text:
\section{Introduction}
The study of strongly correlated fermionic superfluids is of interest in the context of solid
state materials, such as cuprate \cite{WenLee} and pnictide superconductors,\cite{norman08}
as well as ultracold
atomic gases in optical lattices.~\cite{Bloch08,lehur09}
One of the key goals of condensed matter physics is to understand
the nature of single particle and collective excitations in such strongly correlated systems.
Another important problem is to
eludicate the mechanisms by which superfluidity breaks down in these lattice systems ---
such an understanding would shed light on the critical current and
on the nature of vortex cores in strongly interacting superfluids,~\cite{Aeppli01}
which are issues
of significant experimental and theoretical interest.
Moreover, the inverse critical superflow momentum can be shown to be directly related to
characteristic length
scales, associated with
the low-energy excitations in the system and can thus serve as a useful
probe of these excitations. This is especially relevant for neutral cold atom superfluids, in which
the critical flow is unaffected by complications such as disorder effects and current induced
magnetic fields which are present in  solid state superconductors.
While there has been a significant
amount of experimental and theoretical progress in understanding the
breakdown of
superflow for Bose superfluids in an optical lattice,~\cite{Niu01,Altman05,nikuni06,ketterle-bose07} and for Fermi superfluids in the absence of a lattice
potential,~\cite{sensarma06,Ketterle07}
this issue has not been addressed in detail in the context of Fermi
superfluids on a lattice.

In this paper we study the collective modes and the mechanisms for the
breakdown of superflow in the one-band attractive Hubbard model which is a model Hamiltonian
for strongly correlated s-wave Fermi superfluids on a lattice.~\cite{Ranninger90,swz,
trivedi,levin,Scalapino91}
The attractive Hubbard Hamiltonian,
\begin{eqnarray}
\label{eq:1}
H &=& -t \sum_{\langle ij \rangle} (c_{i\sigma}^\dagger c^{\vphantom \dag}_{j\sigma} +
c_{j\sigma}^\dagger c^{\vphantom \dag}_{i\sigma}) - \mu\sum_{i\sigma}n_{i\sigma} \nonumber \\
&-& U\sum_{i}(n_{i\uparrow}-\frac{1}{2}) (n_{i\downarrow}-\frac{1}{2}),
\label{eq:hub}
\end{eqnarray}
describes fermions with spin $\sigma$ hopping, with amplitude $t$, between adjacent sites of a
lattice, and interacting via an on-site attractive
interaction $U$. The chemical potential $\mu$ tunes the fermion density away from half-filling at $\mu=0$.
Summation over repeated spin indices is implicit henceforth, and, unless
stated explicitly, we will measure energies in units of $t$.
We will study this model on the square lattice in
two dimensions (2D) and the cubic lattice
in three dimensions (3D). The ground states of this model include the uniform
superfluid (SF) and a ``checkerboard'' charge density wave (CDW), which is an
insulating crystal.~\cite{Ranninger90}
It is well known that this model has an enhanced ``pseudospin symmetry''
at the point $\mu=0$,~\cite{Zhang90} which leads to a
degeneracy between the SF and the CDW ground states.
Away from $\mu=0$, this degeneracy is lifted and the SF ground
state is energetically more favorable.
One  expects, however, that so long as this pseudospin symmetry is only
weakly broken, the excitation spectrum of the
superfluid will exhibit signatures of proximity to the
CDW ground state. One also
expects that once superflow is imposed, there will be
a critical flow velocity beyond which the kinetic energy of superflow will
overcome the energy difference between the stationary SF and the CDW.
The superflow in this case is limited not by the pairing gap, which may be large, but by this
energy difference, which becomes very small close to half-filling.
This situation is similar to the one encountered, for example, in cuprate superconductors,
where  the
superconducting ground state in the underdoped region of the phase diagram is proximate
to the antiferromagnetic Mott insulator.~\cite{MacDonald08}  Similar physics may also be relevant to
Cu$_x$TiSe$_2$ where there is a competition between charge order and superconductivity.~\cite{morosan06}

This general idea motivates us to study the collective modes as well
the stability of superflow in the SF phase of the attractive Hubbard model.
We begin by studying the collective mode spectrum in the absence of superflow
and find that there is indeed a large window of interactions
and density where the collective mode spectrum exhibits, in addition
to a sound mode, a well-defined roton
minimum at wavevectors ${\boldsymbol \Pi} \equiv (\pi,\pi)$ (in 2D) or ${\boldsymbol \Pi}
\equiv (\pi,\pi,\pi)$ (in 3D) arising
from proximity to the CDW.
The existence of such a roton
minimum has been pointed out in earlier work.\cite{kostyrko,alm,danshita}
We present our results for the values of sound velocity and roton gap as functions of
fermion filling and interaction strength.
These results of the collective
mode spectrum could potentially be verified in studies of collective modes
in the superfluid phase of cold Fermi gases in an optical lattice.
We then turn to a study of the stability of uniform superflow in the
Hubbard model.

We elucidate a variety of superflow-breakdown mechanisms - depairing, Landau instabilities
and dynamical instabilities. We observe Landau instabilities of the collective mode at incommensurate
wavevectors as has also been
reported in Ref.[\onlinecite{danshita}]. More importantly, we discover
dynamical instabilities involving checkerboard or incommensurate stripe-like density
modulations which are distinct from previously studied dynamical instabilities of
Bose superfluids.~\cite{Niu01,Altman05} We show that the checkerboard dynamical instability can be
viewed as the result of the softening of the roton mode and that this instability
has a simple analog in the strong coupling pseudospin model,~\cite{Burkov08}
which we discuss.
In contrast, the incommensurate CDW instability occurs only at intermediate coupling
strengths
and has no analog in the strong-coupling pseudospin model. We explain this instability
as a finite momentum pairing instability
of Bogoliubov quasiparticles, somewhat analogous to the Halperin-Rice exciton
condensate instability in
indirect band-gap semiconductors.\cite{Halperin1968}
These
dynamical instabilities could be experimentally probed by creating a
``running'' optical lattice as has been done for Bose
superfluids.~\cite{ketterle-bose07}

The existence of
the checkerboard dynamical instability was discussed earlier by us from the strong coupling
perspective.~\cite{Burkov08}  The present paper goes considerably beyond our earlier work.
The formalism that we use (the generalized random
phase approximation or GRPA), allows us to address the
entire range of interaction strengths from weak to strong coupling. We have checked that
the GRPA results smoothly match on to the earlier analysis of the strong coupling
pseudospin model and we present these comparisons where appropriate. Within this
formalism we have obtained dynamical phase diagrams of the flowing superfluid over
a wider range of interactions than in our previous study and uncovered the incommensurate
CDW instability which was not reported in our earlier paper.

We have tried to make this paper fairly self-contained.
We begin
in Section II with an overview of some key general results on the Hubbard model.
Section III focuses on the formalism which we use in the remainder of the paper.
We then turn, in Section IV, to results for the collective mode spectrum in the absence
of superflow. Section V contains the results for the instabilities of the flowing
superfluid and the phase
diagram showing the leading instabilities as a function of interaction and fermion
density. Section VI contains a discussion of the experimental observability of our results in
fermionic cold atom systems as well as possible implications of these results for vortex core physics
in strongly correlated superfluids. We end, in Section VII, with a summary of our results and
possible avenues for future research. Appendix A contains the Fourier transform convention which
we use, Appendix B gives details of the GRPA formalism, and the
mean field theory of the ``flowing supersolid'' state is discussed in Appendix C.

\section{Overview of the Model}
Before we turn to a study of collective modes and the stability of superflow using
the Hubbard model Hamiltonian, we begin by reviewing some well known facts
about the model
which will serve to set the stage for our discussion in the remainder of this paper.

\subsection{Pseudospin Operators and Pseudospin Symmetry}

We start by defining pseudospin operators, first introduced in the context of
the theory of superconductivity by Anderson:~\cite{Anderson58}
\begin{eqnarray}
\label{eq:2}
T_i^+ &=& \eta_i c_{i\uparrow}^\dagger c_{i\downarrow}^\dagger, \nonumber \\
T_i^- &=& \eta_i c_{i\downarrow}c_{i\uparrow}, \nonumber \\
T_i^z&=&\frac{1}{2}(c_{i\sigma}^\dagger c^{\vphantom \dag}_{i\sigma}-1),
\end{eqnarray}
where $\eta_i=+ 1$ on
one sublattice and $\eta_i=-1$ on the other sublattice of the square or cubic lattice.
The physical meaning of these operators is clear: $T^{+}_i$ creates a
fermion pair at site $i$, $T^-_i$ annihilates a fermion pair at site $i$,
and $T^z_i$ is the pair number operator.
It is straightforward to check that these operators obey usual spin commutation
relations. Furthermore, provided that $\mu=0$,
the global pseudospin operators,
\begin{eqnarray}
\label{eq:3}
T^z &=& \sum_i T^z_i, \nonumber \\
T^{\pm} &=& \sum_i  T^{\pm}_i,
\end{eqnarray}
all commute with the Hubbard Hamiltonian in Eq.~(\ref{eq:hub}) so that there is a global
pseudospin SU(2) symmetry at this special point.~\cite{Zhang90} This is in addition to the ordinary global
spin SU(2) symmetry which is present for any value of $\mu$.
Pseudospin language is often very convenient and intuitive, in particular when discussing symmetry properties
and collective modes of paired-fermion superfluids.

\subsection{Ground States and Degeneracies}
Quantum Monte Carlo simulations~\cite{Scalapino91,swz,trivedi}
have shown that the ground state of the attractive
Hubbard model is a uniform SF
for generic values of $U/t$ and $\mu/t$. The uniform SF has
an order parameter $\langle c_{i\uparrow}^\dagger c_{i\downarrow}^\dagger\rangle
\sim \Delta {\rm e}^{i\varphi}$ where a choice of the
phase, $\varphi$, of the order parameter corresponds to a spontaneously broken
symmetry. In terms of pseudospin operators, this means that
$\langle T^+_i \rangle \sim \eta_i \Delta {\rm e}^{i\varphi}$.
For $\mu=0$, the pseudospin symmetry discussed above leads to the conclusion that
all states related to this SF ground state by a global pseudospin rotation
are also valid ground states and are characterized by an order parameter
${\bf N} = \eta_i \langle {\bf T}_i \rangle$ which can point to any location on the Bloch sphere.
The uniform SF state has ${\bf N}$ pointing to locations on the equator.
The state with ${\bf N}
\sim \pm \Delta \hat{z}$, which points to the north/south pole of the Bloch sphere,
corresponds to a checkerboard charge density wave state (CDW) of pairs.
Other locations on the Bloch sphere correspond to states with coexisting CDW and SF orders.

\subsection{Strong coupling pseudospin Hamiltonian}
In the limit of large $U/t$, each site on the lattice
will have either no fermions or a pair of tightly
bound fermions. Thus, any allowed fermion configuration will have $T^z_i = \pm 1/2$
at each site and the kinetic
energy will lead to tunneling between such ``classical'' Ising configurations of the
pseudospin. The pseudospin dynamics for $U/t \gg 1$ is then described by an
effective pseudospin Hamiltonian:
\begin{equation}
\label{eq:4}
H_{\rm eff} = J  \sum_{<ij>}{\bf T}_i \cdot {\bf T}_j - \mu \sum_i T_i^z,
\end{equation}
where $J = 4 t^2 /  U$.
For $\mu=0$, this Hamiltonian reduces to a pseudospin Heisenberg model which manifestly
exhibits the pseudospin SU(2) symmetry.

\section{Formalism}
The most general problem we would like to tackle, motivated by the issues discussed in the introduction,
is to understand the collective mode spectrum of a flowing superfluid.
We begin by setting up a two-pronged approach to attack this problem.
At small values of $U/t$,
we build upon the mean field theory of the flowing superfluid to obtain the collective
mode spectrum using GRPA. At strong coupling, we
study the pseudospin model using Holstein-Primakoff spin wave theory. We find that
the GRPA results smoothly match on to the results from spin wave theory of the pseudospin
model as we increase $U/t$, showing that the GRPA correctly captures aspects of the strong
coupling limit. This suggests that the GRPA may be a very good approximation to compute
the collective mode spectrum and address the breakdown of superflow over the
entire range of couplings.

\subsection{Generalized Random Phase Approximation for the Attractive Hubbard Model}
We begin by constructing the mean field theory of the uniform
superconducting state of the attractive Hubbard model in the presence of nonzero superflow.
The mean field Hamiltonian in the presence of a supercurrent is obtained by
forming Cooper pairs with nonzero momentum,
\begin{eqnarray}
\label{eq:5}
H_{\rm MFT} &=& -t \sum_{\langle i j \rangle} (c_{i\sigma}^\dagger c^{\vphantom \dag}_{j\sigma} +
c_{j\sigma}^\dagger c^{\vphantom \dag}_{i\sigma}) - \mu\sum_{i\sigma}n_{i\sigma} \nonumber \\
&-& \Delta_0 \sum_{i} \left( {\rm e}^{i {\bf Q} \cdot {\bf r}_i} c^{\dagger}_{i\uparrow}
c^{{\dagger}}_{i\downarrow} + {\rm e}^{-i {\bf Q} \cdot {\bf r}_i} c^{\phantom\dagger}_{i\downarrow} c^{\phantom\dagger}_{i\uparrow}
 \right),
\end{eqnarray}
where we have set  $U \langle c^{\phantom\dagger}_{i\downarrow} c^{\phantom\dagger}_{i\uparrow}\rangle =
\Delta_0 {\rm e}^{i {\bf Q} \cdot {\bf r}_i}$ and absorbed the uniform Hartree shift into
the chemical potential.
In momentum space, the mean field Hamiltonian takes the form
\begin{eqnarray}
\label{eq:6}
H_{\rm MFT} \!\!=\!\! \sum_{{\bf k}} \xi_{\bf k} c_{{{\bf{k}}} \sigma}^{\dagger} c^{\phantom\dagger}_{{{\bf{k}}} \sigma}
\!\!-\!\! \Delta_0 \!\!\sum_{{{\bf{k}}}}\! \left(\!c^{\dagger}_{{{\bf{k}}} \uparrow} c^{\dagger}_{-{{\bf{k}}} + {{\bf{Q}}} \downarrow} \!+\!
c^{\phantom\dagger}_{-{{\bf{k}}} + {{\bf{Q}}} \downarrow} c^{\phantom\dagger}_{{{\bf{k}}} \uparrow}\!\right),
\end{eqnarray}
where
$\xi_{{\bf{k}}} \equiv-2t \epsilon_{{{\bf{k}}}}-\mu$, with
$\epsilon_{{\bf{k}}} \equiv \sum_{i=1}^{d} \cos(k_{i})$ ($d=2,3$ is the dimensionality of the lattice).

We can diagonalize $H_{\rm MFT}$
by defining Bogoliubov quasiparticles (QPs), $\gamma$, via
\begin{eqnarray}
\label{eq:7}
 \left(\begin{array}{c}
 c_{{{\bf{k}}} \uparrow} \\ c_{-{{\bf{k}}} + {{\bf{Q}}} \downarrow}^\dagger
 \end{array}\right)=\left(\begin{array}{cc}
u_{{\bf{k}}}({{\bf{Q}}}) & v_{{\bf{k}}}({{\bf{Q}}}) \\ - v_{{\bf{k}}}({{\bf{Q}}}) & u_{{\bf{k}}}({{\bf{Q}}})
\end{array}\right)\left(\begin{array}{c} \gamma_{{{\bf{k}}} \uparrow} \\ \gamma_{-{{\bf{k}}} + {{\bf{Q}}} \downarrow}^\dagger
\end{array}\right).
\end{eqnarray}
For simplicity of notation, we will refer to the Bogoliubov transformation coefficients
above as $u_{{\bf{k}}}, v_{{\bf{k}}}$ with the implicit understanding that they depend on ${{\bf{Q}}}$.
Parametrizing
$ u_{{\bf{k}}} \equiv \cos(\theta_{{\bf{k}}})$,
$v_{{\bf{k}}} \equiv \sin(\theta_{{\bf{k}}})$,
and demanding that the transformed Hamiltonian be diagonal
leads to the condition
\begin{equation}
\label{eq:8}
\tan(2\theta_{{\bf{k}}}) = \frac {\Delta_0}{\frac{1}{2} (\xi_{{\bf{k}}} + \xi_{-{{\bf{k}}} + {{\bf{Q}}}})}.
\end{equation}
Defining
\begin{equation}
\label{eq:9}
\Gamma_{{\bf{k}}} = \sqrt{\frac{1}{4} (\xi_{{\bf{k}}} + \xi_{-{{\bf{k}}} + {{\bf{Q}}}})^2 + \Delta^2_0},
\end{equation}
we find that the Bogoliubov transformation coefficients must satisfy the relations
\begin{eqnarray}
\label{eq:10}
u^2_{{\bf{k}}} &=& \frac{1}{2} \left(1 + \frac{\xi_{{\bf{k}}} + \xi_{-{{\bf{k}}} + {{\bf{Q}}}}}{2 \Gamma_{{\bf{k}}}} \right), \nonumber \\
v^2_{{\bf{k}}} &=& \frac{1}{2} \left(1 -  \frac{\xi_{{\bf{k}}} + \xi_{-{{\bf{k}}} + {{\bf{Q}}}}}{2 \Gamma_{{\bf{k}}}} \right), \nonumber  \\
u_{{\bf{k}}} v_{{\bf{k}}}  &=& \frac{\Delta_0}{2 \Gamma_{{\bf{k}}}}.
\end{eqnarray}
In terms of the Bogoliubov QPs, the mean field Hamiltonian finally
takes the form
\begin{eqnarray}
\label{eq:11}
H_{\rm MFT}=E_{\rm GS}+\sum_{{{\bf{k}}}} E_{{\bf{k}}} \gamma_{{{\bf{k}}} \sigma}^{\dagger} \gamma^{\phantom\dagger}_{{{\bf{k}}} \sigma},
\end{eqnarray}
where $E_{\rm GS}$ denotes the ground state energy of $H_{\rm MFT}$
and $E_{{\bf{k}}}$ denotes the Bogoliubov QP dispersion given by:
\begin{eqnarray}
\label{eq:12}
E_{{\bf{k}}} &=& \Gamma_{{\bf{k}}} + \frac{1}{2} \left(\xi_{{\bf{k}}} - \xi_{-{{\bf{k}}} + {{\bf{Q}}}} \right), \nonumber \\
E_{\rm GS} &=& \sum_{{\bf{k}}} \left(\xi_{{\bf{k}}} - E_{{\bf{k}}} \right).
\end{eqnarray}
Demanding self-consistency of the mean field theory yields the gap and number
equations:
\begin{eqnarray}
\label{eq:13}
\frac{1}{U}&=& \frac{1}{N}\sum_{{{\bf{k}}}} \frac{1}{2 \Gamma_{{\bf{k}}}} (1 - n_{F}(E_{{{\bf{k}}}}) - n_{F}(E_{-{{\bf{k}}} + {{\bf{Q}}}})), \nonumber \\
f&=&\frac{2}{N}\sum_{{{\bf{k}}}} \left[u_{{\bf{k}}}^2 n_{F}(E_{{{\bf{k}}}}) + v_{{\bf{k}}}^2 (1 - n_{F}(E_{-{{\bf{k}}} + {{\bf{Q}}}}))\right], \nonumber \\
\end{eqnarray}
where $f$ is the filling, i.e. the average number of fermions per site,
and $N$ is the total number of sites.
For given $U$, filling $f$ and flow momentum ${{\bf{Q}}}$, these equations can be
solved to obtain the SF order parameter $\Delta_0$ and the QP spectrum.

Going beyond mean field theory, we need to include fluctuations of the density
and the superfluid order parameter, which we will do within GRPA.
We begin by considering perturbing external fields that couple to the density
and order parameter modulation operators as:
\begin{eqnarray}
\label{eq:14}
H'_{\rm MFT} &=&H_{\rm MFT} -\!\! \sum_{i} [ h_{\rho}(i,t)\hat{\rho}_i \nonumber \\
&+& h_{\Delta}(i,t)\hat{\Delta}_i e^{i {{\bf{Q}}} \cdot {{\bf{r}}}_i} + h_{\Delta}^{*}(i,t)\hat{\Delta}_i^\dagger
e^{- i {{\bf{Q}}}  \cdot {{\bf{r}}}_i}],
\end{eqnarray}
where
\begin{eqnarray}
\label{eq:15}
\hat{\rho}_i &=& \frac{1}{2} c_{i\sigma}^\dagger c^{\phantom\dagger}_{i\sigma}, \nonumber \\
\hat{\Delta}_i &=& c_{i\downarrow} c_{i\uparrow}.
\end{eqnarray}
Going to momentum space,
\begin{eqnarray}
\label{eq:16}
H'_{\rm MFT}=H_{MFT} - \frac{1}{N}
\sum_{{{\bf{K}}}}\,\, h_\alpha({{\bf{K}}},t) \hat{O}^{\dagger}_{\alpha}({{\bf{K}}}),
\end{eqnarray}
where $\alpha = 1,2,3$, summation over $\alpha$ is implicit, and $\hat{\bf O}^{\dagger}({{\bf{K}}}) \equiv \{\hat{\rho}^{\phantom\dagger}_{-{{\bf{K}}}}, \hat{\Delta}^{\phantom\dagger}_{-{{\bf{K}}} + {{\bf{Q}}}},
\hat{\Delta}_{{{\bf{K}}} + {{\bf{Q}}}}^\dagger\}$
is the vector of fermion bilinear operators corresponding to density and superfluid order parameters
at nonzero momenta, with the order parameters given by:
\begin{eqnarray}
\label{eq:17}
\hat{\rho}^{\phantom\dagger}_{{{\bf{K}}}}&\equiv& \frac{1}{2}\sum_{{{\bf{k}}}} c_{{{\bf{k}}} \sigma}^\dagger c^{\phantom\dagger}_{{{\bf{k}}} + {{\bf{K}}} \sigma}, \nonumber  \\
\hat{\Delta}^{\phantom\dagger}_{{{\bf{K}}}}&\equiv& \sum_{{{\bf{k}}}} c^{\phantom\dagger}_{-{{\bf{k}}} + {{\bf{K}}} \downarrow} c^{\phantom\dagger}_{{{\bf{k}}} \uparrow}, \nonumber \\
\hat{\Delta}^{\dagger}_{{{\bf{K}}}}&\equiv& \sum_{{{\bf{k}}}} c^{\dagger}_{{{\bf{k}}} \uparrow} c^{\dagger}_{-{{\bf{k}}} + {{\bf{K}}} \downarrow}.
\end{eqnarray}
The perturbing fields correspond to
\begin{eqnarray}
\label{eq:18}
h_1({{\bf{K}}},t)&=&h_\rho({{\bf{K}}},t), \nonumber \\
h_2({{\bf{K}}},t)&=&h_\Delta({{\bf{K}}},t), \nonumber \\
h_3({{\bf{K}}},t)&=&h_\Delta^*(-{{\bf{K}}},t).
\end{eqnarray}
We define the susceptibility matrix as
\begin{equation}
\label{eq:19}
 \langle\hat{O}_{\alpha}({{\bf{K}}},\omega)\rangle = \chi_{\alpha\beta}^{0}({{\bf{K}}},\omega) h_{\beta}({{\bf{K}}},\omega).
\end{equation}
The derivation of this bare susceptibility matrix as well as the explicit expressions for
the elements of this matrix are given in Appendix B.

\begin{figure}
\includegraphics[width=2.8in]{cmqp.eps}
\caption{Illustrative example of the collective mode dispersion and quasiparticle-pair
continuum in 2D, for zero superflow (${{\bf{Q}}}=0$) with $U/t=3$ and $f=0.2$ fermions per site,
along the contour displayed in the inset.}
\label{fig:rpaexample}
\end{figure}

In order to account for interaction effects at the GRPA level, we note that the interaction
effectively induces internal fields which renormalize the applied external field so that the
susceptibility within the GRPA can be expressed as:
\begin{equation}
\label{eq:20}
\chi^{GRPA}_{\alpha\beta}=(1-U \chi^{0} D)^{-1}_{\alpha\nu}\chi^{0}_{\nu\beta}.
\end{equation}
Here, $D={\rm diag} \{ 2,1,1 \}$ is a diagonal
matrix that encodes the decoupling of the interaction Hamiltonian. The derivation
of this expression in explained in Appendix B.
This GRPA susceptibility will diverge when the determinant ${\rm Det}(1-U\chi^0 D)$
becomes zero (or equivalently, one of the eigenvalues of this matrix vanishes). We
identify the locus of real frequencies, $\omega\equiv \omega({{\bf{K}}})$, where this happens, as
the dispersion of a sharp (undamped) collective mode.

Fig.~\ref{fig:rpaexample} provides an illustrative example of the collective mode spectrum
obtained using the GRPA in 2D with $U/t=3$ and a filling $f=0.2$ fermions per site in
the absence of superflow ($Q=0$). We find a linearly dispersing superfluid
``phonon'' mode
at small momenta and low energy. The slope of this linear dispersion is the sound velocity.
The collective mode disperses as a function of momentum
over the Brillouin zone and exhibits an extremum at the edges of the Brillouin zone. In
the example here, it is a maximum at ${{\bf{K}}}=(\pi,\pi)$, but with increasing interaction strength
and at fillings closer to $f=1$, the spectrum exhibits a minimum at ${{\bf{K}}}=(\pi,\pi)$ arising
from strong short distance density correlations.
At high energies, there is an onset of a two-quasiparticle continuum
where the collective mode can decay by creating two Bogoliubov quasiparticles with
opposite spins in a manner which conserves energy and momentum.
Once the collective mode energy
goes above the lower edge of the two-particle continuum of Bogoliubov QP excitations,
it will cease to be a sharp excitation and acquire a finite lifetime. The collective mode energy as well as the
pair continuum change in the presence of superflow as discussed subsequently.

\subsection{Strong Coupling Limit: Spin Wave Analysis of Pseudospin Model}

In the pseudospin model, a state with
nonzero supercurrent is obtained by imposing a phase twist on the non-flowing
ground state $\vert 0\rangle $ as
\begin{equation}
\label{eq:21}
\vert {{\bf{Q}}} \rangle = \exp\left[-i \sum_i T^z_i {{\bf{Q}}} \cdot  {{\bf{r}}}_i\right]\vert 0\rangle.
\end{equation}
Equivalently, we can make a unitary transformation to work with the Hamiltonian
\begin{eqnarray}
\label{eq:22}
\nonumber H_{\rm eff} ({{\bf{Q}}}) &=& J \sum_{\langle ij \rangle}[T_{i}^{z} T_{j}^{z} +
(T_i^x T_j^x +T_i^y T_j^y)\cos({{\bf{Q}}} \cdot {{\bf{r}}}_{ij}) \\
&-& (T_i^x T_j^y \!-\! T_i^y T_j^x) \sin({{\bf{Q}}} \cdot {{\bf{r}}}_{ij})]
 - \mu \sum_{i} T^z_i, \nonumber \\
\end{eqnarray}
where ${{\bf{r}}}_{ij} \equiv {{\bf{r}}}_i - {{\bf{r}}}_j$. This amounts
to transforming to a reference frame where the superfluid is at rest.

The classical ground state $|0\rangle_c$,
which is
the ground state obtained under the assumption that the pseudospins are classical vectors,
is equivalent to the mean-field ground state of the interacting fermion model and
can be parametrized by specifying the classical
vector ${\bf T}^c_i$ at each lattice site. This is given by
\begin{equation}
\label{eq:23}
{\bf T}^c_i \equiv S (\eta_i \sin\theta, 0, \cos\theta),
\end{equation}
where the pseudospin magnitude $S=1/2$.
The angle $\theta$ is related to the filling, $f$, defined earlier, as
\begin{equation}
\label{eq:24}
f -1 = \cos\theta.
\end{equation}
Working at fixed filling, the chemical potential $\mu$ is given by
\begin{equation}
\label{eq:25}
\mu = 2 J S \cos(\theta) (\epsilon_0 + \epsilon_{{\bf{Q}}}),
\end{equation}
where $\epsilon_{{\bf{Q}}} \equiv \sum_{i=1}^{d} \cos(Q_{i})$ as before.

\begin{figure}
\includegraphics[width=2.8in]{HPRPA.eps}
\caption{(Color online) Collective mode energy at zero superflow
in 2D at strong coupling, $U/t=15$. The dispersion is shown along the indicated
contour in the Brillouin zone for different fillings: (a) $f=0.8$ fermions
per site and (b) $f=1.0$ per site. The GRPA result (solid line) is in good agreement
with the Holstein-Primakoff spin wave result (dashed line, HP) for the strong coupling pseudospin
model. The roton minimum has a small gap at $f=0.8$ but becomes a gapless mode at
$f=1.0$ due to the pseudospin SU(2) symmetry discussed in the text.}
\label{fig:hprpa}
\end{figure}

We compute the excitations about the ground state $\vert 0\rangle_c$ for the Hamiltonian
$H_{\rm eff}({{\bf{Q}}})$ using a Holstein-Primakoff (HP)
approach~\cite{auerbach} by rotating to new pseudospin operators $\tilde{T}$ given by
\begin{eqnarray}
\label{eq:26}
 \nonumber \tilde{T}_i^z &=& T_i^z \cos(\theta) + \eta_i T_i^x \sin(\theta), \\
 \nonumber \tilde{T}_i^x &=& -T_i^z \sin(\theta) + \eta_i T_i^x \cos(\theta), \\
\tilde{T}_i^y &=& \eta_i T_i^y,
\end{eqnarray}
and expressing $\tilde{T}$ in terms of HP bosons.
Expanding the Hamiltonian to ${\cal O}(S)$, we find
\begin{eqnarray}
\label{eq:27}
H&=& E_{\rm c} + \delta E_{\rm q}
+ \sum_{{{\bf{K}}}} \omega_{{\bf{K}}} ({{\bf{Q}}}) b^\dagger_{{\bf{K}}} b^{\phantom\dagger}_{{\bf{K}}},
\end{eqnarray}
where
\begin{eqnarray}
\label{eq:28}
E_{\rm c} \!\!&\!\!=\!\!&\!-NJS^2 [\cos^2\theta\epsilon_{0} +(1+\cos^2\theta)\epsilon_{{{\bf{Q}}}} ], \nonumber \\
\delta E_{\rm q} \!\!&\!\!=\!\!&\!\frac{JS}{2}\!\sum_{{{\bf{K}}}} [\epsilon_{{{\bf{K}}}}\! \sin^2\!\theta \!-\!
\frac{
(1\!\!+\!\!\cos^2\!\theta)}{2}\!
(\epsilon_{{{\bf{K}}}\!+\!{{\bf{Q}}}}\!\!+\!\!\epsilon_{{{\bf{K}}}\!-\!{{\bf{Q}}}})],
\end{eqnarray}
represent, respectively, the classical ground state energy and the leading quantum correction
to the ground state energy.
The spin-wave dispersion $\omega_{{\bf{K}}} ({{\bf{Q}}})$ is given by
\begin{eqnarray}
\label{eq:29}
\omega_{{\bf{K}}} ({{\bf{Q}}})&=& 2JS (\beta_{{\bf{K}}} ({{\bf{Q}}})+\sqrt{\alpha_{{\bf{K}}}^2 ({{\bf{Q}}})-\gamma_{{\bf{K}}}^2 ({{\bf{Q}}})}),
\end{eqnarray}
with
\begin{eqnarray}
\label{eq:30}
\alpha_{{\bf{K}}} ({{\bf{Q}}})\!\!&=&\!\! \!\epsilon_{{\bf{Q}}}\!+\!\frac{\sin^2 \theta}{2}\epsilon_{{\bf{K}}}\!-\!\frac{(1\!+\!\cos^2 \theta)}{2}
\!\left(\!\!\frac{\epsilon_{{{\bf{K}}} + {{\bf{Q}}}}\!+\!\epsilon_{{{\bf{K}}} - {{\bf{Q}}}}}{2}\!\!\right)\!, \nonumber  \\
\beta_{{\bf{K}}} ({{\bf{Q}}})&=&\frac{1}{2} \cos\theta\left(\epsilon_{{{\bf{K}}} - {{\bf{Q}}}} - \epsilon_{{{\bf{K}}} + {{\bf{Q}}}}\right), \nonumber \\
\gamma_{{\bf{K}}} ({{\bf{Q}}})&=&\frac{1}{4} \sin^2\theta \left(2 \epsilon_{{\bf{K}}}  +
\epsilon_{{{\bf{K}}} + {{\bf{Q}}}} + \epsilon_{{{\bf{K}}} - {{\bf{Q}}}} \right).
\end{eqnarray}

An illustration of the collective mode dispersions obtained using this approach is
shown in Fig.~\ref{fig:hprpa} at a strong coupling value of the interaction $U/t=15$
for two different fillings, $f=0.8,1.0$ fermions per site.
We find that these dispersions are in good agreement with those obtained using the GRPA
which serves to show that the GRPA also correctly captures aspects of the strong coupling
limit. As seen from the dispersion, there is a linearly dispersing superfluid
``phonon'' mode
at small $K$. In addition, for $f=0.8$, we find a low energy
``roton minimum'' at $K=(\pi,\pi)$ which arises from strong local pair density correlations in the
superfluid ground state. For $f=1.0$, the gapless roton mode at $K=(\pi,\pi)$ is a manifestation
of the pseudospin SU(2) degeneracy, discussed earlier, between the crystalline
CDW ground state and the uniform superfluid ground state.

\section{Collective Modes in the Stationary Superfluid}

\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{sv.eps}
\caption{(Color online) The sound mode velocity, $v_s$, as a function of the fermion filling $f$
in (a) 2D and (b) 3D for $U/t=3$ (circles) and $U/t=15$ (diamonds). Solid line is a guide to the eye.
The dashed lines are the weak coupling result,
$v_s = (v_F /\sqrt{d})[1-UN(0)]^{1/2}$, from Ref.[\onlinecite{Randeria94}] for $U/t=3$, with
$N(0)$ being the non-interacting density of states (per spin) at the Fermi level.
The dotted line indicates the Holstein Primakoff spin-wave result for $U/t=15$.
The inset to (b) shows the expected $t/U$ scaling of $v_s/t$ for $U/t \gg 1$ in 3D.}
\label{fig:vsound}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{rot2d3d.eps}
\caption{(Color online) The energy of the collective mode at (a) $(\pi,\pi)$ in 2D and (b) $(\pi,\pi,\pi)$ in
3D
for different interaction strengths. The dashed (solid) lines indicate
that the mode energy corresponds to a local maximum (minimum) of the dispersion.
The inset shows the collective mode energy in 3D at $(\pi,\pi,\pi)$ at a filling of $f=0.8$ fermions per site,
as a function of $t/U$. Inset shows a comparison of the GRPA result (crosses)
with the strong coupling spin-wave theory
result (dashed line).}
\label{fig:Eroton}
\end{center}
\end{figure}

We have seen that the collective mode spectrum obtained using the GRPA or
the strong coupling analysis has a superfluid phonon mode at small momenta
and a roton minimum at ${{\bf{K}}}=(\pi,\pi)$ in 2D or at ${{\bf{K}}}=(\pi,\pi,\pi)$ in 3D. The
phonon mode is a reflection of the fact that the superfluid ground state breaks
a global U(1) symmetry -- it is thus the long wavelength Goldstone mode
associated with this symmetry breaking.
The roton mode arises from strong local density correlations,
analogous to the roton mode in $^4$He. However, unlike in $^4$He, the rotons
only exist at this commensurate wave vector on the lattice
and do not form an incommensurate
ring of wave vectors. Further, the roton mode becomes gapless at $\mu=0$ and
this gaplessness arises from a pseudospin SU(2) symmetry which leads to a
degeneracy between the superfluid and CDW ground states as
discussed earlier.

\subsection{Sound Velocity}
Fig.~\ref{fig:vsound} shows the sound mode velocity $v_s$, extracted using the GRPA,
over a range of densities and interaction
strengths in 2D and 3D. In the limit of low filling and weak interaction,
our calculations of $v_s$ agree with the results of Belkhir and Randeria,~\cite{Randeria94}
although strong finite size effects and a small pairing gap limit our numerical
exploration of the very weak coupling regime $U/t\ll 1$.
At strong coupling, $U/t \gg 1$, our results from the GRPA are in good agreement with the
spin wave result $v_s= (4t^2/U) \sqrt{d} \sqrt{2f-f^2}$. The linear scaling of $v_s/t$
with $t/U$, expected for $U/t \gg 1$, is illustrated in the inset of Fig.~\ref{fig:vsound}(b).

\subsection{Roton Gap}
Fig.~\ref{fig:Eroton} shows the energy of the collective mode at $K=(\pi,\pi)$ in 2D and $K=(\pi,\pi,\pi)$ in 3D.
For low fillings and weak to intermediate coupling strengths, the
mode energy shows a maximum at this momentum. For strong enough interactions, or for
fillings near $f=1.0$, however, the mode energy exhibits a local minimum and can
be justifiably identified as a roton mode. The roton gap clearly scales as $t/U$ for
$U/t \gg 1$ as seen from the inset of Fig.~\ref{fig:Eroton}b. Further, the roton energy
goes to zero linearly as $f\to 1$. This is due to the chemical potential $\mu$, which tunes the
filling $f$ away from half-filling, and explicitly breaks the pseudospin SU(2) symmetry
present at $f=1$.

\section{Superflow Instabilities}

Superflow instabilities are easily understood in the context of bosonic superfluids possessing Galilean invariance. Superflow can then be simply thought of as a Galilean boost performed on a stationary SF. The excitations in the flowing frame are the same as that of the stationary SF, except their energies undergo a Doppler shift. At a certain critical flow velocity, the Doppler shift renders the excitations gapless at some momentum. These gapless excitations are then populated, flow energy is transferred to these bosonic excitations, and superfluidity is lost. The critical velocity is given by $v_{\rm crit} = {\rm min}(\omega_q/q)$, where $\omega_q$ is the energy of an excitation of the stationary SF at momentum q.

Our system, a paired fermionic superfluid on a lattice,
is much richer. The excitations are fermionic Bogoliubov quasiparticles and a bosonic collective mode,
corresponding to coupled
quantum-mechanical fluctuations of the superfluid order parameter and the density.
The effect of flow is now more than simply introducing a Doppler shift in excitation energies since the system explicitly breaks Galilean invariance.
The SF order parameter and the quasiparticle dispersion of the mean field theory are strongly renormalized. In addition, the collective mode dispersion also changes strongly. This is easily seen in the strong coupling
pseudospin model, where the dispersion of spin wave collective modes is given by Eq.(\ref{eq:29}).
The term $\beta_{{\bf{K}}} ({{\bf{Q}}})$ in this equation is the Doppler shift and clearly depends on the flow momentum.
However, the terms $\alpha_{{\bf{K}}} ({{\bf{Q}}})$ and $\gamma_{{\bf{K}}} ({{\bf{Q}}})$ are strong functions of the flow momentum too. Further, on the lattice as opposed to the continuum,
the Doppler shift vanishes at momentum points corresponding to the Brillouin zone edges. The
only effect of superflow in this case is to cause a strong dispersion renormalization via
its effect, at strong coupling, on $\alpha_{{\bf{K}}}({{\bf{Q}}}),\gamma_{{\bf{K}}}({{\bf{Q}}})$.

Due to these effects of imposed superflow, our system undergoes three broad kinds of instabilities, which we call ``depairing", ``Landau" and ``dynamical".
Below, we describe each of these and map out a stability ``phase diagram" as a function of dimensionality, interaction strength and density, indicating the critical flow momentum beyond which the uniform
flowing superfluid is unstable. Here we will restrict our attention to superflow momenta
${{\bf{Q}}}=Q_x\hat{x}$.

\subsection{Depairing Instability}
The system undergoes a depairing instability when the self-consistently calculated SF order parameter, $\Delta$, vanishes in the mean field theory of the flowing SF. At the critical flow momentum, the energy cost of  flow outweighs the condensation energy gain, and the system goes into an unpaired
normal state.

This instability is close to, but not identical with, the quasiparticles becoming gapless due to the flow-induced Doppler shift. In 2D, we find that these two phenomena occur at the same value of the flow momentum.
In 3D however, superfluidity persists beyond the point where quasiparticles become gapless. There is a small window of {\em gapless superfluidity},
where negative energy Bogoliubov quasiparticle states are occuppied. This is analogous to what has been claimed to occur, for instance, in superfluid
$^3$He in the presence of superflow.~\cite{volovik}
In this case, although there are negative energy quasiparticle excitations, the system cannot
arbitrarily lower its energy by occupying such states since the Bogoliubov excitations are
fermionic. The gapless superfluid thus survives as a stable intermediate phase in 3D.~\cite{Wei09}

\subsection{Landau Instability}

A Landau instability occurs when the collective mode energy hits zero and becomes negative, as shown in Fig.~\ref{Landau}. In the strong coupling limit, with the collective mode dispersion given
in Eq.(29), this case is described by $\alpha_{{\bf{K}}}({{\bf{Q}}}) \geq \gamma_{{\bf{K}}}({{\bf{Q}}})$ and $\beta_{{\bf{K}}}({{\bf{Q}}}) <
-\sqrt{\alpha_{{\bf{K}}} ^2({{\bf{Q}}}) + \gamma_{{\bf{K}}}^2 ({{\bf{Q}}})}$. In the GRPA approach, we find that the renormalized susceptibility, $\chi^{GRPA}$, diverges for two real negative frequencies. These frequencies match smoothly onto the results of the spin wave frequencies at strong coupling.

As discussed in Ref.[\onlinecite{Burkov08}], Landau instability is not an instability of linearized dynamics of small fluctuations around the
uniformly flowing state, since some form of mode coupling
is necessary to transfer the energy of the superflow into these negative-energy modes.  Its full theoretical description is thus rather complicated.
Here we restrict ourselves to only finding the critical flow momentum for this instability.

\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{Landau.eps}
\caption{Landau instability - Collective mode spectrum (from GRPA) in 2D with $U/t=7$ at a filling of
$f=0.4$ fermions per site. Collective mode dispersion is
shown at (a) zero flow, (b) just below, and (c) just above, the critical flow momentum for the
Landau instability. The $+ (-)$ sign indicates the mode is at wavevector $+{{\bf{K}}} (-{{\bf{K}}})$ which has an
$x$-component
along (opposite to) the flow direction so that it is Doppler shifted down (up) in energy.
The collective mode frequency becomes negative at an incommensurate wavevector.}
\label{Landau}
\end{center}
\end{figure}

We find that the Landau instability occurs either at small momenta, corresponding to the dispersion of the sound mode going below zero, or at some finite incommensurate momentum. In the case of low filling, unless preempted by depairing, we see a Landau instability of the sound mode. For moderate values of U and the filling, we see Landau instabilities at large incommensurate momenta (as in Fig.~\ref{Landau}). As the filling is reduced, this incommensurate
wavevector moves towards the Brillouin zone centre, so that in the low density limit, it is the long wavelength sound mode that becomes unstable. Similar incommensurate instabilities have also been found in Ref.[\onlinecite{danshita}].

\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{dyncomm.eps}
\caption{Dynamical Instability (Commensurate)- Collective mode spectrum (from GRPA) for $U/t=5$, filling $f=0.8$ fermions per site on a 2D square lattice.
The $+ (-)$ sign indicates the mode is at wavevector $+{{\bf{K}}} (-{{\bf{K}}})$ which has an
$x$-component
along (opposite to) the flow direction.
As the flow momentum Q is increased, the collective mode frequency at ${{\bf{K}}}=(\pi,\pi)$ decreases until it hits zero and becomes complex.
This gives rise to a dynamical instability associated with the ``checkerboard"
CDW order. The part of the dispersion, around $(\pi,\pi)$, which corresponds to unstable modes is not shown.}
\label{dyncomm}
\end{center}
\end{figure}

\subsection{Dynamical Instability}

The underlying lattice potential allows for a new possibility for the
breakdown of superflow, namely through dynamical instabilities.~\cite{Niu01}
This corresponds to the eigenfrequency of the collective mode dispersion in our system becoming zero, and subsequently becoming complex.

The concept of dynamical instability is particularly simple to understand for weakly interacting bosons.
In this case, the instability coincides with the point where
the effective mass of the bosons changes sign as a function of the superflow momentum,~\cite{Niu01} leading to
runaway growth of phase and density fluctuations which eventually
destroy superfluidity.

For strongly interacting bosons the situation is more interesting.
It has been shown~\cite{Altman05} that with increasing interaction strength at a commensurate
filling (integer number of bosons per site), the dynamical instability occurs at a smaller
and smaller superflow momentum. The critical flow momentum eventually tends to zero at the equilibrium superfluid to
Mott insulator transition, scaling as the inverse of the divergent correlation length
associated with this
Mott transition.

Here, in the fermionic analog of the problem of the critical superflow in a strongly correlated superfluid,
we find, in addition,
an entirely different kind of dynamical instability, associated with emergent density-wave
order for a large range of fillings, wherein the mode energies become complex at nonzero wavevectors.
In the strong coupling limit, this happens when $|\alpha_{{\bf{K}}} ({{\bf{Q}}})|$ becomes smaller than $|\gamma_{{\bf{K}}} ({{\bf{Q}}})|$. In the GRPA approach, at a dynamical instability, we find no real frequency at which any eigenvalue
of the inverse GRPA susceptibility vanishes.
The GRPA results for the critical flow momentum ${{\bf{Q}}}$, and the wavevector ${{\bf{K}}}$
at which the dynamical instability occurs, match smoothly onto the spin wave results at strong coupling.
In this limit it is easy to check analytically that
the dynamical instability associated with this emergent charge order is not preempted by a Landau instability.
The appearance of complex collective mode frequencies indicates an
exponential growth of density fluctuations around the uniformly flowing state, at a wavevector ${{\bf{K}}}$.

We find two kinds of such dynamical instabilities --- associated with commensurate (Fig. \ref{dyncomm}) or
incommensurate (Fig. \ref{dynincomm}) charge order.
We see the former for a large range of fillings in the vicinity of  $f=1$, for all interaction strengths. The wavevector
at which this instability happens is at the  Brillouin zone
corner - $(\pi,\pi)$ in 2D and $(\pi,\pi,\pi)$ in 3D. This corresponds to an instability towards a ``checkerboard" CDW state,
with a density modulation of opposite sign on the two sublattices of the square or cubic lattice.
The incommensurate dynamical instability, on the other hand, occurs at wavevectors corresponding to various incommensurate ordering patterns and it arises as follows.
In the presence of superfow, the dispersion of the Bogoliubov quasiparticles is renormalized and Doppler
shifted, giving rise to multiple minima separated by a nonzero wavevector.
This leads to a peak in the
bare mean field susceptibility, $\chi^0$ at this momentum.
The interaction renormalizes this peak into a divergence and leads to a finite momentum pairing instability
of Bogoliubov quasiparticles.
This dynamical incommensurate instability is thus a non-trivial, interaction-driven phenomenon, which only
occurs at intermediate coupling strength.
This phenomenon is somewhat analogous to the formation of excitonic condensates in indirect band-gap
semiconductors.~\cite{Halperin1968}

\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{dynincomm.eps}\\
\caption{Dynamical Instability (Incommensurate) - Collective mode spectrum (from GRPA) for $U/t=3$, filling $f=0.6$ fermions per site on a 2D square lattice.
The $+ (-)$ sign indicates the mode is at wavevector $+{{\bf{K}}} (-{{\bf{K}}})$ which has an
$x$-component
along (opposite to) the flow direction.
As the flow momentum $Q_x$ is increased beyond $0.2\pi$, the collective mode frequency becomes complex at an incommensurate wavevector. The complex frequencies in the unstable region are not shown in the figure.}
\label{dynincomm}
\end{center}
\end{figure}

\subsection{Stability phase diagrams}
Taking these instabilities into account, we map out superflow stability phase diagrams in 2D (Fig. \ref{2dpd}) and 3D (Fig. \ref{3dpd}).
These plots show the first instability that is encountered as imposed flow is increased, for different values of filling. Values of $U$ for the plots
have been chosen so as to illustrate all possibilities. We see that the commensurate dynamical ``checkerboard" CDW  instability comes into play around
$f=1/2$ for all values of the interaction strength and is the dominant instability all the way to $f=1$, where the critical flow momentum vanishes,
reflecting the degeneracy between the SF and CDW states.

In the low density limit, the system is similar to a continuum Fermi gas. In 3D, the density of states at the Fermi level vanishes.
At low interaction strength, this leads to the pairing gap $\Delta$ being exponentially suppressed. The sound velocity, on the other hand, is proportional to the Fermi velocity, so that
$v_s \sim f^{1/3}$. This leads to a rather sharp drop of the sound velocity as $f \to 0$ but the
gap drops to zero much faster.
Therefore, in the 3D case, at weak interaction and low filling, a small imposed flow will drive $\Delta$ to zero before the flow velocity
exceeds the sound velocity, leading to a depairing instability as can be be seen in Fig. \ref{3dpd}.

\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{2dpd.eps}
\caption{(Color online) Stability phase diagram for a 2D square lattice case with (a) $U/t=3$ and (b) $U/t=5$. For every filling, the plot shows the first instability that is encountered as the flow is increased. The solid (blue) line in the low density limit in (a) indicates the region where we expect to see a Landau instability, but finite size effects prevent us from accessing the area.
The other transitions in the figure correspond to Depairing (circles, dotted line), Incommensurate Dynamical Instability
(diamonds, solid line), Commensurate Dynamical Instability (squares, dashed line), Landau Instability (triangles,
dash-dotted line). At very large interaction, the diagram looks similar to $U/t=5$, except that the incommensurate
dynamical instabilities disappear.}
\label{2dpd}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=2.8in]{3dpd.eps}
\caption{(Color online)
Stability phase diagram for a 3D cubic lattice with (a) $U/t=5$ and (b)$U/t=15$. For each filling, the instability first encountered as the flow is increased is shown.
The transitions in the figure correspond to Depairing (circles, dotted line), Incommensurate Dynamical Instability
(diamonds, solid line), Commensurate Dynamical Instability (squares, dashed line), Landau Instability (triangles,
dash-dotted line).}
\label{3dpd}
\end{center}
\end{figure}

In contrast, in the low density continuum limit in 2D,
the density of states goes to a constant, which means the pairing gap stays finite
as $f \rightarrow 0$. With imposed flow, the first instability that one encounters is then the Landau instability, which will
happen when the flow velocity exceeds the sound velocity which scales as $v_s \sim f^{1/2}$.
We have not been able to numerically uncover the Landau instability in this regime
due to severe finite size effects.

At intermediate densities, and at intermediate values of the interaction strength, the leading instability is the incommensurate dynamical
instability, corresponding to the emergence of an incommensurate CDW, characterized by a wavevector which depends on the
density and the interaction strength.
This instability does not happen in either the strong-coupling limit, where the Anderson pseudospin description is appropriate,~\cite{Burkov08}
nor in the weak-coupling BCS superfluid limit, where the instability at intermediate densities is due to depairing.
This dynamical instability is a nontrivial intermediate-coupling phenomenon and is one of the most interesting results reported
in this work.

An interesting question that arises in relation to the emergent CDW-driven dynamical instabilities, is the fate of the system
beyond the critical flow momentum.
We show, in Appendix C, that beyond the commensurate CDW dynamical
instability threshold the system behaves as a ``flowing supersolid'' at the mean field level. However, we argue
that fluctuations beyond
mean field theory are expected to destroy this flowing supersolid. An explicit example
of this,
in the strong coupling limit, has been discussed by us earlier.~\cite{Burkov08}
We expect the same to be true in the case of the incommensurate CDW dynamical instability.

\section{Experimental Implications}

Assuming that future experiments will be able to produce attractive cold Fermi gases in the lowest band of an
optical lattice, it is reasonable to expect that the ground state and low energy excitations of such a system
will be well described by the attractive Hubbard model.\cite{lehur09,levin}
Our results on the collective mode excitations of the Hubbard model, specifically the roton gap and the
sound velocity as a function of filling and interaction strength, could then be verified experimentally.

Turning to superflow instabilities,  the theoretically predicted Landau and dynamical instabilities of Bose
superfluids \cite{Niu01,Altman05,nikuni06}
have been experimentally observed using cold bosonic atoms in an optical lattice.~\cite{ketterle-bose07}
As shown there, a flow in a cold atom system can be
induced by frequency-detuning one pair of the counterpropagating laser beams that form the lattice. This creates a ``running" optical lattice
potential in the corresponding basis direction, i.e. the lattice is effectively moving with respect to the stationary potential of the trap, holding
the atomic cloud.
The effect of this on the bosons is easy to understand if one goes to the reference frame of the moving optical lattice.
In this frame, the bosons must condense in a state of non-zero lattice momentum. If the optical lattice is
moved sufficiently slowly, we expect the initial zero-momentum SF ground state of the bosons to adiabatically
acquire a phase gradient, thus increasing the superflow momentum until dissipation sets in and destroys the
condensate upon crossing a Landau or dynamical instability.

The instabilities of superflow for fermions loaded in an optical lattice may similarly be tested in cold atom
experiments. This has been clearly demonstrated
in a pioneering experiment of Ref.~[\onlinecite{Ketterle07}] which studied the superfluid phase of fermions
in a harmonic trap and showed that the critical velocity smoothly interpolates between the pair breaking
velocity in the BCS regime to the Landau critical velocity in the BEC regime.~\cite{sensarma08} In a deep optical lattice, provided
that the superflow does not lead to excitations into the higher bands,
the superfluid phase
should be reasonably well described by the Hubbard model in the lowest
band. In
this case, one may be able to probe the various dynamical phase diagrams proposed in this paper. Even
before reaching the instability, one may be able to use Bragg spectroscopy \cite{aspect03} to probe the
collective density fluctuation spectrum, and observe the roton softening with increasing lattice velocity
which would enable one to test the mechanism by which the superflow breaks down.

Our results are also of relevance for the vortex-core physics in strongly-correlated lattice superfluids.
Considering the vortex core as simply the region where the current has exceeded the  value beyond which
the uniform superfluid phase is no longer stable, we would expect the dynamical instabilities discussed in
this paper to lead to commensurate or incommensurate density orders in the vortex core.

\section{Conclusions}
In conclusion, we have studied the SF ground state of the negative-U Hubbard model on a square and cubic lattice.
Using GRPA, we have studied the collective mode spectrum at all values of the interaction strength and density.
We have shown that strong density correlations at increasing filling lead to the appearance of roton features in the
collective mode spectrum, which signal developing competition between uniform SF and density-ordered CDW states.
When superflow is imposed on the system, depairing and Landau instabilities arise. In addition, this competition
leads to dynamical instabilities corresponding to
roton mode softening. This dynamical instability occurs at a commensurate wavevector
corresponding to the corner of the Brillouin zone. This instability is easily described in the strong coupling pseudospin
language and is a consequence of the degeneracy between SF and
``checkerboard" CDW ground states at $f=1$.
Another very interesting dynamical instability occurs at an incommensurate wavevector and has no strong coupling analog.
It occurs over a range of densities near $f=1/2$ and at intermediate values of the interaction strength.
Our results may be verified in experiments with ultracold fermions loaded on an optical lattice.
Interesting avenues to explore in the future would include extending this work to other lattice
geometries~\cite{zhao06} and to study superflow instabilities in multiband superfluids.~\cite{burkov09}

\acknowledgments
This research was supported by NSERC of Canada. AP acknowledges support from the Sloan Foundation,
the Ontario ERA, and the Connaught Foundation. We thank the authors of Ref.[\onlinecite{danshita}] for
pointing out an error in Fig.~7 in a previous version.

redpajama_set_name: RedPajamaArXiv


MILLER, Jack Julius - Died at his home in Woodside on May 28, 2002 surrounded by family. He was born in San Francisco on January 1, 1915. His parents Fannie and Jack Miller lived in Stockton where he went to high school and was student body president. He graduated from the University of Oregon and the George Washington Law School. At the age of 22 he founded the law firm of Miller and Kroloff in Stockton. Later he moved to San Francisco and opened the firm of Jack Miller, Richard Katz and Robert Harlem. Jack was a bon vivant, an avid skier and enjoyed travelling with his wife Helen to Europe and Sun Valley, Idaho where he spent many years hiking, riding and skiing. He is survived by his loving wife of 39 years, Helen McCulloch; his stepdaughter Diana Citret, son-in-law Mark Citret, grandsons Toby and Cole Citret of Daly City, and stepdaughter Pamela Crane of Woodside. Jack's sisters Sylvia Cooper and Geri Berkman live in Stockton. A private memorial is planned. Contributions in his memory may be made to Drs. Without Borders or to Pathways Hospice.


text:
Home / Alphabet / google / Jeremy Phillips / Alphabet: Google spells out its reasons -- but does this also spell trouble?
Alphabet: Google spells out its reasons -- but does this also spell trouble?
Jeremy Saturday, August 15, 2015 - Alphabet, google, Jeremy Phillips
The news that online search-and-services giant Google was cultivating an alter ego for itself by opting for the name Alphabet for restructured holding company received less attention than one might have expected, probably because (i) it's August and so many people are on holiday (or just pretending) and (ii) Google have sprung so many surprises on us in recent years that we have lost our ability to find anything Google does newsworthy any more. Be that as it may, this Kat thanks Katfriend and one-time guest Kat Miri Frankel for penning the following note:
Google is no stranger to IP law. In the US alone, it holds
· 78 registered trade marks for Google word and style marks
· More than 400 registered trade marks for word and style marks across all of its properties (e.g. Nest, YouTube, Songza)
· More than 1,500 registered patents
· More than 2,600 patent applications
In addition, the company has conducted consumer trade mark education to prevent genericide, reminding people that they are "searching the web on Google," not "googling." Might this have been in Google's mind when it chose to restructure its business by creating a new parent holding company with the non-too-distinctive name Alphabet? Many companies already trade under the name Alphabet or something similar to it, including a subsidiary of BMW, which owns the URL www.alphabet.com (Google's Alphabet will use the URL www.abc.xyz). As reported here:
"The name isn't just causing waves with BMW. On Wall Street, there is an Alphabet Funds. Lots of midsize and small companies also use the name Alphabet. There is an Alphabet Energy in Hayward, Calif.; an Alphabet Record Company in Austin, Tex.; an Alphabet Plumbing in Prescott, Ariz.; and numerous preschools, inns and restaurants with some variation of the name."
While some have joked that perhaps Google forgot to google search on Google for the name Alphabet before selecting the name, this guest Kat does not doubt that Google was aware of the many pre-existing Alphabet businesses, including BMW's, but chose to proceed nonetheless.Larry Page explains why they chose this name in his letter on Alphabet's website:
"We liked the name Alphabet because it means a collection of letters that represent language, one of humanity's most important innovations, and is the core of how we index with Google search! We also like that it means alpha‑bet (Alpha is investment return above benchmark), which we strive for!"
Google's new corporate HQ?
Will Google (I mean Alphabet) be able to use this name without significant risk of trade mark disputes? Probably. While Alphabet is not generic or descriptive of the goods or services, nor is it hugely distinctive as a trade name. The sheer number of businesses using Alphabet as a trade name or as part of a trade name minimizes the risk of consumer confusion: consumers are used to seeing multiple Alphabets in various industries. In addition, Alphabet itself will not be providing commercial goods or services; its subsidiaries, including Google, Nest, and Calico, will be consumer brands housed under the holding company umbrella of Alphabet. As Larry Page explains,
"we are not intending for this to be a big consumer brand with related products—the whole point is that Alphabet companies should have independence and develop their own brands."
Subsidiary [x] may be developing a self-driving car using Google software, but BMW can rest assured that [x]'s self-driving car is unlikely to be marketed under the competing brand name of Alphabet.
Alphabet: Google spells out its reasons -- but does this also spell trouble? Reviewed by Jeremy on Saturday, August 15, 2015 Rating: 5
Unknown Saturday, 15 August 2015 at 23:47:00 GMT+1
The announcement of Alphabet reminds me of some incidents in the past:
1. When I was with the predecessor of Ladas & Parry, acting for Polaroid Corporation, its founder Edwin Land announced a new camera which he said he would call ALADDIN. We went crazy filing applications to register ALADDIN for cameras and film in every jurisdiction we could, as quickly as we could (using Telex and cable at that time), and then making searches and dealing with whatever problems were encountered. ALADDIN was never used, and the camera and film went under the code name used in the secret development phase - SX-70, which we registered everywhere. ALADDIN was never used.
2. The Coca-Cola Company surprised everyone with NEW COKE, with a change in their traditional flavor. It was an absolute disaster and the company quickly introduced COKE CLASSIC, which was the same juice as before, and that is what they sell today. They wished that they had heeded the maxim: "If it ain't broke, don't fix it."
3. United Airlines changed its name to a three letter word, possibly EOS, the goddess of the dawn. I did not work. As Rance Crain, of Crain' Chicago Business, said, they asked the chairman to parachute out of the cockpit of the company.
4. As a part of the outside worldwide IP counsel team for a major corporation, we were surprised when the company announced a name change to something entirely new, without asking any advice whatsoever.
Anonymous Sunday, 16 August 2015 at 01:00:00 GMT+1
Nothing spotted yet in the DPMA database.
But there is a prior trademark "ABC/XYZ" to one "PSI Metals Non Ferrous GmbH" in Aachen, which includes "Software" in Nice class 09.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to fix strange "Undefined offset"? (laravel) I have these codes below, it always return undefined offset error.
I get the data value using dd it has value in index 1.
array:3 [▼
  0 => "Title 1"
  1 => "Content1"
  2 => "Subcontent1"
]

But when I run error returned:

Undefined offset: 1

My codes
class BlogsImport implements ToModel
{
    use Importable;

    /**
    * @param array $row
    *
    * @return \Illuminate\Database\Eloquent\Model|null
    */
    public function model(array $row)
    {
        // dd($row);
        return new Blogs([
            'title' => $row[0],
            'content' => $row[1],
        ]);
    }
}

A: change your code this way
class BlogsImport implements ToModel
{
    use Importable;

    /**
     * @param array $row
     *
     * @return \Illuminate\Database\Eloquent\Model|null
     */
    public function model(array $row)
    {
        return new Blogs([
            'title' => $row[0] ?? '', // or $row[0] ?? 'default_value'
            'content' => $row[1] ?? '', // or $row[1] ?? 'default_value'
        ]);
    }
}

A: I figure it out by using dump instead of dd. It turns out that there are some empty cell.
thanks to @JustCarty for the hint.
public function model(array $row)
{
    if ($row[0]) {
        return new News([
            'title' => $row[0],
            'content' => $row[1],
        ]);
    }
}

redpajama_set_name: RedPajamaStackExchange


text:
Drucat este o comună în departamentul Somme, Franța. În 2009 avea o populație de 867 de locuitori.

Note

Vezi și
 Lista comunelor din Somme

Comune din Somme

redpajama_set_name: RedPajamaWikipedia


text:
YA Is A Useless Genre
I bet you came here ready to have a fight. But please, let me assure you that I have no beef with "YA" books. I just think it's a rating, not a genre.
YA stands for "Young Adult" and was initially used by bookstores and libraries to provide a space for readers who were no longer children looking for picture books or reading levels for single digit age groups, but who might not be ready to read just everything the bookstore might have to offer. The idea is that 10-14 year olds don't have to go to the children's section and embarrass themselves among their peers, but their parents can be more or less certain that graphic sex or violence will not be shoved under their kids' noses.
All well and good. This is a huge portion of the market, not just because so many people in this age group are spending their pocket money on books, but also because so many adults, too, through nostalgia and a similar yearning for books that are just fun and engrossing without being gross, really like to peruse books aimed at this group.
The issue is that through a series of events including mimicry, prolific authors, and reader expectations, people have started to identify this rating system with a genre. There are tropes that are quite common for this age group: coming of age stories, teens conquering the establishment, love triangles, and also writing styles we associate–like flowery or "purple" prose, more "surface level" or simplistic writing styles that are very easily consumed etc.
And this is where things start to go haywire.
Coming of age stories, like Midnight's Children by Salman Rushdie? Is that YA?
Teens conquering the establishment, like Red Sister by Mark Lawrence?
Love triangles, like Deathless by Cat Valente?
Purple prose like Fafhrd and the Grey Mouser by Fritz Leiber?
Simple writing like Sandman Slim by Richard Kadrey?
It is utter tomfoolery to say that things that are remotely similar to those tropes are YA simply because they include those elements. We can think of a name for a subgenre for writing in this style, but boy howdy would it be a mistake to give a lot of books that could qualify to tweens. Even books that check off all the boxes, like RF Kuang's award nominated Poppy War and Suzanne Collins' "Hunger Games" are not the sort of thing I'd give to a 12 year old unsupervised, though they are things I think make use of the emotive genre's writing style.
Simple writing isn't necessarily bad writing. People drool on themselves looking at Picassos even though I think we all know a few motivated kindergartners who can employ similar use of color and lines. Coming of age stories can be extremely relevant to full adults and not really suitable at all for people who are still in the midst of their transformation. The movie Closer has some of the most notable love triangles I've ever seen, but I think I'd have a heart attack if that was enough reason for it to be in the "Teen Drama" section of Netflix that my baby cousin might peruse, alongside My Hero Academia.
So what is it that makes these books feel like they belong to a genre other than some version of adult fantasy?
Like I mentioned earlier, there is a sort of genre-esque element in combinations of attributes, and I'd love for someone to name it. Purple prose fantasy or Emotive fantasy or something might capture it well enough without accidentally shocking the hell out of some 11 year olds (and let's be honest, more so their parents.) There is a sort of style to this, like there's a style including graphic body horror in grimdark books. There are rhythms, like we have with the rise and fall of the story in a quest fantasy. And it makes sense that we'd see those similarities and label a box for them.
I think a lot of this is perception, not just of the style of writing, but about the people who read stories like this, and I think a lot of that has to do with sexism, racism, and homophobia. We think of twitterpated girls reading Twilight, and the unsophisticated reader (who in our imagination might again be a woman) who just "doesn't know better" who might prefer Harry Potter to Lord of the Rings. But liking a certain type of story, or having a certain type of identity certainly isn't indicative of actual quality (or lack thereof), right? It would be asinine to say that things women and girls like are bad just because they're liked by women and girls, right? It would make no sense to condemn someone for liking 1980s cult classic movies more than JJ Abrams works, would it?
But that sure is the sentiment. That is the perception of people who can find enjoyment in things that are less literary, or books which might indulge a bit in more emotive descriptions. Unless, of course, the work is by a man for men and boys. And then it's just your run of the mill superhero or quest fantasy. Then the prose is beautiful, the love interests romantic, and any failings in quality hearken back to an earlier time which we apparently must just allow because "things were different." There seems to be a rather obvious gap between what is a classic, and what is for a child.
This might feel like a hard pill to swallow. But name me a man whose works are definitely considered YA who did not actively say that he was writing for children or teens. The magical realism and mythical based works from African and South American authors, how many of them are considered "serious" or "adult" fiction?
This is not to say that people who don't like these books, or who would give anything like this to their tween family and friends are bad people. And possibly it's just because we don't have a good subgenre name available for this sort of book, so we lump it with things that are similar. But there's a lot of bias against "YA" for adults, which makes it difficult for those books to win awards, become selected for book clubs, and generally to find audiences perhaps better suited to them than 10-14 year olds, which then condemns the author to needing a pen name, needing to actually write for that age group, thereby fulfilling the prophecy, or content themselves with mediocre careers lacking a solid audience.
And that's a bullshit choice.
I urge you all to help me think of an appropriate subgenre name that allows us to move away from shelving books about war atrocity or baser aspects of human interactions as "YA", and to think through what it is that you would actually feel confident giving to teens for their reading pleasure, versus a writing style you just don't personally enjoy. How does that change a book's classifications, if at all?
Author Rant, Writing
fantasy, genre fiction, homophobia, racism, sexism, summoner sisters, ya, young adult
← Impostor Syndrome Is A Con
Grateful That You're Great! →


text:
IT 150 - Using Microcomputer Productivity Tools
Description: Introduction to typical microcomputer-based productivity tools such as word processing, spreadsheets, databases, and presentation graphics. Lecture and lab. Not for credit IT major/minor, or if had BTE 125.
001 - Lecture: Online 6 week(s), Cheri Wolf 3 hour(s)
Building: Online, contact instructor
Instructor Name: Cheri Wolf
001 - Lecture: M W 10:00 AM - 10:50 AM 16 week(s), Cheri Wolf 3 hour(s)
Building: Center for the Visual Art 151 (CVA 151)
002 - Laboratory : M W 11:00 AM - 11:50 AM, 16 weeks, Instructor to be announced
003 - Laboratory : M W 12:00 PM - 12:50 PM, 16 weeks, Instructor to be announced
004 - Laboratory : M W 1:00 PM - 1:50 PM, 16 weeks, Instructor to be announced
009 - Laboratory : T Th 9:00 AM - 9:50 AM, 16 weeks, Instructor to be announced
010 - Laboratory : T Th 10:00 AM - 10:50 AM, 16 weeks, Instructor to be announced
Building: Old Union 132 (OU 132)
Instructor: Instructor to be announced
013 - Laboratory : T Th 12:00 PM - 12:50 PM, 16 weeks, Instructor to be announced
014 - Laboratory : T Th 1:00 PM - 1:50 PM, 16 weeks, Instructor to be announced
019 - Laboratory : W F 9:00 AM - 9:50 AM, 16 weeks, Instructor to be announced
020 - Laboratory : W F 10:00 AM - 10:50 AM, 16 weeks, Instructor to be announced
This course is tentatively planned for Summer 2020


text:
Q: Issue with react component and gatsby JS I am trying to create a portfolio website using Gatsby js and I am trying to use the card component from the gatsby-plugin-material-ui package. I have the card in a separate file in my components folder and I export it for use in my actual project page file. When I load up the website on localhost everything loads up fine except for when I hit the project button to navigate to the project page. Once I try to access the page I get this error
Error in function createFiberFromTypeAndProps in ./node_modules/react-dom/cjs/react-dom.development.js:25058

Element type is invalid: expected a string (for built-in components) or a class/function (for composite components) but got: object. You likely forgot to export your component from the file it's defined in, or you might have mixed up default and named imports. Check the render method of Cards.

Here is the code where I create and export the component
import * as React from "react"
import Card from 'gatsby-plugin-material-ui';
import CardContent from 'gatsby-plugin-material-ui';
import CardMedia from 'gatsby-plugin-material-ui';
import Typography from 'gatsby-plugin-material-ui';
import { CardActionArea } from 'gatsby-plugin-material-ui'

const Cards = () => {

    return (

            <Card sx={{ maxWidth: 345 }}>
                <CardActionArea>
                    <CardMedia
                        component="img"
                        height="140"
                        image="../images/space.jpg"
                        alt="Gagan Toronto"
                    />
                        <CardContent>
                            <Typography gutterBottom variant="h5" component="div">
                                Space Card
                            </Typography>
                            <Typography variant="body2" color="text.secondary">
                                This is just a placeholder card
                            </Typography>
                        </CardContent>
                </CardActionArea>
            </Card>

    )

}

export default Cards

and here is the code where I import the component to use for my project page
import * as React from "react"

//components used across site
import Header from "../components/header"
import Cards from "../components/Cards"

const ProjectPage = () => (

    <Cards />

  )

export default ProjectPage;

I feel like its a problem with how I am exporting the component but I am not sure what it is, any help would be greatly appreciated :)

A:
I feel like its a problem with how I am exporting the component but I
am not sure what it is, any help would be greatly appreciated :)

Indeed, it seems that some of your imports of Cards component are wrong. I assume that the one that is causing the issue is CardActionArea but it would be nice if you can detect which one is by commenting them all and importing one by one.
I'd try importing like:
import CardActionArea from 'gatsby-plugin-material-ui'


Personalize this apparel mockup featuring a young trendy woman wearing her favorite tee in a VW minivan just living the van life. This mockup is ready to use, all you have to do is place your image over the photo template and our mockup generator will have your asset in no time. Try more photorealistic mockups like this t-shirt mockup to make an impact on your clients! To see another T-Shirt Mockup click here! Now see this amazing Guide On Tees Mockups!


text:
def popen(command, mode='r', bufsize=-1):
    pass

class _wrap_close:
    def close(self):
        pass

redpajama_set_name: RedPajamaGithub


Hutt St Centre - a place of hope and opportunity, helping people facing homelessness to rebuild their lives, without judgement.
Catherine House - providing supported accommodation services to women experiencing homelessness in South Australia, who are 18 years and older, and unaccompanied by children.
MOSH - Minimisation of Suicide Harm offers psychosocial support via its two safe spaces where anyone is welcome on a drop in basis.
Baptist Care SA - actively supporting people experiencing homelessness and South Australia's most disadvantaged by supporting at-risk and vulnerable people to have access to the basics.
Art pieces will be displayed at various cafe's in the City South Precinct of Adelaide during Fringe where many community services are located.
Photograph of Scott sculpting Bushy from Hutt St Centre, being filmed by Harvey from Real Deal Productions. Bushy is sharing his story for film whilst being sculpted.
People don't want charity, they want opportunity.
We are all human, we all have a story and we all have value regardless of our situation in life.
This is exactly what we had in mind when a group of us came together to create The Art of Being Human for Adelaide Fringe 2019.
Through this platform we hope to bring opportunity and hope for those experiencing homelessness, to be seen, to be heard, to be valued and respected as members of our community, helping in the healing process and a pathway back to independence.
Serafina, Founder and Creator of The Art of Being Human and Co-founder of Teaspoons of Change, lives and works in the community having many conversations and connections with those experiencing homelessness. She sees daily the struggle that people go through, yet amidst the struggle is beauty and resilience, a desire to be acknowledged and understood, to not be judged but listened to and accepted for who they are.
Scott, Artist, Sculptor and Co-founder, Co-creator of The Art of Being Human. Scott see's healing through art, he has a unique ability to capture the essence of a person through sharing their story and portraying this is sculpture form, giving life to the persons journey, turning pain into art.
Elizabeth, Founder Real Deal Productions and Co-creator of The Art of Being Human, Elizabeth is a story teller through film, she professionally draws a person into a safe space, breaking down the barriers to reveal the person inside and the significance of their life story, giving inspiration and meaning to their journey, validating their place in the world.
Michael, Photographer and Co-creator, Michael captures the beauty of people through his lens and turns each person into a work of art, every line, every wrinkle, every smile is unique. Michael has the ability to make people feel relaxed and comfortable in front of the camera where they let go and it's in that moment Michael captures their art of being themselves. A true expression of The Art of Being Human.
We would love to see this exhibition be a great success in 2019 and to grow and be a feature at Adelaide Fringe every year, but we need your help to make it a success.
The brains and hands of The Art of Being Human have all volunteered our time and resources and would love others, who are able to contribute financially, to enable us have an positive impact on the people and organisations associated with this creative concept.
We would love your help to do this!
Donate to our crowdfunding campaign by selecting a reward from the list on the right.
To pay the people who are sitting for us to be sculpted and share their stories for the film.
To reimburse some of the costs of materials for the sculptures.
To print some of the incredible photographs of the artists and the people being sculpted.
To cover additional costs involved in creating the sculptures, film and photographs plus some nibbles and wine at our screening.
If we raise more than what is required to cover our costs the remainder will be donated to the 4 charities involved.
Have your Portrait Bust Sculpted by Sculptor Scott Eames. Scott will sculpt you (or a friend/colleague/family member) during a live sitting, it will then be cast for you to keep.
Michael Errey Photography will capture this experience for you on camera with photo's for you to keep.


Writing a resume that gets results is critical in landing job interviews. And you have to know what to include as well as exclude in order to make your resume great. So here are 11 things to avoid when writing your resume.
1. Don't make your resume unnecessarily long.
Keep you reader interested in five- to 10-second increments by showcasing your experience in as little space as possible (without sacrificing your achievements). For most job seekers with more than 10 years of experience, a two-page resume is fine. Chances are, whatever is on subsequent pages, may not get noticed if the reader is not engaged.
2. Don't leave it to the reader to figure out what you do.
Begin your resume with a targeted title and summary punctuating what you do and how you do it. Make it very easy for the reader to discern what you do with no effort on their part.
3. Don't forget to make your resume searchable.
Include key search terms in your summary to (a) make your resume searchable and discoverable by hiring managers seeking out your expertise, and (b) help reiterate your competencies.
Use measurable achievements, focused language, and deliberate phrasing throughout your resume to highlight your experience. Trying to be all things to all people will get you nowhere.
5. Don't use a list of tasks.
Throughout your resume, use action driven words and results-driven bullets. Avoid a list of tasks wherever possible, and don't start a resume bullet with the words "Duties included." That's so 2011.
Candidates are hired now because they can save money, make money, or streamline processes to make money faster. Demonstrate in your resume bullets how you have accomplished this—regardless of your position. Don't start a bullet with the phrase "Responsible for." Instead, be a candidate of action.
First person (I) or third person (he/she) pronouns are not needed in a resume. Stick to using third person verbs only and ditch the pronouns. Save the personalization for your personal statements, cover letters, and bios.
8. Don't miss typos and grammatical mistakes.
Be sure to proofread your resume for typos and grammatical errors. While this may seem obvious, missing errors can happen even to the best candidates. Reading your resume aloud and having a trusted colleague review your resume will help catch errors that spellcheck misses.
9. Don't only write for the ATS (application tracking system).
Roughly 65 to 70 percent of hires happen through employee referrals, social media, and personal contacts when you analyze research from JobVite, SHRM, and CareerXRoads. So be sure to write for people and not just the systems that house resumes. If you're job searching right, you should be sending your resume to people more than you are submitting to job postings.
Use a conservative line of color or a varied size font to help move the eye through the document. Be visually daring with a tasteful text box with pertinent data to draw attention to key accomplishments. Don't be afraid to use a modest selection of visual graphic elements, like color and font treatment, to make the resume stand out among a sea of resumes seen by a recruiter.
11. Don't make it all about you.
Be sure you don't simply list everything you ever did on your resume without thinking of the recipient of your resume. Instead, cater your achievements to demonstrate how you can solve a problem or capitalize on an opportunity for the prospective employer. Make your document about how you can be a solution for them instead of being all about you and your needs.


text:
"Memory I Don't Mess With" is a song by American country music singer Lee Brice. It was released on October 15, 2020, as the second single from his fifth studio album Hey World. Brice wrote the song with Billy Montana and Brian Davis, and it was produced by Ben Glover and Kyle Jacobs.

Background
Brice told PopCulture.com that he wanted to put out something rocking to get people ready to come back to shows.

Brice stated that the song is "about exactly what it says. It's one of those memories, it's one of those things that you can't let yourself get too close to because it just pulls you in too deep, the memory's always there. The pictures are always there in your head. But that's the one you've gotta let alone, and let just be a memory."

Music video
The music video was released on November 3, 2020. It depicts a young couple's love story.

Charts

Weekly charts

Year-end charts

Certifications

References

2020 singles
2020 songs
Lee Brice songs
Songs written by Lee Brice
Songs written by Billy Montana
Curb Records singles


text:
Twenty Years After 9/11: The Threat in Africa The New Epicenter of Global Jihadi Terror.
Author Bacon, Tricia
The metrics are grim. In 2020, over 13,000 people were killed in nearly 5,000 acts of violence. Seventeen designated Foreign Terrorist Organizations affected at least 22 countries. (a) Such statistics might well be expected to come from South Asia (Afghanistan or Pakistan) or the Middle East (Iraq and Syria), regions that have historically served as the central locations of violence from salafijihadi groups linked to al-Qa'ida or the Islamic State. Instead, these statistics reflect the current state of jihadism on the African continent. Once a theater seen by many as peripheral, the continent has emerged as the new center of gravity for jihadism.
The presumed marginality of Africa in U.S. national considerations--the continent having long been considered a "backwater" in the United States' security calculus (1)--ceased to hold at the beginning of the U.S.-led "Global War on Terror." As counterterrorism became the United States' top national security priority, post-2001, fighting terrorism came to define U.S. relationships with African governments. Specifically, U.S.-led efforts initially sought to stymie the ability of international jihadis, presumed to be fleeing from Afghanistan, to exploit the "undergoverned" spaces in Africa to serve as havens for their activities. (2) This new U.S. outlook spurred a flurry of new initiatives. In the Sahel, in 2002, the United States launched the Pan-Sahel Initiative, intended to train and equip six company-sized partner nation rapid-reaction counterterrorism forces--three in Mali and one each in Chad, Mauritania, and Niger--with the goal of enhancing regional cooperation, securing borders, tracking terrorist groups' movement, and deterring the establishment of jihadi terrorist safe havens in the Sahel. (3) Driven by a similar concern on the other side of the continent, the United States erected its first and only permanent military base on the continent in Djibouti in 2002. In 2003, it then initiated the East Africa Counterterrorism Initiative, which focused on improving police and judicial counterterrorism capabilities in Djibouti, Eritrea, Ethiopia, Kenya, Tanzania, and Uganda. (4) Both initiatives grew in size and scope over time.
Reality proved that the U.S. focus on preventing the migration of terrorists to Africa was a miscalculation. Rather than global jihadis fleeing Afghanistan and finding haven in Africa, instead locally minded Islamist and jihadi groups began to coalesce and proliferate within Africa, eventually entering the orbit of al-Qa'ida and later the Islamic State. This reality led to a U.S. reconceptualization of the counterterrorism challenge it faced: it began a shift away from seeing Africa as primarily a haven for non-African jihadis and instead, toward countering homegrown, African jihadi groups in their own right. By 2007, the U.S. Department of Defense stood up its own combatant command for the continent, Africa Command (AFRICOM), which, though based in Germany, was a recognition of "the growing strategic importance of Africa" and the need to develop enduring partnerships on the continent. (5) By 2019, the number of U.S. military personnel on the continent had more than doubled from 2008, (6) and the number of military exercises, programs, and engagements there had risen dramatically. (7) As of August 2021, the United States has approximately 5,100 U.S. service members and about 1,000 Defense Department civilians and contractors in AFRICOM's 15 'enduring' bases and 12 less-permanent 'non-enduring' or 'contingency' bases. (8) The majority of U.S. forces are located in Djibouti with an additional 2,000 soldiers conducting training missions in some 40 countries around the continent. (9)
However, beginning in 2018, another re-posturing was under way. With the 2018 release of the U.S. National Defense Strategy, the Department of Defense articulated that "Inter-state strategic competition, not terrorism, is now the primary concern in U.S. national security." (10) Competing with so-called "great powers" or "near-peer competitors," namely China and Russia, became the new U.S. priority, not the threats posed by al-Qa'ida or the Islamic State. That proclamation proved to be actionable on the continent. In December 2020, the United States declared that it would move all of its troops out of Somalia (11)--there to train, advise, and assist in the effort against al-Shabaab--a move emblematic of the broader Zeitgeist of fatigue with "forever wars" motivated by counterterrorism in Iraq but especially Afghanistan. France has conveyed a similar weariness: in July 2021, it announced that it would scale back its Barkhane counterterrorism mission in the Sahel. (12) But as priorities shift 20 years after 9/11, have U.S. and international efforts against African jihadi actors been effective?
To the contrary. Twenty years after 9/11, jihadi violence on the African continent has experienced a meteoric rise, putting African civilians, African states, as well as U.S. and especially partner interests on the continent in far greater danger than before September 11th. Despite the efforts to minimize jihadi violence, 20 years after 9/11, the African continent is the new leading epicenter of jihadi terrorism in the world today. Alarmingly, the jihadi threat in Africa has not merely worsened: it has reached historically unprecedented levels at the same time that the United States and its partners' appetite to counter it has waned, creating a perfect storm for the situation to further deteriorate. Even as the U.S. posture shifts away from combating jihadi terrorism in Africa, the authors argue that the United States and international community cannot turn their attention from the dire situation. And yet, more of the same is clearly not the solution.
This article proceeds in four main sections. In the first section, the authors outline how, 20 years after 9/11, the prevalence of violence from African jihadi groups has risen dramatically to never-before-seen levels. In the second section, they highlight four interconnected phenomena underpinning the rise of jihadi violence on the continent and discuss why these factors will persist. The third section makes the argument about why the United States should care about the growing jihadi threat in Africa. The authors conclude in the fourth section by proposing policy changes to countering jihadism and the threat it poses within Africa.
The New Epicenter of Jihadi Terror
The prevalence of jihadi violence on the African continent has spiked dramatically in the 20 years since the Global War on Terror began. (13) To be sure, Africa was no stranger to jihadism during the 1990s. For its part, Sudan played host to Usama bin Ladin from 1991 to 1996, where he lived after his fallout with the Saudi royal family. Egyptian jihadis, most notably Ayman al-Zawahiri, bin Ladin's future deputy and successor, joined bin Ladin in Sudan to launch a campaign against Egypt. (14) Nearby, al-Qa'ida had trained Somali militants in the early 1990s and encouraged them to target the U.S. presence during Operation Restore Hope, the U.S.-led and U.N.-backed humanitarian-focused security mission in 1992-1993. (15) Most notably, in 1998, al-Qa'ida orchestrated dual bombings of the U.S. embassies in Nairobi, Kenya, and Dar es Salam, Tanzania, collectively killing 224 (of which 12 were Americans). (16) And, for its part, Algeria was the site of a bloody campaign by jihadis against the state throughout the 1990s. However, by the end of the 1990s, things had quieted, relatively. With bin Ladin's departure, much of al-Qa'ida had moved to Afghanistan, and the Egyptian and Algerian governments had made major gains against their jihadi adversaries. To that end, in 2000, the year before 9/11, the U.S. State Department did not include any jihadi attacks in Africa among its list of "Significant Terrorist Incidents." (17) (b)
Fast forward 20 years, and the story has changed drastically: jihadi violence in Africa has seen a profound rise, particularly in the past decade. According to ACLED, 2020 saw 4,958 violent attacks perpetrated by African jihadi actors. (18) In the last decade alone, jihadi violence on the continent has increased 17-fold: the 4,958 violent events in 2020 stand in stark contrast to "only" 288 violent jihadi events in 2009 according to ACLED. (19) A rise in attacks has led to a concurrent rise in deaths. African jihadi groups were responsible for an estimated 13,059 deaths in 2020 alone. (20) These occurred primarily across five major theaters of instability: Lake Chad, the Sahel, Egypt, Somalia, and Mozambique. (21) Twenty-two African countries--nearly half of the continent--now faces violence from jihadi groups. (22)
This profound rise in jihadi violence over the past 20--but especially past 10--years has catapulted the continent into the new global epicenter of jihadi violence, a lamentable position that has become clear 20 years after 9/11. For its part, START's 2020 overview on the state of global terrorism underscored that seven of the 10 countries with the greatest increases in terrorism in 2019 were in Africa. In the same year, the continent had the second highest number of terrorism-related deaths in the world, following only South Asia. (23) Yet the arrival of the African continent as the greatest global generator of jihadi violence arguably came in the summer of 2021. In June 2021, the Global Coalition to Defeat the Islamic State surprised those not paying attention by declaring Africa as the new global priority region in which to combat the Islamic State; it proposed the creation of a new task force to combat Islamic State groups there and emphasized the importance of bringing in new African members into the anti-Islamic State coalition. (24) The next month, July 2021, saw the African Center for Strategic Studies note that the past year over year review of violence by African Islamist groups showed an unprecedented, record-setting...


The following video is from a talk given by Alan Sears at Middle Tennessee State University in Murfreesboro, TN; the talk was sponsored by the Middle Tennessee branch of Solidarity. For more discussion of these issues, see Alan's essay "Queer in a Lean World," which has been published as a Solidarity pamphlet, and this interview with Alan by the creators of Black Sheep Podcast. Alan is also currently working on a new book on queer politics.
Alan Sears is a member of Toronto New Socialists and a Professor of Sociology at Ryerson University. He is author or co-author of several books, including his most recent, The Next New Left: A History of the Future, and a long time activist in the LGBTQ and Palestine solidarity movements. His essay "Queer in a Lean World" has been published as a Solidarity pamphlet.
You can learn more about the Middle Tennessee branch of Solidarity on their Facebook page.


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Thoughts on Michael Jordan's Documentary: The Last Dance
The Last Dance, what was your favorite part of the documentary series? While Michael Jordan's series left chills down everyone's spine, we wanted to ask some of our basketball coaches what their thoughts were on it. Enjoy!
David Lewis: Associate Head Coach at Angelo State University (NCAA Division II).
Charlie Mason: Head Coach at New England College (NCAA DIII).
Lee Sartor: Head Coach at Erskine College (NCAA Division II).
Vince Kmiec: Assistant Coach at North Central College (NCAA Division III).
Jason Posser: Assistant Coach at Queens College.
Coach Lewis' initial thoughts on the "The Last Dance" documentary is that Dennis Rodman should have had more of a spotlight than he was given with the Bulls organization. Coach Lewis loved Dennis Rodman because he did whatever it took to be his best. He believes that a lot of people didn't realize how amazing his work ethic was, due to his off the court lifestyle, but he was seriously a student of the game. He was good at his 'craft' and was not shy at all on the court. Lewis is always rooting for Rodman because he is also a Texas guy, along with his basketball roots coming from a small college like his.
Coach Mason loved working MJ's camps at Elmhurst College. He discusses in detail how Michael Jordan was a stand out at these camps above even the most elite athletes. "Pick up games of the century" as Coach Mason called it, these basketball camps were some of the best camps Coach Mason has ever been apart of in his career.
Vince Kmiec
Being a Chicago-native, Coach Vince believes that the generation of the Chicago Bulls was the best sports team that's ever been. His mother was a 'Bulls Luvabulls' in 1987-1991, before the Bulls could 'figure it out' but it was still a cool experience. Coach Vince also mentions how interesting it is to see the controversy within the comparisons from MJ and the 1990's Chicago Bulls, to the newer eras that have risen since.
Jason Posser
Coach Posser loved the docu-series with the 1990's Chicago Bulls. When asked if there was a 'part 2' of the series with another NBA basketball team, who would it be? If he could pick any other team, he'd choose the Golden State Warriors so that he could learn more about Steve Kerr and Steph Curry.
The Last Dance will always be one of the best documentary series in sports history - just like the 1990's Chicago Bulls.
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Acanthormius dentatus is een insect dat behoort tot de orde vliesvleugeligen (Hymenoptera) en de familie van de schildwespen (Braconidae). De wetenschappelijke naam van de soort werd voor het eerst geldig gepubliceerd door Granger in 1949.

dentatus


DNA Res. 1998 Dec 31;5(6):355-64.
Nagase T1, Ishikawa K, Suyama M, Kikuno R, Hirosawa M, Miyajima N, Tanaka A, Kotani H, Nomura N, Ohara O.
Kazusa DNA Research Institute, Kisarazu, Chiba, Japan.
In this paper, we report the sequences of 100 cDNA clones newly determined from a set of size-fractionated human brain cDNA libraries and predict the coding sequences of the corresponding genes, named KIAA0819 to KIAA0918. These cDNA clones were selected on the basis of their coding potentials of large proteins (50 kDa and more) by using in vitro transcription/translation assays. The sequence data showed that the average sizes of the inserts and corresponding open reading frames are 4.4 kb and 2.5 kb (831 amino acid residues), respectively. Homology and motif/domain searches against the public databases indicated that the predicted coding sequences of 83 genes were similar to those of known genes, 59% of which (49 genes) were categorized as coding for proteins functionally related to cell signaling/communication, cell structure/motility and nucleic acid management. The chromosomal locations and the expression profiles of all the genes were also examined. For 54 clones including brain-specific ones, the mRNA levels were further examined among 8 brain regions (amygdala, corpus callosum, cerebellum, caudate nucleus, hippocampus, substantia nigra, subthalamic nucleus, and thalamus), spinal cord, and fetal brain.


WATCH OUR SHOW HIGHLIGHTS FROM IDS TORONTO 2018!
The Interior Design Show is Canada's premier showcase of industry leading design professionals, new product innovation and thought-provoking creative concepts. Held every year in Toronto and Vancouver, exhibit halls are transformed into action-packed hubs of emerging and established designers and design savvy consumers. IDS Toronto is an exciting show with the unveiling of our new products for the year, awe-inspiring art installations and interactive experiences.
Whether you are in the inspiration or planning phase for your home renovations, IDS is a must-see show for you. The Interior Design show is bursting with design talent and creative companies promoting their finest design-focused products and services. Even if you don't have a specific project in mind, this is a fantastic show for anyone looking for creative inspiration or for someone who has a deep appreciation of the arts. Feel the pulse of Toronto's thriving design community at IDS Toronto!
IDS attracts both local and national talent as well as speakers and vendors from around the world and helps to shape the interior design industry for the coming year. This renowned show brings together forward thinking, trend setting products coupled with powerhouse talent. Stay tuned for more updates and highlights about what you can expect from BLANCO at IDS and make sure you block these events off in your calendars!