text:
\section{Introduction}
\label{section:Intro}

Development of nanotechnology achieved during the last decade allowed the design of semiconductor microcavities possessing extremely high Q--factors (more than 10000). This opened new opportunities in creation and investigation of fundamental properties of Bose-Einstein condensates (BEC) of exciton-polaritons -- hybrid light-matter quasiparticles emerging in the regime of strong coupling~\cite{Hui}. Although real thermodynamic equilibrium in polariton condensates is never achieved and thus they are fundamentally different from atomic BECs, they exhibit main properties inherent to weakly interacting quantum Bose-gases.  Among them is superfluidity ~\cite{Carusotto,AmoSuperfluidity}, formation of quantized vortices ~\cite{Lagoudakis} and solitons \cite{Sich}, Josephson oscillations and macroscopic self trapping \cite{AmoSelfTrapping}, spin-Hall effect ~\cite{OSHENPhys} and others. The peculiarity of spin structure of polaritons combined with strong polariton-polariton interactions and large coherence lengths makes possible the generation of coherent bosonic spin currents ~\cite{Shelykh-SST-2010}. This opens a new research field of light-mediated spin effects and paves the way for their implementation in optoelectronics, e.g. for the creation of all-optical integrated circuits ~\cite{Ballarini,Sturm-Nature-2014}.

Polariton systems possess sevaral advantages with respect to systems based on cold atoms. First, extremely small mass of the polaritons (about $10^{-5}$ of the mass of free electrons) makes critical temperatures of the observation of quantum collective effects surprisingly high (from a few Kelvin for GaAs based structures to room temperature for GaN structures \cite{RoomTemp}). Besides, polariton condensates allow reasonably simple manipulation by application of the external electric and magnetic fields ~\cite{Schneider, SchneiderSciRep}. This plays essential role in study of the fundamental properties of exciton-polariton condensates. In general case magnetic field affects exciton-polariton emission energy, linewidth and intensity due to the exciton energy shift caused by Zeeman splitting in circular polarizations, modification of the exciton-photon coupling strength \cite{ShelykhIvchenko}, and modification of  scattering process with acoustic phonons ~\cite{Pietka}.  Strong spin anisotropy of polariton-polariton interactions, however, make those dependencies in the non-linear regime highly non-trivial.  In particular,  in Ref.~\cite{Rubo-PLA-2006} it was shown that below some critical value $B_{c}^{} $ of magnetic field  depending on polariton concentration the so-called full paramagnetic screening (also known as spin Meissner effect) occurs. Its signature is independence of the photoluminescence energy on the magnetic field. The latter however affects the polarization of the emission. Its ellipticity gradually changes until the value $B_{c}^{}$ is reached. At this point emission becomes fully circular polarized and Zeeman splitting re-establishes. Main efforts of recent experimental studies of exciton-polariton condensates in magnetic field have been successfully directed towards confirmation of these seminal peculiarities, cf. ~\cite{Pietka,Larionov-PRL-2010,Sturm-PRB-2015,Walker-PRL-2011,Fisher-PRL-2014}.

The interplay between polarization splitting and anisotropic polariton-polariton interactions becomes more tricky in anisotropic cavities when  additional energy splittings in linear polarizations (TE-TM splittings) is present in addition to the Zeeman splitting~\cite{Shelykh-Superlattices}. The situation becomes even more interesting when polaritons are confined in non-simply connected region, e.g. inside ring resonator. In this case the direction of the effective magnetic field provided by TE-TM splitting becomes position-dependent which combined together with magnetic field induced Zeeman splitting leads to the appearance of the geometric Berry phase responsible for generation of synthetic U(1) gauge field for polaritons and possibility of observation of optical analog of Aharonov-Bohm effect ~\cite{Shelykh-PRL-2009}. It should be noted that exciton-polariton spinor BEC in a ring geometry have been experimentally demonstrated by several groups ~\cite{Sturm-Nature-2014, Larionov-PRL-2010, Sturm-PRB-2015, Walker-PRL-2011, Liu-PNAS-2015}. However, polarization properties of interacting spinor polaritons in the rings were not subject of theoretical investigation up to now for the best of our knowledge. On the other hand, the presence of artificial U(1) gauge potential can lead to the onset of the persistent current in the system, i.e. its ground state can be a vortex-type solution. The investigation of the analogs of spin Meissner effect for such states with quantized angular momentum is a fundamentally interesting problem which can in principle lead to applications such as polariton analogue of flux qubits.

Spinor vortex-type solutions in 2D systems were analyzed in Ref.~\cite{Rubo-PRL-2007}. It was shown that besides normal vortices for which both circular polarization components have same non-zero quantized angular momentum, half-vortex solutions for which one of the circular polarizations is not rotating. Half-vortices had been detected experimentally~\cite{Lagoudakis-Science-2009}, but their stability in 2D condensate in presence of TE-TM splitting remained a topic of a debate~\cite{Flayac-PRB-2010, Solano-Rubo-comment, Flayac-reply}. The current view is that small TE-TM splitting does not destroy half-vortices leading only to their warping ~\cite{Solano-PRB-2014}. However, large TE- TM splittings can make in principle half vortex solutions instable. This situation may become relevant when polaritons are confined in the ring, where relevant splittings can reach the values of 1-2 meV for ring thicknesses about 1 micron \cite{Kuther-PRB-1998}.  Recent experimental work on half-vortices in a ring geometry~\cite{Liu-PNAS-2015} demonstrated that polarization patterns and density profiles could not be explained by the existing theory, which led the authors to a conclusion that their experimental configuration corresponds to some spurious superposition of certain "elementary" half-vortex states.

The aim of this paper is to provide a complete theory of interacting non-simply connected polariton BEC, introduce a concept of topological spin Meissner effect and describe rich variety of the states of the condensate both in presence and absence of the TE-TM splitting. In particular, we provide detailed analysis of half-vortex states in the ring and show that superpositions of elementary half-vortex states reported experimentally in Ref.~\cite{Liu-PNAS-2015} appear naturally in the developed theory.

The paper is organized as follows. In Section II we present the model of exciton-polariton condensate in a ring and introduce various types of emerging solutions. In Section III we introduce the topological spin Meissner effect in the case when TE-TM splitting is absent. In Section IV and V we extend the concepts and solutions obtained in the previous section to the general case of finite TE-TM splitting. More specifically, Section IV deals with topological spin Meissner states in the form of constant amplitude solutions and in Section V we study states which spontaneously break the rotational symmetry of the system due to presence of TE-TM splitting. Section VI contains discussion of the experimental relevance of our parameters.

\section{Model and classification of solutions}
\label{sec:model}

Interacting polaritons trapped in a quasi one-dimensional ring resonator  can be described by the following system of dimensionless Gross-Pitaevskii equations (see Appendix~\ref{app:tetm}),
\begin{equation}
\begin{dcases}
i\dot\psi_+= - \partial_x^2\psi_+ + \left( |\psi_+|^2 + \alpha |\psi_-|^2 \right)\psi_+ \\
\quad\quad\quad\quad\quad\quad\quad + \Omega\psi_+  + \kappa e^{-2 i x} \psi_-,
\\
i\dot\psi_- = - \partial_x^2\psi_- + \left( |\psi_-|^2 + \alpha |\psi_+|^2  \right)\psi_-
\\
\quad\quad\quad\quad\quad\quad\quad - \Omega\psi_- + \kappa e^{2 i x} \psi_+.
\end{dcases}
\label{GP-ring}
\end{equation}
Here, $\psi_{\pm}$ are the components of the  exciton-polariton spinor wavefunction $\pmb\psi\equiv\{\psi_+,\psi_-\}$ in the basis of circular polarizations satisfying $\psi_{\pm}(t,x)=\psi_{\pm}(t,x+2\pi)$, parameter $\alpha<0$ characterizes attractive interaction of the cross-polarized polaritons, $\Omega$ is half of Zeeman splitting of a free polariton state (which we will refer to as just "magnetic field") and $\kappa$ is half of the momentum independent TE-TM energy splitting. Parameters $\Omega$ and $\kappa$ are dimensionless and scale in units of $\hbar^2/(2 m^* R^2)$, where $R$ is the ring radius  and $m^*$ is the exciton-polariton effective mass.
We use the dimensionless  particles density per unit length, $\rho\equiv \frac{1}{2\pi}\int_0^{2\pi}\left(|\psi_+|^2+|\psi_-|^2\right)dx$, as a parameter controlling strength of the polariton-polariton interactions.

To study stationary states of the system~\eqref{GP-ring} we use the substitution
\begin{equation}
\psi_{\pm}(t,x) =
\psi_{\pm}(x) e^{-i\mu t}.
\end{equation}
We treat $\mu$ as an unknown variable corresponding to the energy blue shift of a photoluminescence line of the condensate in a steady state~\cite{Kavokin-Microcavities},  found for a given $\rho$. It is also identical to the chemical potential
parameter used in the literature on the Gross-Pitaevskii model and Bose-Einstein condensation. When time-dependent problems are treated $\mu$ gives a frequency of the rotating frame in which dynamics of physical quantities is captured.

The system of equations~\eqref{GP-ring} inherits properties of the nonlinear Schr\"{o}dinger equation. Similar to the nonlinear Schr\"{o}dinger equation~\cite{Carr-I}, an important class of stationary solutions of Eqs.~\eqref{GP-ring} are constant amplitude solutions, which correspond to polarization vortices with the homogeneous density distributions along the ring. Those can be sought in the form
\begin{equation}
\psi_{\pm}(x)=\chi_{\pm}e^{im_{\pm}x},\quad\text{for}\quad~\kappa =0.
\label{const-amp-sols}
\end{equation}
and
\begin{equation}
\psi_{\pm}(x)=\chi_{\pm}e^{i(n\mp 1)x},\quad\text{for}\quad~\kappa\ne 0
\label{const1}
\end{equation}
Here, $\chi_{\pm}$ are $x$-independent amplitudes.
Therefore in the presence of the TE-TM splitting vortex winding numbers in two components of the spinor must
differ by $2$, while these winding numbers $m_{\pm}$ are arbitrary integers for ${\kappa=0}$.
In the  limit of noninteracting polaritons $\mu$ gives the energy spectrum and the existence of solutions \eqref{const1} and \eqref{const-amp-sols} requires
\begin{equation}
\mu =\mu_\pm^{(0)},\; ~\mu_\pm^{(0)}= m_{\pm}^2 \pm \Omega
\label{linear-zero}
\end{equation}
and
\begin{eqnarray}
 && \mu =\mu_\pm,\; ~\mu_\pm = 1+n^2 \pm \sqrt{(2n-\Omega)^2+\kappa^2},
\label{linear-kappa}
\end{eqnarray}
respectively.
It is clear that the vortex energies in the no-interaction limit vary  linearly  with the applied magnetic field $\Omega$ for $\kappa= 0$, While for $\kappa\ne 0$, one deals with the typical anticrossing behavior in the proximity of the points $\Omega=2n$ (see detailed discussion and figure in Sec.~IV).

When nonlinear effects are included, the  energies acquire the corresponding nonlinear shifts proportional to~$\rho$, but not only this. Spin anisotropy of nonlinear interaction $\alpha\ne 1$ makes it possible for the mixed ($\chi_{+}\ne 0, \chi_-\ne 0$) vortex states to loose the dependence of their energies on the applied magnetic field. Such behavior of exciton-polariton condensate in thermodynamic equilibrium is known as spin Meissner effect (for introduction to the spin Meissner effect see Section~\ref{section:Intro} and original Refs.~\cite{Rubo-PLA-2006, Shelykh-Superlattices, Larionov-PRL-2010}).
However, in contrast to the previously studied spin Meissner effect, properties of such vortex states and their domain of existence are defined by the vortex winding numbers.
To highlight the dependence on the winding numbers  we term the vortex states whose energies either exactly or approximately lose dependence on the magnetic field -- {\it topological spin Meissner states} (TSM states). As it will be shown below TSM states are ubiquitous feature of our model.
Note, that vortices  with  $m_+=\pm 1$, $m_-=0$ and $m_+=0$, $m_-=\pm 1$ are so-called half-vortices in the terminology used in \cite{Rubo-PRL-2007,Lagoudakis-Science-2009,Flayac-PRB-2010,Solano-Rubo-comment,Flayac-reply,Solano-PRB-2014}. We will show that half-vortices also exhibit topological spin Meissner effect.

We study nonlinear solutions for $\kappa=0$ in Section III. Importantly, solutions~\eqref{const-amp-sols} with $m_--m_+\ne 2$,  do not disappear as we introduce $\kappa\ne 0$, they simply develop  inhomogeneous density profiles and thus are associated with the breaking the rotational symmetry. We study the $\kappa\ne 0$ case in details in  Section IV. Note, that the system of Eq.~\eqref{GP-ring} even with $\kappa=0$  has various soliton-like solutions with the inhomogeneous density profiles, see, e.g.~\cite{Carr-I}. These solitons can continue to exist for non-zero $\kappa$ as well. In order to limit the scope of the present work we leave these type of inhomogeneous solutions for future studies.

\section{Topological spin Meissner effect: Zero TE-TM splitting.}
\label{section:zero}

\subsection{Stationary Solutions}
We first focus on the case when the TE-TM splitting is absent, $\kappa=0$.
Substituting~ Eq.~\eqref{const-amp-sols} to~\eqref{GP-ring},
we have,
\begin{eqnarray}
\nonumber && \left[-\mu+m_+^2 + \chi_{+}^2+\alpha\chi_{-}^2 +\Omega\right]\chi_{+}=0,\\
&& \left[-\mu+m_-^2 + \chi_{-}^2+\alpha\chi_{+}^2 -\Omega\right]\chi_{-}=0.
\label{chi-eq-general}
\end{eqnarray}
Because the phases of $\psi_\pm$ are arbitrary, in this section we will assume $\chi_-,\chi_+\ge 0$ without loosing the generality, however the relative phase of the amplitudes will play an important role when we will be dealing with the case of non-zero TE-TM splitting in the next section.
One obvious class of solutions of Eqs.~\eqref{chi-eq-general} comes by setting either $\chi_{+}$ or $\chi_{-}$ to zero: this gives two solutions with amplitudes $(\chi_{+},\chi_{-})=(\sqrt{\rho},0)$ and $(\chi_{+},\chi_{-})=(0,\sqrt{\rho})$
and chemical potentials,
\begin{equation}
\mu_+^{(0)}=m_+^2+\rho+\Omega
\label{mu-plus}
\end{equation}
and
\begin{equation}
\mu_-^{(0)}=m_-^2+\rho-\Omega,
\label{mu-minus}
\end{equation}
respectively. Energies of these solutions  either increase or decrease with $\Omega$, depending on whether the polariton spin is parallel or anti-parallel to the applied magnetic field.

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
$
\begin{array}{c}
\begin{picture}(35,23)
\put(-1,0){\includegraphics[width=3.4in]{Omega-mu.png}}
\put(2,20.5){(a)}
\end{picture}
\\
\begin{picture}(35,23)
\put(-1,0){\includegraphics[width=3.4in]{rho-mu.png}}
\put(2,21){(b)}
\end{picture}
\end{array}
$
\caption{\label{fig:families}
(Color online) Families of constant amplitude solutions~\eqref{const-amp-sols} in zero TE-TM splitting ($\kappa=0$) on the diagrams  $\mu$ vs $\Omega$ at fixed $\rho=0.5$ (a) and $\mu$ vs $\rho$ at fixed $\Omega=0$. Red and blue solid lines are pure circular polarization vortices characterized by winding numbers $m_\pm$ and chemical potential given by Eqs.~\eqref{mu-plus} and~\eqref{mu-minus}. Black lines are topological spin Meissner (TSM) states specified by a pair of winding numbers $(m_+,m_-)$. Each TSM state appear from interaction of circular polarization vortices and if nonlinearity is increased from zero arise from the corresponding intersections of the red and blue lines on Figure (a). Figure (b) shows how TSM states appear on the diagram $\mu$ vs $\rho$ when nonlinearity parameter is increased continuously from $\rho=0$ (linear case) to $7$ in zero magnetic field $\Omega=0$.
In both cases $\alpha=-0.05$.
}
\end{center}
\end{figure}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
\includegraphics[width=3.4in]{largescale.png}
\caption{\label{fig:largescale}
(Color online)
Constant amplitude solutions~\eqref{const-amp-sols} in zero TE-TM splitting ($\kappa=0$) on the diagrams  $\mu$ vs $\Omega$ at different values of the nonlinearity parameter $\rho=1$ (a), $3$ (b) $8$ (c) and $15$ (d).
Topological spin Meissner (TSM) states are marked by black lines resting on the corresponding red and blue diagonal lines corresponding to the pure circular polarization vortices. The intervals of magnetic field where a given TSM state exists grow with increasing the nonlinearity $\rho$. More details are provided in Fig.~\ref{fig:families} where the specific TSM states are marked by their winding numbers. In all cases $\alpha=-0.05$.
}
\end{center}
\end{figure}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
$
\begin{array}{cc}
\begin{picture}(16,18)
\put(-2,0){\includegraphics[width=1.75in]{dm2-region.png}}
\put(1,3){(a)}
\end{picture}
&
\begin{picture}(16,18)
\put(-1,0){\includegraphics[width=1.75in]{dm3-region.png}}
\put(2,3){(b)}
\end{picture}
\\
\begin{picture}(16,18)
\put(-2,0){\includegraphics[width=1.75in]{dm4-region.png}}
\put(1,3){(c)}
\end{picture}
&
\begin{picture}(16,18)
\put(-1,0){\includegraphics[width=1.75in]{dm10-region.png}}
\put(2,3){(d)}
\end{picture}
\end{array}
$
\caption{\label{fig:z-rho}
(Color online)
Instability regions of topological spin Meissner (TSM) states with $|\Delta m|=2$ (a), $|\Delta m|=3$ (b), $|\Delta m|=4$ (c) and $|\Delta m|=10$ (d) as given by the analytical formulas~\eqref{rhoc} and~\eqref{Delta-rho}. Dark and light blue areas are instability regions in parameter space $(z,\rho)$ arising at different angular harmonics $|l|$ for two different values of the parameter $\alpha$: $-0.05$ and $-0.2$, correspondingly. The hatched area marks the region $|z|>\rho$ where solutions in the form of TSM states do not exist (see~\eqref{condition-Meissner}).
}
\end{center}
\end{figure}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure*}
\begin{center}
$
\begin{array}{cc}
\begin{picture}(35,22)
\put(0,0){\includegraphics[width=3.5in]{instab-1+1_rho3.png}}
\put(3,0){(a)}
\end{picture}
&
\begin{picture}(35,22)
\put(0,0){\includegraphics[width=3.5in]{instab-1+2_rho8.png}}
\put(3,0){(c)}
\end{picture}
\\
\begin{picture}(35,22)
\put(0,0){\includegraphics[width=3.5in]{instab-2+2_rho7.png}}
\put(3,0){(b)}
\end{picture}
&
\begin{picture}(35,22)
\put(0,0){\includegraphics[width=3.5in]{instab-1+2_rho5.png}}
\put(3,0){(d)}
\end{picture}
\end{array}
$
\caption{\label{fig:instab-dynamics}
(Color online)
Numerically calculated dynamics arising when an unstable states in the instability regions shown in Fig.~\ref{fig:z-rho} are distorted by a small initial perturbation.
(a) $|l|=1$ instability in state $(-1,1)$ at $\rho=3$. (b) $|l|=3$ instability in state $(-2,2)$ at $\rho=7$. (c) $|l|=1$ and (d) $|l|=2$ instabilities in state $(m_+,m_-)=(-1,2)$ at $\rho=8$ and $\rho=5$, correspondingly. The onset of the modes with number of peaks equal to the angular momentum $|l|$ of the unstable mode is well visible, in agreement with Fig.~\ref{fig:z-rho}.
In all cases $z=0$ for each state, $\alpha=-0.05$ and TE-TM splitting is absent $\kappa=0$. Video of the animated dynamics is available in the Supplementary Material.
}
\end{center}
\end{figure*}
\fi

The other distinct class of solutions corresponds to the case when the both components have non-zero  densities,  $\chi_+,\chi_-\neq 0$. Equating the expressions in square brackets in~\eqref{chi-eq-general} and using the normalization $\chi_+^2+\chi_-^2=\rho$ we find
\begin{equation}
\chi_{+}^2=\frac{\rho-z}{2}, \quad
\chi_{-}^2= \frac{\rho+z}{2}
\label{chi-Meissner}
\end{equation}
where
\begin{equation}
z=\frac{2(\Omega-\Omega_{c})}{(1-\alpha)}
\label{z}
\end{equation}
and
\begin{equation}
\Omega_{c}=\frac12 (m_-^2-m_+^2)=0,\pm\frac12,\pm\frac32,\pm 2...
\label{Omc}
\end{equation}
Solutions~\eqref{chi-Meissner} depend on the magnetic field via parameter~$z=\chi_{-}^2-\chi_{+}^2$ subjected to the condition $|z|< \rho$. Therefore, solutions exist only in a limited interval of $\Omega$ given by
\begin{equation}
\left|\Omega-\Omega_{c}\right|<\rho\,\frac{(1-\alpha)}{2},
\label{condition-Meissner}
\end{equation}
and centered around $\Omega_c$. Note that $\Omega=\Omega_c$ ($z=0$) for different $m_+$, $m_-$ are degeneracy points at which energies~\eqref{mu-plus} and~\eqref{mu-minus} of the  circularly polarized solutions coincide.

Substituting~\eqref{chi-Meissner} to~\eqref{chi-eq-general} we find chemical potential for the mixed polarization states~\eqref{chi-Meissner},
\begin{equation}
\mu^{(0)} = \frac12 \left[ m_+^2+m_-^2 + \rho (1+\alpha) \right].
\label{mu-Meissner}
\end{equation}
Thus, chemical potential $\mu=\mu^{(0)}$ in this case does not depend on the magnetic field and according to the terminology introduced in Section~\ref{sec:model}, these are the topological spin Meissner (TSM) states.
Eqs.~\eqref{chi-Meissner},~\eqref{condition-Meissner} and~\eqref{mu-Meissner} are a generalization of the conventional spin Meissner effect to the case when spinor components possess non-trivial phase winding
and therefore can be called {\it topological} spin Meissner effect (TSM effect).
The conventional spin Meissner effect~\cite{Rubo-PLA-2006, Shelykh-Superlattices, Larionov-PRL-2010} arises when spin-dependent polariton-polariton interactions compensate Zeeman splitting. Such compensation is possible until the fully circularly polarized state is reached. In the TSM effect Zeeman splitting in compensated by the combined action of both polariton-polariton interactions and circulation of the exciton-polariton condensate described by the winding numbers $m_+$, $m_-$.
Note, that for TSM states to exists the offset of the magnetic field $\Omega$ from its critical values $\Omega_c$ should be small enough, see Eq.~\eqref{condition-Meissner}.
The graphs of $\mu$ vs $\Omega$ and $\mu$ vs $\rho$ for the families of solutions~\eqref{mu-plus},~\eqref{mu-minus} and~\eqref{chi-Meissner} are shown in Figs.~\ref{fig:families}a and \ref{fig:families}b, respectively. The linear spectrum~\eqref{linear-zero} is recovered in the limit $\rho\to 0$, see Fig.~\ref{fig:families}b.

At a fixed magnetic field the condition~\eqref{condition-Meissner} defines the minimal value of the nonlinearity parameter $\rho$ which is needed to observe a given TSM state. Eqs. \eqref{Omc} and~\eqref{condition-Meissner} at $\Omega=0$ give the existence criterion  for a TSM state with the winding numbers~$m_+$ and~$m_-$,
\begin{equation}
\rho > \rho_{min}(m_+,m_-) \equiv \frac{|m_-^2 - m_+^2|}{1-\alpha}.
\label{rhomin}
\end{equation}
Therefore, more and more TSM states arise as nonlinearity is gradually increased from zero. The appearance of TSM states with increasing nonlinearity is seen on the  $\mu(\rho)$-plot on Fig.~\ref{fig:families}b and $\mu(\Omega)$-plot on Fig.~\ref{fig:largescale}. As seen from  Fig.~\ref{fig:families}b TSM states with $|m_+|=|m_-|$ start off straight from the linear spectrum ($\rho=0$) while those with $|m_+|\neq |m_-|$ require a finite value of nonlinearity given by~\eqref{rhomin}. In presence of large nonlinearity TSM states are the lowest energy states of the system and arrange into a system of energy levels as shown on Fig.~\ref{fig:largescale}.

An illustrative example of TSM effect is the behavior of half-vortices in the presence of magnetic field. Similar to half-vortices in a 2D system~\cite{Rubo-PRL-2007,Flayac-PRB-2010}, {\it half-vortices in a ring} have zero phase winding number of one component and a simple vortex in the other component. These are four distinct states $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$. These are essentially nonlinear states which cease to exist in the linear limit, see Fig.~\ref{fig:families}b. In zero magnetic field half-vortices may only be observed for the nonlinearities stronger than the critical value $\rho > 1/(1-\alpha)$ given by the formula~\eqref{rhomin}. On the other hand, if magnetic field is tuned to $\Omega=\Omega_c=\pm 1/2$ even a very small nonlinearity would be enough to create a half-vortex.

In order to compare our findings  with existing experimental studies of half-vortices in rings \cite{Liu-PNAS-2015} we change into the basis of linear polarization. At $z=0$ constant amplitude solution~\eqref{const-amp-sols} with amplitudes defined by~\eqref{chi-Meissner} takes the form
\begin{equation}
\psi_{\rm lin}(z=0) = \sqrt{\rho}\, \exp{\left[i \frac{m_++m_-}{2} x\right]}
\begin{pmatrix}
\cos{\frac{\Delta m}{2} x}\\
\sin{\frac{\Delta m}{2} x}\\
\end{pmatrix}
\label{hv-pure}
\end{equation}
where two components of the spinor in the basis of linear polarization are $\psi_{\rm lin,1}=(\psi_+ + i \psi_-)\sqrt{2}$ and $\psi_{\rm lin,2}=(\psi_+ - i \psi_-)/\sqrt{2}$.
Note, that the implicit choice of the relative phase of the spinor components $\chi_+=\chi_-$ made here is arbitrary: half-vortex states with different relative phases of $|\chi_+|$ and $|\chi_-|$ are connected by a simple shift of coordinate as will be shown in Section~\ref{section:sb}. For $\Omega$ away from $\Omega_c$ the expression for a half-vortex in linear polarization becomes more involved.
A simple expression may be obtained assuming $z/\rho \ll 1$,
\begin{equation}
\psi_{\rm lin} \approx \psi_{\rm lin}(z=0)
+i\frac{z}{\rho}
\begin{pmatrix}
\sin{\frac{\Delta m}{2} x}\\
-\cos{\frac{\Delta m}{2} x}\\
\end{pmatrix}
\label{psihv}
\end{equation}
The formula~\eqref{psihv} can explain the experimental result~\cite{Liu-PNAS-2015} where a superposition of half-vortex states was observed. According to Ref.~\cite{Liu-PNAS-2015} the anzatz in the form of superposition of states of the type~\eqref{hv-pure}  was used to fit the experimental data.
This puzzling result could not be explained by the existing theory but created an uncertainty of why half-vortices prefer a superposition in favor of a pure state~\eqref{hv-pure}. Assuming the experiments were done in zero magnetic field we find that $|z|={1/(1-\alpha)\neq 0}$. Therefore, formula~\eqref{psihv} makes it clear that a pure half-vortex states in the form~\eqref{hv-pure} can not be observed in zero magnetic field. Furthermore, if a pure half-vortex state~\eqref{hv-pure} is to be observed, a non-zero magnetic field with magnitude equal exactly to $|\Omega|=1/2$ should be applied. As far as we know, this has not been done in the existing experimental studies of exciton-polariton states in a ring.

\subsection{Stability of spin Meissner states}

To analyze stability we consider small time-dependent perturbations $\varepsilon_{\pm}(x,t)$ around vortices:
\begin{equation}
\psi_{\pm}(x,t)=\left[\chi_{\pm}+\varepsilon_{\pm}(x,t)\right]e^{i m_{\pm} x}.
\label{psi-eps}
\end{equation}
Substituting~\eqref{psi-eps} into~\eqref{GP-ring} at $\kappa=0$ we get a system of linear equations for $\varepsilon_{\pm}(x,t)$,
\begin{multline}
i\dot\varepsilon = -\varepsilon_{\pm}'' - 2i m_{\pm} \varepsilon_{\pm}'
 + \chi_{\pm}^2(\varepsilon_{\pm}+\varepsilon_{\pm}^*)+
\alpha\chi_{\pm}\chi_{\mp}(\varepsilon_{\mp}+\varepsilon_{\mp}^*) =0
\end{multline}
Expanding $\varepsilon_{\pm}(x,t)$ into Fourier series in $x$, see, e.g., Ref. \cite{Skryabin-PRA-2000},
\begin{equation}
\varepsilon_{\pm}(x,t) = \sum_{l=0}^{\infty} U_{\pm,l}(t) e^{ilx} + V_{\pm,l}^*(t) e^{-ilx}
\label{eps-Fourier}
\end{equation}
we get a set systems of equations on $\mathbf{W}_l(t)=(U_{+,l},V_{+,l},U_{-,l},V_{-,l})$, decoupled for different integer~$l$,
\begin{equation}
i\dot{\mathbf{W}_l} = \hat{\eta}\hat{\mathcal{H}_l}\mathbf{W}_l
\end{equation}
where
\begin{equation}
\hat\eta=
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1
\end{pmatrix}
\label{eta}
\end{equation}
and
\begin{equation}
\hat{\mathcal{H}_l}=\begin{pmatrix}
d_+ & \chi_+^2 & \alpha\chi_+\chi_- & \alpha\chi_+\chi_-
\\
\chi_+^2 & \tilde{d}_+ & \alpha\chi_+\chi_- & \alpha\chi_+\chi_-
\\
\alpha\chi_+\chi_-  & \alpha\chi_+\chi_- & d_- & \chi_-^2
\\
\alpha\chi_+\chi_- & \alpha\chi_+\chi_-  & \chi_-^2 & \tilde{d}_-
\end{pmatrix}
\label{H0}
\end{equation}
where $
d_\pm \equiv
l^2 + 2l m_\pm + \chi_{\pm}^2$, $\tilde{d}_\pm \equiv
l^2 - 2l m_\pm + \chi_{\pm}^2$.
Assuming $\mathbf{W}_l= \mathbf{w}_l e^{-i\lambda t}$, $\mathbf{w}_l\equiv (u_{+,l},v_{+,l},u_{-,l},v_{-,l})$  we get an eigenvalue problem:
\begin{equation}
\hat\eta \hat{\mathcal{H}_l}\mathbf{w}_l=\lambda \mathbf{w}_l
\label{etaH-problem}
\end{equation}
The solution is spectrally unstable if there is at least one eigenvalue with positive imaginary part $\Im\lambda > 0$. Because of equality ${\rm Tr}[(\hat{\eta}\hat{\mathcal{H}_l})^\dagger]={\rm Tr}[\hat{\eta}\hat{\mathcal{H}_l}]$ the eigenvalues of the matrix $\hat{\eta}\hat{\mathcal{H}_l}$ come in complex conjugated pairs.

In the special case $\Omega=\Omega_{c}$ the eigenvalue problem~\eqref{etaH-problem},~\eqref{H0} allows simple analytical solution. For $\Omega=\Omega_{c}$ we have, according to~\eqref{chi-Meissner}, $\chi_+=\chi_-=\sqrt{\rho/2}$. Substituting to~\eqref{etaH-problem},~\eqref{H0} and solving the eigenvalue problem we get four eigenvalues
\begin{multline}
\lambda= l \Big( m_+ + m_-
\\
\pm \sqrt{\Delta m^2+l^2+\rho\pm
\sqrt{4\Delta m^2(l^2+\rho)+\alpha^2\rho^2}}\,\Big)
\label{lambdas-z0}
\end{multline}
where
\begin{equation}
\Delta m \equiv m_- - m_+
\end{equation}
From~\eqref{lambdas-z0}, the unstable regions can be easily found,
\begin{equation}
|l^2 + \rho - \Delta m^2| < \alpha\rho
\label{instab-at-Omc}
\end{equation}
It is straightforward to see from~\eqref{instab-at-Omc} that there is no instability in the case $\alpha=0$ when interaction between the circular components is absent.
However, for  finite $\alpha$ we have an unstable region in $\rho$
extending up to $\Delta\rho\approx |\alpha|\rho_c$ either side from  $\rho_{c}= \Delta m^2-l^2$.

We analyze stability for arbitrary values of $\Omega$ we use the perturbation theory in parameter $\alpha$.
We take into account the dependence of $\chi_+$ and $\chi_-$ on $\alpha$ exactly using their expressions~\eqref{chi-Meissner}, while we treat perturbatively only those part of~\eqref{H0} depends on $\alpha$ explicitly.
\begin{equation}
\mathcal{H}_l=\mathcal{U}_l+\alpha \mathcal{V}(\alpha)
\end{equation}
\begin{equation*}
\mathcal{U}_l=\begin{pmatrix}
d_+ & \chi_+^2 & 0 & 0
\\
\chi_+^2 & \tilde{d}_+ & 0 & 0
\\
0  & 0 & d_- & \chi_-^2
\\
0 & 0  & \chi_-^2 & \tilde{d}_-
\end{pmatrix}
,\;\;
\mathcal{V}(\alpha)=\chi_+\chi_-\begin{pmatrix}
0 & 0 & 1 & 1
\\
0 & 0& 1 & 1
\\
1  & 1 & 0 & 0
\\
1 & 1  & 0 & 0
\end{pmatrix}
\end{equation*}
For eigenvalues of matrix $\eta\mathcal{U}_l$ we find simple expressions
\begin{equation}
\begin{split}
\lambda_{1,2}^{+} = l\left[2m_+ \pm \sqrt{l^2+2\chi_{+}^2}\right], \\
\lambda_{1,2}^{-} = l\left[2m_- \pm \sqrt{l^2+2\chi_{-}^2}\right].
\end{split}
\label{lambdas}
\end{equation}
Eigenvalues of $\eta\mathcal{H}$ with non-zero imaginary part may appear around degeneracy points of $\eta\mathcal{U}_l$. In the trivial case $\Delta m=0$ the degeneracies may only appear when $\chi_-^2=\chi_+^2$, i.e. at $z=0$ which is the case studied above: the eigenvalues of matrix~\eqref{H0} are given by~\eqref{lambdas-z0}. As seen from Eq.~\eqref{lambdas-z0} this does not lead to any instabilities as soon as $|\alpha|\le 1$.
In follows we will assume $\Delta m\neq 0$. Equating the eigenvalues $\lambda_{1,2}^{+}$ and $\lambda_{1,2}^{-}$ we get for degeneracies  $\lambda^{+}_1=\lambda^{-}_2$ ($\Delta m >0$) and $\lambda^{+}_2=\lambda^{-}_1$ ($\Delta m <0$) realized when
$|z|\le 2\Delta m^2$,
\begin{equation}
\rho_c(z) = \Delta m^2 - l^2 + \frac{z^2}{4\Delta m^2}
\label{rhoc}
\end{equation}
Applying the perturbation theory for non-Hermitian operators~\cite{Sternheim-1972} we find the first order corrections to the degenerate eigenvalues. At $\rho=\rho_c$ these are
\begin{equation}
\Delta \lambda = \pm \frac{i \alpha l \chi_+ \chi_- }{\left[(l^2+2\chi_+^2)(l^2+2\chi_-^2)\right]^{1/4}}
\label{dlambda}
\end{equation}
and for $l\neq 0$ lead to instability due to appearance of an eigenvalue with positive imaginary part.

Treating the deviation $\rho-\rho_c$ from $\rho_c$ as a perturbation we find the size of the instability interval $(-\Delta\rho,\Delta\rho)$ centered around $\rho_c$ given by the formula
\begin{equation}
\Delta\rho = \left(  \frac{ 8 \alpha^2 \chi_+^2\chi_-^2 \sqrt{l^2+2\chi_+^2}\sqrt{l^2+2\chi_-^2}}{l^2+\chi_+^2+\chi_-^2 + \sqrt{l^2+2\chi_+^2} \sqrt{l^2+2\chi_-^2}} \right)^{1/2}
\label{Delta-rho}
\end{equation}
where $\chi_+$, $\chi_-$ are evaluated using Eq.~\eqref{chi-Meissner} at $\rho=\rho_c$.

One may check that the other two degeneracies, ${\lambda^{+}_1=\lambda^{-}_1}$ and $\lambda^{+}_2=\lambda^{-}_2$ which realize when $|z|\ge 2\Delta m^2$ do not lead to imaginary eigenvalues and, therefore, do not cause instabilities.

Formula~\eqref{rhoc} together with condition $|z|<\rho$ imply that the four half-vortices $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$ have no instability regions, i.e. linearly stable for all values of $\rho$. The first non-trivial case arises for states with $|\Delta m|=2$. Instability regions for states with $|\Delta m|=2, 3, 4$ and $10$ are shown in a parameter space of $\rho$ vs $z$ in Fig.~\ref{fig:z-rho}.

We estimate the dynamical effect of the unstable mode on distribution of the spinor components. Assuming the unstable dominates other modes but still can be treated as a perturbation around the stationary solution, we write
\begin{equation}
\varepsilon_{\pm} \approx \left[ u_{\pm} e^{ilx-i\lambda_r t} + v^*_{\pm} e^{-ilx+i\lambda_r t}\right] e^{\lambda_i t}
\label{eps-unstable}
\end{equation}
where we explicitly separated the real $\lambda_r =\Re \lambda$ and imaginary $\lambda_i =\Im \lambda$ parts of the eigenvalue causing the instability.
Substituting~\eqref{eps-unstable} to the expressions for densities
$|\phi_{\pm}|^2 = |\chi_{\pm}+\varepsilon_{\pm}(x,t)|^2$
we see that the growing unstable mode modulates densities of the spinor components in the form of a propagating wave with phase velocity $v=\lambda_r/l$.
The real part $\lambda_r$ can be estimated from expressions~\eqref{lambdas}. At degeneracies $\lambda^{+}_1=\lambda^{-}_2 \equiv \lambda$ and $\lambda^{+}_2=\lambda^{-}_1 \equiv \lambda$ we find for the phase velocity at $\rho= \rho_c(z)$,
\begin{equation}
v = m_+ + m_- - \frac{z}{2\Delta m}
\label{v}
\end{equation}
Thus, $v$ is determined by the total angular momentum $m_+ + m_-$ of a TSM state.

We use the Split Step (Fourier) method to numerically analyze dynamics of the unstable states. The $(-1,1)$, $(-2,2)$ and $(-1,2)$ cases are shown in Fig.~\ref{fig:instab-dynamics}.
The onset of unstable mode with number of peaks or deeps is equal to the angular harmonics $l$ in agreement with the corresponding instability regions in Fig.~\ref{fig:z-rho}.
The instability develops as a wave of density modulations. The unstable modes with $l=1$ and $l=2$ around the state $(-1,2)$ cause the density modulation to rotate with phase velocity $v\approx 1$, see Figs.~\ref{fig:z-rho}c,d. While density modulations in state $(-1,2)$ move anticlockwise (increasing $x$),  density modulations in state $(-2,1)$ moves clockwise in agreement with the opposite sign of the phase velocity in~Eq.~\eqref{v} (see Supplementary Material).

\section{Constant-amplitude spin Meissner states in presence of TE-TM splitting.}
\label{section:const-amp}

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
$
\begin{array}{c}
\begin{picture}(35,23)
\put(-1,0){\includegraphics[width=3.4in]{linear-spectrum.png}}
\end{picture}
\end{array}
$
\caption{\label{fig:linear}
Linear spectrum in presence (solid lines) and absence (dashed lines) of TE-TM splitting. TE-TM splitting results in anticrossings at $\Omega=\Omega_c\equiv 2n$, $n=0,\pm 1,...$
for branches $m_\pm=n\pm 1$, see Eq.~\eqref{linear-kappa}.
}
\end{center}
\end{figure}
\fi

\iffigures
\begin{figure*}
\begin{center}
\includegraphics[width=7in]{rho3-instab.png}
\caption{\label{fig:rho3}
(Color online)
The graphs show evolution of constant-amplitude solutions of the system~\eqref{GP-ring} with increasing TE-TM splitting for different values of the phase winding number $m$. Black, blue, red and green lines correspond to the states with $|n|=0,1,2$ and $3$. TE-TM splitting parameter~$\kappa$ is increased from 0.0 (a) to 0.1 (b), 0.5 (c), 1.0 (d) and 2.0 (e). The nonlinearity parameters are $\rho=3$ and $\alpha=-0.05$. Splitting of constant amplitude TSM states into stable and unstable branches is visible on Fig.b when non-zero $\kappa$ is introduced. For visibility purposes unstable regions are marked by solid lines and stable regions are marked by dotted lines. Note that only constant amplitude solutions are shown here.
}
\end{center}
\end{figure*}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
$
\begin{array}{c}
\begin{picture}(35,23)
\put(-1,0){\includegraphics[width=3.4in]{rho-mu-const-ink.png}}
\end{picture}
\end{array}
$
\caption{\label{fig:rho-mu}
Splitting of the spin Meissner states $\Delta m=2$ under a non-zero $\kappa$: states $(-1,1)$, $(-2,0)$ and $(0,2)$ shown here split into symmetric ($\chi_+\chi_->0$) and antisymmetric ($\chi_+\chi_-<0$) branches marked by `s' and `a' after the brackets with a pair of winding numbers $(m_+,m_-)$. Dashed line and solid lines correspond to $\kappa=0$ and $\kappa=0.1$, respectively. Other parameters are $\alpha=-0.05$ and $\Omega=0$. For simplicity, only constant amplitude solutions are shown here.
}
\end{center}
\end{figure}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
$
\begin{array}{cc}
\begin{picture}(17,20)
\put(-1,0){\includegraphics[width=1.75in]{lambda-rho3.png}}
\end{picture}
&
\begin{picture}(17,20)
\put(-1,0){\includegraphics[width=1.75in]{lambda-rho8.png}}
\end{picture}
\end{array}
$
\caption{\label{fig:lambda-unstable}
(Color online)
Imaginary part of the eigenvalues of matrix~\eqref{H-with-kappa} as a function of $\kappa$, evaluated at the unstable branch of the state $(-1,1)$ and $\Omega=0$. Black, dashed and dashed-dotted lines correspond to instabilities caused by angular harmonics $l=0$, $1$ and $2$, respectively. Dotted line is the theoretical estimate given by the first term in formula~\eqref{lambda-expansion} with~\eqref{lambda02}.
}
\end{center}
\end{figure}
\fi

\iffigures
\setlength{\unitlength}{0.16in}
\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{instab-1+1_rho3_kappa01.png}
\caption{\label{fig:stability-1}
(Color online)
Numerically calculated dynamics of instability arising when unstable state $(-1,1)$ at $\rho=3$, $\Omega=0$ is distorted by a small initial perturbation in presence of TE-TM splitting $\kappa=0.1$. The onset of $|l|=0$ mode is visible in agreement with Figure (a). Video of the animated dynamics is available in the Supplementary Material.
}
\end{center}
\end{figure}
\fi

We now going to look at how the presence of a non-zero TE-TM splitting
influences TSM states and the topological spin Meissner effect.

It is convenient to eliminate the explicit dependence on $x$ from the TE-TM splitting term, which is achieved through the substitution
\begin{equation}
\psi_{\pm} =
\phi_{\pm}(x) e^{\mp i x}.
\end{equation}
With this substitution we get the following auxiliary system of equations which takes a  rotationally invariant form
\begin{eqnarray}
\nonumber && i\dot\phi_+ = \left[ \hat D_{+} +|\phi_+|^2 + \alpha |\phi_-|^2 \right] \phi_+    + \kappa \phi_-
\\
&& i\dot\phi_- = \left[ \hat D_{-} +|\phi_-|^2 + \alpha |\phi_+|^2 \right] \phi_-    + \kappa \phi_+
\label{GP-phi}
\end{eqnarray}
where
$\hat D_{\pm}=1-\mu-\partial_x^2 \pm 2i\partial_x \pm \Omega$. The form of equation~\eqref{GP-phi} is explicitly invariant under rotations, i.e. a shift of coordinate $\mathcal{\hat R}(\Delta x)\phi_\pm(x)=\phi_\pm(x+\Delta x)$.

In the linear regime and $\kappa\neq 0$, the stationary version of the system~\eqref{GP-phi} is a linear system of equation with constant coefficients which has solutions in the form of exponentials
\begin{equation}
\phi_{\pm}(x)=\chi_{\pm}e^{inx},~\kappa\ne 0
\label{const-amp-n}
\end{equation}
with the winding number $n$ for both components and amplitudes
\begin{equation}
\chi_+ = \sqrt{\frac{\rho}{1+\xi^2}}, \quad
\chi_- = \xi\,\sqrt{\frac{\rho}{1+\xi^2}}
\label{lin-sol-kappa}
\end{equation}
where $\xi\equiv =\chi_-/\chi_+$ is given by
\begin{equation}
\xi=\xi_\pm,\quad \xi_\pm \equiv \frac{2n - \Omega}{\kappa} \pm \sqrt{1+\left(\frac{2n - \Omega}{\kappa}\right)^2}
\end{equation}
Energies of the solutions~\eqref{lin-sol-kappa} are
\begin{equation}
\mu = 1+n^2 \pm \sqrt{(\Omega-2n)^2+\kappa^2}.
\label{spectrum-linear}
\end{equation}
Note, that the winding numbers in the $\psi_{\pm}$ representation (Eq.~\eqref{GP-ring}) are given by
\begin{equation}
m_{\pm}=n\mp 1, \quad \Delta m=2
\end{equation}
The linear spectrum~\eqref{spectrum-linear} is plotted on Figs.~\ref{fig:linear} for the cases of zero ($\kappa=0$) and non-zero ($\kappa=0.1$) TE-TM splitting. The splitting of energy levels at $\Omega=\Omega_c\equiv 2 n$ caused by TE-TM splitting is seen on Fig.~\ref{fig:linear}b. The avoided crossings arrange in a parabolic pattern $\mu(\Omega_c)\sim \Omega_c^2$ as given by the formula~\eqref{spectrum-linear}.

We now look into the nonlinear case. Constant amplitude solutions of the nonlinear system~\eqref{GP-phi} have the same form~\eqref{const-amp-n}
where the amplitudes~\eqref{lin-sol-kappa} are defined by real roots of the 4th order algebraic equation on $\xi$,
\begin{multline}
\kappa(\xi^4-1)-\rho\left(1-\alpha\right)(\xi^3-\xi) \\
+2(\Omega-2n)(\xi^3+\xi)=0
\label{ring-sol-nln}
\end{multline}
which, in general, may have 4, 2 or 0 real roots.
The bifurcations between pairs of real and complex roots when changing the magnetic field and
strength of TE-TM splitting $\kappa$ can be traced on Fig.~\ref{fig:rho3}. The figure shows evolution of the constant amplitude branches with changing strength of TE-TM splitting $\kappa$ at a fixed nonlinearity $\rho=3$. The TSM branches exist for small $\kappa$ keeping their magnetic field-independent form of chemical potential. On increasing $\kappa$, the topological spin Meissner effect in the lower branch gradually comes to a naught acquiring a parabolic dependence on $\Omega$, while the top branch disappears completely at high $\kappa$.

Although, the exact solutions can be found by solving the equation~\eqref{ring-sol-nln} it is instructive to find their explicit expressions in the limit of small $\kappa$ (see Supplementary Information). In the first order in $\kappa$ we find
for TSM states $\Delta n =0$,
\begin{equation}
\mu = \mu^{(0)} + \frac{\kappa\,\rho}{2\chi_+\chi_-} + O(\kappa^2)
\end{equation}
where zero order term $\mu^{(0)}$ is given by~\eqref{mu-Meissner}
and ${\chi_+\chi_->0}$, ${\chi_+\chi_-<0}$ are two distinct branches of solutions. Splitting of the TSM states $\Delta m =2$ into symmetric ($\chi_+\chi_->0$) and antisymmetric ($\chi_+\chi_-<0$) when a non-zero $\kappa$ is introduced, is shown on the Fig.~\ref{fig:rho-mu}
In zero magnetic field and absence of interactions, $(-1,1)s$ and $(-1,1)a$ are two lowest TE and TM modes in a ring, while higher order states such as $(-2,0)s$, $(0,2)s$ and $(-2,0)a$ and $(0,2)a$ are propagating TE and TM modes with non-zero wavevector.
In the linear limit $\rho\to 0$ the splitting of the $(-1,1)$ state is the avoided crossings given by linear spectrum~\eqref{spectrum-linear} and shown on Fig.~\ref{fig:linear}.

We analyze analytically stability of constant amplitude TSM states solving perturbatively the eigenvalue problem for operator $\hat L = \eta \hat H$, where operator $\hat H$ is given in Appendix~\ref{app:stab-th}. Because the constant amplitude TSM states are split in two branches when non-zero $\kappa$ is present, perturbation expansion for eigenvalues of the operator
$\hat L=\hat L_0+\kappa\hat L_1+O(\kappa^2)$ will involve powers of $\kappa^{1/2}$,
\begin{equation}
\lambda=\kappa^{1/2}\left[ \lambda_0 +O(\kappa)\right]
\label{lambda-expansion}
\end{equation}
where for $\lambda_0$ we find (see details in Appendix~\ref{app:stab-th})
\begin{equation}
\lambda_0^2 = \mp 2(1-\alpha)\sqrt{\rho^2-z^2}
\label{lambda02}
\end{equation}
for states $\chi_+ \chi_- >0$ and $\chi_+\chi_-<0$, correspondingly. Therefore, for $\kappa>0$ the state with $\chi_+=\chi_-$ is unstable and state $\chi_+=-\chi_-$ is stable (for $\kappa<0$ the situation reverses).
For the unstable mode with eigenvalue given by~\eqref{lambda02} we find
\begin{equation}
\varepsilon_{\pm} \sim \pm \chi_\pm e^{inx} = \pm \phi_{\pm}(x)
\label{l0-mode}
\end{equation}
i.e. the unstable mode~\eqref{l0-mode} is homogeneous over the ring.

We compare the theoretical estimate~\eqref{lambda-expansion}, ~\eqref{lambda02} of the imaginary part of the eigenvalue $\lambda$ causing the instability to its exact value obtained by numerical diagonalization of matrix~\eqref{H-with-kappa} for different angular harmonics $l$. The results of comparison for dependence of ${\rm Im}\lambda$ on $\kappa$ is presented in  Figs.~\ref{fig:lambda-unstable}a,b for two different values of nonlinearity parameter $\rho$.
The theoretical results~\eqref{lambda-expansion}, ~\eqref{lambda02} obtained by perturbation expansion agree with numerical calculation for $l=0$ at small $\kappa$.
 A small region of instability caused by angular harmonics $l=1$ is also visible in Figure~\ref{fig:lambda-unstable}a for small values of $\kappa$ and is caused by the instability region $l=1$ on Fig.~\ref{fig:z-rho}a (see Fig.~\ref{fig:z-rho}a at $\rho=3$). With increasing strength of TE-TM splitting a state becomes unstable with respect to several harmonics simultaneously.

In our numerical analysis of stability we solve the eigenvalues problem of the operator $\hat{\eta}\hat{\mathcal{H}}_l(\kappa)$, where (see Appendix~\ref{app:stab-num} for details)
\begin{multline}
\hat{\mathcal{H}}_l(\kappa)=
\\
\begin{pmatrix}
d_+ & \chi_+^2 & \alpha\chi_+\chi_- +\kappa & \alpha\chi_+\chi_-
\\
\chi_+^2 & \tilde{d}_+ & \alpha\chi_+\chi_- & \alpha\chi_+\chi_- +\kappa
\\
\alpha\chi_+\chi_- +\kappa & \alpha\chi_+\chi_- & d_- & \chi_-^2
\\
\alpha\chi_+\chi_- & \alpha\chi_+\chi_- +\kappa & \chi_-^2 & \tilde{d}_-
\end{pmatrix},
\label{H-with-kappa}
\end{multline}
$\hat\eta$ was defined above and the diagonal elements $d_{\pm}, \tilde{d}_{\pm}$ are given by
\begin{multline}
d_\pm \equiv
l^2 + 2l m_{\pm} + \chi_{\pm}^2 \\
+ \left[-\mu+m_{\pm}^2+ \chi_{\pm}^2+\alpha\chi_{\mp}^2 \pm\Omega\right]
\end{multline}
\begin{multline}
\tilde{d}_\pm \equiv
l^2 - 2l m_{\pm} + \chi_{\pm}^2 \\
+ \left[-\mu+m_{\pm}^2 + \chi_{\pm}^2+\alpha\chi_{\mp}^2 \pm\Omega\right].
\end{multline}
Matrix \eqref{H-with-kappa} is a generalization of~\eqref{H0} to nonzero $\kappa$

The solution is spectrally unstable if there is at least one eigenvalue with positive imaginary part $\Im\lambda > 0$. The results of the stability analysis are shown on Figure~\ref{fig:rho3}. Dashed lines mark stable regions and solid lines mark unstable regions (stability analysis with respect to the individual harmonics $l$ can be found in the Supplementary Material).
As seen from Fig.~\ref{fig:rho3}, the constant amplitude TSM states are split into
a stable (bottom) and unstable (top) branches, in agreement with~\eqref{lambda02}.
At $\kappa=0$ the instability is caused by angular harmonics $l=1$ which corresponds to the instability region shown in Fig.~\ref{fig:z-rho}a while for non-zero $\kappa$ the $l=0$ mode appears as seen in Fig.~\ref{fig:lambda-unstable}a.

Finally, we analyze dynamics of instability arising due to the unstable mode in presence of TE-TM splitting. The results of our time-dependent numerical calculations are shown on Fig.~\ref{fig:stability-1}. As seen from the Figure presence of TE-TM splitting leads to a homogeneous instability mode in agreement with formula~\eqref{l0-mode}.

\section{Symmetry breaking spin Meissner states}
\label{section:sb}

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
$
\begin{array}{c}
\begin{picture}(35,23)
\put(-1,0){\includegraphics[width=3.4in]{kappa-mu-rho3.png}}
\end{picture}
\end{array}
$
\caption{\label{fig:kappa-mu-rho3}
Numerical continuation of states in parameter $\kappa$ to the region of non-zero TE-TM splitting. Solid lines mark constant amplitude solutions, dashed and dotted lines mark symmetry-breaking states. Splitting of the state $(-1,1)$ into two branches can be seen. Contrary to the constant amplitude states, symmetry-breaking states do not split into branches but develop inhomogeneous density profiles. The $(1,1)$, $(-1,-1)$ develop instabilities at $\kappa\approx 0.3$  which lead to breaking of the branch. Other parameters are $\alpha=-0.05$ and $\Omega=0$.
}
\end{center}
\end{figure}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
\includegraphics[width=3.4in]{images-rho3.png}
\caption{\label{fig:images-rho3}
(Color online)
Numerical continuation of TSM states given by Eqs.~\eqref{chi-Meissner} at $\kappa=0$ to the domain of non-zero TE-TM splitting. Evolution of the densities
of the spinor components of symmetry-breaking states during the continuation is shown with parameter $\kappa$ increasing from left to right. Polar plots correspond to the branches of the symmetry-breaking states on Fig.~\ref{fig:kappa-mu-rho3} with the same nonlinearity strength $\rho=3$. Parameters of the calculations are $\Omega=0$ and $\alpha=-0.05$. States $(-1,0)$ and $(0,-1)$ are not shown as they evolve in the same way as $(0,1)$ and $(1,0)$, respectively, with spinor components interchanged. Dashed lines and fainter colors mark unstable states.
}
\end{center}
\end{figure}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
\includegraphics[width=3.4in]{Omega-mu-rho3.png}
\caption{\label{fig:Omega-mu-rho3}
(Color online)
Topological spin Meissner effect in presence of TE-TM splitting.
As seen from the Figure energies of symmetry-breaking TSM states (solid lines) depend weakly on the magnetic field even in the presence of significant TE-TM splitting $\kappa=1$ as in the calculation presented here. Note, that spin Meissner effect is not exact for states with non-zero net angular momentum $m_+ + m_-$ while it holds better for states with zero angular momentum such as $(0,0)$ and $(1,-1)$. Dashed lines mark constant amplitude solutions.
Nonlinearity parameters $\rho=3$ and $\alpha=-0.05$.
}
\end{center}
\end{figure}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure*}
\begin{center}
$
\begin{array}{cc}
\begin{picture}(35,35)
\put(-1,0){\includegraphics[width=3.5in]{hv-1+0_rho3.png}}
\put(0,0){(a)}
\end{picture}
 &
\begin{picture}(35,35)
\put(0,0){\includegraphics[width=3.5in]{hv+1+0_rho3.png}}
\put(1,0){(b)}
\end{picture}
\end{array}
$
\caption{\label{fig:hvs}
(Color online)
Topological spin Meissner effect in half-vortices $(-1,0)$ (a) and $(1,0)$ (b): the effect of magnetic field is balanced by changing densities of the spinor components of a vortex state, keeping the energy of the state nearly constant (see Fig.~\ref{fig:Omega-mu-rho3}).
Polar plots of numerically calculated density distributions are presented. Pink and blue colors represent densities $|\phi_+|^2$ and $|\phi_-|^2$, correspondingly. TE-TM splitting $\kappa$ takes values, from top to bottom: $0$, $0.3$ and $0.6$ in (a) and $0$, $0.5$ and $1.0$ in (b). Magnetic field is given by $z/\rho=-0.5$, $0$ and $0.5$ from left to right, with $z$ defined by eq.~\eqref{z}. Nonlinearity parameters $\rho=3$ and ${\alpha=-0.05}$. Dashed lines and fainter colors mark unstable states.
}
\end{center}
\end{figure*}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{star_rho3.png}
\caption{\label{fig:star}
(Color online)
Numerically continued symmetry-breaking TSM state $(4,-4)$ of higher order rotational symmetry.  Pink and blue colors represent densities $|\phi_+|^2$ and $|\phi_-|^2$, correspondingly. The values of $\kappa$ are $2$, $4$ and $6$ from top to bottom. Magnetic field is given by $z/\rho=-0.5$, $0$ and $0.5$ from left to right, with $z$ defined by eq.~\eqref{z}. Balancing of the magnetic field by density distributions of the spinor components in a polarization vortex (topological spin Meissner effect) is seen here. This leads to all states having energies independent of the value of magnetic field. Nonlinearity parameters $\rho=3$ and ${\alpha=-0.05}$. All of the displayed configurations were found to be stable with respect to linear perturbations.
}
\end{center}
\end{figure}
\fi

\iffigures
\setlength{\unitlength}{0.1in}
\begin{figure*}
\begin{center}
$
\begin{array}{cc}
\begin{picture}(35,22)
\put(0,0){\includegraphics[width=3.5in]{hv-1+0_z-05.png}}
\put(3,0){(a)}
\end{picture}
&
\begin{picture}(35,22)
\put(0,0){\includegraphics[width=3.55in]{hv+1+0_z+05_kappa05.png}}
\put(3,0){(c)}
\end{picture}
\\
\begin{picture}(35,22)
\put(0,0){\includegraphics[width=3.5in]{hv-1+0_kappa06_z+05.png}}
\put(3,0){(b)}
\end{picture}
&
\begin{picture}(35,22)
\put(0,0){\includegraphics[width=3.5in]{hv+1+0_z00_kappa10.png}}
\put(3,0){(d)}
\end{picture}
\end{array}
$
\caption{\label{fig:instab-hvs}
(Color online)
Dynamics of instabilities arising when an unstable half-vortices shown on Figures~\ref{fig:hvs} are distorted by a small initial perturbation: state $(-1,0)$ at (a) $z/\rho=-0.5$, $\kappa=0.6$  and (b) $z/\rho=0.5$, $\kappa=0.6$ (b) with density profile on Fig.~\ref{fig:hvs}a and state $(1,0)$ at (c) $z/\rho=0.5$, $\kappa=0.5$ and (d) $z/\rho=0$, $\kappa=1$ with density profile on Fig.~\ref{fig:hvs}b. Videos of animated dynamics are available in the Supplementary Material.
}
\end{center}
\end{figure*}
\fi

In the previous section we focused our attention to the TSM states with the constrained phase winding numbers $\Delta m = 2$: these are constant-amplitude solutions at non-zero TE-TM splitting. In this section we will investigate the fate of the more general states~\eqref{const-amp-sols} with arbitrary $m_+$, $m_-$. It turns out that solutions with $\Delta m \neq 2$ do not disappear in the presence of non-zero $\kappa$ but instead develop inhomogeneous profiles.

At $\kappa=0$ constant amplitude solutions of the auxiliary system of equations~\eqref{GP-phi} are expressed via solutions~\eqref{const-amp-sols} studied in Section~\ref{section:zero},
\begin{equation}
\phi_{\pm}(x)=\chi_{\pm}e^{i(m_{\pm}\pm 1) x}
\label{phi-pm}
\end{equation}

Important aspects of the continuation of TSM states from $\kappa=0$ to $\kappa\ne 0$
can be understood if we consider changes in the symmetry properties of the model equations and of the solutions themselves. Eqs.~\eqref{GP-phi} with $\kappa=0$ are invariant under the following three transformations: rotation of the total phase of the spinor, rotation of the relative phases of the spinor components and shift of the azimuthal coordinate $x$.
However, not all of these operations are independent. Consider shift of the coordinate $x\to x + \Delta x$,
\begin{equation}
\mathcal{\hat R}\left(\Delta x\right)
\begin{pmatrix}
\chi_+ e^{i n_+ x}
\\
\chi_- e^{i n_- x}
\end{pmatrix}
= e^{i \Delta x (m_+ + m_-)/2 }
\begin{pmatrix}
\chi_+ e^{i n_+ x - i \delta/2}
\\
\chi_- e^{i n_- x + i \delta/2}
\end{pmatrix}
\label{rotation}
\end{equation}
where $\delta=(\Delta m-2)\Delta x$ and $n_\pm = m_\pm \pm 1$. Thus the shift of the coordinate of the constant amplitude solutions~\eqref{phi-pm} is equivalent to rotation of the total phase, if $\Delta m = 2$, or, rotation of both total and relative phases, if $\Delta m \neq 2$.
When  TE-TM splitting is not zero, $\kappa\ne 0$, then the symmetry of the model with respect to the shift of the relative phase is broken. This affects very differently solutions with $\Delta m =2$ and $\Delta m\neq 2$. Those, with $\Delta m =2$ remain invariant with respect to rotations in $x$, which are equivalent to the corresponding shift in the still present total phase. Meanwhile former $\Delta m\ne 2$ solutions develop inhomogeneous density profiles as their rotational symmetry becomes broken.
Thus, for $\kappa\ne 0$, $\Delta m=2$ solutions have one broken symmetry (total phase) and one Goldstone mode associated with it and $\Delta m\ne 2$ solutions have two broken symmetries and two Goldstone modes.
While for $\kappa=0$, both types of solutions have two broken symmetries (in the total and relative phases). Because the number of Goldstone bosons does not change for $\Delta m\ne 2$ solutions, they do not branch as we introduce $\kappa\ne 0$, while $\Delta m=2$ split into branches, see Fig.~\ref{fig:kappa-mu-rho3}.

Numerical continuation of the solutions~\eqref{const-amp-sols} in parameter $\kappa$ to the domain of non-zero TE-TM splitting is presented on Fig.~\ref{fig:kappa-mu-rho3} for a fixed nonlinearity parameter $\rho=3$. At $\kappa=0$ energies of the solutions are given by Eqs.~\eqref{mu-plus},~\eqref{mu-minus} and~\eqref{mu-Meissner}, see also a cross section of Fig.~\ref{fig:families}b at $\rho=3$.
With increasing TE-TM splitting parameter the solutions with winding numbers $(m_+,m_-)$, $\Delta m\neq 2$ continue to exist developing inhomogeneous density profiles. Splitting of the constant amplitude states with $\Delta m=2$ into two branches is also visible in Fig.~\ref{fig:kappa-mu-rho3}. Contrary to the constant amplitude states, symmetry breaking states do not split into branches.
Indeed, as can be seen from formula~\eqref{rotation}, states with different relative phase of the spinor components at $\kappa=0$ seed the same solution at $\kappa \neq 0$, apart from the coordinate shift and a common phase factor. Snapshots of evolution of densities of the spinor components under continuously changing $\kappa$ is shown on Fig.~\ref{fig:images-rho3}.

Symmetry-breaking states can be classified according to which states they can be continued from by increasing TE-TM splitting $\kappa$ from zero because they inherit topological properties of the seeding solutions, i.e. their two phase winding numbers. Indeed, the two topological invariants which could be used to characterize a symmetry-breaking TSM state are
\begin{equation}
\frac{1}{2\pi i} \int_0^{2\pi} \frac{\partial_x\phi_\pm}{\phi_\pm} dx
\end{equation}
coincide with $m_\pm$ of the constant amplitude TSM state at $\kappa=0$ to which it can be continued given that neither of the components turned to zero during the continuous transformation.

To analyze analytically TSM states with broken rotational symmetry in presence of TE-TM splitting we use perturbative approach and consider a small distortion $\varepsilon_\pm(x)$ of the shape of the TSM state~\eqref{chi-Meissner},
\begin{equation}
\phi_\pm(x) = \left[\chi_\pm + \kappa \varepsilon_\pm(x) + O(\kappa^2)\right]e^{i (m_\pm \pm 1) x}
\end{equation}
\begin{equation}
\mu = \mu^{(0)} + \kappa \mu^{(1)} + O(\kappa^2)
\end{equation}
where $\varepsilon_{\pm}(x)$ are complex functions and $\chi_\pm$ are amplitudes of the TSM state at $\kappa=0$.
Substituting to~\eqref{GP-phi} we get the system of equations on $\varepsilon_{\pm}(x)$,
\begin{multline}
-\varepsilon_{\pm}'' - 2i m_{\pm} \varepsilon_{\pm}'
 + \chi_{\pm}^2(\varepsilon_{\pm}+\varepsilon_{\pm}^*)+
\alpha\chi_{\pm}\chi_{\mp}(\varepsilon_{\mp}+\varepsilon_{\mp}^*)
\\
=\mu^{(1)} \chi_{\pm} - e^{\pm i (\Delta m-2) x} \chi_{\mp}
\label{epsilon}
\end{multline}
Assuming $\Delta m\neq 2$, we may seek for solutions of~\eqref{epsilon} in the form,
\begin{equation}
\varepsilon_{\pm} = A_{\pm}e^{\pm i(\Delta m-2)x} + B_{\pm}e^{\mp i(\Delta m-2)x} + C_{\pm}
\label{ABC}
\end{equation}
where coefficients $A_{\pm},B_{\pm}$ and $C_{\pm}$ can be chosen real. Substituting to~\eqref{epsilon} and using the normalization condition we get two decoupled systems of equations
\begin{equation}
\hat{\mathcal{H}}_{\Delta m-2}\mathbf{W} = \mathbf{R}
\label{PW}
\end{equation}
\begin{equation}
Q\mathbf{C} = 0
\label{QC}
\end{equation}
where $\mathbf{W}=(A_{+},B_{+},A_{-},B_{-})^T$, $\mathbf{C}=(C_{+},C_{-},\mu^{(1)})^T$, $\mathbf{R}=(-\chi_{-},0,-\chi_{+},0)^T$, matrix $\hat{\mathcal{H}}_{\Delta m-2}$ is given by Eq.~\eqref{H0} for $l=\Delta m-2$ and
\begin{equation}
Q=\begin{pmatrix}
2\chi_{+}^2 & 2\alpha\chi_+ \chi_- & -\chi_+ \\
2\alpha\chi_+\chi_- & 2\chi_{-}^2 & -\chi_- \\
\chi_+ & \chi_- & 0
\end{pmatrix}
\end{equation}
The determinant of matrix~$Q$ is $\det Q = 4\chi_+^2\chi_-^2(1-\alpha)$
which is non-zero as $\chi_-,\chi_+\neq 0$. Therefore, for the TSM states, only trivial solution to the system~\eqref{QC} exists, i.e. $C_1=C_2=\mu^{(1)}=0$. Because $\mu^{(1)}=0$ and l $\mu^{(0)}$ does not change with the magnetic field, energy $\mu$ of the symmetry-breaking solutions originated by TSM states is also independent of the magnetic field, i.e. {\it symmetry-breaking solutions originated by topological spin Meissner states remain spin Meissner states}, at least in the first order in TE-TM splitting.
Therefore, from~\eqref{mu-Meissner} for symmetry-breaking TSM states in presence of TE-TM splitting we have
\begin{equation}
\mu = \frac12 \left[ m_+^2+m_-^2 + \rho (1+\alpha) \right] + O(\kappa^2).
\end{equation}
To find the density profiles of the symmetry-breaking solutions we solve the system~\eqref{PW}. For small $\kappa$ perturbative solutions~\eqref{ABC} are in good agreement with our numerical calculations (see Supplementary Information for comparison between the theoretical and numerical results).

The influence of TE-TM splitting on the topological spin Meissner effect in shown on Figure~\ref{fig:Omega-mu-rho3}. As seen from the Figure, energies of symmetry-breaking TSM states depend weakly on the magnetic field even in the presence of significant TE-TM splitting $\kappa=1$. Notice that spin Meissner effect is not exact for states with non-zero net angular momentum $m_+ + m_-$ while it holds better for states with zero angular momentum such as $(0,0)$ and~$(1,-1)$.

Studies of topological spin Meissner effect in half-vortices $(-1,0)$ and $(1,0)$ are shown on Fig.~\ref{fig:hvs}a and b. As seen form the Figure, the magnetic field is balanced by the densities of the spinor components of a vortex state, with energy of the state remaining nearly constant (cf. Fig.~\ref{fig:Omega-mu-rho3}).
Fig.~\ref{fig:Omega-mu-rho3}a and b shows polar plots of the numerically calculated densities $|\phi_+|^2$ and $|\phi_-|^2$ for a fixed nonlinearity parameter $\rho=3$. Dashed lines and fainter colors mark spectrally unstable states.
The other two half-vortices, $(0,1)$ and $(0,-1)$ coincide with $(1,0)$ and $(-1,0)$ when two circularly polarized are interchanged and direction of the magnetic field is reversed.

As seen from the analytical formula~\eqref{ABC}, quantity $\Delta m - 2$ defines order of the discrete rotational symmetry in the density distribution of the states (i.e. number of peaks or deeps): in $(1,0)$ and $(0,-1)$ it has a three-fold rotational symmetry while in states $(0,1)$ and $(-1,0)$ the density distribution has "1-fold" symmetry (i.e. no rotational symmetry).
Evolution of numerically calculated density distributions for a higher order state $(4,-4)$ with changing TE-TM splitting parameter $\kappa$ is shown on Fig.~\ref{fig:star}. Due to $|\Delta m - 2|=10$, its density distribution is 10-fold rotationally symmetric.

We investigate stability of the symmetry-breaking states numerically evaluating the eigenvalues of the discretized Hessian matrix (the continuous version of the Hessian matrix is given by Eq.~\eqref{Hessian} in Appendix~\ref{app:stab-th}). The unstable states resulting from this analysis are marked by dashed lines and faint colors in Figures~\ref{fig:hvs}a and~\ref{fig:hvs}b. No instabilities were found among the configurations displayed in Figure~\ref{fig:star}.
We analyzed the unstable states of half-vortices marked by dashed lines and faint colors in Fig.~\ref{fig:hvs}. The arising dynamics when an unstable state is disturbed by a small perturbation is shown on Fig.~\ref{fig:instab-hvs}.
As seen from the Figure, in the initial stage the densities patterns are nearly constant as it takes time for the instability to develop. In the next stage when instability has grown large enough, quasi-periodic patterns appear which indicate an onset of propagating waves which modulate densities of the circular polarized components. Videos of the propagating density modulations is available as Supplementary Material.

\section{Discussion}

To conclude, we have shown that exciton-polariton condensate placed into a trap of non-simply connected geometry may exhibit states whose energies are independent of the applied magnetic field. Properties of these states are dictated by the topology of the condensate wavefunction, i.e. two phase winding numbers of its spinor components.
We analyzed the stability of these topological spin Meissner states and indicated the range of parameters where such states may exist and are stable. These findings helped us to shed the light onto the properties of half-vortices in a ring and gave a clue in understanding of the recent experiments.  We analyzed the effect of TE-TM splitting on the topological spin Meissner states and found that the stable states exist even in presence of significant TE-TM splitting strengths. Finally, we found that a certain class of TSM states exist which breaks rotational symmetry in presence of TE-TM splitting by developing inhomogeneous densities.

The range of parameters discussed in this paper can be reached experimentally. Depending on the size of the ring and detuning, the characteristic energy $\hbar^2/2 m ^* R^2$ may be varied in a broad range of energies. For a ring diameter $10\;\mu m$
the unit energy can be varies from $4\;\mu eV$ to $40\;\mu eV$, depending on the detuning.
Therefore, both small $\Omega\sim 10$ and higher and well accessible in experiments.
The effect of TE-TM splitting can be made significant, if desired.
In a $1~\mu m$ waveguide TE/TM splitting can be as high as $\sim1\;\rm meV$~\cite{Dasbach-PRB-2005,Kuther-PRB-1998} which allows to reach $\kappa\sim 10$ and even $\kappa\sim 100$, in normalized units. On the other hand $\kappa$ can be made negligibly small by choosing larger ring widths, controlling detuning~\cite{Duff-PRL-2015} and properties of distributed Bragg reflector~\cite{Panzarini-PRB-1999}.

redpajama_set_name: RedPajamaArXiv


AMD's Low End 28Nm GPU's to arrive in 2011.
AMD's Low End 28Nm GPU's to arrive in 2011.
A recent mobile roadmap slide leak has revealed that AMD's next upcoming 28Nm low end GPU's will arrive in Quarter of 2011 and will be available by Late 2011, Higher end chips would also arrive by early 2012. However, 28Nm is only limited to the Lower End cards.
The Lower End cards consist of the Chelsea and Thames Chip, Both are 28nm and 128-bit but the Thames feature twice the performance as Seymour (Radeon HD 6000M). Both these chips will arrive in Q4 2011. Higher end models would also arrive by the same time but won't be built on 28nm process.
Heathrow instead will feature both 128 and 192 bit versions and would be released in Q4 2011 while Wimbledon will feature 256-bit on a 28nm process and will be released in Q2 2012.


Tear Film LIPIDS are of utmost Importance for the Function of the TEAR FILM and thus for the Ocular Surface as a whole. The MEIBOMIAN GLANDS inside the eye lids produce lipids. The Meibomian Lipids form an OIL at body temperature. The Meibomian Oil forms the superficial phase of the tear FILM - The tear FILM LIPID LAYER (TFLL). The Meibomian OIL in the Tear Film has important functions for the STABILITY of the Tear Film. Meibomian oil RETARDS the EVAPORATION of the aqueous phase. Meibomian OIL provides a HOMOGENEOUS and perfectly SMOOTH SURFACE for the Tear FILM for perfect light refraction and visual acuity. The lipid production for the tear film is achieved by a large number of individual Meibomian glands. Every single Meibomian Gland has its own orifice. Meibomian Glands have some structural characteristics in contrast to other sebaceous glands such as a long slender gland body. Therefore, SECRETION and DELIVERY of the oil occur in separate regions in Meibomian Glands. SECRETION is the production of the oil deep down inside the Glands. DELIVERY on the other ´end´ describes the process of driving out or delivering the oil from the orifice onto the lid margin. The Differentiation between SECRETION and Delivery explains Obstruction as a main pathology of the Meibomian Glands.
Tear Film LIPIDS are of utmost Importance for the Function of the TEAR FILM and thus for the Ocular Surface as a whole.
The Meibomian glands are lipid producing sebaceous glands in the eye lids. The lipids are very important for the homeostasis of the tear film and for vision.
The TEAR FILM LIPID LAYER is the outer surface of the tear film, it improves tear film stability through retarding the evaporation of the aqueous main phase and provides a smooth homogeneous surface for refraction of incoming light.
Due to the important functions of the Meibomian Lipids for the TEAR FILM ... and since the tear film is the most important requirement for health and integrity of the Ocular Surface ... it is no surprise, that Meibomian Gland Dysfunction (MGD) is the main cause for Dry Eye Disease.
The Meibomian glands are located inside the tarsal plates of both the upper and lower eye lids.
They produce lipids that are liquid at body temperature and thus form an OIL.
Healthy Meibomian Glands deliver a clear liquid oil (arrow) through their orifice onto the lid margin. This can be tested by a technique termed ´Diagnostic Expression´ with very mild pressure from the outside - every orifice should then deliver a little puddle of oil (arrow).
The Tear Film in front of the transparent cornea (the pre-corneal tear film) is a layered structure in which typically three layers are discriminated: (1) at the outside surface to the ambient air is, as just explained in the text to the left, the thin layer of Meibomian Oil (yellow). The oil covers (2) the aqueous main phase of the tear film that is conceivably to some extent mixed with Mucins that constitute in a concentrated for the (3) inner Mucin (pink) Layer that is locate directly on and produced by the cells of the surface epithelium of the ocular surface. The thin lipid layer on top is in the schematic drawing shown much too broad - because, in the real size relation, it would be almost impossible to see it in cross section through a tear film.
The main functions, among probably many others, are the retardation of evaporation of the aqueous phase, the water of the tear film, by sealing the tear film like a lid on a pot of warm water. A second important function is to provide a very smooth and homogeneous surface for the air-to-tear interface that is the main refractory surface of the eye. It provides about 3/4 of the refractive power of the eye.
... WHY ARE THEY CALLED GLANDS IN THE PLURAL FORM ?
The extended long gland bodies of the MEIBOMIAN GLANDS can be seen through the conjunctiva when the eye lid (here the lower one) is slightly everted. The individual orifices (indicated here by open arrows) are typically arranged in a single row, very close to the posterior lid border - immediately in front of the anterior rim of the tear meniscus.
Other sebaceous glands are typically associated with hairs, as e.g. all the hairs in the skin have associated sebaceous glands. Sebaceous glands, particularly those in the skin, are typically short and stout with a short excretory duct that delivers the produced oil into the hair follicle around the hair shaft and onto the skin.
acini are filled by secretory cells termed Meibocytes.
and the main therapeutic challenge is the removal of obstruction in order to get the lipid secretion flowing again.
WHAT IS a SEBACEOUS GLAND ?
Secretion and Delivery of the Meibomian Glands - what is that ?
Due to the explained long extension of the Meibomian glands the secretory Meibocytes in the spherical acini are typically far away from the orifice at the posterior lid margin.
In an obstructed gland (right) the delivery of the secreted oil is blocked and therefore the increasing pressure inside the gland leads to a dilatation of the ducts and to a pressure atrophy of the secretory acini.
The Differentiation between SECRETION deep down in the Gland that fills the ductal system with oil and the eventual DELIVERY from the ductal system out of the gland onto the lid margin allows to better understand obstructive MGD.
Obstruction of the terminal part of a Meibomian Gland will necessarily lead to a stasis of Meibomian oil inside the gland and, since the secretion in the acini continues, the pressure inside the gland will increase. This will necessarily lead to the potential onset of pressure-dependent atrophy of the gland with dilatation of the ducts and destruction of the secretory tissue.


text:
How to save green tech from crashing
There's a real risk that innovation in green technologies will slow for lack of money and interest. Here's why we need a leaner, meaner green tech 2.0.
Martin LaMonica
Aug. 26, 2011 4:00 a.m. PT
Entrepreneur Brad Hines' first solar company crashed and burned, but he's already started another venture, a story which offers cues on how the entire green-tech sector can reinvent itself in tough times.
In 2005, Hines left his work as a NASA engineer to build a radically new and powerful type of rooftop solar collector. But even after taking in tens of millions of dollars from top-shelf investors, the company, Soliant Energy, ultimately failed and its intellectual property was sold off, reportedly at bargain basement prices.
Now Hines is at work on another solar startup called Thermata with a whole new approach. Rather than try to reinvent the solar panel, he's using existing concentrating solar thermal technology to address a very specific business need--making steam for industrial boilers. Getting funding is a completely different game than it was in 2005, too.
"It was a time when a naive new entrepreneur could have a good idea and get funded. In this market, just a good idea won't get you funded. You need a lot more." said Hines. "And in solar, the days of quantum changes in cost and performance are over...It isn't like microprocessors where you can keep doubling the speed."
Welcome to green tech 2.0. In the first wave, venture capitalists threw billions of dollars at technologies with potential to disrupt the energy industry and stall climate change. Now, entrepreneurs need to be far more savvy about raising money and smarter about how to build a business. In short, people who want to do good by the planet and do well financially need a new playbook.
What's at stake isn't just whether the next Google will come out of the green-tech sector. It's not a cure-all, but innovation in clean-energy technologies can help perk up the U.S. economy while addressing environmental problems and cutting the country's fossil fuel use. If businesses--the engines of green-technology development--are starved of investment and public interest, the pace of innovation could slow down.
The good news is that entrepreneurs are still being attracted to clean-energy technologies, often motivated by a desire to work on something meaningful. But the realities of venture capital mean that science experiments most likely won't get funded by those sources anymore. Projected cuts in federal spending don't bode well for a big influx of research funding either.
Instead, funding could come from multiple sources, including VCs, and innovation may be as much about business models as technology. For example, some of the most successful startups in solar are installers which offersolar leases, a financial innovation that's helping make rooftop solar panels accessible. Even with a big technical advance, say in energy storage or fuel cells, people need to be very calculating about how to get a commercial foothold.
"The first wave of clean tech belonged to the material scientists. The next wave belongs to the entrepreneurs," said green-tech investor and industry observer Rob Day. "There's plenty of technology innovation that would make an improvement on the status quo...The biggest challenges are actually commercializing it."
Ambitious green companies should apply their technology to solve a specific problem and bundle it up in a way that's easy and low-risk for customers to adopt, said Rob Day. The computer industry learned that years ago: instead of selling a PC, IBM reinvented itself selling packaged solutions which include hardware, software, and services--a strategy many others have copied.
Breakthrough versus slow and steady
Revisting just a sampling of ambitious green-tech startups from last decade today shows that things haven't always worked out as planned. Venture-backed companies don't fully represent all innovation in green technologies, as large companies and research labs are known to innovate as well. But the wave--some would call it a bubble--of VC activity created high expectations and buzz around the idea of making money by being green.
The solar industry is seeing the dangers of competing in an industry where you're creating a commodity product--a flow of electrons. Dozens of Silicon Valley solar startups bet on thin-film solar cells, which remains viable technology. But the onrush of Chinese manufacturers in traditional silicon solar cells has lowered the cost per watt by over 50 percent in the last two years, making it difficult for companies without a technical edge and low-cost manufacturing to survive. Evergreen Solar filed for bankruptcy protection this month, but analysts say that there will be tough times and consolidation among the others.
Another popular investment target, biofuels have failed to scale beyond pilot stage or meet government mandates. A dismal example is Range Fuels which last year secured $162 million in local, state, and federal subsidies for a plant to make ethanol from locally harvested wood chips. But the plant shut down earlier this year after technical challenges couldn't be overcome.
• For VCs in green tech, it's time for Plan B
• Once booming, green-tech investing shifts down
• Do new water technologies stand a chance?
• Why green-tech startups depend on big business
• Harsh lessons from Evergreen Solar flame-out
Similarly, algae companies offered the compelling idea of making diesel fuel from algae fed only sunlight and carbon dioxide from big polluters, such as power plants. Some companies, including GreenFuel Technologies, have failed or scaled back their plans, which were not helped by gyrating oil prices. Many researchers contend now that another 10 years of research and development are required before algae can realistically substitute for oil at scale.
For sure, there continue to be some real potential technology breakthroughs. Solar startup 1366 Technologies plans to build a demonstration plant of a solar technology that can substantially lower manufacturing costs. Another is Joule, which is using microorganisms to make diesel fuel, or PolyPlus Battery which is developing a battery capable of giving an electric car hundreds of miles of range.
But given some of the missteps, it's no surprise then that venture capitalists are reining in investment, particularly on early-stage ventures. Instead, they're putting money into their current companies or, in some cases, phasing out their activities in green tech.
Fat niches
Even though there have been disappointments--and perhaps more coming--it would be a mistake to write off green tech as a failed investment thesis.
For starters, a number of green-tech businesses have successfully grown and made an impact. It's too early to say whether Tesla Motors will become a sustainable, profitable business but its bet on electric vehicles has shaken up the auto industry status quo, from General Motors to Toyota. Less high-profile are algae company Solazyme, smart-grid networking specialist SilverSpring Networks, and solar-panel microinverter company EnPhase Energy. They are among the green-tech companies to build a substantial business and file to go public.
Also, investors have gotten smarter. Venture capitalists simply aren't equipped to fund a venture that will require five years of R&D and a billion dollars to build a factory. VCs are still placing bets in this area, but more funding will need to come from other investors willing to stick around for the long term. Depending on the need, sources can be family funds, government research programs, and potentially late-stage private equity funds. Large corporations, such as General Electric or Chevron, are particularly crucial for commercializing products at scale or, as Google does, acting as a customer.
ARPA-E researchers dig deep for energy innovation (photos)
Entrepreneurs, meanwhile, need to focus on solving one problem very well. That's not to say that people shouldn't be ambitious and shoot for the moon. But change comes slowly in energy--the laws of thermodynamics often apply more than Moore's Law. So don't expect to duplicate the valuation and impact of a Facebook or Groupon in a few short years.
Targeting specific niches may sound less exciting or ambitious. But in energy, a niche can be a huge market. A couple of examples: Amyris is one of many synthetic biotech companies targeting biofuels, but it is making industrial chemicals first because that's a quicker route to scale up. Digital Lumens has made a clever commercial LED light fixture targeted only at warehouses. That alone is a substantial market and success there could lead to a springboard to other commercial applications.
"We're going after one slice of the market and repurposing a couple of engineering inventions to address that market. And we'll be out in a year and a half. There's still money to be made in boring businesses."
--Brad Hines, CEO, Thermata
Make no doubt, environmental regulations and government industrial policies make a major difference, as seen in China's rapid advance in wind and solar. But prospects for bold legislation on clean energy seem dim and even investments in research and infrastructure, such as the power grid, could be scaled back. The most crucial programs for nurturing innovations are in research and development, such as ARPA-E, and the federal loan guarantee program, where the Department of Energy backs large-scale rollouts of new technology, such as a utility-scale renewable-energy project.
Green-tech innovators need to borrow heavily from the digital world as well. One approach that's not fully explored, called the Cleanweb by investor Sunil Paul, is using the power of social media and mobile technology to help consumers and businesses be more efficient and lead a greener lifestyle.
To be clear, technical breakthroughs in basic science are very much needed and could dramatically change how we use energy. But there are many ways to use IT and other existing technologies to use natural resources--water, energy, minerals, land--more efficiently.
That's exactly what Brad Hines is doing at his new company. Thermata is coupling off-the-shelf solar technology with clever software to help big industrial companies save money they'd otherwise spend on natural gas for steam. He doesn't plan to make the product himself but outsource manufacturing to keep down costs and rely on big corporate partners to help get initial customers.
"We're going after one slice of the market and repurposing a couple of engineering inventions to address that market. And we'll be out in a year and a half," Hines said. "There's still money to be made in boring businesses."
Green Tech Google
Discuss: How to save green tech from crashing

redpajama_set_name: RedPajamaCommonCrawl


The dissertation is a final project before getting a doctoral degree. In some universities, students also write dissertations on their undergraduate level in order to get their degrees signifying the reach of a higher educational level. While you might be already brainstorming how to structure and fill in with relevant reasoning as well as valid evidence each and every section of such a long academic work, there are also another questions to consider before you sit down and write the first draft of your dissertation file. For example, before any dissertation is completed, students often write a dissertation proposal and submit it to their mentor. In case if this paper is accepted, they can move on with their further research the way that they have been doing it until they handed in the proposal. If you read some good dissertation proposal examples, you will understand what this document is all about. Many scholars consider it equally important with the final dissertation itself because a good proposal portrays your objectives, methodology of research, questions, literature, and many other things to know before you can be sure that every method you use is a good idea for writing a dissertation on its basis.
Regardless of whether the dissertation proposal is mandatory in your university or not, writing it before you start working on your dissertation is always useful because by doing so you can take a look at the whole research without having to write hundreds of pages of scientific content. Just read an example of a good dissertation proposal to see how it can wonderfully portray all the significant stages of your research like setting up the main questions, refining the problem that needs to be resolved, as well as finding ways to tackle it.
Submitting a dissertation proposal to your mentor can be not such an easy task, nevertheless, it is important for your final result. Because how can you get your post-graduate degree if your dissertation won't be considered truly useful for the academic community that you belong to? That's why while hearing your mentor criticizing your write-up might be not a pleasant practice, it is absolutely necessary for your further success. Similarly, to give you a helping hand, our professionals with academic experience and degrees in various fields write a dissertation proposal example to help you to get through with this stage of working on your dissertation.
Since a dissertation proposal structure is somewhat similar with the dissertation itself, you get a great training in order to test your research questions, problems that you discuss, and methods that you apply for the research way before the final work is done. The main advantage of this is that you can correct all of that factors as you go and it would be much easier than editing all the huge dissertation after you have already written the first draft of the whole thing. If you are still not sure how to write your proposal, it is fine. Everyone does it for the first time. Just check out our page about how to write a dissertation proposal examples and so many things will become clearer to you right away.


Hi all, I've just joined so forgive me if I go on a bit. I was diagnosed with Lung Cancer last March after having so many scans on various parts of my body I was operated on on April 11th. Whilst they had me open they found it had spread to the lymph nodes so they removed what they could and 6 weeks later I started a 3 Month course of Chemo to clear /Stop it spreading, then 8 weeks after that I had a 20 days course of Radiotherapy. Through it all managed really well no sickness just a bit of fatigue. I went back to work after the Chemo and I've been fine ??. I had my 1st scan and it was CLEAR so I should be ok now ....Right ???? Well that's the thing I get up everyday I dress,put make-up on, do my hair and smile. I am supporting a friend and her mum who is now going through the same thing. I'm doing everything I was doing before but I feel like a part of me has gone. I now feel anxious, scared, I over think everything and I'm starting to turn into a hypochondriac on the surface im coping and all is well but really I feel like I'm sinking. Sorry if this is a bit depressing but just wanted to know if anyone else felt like this ??
First of all, I do hope other online members will share their experiences with you - as what you're feeling is normal, although it may not feel it.
You have been through a life changing, and for a time, life threatening experience. It's as big as it gets. Those around you may have thought that the 'all clear' was a sign to leap up and down with joy. What they don't perhaps realise, is that you are living with the fear of it coming back. You've also been through a gruelling amount of treatments which will have taken their toll.
It takes time for the 'high alert' button to calm down - you'll understandably be wondering if any new aches or pains are cancer related, even when it's most likely not the case. You've changed - and what you've been through has taken away certainty.
So...be kind to yourself. It sounds like it's about the anniversary of your original diagnosis, and your emotions and anxiety levels may be enhanced at this time. You're also supporting a freind, and that can bring back feelings you had about your own diagnosis and treatment. Many people have told us that the book 'The Cancer Survivor's Companion: practical ways to cope with your feelings afer cancer' helps clarify things.
If you live near one of our Maggie's Centres, you'd be welcome to come in and ask about our 'Where Now?' course, which is exactly for this post treatment period. You may also benefit from a one to one discussion and support from one of their cancer support specialist team.
I'll drop you a line to introduce myself, and you're welcome to message Robyn or myself to talk more about how you feel. Feelings which are natural...and you're certainly not alone in experiencing them.


Shark Eyes Sticker Black is a visual shark deterrent for all hard smooth surfaces and watercraft.
Designed to stick to dive fins, kayak, ski, surfboard, bodyboard, SUP, kite surf board, windsurf board, canoe, etc.
It's affordable and easy to use. Please note the bigger the sticker the better the deterrent!
Black background enhances the colours of the eyes on a black or dark coloured background.
Small stickers are 210mm wide.
Medium stickers are 290mm wide x 91mm high.
Large stickers are 410mm x 128mm.
Decals are UV/water resistant and Australian made.
They are designed as a super sticky, high-tac, vinyl durable sticker.
It is vital that you make sure the surface is very clean and completely free of all sand and salt when sticking. This helps you get maximum life from your sticker.


text:
class OSGFPhysicsWorld;
class OSGFDrawablePhysicsBody
	:public OSGF3DDrawableComponent
{
public:
	OSGFDrawablePhysicsBody(Game& game,
		OSGFPhysicsWorld& world);
	virtual void Update(double dTime);
	virtual void Render()const;
	virtual void Initialize();
	virtual void SetTranslation(float x,float y,float z)
	{
		SetTranslation(btVector3(x,y,z));
	}
	virtual void SetTranslation(const D3DXVECTOR3& v)
	{
		SetTranslation(v.x,v.y,v.z);
	}
	virtual void SetTranslation(const D3DXMATRIX& m)
	{
		SetTranslation(m._41,m._42,m._43);
	}
	void SetTranslation(const btVector3 &v)
	{
		mPos = v;
		CreateMotionSatate();
	}
	virtual void SetScaling(float x,float y,float z)
	{
		SetScaling(btVector3(x,y,z));
	}
	void SetScaling(const btVector3& v)
	{
		mShape->setLocalScaling(v);
		CalculateInertia();
		mBody->setCollisionShape(mShape);
	}
	virtual void SetScaling(const D3DXVECTOR3& v)
	{
		SetScaling(v.x,v.y,v.z);
	}
	virtual void SetScaling(const D3DXMATRIX& m)
	{
		SetScaling(m._11,m._22,m._33);
	}
	virtual void SetRotation(float x,float y,float z)
	{
		SetRotation(btQuaternion(y,x,z));
	}
	void SetRotation(const btQuaternion &q)
	{
		mRotation = q;
		CreateMotionSatate();
	}
	virtual void SetRotation(const D3DXQUATERNION &q)
	{
		SetRotation(BulletQuatFromDX(q));
	}
	virtual void RotateX(float a)
	{
		btQuaternion b(0,a,0);
		SetRotation(mRotation * b);
	}
	virtual void RotateY(float a)
	{
		btQuaternion b(a,0,0);
		SetRotation(mRotation * b);
	}
	virtual void RotateZ(float a)
	{
		btQuaternion b(0,0,a);
		SetRotation(mRotation * b);
	}
	void SetMotionState(btDefaultMotionState* m)
	{
		SafeDelete(mMotionState);
		mMotionState = m;
	}
	btDefaultMotionState* GetMotionState()const
	{
		return mMotionState;
	}
	btRigidBody* GetRigidBody()
	{
		return mBody;
	}
	void SetDrawData(OSGF3DDrawableComponent* drawData)
	{
		mDrawData = drawData->Clone();
		if(GetLight())
			mDrawData->SetLight(GetLight());
		if(GetCamera())
			mDrawData->SetCamera(GetCamera());
	}
	const OSGF3DDrawableComponent* GetDrawData()const
	{
		return mDrawData;
	}
	virtual void SetCamera(const OSGFCamera* cam)
	{
		OSGF3DDrawableComponent::SetCamera(cam);
		if(mDrawData)
			mDrawData->SetCamera(cam);
	}
	virtual void SetLight(const Light* light)
	{
		OSGF3DDrawableComponent::SetLight(light);
		if(mDrawData)
			mDrawData->SetLight(light);
	}
	void SetMass(btScalar mass)
	{
		mMass = mass;
		CalculateInertia();
	}
	btScalar GetMass()const
	{
		return mMass;
	}
	virtual void SetRotation(const D3DXMATRIX& m);
	void SetCollisionShape(btCollisionShape* shape)
	{
		mShape = shape;
		CalculateInertia();
	}
	btCollisionShape* GetCollisionShape()
	{
		return mShape;
	}
	OSGFPhysicsWorld& GetPhysicsWorld()
	{
		return mWorld;
	}
	~OSGFDrawablePhysicsBody();
protected:
	OSGFPhysicsWorld & mWorld;
private:
	void CreateMotionSatate()
	{
		SafeDelete(mMotionState);
		mMotionState = new btDefaultMotionState(btTransform(mRotation,mPos));
		mBody->setMotionState(mMotionState);
	}
	void CalculateInertia()
	{
		mShape->calculateLocalInertia(mMass,mInertia);
	}
	btRigidBody* mBody;
	btCollisionShape* mShape;
	btDefaultMotionState* mMotionState;
	btQuaternion mRotation;
	btVector3 mPos;
	btVector3 mInertia;
	btScalar mMass;
	OSGF3DDrawableComponent* mDrawData;
};

redpajama_set_name: RedPajamaGithub


text:
Yoldiella jeffreysi är en musselart som först beskrevs av Hidalgo 1877.  Yoldiella jeffreysi ingår i släktet Yoldiella och familjen Yoldiidae. Inga underarter finns listade i Catalogue of Life.

Källor

Musslor
jeffreysi

redpajama_set_name: RedPajamaWikipedia


text:
Education & Outreach,Events,grants,news,programs,Volunteering
The Yellow Dog Watershed Preserve's Nature School has been awarded a $4000.00 grant from the State of Michigan Council for Arts and Cultural Affairs (MCACA) to host a three-day Traditional Arts Gathering along the Yellow Dog River west of County Rd 510 on May 23, 24, and 25. This grant was one of 561 applications to compete for MCACA fiscal year 2019 funding. More…
featured, featured image
Events,Land Preservation,news
It is with great excitement and honor to announce the unveiling of YDWP's newest protected lands. As of April 11, 1,000 acres on the Yellow Dog Plains will forever be known as the Mudjekewis Wildlife Refuge. The land was the legacy of C. Fred Rydholm and it was donated to our organization for the purpose of preserving wild spaces, building community awareness of the importance of nature in our lives, and keeping a bit of history alive.
To celebrate, YDWP will be hosting the inaugural hike to the Wildlife Refuge on May 26th. Participants will get to tour the grounds, hear some good "Fred" stories, and enjoy the legacy that is now here for public benefit. Space is limited to email emily@yellowdogwatershed.org to secure your spot! Visit our webpage for more information, pictures, and maps.
The Yellow Dog Watershed Preserve invites those who love the outdoors to join us for the unveiling of a new land monitoring program on June 2 in Marquette. The event aims to inform participants on how to help be caretakers of YDWP's protected properties. In all, the organization owns and protects 1,340 acres in northern Marquette County. Originally, staff members at the Preserve monitored the properties but as the momentum grows to increase the amount of protected lands in our area, it is time to engage the community in aiding in this task. More…
news,Sulfide Mining
County Road 595 Appeal Concludes
The battle between the U.S. Environmental Protection Agency and the Marquette County Road Commission concluded on March 20th in the U.S. 6th Circuit Court over County Road 595, the proposed road leading from Eagle Mine to Humboldt Mill. For those who need a refresher, the permit for CR 595 was submitted to the MDEQ in 2012 and as part of the review process, the U.S. EPA issued objections to the project based on the Clean Water Act. Long story short, the MDEQ did not issue the permit and the process then transferred to the U.S. Army Corps of Engineers.
county road 595
Land Preservation,news,programs
Another 240 Acres Preserved
Our community can now count another 240 acres of land in the Yellow Dog watershed as permanently protected and forever open to the public. YDWP recently became the new owners and caretakers of the property which is along the Yellow Dog River in Champion Township. It is a magnificent parcel that we are honored to hold.
The new property is directly adjacent and upstream from our Pinnacle Falls Tract, which is a 40 acre parcel that was purchased by YDWP in 2002. The property originally was owned by the Mead Paper Company who sold it to James and Barbara Kidder in 2000. The Kidders held it for 16 years as part of a larger holding of over 800 acres. "We're glad YDWP is the new owners and that the transfer is complete," states Jim and Barbara Kidder. The Nature Conservancy will hold a Conservation Easement on the property, which will provide additional protection. "Working with partner organizations such as Yellow Dog Watershed Preserve is crucial in The Nature Conservancy efforts to preserve biological diversity across the whole of Earth. These groups are essential and expand our work at the local, grass roots level for they can inspire and engage local communities to connect people with nature in an effort to preserve our disappearing landscape," says Jeff Knoop, U.P. Director of Land Protection for the Nature Conservancy. More…
land protection program, Pinnacle Falls, yellow dog river
Education & Outreach,news,Sulfide Mining
Watershed Mineral Exploration Update
9 Jun , 2016
Chauncey Moran, Yellow Dog RIVERKEEPER, recently conducted an aerial survey of the Salmon Trout and Yellow Dog River watersheds, scanning the landscape and waterways for mining activity and other land use issues. Both watersheds were surveyed via airplane and Moran reports that there are 7 total exploratory drilling operations that are currently active. All 7 of those are occurring directly east of Eagle Mine within a relatively short distance of the portal. This location has been called Eagle East and represents a plausible location for additional mineral deposits. No other active drilling elsewhere in the watersheds was noted during the flight. YDWP will continue to fly and conduct monitoring of this activity in order to inform concerned community members. More…
aerial surveys, Eagle Mine, featured, salmon-trout river