiments provide evidence of SSP
in dipolar bosons in a number of ways.
Strong global phase coherence was found in
the SSP realization of Ref. \cite{chomx}
as opposed to the lack of it in the
isolated droplet phase where no superfluid flow
is present between adjacent droplets in the NSP.
Similarly, robust phase coherence across a
linear array of quantum droplets
was observed in Ref. \cite{bott19}.
Stable stripe modulations have also been observed
in dipolar quantum gases \cite{wenzel,tanzi}.
A partial phase coherence is suggested in Ref. \cite{tanzi},
thus indicating possible SSP behavior.
The characteristic symmetry breaking of a SSP was observed
through the appearance of compressional oscillation
modes in a harmonically trapped dipolar condensate \cite{tanzi_new}.
In another work \cite{tanzi19}, the reduction of
the moment of inertia under slow rotation --previously predicted for
a dipolar SSP \cite{stringa19}-- has been measured.
The response of the dipolar SSP has been
studied experimentally in Ref. \cite{ferla19}, where
the out-of-equilibrium superfluid flow
across the whole system was revealed by a rapid re-establishment of
global phase coherence after a phase-shattering excitation was applied. Instead, no such
re-phasing was observed in the NSP, where tunneling between adjacent droplets is suppressed.

Most of the recent evidence of supersolid
behavior in dipolar gases is based on
the identification of two main features of a SSP,
i.e. (i) a non-zero non-classical translational/rotational inertia \cite{sep_joss_rica}
and (ii) the appearance  of the
Nambu-Goldstone gapless mode \cite{pfau} corresponding to phase fluctuations
--besides the phonon mode associated to density fluctuations which results from
the translational discrete symmetry of the system.
But no evidence of another hallmark of
superfluidity has been gathered so far,
namely the presence of quantized vortices \cite{stringa19}.
Experimental realization and detection of quantized vortices would provide
a direct evidence of global coherence
in the SSP of a dipolar system.

The structure of vortex arrays in a trapped supersolid system
rotating in the plane orthogonal to the polarization axis
has been theoretically studied in Ref. \cite{stringa20}, where
vortex nucleation was found to be
triggered, as in standard condensates,
by the softening of the surface quadrupole mode.
For larger values of the angular velocity of the rotating trap
an array of regularly arranged vortices appears,
which coexists with the triangular geometry of the supersolid
lattice and persists during the free expansion of the
atomic cloud.
The vortex cores are localized in
the interstitial regions between droplets where the
local particle density can be very small.
As a consequence, the critical angular velocity for the nucleation
of an energetically stable vortex in the SSP has been found
to decrease with respect to that in the SFP \cite{stringa20}.
It was shown in Ref. \cite{arxiv}  that a vortex-hosting trapped SSP
could be obtained by starting
from the vortex-hosting superfluid phase  and
tuning the interaction strength to that of
the SSP. There it was also shown that evidence
for the presence of vortices in the SSP could be
obtained from interference features of the expanding cloud.

Previous theoretical studies of vortices in a SSP
have been performed for dipolar bosons in full confinement
in a rotating trap, i.e. in a harmonic trap along the three
spatial directions. In order to achieve an understanding
of the intrinsic properties of a vortex in the SSP,
in the present work we study isolated vortices in
the SSP of an extended dipolar BEC, i.e. free
from the radial harmonic trap in the plane perpendicular
to the polarization direction but subject to a harmonic
confinement along it. The use of periodic boundary conditions
in the plane dictates that the crystalline structure of
the SSP is fixed in space and vortices are imprinted by
suitable phase factors. We find that the isolated vortices
are stable excitations of the system.

We compare the properties of a vortex in the SFP of
dipolar bosons with those in the SSP and study
how the vortex energy changes depending on the
location of the vortex core in the droplet network.
We find that a vortex dipole, i.e. two vortices of opposite
chirality, has a completely different
behavior in the SSP than in the SFP.
A vortex dipole in the SFP uniformly translates
keeping a fixed inter-vortex distance unless the two
vortices are very close at distances of order of the
healing length \cite{donnelly,roberts}.
At variance,
as a consequence of the spatial modulation
of the condensate, we find that in the SSP the dipole does not translate and
it is no longer a
stable excitation of the system; rather, the two vortices
of opposite chirality feel an attractive interaction
so that they approach each other until they
annihilate each other and their energy is
transferred into bulk excitations.

This paper is organized as follows.
In Sect. II we present the theoretical model and outline the
numerical methods used in this study.
The ground state of dipolar
bosons in the SFP and in the SSP is studied in Sect. III. We present the
results for a single vortex in Sect. IV, and
discuss the results for a vortex dipole in Sect. V.
Finally, Sect. VI contains a summary
of our study.

\section{Method}

A standard approximation for dipolar bosonic atoms with
mass $m$ and magnetic moment $\mu$  at $T=0$ is represented by a macroscopic wave function
$\phi({\bf r})$ that obeys the extended Gross-Pitaevskii equation (eGPE) \cite{santos_wachtler}:

\begin{align}
 H\phi({\bf r})\equiv
& \left\{-\frac{\hbar^2}{2m}\nabla^2+V_t({\bf r})+g|\phi({\bf r})|^2+
\gamma(\epsilon_{dd})|\phi({\bf r})|^3+ \right.
\nonumber
\\
  & \left. \int d{\bf r'}|\phi({\bf r'})|^2V_{dd}({\bf r}-{\bf r'})
   \right\}\phi({\bf r})=\varepsilon  \phi({\bf r})
 \label{acca}
\end{align}
Here $g=4\pi\hbar^2a_s/m$, $a_s$ being the s-wave scattering length,
$V_{dd}({\bf r})=\frac{\mu_0\mu^2}{4\pi}\frac{1-3\cos^2\theta}{r^3}$
is the dipole-dipole interaction between two identical magnetic dipoles
aligned along the z axis
($\theta$ being the
angle between the vector ${\bf r}$ and the polarization direction $z$), and
$\mu_0$ is the permeability of the vacuum.
$V_t({\bf r})$ is the trapping potential.
The number density of the system is $\rho ({\bf r})=|\phi ({\bf r})|^2$.
The $\gamma(\epsilon_{dd})$ term
is the beyond-mean-field (Lee-Huang-Yang, LHY) correction \cite{lima_pelster},
where $\gamma(\epsilon_{dd})=
\frac{32}{3\sqrt{\pi}}ga_s^{3/2}F(\epsilon_{dd})$,
$\epsilon_{dd}=\frac{\mu_0\mu^2}{3g}$
being the ratio between the strengths of the dipole-dipole and
contact interactions, and
$F(\epsilon_{dd})=\frac{1}{2}\int_0^{\pi}d\theta
\sin\theta[1+\epsilon_{dd}(3\cos^2\theta-1)]^{5/2}$.
The chemical potential $\varepsilon$ is determined
by the normalization condition
$\int |\phi ({\bf r})|^2 d{\bf r}=N$, $N$ being the total number of dipoles.

We solve the above equation by propagating  it
in imaginary time, if stationary states are sought, or by
propagating in real-time its time-dependent
counterpart $i\hbar \partial \phi/\partial t=H\phi$ to
simulate the dynamics of the system,
starting from a suitable initial wave function specified in the
following Sections.

To compute the spatial derivatives
appearing in Eq. (\ref{acca})
we used an accurate 13-point finite-difference formula \cite{Anc17}.
The convolution integral in the potential energy term
of Eq. (\ref{acca}) is efficiently evaluated
in reciprocal space by using Fast Fourier transforms,
recalling that the Fourier transform of the dipolar interaction is
${\tilde V}_{\bf k}=(\mu _0 \mu/3)(3\,cos^2\alpha -1)$,
where $\alpha$ is the angle between ${\bf k}$ and
the $z$-axis \cite{lahaye}.
The spatial mesh spacing and time step are chosen such that
during the time evolution excellent conservation
of the total energy of the system is guaranteed.
We verified that the
dimension of the simulation cell along the $z$-direction,
where the harmonic confinement is active,
is wide enough to make negligible
the effect, on the energy values
and density profiles, of the
dipole-dipole interaction between periodically repeated images.

The investigated system is made by $N \sim 1.3-1.4\times 10^6$ $^{164}$Dy atoms
(the chosen $N$ value slightly changes depending upon
the value of $a_s$, as discussed in the following)
subject to a harmonic trapping potential along the $z$ axis,
with frequency $\omega_z=150 \times 2\pi \,$ Hz.
The harmonic length associated to this trapping potential is
$a_{ho}=\sqrt{\hbar^2/m\omega_z}=1.2112\times 10^4\,$a$_0=0.64\,\mu m$.
The dipolar gas is thus extended in the $x-y$ plane
(with periodic boundary conditions), and flattened in
the direction of the dipole polarization.
Typical transverse sizes
are $L_x,L_y\sim 30\,\mu m$.

A linear, singly quantized vortex excitation in
the extended dipolar system with axis in
the $z$-direction and core in the position $(x_v,y_v)$,
can be generated by the ``imprinting procedure'' \cite{Anc17},
i.e. we compute the lowest energy state
obtained by  starting the imaginary time evolution from
the wave function
\begin{equation}
\phi_0(\mathbf{r})=\rho_0^{1/2}(\mathbf{r})\, \left[ {(x-x_v)+\imath (y-y_v)
\over \sqrt{(x-x_v)^2+(y-y_v)^2}}  \right]
\label{imprint}
\end{equation}
where $\rho_0(\mathbf{r})$ is the
ground-state density of the vortex-free dipolar system.
During the imaginary time evolution,
the vortex position and vortex core structure
change to provide at convergence the lowest energy
configuration.

The flow field of a linear vortex has a long-range character,
$\sim 1/r$, where $r$ is the distance from the vortex axis.
We have imposed antiperiodic boundary conditions \cite{Pi07} in order
that the condition of no flow across the boundary of
the computational cell  is satisfied. This is equivalent
to sum over the phases of an infinite array of vortex-antivortex,
i.e. a vortex of opposite chirality is present in each
nearest neighbor cell of the computation cell \cite{sadd}.
The sign of the circulation can be changed by changing the sign of the complex
$\imath$ in Eq. (\ref{imprint}); this equation can be
easily generalized to accomodate a vortex array made of an arbitrary number
of vortices and/or antivortices \cite{Anc18}.
A generalization to the case of a pair of vortices with opposite circulation
(vortex dipole) will be used in Sec. V.

\section{Ground-state in dipolar bosons}

We have taken as a case of study a dipolar bosons system made of $^{164}$Dy atoms.
The relative strength of the dipolar interaction
over the short-range one is defined by the dimensionless parameter
$\epsilon _{dd}=a_{dd}/a_s$, where $a_{dd}=132$\,a$_0$ is the
dipolar length for $^{164}$Dy atoms \cite{ferla19}.
The behavior of dipolar bosons depends crucially on the
value of the $\epsilon_{dd}$ ratio, whose value
can be experimentally varied by changing $a_s$.

When $\epsilon_{dd}$ is
below a threshold value $\epsilon ^c_{dd} $ the
system at $T=0$ K is in a standard superfluid state,
in our case it is uniform in the $x-y$ plane and non-uniform
in the $z$-direction due to the trap in that direction.
When $\epsilon _{dd}$ is above the threshold the system spontaneously
develops a periodic density modulation in the $x-y$ plane, entering
the SSP.

As $\epsilon_{dd}$ is increased toward the threshold,
one can get a signature of the SFP to SSP transition from
the excitation spectrum of the superfluid, ${\cal E}(k)$.
We first obtain the ground state SFP wave function
by propagating in imaginary
time a starting wave function which is just a constant in the $x, y$
variables times the ground state wave function for free
particles in the $V_t(z)$ trap.
Next, using the method described in Ref. \cite{ancilotto}, we solve
the Bogoliubov-de Gennes
equations for superfluid states corresponding to different $\epsilon _{dd}$ values.
For small $\epsilon _{dd}$ the dispersion relation ${\cal E}(k)$ is structureless,
starting as a linear function of $k$ corresponding to phonon excitations which evolves toward
a quadratic dispersion at large $k$ corresponding to  free particles.
Increasing $\epsilon _{dd}$, the excitation spectrum
changes shape, and close to $\epsilon^c_{dd}$ ${\cal E}(k)$
develops a phonon-maxon-roton structure as shown in Fig. \ref{fig1-SM}.
This is due to the interplay between confinement in the
$z$-direction and the interatomic interaction consisting
of the short range interaction plus the long-range
interaction of the dipoles oriented in the direction of the confinement \cite{lewe,odell}.

The four curves in Fig. \ref{fig1-SM}
show the excitation spectrum of the SFP for $a_s=97.8,\, 98,\, 98.5$, and 99\,a$_0$,
which correspond to a narrow range of $\epsilon _{dd}$ values between
$\epsilon _{dd}= 1.35$ (corresponding to $a_s=97.8$\,a$_0$) and $\epsilon _{dd}= 1.33$
(corresponding to $a_s=99$\,a$_0$).
The vanishing of the roton gap in the present case occurs just below $a_s=97.8$\,a$_0$,
corresponding to the value $\epsilon^c_{dd}\sim 1.35$.
Notice that this value depends on the strength of confinement
in the $z$-direction and on the mass of the bosons.
Beyond the threshold the system becomes non-uniform in the $x$ and $y$-directions.

\begin{figure}[!]
\centerline{\includegraphics[width=1.0\linewidth,clip]{disp_rel_SF.pdf}}
\caption{Dispersion relation ${\cal E}(k)$ of the extended SFP
as a function of the wave vector $k$ in the $x-y$ plane for different
values of $a_s$. ${\cal E}(k)$ is in units of the
harmonic energy $\omega_z$ and $k$ is in units of the
inverse of the harmonic length $a_{ho}=\sqrt{\hbar/(m\omega_z)}$.
}
\label{fig1-SM}
\end{figure}

We obtain the ground state wave function in the SSP
by evolving in imaginary time a starting wave function that is
a superposition of gaussian profiles
in the $x-y$ plane, located at the sites of a triangular lattice, modulated
along the $z$-direction by the ground state of free
particles in the harmonic trap in that direction.
In the following, most results are obtained with $\epsilon_{dd}$
in the $1.39<\epsilon _{dd}<1.48$ range, where the system
shows spontaneous density modulation with supersolid behavior.
The scattering lengths associated to this interval
are $95\,a_0 > a_s > 89\,a_0$.

The modulated system obtained in this way
consists of a planar ordered
array of ``droplets'' with triangular symmetry,
elongated in the direction of the polarization axis,
each containing $\sim 15{,}000$ atoms.
To good approximation, these cigar-shaped droplets
are circular in the $x-y$ plane and with an extension
in the $z$-direction  which is typically 4-5 times larger than their diameter
in the $x-y$ plane, as shown in Fig. \ref{fig1} for $a_s=93\,a_0$.

An almost uniform fluid occupies the
space between droplets, and its density
is much smaller than that of the droplets.
Global phase coherence is maintained between adjacent
droplets due to this low-density superfuid background
which allows tunneling of atoms from one droplet to another.
Typical average densities in the $z=0$ plane of symmetry
are $\rho _{av}\sim 3.5\times 10^{-11}\,a_0^{-3}$.

\begin{figure}[t]
\centerline{\includegraphics[width=1.1\linewidth,clip]{rho_93_views.pdf}}
\caption{Supersolid phase for the extended system with $a_s=93\,a_0$.
Upper panel:
number density $\rho$ in a $x-z$ plane passing
through a row of droplets, in units of $a_{ho}^{-3}$.
Lower panel: density in the $x-y$ plane (integrated along the
$z$-direction) in units of $a_{ho}^{-2}$.
Lengths are in $\mu m$.
}
\label{fig1}
\end{figure}

The triangular packing of droplets is characterized by a lattice constant
$a$, corresponding to the nearest-neighbor distance
between two adjacent droplets.
The periodic boundary conditions impose a
constraint on the value of $a$ such that an
integer number of crystalline cells
fits in the size of the computation box.
For a given value of the scattering length $a_s$,
we preliminary determine the
equilibrium value for $a$ by computing the energy per atom
for different values of $a$ and look for the minimum
energy value.
An example of such calculation is shown in Fig. \ref{fig2},
where the energy per atom is shown --for the $a_s=93\,a_0$ case--
as a function of the lattice constant $a$.
Fig. \ref{fig3} shows the equilibrium values of the lattice constant
for different choices of $a_s$.

\begin{figure}[t]
\centerline{\includegraphics[width=1.0\linewidth,clip]{e_vs_a_lattice.pdf}}
\caption{ Energy per atom vs. $a$ for the case $a_s=93\,a_0$.
}
\label{fig2}
\end{figure}

\begin{figure}[t]
\centerline{\includegraphics[width=1.0\linewidth,clip]{lattice_vs_as.pdf}}
\caption{ Equilibrium value of the lattice constant $a$
as a function of the scattering length $a_s$.
}
\label{fig3}
\end{figure}

All the modulated structures investigated here
have supersolid character.
In order to prove this we checked for the presence
of a finite non-classical translational inertia (NCTI)
in the system.
This is done by solving for stationary states the
real-time version of Eq. (\ref{acca}) in the
comoving reference frame with uniform velocity $v_x$, i.e.
\begin{equation}
 i\hbar\frac{\partial}{\partial t}\phi({\bf r})=
\left(H+i\hbar v_x\frac{\partial}{\partial x}\right)\phi({\bf r})
\end{equation}
Following Ref. \cite{sep_joss_rica}, we define the superfluid fraction $f_s$
as the fraction of particles that
remains at rest in the comoving frame:
\begin{equation}
 f_s=1-\lim_{v_x\to 0} \frac{\langle P_x \rangle}{N m v_x}
\eeq
where $\langle P_x \rangle =-i\hbar \int d \mathbf{r}\, \phi ^{\ast} \partial \phi /\partial x$
is the expectation value of the momentum and $Nmv_x$ is
the total momentum of the system if all atoms were moving.
We have verified that, in the low-velocity limit, the same value for $f_s$
is obtained by applying instead the boost along the $y$-direction.
We remark that the superfluid fraction computed in this way
should not be confused with the total superfluid fraction
because each droplet is superfluid but do not
necessarily contribute to the superfluid response of the overall system.

The calculated average superfluid fraction is shown
in Fig. \ref{fig4} as a function of $a_s$.
We also show in the same figure the ratio
between the density $\rho _{min}$ of the superfluid background
in the interstitial site between three adjacent droplets
and the maximum density of the clusters in the $z=0$ plane of symmetry,
$\delta = \rho _{min}/\rho _{max}$.
Not surprisingly, the lower the ratio, the
smaller the associated superfluid fraction.
We find that the SFP $\rightarrow $ SSP transition
occurs around the value $a_s=97\,a_0$ (an exact determination
from the simulation is problematic: close to the transition
residual, very low modulations in the density
take overly long imaginary times to be reduced).
The vertical line in Fig. \ref{fig4} shows the estimated
location of
the SFP $\rightarrow $ SSP transition. For values of $a_s$
to the right of this line, the lowest energy structure is
a homogeneous (in the $x-y$ plane) SFP with $f_s=1$.

In the SSP regime, when the value of $a_s$ is progressively decreased the
background density becomes steadily smaller and eventually the
system is no longer supersolid, being composed
of isolated droplets with incoherent phases \cite{chomx}.
We cannot follow this transition from a supersolid
to this ``normal solid'' made of droplets (the NSP in our notation)
because the theory we use, which is based on a single order parameter
implying a global phase coherence across the whole system, is not
fully appropriate when the different droplets become incoherent.

\begin{figure}[t]
\centerline{\includegraphics[width=1.0\linewidth,clip]{legget_new.pdf}}
\caption{ Upper curve: superfluid fraction $f_s$. Lower curve:
ratio $\delta = \rho _{min}/\rho _{max}$.
The vertical line shows the location of the SFP $\rightarrow$ SSP transition.
}
\label{fig4}
\end{figure}

\section{Single vortex in dipolar bosons}

We have studied singly quantized vortices in our
extended system for different values of the scattering length $a_s$.
In the SFP, the density profile is featureless for large $a_s$ values,
quadratically vanishing at the position of the vortex core
and monotonously approaching the bulk density at large distances,
as in usual BEC systems. As $a_s$ is decreased (and thus $\epsilon _{dd}$ increases)
the density profile develops damped oscillations near the vortex core \cite{Yi06}.
This is shown in Fig. \ref{fig2-SM}, where we display the equilibrium vortex-hosting density profiles
in the SFP for the same $a_s$ values as in Fig. \ref{fig1-SM}.
The smaller the $a_s$, the higher the density peaks around the vortex core.
The oscillations become  more pronounced as the instability limit $\epsilon^c _{dd}\sim 1.35$ is approached.
The wave vector of the oscillations in the vortex-hosting density profile
is close to the roton $k_R$.
These oscillations are similar to those predicted in superfluid $^4$He which
have been described as a cloud of virtual rotons \cite{reatto}.

\begin{figure}[!]
\centerline{\includegraphics[width=0.95\linewidth,clip]{vort_SF.pdf}}
\caption{Radial vortex density profiles in the SFP for different $a_s$ values.
}
\label{fig2-SM}
\end{figure}

As we have discussed previously,
when $a_s$ becomes smaller than a critical value the system
enters the supersolid phase, as in the structure shown in
Fig. \ref{fig1}.
In this phase, the lowest energy position of the vortex core is
in the interstitial region between three adjacent droplets, where the
local particle density $\rho ({\bf r})$ is very small, so that
the energy cost of the additional
depression of $\rho ({\bf r})$ near the vortex core
(remember that the local density has to vanish at the vortex core) is minimized and
the kinetic energy of the superfluid flow around the vortex core also decreases
\cite{stringa19}.
To visualize the vortex core structure, the map of the
difference between the densities of the configuration with and
without vortex at the interstitial site for the $a_s=93\,a_0$ case
is shown in the upper panel of Fig. \ref{fig4b}.
It appears that the core is very deformed
following the symmetry of the interstitial site \cite{rocc}.
We have found a similar vortex core structure for other values of the scattering length.

The phase associated to the vortex
is shown in the lower panel of Fig. \ref{fig4b}.
Notice the hexagonal texture which is a consequence of the
triangular lattice of clusters in the vortex-hosting SSP.
As shown in Fig. \ref{fig11},
this texture becomes prominent in the plot of
the modulus of the velocity field $|{\bf v}({\bf r})|$, where
${\bf v}({\bf r})=(\hbar/m)\nabla _R\theta({\bf r})$ with
$\theta $ being the phase of the order parameter $\phi$,
and $\nabla _R\equiv (\partial/\partial x,\partial/\partial y)$.
It appears that $|{\bf v}({\bf r})|$ completely
differs from the $1/r$ behavior in a superfluid, being instead
strongly anisotropic and modulated, and very small
inside the droplets. We will show below how
this is affecting the kinetic energy of the vortex.

We show in Fig. \ref{stream1} the velocity streamlines in the $z=0$ symmetry plane
for the case described above, which are remarkably different from the
simple tangential flow of a vortex in a homogeneous superfluid.
We find that the circulation calculated around a closed path encircling the
vortex core is $\oint {\bf v}\cdot {\bf dl}=(2\pi \hbar /m)n$, with $n=1.01\pm 0.01$, and
where the error reflects slightly different
outcomes depending on the chosen size of the closed path around the vortex core.

\begin{figure}[t]
\centerline{\includegraphics[width=0.8\linewidth,clip]{panel_dens_phase.pdf}}
\caption{ Upper panel:
Density difference
($\rho _{vortex} -\rho_{no vortex})/\rho_{no vortex}$ in the $z=0$ symmetry plane
for the vortex at the interstitial site when $a_s=93\,a_0$.
Lengths are in $\mu m$.
Lower panel:
Phase of the vortex state shown in the upper panel.
Lengths are in $\mu m$.
}
\label{fig4b}
\end{figure}

Besides the stable equilibrium vortex position at
interstitial sites of the triangular droplet lattice,
we have found that the saddle point between two adjacent droplets is a
(meta)stable equilibrium vortex position, with slightly
higher energy than the interstitial equilibrium one.
The map of the difference between the
densities of the configuration
with and without vortex at the saddle point for the
$a_s=93\,a_0$ case is shown in  Fig. \ref{fig6}.
Again, the density depression due to the vortex
is very anisotropic and reflects the local symmetry of the system.

\begin{figure}[t]
\centerline{\includegraphics[width=1.2\linewidth,clip]{velocity_mod.pdf}}
\caption{
Modulus of the velocity field (in units of $a_0 \omega_z$)
on the
$z=0$  plane of symmetry for $a_s=93\,a_0$.
Lengths are in $\mu m$.
}
\label{fig11}
\end{figure}

\begin{figure}[t]
\centerline{\includegraphics[width=0.9\linewidth,clip]{streamlines_new.pdf}}
\caption{Velocity streamlines for the SFP state for $a_s=93\,a_0$.
The vortex core is in the origin.
Superimposed to the field lines is the density in the $z=0$ plane, in units of
$a_{ho}^{-3}$.
}
\label{stream1}
\end{figure}

We also considered the possibility of a third equilibrium position
with the vortex line piercing a droplet parallel to the
polarization axis and passing through its center.
Such configuration has proven however to be unstable, and the vortex line
is expelled during the imaginary time minimization,
ending in the lowest energy interstitial site configuration.
A similar unstable behavior has been observed for a vortex
aligned with the polarization axis at
the center of self-bound dipolar droplets \cite{macri}.

\begin{figure}[t]
\centerline{\includegraphics[width=1.4\linewidth,clip]{rho_93_vort_diff_saddle.pdf}}
\caption{Density difference
($\rho _{vortex} -\rho_{no vortex})/\rho_{no vortex}$ in the $z=0$ symmetry plane
for the vortex at the saddle-point site when $a_s=93\,a_0$.
Lengths are in $\mu m$.
}
\label{fig6}
\end{figure}

The difference between the energies of the saddle-point and interstitial site
configurations represents
the barrier for displacing a vortex through the supersolid
lattice. The calculated energy barriers per unit length
are shown in Fig. \ref{fig7} for different values of $a_s$.
Here $L$ in the length of the vortex line along the $z$-direction,
which we take as $L=\sqrt{\int \bar{\rho}(z) z^2 dz /\int \bar{\rho } (z) dz   }$,
where $\bar{\rho}(z)\equiv \int \rho (x,y,z)dx dy/(L_x L_y)$. We find
$L\sim 6.7\,\mu m$.
We notice the strong decrease of the energy
barrier as the density of the superfluid background diminishes.

\begin{figure}[t]
\centerline{\includegraphics[width=1.1\linewidth,clip]{barrier.pdf}}
\caption{ Energy barrier per unit length in the $z$-direction
between interstitial and saddle-point configurations
as a function of $a_s$.
}
\label{fig7}
\end{figure}

Although the very deformed and anisotropic shape of the vortex core in the SSP as
shown in Fig. \ref{fig4b} hinders a clear definition of the vortex core extension,
one can estimate it by performing
an angular average of the density (on the $z=0$ symmetry plane) over the polar angle
around the interstitial site where the vortex is placed.
We show this average in the lower panel of Fig. \ref{fig9}
for the state with and without vortex corresponding to $a_s=93\,a_0$;
the upper panel  shows such angular average for
different values of $a_s$.
The almost overlapping horizontal lines show the average density in the
$z=0$ plane for these $a_s$ values.
The radius $R_c$ of the averaged cavity is
measured at half the value
of the depression of the local density at the vortex axis, i.e.
from the condition
$\rho _v(R_c)-\rho_0(R_c)=\rho_0 (0)/2$,
where $\rho _v$ and $\rho _0$ are the average density profiles
with and without vortex, respectively.
The cavity radius so calculated ranges
from $R_c\sim 23{,}000\,a_0$ for $a_s=95\,a_0$ to
$R_c\sim 35{,}000\,a_0$ for $a_s=89\,a_0$.
The calculated values are reported in Table I.
For comparison,
the distance between the vortex core position and
the neighboring droplets center is $37{,}000\, a_0$ (for the $a_s=93\,a_0$ case).

\begin{figure}[t]
\centerline{\includegraphics[width=1.0\linewidth,clip]{rho_radial.pdf}}
\caption{
Upper panel: angular average on the $z=0$ symmetry plane, of the number density around a vortex.
 From top to bottom --at the first maximum--
$a_s=89, 90, 91, 93$, and $95\,a_0$. The  nearly overlapping horizontal lines show the average densities
on the same plane.
Lower panel: angular average of the number density with and without vortex, for the $a_s=93\,a_0$ case.
}
\label{fig9}
\end{figure}

A measure of the local vortex excitation energy per
unit length of a vortex of length $L$ is
the integrated vortex kinetic energy \cite{reatto,rossi}, which can be
defined as follows:

\begin{equation}
\epsilon_v(R) =\frac{1}{L} [E_{kin}^v(R)-E_{kin}(R)]
\end{equation}
where $E_{kin}^v,E_{kin}$ are the kinetic energies
within a cylinder of radius $R=\sqrt{x^2+y^2}$ and length $L$
(with and without a vortex line along its axis)
as a function of the distance $R$
from the vortex axis (here the $z$-axis).

Neglecting the $z$-dependence of the energy we compute
$\epsilon _v(R)$ by averaging
the kinetic energy densities on the
$z=0$ symmetry plane.
$E_{kin}(R)/L$ is thus given by the integral
$2\pi \int _0 ^R \epsilon (R')R' dR'$,
where $\epsilon (R)$ is the
angular average, performed in the $z=0$  plane, of
$ (\hbar ^2/2m)|\nabla_R \phi|^2$, where
$\nabla_R \equiv (\partial/\partial x,\partial/\partial y)$. Similar expressions hold for
$E^v_{kin}(R)/L$.

We notice that the classical hydrodynamical counterpart of $\epsilon _v(R)$
for a vortex of circulation $\kappa$ in an incompressible fluid of density $\rho _0$  is

\begin{equation}
\epsilon _v ^{hydro}(R)=\frac {\kappa ^2}{4\pi} m
\rho_0 \left[ln\left(\frac{R}{d_v}\right)+\delta \right]
\label{hydro_class}
\end{equation}
where $d_v $ is the vortex core radius and $\delta$
depends on the core model  ($\delta=0$
for the hollow core model and $\delta=1/4$ for a
core in rigid rotation) \cite{donnelly}. The parameter $\delta$ in the previous
equation can be absorbed in the logarithmic
term; using the quantum value for the circulation, $\kappa=h/m$,
for a singly-quantized vortex $\epsilon_v ^{hydro}$ reads

\begin{equation}
\epsilon _v ^{hydro}(R)=\frac{\hbar^2}{m}\pi
\rho _0 \, ln\left(\frac{R}{\lambda}\right)
\label{hydro}
\end{equation}
where $\lambda =d_ve^{-\delta}$ is the core parameter.

We have used the above expression to fit the
calculated $\epsilon _v(R)$ for the supersolid structures
investigated here with fitting parameters $\rho _0$ and $\lambda$.
We show in Fig. \ref{fig10}, for different values of $a_s$, the calculated $\epsilon_v(R)$
values (filled squares)
and the best fit obtained by using the hydrodynamical approximation Eq. (\ref{hydro}) (solid line).
As expected, the approximation breaks down at distances approaching
the vortex core, as there the local density becomes very small.
One can notice from the figure that the computed
kinetic energy has some weak oscillations whose
locations are related to the distances from the interstitial site where the vortex is placed, to
the different droplet shells.
For comparison, we also show in Fig. \ref{fig10}
the results for the vortex-hosting SFP corresponding to two $a_s$ values (open squares).

We report in Table I the calculated values for
the core parameter $\lambda$ and prefactor $\rho _{0}$
as obtained from the fit.
At variance with the SFP, for which the calculated value of $\rho _{0}$
coincides with the average density on the $z=0$ plane (4th column
in the Table), it does not for the SSP: here the
calculated $\rho _{0}$ values show a marked decrease with $a_s$.
One may argue that these values should be related to the superfluid halo
embedding the droplets, and thus to
the (minimum) density in the interstitial regions
between droplets (5th column in the Table):
this is not the case though, since the minimum density values
decrease with $a_s$ much faster than $\rho _{0}$.
We also show in the Table the vortex
``core'' energy, defined as $E _v(R_c)$, where the core radius $R_c$
has been estimated as described before.

The core parameter $\lambda$ and core energy $E_c$
depend only weakly on the density in the SFP whereas
the core radius $R_c$ increases as $a_s$ decreases,
i.e. as the dipolar interaction becomes more important.
In the SSP there is a clear change in the dependence of
these vortex parameters as a function of $a_s$.
As mentioned before, $\rho_0$ is a strongly decreasing
function of $a_s$. Both $\lambda$ and $R_c$
increase significantly for decreasing $a_s$, and
this means that the vortex size expand as the
value of the low density halo decreases.
The core energy has the opposite dependence on $a_s$, i.e.
$E_c$ decreases as $a_s$ does,
indicating that the decrease of the halo
density has a stronger effect on the $E_c$ value than
the increase of the vortex core radius.

\begin{table*}[!]
  \begin{center}
    \label{tab:table1}
        \begin{tabular}{ccccccccccccc}
\hline
\hline
      $a_s \,(a_0)$ &$\quad$&$\rho _{0}\,(a_0^{-3}) $ &$\quad$& $\lambda \,(a_0)$ &$\quad$& $\rho _{av}\,(a_0^{-3})$ &$\quad$& $\rho _{min}\,(a_0^{-3})$ &$\quad$& $R_c\,(a_0)$
      &$\quad$& $E_c$  (nK/$a_0)$ \\
      \hline
  89\,(SSP) & $\quad$&$1.23\times 10^{-11}$ &$\quad$& 24{,}360 &$\quad$& $3.45\times 10^{-11}$ &$\quad$& $4.36\times 10^{-13}$ &$\quad$& $35{,}221$
  &$\quad$& $2.27\times 10^{-2}$\\
  90\,(SSP)  &$\quad$&$1.82\times 10^{-11}$ &$\quad$& 23{,}160 &$\quad$& $3.46\times 10^{-11}$ &$\quad$& $7.97\times 10^{-13}$ &$\quad$& $32{,}704$
  &$\quad$& $3.18\times 10^{-2}$\\
  91\,(SSP)  &$\quad$&$2.56\times 10^{-11}$ & $\quad$ &21{,}790 &$\quad$& $3.47\times 10^{-11}$ &$\quad$& $1.40\times 10^{-12}$ &$\quad$& $30{,}474$ &$\quad$& $4.33\times 10^{-2}$ \\
  93\,(SSP) &$\quad$& $4.56\times 10^{-11}$ &$\quad$& 20{,}180 &$\quad$& $3.49\times 10^{-11}$ &$\quad$& $4.01\times 10^{-12}$ &$\quad$& $26{,}130$
  &$\quad$& $6.50\times 10^{-2}$\\
  95\,(SSP) &$\quad$& $6.44\times 10^{-11}$ &$\quad$& 15{,}420 &$\quad$& $3.51\times 10^{-11}$ &$\quad$& $1.09\times 10^{-11}$ &$\quad$& $22{,}644$
  &$\quad$& $8.40\times 10^{-2}$\\
  \hline
 98\,(SFP) &$\quad$& $3.54\times 10^{-11}$ &$\quad$& 7{,}484  &$\quad$& $3.53\times 10^{-11}$ &$\quad$& ---  &$\quad$&$17{,}755$ &$\quad$& $5.56\times 10^{-2}$\\
 100\,(SFP) &$\quad$& $3.52\times 10^{-11}$ &$\quad$& 7{,}777  &$\quad$& $3.52\times 10^{-11}$ &$\quad$& ---  &$\quad$&$16{,}470$ &$\quad$& $5.72\times 10^{-2}$\\
 105\,(SFP) &$\quad$& $3.50\times 10^{-11}$ &$\quad$& 7{,}612  &$\quad$& $3.50\times 10^{-11}$ &$\quad$& --- &$\quad$& $14{,}350$ &$\quad$& $5.97\times 10^{-2}$\\
 110\,(SFP) &$\quad$& $3.48\times 10^{-11}$ &$\quad$& 7{,}200  &$\quad$& $3.48\times 10^{-11}$ &$\quad$& --- &$\quad$& $12{,}750$ &$\quad$& $6.03\times 10^{-2}$\\
 \hline
 \hline
    \end{tabular}
        \caption{
Parameters $\rho_0$ and $\lambda$ of the fit of the vortex kinetic energy
with the hydrodynamic form Eq. (\ref{hydro}) for some values of $a_s$
in the superfluid (SFP) and in the supersolid (SSP).
Other reported quantities are the average density
$\rho_{av}$ on the $z=0$ plane, the density $\rho_{min}$
at the interstitial site of the supersolid, the radius $R_c$
of the vortex cavity, and the vortex core energy $E_c$.
}
  \end{center}
\end{table*}

An important issue is the nature of the
low energy excitations of a vortex in the SSP.
We already mentioned a difference between
a vortex in a superfluid and in a supersolid:
in a homogeneous superfluid a vortex can move
without restriction and its low energy
excitations are Kelvin waves whose
energy in an unbounded fluid vanishes as the wave
vector $k$ goes to zero \cite{lamb,donnelly}.
We have shown that in a supersolid a vortex has a discrete set
of equilibrium positions and an energy
barrier is present for its motion from
one site to another. Therefore, the vortex
is a localized excitation in
the plane perpendicular to the vortex axis
and this should modify the properties of the Kelvin
waves, in particular their spectrum
should display an energy gap.
This gap has a completely different origin from the
small energy gap of Kelvin waves when bosons
are in a trap \cite{fetter1}.

There is the possibility that the
low energy excitations of a vortex
in a supersolid have a quite different character
when the droplets are very elongated
along the $z$-axis. Such excitations
could be kinks along the vortex
axis such that the vortex core moves
from one equilibrium site to another site along the axis.
The energy of the vortex
in a SSP is a periodic function of position
and this plays a role similar to  the
Peierls potential acting on a dislocation
in a crystalline solid.
Kinks are indeed the low energy
excitations of a dislocation in solid $^4$He \cite{beamish}.
The study of the excitations of a
vortex in a supersolid is an interesting
problem but it is quite outside the scope of the present paper.

\begin{figure}[t]
\centerline{\includegraphics[width=1.0\linewidth,clip]{ekin_loc.pdf}}
\caption{Integrated vortex kinetic energy $E_v(R)$ as a function of
the distance from the vortex core.
Filled squares, from top to bottom: $a_s=95,93,91,90,89\,a_0$ (SSP).
Open squares: $a_s=110, 100\,a_0$ (SFP).
The lines are the hydrodynamical form fit.
}
\label{fig10}
\end{figure}

\section{Vortex dipole in dipolar bosons}

As discussed in Sect. IV, vortices in a dipolar supersolid
have well defined equilibrium positions in the
``solid'' structure so that the vortex energy  is a periodic function of its position.
Therefore, a number of properties are expected to differ
from those in a superfluid.
In this Section we study one of them, namely the dynamics of
a vortex dipole, i.e. two counter-rotating straight parallel vortices.
Vortex dipoles have been created and observed in BECs confined
by parabolic potentials \cite{dipo_bec}, and
the motion of vortex dipoles in superfluid $^4$He has been
simulated within the time-dependent DFT approach \cite{dipole}.

In classical hydrodynamics of incompressible fluids \cite{lamb}
a vortex dipole is a stable entity that moves with a constant
velocity that is perpendicular to the plane containing the
vortices and inversely proportional
to the distance $d$ between them.
The same behavior holds in superfluid systems
when $d$ is much larger than the healing length
(which in the present case can be estimated from the
vortex core sizes in Fig. \ref{fig2-SM});
in the superfluid,
the vortex dipole propagates with a constant
velocity $v_d=\hbar/(m\,d)$ \cite{donnelly}.

We have addressed the dynamics of a vortex dipole in the extended
dipolar bosons system. To this end, we have first
prepared the initial vortex dipole configuration. This
can be done generalizing as follows Eq. (\ref{imprint}) in Sec. II:
\begin{eqnarray}
\phi_0(\mathbf{r})&=&\rho_0^{1/2}(\mathbf{r}) \left[ {(x-x_v)+\imath (y-y_v) \over \sqrt{(x-x_v)^2+(y-y_v)^2}}  \right.
\nonumber
\\
&&\left.{(x-x_{\bar{v}})-\imath (y-y_{\bar{v}}) \over \sqrt{(x-x_{\bar{v}})^2+(y-y_{\bar{v}})^2}}
 \right]
\label{imprintdipole}
\end{eqnarray}
where $(x_v, y_v)=(d/2, 0)$ and $(x_{\bar{v}}, y_{\bar{v}})=(-d/2,0)$
We evolve this wave function in
imaginary time until a stationary configuration is reached. Since
the total circulation is zero, usual periodic boundary conditions
can be used.

It should be noticed that the dipole wave function is not
orthogonal to that of the ground state. Hence, a very long
imaginary time evolution would bring the
initial state $\phi_0(\mathbf{r})$ to the ground
state of the system, i.e. the vortex dipole would disappear.
However,
 we find that after a relatively short imaginary time
$\tau_c$ the initial phase settles down to a quasi-stationary state
where the vortex cores are fully developed while remaining in their
initial positions
(whereas orders of magnitude longer imaginary times would be
needed to recover the vortex-free ground state).
At this point we initiate the evolution  of the wave function in real time.
By using different values of $\tau_c$ we have verified
that the real time evolution does not depend on the used  $\tau_c$  value.

\begin{figure}[t]
\centerline{\includegraphics[width=1.1\linewidth,clip]{panel_SF.pdf}}
\caption{Vortex-antivortex dynamics in real time,
for the case $a_s=110\,a_0$ (SFP). The density, integrated along the $z$-axis,
is shown in units of $a_{ho}^{-2}$.
Snapshots are taken, from top to bottom and from left to right,
at $t=0, 24, 48, 72, 96$, and $120$\,ms.
Lengths are in $\mu$m.
}
\label{fig12}
\end{figure}

 the vortex dipole translates with constant velocity and this velocity is of order of that given by classical hydrodynamics,

We have first studied the vortex dipole in the SFP. In this case we
find that the vortex dipole translates
with a constant velocity, and this velocity is of order
of that given by classical hydrodynamics,
$v_{d}=\hbar/(m\,d)$.
Notice that the observed velocity
can differ from the hydrodynamic result for
two reasons, because $d$ is not very large compared
to the healing length and because the periodic
boundary conditions modify the streamlines
of the dipole near the cell boundaries
with respect to those in an unbounded fluid \cite{roberts}.

The real time evolution of a vortex dipole in the SFP for $a_s=110\,a_0$ is shown in Fig. \ref{fig12},
in the form of snapshots  of the density taken at increasing times during the dynamics.
The vortex dipole has been prepared with an initial
vortex-antivortex distance $d=6.5\,\mu m$.

\begin{figure}[t]
\centerline{\includegraphics[width=1.1\linewidth,clip]{panel_SS.pdf}}
\caption{Vortex-antivortex dynamics in real time,
for the case $a_s=93\,a_0$ (SSP). For clarity, the logarithm of the $z$-averaged density
(in units of $a_{ho}^{-2}$) is shown.
Snapshots are taken, from top to bottom and from left to right,
at $t=0, 27, 36, 41, 48$, and $75$\,ms.
Lengths are in $\mu$m.
}
\label{fig13}
\end{figure}

A completely different behavior is found in the SSP: no
translation of the dipole takes place, and
the two vortices approach until they annihilate
each other. The dynamics of a vortex dipole in the SSP
is displayed in Fig. \ref{fig13}, where it appears that the
vortex-antivortex pair undergo annihilation in a very short time.
Notice that a logarithmic scale is used in this figure
in order to highlight the vortex positions, which would
be otherwise invisible in an ordinary plot.
The results presented in Fig. \ref{fig13}
are for $a_s=93\,a_0$ and initial intervortex distance
$d=4a$, where $a=3.44\,\mu m$ is the lattice parameter of the SSP for this $a_s$ value.
We have found that similar dynamics of the vortex dipole occur
for other values of $d$ and $a_s$ corresponding to a SSP.
Therefore, the disclosed annihilation process of
two counter-rotating vortices appears to be generic
for the SSP of dipolar bosons.

By following the snapshots in Fig. \ref{fig13}
one can see that the recombination process proceeds by a set of jumps
from one interstitial site to a neighboring one and that in the jump
process each vortex becomes elongated by extending over the two
neighboring sites as well as over the in-between saddle point.
Since the snapshots represent the projected density on
the $x-y$ plane one can ask if the elongation of the vortex during
the jump is due to propagation of kinks along the vortex axis
as discussed in the previous Section or if the vortex rigidly
translates from one site to another one. We find that this
second case indeed occurs since we have verified that the density
in single $x-y$ planes
has exactly the same dynamics of the projected one.

\begin{figure}[t]
\centerline{\includegraphics[width=1.0\linewidth,clip]{spectra_93.pdf}}
\caption{Energy spectrum (in units of $1\times 10^{-18}$ hartree $\times a_0$)
before (top) and after (bottom) the vortex dipole
annihilation in the SSP with $a_s=93\,a_0$.
Solid lines, incompressible spectrum; dashed lines, compressible spectrum.
The meaning of the vertical lines is explained in the text.
}
\label{fig18}
\end{figure}

The energy released immediately after the annihilation goes into excitations
of the droplet array, which appears to be deformed
and oscillating after the recombination event.
Such change in the character of the excitation energy can be characterized by computing
the spectral density of the kinetic energy  of
the superfluid velocity field, decomposing it into compressible
and incompressible parts \cite{Nor97,tsubota} by utilizing the
Helmholtz decomposition theorem.
We show in Fig. \ref{fig18} these
two contributions as a function of the wavevector in the $x-y$
plane $k$, where we have taken a time average
over the first $40\,ms$ of the dynamics, which corresponds to times before
vortex annihilation, and over $20\,ms$ after vortex
annihilation.
The vertical lines show three relevant wave vectors:
$k_L=2\pi/L$ where $L=\sqrt{L_xL_y}$ is the average size
of the simulation cell, $k_R=2\pi/a$ is the
wave vector corresponding to the periodic arrangement of droplets
($a$ being the droplet-droplet nearest-neighbor distance
in the SSP), and $k_D=2\pi/D$ where $D$ is the average droplet diameter.
One can see that when the vortex dipole is present (first time interval),
the main contribution to the spectral density is in the
incompressible component at rather small $k$.
After the vortex dipole annihilation, the main contribution is
transferred into the compressible component at
$k$ values $\sim k_R$ which
corresponds to the
$k$ value associated with
solid order. A similar
roton burst has been observed in the annihilation of vortex dipoles in
superfluid $^4$He nanodroplets \cite{escartin}.

\section{Summary}

It is now well established that dipolar bosons, in addition
to a superfluid state, can be in a supersolid state
in which the local density of the condensate is modulated in the plane
perpendicular to the polarization direction.
This phase appears when the strength of the dipolar
interaction is strong enough compared to
the contact interaction. The supersolid state
consists of a regular array of droplets and a
low-density superfluid background which occupies the space between them.

In order to determine the intrinsic
properties of vortices in the SSP we have studied
the system without trapping potential in the $x-y$ plane
but with suitable boundary conditions in the computation
cell, and a harmonic trap is present only in the $z$-direction,
the polarization direction.
We have determined the equilibrium lattice parameter and
the superfluid fraction of this extended system of dipolar bosons
for different values of the s-wave scattering length.
We have found that a singly quantized, linear vortex
along the polarization direction is a stable
excitation of the system in a stationary state.
A fundamental property of a vortex in the supersolid
state is that it has equilibrium positions in
the crystalline structure of the bosons, in our
case at the interstitial sites of the triangular droplets lattice.

We have characterized the density profile of the vortex, its velocity field and the
kinetic energy as function of the distance from the vortex core.
The velocity field strongly
differs from the simple circular
flow in a superfluid; besides,
most of the contribution to the quantum of circulation in the SSP
is due to the low-density background, the droplets
give only a small contribution to it.

As the ratio of the dipolar to the contact interaction increases,
the vortex core expands and the core energy decreases.
We have computed the energy barrier that a
vortex has to surmount to move from an equilibrium site to a neighboring
one. The existence of this energy barrier
--and the density modulation due to the presence of droplets--
represents a major difference from a uniform superfluid
in which a vortex is free to move when it is far from the
boundaries.
As a consequence, the vortex dynamics in a
supersolid is expected to be strongly modified
with respect to that in a
superfluid. In a SSP such dynamics should be characterized
by oscillations around the equilibrium position
and by jumps from one site to another.

We have explored one particular aspect of the
vortex dynamics, namely the behavior of a vortex dipole made of
two counter-rotating parallel vortices. In a uniform superfluid
the vortex dipole is a stable entity which rigidly
translates with constant velocity. We have found that in the SSP
the vortex dipole is an unstable excitation,
the two vortices of opposite chirality do not translate but
they approach each other by a series of jumps until they
annihilate and the excitation energy goes into bulk
excitation of the supersolid. We believe that this is
just an example of new features of the vortex dynamics
in a supersolid. For instance Kelvin waves, the basic
excitations of a vortex, should be strongly modified in the SSP with
respect to the SFP and a new kind of excitation
might arise, at least when the droplets are elongated enough,
consisting of kinks of the vortex core transferring the
core of the vortex from one equilibrium site to another.

Finally, it should also be noticed that the periodic
boundary conditions in the plane perpendicular to the polarization direction
impose a constraint on the spatial order
of the droplets, so that they cannot rotate even
when a vortex is present. However, we believe that this
is not an artifact of boundary conditions.
In fact, we have verified that a single
vortex is also a stable excitation when  the system is
confined in the $x-y$ plane by a cylindrically symmetric
trap. In this case, the droplets do not rotate either.
This behavior is at variance with the state of
dipolar bosons in a rotating trap which induces
single or multiple vortices \cite{stringa19,stringa20,arxiv}
with the whole supersolid structure set in rotation.
It will be interesting to study the behavior of a
vortex dipole for other systems in a SSP, especially
if the spatial order is different from that of dipolar
bosons.

\begin{acknowledgments}
One of us (F.A.) is indebted to Elena Poli for
contributing to the early stages of this work.
This work has been supported by Grant No.  FIS2017-87801-P (AEI/FEDER, UE) (M.B., M.P.).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


There are so many different methods of coping.
Usually the "negative" emotions (ones that cause pain so, in most people's minds, are viewed as bad) are the ones that are internalized.
For me, especially in 2008, I've found myself internalizing what I'm going through. There had come a point in my life— sometime during this year, kind of in the midst of all of the turmoil of unexpected circumstance—where I just felt like my processing it needed to come to a halt because I felt that the conversation around what was challenging me actually no longer served me.
I know that, for myself, talking about lost loved ones, troubles at work, challenging times and so on can be served through conversations with others.
But I also know that some of my most valuable learning has stemmed from my silence in an experience, and my reflection on how I'm actually feeling on the inside and what parts of me are affected.
There were so many parts of my day that were filled with people asking how I was doing, how I was coping, where I was in my grieving process, who I was talking to about it…and I found myself truly exhausted. And going to the other extreme—talking about none of what I'm going through. I don't necessarily feel as though that should be the primary exhibition of coping, but I do feel that it should be listened to when needed. We as humans weren't meant to constantly process and be vocal.
Sometimes being internally quiet will allow us, as individuals, to discover what our needs are and what it is that is causing us sadness.
Another reason I feel that I have internalized so much of it is because there are so many people in my community who are so brutally aware of all of the things that are occurring that I am consistently questioned. And I don't want to just identify as someone who is going through all of these things, and/or someone who is always suffering emotionally.
I do feel, though, it's dangerous to be unbalanced in navigating through life and its unpredictability. As with most things in life, everything in moderation. I don't feel as though excessively verbalizing or internalizing is conducive to prosperity. But what I do believe is that the best thing we can do, in any circumstance—whether it be strange, scary, exciting—is to listen.
Listen to yourself and you will know when it is time to speak, and when it is time to just reflect.
So in my circumstance, I feel as though internalizing some of my storms is of benefit to both myself and those people around me. Like I said, there's an argument for communicating calm seas—even when turbulence is the only internal weather report.


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Japanese Americans leaving camp by bus
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National Archives and Records Administration Collection
Original WRA caption: Gila River Relocation Center, Rivers, Arizona. Aboard the Greyhound bus, which will take them to their former homes in California, these evacuees are anxious to be off. On September 15, two weeks before the Canal Camp at Rivers, Arizona, was to close, only 635 people remained and 370 of these had bus or train reservations for the following week (the Canal Camp once had more than 5,000 residents). Most of the people are going out by special Greyhound buses. Their property, crated for freight shipment, is picked up at their homes and stored in project warehouses until it is loaded on the heavy trucking vans. Before the relocators leave the Center they secure their travel vouchers and their ration books from the Leave Office and get their special Relocation Grant from the Agent Cashier. Farm machinery, once used in the production of vegetables and melons for Gila and other centers, now is assembled ready to be moved from the Center. All the livestock is gone and buildings and fences are being torn down.
Gila River concentration camp, Arizona
OBJECT GENRE
OBJECT FORMAT
World War II -- Leaving camp -- Returning home
Iwasaki, Hikaru (photographer)
Courtesy of the National Archives and Records Administration
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Home / Business / Comment: What's so great about the High Street?
Sure, the footfall is there, but the majority are shopping for clothes, make up, groceries. The enthusiast will travel, argues Mark Sutton.
Comment: What's so great about the High Street?
This month's dealer profile is one I've been meaning to do for a while. Housed inside a facility I regularly travel to on a rainy weekend, 'Absolute' – in the XC centre in Hemel Hempstead – has seen my custom several times and not because I'd travelled with a purchase in mind. Why then?
The simple answer, as wreckless as it sounds, is simply because it's there, in that location and it's right by the front door, so it'd be rude not to have a quick browse. This catches me out on every single visit and on several occasions I've come away with anything from coats to non-essential parts for my bike. If you consider that I'm just one of the 600-odd people passing through 'The XC' on a weekend you can see how all of a sudden a prime location begins to pay for itself.
This leads me to question whether the High Street really is the best place for an enthusiast dedicated bike shop, or any outdoor sports shop, for that matter. Unless you're protesting, chances are you're not going to be camping on the High Street, so why buy a tent there? Somewhere there's a store on a campsite not stressing about where the next sale is coming from.
There's a reason why stores like Rutland Cycles, for example, so quickly earn a strong reputation in the trade. Is it because of customer service? Could be. But I'd hazard a guess that it's more likely down to the fact that everyone who rides a bike or rows a boat has been there. Can the same be said for many town centre stores?
Footfall associated with town centres is of course important, as is converting newbies to cycling. But if you're selling mid-to-top-end mountain bikes, only a minute fraction of new customers passing your door that didn't know about you previously will enter the shop, let alone purchase.
Now I'm not suggesting road stores pitch themselves at the scenic peak of a country lane, but location is without doubt important for complete bike sales. Much the same as you'd stop at a pub or cafe in a prime location, it's well worth taking a look at the local geography, routes that are popular among cyclists and analysing if there's a 'hotspot' that most will pass on a ride.
The other perk of non-High Street locations is the overheads. Let me guess, your landlord has been less than kind in his demands this year? It's certainly a key reason why BikeBiz's local was recently forced to consolidate its business. Not a month or two later and we've a fresh store in town, pitched up 50 foot from a skate park and a new supermarket.


Your baby's celebration bubble will be added to our Born at the Women's Celebration Wall, in the main foyer of the hospital here in Parkville.
By celebrating your baby's birth in this way you will help raise much needed funds for the Women's Foundation to support the vital work of the Women's, Australia's first and largest specialist hospital for women and newborn babies. Your donation to the Women's Foundation today will help deliver advancements in health research, technology, programs and equipment for women and newborns.


Based on a toy property that kind of doesn't lend itself to one, Monster in My Pocket is kind of fun game, but with some big issues.
The Teenage Mutant Ninja Turtles got their first (and only) fighting game with TMNT Tournament Fighters. It's still puzzling why they never got another.
One of the best shooters on the Nintendo 64, Turok hasn't been seen much in recent years. So how does the original hold up?
The Teenage Mutant Ninja Turtles series continued onto the SNES with Turtles in Time. And it's still one of the best games the franchise has produced.
Everybody knows Sonic as the mascot for the Sega Genesis, but did you know that he graced Sega's portable as well? As you can imagine, it's a completely different game.
Sonic the Hedgehog is in for some pretty great months.
The beginning of the three-dimensional age for videogames was the beginning of the end for a few long running and well loved franchises.
A wrestling game without a license that plays more like a fighting game? Yep, that's Saturday Night Slam Masters.
Are you ready for an adventure?! With Sonic?! Yes, we're looking all the way back to the birth of the Dreamcast for a Retro Review of Sonic Adventure.
The third and final numbered installment of the classic Teenage Mutant Ninja Turtles NES series is another fun outing for the franchise.


Projects: Databases – organize small and big amounts of data into usable information, link data and generate reports analyzing and listing the data.
Genealogical Tree of the families Mossakowski and Reimann, with pictures and further texts. Graphical display.


He was standing by the Nile, when out of the river there came up seven cows, sleek and fat, and they grazed among the reeds.
After them, seven other cows, ugly and gaunt, came up out of the Nile and stood beside those on the riverbank. And the cows that were ugly and gaunt ate up the seven sleek, fat cows.
Seven heads of grain, healthy and good, were growing on a single stalk. After them, seven other heads of grain sprouted-thin and scorched by the east wind. The thin heads of grain swallowed up the seven healthy, full heads.
Then Pharaoh woke up; it had been a dream.
We all must face that question at some point in our lives though.
Joseph has been in prison for years at this point and up comes the opportunity of a lifetime. He has the opportunity to interpret Pharaoh's dreams. If Joseph interprets well then his life changes. If Joseph interprets poorly then it's back to prison for Joseph.
That's great faith in a big God.
This great faith leads Joseph to promotion that begins to reveal what God's plan for Joseph is.
At the beginning of Genesis 41, Joseph is stuck in prison. However, at the end he is the prime minister of Egypt.
How much of this did he see in advance? Zero!
Who did this? God did!
Joseph's story is a story of redemption where God is the hero.
Here is a man who was sold into slavery, wrongfully convicted, stuck in prison but became royalty.
Why? Because God is a big God who does big things in our life!
How does your view of God need to be adjusted so that you can trust God with every aspect of your life?


text:
Left and Right agree: Gay "marriage" brings in "choking," "authoritarian" social "conformism"
"Never since the Progressive Era has there been a social movement as elite-driven as the one for gay marriage," said one critic
Thu May 2, 2013 - 6:01 pm EST
By Hilary White, Rome Correspondent
ROME, May 2, 2013 (LifeSiteNews.com) – While pro-family advocates have noted the incredible rate at which the homosexualist movement has gained the political upper hand in the last five years, others are voicing concern about the shrinking "social space" in the public discourse for those who still object to "gay marriage."
Some pundits on the left and the center-right are warning that as the homosexual movement gains legislative ground, the freedom to object is narrowing and those holding out against it are increasingly under threat. The stunning advances made by the "gay movement" some are saying, is not a triumph for social freedom, but for an increasingly brutally enforced social conformism.
Left-libertarian British journalist Brendan O'Neill warned in a column in Spiked, that in 20 years of writing on political issues, "I have never encountered an issue like gay marriage, an issue in which the space for dissent has shrunk so rapidly, and in which the consensus is not only stifling but choking."
The change in public opinion, O'Neill added, can best be described as a "conformism, the slow but sure sacrifice of critical thinking and dissenting opinion under pressure to accept that which has been defined as a good by the upper echelons of society: gay marriage."
Click "like" if you want to defend true marriage.
"In truth, the extraordinary rise of gay marriage speaks, not to a new spirit of liberty or equality on a par with the civil-rights movements of the 1960s, but rather to the political and moral conformism of our age; to the weirdly judgmental non-judgmentalism of our PC times; to the way in which, in an uncritical era such as ours, ideas can become dogma with alarming ease and speed; to the difficulty of speaking one's mind or sticking with one's beliefs at a time when doubt and disagreement are pathologised."
Elsewhere, O'Neill described the movement as an "iron fist in a velvet glove," and denounced the Conservative government's "authoritarian instinct" in forcing objectors to comply. He said it is the only issue on which, after having argued against it from a "liberal" perspective, he has received death threats.
Christopher Caldwell, a writer for the Weekly Standard, wrote for the Claremont Institute website last month, "The most troubling aspect of the gay-marriage movement is that, more than any social movement in living memory, more than feminism at its bra-burning peak in the 1970s, it aims not to engage in lively debate but to shut it down."
The internet has made it much easier to shut down this debate, Caldwell added. "Anyone who expresses the slightest misgivings about gay marriage can become the object of boycotts, blacklists, and attempts to get him fired."
"It is certainly worth asking why, if this is a liberation movement, it should be happening now, in an age not otherwise gaining a reputation as freedom's heyday."
"Half the country cannot even fathom the logic of it," Caldwell said. "Until about a decade ago, the public was nearly unanimous in considering it a joke."
"In a decade, gay marriage has gone from joke to dogma," and now both activists in the homosexualist movement and their supporters in governments and media consider "gay marriage" to be a fait accompli and only a matter of time before it is universal throughout the west.
O'Neill quotes polls in the U.S. that found 58 percent of Americans support "gay marriage", compared with just 37percent a decade ago; and a British poll in which 62percent in support compared to 31percent against 10 years ago.
O'Neill adds that in that time, it has become increasingly dangerous for anyone in politics or elsewhere to object. "Opponents of gay marriage are now treated by the press in the same way queer-rights agitators were in the past: as strange, depraved creatures, whose repenting and surrender to mainstream values we await with bated breath."
In his review of the book From the Closet to the Altar: Courts, Backlash, and the Struggle for Same-Sex Marriage, by Michael J. Klarman, Caldwell notes the effect of the media's promotion and the changes in law have had on public opinion. He wrote, "A barrage of judicial activism on one issue can soften up voting public's resistance on others."
"Marriage litigation has been a bonanza," he added. "Judicial fiat put a halo of normalcy around gay marriage where none had existed before…When elites rally unanimously to a cause, it can become a kind of common sense." No one, he said, wants to be seen to be more conservative than his neighbours, and "the elite view thus becomes a self-fulfilling prophecy."
O'Neill wrote that while the movement normally gets cast as a continuation of the American civil rights movement of the 1950s and '60s, "it's better understood as a continuation, and intensification, of the modern state's desire to get a foot in the door of our private lives and to assume sovereignty over our relationships.
"From the get-go, the depiction of the campaign for gay marriage as a liberty-tinged movement for greater equality was questionable to say the least," he said.
Where "gay marriage" has been installed, it has largely been the result of the deliberate suppression of the democratic process and the work of the judiciary, the "legal elite."
Caldwell writes, "Never since the Progressive Era has there been a social movement as elite-driven as the one for gay marriage. No issue divides the country more squarely by class. Opponents of California's anti-marriage Proposition 8 have come to include virtually all of Hollywood, Apple, Google, Amazon, and the White House."
O'Neill agrees, saying, "Grassroots public protesting for the right of homosexuals to marry was notable by its absence. Instead, this has been a movement led by lawyers and professional activists, backed by the CEOs of hedge-fund corporations and newspapers of record such as The Times."
Caldwell quotes research that found public support for gay marriage has been increasing by four points a year since 2009. "Public opinion does not change this fast in free societies. Either opinion is not changing as fast as it appears to be, or society is not as free," he concludes.
same-sex 'marriage'
Left and Right agree: Gay "marriage" brings in "choking,"… Left and Right agree: Gay "marriage" brings in "choking," "authoritarian" social… News By Hilary White, Rome Correspondent


text:
Mus pahari é uma espécie de roedor da família Muridae.

Pode ser encontrada nos seguintes países: China, Índia, Laos, Myanmar, Tailândia e Vietname.

Mus
Mamíferos descritos em 1916

redpajama_set_name: RedPajamaWikipedia


You will need to provide an email address to download the fix.
Recently it was brought to my attention that you could end up in a situation where a reinstall of Inventor might take forever to finish without any visible signs of activity or progress.
If you would leave the reinstall run over night (or for several hours), the process finishes successfully.
The problem is that not many users will have the patience to let the process sit and run for so long and they will probably kill the reinstall process.
The reason why I wrote this blog post is to just ask you to be patient and let the process take its time because otherwise you might end up with a corrupt Inventor installation.
Hit the Add button and enter the network location (that requires login credentials) of the service pack you want to add.
The result is that the installation process will seemingly hang because of a Microsoft defect that causes the installer to do a check on the credentials for every file that is part of the deployment image.
Every check takes a couple of seconds, multiplied with a large amount of files, gives you this long delay that might take hours.
Note that when you try to click the Cancel button or the Exit "X" button, there is also no response which tempts users to kill the process from Task Manager.
If you prefer to continue to work with a deployment image that has the "Remember my credentials" option unchecked, disconnect the machine from the network before you try to reinstall Inventor.
A dialog asking for the source file location will pop up and allows you to input the correct location and the reinstall will proceed successfully without excessive delays.
We will update this blog post once Microsoft makes a fix available.
I have a related issue - I am unable to append a SP to my deployments - have you seen this before?
Do you get an error message or is the SP nowhere to be seen in the Windows control panel or in the About dialog? Did you check you log files in %temp% for any clue?
This is a great website, might you be interested in doing an interview regarding just how you developed it? If so e-mail me!


Southbridge Credit Union based in Southbridge, Massachusetts, is offering some of the lowest mortgage rates currently available anywhere in MA. To give you an example right now the national average conforming 30 year mortgage rate is at 3.69 percent but Southbridge is advertising conforming 30 year mortgage rates and refinance rates at 3.50 percent with zero mortgage points.
You probably won't find any credit union or bank mortgage rates lower than this rate with zero points for a 30 year loan.
Wow, that is a really good rate, you can find other lenders offering 30 year mortgage rates today at 3.50 percent but you'll have to pay points up front. Points allow you to get a lower mortgage rate or refinance rate by paying money at closing. One point is equal to one percent of the loan.
Southbridge Credit Union is also offering other fixed conventional mortgage rates including 20 year mortgage rates which are currently at 3.375 percent also with zero points. Like I always say if you can afford the higher mortgage payments each month that come with a shorter term loan you should go for it.
15 year mortgage rates today at Southbridge Credit Union are even lower than 30 year rates and 20 year rates. Current mortgage rates on 15 year conforming loans are advertised at 2.875 percent with zero mortgage points.
Today's mortgage rates on 10 year fixed rate mortgages are advertised at 2.99 percent with zero points. Most people won't be able to afford a 10 year loan but if you can the amount of money you'll save as compared to a 30 year loan is unbelievable.
If you want an even lower rate you can also get mortgage rate quotes from Southbridge Credit Union by contacting them directly.
Southbridge Credit Union has adjustable mortgages available with some of the best rates around. For example, current mortgage rates on 1 year adjustable home loans are advertised at 2.375 percent. 3 year rates are advertised at 2.875 percent.
5 year adjustable rates are advertised at 3.25 percent and 7 year adjustable rates are advertised at 3.50 percent. All the adjustable home loans are amortized over a 30 year period. There are also annual caps of 2 percent and lifetime caps of 6 percent.
As with any credit union you'll have to join to apply for a loan and as with any credit union membership isn't opened to everybody. Membership in the Southbridge Credit Union is available to people, businesses and organizations that are either residing, working or operating within the Massachusetts counties of Worcester, Bristol, Franklin, Hampshire, Norfolk, Hampden and Middlesex, and within a 25-miles radius of our the credit union's main branch in Southbridge.


text:
Alfred Ernest Levesque
DEXTER, ME - Alfred Levesque died on November 2 in Dexter, ME after a long battle with liver disease and diabetes. He was 69.
Mr. Levesque was born on Halloween in 1949 to Alfred J. and Eleanor M. (neè Battige). He spent the early years of his life in Agawam before proudly enlisting in the Army. After an honorable discharge, he moved to Boston, where he enjoyed driving a checkered taxi. He then moved to Cape Cod, where he met Mendy Whelden. They were married in 1979, and had two children before divorcing in 1989. Until 2018, Cape Cod was his home.
He was a devoted father, a generous friend to many, and always looked for the simple and silly things in life.
Alfred lived his life with a passion for music, dancing no matter who was looking, poetry, and any cringe worthy dad joke.
Mr. Levesque is survived by his son and daughter-in-law, Joseph and Kiersten; his daughter, Reneè; his siblings, Ernest, Elaina, and Thomas; and many nieces and nephews. He is preceded in death by his parents and a younger brother, Richard.
A funeral mass will be held at 11 am, November 30 in Hyannis at the Saint Francis Xavier church, followed by a reception at Hearth n Kettle in South Yarmouth.


Alex Kruger has been teaching Permaculture for 20 years in Southern Africa. Her passion is reconnecting people to nature and creativity – empowering people to live sustainably and responsibly through solutions-based thinking and actions, while having a lot of fun. One of her favorite paths to this is through designing and growing food gardens as well as cooking and preserving food and saving heritage seed. There is a lot to discover about yourself through your garden, and she is excited to share this journey with you.
Watch this space or contact Alex directly to find out about the next up and coming workshop.


text:
IFS: Estonian Ministry of Creative Affairs
Photograph by Agnese Sanvito.
by Osman Ahmed
For former dancer Kristina Viirpalu, her Estonian heritage defines her work. Each of her dresses are made using the Estonian handwork technique in traditional patterns which have been passed down through generations over the last 200 years. "Everything is handmade and made-to-order and a dress can take up to five months to create," she says. "My grandmother and mother were both seamstress so I've always felt as though dressmaking is in my blood."
Estonia-born, London-based menswear designer Kristian Steinberg's third collection is on display and he hopes this opportunity will help him develop in the early stages of his business endeavour. "It's very exciting for the public to be able to come in and see our work as it gives them a chance to step into London Fashion Week. It's a chance for me to be a part of something bigger and get buyers and press interested in my work."
"It is about the relationship between high-tech and natural, mixing the harder plastics and flameproof fabrics with the softer wools and furs," explains Steinberg. "It started from looking at the Brutalist architecture in London and translating that into the clothes."
The exhibition, which takes place at The Horse Hospital in Bloomsbury, also explores the relationship between fashion and film. Curator Helen Sirp took the opportunity to make the exhibition the final piece in the jigsaw for her Central Saint Martins MA degree in Narrative Environments.
"I curated the International Fashion Showcase last year for Estonia and I loved it. This time it is a part of my final project for my MA in Narrative Environments at Central Saint Martins, so
"I'm exploring the ways in which we cane express fashion in a spatial manner," she explains. "I've done it in a way that displays the pieces in a fashion film concept as we have a video playing throughout the exhibition. It tells the story of the designers' creations as characters and the fairy-tale stories surrounding them."


After a hard fought game on the field, relaxation time in the bath and in the bar brought out the singing voices. Here we list a number of the well known songs.
If I were the marrying kind.
I don't want to be a soldier.
Sing us another one do……..


Tucked away in quiet woodland, Springfort Hall Hotel is a peaceful 18th-century manor located 6 km from Mallow Town.
On a 500-acre estate surrounded by lush green valleys, salmon-stocked rivers, and trees representing the battle of Waterloo, this 18th-century Georgian country house offers luxurious, elegant accommodation and stunning cuisine.
With a 20-metre swimming pool, a leisure club, and a bistro, The Hibernian Hotel is situated in the centre of Mallow. Guests can enjoy a modern bar and a coffee dock, just 30 minutes' drive from Cork.
The Blarney Woollen Mills Hotel is just 500 metres from the world famous Blarney Castle and Blarney Stone. It offers a bar and restaurant, and a fitness suite. Rooms feature elegant furnishings as well as a flat-screen TV and free WiFi.
Just 200 yards from Blarney Castle and, 5-minutes' walk from the world famous Blarney Stone. It offers spacious rooms with satellite TV, a restaurant and traditional bar.


Fig-lab Finger Five!, originally uploaded by datadub.
Coming Very Soon to a Hand Near YOU!!!
Giving a presentation on the first five years of Fig-lab at Pecha Kucha tonight!
IMG_0025, originally uploaded by datadub.


text:
Phnom Penh Post - Workers' minimum wages on table
Workers' minimum wages on table
Mom Kunthear | Publication date 15 August 2021 | 20:04 ICT
Garment workers leave their factory during lunch break in Phnom Penh's Meanchey district. Heng Chivoan
Mom Kunthear
The National Council on Minimum Wage (NCMW) will begin negotiating the 2022 minimum wages for workers in the textile, garment and footwear sectors in September, according to Ministry of Labour and Vocational Training spokesman Heng Sour.
Worker unions and labour rights groups believe that the minimum wage in the sector should increase from $192 to $200 a month.
Sour told The Post on August 15 that the tripartite minimum wage negotiations for 2022 were scheduled for September and may be conducted online.
Each year in July, the NCMW begins internal discussions on the minimum wage. Bilateral and multi-party discussions often take place informally in August and the formal meetings to decide the minimum wage are then held in September.
Cambodian Labour Confederation president Ath Thorn told The Post on August 15 that the unions were conducting research and gathering data in order to reach a figure that they would request as the wage increase for workers in 2022.
"So far, the unions have not yet set any figures for the negotiations because we are still doing the research. I can see that negotiations for this minimum wage are going to be hindered by the Covid-19 crisis, but we are ready to talk," he said.
Ath Thorn added that the unions will try their best to get a wage increase for all workers. They previously suggested a $200 minimum wage for 2022.
Fa Saly, president of the National Trade Union Confederation, said he expects the minimum wage in 2022 to be increased to $200.
"I expect that the government as well as employers will increase the minimum wage in 2022 to at least $200 a month and I am optimistic that it will reach this level," he said.
He added that workers were still experiencing great difficulties due to the Covid-19 pandemic but exports seemed to be improving, which is a positive sign.
Sorn Sok, a 30-year-old worker at CG Glory Manufacturing in Phnom Penh's Meanchey district, said he expected to receive higher wages in 2022 than he does this year.
"If the wage increase in 2022 is lower than what we get this year, I will be disappointed because that is too little for us. But we will still accept it because everyone has the same problem these days: Covid-19," he said.
According to the ministry, during the negotiations all sides use legal arguments, scientific research and economic criteria like the rate of inflation and the cost of living, productivity, competitiveness, labour market conditions and levels of profit in the sector as the basis for their arguments about the minimum wage.
This year, factory workers in the textiles, garments and footwear sectors have received a $2 minimum wage increase from $190 to $192.
Contact author: Mom Kunthear
PM adds $2 to minimum wage agreed for next year
Tripartite talks over minimum wage start
Dates set for minimum wage raise discussion
Unions to request 11.6% raise to minimum wage
Minimum wage for workers up to $192 for 2021
Minimum wage to be decided
The National Council on Minimum Wage will decide on a wage increase for workers in the garment, footwear, textile and travel product sectors at a
Minimum wage talks set to roll
Trade unions, GMAC and ministry set for minimum wage negotiation
GMAC seeks delay of 2021 wage talks
Minimum wage talks loom


It's unbelievable how each year, since the beginning of high school, my perfectly laid plans for myself have unavoidably deteriorated. When I entered my freshman year, I had aspirations of being a doctor (something I had wanted to be since I was a child); I was even taking Latin to help with the medical jargon. Now, here I am entering my senior year in high school without any idea of what I want to do. Now seems to be the time to start taking life seriously and making responsible, educated choices.
Looking back on it now, I realize that I cannot become the doctor I wanted to be. I have come to realize that, that dream was not only my own, but a dream of my family. My parents often talked of me becoming a doctor and although their enthusiasm continued throughout my childhood and early adolescence, mine slowly diminished; until finally I realized I did not want to become a doctor. I remember how hard it seemed to tell my parents of my decision, I felt as if I was letting them down, but I eventually came to realize that they wanted me to do what made me happy. I am not definite why I changed my mind in regard to being a doctor, I had the grades, the drive, and the willingness to make certain sacrifices, but somewhere I was missing something and I felt that I would not be satisfied in a medical career. So I started thinking about what I wanted to do; I went from teaching to law enforcement, computer programming to astronautics and numerous other professions. Now I come to a crossroad in my life where I must choose what to do with my future, choose what will make me happy.
I have always wanted the typical "˜American Dream' to have a husband I am in love with, a stable job, loving children, and a house in the suburbs. But now I realize that there are so many other steps I need to take in order to achieve these so-called goals. This includes graduating from high school and college, finding that special someone, and finding that perfect job. I have begun to realize that I have yet to begin my life; everything up until now has been practice, as if I have been in a cage and it is only now that I am beginning to break free and do things for myself.
My one goal for the future is to be happy regardless of what I choose to do. I want to be satisfied with my decisions, to be able to accept and forgive, and most of all to be able to live up to the expectations I have for myself. I realize I cannot set my goals and dreams on the basis of others and I need to achieve things for myself. If I live my life for the people around me, I will never truly be happy. I will only accomplish my goal in being happy when I am able to live my life for myself and still able to provide love and support to others.
I realize I don't have to have all the answers rights now, and that it's probably better that I don't. I have also figured that no matter what I do in life, I will never be completely satisfied; and that's how it should be, always looking for something more, always striving for something better.
On a separate sheet of paper, write a well-developed, one page essay describing why you should receive a tuition scholarship. Include your academic goals, career aspirations, and reasons for selecting the University of Maine. Please supply any additional information about yourself you believe would be useful to the selection committee.
I am not really confident with this essay as I feel I don't have a strong theme for it and I am not sure if I have my academic goals are clear in this essay.Any criticisms are welcomed.Thank you so much.
My parents have never been to college. I knew they had the same dream as me, the same passion as me to pursue their studies in the tertiary education. Due to the fact of financial difficulties at that time, they never had that chance to do what they wanted. Instead they accepted the duty of being the ones to care for their family. They entered the the working world after high school. They never did tell me about their most precious desires at that time, but I have always known. My parents have always made sure that I never compromised my education. They tried their best to help me with my homework and made sure that I would get the help I needed. They would always say to me that, "No matter what, you'll be getting into the college to do what you want." I intend to accomplish their dream of seeing their daughter going attending college.
I am an obstinate learner who is always longing for knowledge. Coming from a highly competitive high school, I have become someone who likes to challenge myself. I have taken rigorous subjects in school including three Science electives to help me prepare for my pre-medical studies. I have always been a dedicated student, working diligently to always make sure that I am the top of my class. My high school experience was a stimulating adventure where I was always on the hunt for knowledge and ways to improve my intellectual abilities. Thus, Maine University will be my stepping stone for my quest to excel in my pre-medical studies and enter an excellent medical school. The thirst for adventure has not abated for me, in fact, as I prepare for Maine University, the lure of greater achievement and deeper knowledge beckons.
During one of my duty hours as a first aid volunteer, I had to handle a case where the patient had a sudden asthma attack and my first aid knowledge helped her abate the attack and saved her life. I did not just use my first aid knowledge to help her, in fact, held on to her hand and told her to not give up and keep fighting. I realized that by giving her hope to go on, I had helped her to survive. That moment was a turning point in my life. I had found my true calling of being a doctor. The power of making a difference in a person's life is the best gift I could give a person. Throughout the five years of being a first aid volunteer under St. John Ambulance Malaysia, the thing that struck me most is that I was able to witness a person getting better after I treated them. It gave me the highest possible satisfaction as I have made an impact in that person's life. I personally aspire to be someone who brings impact into my community and let that impact spread through. I learned that a person needs love, concern, mercy, sympathy and compassion to recover. As a first aid volunteer, I have discovered that the human touch can trigger hope and healing within an ailing person. By being a doctor, I am able to continue what I love doing the most; making a difference, aiding recovery and even saving lives.
When I discovered Maine University through meticulous research on the internet, I realized that this was the perfect place for me to complete my undergraduate student. As an international student, I have yearned to be part in the United State advanced education system in which Maine is the one university that is able to fulfill my educational pursuits. I have never visited Maine University before, but no other university was able to bring me the same contentment.


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Provincial Gazette for Free State No. 94, 01 November 2019
Summary 20191101 - Provincial Gazette for Free State No. 94 of 01-November-2019, 91 page(s), 3.2Mb
Publication Provincial Gazettes (Free State)
91 pages for Gazette No. 94
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