text:
The Heroes
Georg Baselitz, The Great Friends, 1965, oil on canvas, 98.4 x 118.1 inches (250 x 300 cm) © Georg Baselitz 2016, courtesy Museum Ludwig, Cologne. Photo by Frank Oleski, Köln
"Georg Baselitz: The Heroes" at the Städel Museum, Frankfurt, Germany.
On view June 30 through October 23, 2016.
The Städel Museum is presenting Georg Baselitz's famous "Heroes" in a comprehensive monographic special exhibition fifty years after the paintings' making. Georg Baselitz (b. 1938) definitely ranks among the most influential painters and sculptors of our time. His powerful workgroup of "Heroes" and "New Types" is regarded as a key achievement of 1960s German art all over the world. In the exhibition curated by Max Hollein, it is being shown on a large scale for the first time. Some seventy paintings and works on paper are on view: their aggressively and defiantly painted monumental figures have lost nothing of their ambiguous, portentous and vulnerable quality to this day. Baselitz's "Heroes" are raddled soldiers, resigned painters, marked by their latent failure as well as by their uncertain future. The fragility and contradictory nature of the "Heroes" in terms of contents find their formal equivalent. The consistently frontal depiction and central placement of the clearly outlined figures contrast with the wildness of the artist's palette and the vehemence of his painting style. Loans from important international museums and private collections offer the public a wide-ranging view of these icons of German postwar art conceived by the only twenty-seven-year old artist in a spurt of explosive productivity in 1965 and 1966. After its début at the Städel Museum in Frankfurt, the major exhibition will travel to the Moderna Museet Stockholm, the Palazzo delle Esposizioni in Rome, and the Guggenheim Museum Bilbao.
"The 'Heroes' are both a landmark and a fervent pivot in Georg Baselitz's oeuvre. They have sprung from a deep, inner necessity in a deliberate confrontation with pressing, charged subjects and unfold a timeless reflection on the artist's existence as such. Giving expression to strikingly visualized and self-felt isolation, uprooting, and lack of orientation, the works render the artist's precarious state of experience in a broken world, establishing a paradigmatic image of his condition", says Max Hollein, curator of the show.
In 1965, Georg Baselitz found himself faced with an order destroyed in multiple ways: twenty years after the end of World War II, ideologies and political systems, as well as artistic styles, were up for discussion. This lack of systems of order was very much in keeping with the artist's own nature: appropriation through artistic categorization is something that has remained foreign to him throughout his life. Fundamentally skeptical, Baselitz emphasized the equivocal aspects of his time. Breathing failure and resignation, his monumental "Heroes" in their tattered battle dress evince an accordingly contradictory character. That the artist devoted himself to the subject of "Heroes" or "Types" at all at that time was a provocation in itself. (Male) heroism and its one-time exponents had been called into question by the war and the postwar period. Figures from a presumably buried past are brought back to life, picturing a reality which was anything but welcome in the German Federal Republic's success story of the economic miracle – and this in the supposedly obsolete form of figurative painting. Yet the artist was concerned with far more than general issues of society. In numerous role depictions – the spectrum includes the "New Type", the rebel/partisan with historical/political connotations, the spiritual shepherd and the painter adopting a position – Baselitz visualized his individual stance and his personal path as a painter. In a staggering act of self-assertion and identity definition that ran contrary to the prevailing currents of the time, he reflected on his own position vis-à-vis the society in which he lived. "I've carried out a lot of experiments in fifty years. But I don't think the 'Heroes' require any further coaching", comments Georg Baselitz on the "Heroes" and "New Types" workgroup.
"The Städel's exhibition is presenting Baselitz's 'Heroes' across two floors of its Exhibition Building in an interplay of empty spaces and zones of concentration," says Eva Mongi-Vollmer, co-curator of the exhibition. Special emphasis has been placed on the impact of the individual paintings and drawings. Thanks to the varied wall colors and the strongly rhythmical – and frequently surprising – presentation, a visit to the exhibition heightens the perception of the works and sensitizes the viewer.
Baselitz's "Heroes" and "New Types" with their colossal bodies and extremely small heads are always positioned in the very center of the picture. They stagger or, sometimes clumsily, sometimes in complete control, stalk through the pictorial space. In accordance with their maltreated bodies, their bleak surroundings suggest devastation: houses on fire, trees stripped of their leaves, thrown-up mounds of earth. The vagrant "Heroes" are furnished with a repertoire of recurring objects they carry with them: field packs, palettes and brushes or instruments of torture. Despite the repetitive format of 162 × 130 centimeters, each of the works strikes us with an expression all of its own, which strongly depends on the chosen method of painting and the colors employed. The loose chronological sequence of works in the presentation testifies to Baselitz's gradual breaking away from his motif. It is only a short way from there to his later inversion of the subject.
Baselitz began the "Heroes" and "New Types" workgroup during the period he spent at the Villa Romana in Florence on a grant. After returning to West Berlin, he continued developing the theme. The much-discussed history of Baselitz scandals that had begun in 1963 with the show at the Galerie Werner & Katz was now drawing to a close. Within the œuvre of the artist's early years, the "Hero" paintings represent a special turning point and can today be regarded as a historical document. These works did not fall into line with the artistic tendencies of their time – whether the ZERO group's vision of the future, the French or American approaches to abstraction or the variations on German post-war Informel. Even twenty years after the end of the war, they did not content themselves with a superficial feeling of a new beginning. And even if the "Heroes" and "New Types" adhere to recurring elements in terms of motif, they are monstrous, damaged and forceful in their painterly formulation. They represent an important stance within post-1945 German art.
Born in Deutschbaselitz in eastern Saxony on 23 January 1938, Georg Baselitz began his studies at the College of Fine Arts in Berlin-Weissensee (East). After he had been expelled from the school because of "sociopolitical immaturity" after two semesters, he continued his studies in Berlin-Charlottenburg (West) in 1957. First travels abroad led him to Amsterdam and Paris. 1961 saw the beginning of his exhibition activities together with Eugen Schönebeck in an unoccupied house, on the occasion of which the first "Pandemonium Manifesto," as it came to be called, was published. The following gallery shows were controversially received. In 1966 Baselitz left Berlin for a town in Rhine-Hesse near Worms. He painted his first picture showing its motif upside down in 1969 – a decision he remained true to throughout his further career. As he became more and more renowned, he increasingly presented his works in exhibitions abroad; in 1980 he and Anselm Kiefer were invited to represent the Federal Republic of Germany at the 39th Venice Biennale. Much-acclaimed exhibitions in various countries such as Great Britain and the United States followed. Baselitz continued his teaching activities, which he had begun in Karlsruhe in 1978, in Berlin from 1983 to 1988 and from 1992 on. Numerous retrospectives, honors, awards, and honorary professorships still pay tribute to the outstanding relevance of his work.
Schaumainkai 63
T. +49.69.605.098-200
E. info@staedelmuseum.de
Website: Städel Museum | Georg Baselitz


text:
Q: Angular directive parent scope I'm building a directive that decorates a table header item and sort the data upon the click event. I cannot apply the changes on the data model to the parent scope using the directive scope.
vm.data is an array on the parent scope that contains the data I want to sort in the directive.
After the click the data object in the directive has changed but the parent is still in the same order.
I dont want to access the parent scope using $parent, What I'm missing ??
Browser

directive code:
angular
    .module("app")
    .directive("sortable", ['lodash', sortableDirective]);

function sortableDirective(lodash) {
    return {
        restrict: "A",
        scope:{
            data:"="
        },
        controller:function($scope){

        },
        link: function (scope, element, attributes) {

            var sorted = undefined;
            var col = attributes['sortable'];
            var oldClass = 'sorting'

            attributes.$$element.addClass(oldClass);
            $(element).on("click", sort);

            function changeClass(){
                if(sorted=='asc'){
                    attributes.$$element.removeClass(oldClass);
                    attributes.$$element.addClass('sorting_asc');
                    oldClass = 'sorting_asc';
                }
                else if(sorted=='desc'){
                    attributes.$$element.removeClass(oldClass);
                    attributes.$$element.addClass('sorting_desc');
                    oldClass='sorting_desc';
                }

            }

            function sort() {

                if (sorted == 'asc') {
                    sorted = 'desc';
                }
                else {
                    sorted = 'asc';
                }

                scope.data = lodash.sortBy(scope.data, function (o) {
                    return o[col];
                });

                if (sorted == 'desc') {
                    lodash.reverse(scope.data);
                }

                changeClass();
            }
        }
    };
}

A: This is because you are using jQuery to listen to change on the element. So just change this line:
$(element).on("click", sort);

to
element.on("click", sort);

The 2nd attribute i.e. element is already an instance of jQlite if jQuery is not available and will be an instance of jQuery if jQuery is available.
In any case, there is a method available .on which will be executed on the value change. Since you again wrapped it to $(), the Angular was not getting notified of the change in the data.
Edit:
On the 2nd walk through of your code, I see the actual problem. You are reassigning the complete scope.data in the sort() method which is breaking the pass by reference behavior of Javascript (or in any OOPS programming).
The pass by reference will only work if you continue to modify your SAME reference variable. Noticed the word, SAME?? By writing scope.data = lodash.sortBy(scope.data, function (o) {}) you removed the reference of the actual data passed to the directive. Hence the values are not updated.
So to fix this problem, you have a few options:

*

*Change your sorting code to not reassign the complete scope.data variable RECOMMENDED (use inbuilt sort method)

*Pass the modified data to the parent scope using scope.$emit()

*Or use the $parent property which you don't want to use

A: The bidirectional binding will update the parent on each digest cycle but the click handler needs to invoke that digest cycle with $apply:
    link: function (scope, element, attributes) {

        var sorted = undefined;
        var col = attributes['sortable'];
        var oldClass = 'sorting'

        element.addClass(oldClass);

        //REPLACE this
        //$(element).on("click", sort);
        //
        //WITH this
        element.on("click", function (e) {
            sort();
            scope.$apply();
        });

        function changeClass(){
            if(sorted=='asc'){
                element.removeClass(oldClass);
                element.addClass('sorting_asc');
                oldClass = 'sorting_asc';
            }
            else if(sorted=='desc'){
                element.removeClass(oldClass);
                element.addClass('sorting_desc');
                oldClass='sorting_desc';
            }

        }

The ng-click directive automatically invokes $apply but when the click event is handled by AngularJS jqLite, the code needs to notify the AngularJS framework.
From the Docs:

$apply([exp]);
$apply() is used to execute an expression in angular from outside of the angular framework. (For example from browser DOM events, setTimeout, XHR or third party libraries). Because we are calling into the angular framework we need to perform proper scope life cycle of exception handling, executing watches.

-- AngularJS $rootScope.scope API Reference -- $apply

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}

The last few years saw significant progress in the study of gapless systems with non-trivial topology, mainly fueled by the successful generalization of the well established phenomenology of topological insulators (TI) to gapless systems.
In a TI, where the band structure of an insulator supports a topological non-trivial state, the bulk remains gapped, whereas counterpropagating edge states are present on the surface~\cite{Hasan2011,Qi2011}.
Topological phases are however not constrained to materials with a bulk gap,
gapless topological phases can be realized in systems with protected Fermi points and superconductors with nodes~\cite{Matsuura2013}.
In these materials, the simultaneous presence of non-trivially realized global symmetries and gapless, most often relativistic fermions offers a fertile ground for numerous exotic quantum phases~\cite{Turner2013,Vafek2014,Metlitski2014}.

A well-studied type of gapless topological system is the Weyl semimetal (WSM)~\cite{Burkov2011,Wan2011}, which features an even number of non-degenerate Fermi surface points with linear Dirac cone dispersion. In these systems, the lifting of degeneracy is facilitated by breaking of time-reversal symmetry. The chiral Weyl cones are sources and sinks of Berry flux, leading to non-trivial topological properties.

WSM exhibit a rich phenomenology, including the chiral anomaly~\cite{Nielsen1983,Son2012, Chen2013}, a large negative magnetoresistance~\cite{Son2013,Burkov2014,Gorbar2014}, as well as special protected surface states (Fermi arcs)~\cite{Wan2011,Baum2015}.
Experimentally, TaAs~\cite{Xu2015,Lv2015} and NbAs~\cite{Xu2015a} have been found to harbor a WSM,
some experiments have also seen signatures of Fermi arcs~\cite{Xu2015a,Lv2015a} and the detection of the chiral anomaly has been reported~\cite{Zhang2016}.

Quite some attention was attributed to the effects of disorder in a WSM.
Since short-ranged impurities are irrelevant in three dimensions, an instability towards a diffusive metal is only expected at a finite impurity density~\cite{Fradkin1986,Fradkin1986a}.
The associated quantum critical point has been studied intensely, both with respect to critical scaling laws~\cite{Goswami2011,Roy2014,Syzranov2015,Sbierski2015,Liu2016,Syzranov2016} and the wider phase diagram~\cite{Sbierski2014,Ominato2014,Kobayashi2014,Pixley2015,Pixley2016,Bera2016,Roy2016}.
In the limit of weak disorder, the calculation of the dc-conductivity was done early on~\cite{Burkov2011,Burkov2011a,Hosur2012}, later the focus shifted to the exploration of magnetoconductance~\cite{Son2013,Burkov2014,Gorbar2014,Burkov2015,Klier2015,Altland2015,Ramakrishnan2015,Khalaf2016}.

Directly at charge neutrality (chemical potential $E=0$), the density of states (DOS) vanishes and with it the effects of any weak disorder.
A popular choice to model this situation is a Gaussian distribution for the disorder, which in turn allows to calculate the self-energy and thus the scattering rate $\Gamma$ with the help of the self-consistent Born approximation (SCBA)~\cite{Burkov2011,Hosur2012,Ominato2014,Sbierski2014,Tabert2016}.
For the dc-conductivity, the Boltzmann approach then yields $\sigma_{dc}\sim E^2/\Gamma$~\cite{Burkov2011,Hosur2012,Lundgren2014}. Together with the quadratic energy dependence of the DOS ($\rho\sim E^2\sim\Gamma$) this leads to a nonzero conductivity which diverges with decreasing disorder.
With the Kubo formalism, the conductivity contains an additional term proportional to the scattering rate~\cite{Ominato2014,Tabert2016}. It can be shown that irrespective of the precise form of $\Gamma(E)$, a generic expression holds for $\sigma_{dc}$ in the regime of weak disorder~\cite{Tabert2016},
\begin{align}
\sigma_{dc}&=\sigma_0\frac{E^2+3\Gamma(E)^2}{\Gamma(E)}.
\label{eq:genericdccond}
\end{align}
A few studies have also elaborated on the $\omega$ and T-dependence of the conductivity within SCBA~\cite{Burkov2011,Hosur2012,Rosenstein2013,Ashby2014,Tabert2016}.

This simple picture of weak disorder does however not differentiate between a small impurity density $n_i$ and weak impurities with a small scattering potential $V$~\cite{Nandkishore2014}. In particular, it cannot account for rare cases where the Weyl fermions are repeatedly (resonantly) scattered.
Recently another issue was pointed out~\cite{Biswas2014}. The WSM features an anisotropic scattering amplitude, which results in a difference between the quasiparticle lifetime and transport lifetime. This is due to the intrinsically relativistic dispersion, leading to the interlocking of spin with rotational degrees of freedom. This property affects the coefficients of various response functions, but was not considered in the initial treatments of disordered WSMs. Here, we address the problem of resonant scattering within a microscopic calculation, which fully accounts for the mentioned idiosyncracies of Weyl semimetals.

As a side remark, we point out that charged impurities with a long-ranged impurity potential lead to an entirely different situation. This kind of disorder is  relevant and leads to the formation of charge puddles~\cite{Skinner2014}. Therefore, even for $E=0$, a zero DOS is avoided and a minimal dc-conductivity proportional to $n_i^{1/3}$ is found for the low temperature limit~\cite{Skinner2014,Ominato2015,Rodionov2015}. However, charged impurities themselves shift the chemical potential away from $E=0$, making it difficult to tune a WSM with dopants to the degeneracy point.

The focus of this work are the effects of resonant scattering, which appears in the presence of strong but dilute short-ranged impurities. In this case, if the impurity potential is tuned to a resonance, the total scattering cross section of an impurity center diverges.
This behavior, also termed ``rare-region effect'' is the leading contribution to electronic transport close to the Weyl point and was identified as a way to avoid the quantum critical point between WSM and diffusive metal~\cite{Nandkishore2014,Pixley2016a}.
In other words, in the presence of short-ranged disorder, even the perfectly tuned WSM does not retain a vanishing density of states (DOS) down to the smallest energies. As short-ranged impurities are irrelevant, this effect is non-perturbative in disorder strength and is not visible within the SCBA. In the analogous two-dimensional case of graphene, disorder is marginal and a finite DOS is recovered already from the SCBA~\cite{Ostrovsky2006}.

The importance of rare regions for WSMs was first pointed out by \citeauthor{Nandkishore2014}~\cite{Nandkishore2014} by demonstrating the existence of power-law bound states. In particular, they derived the phase shift of the associated scattering problem and obtained a exponentially small DOS for the case of a short-range correlated disorder distribution with zero mean.
A more recent numerical study by \citeauthor{Pixley2016a}~\cite{Pixley2016a} investigated the effect of resonant scattering using a Gaussian disorder model in a finite system. They concluded that a finite DOS is indeed recovered in the thermodynamic limit but might be very small.

While the importance of resonant scattering is established for a WSM tuned to the Weyl point, the corresponding calculation of the associated self-energy, DOS and conductivity has not been done.
This extends to the question of length scales and the universality of the results, both of which was only sketched in~\cite{Nandkishore2014}.
On a more technical level, just like for weak impurities the scattering amplitude is anisotropic, and the conductivity is expected to be renormalized by the contribution of particle hole ladders.
This makes it necessary to correctly capture the diffusion pole by using an approximation which preserves charge conservation.

Here, we calculate self-energy, DOS and dc-conductivity in the presence of strong impurities for $T=0$ by solving the scattering problem of a rectangular impurity potential. Central in the derivation is the determination of the self-consistent self-energy and scattering T-matrix, which we determine in accordance with the Ward identities.
Preserving the diffusion pole, a universal form of the dc-conductivity emerges, which spans the entire parameter regime from weak to strong impurity centers.

As the main results, we show that scattering from rare regions leads to a self-energy proportional to the square root of the impurity density $n_i$, in sharp contrast to the perturbative regime given by SCBA. This affects the estimates for the energy windows in which a WSM crosses over from effectively weak to strong scattering, resulting in significantly different predictions for the mean free path close to the Weyl point.

The remaining sections are structured as follows. Section~\ref{sec:II} contains a brief exposition of the microscopic scattering problem of Weyl fermions hitting an impurity with a rectangular potential barrier. Then, in section~\ref{sec:III}, the self-energy is determined within the self-consistent T-matrix approach. For this, we resolve both the energy and momentum dependence of the scattering T-matrix close to resonance. The calculation of DOS and dc-conductivity are presented in~\ref{sec:IV}, focusing on the impact of the self-consistent self-energy and of the renormalized current vertex. We conclude in section~\ref{sec:V}.

\section{Weyl fermions scattered by a potential barrier}

\label{sec:II}

In this section we derive the scattering cross section of a Weyl fermion
encountering a spherical potential energy barrier.
The system of our interest is defined in a spherical coordinate with the radial vector $\bm{r}$,
polar angle $\theta,$
and azimuthal angle $\varphi$.
We consider a Weyl fermion of energy $E$ incident in the direction of $\mathbf{v}$ (angles $\zeta$, $\eta$) and scattered by a spherically symmetric potential of strength $\mathcal{V}(r)$, where $\mathcal{V}(r)$ has
a nonzero value $V$ for $r<b$, and $0$ otherwise.
The direction of the scattered wave is labeled as $\mathbf{v'}$,
with energy $E'$ and angles $\zeta'$, $\eta'$.
While we treat the single scattering of a plane wave which is hitting the impurity, the direction of the outgoing wave $\mathbf{v'}$ is equal to the direction of observation $\hat{\bm{r}}=\frac{\bm{r}}{r}$. For elastic scattering it is additionally $E=E'$, later on in section~\ref{sec:III} intermediate states will be taken into account.

\subsection{Helicity eigenbasis}

The scattered states can be obtained by solving the Schr\"odinger equation
with the Hamilton operator
\begin{align}
H & =v\bm{\sigma}\cdot\bm{p}+\mathcal{V}(r)\label{eq:hami}
\end{align}
where $\bm{\sigma}$ is the vector of Pauli matrices. Henceforth we
will use the convention that $\hbar v=1$ and only restore units when
necessary.

Eigenstates of a clean system can be represented as plane waves, which are normally labeled by the momentum vector $\bm{k}$ and the chirality parameter $\lambda=\pm1$ (corresponding to the projection of spin onto the momentum direction).
The wave function of such a state can be written as
\begin{align}
|\bm{k},\lambda\rangle & =e^{i\bm{k}\cdot\bm{r}}|\lambda\hat{\bm{k}}\rangle\shortintertext{where}|\bm{n}\rangle & =\left(\begin{array}{c}
\cos(\theta/2)\\
\sin(\theta/2)e^{i\phi}
\end{array}\right)
\end{align}
The ket notation is used for the spinor in the direction $\bm{n}$ with the polar and azimuthal angles $\theta$ and $\phi$.
The eigenstate energy is $E=\lambda|\bm{k}|$.
Instead of using the momentum $\bm{k}$ and chirality $\lambda$ to label states,
here we use an alternative representation in terms of energy $E$ and a unit velocity vector $\mathbf{v}$ as follows
\begin{align}
|E,\mathbf{v}\rangle & =e^{iE\mathbf{v}\cdot\bm{r}}|\mathbf{v}\rangle.\label{eq:helicitybase}
\end{align}
This representation is known as helicity basis in the study of relativistic
systems, and it is related to the momentum representation by $\bm{k}=E\mathbf{v}$
and $\lambda=\mathrm{sgn}\, E$. The normalization has the form
\begin{align}
\langle E,\mathbf{v}|E',\mathbf{v'}\rangle & =\frac{(2\pi)^{3}}{E^{2}}\delta(E-E')\delta(\mathbf{v}-\mathbf{v'}).
\end{align}

As the clean Weyl Hamiltonian $H=v\bm{\sigma}\cdot\bm{p}$ conserves the total angular momentum $\bm{J}$, one can also construct eigenstates using the total angular momentum number $j$ and the z-axis component $j_{z}$ as quantum numbers, yielding spherical waves
\begin{align}
|E,j,j_{z}\rangle^{(1,2)} & =\frac{E}{\sqrt{2}}\biggl[h_{j-1/2}^{(1,2)}(Er)|\phi_{j,j_{z}}^{+}\rangle\nonumber \\
 & \qquad+ih_{j+1/2}^{(1,2)}(Er)|\phi_{j,j_{z}}^{-}\rangle\biggr].
\end{align}
Here, $h_{l}^{(1,2)}(x)$ are the spherical Hankel functions and $|\phi_{j,j_{z}}^{\pm}\rangle$ are spherical spinors.
The index $(1,2)$ corresponds to waves which propagate outwards or inwards form the origin, respectively. The normalization is chosen such that the total particle flux in these states equals unity.
Details of the spherical wave basis $|\phi_{j,j_{z}}^{\pm}\rangle$,
the construction of an eigenbasis for $H$ in the clean system and the partial wave expansion for the scattering problem can be found in appendix~\ref{sec:appA}.
In the following, we focus on the solution for a rectangular impurity potential, $\mathcal{V}(r)=\Theta(b-r)V$.

\subsection{Scattering at a rectangular potential}

\label{sec:partialwave}

For the eigenstates of the Hamiltonian Eq.~(\ref{eq:hami}) we make
the ansatz

\begin{align}
|\psi_{Ejj_{z}}\rangle & =f(r)|\phi_{j,j_{z}}^{+}\rangle-ig(r)|\phi_{j,j_{z}}^{-}\rangle,
\end{align}
which yields two coupled differential equations for the radial functions
$f$ and $g$. The solutions are given in terms of spherical Bessel
functions of the first and the second kind, $\mathscr{\textrm{j}}_{l}(x)$
and $\mathscr{\textrm{n}}_{l}(x)$. Constructing the total wave function
of the scattering problem $|\psi\rangle$ as a sum over all angular
parts $|\phi_{j,j_{z}}^{\pm}\rangle$, one obtains in presence of
the impurity \looseness=-1

\begin{widetext}
\begin{equation}
|\psi\rangle=\begin{cases}
\sum_{jj_{z}}A_{jj_{z}}\left[\mathscr{\textrm{j}}_{j-1/2}(\bar{E}r)|\phi_{j,j_{z}}^{+}\rangle-i\mathscr{\textrm{j}}_{j+1/2}(\bar{E}r)|\phi_{j,j_{z}}^{-}\rangle\right]\;, & r<b\\
\sum_{jj_{z}}A'_{jj_{z}}\left[\mathscr{\textrm{j}}_{j-1/2}(Er)|\phi_{j,j_{z}}^{+}\rangle-i\mathscr{\textrm{j}}_{j+1/2}(Er)|\phi_{j,j_{z}}^{-}\rangle\right]\;\\
\quad+B'_{jj_{z}}\left[\mathscr{\textrm{n}}_{j-1/2}(Er)|\phi_{j,j_{z}}^{+}\rangle-i\mathscr{\textrm{n}}_{j+1/2}(Er)|\phi_{j,j_{z}}^{-}\rangle\right], & r>b,
\end{cases}\label{eq:totalwavefunction}
\end{equation}
\end{widetext}

where we introduced $\bar{E}=E-V$. We point out that $|\psi\rangle$ must
be regular at the origin, therefore there is no coefficient $B_{jj_{z}}$
for $r<b$. The coefficients $A_{jj_{z}}'$ and $B_{jj_{z}}'$ are
related to $A_{jj_{z}}$ by matching the wave functions at $r=b$,
yielding
\begin{align}
A_{jj_{z}}' & =-A_{jj_{z}}N_{j}(\bar{E},E)\left(Eb\right)^{2},\label{eq:A'}\\
B_{jj_{z}}' & =A_{jj_{z}}M_{j}(\bar{E},E)\left(Eb\right)^{2},\label{eq:B'}
\end{align}
where we have used the fact that $\mathscr{\textrm{j}}_{n+1}\left(x\right)\mathscr{\textrm{n}}_{n}\left(x\right)-\mathscr{\textrm{j}}_{n}\left(x\right)\mathscr{\textrm{n}}_{n+1}\left(x\right)=x^{-2},$
and defined two shorthands
\begin{align}
N_{j}(\bar{E},E) & =\mathscr{\textrm{j}}_{j-1/2}(\bar{E}b)\mathscr{\textrm{n}}_{j+1/2}(Eb)\nonumber \\
 & \quad-\mathscr{\textrm{n}}_{j-1/2}(Eb)\mathscr{\textrm{j}}_{j+1/2}(\bar{E}b),\label{eq:N}\\
M_{j}(\bar{E},E) & =\mathscr{\textrm{j}}_{j-1/2}(\bar{E}b)\mathscr{\textrm{j}}_{j+1/2}(Eb)\nonumber \\
 & \quad-\mathscr{\textrm{j}}_{j-1/2}(Eb)\mathscr{\textrm{j}}_{j+1/2}(\bar{E}b).\label{eq:M}
\end{align}

Scattering means that the incident plane wave is disturbed so that
a part of the current is redistributed to other directions $\hat{\bm{r}}\neq\mathbf{v}$,
i.~e. the total wave function can be written as
\begin{align}
|\psi\rangle & =e^{iE\mathbf{v}\cdot\bm{r}}|\mathbf{v}\rangle+f(\mathbf{v},\hat{\bm{r}})\frac{e^{iEr}}{r}|\hat{\bm{r}}\rangle.\label{eq:scatteringwf}
\end{align}
Here, $f(\mathbf{v},\bm{r})$ is the amplitude for the scattering
of the incident wave from $(\cos\frac{\zeta}{2},\sin\frac{\zeta}{2}e^{i\eta})$
basis to $(\cos\frac{\theta}{2},\sin\frac{\theta}{2}e^{i\varphi})$
basis. Matching the coefficients of Eq.~(\ref{eq:totalwavefunction}) to the form of Eq.~(\ref{eq:scatteringwf}) for $r\rightarrow\infty$, one obtains
\begin{align}
f(\mathbf{v},\bm{r}) & =\frac{4\pi}{iE}\sum_{jj_{z}}\biggl[\left\langle \phi_{j,j_{z}}^{+}(\zeta,\eta)|\mathbf{v}\right\rangle \langle\phi_{j,j_{z}}^{+}(\theta,\varphi)|\mathbf{r}\rangle^{*}\nonumber \\
 & \quad\times\frac{M_{j}}{M_{j}+iN_{j}}\biggr].\label{scamplitude}
\end{align}
The scattering cross section can be obtained by integrating out the
angular part of the scattered wave function,
\begin{align}
\sigma & =\int\mathrm{d}\Omega_{\bm{r}}\left|f(\mathbf{v},\bm{r})\right|^{2},\\
 & =4\pi\sum_{j}\frac{2j+1}{E^{2}}\frac{M_{j}^{2}}{M_{j}^{2}+N_{j}^{2}}.\label{eq:specificsigma}
\end{align}
For forward scattering ($\mathbf{v'=v}$), the imaginary part of the
scattering amplitude becomes
\begin{align}
\Im f(\mathbf{v},\mathbf{v}) & =\sum_{j}\frac{2j+1}{E}\frac{M_{j}^{2}}{M_{j}^{2}+N_{j}^{2}},
\end{align}
and therefore the relation between the total scattering cross section
and the imaginary part of the forward scattering amplitude $f(\mathbf{v},\mathbf{v})$
is
\begin{equation}
\sigma=\frac{4\pi}{E}\Im f(\mathbf{v},\mathbf{v}),\label{eq:opticaltheorem}
\end{equation}
which is the optical theorem.

Fig.~\ref{fig:scattercrossectionvsVLow-energy-} shows a plot of
the total scattering cross section $\sigma_{j}$ for the lowest order
of the partial wave expansion ($j=1/2$) as a function of the impurity
strength $Vb$.
\begin{figure}
\includegraphics[width=1\columnwidth]{crosssection}
\caption{
Total scattering cross section at low energy $E=0.5/b$ as a function
of the scaled impurity potential $Vb$, normalized to the value at $Vb=\pi$.
At resonance the total scattering cross section peaks closely to $V_{r}b=n\pi+2Eb$, with $n$ a nonzero integer and a broadening width proportional to $(Eb)^{2}$.\label{fig:scattercrossectionvsVLow-energy-}}
\end{figure}

At small $Vb$, the total scattering cross section is%
\begin{equation}
\sigma_{1/2}^{born}\sim\frac{8\pi}{9}b^{2}(Vb)^{2}(Eb)^{2},\label{eq:bornscatter}
\end{equation}
which constitutes the Born approximation.
When $Vb$ is not small
$\sigma_{1/2}$ shows pronounced resonances whose height and  shape are independent of the order of the resonance. The peaks are located at values $V_{r}$ of the impurity potential fulfilling $V_{r}b=n\pi+2Eb+\mathcal{O}((Eb)^{2})$, where $n$ is a nonzero integer. Close to a resonance the total scattering cross section is
\begin{equation}
\sigma_{1/2}^{res}\sim8\pi b^{2}\frac{\left(Eb\right)^{2}}{\left(Eb\right)^{4}+\left(Vb-2Eb-n\pi\right)^{2}},\label{eq:resscatter}
\end{equation}
which is obtained by first taking $Eb\rightarrow0$.
Such a Lorentzian was also found in~\cite{Nandkishore2014}. Inserting $V_{r}$ into Eq.~(\ref{eq:resscatter}), the total scattering cross section has a maximum at $8\pi/E^{2},$ with a broadening width proportional to $(Eb)^{2}$.
It is important to note that the resonance is not peaked at $Vb=n\pi$, which will become important in the self-consistent analysis in the next section.
At the value $Vb=n\pi$ of the scattering strength the cross section is $\sigma_{1/2}=2\pi b^{2}$, which coincides with the value for hard wall scattering in the classical limit.

\section{Self-consistent T-matrix approach}

While the solution of the scattering problem comprises a universal resonant scattering cross section, the width of the resonance depends on the microscopic UV-cutoff $b^{-1}$. To appropriately take this into account, it is necessary to go beyond on-shell processes and include not only the energy but also the momentum dependence of the scattering amplitude. To this end, we derive in the following an explicit expression for the self-energy in the presence of finite potential scatterers within the self-consistent T-matrix approximation (SCTMA)~\cite{Ostrovsky2006}.

\label{sec:III}
The average Green's function of the clean system is diagonalized
in the basis of Eq.~(\ref{eq:helicitybase}). The effects of disorder are subsumed into a complex self-energy $\Sigma(E)$, which is also diagonalized in the helicity basis since the impurity is spinless. The average Green's function is therefore
\begin{align}
G(E) & =\int\!\frac{\epsilon^2\mathrm{d}\epsilon\mathrm{d}\Omega_{\mathbf{v}}}{2\pi^3}
\frac{|\epsilon,\mathbf{v}\rangle\langle \epsilon,\mathbf{v}|}
{E-\epsilon-\Sigma(E,\epsilon)}.
\label{eq:greensfun}
\end{align}
Here, $\epsilon$
are the matrix elements of the kinetic energy in the helicity basis and $\Sigma(E,\epsilon)$ are the matrix elements of the self-energy. In the same vein, we define $G^{R/A}(E,\xi)=(E-\xi-\Sigma^{R/A}(E,\xi))^{-1}$. The eigenvalues of the bare Green's function are given by $G^{R/A}_0(E,\xi)=(E-\xi\pm i 0^+)^{-1}$.

In Sec.~\ref{sec:tmatrix} the T-matrix is computed, which allows
to calculate the self-energy for repeated scattering from a single impurity.
By making this calculation self consistent in Sec.~\ref{sec:selfconsistent}, we can take into account certain effects from multi-impurity scattering, thus obtaining a non-vanishing imaginary part of $\Sigma$ at the
Weyl point.

\subsection{Single impurity scattering}

\begin{figure}
\includegraphics[width=\columnwidth]{fig2}
\caption{Diagrams used in the calculation of the T-matrix. Single dashed lines are impurity scatterings from the rectangular potential, full lines are Weyl fermions and the doubly dashed line is the impurity interaction from repeated scattering. \label{fig:fullSIS}}
\end{figure}

\label{sec:tmatrix}
When the impurity concentration is low, we can treat the scattering problem of the system as that of repeated scattering off a single impurity, shown graphically in Fig.~\ref{fig:fullSIS}.
The self-energy operator $\Sigma_0$ for single impurity scattering is by definition equal to the product of the concentration $n_{i}$ and a transition operator
$T_0$ as follows
\begin{align}
\Sigma_0(E) & =n_{i}T_0(E),\label{eq:general_SE}
\shortintertext{which becomes obvious when writing}
T_0(E) &=\mathcal{V}+
\mathcal{V}G_{0}(E)\mathcal{V}+\dots\nonumber \\
& =\mathcal{V}+\mathcal{V}G_{0}(E)T_0(E),
\label{eq:general_Tmatrix}
\end{align}
with the bare Green's function operator $G_0(E)$. Eq.~(\ref{eq:general_Tmatrix}) can easily be solved for $T_0$ by iterating it numerically. Here, we instead opt for an analytical solution for the self-energy near to a resonance, which will guide the way to the self-consistent calculation.

The general formalism is as follows. Given the eigenenergies $E_{\lambda}$ and eigenstates $|\psi_{\lambda}\rangle$
of the Hamiltonian Eq.~(\ref{eq:hami}), the transition operator
fulfills
\begin{align}
\mathcal{V}|\psi_{\lambda}\rangle & =T_0(E_{\lambda})|E_{\lambda},\mathbf{v}\rangle.\label{eq:V_Tequlity}
\end{align}

Inserting the identity $\sum_{\lambda}\left|\psi_{\lambda}\right\rangle \left\langle \psi_{\lambda}\right|=1$
into Eq.~(\ref{eq:general_Tmatrix}) and making use of Eq.~(\ref{eq:V_Tequlity})
yields for $\Sigma_0(E)$~\cite{Mahan1990},
\begin{align}
\Sigma_0(E)
&=n_{i}\mathcal{V}
+n_{i}\sum_{\lambda}\frac{T_0(E_{\lambda})
\left|E_{\lambda},\mathbf{v}\right\rangle
\left\langle E_{\lambda},\mathbf{v}\right|
T_0^{\dagger}(E_{\lambda})}
{\left(E-E_{\lambda}\right)}.\label{eq:ultimaSE}
\end{align}

We reiterate that $E$ denotes the on-shell energy, which is identical
to the energy of the incident Weyl fermion in Sec.~\ref{sec:II},
and which is given simply by the chemical potential in the WSM.
As per usual, the denominator in Eq.~\ref{eq:ultimaSE} is not well
defined unless one replaces $E\rightarrow E\pm i0^{+}$, where the signs refer to retarded and advanced self-energies, respectively.

The expectation value of the self-energy can be determined by sandwiching Eq. (\ref{eq:ultimaSE}) with a state $|\xi,\mathbf{v'}\rangle$ with scattering direction $\mathbf{v'}$ and energy $\xi$.
To this end, we write
\begin{align}
\Sigma_0(E,\xi) & \equiv\langle \xi,\mathbf{v'}|\Sigma_0(E)|\xi,\mathbf{v'}\rangle\nonumber \\
 & =n_{i}\langle \xi,\mathbf{v'}|
 \mathcal{V}
 |\xi,\mathbf{v'}\rangle\nonumber \\
 & \qquad+n_{i}\sum_{\lambda}\frac{\left|\langle \xi,\mathbf{v'}|T_0(E_{\lambda})|E_{\lambda},\mathbf{v}\rangle\right|^{2}}{E-E_{\lambda}}.\nonumber \\
 & =n_{i}\mathcal{V}(\xi)+n_{i}\int\frac{\epsilon^{2}\mathrm{d}\epsilon\mathrm{d}\Omega_{\mathbf{v}}}{(2\pi)^{3}}\frac{\left|T_0(\xi,\epsilon,\epsilon)\right|^{2}}{E-\epsilon},\label{eq:offmassselfSEmatrixelements}
\end{align}
where we replaced $\sum_{\lambda}$ by $\int\mathrm{d}^{3}\epsilon/\left(2\pi\hbar v\right)^{3}$.
The angular integration with $\mathrm{d}\Omega_{\mathbf{v}}$ averages over all angles of incidence, which is equal to an average with respect to the position of the impurity.
Writing out all dependencies, the T-matrix elements are defined as
\begin{align}
T_0(\xi,E,E) & =\langle \xi,\mathbf{v'}|
T_0(E)|E,\mathbf{v}\rangle.
\end{align}
This also means that $\Sigma_0(E,\xi)=n_iT_0(\xi,E,\xi)$. For elastic scattering $(E=\xi)$, the T-matrix element can be related to the scattering amplitude as follows
\begin{align}
\langle E,\mathbf{v'}|T_0^{R}(E)|E,\mathbf{v}\rangle & =-\frac{2\pi}{E}f(\mathbf{v},\mathbf{v}'),\label{eq:defT}
\end{align}
which can be proved by use of the Lippmann-Schwinger equation.
From the imaginary part of the scattering amplitude one can therefore refer the scattering cross section.

At this point, it is necessary to assume a certain scattering potential to fill Eq.~\ref{eq:offmassselfSEmatrixelements} with life. Specifying to a rectangular impurity potential and using the spherical wave basis, we obtain immediately
\begin{align}
\frac{\mathcal{V}(\xi) }{V} & =2\pi\sum_{j}\left(2j+1\right)\int_{0}^{b}\!\!\! r^{2}\mathrm{d}r\left[\mathscr{\textrm{j}}_{j-1/2}^{2}(\xi r)+\mathscr{\textrm{j}}_{j+1/2}^{2}(\xi r)\right]\nonumber \\
 & =2\pi b^{3}\sum_{j}\left(2j+1\right)\biggl[\mathscr{\textrm{j}}_{j-1/2}^{2}(\xi b)+\mathscr{\textrm{j}}_{j+1/2}^{2}(\xi b)\nonumber \\
 & \qquad-\frac{2j+1}{\xi b}\mathscr{\textrm{j}}_{j-1/2}(\xi b)\mathscr{\textrm{j}}_{j+1/2}(\xi b)\biggr].\label{eq:averageV}
\end{align}
From Eq.~(\ref{eq:V_Tequlity}), we can easily calculate the matrix elements
$T_0(\xi,E,E)$ with the help of the partial wave expansion of section~(\ref{sec:partialwave}).
One obtains
\begin{align}
T_0(\xi,E,E) & =-i16\pi^{2}Vb^{3}\sum_{jj_{z}}\biggl[\frac{M_{j}(\bar{E},\xi)}{M_{j}(\bar{E},E)+iN_{j}(\bar{E},E)}\nonumber \\
 & \quad\times\frac{\langle\mathbf{v}'|j_z\rangle\langle j_z|\mathbf{v}\rangle }{(\bar{E}b-\xi b)(Eb)^{2}}\biggr],\label{eq:Tmatrixelement}
\end{align}
where we introduced the shorthand $\langle\mathbf{v}'|j_z\rangle=
\left\langle \phi_{j,j{}_{z}}^{+}(\zeta',\eta')|\mathbf{v}'\right\rangle ^{*}$, $\langle j_z|\mathbf{v}\rangle=\left\langle \phi_{j,j{}_{z}}^{+}(\zeta,\eta)|\mathbf{v}\right\rangle$ etc.
We point out that the combination $M_{j}/(M_{j}+iN_{j})$, which similarly
appeared in the scattered wave in Eq.~(\ref{scamplitude}), now contains
unequal arguments in numerator and denominator. Inserting Eq.~(\ref{eq:Tmatrixelement})
into the right hand side of Eq.~(\ref{eq:offmassselfSEmatrixelements})
and integrating out the angular dependence, we arrive at
\begin{align}
\int\!\frac{\epsilon^{2}\mathrm{d}\epsilon\mathrm{d}\Omega_{\mathbf{v}}}{\left(2\pi\right)^{3}}
\frac{\left|T_0(\xi,\epsilon,\epsilon)\right|^{2}}
{\left(E-\epsilon\right)} & =2V^{2}\sum_{j}\left(2j+1\right)I_{j}^{V}(E,\xi),\label{eq:nightmare}\\
I_{j}^{V}(E,\xi) & =\int\!\!\frac{\mathrm{d}\epsilon}{\epsilon^{2}(\bar{\epsilon}-\xi)^{2}(E-\epsilon)}\nonumber \\
 & \quad\times\frac{M_{j}^{2}(\bar{\epsilon},\xi)}{M_{j}^{2}(\bar{\epsilon},\epsilon)+N_{j}^{2}(\bar{\epsilon},\epsilon)}.\label{eq:mainintegral}
\end{align}
In the following we concentrate on the first term in the partial wave expansion, setting $j=\tfrac{1}{2}$ and henceforth dropping this index.
Larger $j$ lead to a vanishingly small integrand in $I^{V}(E,\xi)$ near the quasiparticle pole $\epsilon=E+i0^{+}$, meaning that terms of higher angular momentum are strongly suppressed and can safely be discarded.

\subsubsection{At the limit of $Vb\rightarrow0$}

From this point on all energies occur in combination with $b$, the
size of the impurity. For the remaining analysis we therefore adopt
dimensionless variables $\tilde{E}=\frac{Eb}{\hbar v}$, $\tilde{V}=\frac{Vb}{\hbar v}$,
$\tilde{\Sigma}=\frac{\Sigma b}{\hbar v}$ etc. and $\tilde{T}=Tb^{2}\hbar v$,
$\tilde{n}_{i}=n_{i}b^{3}$, and drop the tildes. The first order
term $\langle\mathcal{V}\rangle$ in Eq.~(\ref{eq:offmassselfSEmatrixelements})
is only trivially dependent on $V$ and yields a real quantity,
\begin{align}
\mathcal{V}(\xi) & =4\pi V\frac{\xi^{2}-\sin^{2}\xi}{\xi^{4}}.
\end{align}
For a vanishingly small $V$, the integral $I^{V}(E,\xi)$ in Eq.~(\ref{eq:nightmare}) is to leading order independent of $V$ and can be done immediately by a contour integration.
Writing out the Bessel functions in Eq.~(\ref{eq:mainintegral}),
one obtains for $V=0$
\begin{align}
I^{0}(E,\xi) & =\frac{\sin^{2}\xi}{\xi^{2}}\int\!\!\frac{\mathrm{d}\epsilon}{\epsilon^{2}(\epsilon-\xi)^{2}(E-\epsilon)\xi^{2}}\bigl[\epsilon \xi\cos\epsilon\nonumber \\
 & \quad-(\xi-\epsilon+\epsilon \xi\cot \xi)\sin\epsilon\bigr]^{2}.
\end{align}
In the complex plane the integrand has poles at $\epsilon=0,E+i0^{+},\xi$.
Performing the integration of real and imaginary part of $I^{0}(E,\xi)$
yields
\begin{align}
\Re[I^{0}] & =\frac{\pi}{2}\frac{\sin^{2}\xi}{\xi^{2}}\biggl[\frac{2}{E-\xi}\left(\frac{1}{\sin^{2}\xi}-\frac{1}{\xi^{2}}\right)\nonumber \\
 & \quad-\frac{2D\cos2E+(D^{2}-1)\sin2E}{(E-\xi)^{2}}\biggr]
 \label{eq:BA1}\\
\Im[I^{0}] & =-\pi\int\!\frac{\mathrm{d}\epsilon\delta(\epsilon-E)}{\epsilon^{2}(\epsilon-\xi)^{2}}\frac{M^{2}(\epsilon,\xi)}{M^{2}(\epsilon,\epsilon)+N^{2}(\epsilon,\epsilon)}\nonumber\\
 & =-\frac{\pi\sin^{2}\xi}{\xi^{2}}\biggl[\frac{D\sin E+\cos E}{E-\xi}\biggr]^{2},
\label{eq:BA2}
\end{align}
where $D=\xi^{-1}-E^{-1}-\cot \xi$.
Thus, in the low energy limit by taking $E\rightarrow0$,
\begin{align}
\Im\Sigma_0(E,\xi)=-4\pi n_{i}V^{2}E^{2}\left[\frac{\sin \xi-\xi\cos \xi}{\xi^{3}}\right]^{2}.\label{eq:SE_correction_BAlimit}
\end{align}
On the mass shell ($\xi=E$) the imaginary part of the self-energy
becomes [cf. Eq.~(\ref{eq:bornscatter})],
\begin{align}
\Im\Sigma_0(E,E) & =-\frac{4\pi}{9}n_{i}V^{2}E^{2},\nonumber \\
\Im\frac{\Sigma_0 b}{\hbar v} & =-\frac{n_{i}b^{3}}{2}\frac{\sigma_{1/2}^{born}}{b^{2}},\label{eq:onmassshell_born}
\end{align}
where we have restored units in the last line for clarity.
This result connects the on-mass-shell self-energy $\Sigma_0$ to the scattering amplitude $f$ and the scattering crossection $\sigma$, thus recovering the optical theorem.
We point out that Eq.~(\ref{eq:SE_correction_BAlimit}) represents $\Im\Sigma$ in the Born approximation including the momentum dependence (encoded in $\xi$). Previous calculations instead regulated the self-energy integral by introducing a cutoff scale~\cite{Hosur2012,Biswas2014}.
The present formulation is advantageous as it allows the direct comparison with experimentally obtained parameters.
Additionally, for weak disorder, the formulas Eqs.~(\ref{eq:BA1},\ref{eq:BA2}) hold for all $E$ and can be used to study the electronic response without further restrictions, which is however outside the present scope.

\subsubsection{Resonant scattering}

For general values of $V$, the integral in Eq. (\ref{eq:nightmare})
is
\begin{align}
I^{V}(E,\xi) & =\frac{\sin^{2}\xi}{\xi^{2}}\int\!\frac{\mathrm{d}\epsilon}{(E-\epsilon)}\nonumber \\
 & \quad\times\frac{\left(\frac{1}{\epsilon-V}-\cot(\epsilon-V)-\frac{1}{\xi}+\cot(\xi)\right)^{2}}{(\epsilon-\xi-V)^{2}}\nonumber \\
 & \quad\times\frac{1}{1+\left(\frac{1}{\epsilon}-\frac{1}{\epsilon-V}+\cot(\epsilon-V)\right)^{2}}.
\end{align}
When the chemical potential is close to the Weyl point ($E\approx 0$), one can expand $V$ around the resonance $V_{r}$ by writing $V=n\pi+2E+\bar{\delta}$
with a small deviation $\bar{\delta}\sim E^{2}$ and a nonzero integer $n$. It is helpful to shift the integration variable $\epsilon\rightarrow\epsilon+2E+\bar{\delta}$,
leading to
\begin{align}
I^{V_{r}}(E,\xi) & =\frac{\sin^{2}\xi}{\xi^{2}}\int\!\frac{\mathrm{d}\epsilon}{(-E-\epsilon-\bar{\delta})}\nonumber \\
 & \quad\times\frac{\epsilon^{2}\left(\frac{1}{\epsilon-n\pi}-\cot(\epsilon)-\frac{1}{\xi}+\cot(\xi)\right)^{2}}{(\epsilon-\xi-n\pi)^{2}}\nonumber \\
 & \quad\times\frac{1}{\epsilon^{2}\left[1+\left(\frac{1}{\epsilon+2E+\bar{\delta}}-\frac{1}{\epsilon-n\pi}+\cot(\epsilon)\right)^{2}\right]}.\label{eq:integral_split}
\end{align}
We seek to evaluate this integral by residues. To this end, we first identify all poles in the upper half-plane of the complex variable $\epsilon$.
In Eq.~(\ref{eq:integral_split}), the denominator in the first line defines the quasiparticle pole, located at $\epsilon=+i0^{+}-E-\bar{\delta}$.
While the second line is singular for a multiple $m$ of $\pi$, i.~e. for $\epsilon=m\pi$ (where $m\neq0,n$), these poles are canceled by zeros of the term in the third line.
\begin{figure}
\centering\includegraphics[width=0.75\columnwidth]{polesandresidues2}
\protect\caption{Heat map of the modulus of the third term in Eq.~(\ref{eq:integral_split})
which shows the poles in the complex plane. The red dots signify the
pole location. In the upper half plane, the two poles closest to the
origin (marked by white arrows) are the ones kept in the approximation
of Eq.~(\ref{eq:mainapproximation}). The parameters used in the
plot are $n=1$ and $2E+\bar{\delta}=0.5$.\label{fig:heatmappoles}}
\end{figure}

We thus only have to take care of the poles introduced by the latter term, which are shown in Fig.~\ref{fig:heatmappoles}.
As it turns out, the leading behavior is recovered if one keeps the pole which is closest to the origin.
A minor complication is the fact that any approximation in the third term in Eq.~(\ref{eq:integral_split}) destroys the cancellation property between the terms of second and third line, which can be circumvented by replacing the second line
by its $\epsilon=0$ limit.
Approximating in the third line $\cot\epsilon\approx1/\epsilon$
and $1/(\epsilon-n\pi)\approx-1/n\pi$, the poles are given by a quartic polynomial.
Putting everything together, it is
\begin{align}
I^{V_{r}}(E,\xi) & \approx\frac{\sin^{2}\xi}{\xi^{2}}\int\!
\frac{\mathrm{d}\epsilon}{(-E-\epsilon-\bar{\delta})}\nonumber \\
 & \quad\times\frac{1}{(\xi+n\pi)^{2}}\nonumber \\
 &\quad\times\frac{1}{\epsilon^{2}+\left(\frac{\epsilon}{\epsilon+2E+\bar{\delta}}+\frac{\epsilon}{n\pi}+1\right)^{2}}\label{eq:mainapproximation}
\end{align}
While this integral can be solved, one can further expand in the pole
closest to the origin, which yields for the third line in Eq.~(\ref{eq:mainapproximation}),
$E^{2}/(E^{4}+4(\epsilon+E+\delta/2)^{2})$, with $\delta=\bar{\delta}+E^2/n\pi$. The end result can then be written as
\begin{align}
\Sigma_0(E,\xi)&=4n_iV^2I^{V_{r}}\approx
n_i\gamma_{0}(E,\delta)t^2(\xi),
\label{eq:integral_result}
\shortintertext{where we introduced the singular amplitude $\gamma_{0}$ and a regular function $t$,}
\gamma_{0}(E,\delta) & =\frac{-4\pi}{\delta-iE^{2}}
\label{eq:integral_resonnance}\\
t(\xi)&=\frac{(n\pi)\sin\xi}{\xi(\xi+n\pi)}.
\end{align}
It is $t(0)=1$. Importantly, the resonant amplitude factorizes with respect to its energy and momentum dependence. As shown in Fig. \ref{fig:resonancezeroorderSE}, the asymptotic
solution is in a good agreement with a numerical integration.
\begin{figure}
\includegraphics[width=1\columnwidth]{bare01} \protect\protect\caption{\label{fig:resonancezeroorderSE}Self-energy matrix elements near to resonance $V\rightarrow n\pi$ as a function of $\xi$.
$\Sigma_0(E,\xi)$ is obtained by numerical integration of Eq.~(\ref{eq:offmassselfSEmatrixelements})
and $\Sigma_{a}$ is plotted by replacing the integral on the right
hand side of Eq. (\ref{eq:nightmare}) by the solution given in Eq.~(\ref{eq:integral_result}).
The parameters used in the plot are: $V=\pi+0.016$, $E=0.01$ and
the impurity concentration $n_{i}=0.01$. The inset shows the behavior of the modulus of the integral Eq.~(\ref{eq:nightmare}) for large $\xi$ (red) compared to the approximation of Eq.~(\ref{eq:integral_resonnance})
(blue). For better visibility parameters less close to resonance were
chosen for the inset plot.}
\end{figure}

We can further check the validity of Eq.~(\ref{eq:integral_resonnance})
by a well defined identity on the mass-shell ($\xi=E$). To this end we use Eq.~(\ref{eq:general_SE}) to express $\Sigma_0(E,E)$ with the known matrix element $T_0(E,E,E)$ [Eq.~(\ref{eq:Tmatrixelement})],

\begin{align}
T_0(E,E,E) & =n_{i}^{-1}\Sigma_0(E,E)\nonumber \\
\frac{4\pi}{iE^{2}}\frac{M(\bar{E},E)}{M(\bar{E},E)+iN(\bar{E},E)} & =\mathcal{V}(E) +\nonumber \\
 & \hspace{-3.7cm}4V^{2}\int\!\frac{\mathrm{d}\epsilon}{\epsilon^{2}(\bar{\epsilon}-\epsilon)^{2}(E-\epsilon)}\frac{M^{2}(\bar{\epsilon},E)}{M^{2}(\bar{\epsilon},\epsilon)+N^{2}(\bar{\epsilon},\epsilon)},\label{eq:Tandsigmaequlity}
\end{align}
From this equality, we can verify both real and imaginary part of the self-energy. For the left-hand side of the identity, we obtain near to resonance
\begin{align}
T_0(E,E,E) & =-4\pi\frac{\tan(V-2E)+iE^{2}}{\tan^{2}(V-2E)+E^{4}}.\label{eq:tmatrixreson}
\end{align}
Using the approximation of Eq.~(\ref{eq:integral_resonnance}), the
right hand side of Eq.~(\ref{eq:Tandsigmaequlity}) yields
\begin{align}
\frac{\Sigma_0(E,E)}{n_i}
&=
\frac{4\pi}{3} V-4\pi
\frac{\delta+iE^2}{\delta^2+E^4}.
\label{eq:tmatrixreson2}
\end{align}
For a finite disorder strength ($V\approx n\pi$), the contribution $4\pi V/3$ in Eq.~(\ref{eq:tmatrixreson2}) from the impurity average $\left\langle \mathcal{V}\right\rangle $ is negligible against the contribution from resonant scattering, which diverges near to the resonance with $\delta^{-1}\sim E^{-2}$.
Keeping in mind that $V-2E\approx n\pi+\delta$, the correspondence between Eqs.~(\ref{eq:tmatrixreson}) and (\ref{eq:tmatrixreson2}) is manifest.

From the employed approximations we can deduce a restriction for $\xi$: Taking the $\epsilon\rightarrow0$ limit of the term in the second line of Eq.~(\ref{eq:integral_split}) is only valid as long as $\xi$ is not too close to a multiple of $\pi$, otherwise the $\cot \xi$ is divergent.
From the mismatch of this term with the other singular terms in the integrand we read of that this affects the integral as long as $|\xi-m\pi|<|2E+\delta|$, which can be seen in the inset plot in Fig.~\ref{fig:resonancezeroorderSE}.

We finally point out that in the limit $\xi\rightarrow\infty$ the integral $I^{V}(E,\xi)$ decays with $\xi^{-3}$, but the singular part as determined in Eq.~(\ref{eq:integral_resonnance}) actually vanishes
with $\xi^{-4}$, meaning that the effects of resonant scattering are only dominant over a finite range of energies $\xi$ (cf.~Fig.~\ref{fig:resonancezeroorderSE}).

From the results in Eqs. (\ref{eq:onmassshell_born}) and (\ref{eq:tmatrixreson2}), we have shown that at zero chemical potential, the imaginary part of the self-energy matrix elements goes to zero, unless exactly at resonance. While one might attempt to guess the impact of resonant scattering purely on scaling arguments, tuning directly to resonance is delicate as it sensitively depends on the order of limits taken. In particular, the vanishing DOS at the Weyl point leads to an very long mean free path, so that taking only a single scattering center into account is no longer enough.
To obtain a nonzero density of states under realistic conditions it is therefore necessary to non-perturbatively include further scattering processes beyond the T-matrix approximation, which is done in the following by the use of a self-consistent calculation.

\subsection{Self-consistent T-matrix approximation}
\label{sec:selfconsistent}
Our goal is to solve the following two equations self-consistently for $T(E)$ and $\Sigma(E)$,
\begin{align}
\Sigma(E)&=n_i T(E)\\
T(E)&=T_0(E)+T_0(E)\left[G(E)-G_0(E)\right]T(E).
\label{SCTmatrix}
\end{align}

Near to a resonance, the self-energy is dominated by the second term in Eq.~(\ref{eq:offmassselfSEmatrixelements}), so that it assumes the approximate form~[Eq.~(\ref{eq:integral_result})]
\begin{align}
\Sigma_0(E,\xi)&=
n_i\gamma_0(E,\delta)t^2(\xi).
\label{eq:sigma0}
\end{align}
As pointed out earlier, in this expression the dependence on $E$ and $\xi$ factorizes, which also entails that the singular part only depends on the on-shell energy $E$. This property allows for a crucial simplification of the self-consistency condition, Eq.~\ref{SCTmatrix} which we discuss now.
The first step is to generalize the procedure used earlier in the calculation of the self-energy.
It is straightforward, but somewhat tedious to show that the  T-matrix $T_0(\xi_1,E,\xi_2)$ can indeed be calculated in the very same fashion. In the limit $E\approx 0$ and close to resonant scattering, the result is
\begin{align}
T_0(\xi_1,E,\xi_2)&=
\gamma_0(E,\delta)t(\xi_1)t(\xi_2)\nonumber\\
&\quad \times 4\pi\sum_{j_z}
\langle j_z|\mathbf{v}_2\rangle\langle \mathbf{v}_1|j_z\rangle.
\label{eq:tmatrix0}
\end{align}
Secondly, we propose for the self-consistent T-matrix $T$ and for the self-energy $\Sigma$ that the singular part $\gamma_0(E,\delta)$ in Eqs.~(\ref{eq:sigma0}), (\ref{eq:tmatrix0}) is replaced by a self-consistent amplitude $\gamma(E,\delta)$, while the momentum dependence on $\xi$ remains the same, e.~g.
\begin{align}
\Sigma(E,\xi)&=
n_i\gamma(E,\delta)t^2(\xi).
\label{eq:sigmasc}
\end{align}
Inserting this assumption into the self-consistency equation [Eq.~(\ref{SCTmatrix})] yields
\begin{align}
\gamma&=\gamma_0+4\pi\gamma_0\gamma\int \frac{\epsilon^2\mathrm{d} \epsilon }{2(2\pi)^3}
t^2(\epsilon)\left(G(E,\epsilon)-G_0(E,\epsilon)\right),
\label{eq:GG0original}
\end{align}
where the arguments of $\gamma_0$ and $\gamma$ have been suppressed. Eq.~\ref{eq:GG0original} is a much simpler integral equation which is only a function of energy.
Under the integral the combination $\epsilon^2t^2(\epsilon)$ is of order one up to a distance $n\pi$ from the origin. The separate integration of both Green's functions in the difference $G-G_0$ is therefore sensitively dependent on the UV behavior of the Green's function, and thus the precise form of $\Sigma$. Most importantly, taking $\Sigma=const.$ in the Green's function $G(E,\epsilon)$ in Eq.~(\ref{eq:GG0original}) leads to an incorrect result. This becomes obvious by forming the common denominator and using Eq.~(\ref{eq:sigmasc}), with the result
\begin{align}
\frac{\gamma}{\gamma_0}&=1+\frac{n_i \gamma^2}{4\pi^2}I_4^{0},
\label{eq:SCreduced}\\
I_4^{0}&=\int \epsilon^2\mathrm{d} \epsilon\ t^4(\epsilon)
G(E,\epsilon)G_0(E,\epsilon).
\end{align}
The integral $I_4^{0}$ contains the much faster decaying combination $\epsilon^2t^4(\epsilon)$, meaning that the self-energy in the denominator of $G$ can now be assumed as  constant. Under the assumption that $E$ and $\Sigma$ are both small, the integration then yields to leading order
\begin{align}
I_4^{0}(E=0,\Sigma=0)=\frac{ 4\pi}{3}+\frac{10}{n^2\pi}.
\label{eq:I40approx}
\end{align}
Upon approaching the resonance for a fixed impurity density $n_i$, $\gamma_0$ diverges. In this case, Eq.~(\ref{eq:SCreduced}) leads to a universal amplitude given by $\gamma^2=-4\pi^2/n_iI_4^0$. Otherwise, but as long as $I_4^{0}$ remains constant, the solution is
\begin{align}
\gamma&=2\frac{1-\sqrt{1-c_\gamma\gamma_0^2}}
{c_\gamma\gamma_0},
\qquad\quad c_\gamma=\frac{n_i I_4^0}{\pi^2}.
\label{eq:SCgeneralgamma}
\end{align}
Inserting the result near resonance into the self-consistent self-energy, one obtains a non-zero imaginary part for any finite impurity density,
\begin{align}
\Im\Sigma(0,\xi)=-2\pi \sqrt{\frac{n_i}{I_4^0}}t^2(\xi).
\label{eq:finalscsigma}
\end{align}
The self-consistent self-energy is thus of order $\sim n_i^{1/2}$ which is indeed small, so that the approximation $\Sigma \approx 0$ which was used to evaluate Eq.~(\ref{eq:SCreduced}) is fulfilled.
The width of the resonant behavior is restricted by $1\sim c_\gamma\gamma_0^2$, which reduces to a condition for detuning $|\delta|\ll \sqrt{n_i}$ at the Weyl point. The self-energy $\Sigma(0,0)\equiv n_i\gamma$ using the function Eq.~(\ref{eq:SCgeneralgamma}) is shown in Fig.~\ref{fig:SCdetuning}.

\begin{figure}
\includegraphics[width=1\columnwidth]{SC04} \protect\protect\caption{\label{fig:SCdetuning}
Real and imaginary part of the self-consistent self-energy (dashed and full lines, respectively) as a function of detuning. The parameters are $n=1$, $E=0$ and $\xi=0$. The approximations limit the applicability of the resonant result to values $\delta\lesssim 1$.
}
\end{figure}

\section{Density of states and conductivity}

\label{sec:IV}

As it became clear in the previous section, multiple scattering leads to a finite imaginary part of the self-energy at the Weyl point. This alone implies a finite DOS and also affects the conductivity on a semiclassical level via the mean free path~\cite{Nandkishore2014}. However, with the knowledge of the momentum dependence we are now in the position to calculate these quantities based on a microscopic model. We note that the integral for the conductivity is more convergent than it was the case for the calculation of the self-energy, thus the generic form, Eq.~(\ref{eq:genericdccond}) is expected to also hold for resonant scattering. Less obvious is the expected impact of the vertex renormalization in this regime.

\subsection{Preliminaries}
Units are restored in this subsection. The density of states is given by
\begin{align}
\rho(E) & =-\frac{1}{\pi}\Im\Tr\int\!\!\frac{\mathrm{d}^{3}k}{(2\pi\hbar)^{3}}G^{R}(E,\bm{k}).
\end{align}
Switching to the helicity basis and performing trace and angular averaging leaves
\begin{align}
\rho(E)%
&=-\frac{1}{2(\hbar v \pi)^3}
\int\epsilon^{2}\mathrm{d}\epsilon\,\Im G^{R}(E,\epsilon).\label{eq:DOSenergies}
\end{align}
The same basis change can be applied to the Kubo formula for the longitudinal conductivity.
Given the electron charge $e$ and the current operator
$\bm{j}=ev\bm{\sigma}$, in the limit of zero frequency and momentum the conductivity is in momentum basis
\begin{align}
\sigma_{\alpha\alpha} & =\frac{\hbar}{\pi}\Tr\int\!\!\frac{\mathrm{d}^{3}k}{(2\pi\hbar)^{3}}\left[j_{\alpha}\Im G^{R}(E,\bm{k})j_{\alpha}\Im G^{R}(E,\bm{k})\right]\nonumber \\
 & =\frac{\hbar}{4\pi}\Tr\int\!\!\frac{\mathrm{d}^{3}k}{(2\pi\hbar)^{3}}\Bigl[2j_{\alpha}G^{A}j_{\alpha}G^{R}-j_{\alpha}G^{R}j_{\alpha}G^{R}\nonumber \\
 & \qquad\qquad\qquad\qquad\quad-j_{\alpha}G^{A}j_{\alpha}G^{A}\Bigr]
 \label{eq:ARconductivity}
\end{align}

In the following we will employ the ladder approximation, which corresponds to the sum of diagrams shown in Fig.~\ref{fig:ladder}.
This approximation neglects the effects of weak localization and allows to resum the series into a single bubble diagram containing a renormalized current vertex.

Due to the momentum structure, we forgo the calculation of the renormalized vertex and instead directly compute the $N$th-order current-current correlation function $\Pi_N$ at zero frequency and momentum. Except for $N=1$, which will be treated separately, the contribution involving both advanced and retarded Green's functions is finite on its own and no UV regulation from the purely retarded or advanced diagrams is required. However, the latter parts are nonzero and cannot be neglected entirely in the quantum limit.

\begin{figure}
\includegraphics[width=\columnwidth]{fig5}
\caption{Ladder approximation for the conductivity. The wiggly lines are attached to current vertices, the thick straight lines are dressed Weyl fermions.
\label{fig:ladder}}
\end{figure}

\begin{figure}
\centering\includegraphics[width=.7\columnwidth]{fig6}
\caption{The two building blocks of the conductivity calculation. The symbols are the same as in Fig.~\ref{fig:ladder}.
\label{fig:PiN}}
\end{figure}

The units for DOS and conductivity are $[\rho]=(\hbar v b^2)^{-1}$ and
$[\sigma_{\alpha\alpha}]=e^{2}/ \hbar b$.

\subsection{Density of states at the Weyl point}

It holds generally that $\Im G^{R}=\Im\Sigma^RG^RG^A$.
For nearly resonant scattering the self-energy is dominated by its imaginary part, since the real part vanishes at resonance [Eq.~(\ref{eq:finalscsigma})]. Thus, we proceed to drop the real part of the self-energy and estimate the DOS as
\begin{align}
\rho(E) & =-\frac{1}{2\pi^{3}}\int\!\epsilon^{2}\mathrm{d}\epsilon\,
\frac{\Im\Sigma^R(E,\epsilon)}{(E-\epsilon)^{2}+[\Im\Sigma^R(E,\epsilon)]^{2}}.
\end{align}
Within the SCTMA approximation for resonant scattering which was detailed in Sec~\ref{sec:III}, the self-energy factorizes with respect to energy and momentum. Consequently, in the integration the dependence on $\epsilon$ of the self-energy is regular, leading to a vanishing integrand near to $\epsilon\approx0$. Thus, the integral is \emph{not} dominated by the contribution of the quasiparticle pole, but by the behavior around $\epsilon=-n\pi$.
Approximating the self-energy in the numerator by its resonant form, $\Im\Sigma^R(E,\epsilon)=n_i\Im\gamma(E,\delta) t^2(\epsilon)$ and in the denominator by its value at $\epsilon=-n\pi$,
\begin{align}
-\Im\Sigma^R(-n\pi,0)=-n_i\Im\gamma(E,\delta)\equiv i s,
\end{align}
which also coincides with the value of $\Sigma$ at $\epsilon=0$, we obtain
\begin{align}
\rho(E)&=
\frac{s}{2\pi^3}I_2(E,s),
\label{eq:DOSresult1}\\
I_2(E,s)&=
\int\!\epsilon^{2}\mathrm{d}\epsilon\, t^2(\epsilon)G^R_s(E,\epsilon)G^A_s(E,\epsilon).
\label{eq:defI2}
\end{align}
In this expression, we introduced a simplified form of the Green's function, $G_s^{R/A}(E,\xi)=(E-\xi\pm i s)$.
It is important to note that similar to  Eq.~(\ref{eq:GG0original}), convergence is provided by the momentum dependence of the self-energy in the numerator.
The evaluation of $I_2$ is written out in appendix~\ref{sec:energyints}.
At the Weyl point and for vanishing impurity density, it is $I_2(0,0)=2\pi$, the density of states is thus of size $\rho(0)\sim \sqrt{n_i}$.
We point out that the condition for resonant scattering was $V_r=n\pi+2E+\mathcal{O}(E^2)$.
Therefore, the leading effect of a change in the chemical potential will be a detuning away from resonance. In particular, if the WSM contains impurities with a given potential which becomes resonant for $E=0$, the DOS develops a characteristic bump,
\begin{align}
\rho(E)&=-\frac{n_i}{\pi^2}\Im \gamma(E,0)\\
&\approx c_\rho \sqrt{n_i}-\frac{c_\rho'E^2}{\sqrt{n_i}}.
\label{eq:DOSLOA}
\end{align}
For an impurity potential $V=\pi$, it is $c_\rho=2(I_4^0)^{-1/2}/\pi=0.23$ and $c_\rho'=(I_4^0)^{-3/2}/4\pi=0.004$. At any finite disorder density $n_i$, there is a small additional shift of the resonance condition in the self-consistent calculation due to the change of $I_4^0$ with nonzero self-energy (cf. appendix ~\ref{sec:energyints}).
In many cases, the disorder is instead assumed to possess a smooth distribution function in the impurity potential strength. Then the bump in the DOS is smeared out, leaving only a bottoming-out of the DOS at a finite value. This was previously seen in numerical calculations using Gaussian or box disorder~\cite{Pixley2016a}.
The DOS in the presence of rare regions is shown in Fig.~\ref{fig:DOSbump}.

\begin{figure}
\includegraphics[width=\columnwidth]{DOS01}
\caption{DOS as a function of chemical potential, where $E=0$ is the Weyl point. The red line indicates the DOS induced by resonant scattering from rare regions of strength $V=\pi$ in the leading order approximation (LOA), Eq.~\ref{eq:DOSLOA}. Plotting Eq.~\ref{eq:DOSresult1} directly yields the gray curve, which is not particle hole symmetric, as the disorder distribution (DD) contains only impurities with the same potential. Here, the resonance is centered around $E=0$ for a choice $V=\pi+0.026$, the deviation from $\pi$ being due to a small shift in the resonance condition for the self-consistent case. A smooth DD would yield something close to the dashed line, where the resonance peak is smeared out. Parameters are $n_i=5\cdot 10^{-4}$, $n=1$.}
\label{fig:DOSbump}
\end{figure}

\subsection{Disorder enhanced conductivity}

While the Green's function is diagonal in the helicity basis, both the T-matrix $T$ and the current operator $j_\alpha$ are not.
Thus, each fermion line in the current bubble can be of  different helicity.
Additionally, each T-matrix introduces a sum over angular indices.
In the following, we compute the conductivity via $\tfrac{1}{3}(\sigma_{xx}+\sigma_{yy}+\sigma_{zz})$ in order to simplify some intermediate steps.
For the self-consistent self-energy at resonance the approximate expression of Eq.~(\ref{eq:finalscsigma}) will be used.

The T-matrix approximately factorizes with respect to both, the angular and the energy dependence of the in- and outgoing states.
This makes it possible to calculate each loop appearing in $\Pi_N$ independently, leaving only two distinct building blocks (Fig.~\ref{fig:PiN}), which  resemble a one-loop current vertex  and a four-point vertex, respectively.

Setting $j=1/2$, for $N=1$, it is necessary to keep all three contributions in Eq.~(\ref{eq:ARconductivity}) from the start to ensure convergence. The current vertices can be resolved straightforwardly:
\begin{align}
&\langle \mathbf{v}|\bm{\sigma}|\pm\mathbf{v}\rangle
\langle \pm\mathbf{v}|\bm{\sigma}|\mathbf{v}\rangle
\nonumber\\&=
\Tr\left[\bm{\sigma}|\pm\mathbf{v}\rangle
\langle \pm\mathbf{v}|\bm{\sigma}|\mathbf{v}\rangle
\langle \mathbf{v}|\right]
\nonumber\\&=
\tfrac{1}{4}\Tr\left[
\bm{\sigma}\cdot \bm{\sigma}\pm
\bm{\sigma}(\bm{\sigma}\cdot\mathbf{v})
\bm{\sigma}(\bm{\sigma}\cdot\mathbf{v})
\right]
\nonumber\\&=\tfrac{1}{4}(6\mp2|\mathbf{v}|^2)=\frac{3\mp 1}{2}.
\end{align}
After the angular integration one thus arrives at
\begin{align}
\Pi_1 & =\frac{1}{6\pi^{2}}\int\!\epsilon^{2}\mathrm{d}\epsilon\Bigl[\left(\Im G^{R}(E,\epsilon)\right)^{2}\nonumber \\
 & \quad+2\Im G^{R}(E,\epsilon)\Im G^{R}(E,-\epsilon)\Bigr],
\end{align}
where retarded and advanced functions have already been recombined. Pulling out the energy integrals, this becomes
\begin{align}
\Pi_1 & =\frac{s^2}{6\pi^{2}}
(I_4''+2I_4''')\\
I_4''&=\int\epsilon^{2}\mathrm{d}\epsilon\,
t(\epsilon)^4[G^R_s(E,\epsilon)]^2[G^A_s(E,\epsilon)]^2\\
I_4'''&=\int\epsilon^{2}\mathrm{d}\epsilon\,
\bigl[t^2(\epsilon)G^R_s(E,\epsilon)G^A_s(E,\epsilon)
\nonumber\\&\quad\times
t^2(-\epsilon)G^R_s(E,-\epsilon)G^A_s(E,-\epsilon)\bigr].
\end{align}
Close to the Weyl point and to leading order in the impurity density $n_i$, we obtain
\begin{align}
I_4''&=\frac{\pi}{2s}+\frac{\pi E^2}{2s^3}\\
I_4'''&=\frac{\pi}{2s},
\end{align}
which results in
\begin{align}
\sigma_1&=\frac{1}{12\pi^2}\frac{E^2+3s^2}{s}
\end{align}
This reproduces the result recently obtained for a generic energy dependent impurity scattering rate model, Eq. (\ref{eq:genericdccond}), including the numerical coefficient~\cite{Tabert2016}. We thus conclude that even for resonant scattering, the conductivity as given by the Kubo formula does not depend on the microscopic details of a short-ranged impurity except through the self-energy $\Sigma(0,0)$.

The ladder term $\widetilde{P}$ with external legs with indices $1,2,3,4$ and zero momentum transfer ($\mathbf{v}_1\xi_1=\mathbf{v}_4 \xi_4$, $\mathbf{v}_2\xi_2=\mathbf{v}_3 \xi_3$) is given by
\begin{widetext}
\begin{align}
\widetilde{P}(\xi_1,\xi_2,\xi_3,\xi_4)
&=n_i^2\langle\xi_1,\mathbf{v}_1|
\langle\xi_4,\mathbf{v}_4|
T(E)G^R(E)T^\dagger(E)T(E)G^A(E)T^\dagger(E)
|\xi_2,\mathbf{v}_2\rangle
|\xi_3,\mathbf{v}_3\rangle
\\
&=n_i|4\pi\gamma|^2
t(\xi_1)t(\xi_2)t(\xi_3)t(\xi_4)
\smashoperator{\sum_{j_{z1},j_{z2},j_{z3},j_{z4}}}{}
\langle\mathbf{v}_1|{j_{z1}}\rangle
\langle {j_{z2}}|\mathbf{v}_2\rangle
\langle\mathbf{v}_3| {j_{z3}}\rangle
\langle {j_{z4}}|\mathbf{v}_4\rangle
\hat{P}_{j_{z1}j_{z2}j_{z3}j_{z4}}
\notag\\
\hat{P}_{j_{z1}j_{z2}j_{z3}j_{z4}}&=
n_i|4\pi\gamma|^2
\int\!\! \frac{\epsilon^2\mathrm{d}\epsilon}{(2\pi)^3}
t(\epsilon)^4G^R(E,\epsilon)G^A(E,\epsilon)
\int \mathrm{d}\Omega_{\mathbf{v}} \langle {j_{z1}}|\mathbf{v}\rangle
\langle\mathbf{v}| {j_{z2}}\rangle
\langle {j_{z3}}|\mathbf{v}\rangle
\langle\mathbf{v}| {j_{z4}}\rangle+\dots
\end{align}
\end{widetext}
Here, the dots indicate that $\epsilon$ and $\mathbf{v}$ can be of dissimilar helicity in the upper and the lower line.
The angular integration yields
\begin{align}
\hat{P}_{1234}&=
(P+2P')\delta_{1,2}\delta_{3,4}
\notag\\&\quad
+(P-P')\delta_{1,4}\delta_{23},
\label{eq:ladderelem}
\end{align}
with
\begin{align}
P&=\frac{n_i|\gamma|^2}{12\pi^2}I_4(E,s)\qquad
P'=\frac{n_i|\gamma|^2}{12\pi^2}I_4'(E,s),
\end{align}
\begin{align}
I_4(E,s)&=\int\! \epsilon^2\mathrm{d}\epsilon\,
t(\epsilon)^4G^R_s(E,\epsilon)G^A_s(E,\epsilon)
\\
I_4'(E,s)&=\int\! \epsilon^2\mathrm{d}\epsilon\,
t(\epsilon)^2t(-\epsilon)^2
G^R_s(E,-\epsilon)G^A_s(E,\epsilon).
\end{align}
Based on the structure of $\hat{P}$ we propose for the ladder sum a kernel of the form
\begin{align}
D\delta_{1,2}\delta_{3,4}
+D'\delta_{1,4}\delta_{2,3}.
\end{align}
The recursion relation for the ladder sum then reads
\begin{align}
& D\delta_{1,2}\delta_{3,4}+
D'\delta_{1,4}\delta_{2,3}
\notag\\&=
\delta_{1,1}\delta_{3,4}
+
\sum_{a,b}
\Bigl[(D\delta_{a,2}\delta_{3,b}+
D'\delta_{a,b}\delta_{2,3})
\notag\\&\qquad\qquad\qquad\qquad\times
\bigl((P+2P')\delta_{1,a}\delta_{b,4}
\notag\\&\qquad\qquad\qquad\qquad+
(P-P')\delta_{1,4}
\delta_{a,b}\bigr)\Bigr],
\end{align}
the solution of which is
\begin{align}
D&=\frac{1}{1-P-2P'}\\
D'&=\frac{P-P'}{(1-3P)(1-P-2P')}.
\end{align}
The ladder is closed with a current vertex on both ends, with the closing loop containing the following integrals,
\begin{align}
\bm{C}_{j_{z1}j_{z4}}&=\biggl[\int \frac{\epsilon^2\mathrm{d}\epsilon}{(2\pi)^3}
[t(\epsilon)^2G^R(E,\epsilon)G^A(E,\epsilon)
\notag\\&\quad
\int \mathrm{d}\Omega_{\mathbf{v}}
\langle {j_{z1}}|\mathbf{v}\rangle
\langle \mathbf{v}|\bm{\sigma}|\mathbf{v}\rangle
\langle \mathbf{v}| {j_{z4}}\rangle+\dots\biggr],
\end{align}
where the dots again denote the summation over different helicities. The scalar product of left and right termination is
\begin{align}
\bm{C}_{14}\cdot
\bm{C}_{23}
&=\left(2\delta_{1,2}\delta_{3,4}
-\delta_{1,4}\delta_{2,3}\right)
\notag\\&\quad\times
\frac{1}{(2\pi)^6}\left(\frac{I_2}{6}+\frac{I_2'}{3}\right)^2.
\end{align}
$I_2$ was already defined in~(\ref{eq:defI2}), $I_2'$ is
\begin{align}
I_2'(E,s)&=\int\!\epsilon^2\mathrm{d}\epsilon\,
t(\epsilon)t(-\epsilon)
G^R_s(E,-\epsilon)G^A_s(E,\epsilon).
\end{align}
The energy integrals $I_4$, $I_4'$, $I_2$ and $I_2'$ are tabulated in appendix~\ref{sec:energyints}. For $E\approx0$ and $s\sim E$ they reduce to
\begin{align}
I_4(E,s)&=\frac{4\pi}{3}+\frac{10}{n^2\pi}\label{eq:diffpole}\\
I_4'(E,s)&=-\frac{2\pi}{3}-\frac{5}{4n^2\pi}\\
I_2(E,s)&=2\pi-\pi s-\frac{6E}{n}+\frac{\pi E^2}{s}\\
I_2'(E,s)&=-\pi+2\pi s
\end{align}
The ladder sum is then
\begin{align}
3\Pi^{AR}&=\frac{n_i|4\pi\gamma|^2}{6(2\pi)^6}\frac{(I_2+2I_2')^2}{1-P-2P'}\\
\Pi^{AR}&\approx\frac{n_i|\gamma|^2}{72\pi^4}\frac{(3\pi s+\pi E^2/s-6E/n)^2}{1-c_I},
\label{eq:laddersum}
\end{align}
with $c_I=5n_i|\gamma|^2/8n^2\pi^3$.
It was shown in Sec.~\ref{sec:selfconsistent} that the amplitude squared of the T-matrix approaches  $\gamma^2=-4\pi^2/n_iI_4^0$ at resonance , which means that for $n=1$ it is $c_I=0.108$. As a consistency check we show that these expressions reproduce the diffusion pole, confirming that the approximations taken for the determination of the T-matrix conserve charge.
For the diffuson, the terminating angular integrations yield a term $\sim\delta_{1,4}\delta_{2,3}$. After performing the trace, the result is thus proportional to
\begin{align}
\frac{1}{1-3P}.
\end{align}
The diffusion pole is recovered for $P=1/3$, which is true if $I_4=I_4^0$, as it is indeed the case to leading order in $E$ and $s$ [Eqs.~(\ref{eq:I40approx}, \ref{eq:diffpole})].
It is also possible to show more generally that the corresponding Ward identity is conserved asymptotically for $E\approx 0$, which is detailed in appendix~\ref{app:ward}.

We also calculated the doubly retarded ladder, which is of comparable size at the Weyl point. The result is
\begin{align}
\Pi^{RR}&=\frac{n_i|\gamma|^2}{72\pi^4}\frac{\left(\frac{6(E+is)}{n}\right)^2}{1-c_I}.
\end{align}

As it becomes clear from Eq.~(\ref{eq:laddersum}), the ladder sum is of order $s^2$ and thus subleading compared to the particle-hole bubble $\Pi_1$ if one approaches the Weyl point, $E\rightarrow 0$. This surprising behavior is not linked to the ladder being perturbatively irrelevant, at resonance each element is indeed of order 1 which results in a finite renormalization by $1-c_I$ in Eq.~(\ref{eq:laddersum}). Instead, a factor proportional to $s$ is introduced by each of the two enclosing current vertices.
This leads to the rather unusual situation that the semimetallicity becomes partially masked close to the Weyl point. In particular, the conductivity is given just by the Kubo formula, like in a metal with isotropic scattering. On the other hand, the diffusion coefficient is depending on $I_4$, where the special properties of the scattering matrix for a Weyl dispersion enters explicitly.

In summary, using a self-consistent self-energy, the vertex renormalization has no effect to leading order in the impurity density, in spite of the fact that the scattering matrix  remains anisotropic.
This finding applies exclusively to the quantum limit, in the limit where the Boltzmann approach works and $E^2/s$ is much larger than $s$, the scattering amplitude is $\gamma_0$ and the ladder leads to the expected replacement of the quasiparticle lifetime $\tau$ by transport lifetime $\tau_{tr}$ in the expression for the conductivity. For resonant scattering and if $E^2 \gg s$, the ratio is
\begin{align}
\frac{\tau_{tr}}{\tau}&=1+\frac{1}{3(1-\frac{n_i|\gamma_0|^2}{12\pi^2}\frac{\pi E^2}{s})}=\frac{3}{2}.
\end{align}
We note that this result has only a limited region of applicability, as resonant scattering eventually becomes negligible for a large enough DOS away from the Weyl point. In particular, there is not necessarily any intermediate regime where the scattering rate due to a T-matrix with amplitude $\gamma_0$ is larger than the one given by the SCBA. Importantly, the same ratio $\tau_{tr}/\tau=3/2$ was earlier obtained within SCBA~\cite{Biswas2014}.

Putting everything together and restoring units, in the presence of resonant scattering the conductivity is at zero frequency given by
\begin{align}
\sigma_{\alpha\alpha}
&=\frac{e^2}{\hbar^2 v}\frac{3s^2+E^2}{12\pi^2 s},
\label{eq:finalconductivity}
\end{align}
where the scattering rate in units of energy is $s=2\pi\hbar v\sqrt{n_ib/I_4^0}$.
This result was obtained using a rectangular impurity potential, but we do not expect that a generic short-ranged potential will change the expression.
This can be concluded from the value of $I_4^0$, which is essentially determined by the large scattering amplitudes at $E=0$ and $Eb=-n\pi\hbar v$ and does in leading order not depend on the microscopic details of the impurity.
However, we point out that the conductivity is not dimensionless in 3D, therefore the length scale $b$ still enters explicitly in Eq.~(\ref{eq:finalconductivity}). For a general impurity potential this length scale is given by the effective size of an impurity site.

Compared to the previous estimate of the mean free path for hopping between rare regions $\ell\sim (n_i b^2)^{-1}$~\cite{Nandkishore2014}, it turned out to be of size $(n_ib)^{-1/2}$. In the presence of disorder with an Gaussian impurity potential distribution this finding implies that the crossover from the SCBA regime to one dominated by resonant scattering happens at comparatively higher scales than previously thought.
As one important consequence we estimate how the conductivity saturates near the Weyl point as a function of decreasing chemical potential. As discussed in the calculation of the DOS, a smooth disorder distribution smears out the effects of detuning. The mean free path due to resonant scattering is shorter than the one given in SCBA once $E<\hbar v(n_ib)^{1/4}$. Continuing to smaller scales, Eq.~\ref{eq:finalconductivity} tells us that the semiclassical term $E^2/s$ is comparable to the part of size $s$ if $s\sim E$, i.e. it is irrelevant once $E<\hbar v\sqrt{n_ib}$. Below this scale, the conductivity approaches a residual value ($n=1$)
\begin{align}
\sigma_{min}&=0.06 N\sqrt{n_i b}\frac{e^2}{\hbar} .
\label{eq:rescond}
\end{align}
So far the focus was on a single Weyl cone ($N=1$), in any real material $N$ is even and can be as large as 24~\cite{Wan2011}.

\section{Conclusion}
\label{sec:V}
We have presented the solution to the scattering problem of chiral fermions in a WSM close to resonant scattering.
The singular amplitude of the T-matrix was found to factorize with respect to energy and momentum dependence, which made it possible to also determine the T-matrix in a self-consistent fashion. Using the self-consistent solution, which is non-analytic in disorder strength, a finite DOS emerges at the Weyl point. Interestingly, the lifting of the semimetallic DOS turns out to be large enough to render vertex corrections to the Kubo estimate of the conductivity to be irrelevant. This was demonstrated in a conserving approximation.

The main consequence for experiment is the predicted residual conductivity at charge neutrality [Eq.~(\ref{eq:rescond})], which scales with the square root of the impurity density and is thus qualitatively larger than previously estimated.

Apart from the consequences for zero temperature presented here, in the future the same calculational scheme can be easily adapted to calculate finite T and finite-$\omega$ transport properties in a microscopic model. This can complement studies with generic scattering models~\cite{Lundgren2014,Tabert2016} by supplying a detailed understanding of the underlying scattering amplitudes. Another interesting possibility is the study of boundary effects, which is important to estimate the stability of topological properties of a dirty WSM.

\section*{Acknowledgments}
We thank E. Khalaf for useful discussions. T. H. is supported by the Minerva foundation.

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You'll Have to Stand to Hear about the Amazing Things Happening with Digital Portfolios
This past November was an exciting month where teachers from Cuba-Rushford and CABOCES, Linda Botens and Justine Lombardi respectively, alongside Betsy Hardy, Staff Specialist for Distance Learning at CABOCES and this year's presenter at iNACOL, delivered a gripping presentation about developing digital portfolios in the classroom. The room was filled to capacity and the audience was spilling out into the hallway to hear about the possibilities for using digital portfolios in the classroom.
In their presentation, "The Paper Extinction and the Rise of the ePortfolio," Linda, Justine, and Betsy showed multiple examples of students' work throughout the years that allowed students to show off their talents to prospective colleges and employers and explore new talents that some students might have otherwise been too shy to share.
Linda Botens shared one of her public speaking student's forays into product endorsement and advertising. Normally, this student is shy, but when given the option to express herself on video, her dramatic personality had the opportunity to be shared with her teacher and peers.
Justine shared her uses of e-portfolios in her online Mythology and Folklore class and World Religions class that she teaches every year as elective options. In her classes, students share papers and projects, alongside art and presentations and have a way to group them all together and share with both teachers and peers. Justine also shared the uses of e-portfolios in the CTE classroom where students working on welding projects can photograph their talents and share them along with pictures of their earned certifications and share with their future employers.
By: Christina McGee, CA BOCES Learning Resources

redpajama_set_name: RedPajamaCommonCrawl


We have pop up booths of three shape-- oval, triangle and cyclinder-- that are designed to meet the trade shows, conferences, conventions and exhibition needs. The pop up booth always has a flat top which can serve as a product display shelf and you can put your products on it for exhibition. With a unique outlook, the square pop up booths are great for both indoor and outdoor events.
It is really effective to attract your potential customers and clients with a square pop up booth as it boasts good-tailored shape and vibrant graphics. An eye-catching and traffic-stopping pop up booth can immediately arise people's curiosity. Also, our trade show stands feature high resolution graphics and splendid life-like images which will entice people around to come to your trade show booth and see the products that you offer. Just as its name indicates, the pop up booth can be popped up to full size with ease and you can set it up with seconds. Meanwhile, this trade show stand is always lightweight and you can move it from one place to another, quickly.
Ask about Fabric Pop Up Booth Square and get answer in email.


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WHAT'S ON AUTHORS P2
WHATS ON AUTHORS P3
Show me the Gold!
THE QUEENSLAND CABARET FESTIVAL: DYNAMIC DUO – DODGY DUETS DONE DECENTLY. The Old Museum, Fortitude Valley, BRISBANE, Saturday, 15 June 2019 from 9.30 p.m. to 10.15 p.m. Tickets: Adult $25, Concession/Student (card holders) $20. Vocal powerhouses, BRIAN CAVANAGH and ALISON ST LEDGER are on a mission to avenge crimes against music to bring you dodgy duets done decently or decent duets done dodgilly – you be the judge. Find information about how to book your tickets at https://www.oldmuseum.org/event/dodgyduets (call the Old Museum Box Office on (07) 3257 4089 if you encounter problems making a booking). (Photo courtesy of The Queensland Cabaret Festival)
This page is devoted to events and activities of interest to writers, illustrators and readers! Go to entries below the quick list for more detailed information.
MARK MANSON is the #1 New York Times bestselling author of The Subtle Art of Not Giving a F*ck, the mega-bestseller that has been translated into over 50 languages and sold more than seven million copies worldwide. Hear Mark talk about his new book Everything is F*cked: A Book About Hope, when he returns to the Brisbane Powerhouse, New Farm, on Thursday, 18 July 2019 at 7.30 p.m.
The Event: VICKY'S WINTER BOOKCHAT
Date and time: Monday, 15 July 2019 at 6.30 p.m.
Venue: Riverbend Books, 193 Oxford St, Bulimba Queensland 4171.
Ticket price: $10.00. Phone: (07) 3899 8555, email info@riverbendbooks.com.au or go online at http://riverbendbooks.com.au and click on Events to book your place(s). PLEASE NOTE: Riverbend does not issue paper tickets for events; your name will be on a list at the door, and they do not issue refunds or exchanges for event tickets except in case of event cancellation.
What's happening: As the days get shorter and the nights get colder, settling in with a good book is compulsory behaviour! To help you make your selections from the ever growing list of good reads Riverbend have in store, SUZY AND VICKY will be hosting some WINTER BOOKCHATS! Come along, have a glass of wine and settle in for a cosy evening of literary chat! Your ticket price includes a drink and 10% off all books.
The Event: MARK MANSON – #1 NEW YORK TIMES BESTSELLING AUTHOR – EVERTHING IS F*CKED TOUR
Date and time: Thursday, 18 July 2019 at 7.30 p.m.
Venue: Powerhouse Theatre, Brisbane Powerhouse, 119 Lamington Street, New Farm QLD 4005. Click on link for a map.
Ticket price: Adult: $50.90 (plus transaction fee of $6.90). Ticket and Book $72, and VIP Meet and Greet $200. Note that this session is suitable for ages 15+. For more information and online bookings go to the Brisbane Powerhouse website at https://brisbanepowerhouse.org/events/2019/07/18/mark-manson.
What's happening: MARK MANSON is the #1 New York Times bestselling author of The Subtle Art of Not Giving a F*ck, the mega-bestseller that has been translated into over 50 languages and sold more than seven million copies worldwide. Mark runs one of the largest personal growth websites in the world, MarkManson.net, a blog with more than one million monthly readers. His work has been published in Time magazine, Forbes, Vice, CNN, and Vox, and many others.
Now Mark Manson returns to Brisbane Powerhouse with his new book, Everything is F*cked: A Book About Hope, which was published on 14 May 2019. Join Mark as he offers up self-help for people who hate self-help – a no-BS brand of life advice and cultural commentary that has struck a chord with people around the world.
The Event: LOCAL HISTORY NIGHT WITH AUTHOR DUNCAN RICHARDSON
Date and time: Thursday, 18 July 2019 from 6.00 p.m. to 7.00 p.m.
Venue: Annerley Library.
Ticket price: FREE to attend, but bookings are essential. Ring the Annerley Library on (07) 3403 1735 to reserve your place(s).
What's happening: Discover the many forgotten voices preserved in letters, diaries, transcripts and memoirs, as presented by DUNCAN RICHARDSON, author of Year of Disasters – Brisbane 1864: Fire, plague, cyclone and flood. This session is presented in conjunction with The Annerley Stephens History Group Inc.
The Event: MEET AUTHOR AMANDA HAMPSON
Date and time: Friday, 19 July 2019 from 10.00 a.m. to 11.00 a.m.
Venue: Carindale Library.
Ticket price: FREE to attend, but bookings are essential. Ring the Carindale Library on (07) 3407 1490 to reserve your place(s).
What's happening: Catch up with author AMANDA HAMPSON to talk about her novel Sixty Summers, a story about three women in their sixties looking back on their lives and taking stock. When Maggie, Fran and Rose met in their youth, they had dreams and ambitions. Forty years later, the three friends are turning sixty, each of them restless and disenchanted with their lives.
Fran works in a second-hand bookshop. Her lover, one of a long line of disappointing men, is drifting away and her future is uncertain. Maggie married into a volatile family. Her beautiful, indulged twin daughters are causing havoc, and her elderly mother-in-law has moved in and is taking charge. Rose has been an offsider for her hopelessly vague but academically brilliant husband and their two sons. Time is running out to find and fulfil ambitions of her own.
In an attempt to recapture the sense of freedom and purpose they once possessed, the three women decide to retrace the steps of their 1978 backpacking trip through Europe, and set off on an odyssey that will test their friendship, challenge their beliefs and redefine the third age of their lives. Books will be available for purchase on the day, or bring your copy from home for Amanda to sign.
The Event: FAIRFIELD WRITERS GROUP BOOK LAUNCH: LOVE EVENTUALLY
Date and time: Saturday, 20 July 2019 from 10.30 a.m. to 12.30 p.m.
Venue: Fairfield Library.
Ticket price: FREE to attend, but bookings are essential. Ring the Fairfield Library on (07) 3403 8615 to reserve your place(s).
What's happening: Join the FAIRFIELD WRITERS GROUP for the launch of their sixth story anthology LOVE EVENTUALLY, an eclectic mix of love stories which extends beyond the traditional romance genre. They promise something for every taste. Books will be available for purchase and signing at the event.
The Event: WORKSHOP: INTRODUCTION TO CREATIVE WRITING WITH THE QWC
Date and time: Saturday, 20 July 2019 from 1.00 p.m. to 3.30 p.m.
Venue: Sunnybank Hills Library.
Ticket price: FREE to attend, but bookings are essential. Ring the Sunnybank Hills Library on (07) 3407 0571 to reserve your place(s).
What's happening: In this INTRODUCTION TO CREATIVE WRITING WORKSHOP with the Queensland Writers Centre (QWC), you will explore creative writing basics and come away with tools and exercises to nurture your inner storyteller. This session is ideal for beginners or those looking to brush up their skills.
THE QUEENSLAND CABARET FESTIVAL: MEGAN MULLALLY AND HER BAND, NANCY AND BETH. Concert Hall, Queensland Performing Arts Centre, SOUTH BRISBANE, Wednesday, 19 June 2019 from 8.00 p.m. to 9.30 p.m. Tickets: $85–$105. The band formally known as NANCY AND BETH is the punk/vaudeville heart-child of Megan Mullally (Will & Grace, Parks & Recreation) and Stephanie Hunt (Friday Night Lights, Californication). The two friends, Nancy and Beth, described by Noel Gallagher as 'definitely not bigger than The Beatles,' pick dope songs and then sing them while dancing and wearing matching costumes. NANCY AND BETH IS… Megan Mullally (vocals, choreography), Stephanie Hunt (vocals), Datri Bean (keyboards, vocals), Joe Berardi (drums), Petra Haden (strings, vocals), Roy Williams (guitar, vocals), and Andrew Pressman (bass, vocals). Book your tickets at https://www.qpac.com.au/event/megan_mullally_19. (Photo courtesy of The Queensland Cabaret Festival)


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Mark Scharf's "The Whispers of Saints" in New York City
October 18, 2016 March 6, 2012 by Baltimore Playwrights Festival
The Oberon Theatre Ensemble will present a staged reading of Mark Scharf's play The Whispers of Saints on Sunday, March 25th at 7 p.m. at The Lambs (America's First Professional Theatre Club – est. 1874) located at 3 W. 51st Street, New York City.
The Whispers of Saints won both Best Play and Best Production in the 2002 Baltimore Playwrights Festival.
BPF Alumna Gets Reading at The Strand 3/10/2012
Our Lady of Sandwich, by Mario Baldessari and Keith Bridges, to be produced in Washington, DC


Hello again golfers and welcome to our latest issue, it's been another mixed week in terms of the weather but at least the evenings are getting a bit lighter and we're nearly through the month of January - which is never the best month for golf in this country! Despite the poor weather it's been a busy week in the shop with new products arriving almost daily and when all this new kit starts arriving it really does mean that the golf season is only just around the corner.
Here's a fun little quiz to start this week's newsletter, everybody knows that a one under par score on a hole is called a birdie, two under is an eagle and three under par is an albatross, but who can tell me what a four under par score is called? It's only possible on a par five hole but it has happened, click here to send me your answers (and no googling!) and the first correct guess will win a nice little prize!
Tim Barnes and Martin Major had been quiet in the Winter League but they came to life on Saturday as they fired a fantastic 25 point total, after a par at the first hole they fired birdies on the next four holes, a par on six was followed by two more birdies so they were stood on the ninth tee at six under par gross - a double bogey on the last hole brought them back down to earth but by then they had more than enough points to take top spot.
There was a log jam of six pairs on 22 points but Pete Wildin and Ken Powell had the best back six to take second and third place went to Antony Green and Dave Johnson.
The next round of the Winter League is coming up this Sunday so the leading positions could well change again!
Please note that the club AGM and winter prize presentation evening has been moved to a new date as it originally clashed with Easter Sunday - it will now be on Sunday 20th March at 6pm with a nine hole competition beforehand.
It's an exciting time of year in the pro-shop as there are boxes arriving every day containing new stock, just this week so far we've had new shoes arrive from Nike and FootJoy, new clothing from Nike (worn by Rory McIlroy last week in Abu Dhabi), new Callaway XR OS irons and hybrids and Scotty Cameron putters. This is just the tip of the iceberg as more and more new kit will be arriving in the next few weeks so look out for lots of new stock in the shop.
This week we thought we would highlight the new Scotty Cameron putters that have arrived, a lot of golfers will spend a fortune on the latest driver but use an old beat up putter that they found at a car boot sale! Remember the old adage 'drive for show, putt for dough' - you will undoubtedly use your putter more than any other club in your bag in a round of golf so it's worth making sure you have the right one!
There is no denying that the Scotty Cameron putters are expensive but are they worth the money? When you consider the number of shots you can save on the green I would argue that the putter is the most important club in your bag so it's worth investing in a decent one! It's no coincidence that many of the world's leading professionals (including Jordan Spieth) put their trust in a Scotty Cameron putter and with their deep milled face they offer a softer feel than any other putter on the market.
A further positive aspect this year is that the range of Scotty Cameron putters has increased so that there are more head shapes available from traditional through to mallet, but also face balanced and heel balanced and even counter balanced designs so that you can really make sure you get the right putter to suit your stroke. If you're not sure about which one will suit you best that is where we come in! We can give you some expert advice to help you chose the right putter and if you do it's virtually guaranteed to help you knock shots off your game!
Call in now to have a look and we would be most happy to take you through the range.
Sharon Dunning had a nice surprise at the weekend as she received this lovely new Motocaddy S1 Pro electric trolley, it was a surprise birthday present from husband Neil and as you can see she looked chuffed to bits to find the new trolley waiting for her in the clubhouse!
A lot has been spoken recently of the 'big three' at the top of our game and how they're going to be battling it out for years to come, but has someone else gate-crashed the party? 27-year-old Rickie Fowler fired himself to fourth in the world rankings after producing some magical golf to claim the Abu Dhabi HSBC Golf Championship last weekend. It seems like he's been around for a long time now, but being in his mid-20s he is still just a young pup with numerous victories and vast experience behind him. Watch out Jordan, Jason and Rory, and one things for sure - it promises to be a great season!
To start the new year our PGA coach Adam Powell has a number of junior golf tuition sessions coming up, Adam has already helped hundreds of kids have a go at golf and his popular Academies return with a new batch of dates along with some FREE taster sessions for children that have never tried golf before and would like to give it a go. There are sessions at various times including evenings, weekends and school holidays, please click on the links to have a look at the full details.
The seniors played an 18 hole Stableford on Tuesday, conditions were windy to say the least but the rain stayed away and conditions did nothing to deter winner Roy Strefford. 19 handicapper Roy has had a great few months with four top-three finishes in his last five seniors competitions and this week he scored 41 points to take the victory. Roy just edged out another form man in Bob Bowen, Bob had two birdie two's in his round but just lost out on first place after a card play-off. In third place was Andy Margeson on 40 points. Next Tuesday the seniors play a nine hole Stableford.
The ladies got back to playing competition golf last week as they played a Thursday 9 hole Stableford and then a Sunday 9 hole Stableford.
On Thursday conditions weren't ideal as there was an icy wind blowing across the golf course, Joyce Graham battle through the cold the best to score 16 points, in second place was Jean Jones on 15 points and in third place was Alison Worrall on 14 points.
Here are Joyce and Jean enjoying the warmth of the shop after their round!
On Sunday it was a lot milder for the ladies and Jean Clare came out as the winner with a 20 point score, made more impressive by the fact that she didn't even score on the first hole! In second place was Sue Gunning on 18 points and in third place was Di Sommerville on 18 points.
The ladies play a match this Sunday in the first round of the Mail on Sunday Classic at home against Frodsham, best of the luck to the ladies involved!