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

A complex extension to quantum mechanics, based on a special class of non-Hermitian Hamiltonians with purely real spectra, was discovered more than two decades ago~\cite{bender1998,bender2002,ali2002,ali2010}. These continuum Hamiltonians on an infinite line shared the property that each of them is invariant under combined operations of parity and time-reversal (${\mathcal {PT}}$); i.e. the Hamiltonian commutes with the ${\mathcal {PT}}$ operator. Over the past decade, it has become clear that ${\mathcal {PT}}$-symmetric systems represent open, classical systems with balanced, spatially or temporally separated gain and loss that are represented by complex real-space potentials~\cite{feng2017,el2018,ozdemir2019}.

The spectrum $E_\alpha$ of a ${\mathcal {PT}}$-symmetric Hamiltonian $H(\gamma)$ is real at small non-Hermiticities $\gamma$ and turns into complex conjugate pairs at large non-Hermiticities. When the spectrum is purely real, we can choose an eigenvector $\ket{\epsilon_\alpha}$ of $H$ to be a simultaneous eigenvector of the antilinear ${\mathcal {PT}}$-operator with eigenvalue +1. When the spectrum is complex, the ${\mathcal {PT}}$ operator acting on $\ket{E_\alpha}$ transforms it into the eigenvector with complex-conjugate eigenvalue, i.e. ${\mathcal {PT}}|E_\alpha\rangle=|E^*_\alpha\rangle$ where $*$ denotes complex conjugation. The transition from a purely real to complex conjugate spectrum occurs when $\gamma=\gamma_\mathrm{PT}$ where $\gamma_\mathrm{PT}$ is called the ${\mathcal {PT}}$-symmetry breaking threshold. At the threshold, the geometric multiplicity of the eigenvalues of the Hamiltonian $H(\gamma_\mathrm{PT})$ is smaller than the algebraic multiplicity. Such a Hamiltonian degeneracy is called an exceptional point (EP) degeneracy, where not only do the eigenvalues become degenerate, but the corresponding eigenvectors also coalesce.

In the past decade ${\mathcal {PT}}$-symmetric systems with balanced gain-loss have been realized in classical wave systems including evanescently coupled waveguides~\cite{ruter2010}, fiber loops~\cite{regensburger2012}, optical resonators ~\cite{chang2014, hodaei2014}, electrical circuits ~\cite{schindler2011,choi2018}, and mechanical oscillators~\cite{bender2013}. However, since the EP degeneracies also occur for Hamiltonians with mode selective dissipation, the dynamics of $\mathcal{PT}$-symmetric Hamiltonians have also been realized in purely lossy classical systems consisting of coupled waveguides, resonators, or electrical circuits~\cite{guo2009,lenmontiel2018}, semiclassical systems with ultracold atoms~\cite{li2019}, and quantum systems~\cite{Wu2019,naghiloo2019,klauck2019}.

In this paper we obtain the ${\mathcal {PT}}$-breaking threshold in a one-dimensional Kitaev model in the presence of one pair of gain-loss potentials $\pm i\gamma$ located on reflection-symmetric sites. Kitaev model is a toy model for a topological superconductor with Majorana fermions as excitations. A Majorana fermion is, by construction, its own antiparticle, i.e. it is a fermion constructed from a electron-hole pair. There are a number of studies on topological superconductors with non-Hermitian, ${\mathcal {PT}}$-symmetric potentials that preserve the translational invariance of the system~\cite{wang2015,li2018,yao2018,yuce2016,klett2017,kawabata2018,li2020,sarma2015,leumer2020}. They focused on the fate of the edge states that are characterized by zero energy. A finite Kitaev chain with a pair of ${\mathcal {PT}}$-symmetric potentials at the edge has also been studied~\cite{yuce2016,kawabata2018,li2020,klett2017}. The result is the emergence of an additional pair of edge state with a non-zero energy eigenvalue~\cite{kawabata2018}. In contrast to these studies on the Majorana edge modes, here we concentrate on the variation of the ${\mathcal {PT}}$-breaking threshold $\gamma_\mathrm{PT}(m_0,N)$ with the location $m_0$ of the gain potential in a Kitaev chain of size $N$.

This paper is organized as follows. In Sec.~\ref{sec:model} we describe the Hermitian Kitaev chain along with its symmetry properties, and then introduce the non-Hermitian perturbation. In Sec.~\ref{sec:results}, we present numerical results for the ${\mathcal {PT}}$-symmetry breaking threshold as a function of different parameters of the Hermitian model and the relative location of the gain potential. We point out key differences among the ${\mathcal {PT}}$-symmetry breaking thresholds for various settings of the on-site potentials $\mu$ and superconducting coupling strengths $\delta$. In Sec.~\ref{sec:N5case} we describe a small-system case of $N=5$ sites and analytically obtain the dependence of the ${\mathcal {PT}}$ threshold when the gain-loss potentials are farthest apart and closest together. In Sec.~\ref{sec:reentrant}, we show that the Kitaev model shows re-entrant $\mathcal{PT}$-symmetric phase, and map out its EP contours. Finally in Sec.~\ref{sec:conclusion}, we conclude by summarizing the results.

\section{\label{sec:model}Tight-binding Model}

The Kitaev model of a one dimensional, p-wave superconducting chain with $N$ sites and open boundary conditions is described by the following Hermitian Hamiltonian,
\begin{eqnarray}
\label{eq:H_fermion}
    H_0 & = & - \mu \sum_{n=1}^N c_n^{\dagger}c_n -J \sum_{n=1}^{N-1} (c_{n}^{\dagger}c_{n+1} + \mathrm{h.c.})\nonumber\\
&+  &i\delta \sum_{n=1}^{N-1}(c_{n}c_{n+1} - \mathrm{h.c.}).
\end{eqnarray}
Here $c^{\dagger}_n$ and $c_n$ are fermionic creation and annihilation operators for site $n$ in the chain, $\mu$ is the on-site potential, $J>0$ is the nearest-neighbor hopping strength, and $\delta>0$ is the amplitude of the (p-wave) superconducting coupling for a Cooper pair that is localized across neighboring sites~\cite{kitaev2001}. The global phase of the superconducting order parameter is fixed at $\pi/2$ to ensure that Eq.(\ref{eq:H_fermion}) is parity-time symmetric, with the parity operator given by
$\mathcal{P}: c_n\rightarrow c_{\bar{n}}$ where $\bar{n}=N+1-n$ is the mirror-symmetric counterpart of site $n$ and the time-reversal operator is given by complex conjugation, $\mathcal{T}=*$.

We rewrite Eq.(\ref{eq:H_fermion}) by using the Bogoliubov-de Gennes representation in terms of the operator-vector $\Psi=(c_1,c^{\dagger}_1,c_2,c^{\dagger}_2 \dots c_N, c^{\dagger}_N)^T$ as
$H_0=\Psi^\dagger H_\mathrm{BdG}\Psi$ where the $2N\times 2N$ matrix $H_\mathrm{BdG}$ in the site-representation is given by
\begin{eqnarray}
H_\mathrm{BdG} & = & -\frac{\mu}{2} \sum_{n=1}^N\ket{n}\bra{n}\otimes \sigma_z \nonumber \\
        &   & -\frac{J}{2} \sum_{n=1}^{N-1}\left(\ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right)\otimes \sigma_z \nonumber \\
        &   & + \frac{i\delta}{2} \sum_{n=1}^{N-1}\left(\ket{n}\bra{n+1} - \ket{n+1}\bra{n}\right)\otimes \sigma_x,
\label{Eq:kiHamil}
\end{eqnarray}
where $\sigma_x,\sigma_z$ are the standard Pauli matrices. For a chain with periodic boundary conditions, translational invariance allows us to transform the site-space Hamiltonian (\ref{Eq:kiHamil}) into momentum space, $\tilde{H}_\mathrm{BdG} =UH_\mathrm{BdG}U^{\dagger}$
with a unitary
\begin{align}
U& =\frac{1}{\sqrt{N}} \sum_{k,n=1}^N e^{-i\ket{p_k}\bra{n}}\otimes\left( e^{i\pi/4}\mathbbm{1}_2+e^{-i\pi/4}\sigma_z\right).
\end{align}
The block-diagonalized momentum-space Hamiltonian is given by $\tilde{H}_\mathrm{BdG}=\sum_{k=1}^N h(p_k)|p_k\rangle\langle p_k|$ where $p_k=2\pi k/N$ are the discrete quasimomenta for a finite chain and
\begin{eqnarray}
h(p) &=&
\begin{pmatrix}
-J\cos{p}-\mu/2 & -i\delta \sin{p} \\
i\delta \sin{p} & J\cos{p}+\mu/2
\end{pmatrix}.
\end{eqnarray}
The bulk energy spectrum of the Hamiltonian $\tilde{H}_\mathrm{BdG}$ is given by
\begin{equation}
E_{\pm}(p)=\pm \sqrt{(J\cos{p} + \mu/2)^2 + \delta^2 \sin^2{p}},
\label{eq:bulk}
\end{equation}
and it shows that in the limit of an infinite chain, $N\gg 1$, the gap in the spectrum vanishes at $p=\pi$ when $\mu=2J$. For the finite chain, the spectrum Eq.(\ref{eq:bulk}) is symmetric about $\delta=0$ because $H_\mathrm{BdG}(-\delta)=\mathcal{S} H_\mathrm{BdG}(\delta)\mathcal{S}^\dagger$ with a unitary operator $\mathcal{S}=\mathbbm{1}_N\otimes\sigma_z$. When the boundary conditions are changed from periodic to open, in the zero chemical potential limit, Majorana zero-modes (fermionic excitations) appear localized on the edges of the chain. These edge modes are robust when gain-loss potentials are introduced on random sites~\cite{yuce2016} or on parity symmetric sites with disorder~\cite{kawabata2018}.

To this toy model with open boundary conditions, we add a pair of balanced gain-loss potentials
$\pm i\gamma$ at mirror symmetric sites $m_0$ and $\bar{m}_0$,
\begin{equation}
i\Gamma= \frac{i\gamma}{2} \left(\ket{m_0}\bra{m_0} - \ket{\overline{m}_0}\bra{\overline{m}_0}\right)\otimes \sigma_z,
\end{equation}
and thereby get a non-Hermitian, $\mathcal{PT}$-symmetric Kitaev chain Hamiltonian
\begin{equation}
\label{eq:hkitaev}
H_\mathrm{K}(\gamma,\delta,\mu) =H_\mathrm{BdG}+i\Gamma.
\end{equation}

\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{kitaevchain.pdf}
\caption{Schematic representation of a Kitaev model with one pair of gain and loss potentials, Eq.(\ref{eq:hkitaev}). Two chains (gray and white) with detuned potentials $\pm\mu$ have nearest neighbor Hermitian tunneling amplitudes $\pm J$, and next-nearest-neighbor Hermitian amplitudes $\pm i\delta$. Due to the presence of two degree of freedom on each site~\cite{harsha2013}, the potential $i\gamma$ on site $m_0$ acts as gain (red) for one and loss (blue) for the other. This schematic can be realized with coupled resonator rings where one can engineer complex, Hermitian tunneling amplitudes.}
\label{fig:kichain}
\end{figure}

Fig.~\ref{fig:kichain} shows schematic representation of a lattice model described by Eq.(\ref{eq:hkitaev}). Although the original model refers to many-body fermionic system with two bands, in its ``single-particle'' form, Eq.(\ref{eq:hkitaev}), it can be interpreted as two, detuned coupled chains with Hermitian nearest-neighbor couplings $\pm J$ and Hermitian, purely imaginary, next-nearest-neighbor couplings
$\pm i\delta$~\cite{harsha2013}. The gain potential on site $m_0$, given by $(i\gamma/2) |m_0\rangle\langle m_0|\otimes\sigma_z$, then stands for gain in one chain and loss in the second chain. This representation of the $\mathcal{PT}$-symmetric Kitaev model can be experimentally implemented in resonator arrays where real and purely imaginary tunneling amplitudes can be easily engineered. In the next section we explore the global phase diagram for the $\mathcal{PT}$-symmetry breaking threshold $\gamma_\mathrm{th}(m_0,\delta,\mu)$.

\section{\label{sec:results} Numerical results for $\gamma_\mathrm{th}(m_0,\delta,\mu)$}

\begin{figure*}
    \centering
    \includegraphics[width=0.9\textwidth]{N20YthDelM0.pdf}
    \caption{Energy eigenvalues (in units of $J$) of an $N=20$ site Hamiltonian $H_\mathrm{BdG}(\delta)$, Eq.(\ref{Eq:kiHamil}) as a function of detuning. (a) $\mu/J=0$, (b) $\mu/J=1$ have topological edge modes, while the system is in the topologically trivial phase at (c) $\mu/J=2$ and (d) $\mu/J=3$. Corresponding $\mathcal{PT}$ threshold values $\gamma_\mathrm{th}/J$ obtained from the Hamiltonian $H_\mathrm{K}$, Eq.(\ref{eq:hkitaev}) are plotted as a function of the gain location $m_0\in[1,N/2]$ and the superconducting order parameter $\delta/J$: (e) $\mu/J = 0$, (f)  $\mu/J = 1$, (g)  $\mu/J = 2$ (h)  $\mu/J = 3$. Most of these features can be understood in terms of Hermitian band structure, panels a-d.}
    \label{fig:PTeven}
\end{figure*}

\begin{figure*}
    \centering
    \includegraphics[width=0.9\textwidth]{N21YthDelM0.pdf}
    \caption{Energy eigenvalues (in units of $J$) of an $N=21$ site Hamiltonian $H_\mathrm{BdG}(\delta)$, Eq.(\ref{Eq:kiHamil}) as a function of detuning. (a) $\mu/J=0$, (b) $\mu/J=1$ have mid-gap states with zero energy, but these are not topological. (c) $\mu/J=2$ and (d) $\mu/J=3$ show emergence of a gapped spectrum. Corresponding $\mathcal{PT}$ threshold values $\gamma_\mathrm{th}/J$ obtained from the Hamiltonian $H_\mathrm{K}$, Eq.(\ref{eq:hkitaev}) are plotted as a function of the gain location $m_0\in[1,(N-1)/2]$ and the superconducting order parameter $\delta/J$: (e) $\mu/J = 0$, (f)  $\mu/J = 1$, (g)  $\mu/J = 2$ (h)  $\mu/J = 3$. Due to the absence of topological edge modes, the $\mathcal{PT}$ threshold behavior at $\mu=0$ is markedly different from that of an even chain, Fig.~\ref{fig:PTeven}(e). At nonzero detuning, the threshold is non-monotonically suppressed with increasing $\delta$.}
    \label{fig:PTodd}
\end{figure*}

The results presented in this section are obtained by diagonalizing $H_\mathrm{K}$ or $H_\mathrm{BdG}$ for Kitaev chains of size $N=20$ (Fig.~\ref{fig:PTeven}) and $N=21$ (Fig.~\ref{fig:PTodd}). They remain qualitatively same for larger chain sizes, and the differences between even and odd parity chains persist in the large $N$ limit, as they do for a simple tight-binding model. All energy scales are measured in units of the tight-binding coupling $J=1$. Figures~\ref{fig:PTeven}(a)-(d) show the the energy eigenvalues $E_n$ for a Hermitian Kitaev chain as a function of the superconducting order parameter $\delta/J$. When $\delta=0=\mu$, we get the cosine-band of a tight-binding model. As the detuning $\delta$ is increased from (a) to (d), the two bands become well-separated. On the other hand, at a fixed detuning, when $\delta$ is increased, the bands develop fan-like linear dispersion, leading to massively degenerate flat bands at $\delta/J=1$ at zero detuning. As the detuning is increased from $\mu=0$, the system develops two crossing points (shown by blue circles in (b)). We also note that zero-energy states are present when $\mu<2J$. At $\mu/J=2$, the superconducting gap closes marking a phase transition to the topologically trivial phase. Here the mid-gap states become a part of the bulk. When $\mu$ is increased further, Fig.~\ref{fig:PTeven}(d), the system is in the trivial superconducting phase and energy spectrum is gapped. The topological, edge-localized zero energy states only occur when $\mu\leq 2J$. At $\mu=0$, these zero-energy states are fully localized on the end sites. When $\mu$ is increased, these states extend into the bulk of the chain, with an exponentially decaying probability density~\cite{leumer2020,sarma2015}. The Hermitian, (near or exact) degeneracies of the Hamiltonian $H_\mathrm{BdG}$ play an important role in determining the threshold gain-loss strength $\gamma_\mathrm{th}$ when a pair of gain-loss potentials is introduced at mirror symmetric sites.

Figures~\ref{fig:PTeven}(e)-(h) show the numerically determined $\mathcal{PT}$-symmetry breaking threshold $\gamma_\mathrm{th}$ for the $N=20$ chain as a function of $m_0$ and superconducting order parameter $\delta$. When $\mu/J=0$ (panel e), we see that $\gamma_\mathrm{th}(m_0,\delta)$ has the characteristic U-shaped behavior~\cite{joglekar2010,agarwal2018} when $\delta=0$ and becomes mostly zero for intermediate locations $m_0\sim N/4$. When $m_0=1$, i.e. when the gain-loss locations are farthest apart, the $\mathcal{PT}$ threshold is maximized to $\gamma_\mathrm{th}=J$, and reflects the non-local robustness that is ubiquitous for systems with open boundary conditions~\cite{joglekar2010,Joglekar2013}. In this case, the states that participate in the $\mathcal{PT}$-breaking process are the mid-band states. As $m_0$ is increased, the threshold decreases and it rises back to $\gamma_\mathrm{th}=J$ when the gain and loss locations are nearest neighbors, i.e $m_0=N/2$. In this situation, all eigenvalues simultaneously and pairwise become complex, giving rise to maximal $\mathcal{PT}$-symmetry breaking~\cite{joglekar2011}.

In contrast to the variation with $m_0$, we find that when the superconducting order parameter $\delta$ is varied, for most locations $m_0$, the $\mathcal{PT}$-threshold is uniformly suppressed from its $\delta=0$ value. The exception is the region $m_0\sim 1$, where, as $\delta$ is increased, we see that the $\mathcal{PT}$ threshold at $\delta=J$ is double its $\delta=0$ value~\cite{kawabata2018}, where the flat bands occur; see Fig.~\ref{fig:PTeven}(a). As $\delta$ is increased further, the threshold dips to zero and then increases reaching a steady, $\delta$-independent value of $\gamma_\mathrm{th}=J/2$. As the detuning $\mu$ is increased from zero, Fig.~\ref{fig:PTeven}(f), there is an overall suppression of the $\mathcal{PT}$-breaking threshold $\gamma_\mathrm{th}$ although the characteristic U-shape behavior as a function of $m_0$ and the non-monotonic behavior as a function of $\delta$ for farthest gain-loss potentials are both retained. These qualitative trends continue for $\mu\leq 2J$, i.e. when the system is in the topological phase.

When the detuning is large, $\mu>2J$, the system enters trivial superconducting phase with no edge localized states, (g) and (h). In this regime, the system consists of two separated bands, and therefore the $\mathcal{PT}$-threshold does not sensitively depend on the detuning. On the other hand, when gain and loss are on nearest neighbor sites, $m_0=N/2$, the threshold is suppressed to zero for $\delta\sim 2J$. This is explained by the level crossings that occur near band edges; see Figs.~\ref{fig:PTeven}(c) and (d).

So are there any differences in the threshold behavior for an odd Kitaev chain? Figure~\ref{fig:PTodd} shows corresponding, representative results for a chain with $N=21$ sites. Panels (a)-(d) show the dispersions of the Hermitian Kitaev chain as a function of $\delta/J$ for increasing detuning values. At zero detuning, panel (a), the band structure looks similar to that in Fig.~\ref{fig:PTeven}(a), but with a key difference: there is no degenerate pair of topological zero energy states. As $\mu$ is increased, the qualitative evolution of the band structure is similar to that of an even Kitaev chain, with the band gap closing at $\mu=2J$ and well-separated two-band structure at higher detuning values. Panel (e) shows the $\mathcal{PT}$ threshold $\gamma_\mathrm{th}/J$ as a function of the gain location $m_0$ and the superconducting order parameter $\delta/J$. Near $\delta=0$, we recover the characteristic U-shaped behavior with a robust threshold $\gamma_\mathrm{th}\sim J$ when $m_0=1$, i.e. the farthest gain and loss pairs.  In contrast, for closest gain-loss locations, i.e. $m_0=(N-1)/2$, the threshold reaches $\gamma_\mathrm{th}\sim J/2$~\cite{joglekar2010,joglekar2011,agarwal2018}. This behavior is seen across the entire range of $\mu/J$; panels (f)-(h).

In a sharp contrast, the behavior of the threshold $\gamma_\mathrm{th}/J$ as a function of the superconducting order parameter $\delta/J$ is markedly different for the zero detuning case, panel (e), vs. the nonzero detuning case, panels (f)-(h). For the latter, the threshold shows a non-monotonic suppression of $\gamma_\mathrm{th}$ with increasing $\delta/J$. When $\mu=0$, on the other hand, we see that the $\gamma_\mathrm{th}$ increases with $\delta/J$, thereby strengthening the $\mathcal{PT}$-symmetric phase. We emphasize that when $m_0=1$---gain and loss localized on the end sites---this enhancement occurs even at $\delta/J=1$. Recall that at $\delta/J=1$, the Hermitian band structure forms flat bands (Figs.~\ref{fig:PTeven}a,~\ref{fig:PTodd}a), leading to a zero threshold irrespective of $m_0$ in the even chain, Fig.~\ref{fig:PTeven}e.

\section{\label{sec:N5case} Analytical approach}

To get better insights into the rich structure of the $\mathcal{PT}$ threshold, we consider the behavior of $\gamma_\mathrm{th}/J$ for nearest-neighbor gain-loss potentials, $m_0=N/2$, as a function of $\mu/J$ and $\delta/J$ for an $N=20$ site chain (Fig.~\ref{fig:N20threshold}(a)).  Apart from the nonzero threshold that occurs in the limit of a non-superconducting, tight-binding chain ($\delta=0$) for any detuning, we see that $\gamma_\mathrm{th}=0$ for large $\delta$ for any $\mu$, and there is beak-shaped region in the $\mu-\delta$ plane with a positive $\mathcal{PT}$ threshold. In the magnified view of the region at small $\delta/J<1$ (Fig.~\ref{fig:N20threshold}(b)), we see significant variations in the ${\mathcal {PT}}$ threshold as we sweep across $\mu/J$. These threshold ``dips'' occur at values of $\mu/J$ where the lowest energy levels in the bulk become degenerate. The white dashed line in Fig.~\ref{fig:N20threshold}(a), separating the zero-threshold region from the positive-threshold regions, is described by equation $\alpha\mu J + |J^2 - \delta^2|$ where $\alpha\sim0.5$ is an $N$-dependent constant. The region $0<\mu/J<2$, $<\delta/J<1$ enveloped in the ${\mathcal {PT}}$ phase boundary shows many ripples with $\gamma_\mathrm{th}>0$  but it decays to zero in the thermodynamic limit.

\begin{figure}[h]
 \centering
 \includegraphics[width=1\columnwidth]{N20Threshold.pdf}
 \caption{ (a) $\mathcal{PT}$-symmetry threshold for a chain with $N=20$ and $m_0=N/2$. The white dashed line, separating the zero-threshold region from the nonzero-threshold region, is empirically fit by equation $\alpha\mu J +|J^2 - \delta^2|=0$ where $\alpha\rightarrow 0.5$ as $N\rightarrow\infty$; at $N=20$, we find that $\alpha = 0.53$. This functional dependence can be obtained by requiring that two adjacent levels in the Hermitian band-structure become degenerate to get $\gamma_\mathrm{th}=0$. (b) closeup of the boxed region near the origin shows multiple ripples in $\gamma_\mathrm{th}$. }
 \label{fig:N20threshold}
\end{figure}

We remind the reader that in the $\delta=0$ tight-binding case with nearest-neighbor gain-loss potentials, all states contribute pairwise to the $\mathcal{PT}$-symmetry breaking~\cite{joglekar2011}. In contrast, in the current set up, only the states near the band edges become degenerate and then complex conjugate. In order to find the asymptotic behavior of the zero-threshold line, we turn to the Hermitian band structure, Eq.~\ref{eq:bulk}. A zero threshold is a result of degeneracy in the consecutive levels, i.e.
$E(q_k)=E_(q_{k-1})$ where $q_k=\pi k/(N+1)$ are the lattice quasimomenta consistent with open boundary conditions. Simplifying the degeneracy criterion gives
\begin{align}
& a_1\mu J + a_2( J^2-\delta^2)=0,\\
& a_1=\cos(q_k)-\cos(q_{k-1}),\\
& a_2=a_1\left[\cos(q_k)+\cos(q_{k-1})\right].
\end{align}
Defining $\alpha=a_1/|a_2|$, we obtain an analytical expression for asymptotic value of $\alpha$. From the energy spectra in Figs.~\ref{fig:PTeven} (a)-(d) and numerical analysis, it follows that regions near $q\sim 0,\pi$ contribute giving $\alpha\rightarrow 0.5$ in the limit $N\rightarrow\infty$.

\begin{figure*}[ht!]
    \centering
    \includegraphics[width=1\textwidth]{N5EnergyThreshold.pdf}
    \caption{$\mathcal{PT}$ threshold for an $N=5$ chain. (a) Schematic of $N=5$ chain with $m_0=1$. (b)-(c) Flow of real part of eigenavalues of the $N=5$ chain as a function of $\gamma$ for farthest gain-loss locations (far left) shows that $\mathcal{PT}$ breaking occurs at an EP3 and the threshold increases monotonically with the superconducting order parameter $\delta$. (d) $\gamma_\mathrm{th}(\mu,\delta)$ shows behavior consistent with Fig.~\ref{fig:N20threshold} including a contour of zero threshold given by $\alpha\mu J+|J^2-\delta^2|=0$. (e) Schematic of $N=5$ chain with $m_0=2$. (f)-(g) Corresponding results for closest gain-loss locations, $m_0=2$, show that $\mathcal{PT}$-breaking occurs at an EP2, and the threshold varies non-monotonically with $\delta$. (h) $\gamma_\mathrm{th}(\mu,\delta)$ map shows features similar to those in panel (d).  The white-daashed line is zero-threshold contour given by $\alpha\mu J+|J^2-\delta^2|=0$ with $\alpha=1.6$.}
    \label{fig:N5EnergyThreshold}
\end{figure*}

Next, to understand the global behavior of the $\mathcal{PT}$ threshold $\gamma_\mathrm{th}(m_0,\mu,\delta)$ in an odd chain, we look towards the smallest nontrivial case with zero detuning, i.e. $N=5$ and $\mu=0$. When $m_0=1$, the doubly-degenerate energy spectrum is analytically tractable and is given by
\begin{align}
 E_n &=\frac{\mp 1}{2\sqrt{2}}\left[4(J^2+\delta^2)-\gamma^2\pm\sqrt{4(J^2-\delta^2)^2+\gamma^4}\right]^{1/2},
\label{eq:m01}
\end{align}
along with two (degenerate) zero eigenvalues, $E_{5,6}=0$. As $\gamma$ is increased, the energy levels $E_{3,4}=-E_{7,8}$ first approach each other, merge with the zero-levels, and then become complex conjugate, thereby giving rise to an exceptional point of order three (EP3). The $\mathcal{PT}$-threshold in this case is given by
\begin{equation}
\gamma_\mathrm{th}(1)=J\left[\frac{3(\delta^2+J^2)^2+4\delta^2 J^2}{2J^2(\delta^2 +J^2)}\right].
\label{eq:N5m1threshold}
\end{equation}
A similar analysis for the case with next-nearest-neighbor gain-loss potentials gives particle-hole symmetric, doubly degenerate spectra
\begin{align}
E_n&=\frac{\mp 1}{2\sqrt{2}}\left[4(J^2+\delta^2)-\gamma^2\pm\sqrt{A}\right]^{1/2},\\
A&=4(J^2-\delta^2)^2+\gamma^4-8\gamma^2(J^2+\gamma^2),
\end{align}
along with two (degenerate) zero eigenvalues, $E_{5,6}=0$. As $\gamma$ is increased, we now find that the levels near the band-edge approach each other and become degenerate, giving rise to an EP2. The $\mathcal{PT}$ threshold, obtained by requiring $E_{1,2}(\gamma_\mathrm{th})=E_{3,4}(\gamma_\mathrm{th})$, is given by
\begin{equation}
\gamma_\mathrm{th}(2)=\left[ 4(J^2+\delta^2)-2\sqrt{ 3\delta^4+10\delta^2J^2+3J^4}\right]^{1/2}.
  \label{eq:N5m2threshold}
\end{equation}
We note that these analytical results are only valid for zero detuning, and for finite detuning $\mu>0$, we have to resort to numerical calculations.

Figure~\ref{fig:N5EnergyThreshold}(a) shows the schematic of an $N=5$ site chain with gain-loss potentials at its ends, i.e. $m_0=1$. Panels (b)-(c) show the flow of the real parts of energy eigenvalues for the model as a function of $\gamma/J$ for different values of $\delta$. We see that increasing $\gamma$ leads to $\mathcal{PT}$-breaking that occurs at the center of the band, giving rise to an EP3. They also show that the threshold increases monotonically with $\delta$, consistent with what is seen in Fig.~\ref{fig:PTodd}e. Panel (d) shows numerically obtained threshold diagram in the $\mu-\delta$ plane.

Figure~\ref{fig:N5EnergyThreshold}(e) shows the configuration with nearest-possible gain-loss potentials, i.e. $m_0=2$. Panels (f)-(g) show the flow of real part of eigenvalues for the model. Increasing $\gamma$ in this case leads to $\mathcal{PT}$-breaking at the band edges, and it has a non-monotonic dependence on the superconducting order parameter $\delta$, also seen in Fig.~\ref{fig:PTodd}e. Panel (h) shows numerically obtained threshold $\gamma_\mathrm{th}(\mu,\delta)$. The similarity of these threshold maps with Fig.~\ref{fig:N20threshold} is striking. By fitting the zero threshold contour to the form $\alpha\mu J+ |J^2-\delta^2|=0$ (dashed white line in panel (h)), we obtain $\alpha=1.6$.

\section{\label{sec:reentrant} Exceptional Lines and rentrant $\mathcal{PT}$ phase}

In one-dimensional lattice models with a single pair of gain and loss potentially, typically, the $\mathcal{PT}$-symmetry breaking occurs monotonically with increasing gain-loss strength $\gamma$. This is true for uniform chains with open~\cite{joglekar2010} or periodic boundary conditions~\cite{scott2012}; the Su-Schrieffer-Heeger, the Aubrey-Andre-Harper or quasi-periodic models~\cite{harter2016,harter2018}; and models with non-uniform, parity-time symmetric tunneling profiles~\cite{joglekar2011.2}, including the perfect-state transfer models. On the other hand, the presence of two or more gain-loss potentials can lead to re-entrant $\mathcal{PT}$-symmetric phase~\cite{joglekar2012,liang2014} where increasing gain-loss strength leads to repeated $\mathcal{PT}$-symmetry breaking and $\mathcal{PT}$-symmetry restoration transitions.

\begin{figure}[h]
    \centering
    \includegraphics[width=\columnwidth]{PTreentrant.pdf}
    \caption{(a) $\mathcal{PT}$ phase diagram in the $\gamma-\delta$ plane for an $N=8$ lattice with $\mu=0$ and $m_0=1$ shows the heat map of $\Lambda\equiv\log_{10}\max_k\Im(E_k)$ where $E_k$ are the $2N$ are the eigenvalues of $H_\mathrm{K}$, Eq.(\ref{eq:hkitaev}). A re-entrant $\mathcal{PT}$-symmetric phase (black) emerges in the range $1\leq\delta/J\leq\sqrt{2}$ as the gain-loss strength $\gamma/J$ is increased. (b) EP2 contours at the $\mathcal{PT}$ boundary and in the $\mathcal{PT}$-broken region show sequential coalescence of eigenvalues. At $\delta/J=1$, due to the presence of robust Majorana modes, a third-order EP emerges at $\gamma/J=2$ (yellow circle). (c)-(d) corresponding results for an $N=24$ lattice shows same qualitative features.}
    \label{fig:PTreentrant}
\end{figure}

In contrast to these models with multiple non-Hermitian terms~\cite{joglekar2012,liang2014}, the Kitaev chain we have considered shows a re-entrant $\mathcal{PT}$-symmetric phase and its subsequent breaking when the gain-loss strength $\gamma$ is increased. This phenomenon occurs for an even chain with $\mu=0$ and $m_0=1$, at moderate superconducting order parameter $1\leq \delta/J\leq \sqrt{2}$, independent of the chain size. In Fig.~\ref{fig:PTreentrant}(a), we plot $\Lambda(\gamma,\delta)=\log_{10}\max_k\Im(E_k)$ where $E_k$ are (purely real or complex-conjugate) eigenvalues of the Hamiltonian $H_\mathrm{K}$ for an $N=8$ chain. The $\mathcal{PT}$-symmetric region is marked by black,  and the rest is $\mathcal{PT}$-symmetry broken region. With $\delta/J\sim 1$, as $\gamma$ is increased, the first ${\mathcal {PT}}$ symmetry breaking near $\gamma/J\sim 0.5$ occurs due to the level-attraction between and coalescence of two highest energy states in the upper band; recall that due to the particle-hole symmetric nature of the spectrum, two lowest energy levels in the lower band concurrently become degenerate. With increasing $\gamma$, subsequent lower energy levels, except the lowest state in the upper band, coalesce in pairs. This sequence of transitions leads to a large number of exceptional points in the $\mathcal{PT}$-symmetry broken region. Further increasing $\gamma$ leads to a reverse process where levels with complex-conjugate energies undergo level-attraction and $\mathcal{PT}$-symmetry is restored. For the lowest-energy states in the upper band (and their chiral counterparts), the re-entrant $\mathcal{PT}$-symmetric phase is accompanied by a qualitative change where the wave-function weight shifts from the bulk to the edges. As $\gamma$ is increased further, the system enters $\mathcal{PT}$-broken region again. This second $\mathcal{PT}$ transition across an EP is driven by coalescence of the near-zero-energy state with state at the bottom of the top band.

To map out the exceptional point contours in the $\delta-\gamma$ plane, we use the (Dirac) inner-product matrix $M_{pq}=|\langle \psi_p|\psi_q\rangle|$ where $|\psi_k\rangle$ is the (Dirac)-normalized right eigenvector of $H_\mathrm{K}$ with eigenvalue $\lambda_k$. The order of the EP is then given by $\max_{p}\sum_{q\neq p}M_{pq}$. Figure~\ref{fig:PTreentrant}(b) shows the contours of exceptional points in the parameter space. In addition to the boundaries of $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken regions, seen in Fig.~\ref{fig:PTreentrant}(a), we see EP contours that denote the cascades of eigenvalue coalescence that occur in the $\mathcal{PT}$-broken region as $\gamma$ is increased. Of particular interest is the contour that starts at $\delta=0$ and $\gamma/J=1$. At point $\delta/J=1$, the system has fully degenerate bands with robust, mid-gap edge states (Fig.~\ref{fig:PTeven}a). Therefore, introduction of the gain-loss potentials leads to a third-order EP at $\gamma/J=2$ (shown by a yellow circle) in the otherwise second-order EP contour. We note that the prominent reentrant $\mathcal{PT}$ phases only occur when the gain-loss potentials are farthest apart, i.e. $m_0=1$ and remain robust only at $\mu=0$ for any even $N$; Figs.~\ref{fig:PTreentrant}(c)-(d) show the phase diagram and EP contours for an $N=20$ Kitaev chain.

\section{\label{sec:conclusion}Conclusion}
In this paper, we have investigated the dependence of the $\mathcal{PT}$-threshold $\gamma_\mathrm{th}$ on the properties of the underlying Hermitian Kitaev model and gain-loss potential locations. We have shown that the threshold profile is rich, with persistent differences between even and odd parity lattices. In particular, we have found that for a zero-detuning chain with odd number of sites, the threshold is enhanced with increasing superconducting order parameter. For an even chain with edge gain-loss potentials and superconducting coupling $\delta \gtrsim 1$, we discover re-entrant $\mathcal{PT}$-symmetric phase, and $\mathcal{PT}$-phase boundaries that contain both second and third order EPs. We have also discussed, briefly, a potential realization of our lattice model with coupled optical resonators. Our results further the understanding of non-Hermitian condensed matter models in the presence of realistically achievable gain and loss.

\bibliographystyle{apsrev4-2}

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

A complex extension to quantum mechanics, based on a special class of non-Hermitian Hamiltonians with purely real spectra, was discovered more than two decades ago~\cite{bender1998,bender2002,ali2002,ali2010}. These continuum Hamiltonians on an infinite line shared the property that each of them is invariant under combined operations of parity and time-reversal (${\mathcal {PT}}$); i.e. the Hamiltonian commutes with the ${\mathcal {PT}}$ operator. Over the past decade, it has become clear that ${\mathcal {PT}}$-symmetric systems represent open, classical systems with balanced, spatially or temporally separated gain and loss that are represented by complex real-space potentials~\cite{feng2017,el2018,ozdemir2019}.

The spectrum $E_\alpha$ of a ${\mathcal {PT}}$-symmetric Hamiltonian $H(\gamma)$ is real at small non-Hermiticities $\gamma$ and turns into complex conjugate pairs at large non-Hermiticities. When the spectrum is purely real, we can choose an eigenvector $\ket{\epsilon_\alpha}$ of $H$ to be a simultaneous eigenvector of the antilinear ${\mathcal {PT}}$-operator with eigenvalue +1. When the spectrum is complex, the ${\mathcal {PT}}$ operator acting on $\ket{E_\alpha}$ transforms it into the eigenvector with complex-conjugate eigenvalue, i.e. ${\mathcal {PT}}|E_\alpha\rangle=|E^*_\alpha\rangle$ where $*$ denotes complex conjugation. The transition from a purely real to complex conjugate spectrum occurs when $\gamma=\gamma_\mathrm{PT}$ where $\gamma_\mathrm{PT}$ is called the ${\mathcal {PT}}$-symmetry breaking threshold. At the threshold, the geometric multiplicity of the eigenvalues of the Hamiltonian $H(\gamma_\mathrm{PT})$ is smaller than the algebraic multiplicity. Such a Hamiltonian degeneracy is called an exceptional point (EP) degeneracy, where not only do the eigenvalues become degenerate, but the corresponding eigenvectors also coalesce.

In the past decade ${\mathcal {PT}}$-symmetric systems with balanced gain-loss have been realized in classical wave systems including evanescently coupled waveguides~\cite{ruter2010}, fiber loops~\cite{regensburger2012}, optical resonators ~\cite{chang2014, hodaei2014}, electrical circuits ~\cite{schindler2011,choi2018}, and mechanical oscillators~\cite{bender2013}. However, since the EP degeneracies also occur for Hamiltonians with mode selective dissipation, the dynamics of $\mathcal{PT}$-symmetric Hamiltonians have also been realized in purely lossy classical systems consisting of coupled waveguides, resonators, or electrical circuits~\cite{guo2009,lenmontiel2018}, semiclassical systems with ultracold atoms~\cite{li2019}, and quantum systems~\cite{Wu2019,naghiloo2019,klauck2019}.

In this paper we obtain the ${\mathcal {PT}}$-breaking threshold in a one-dimensional Kitaev model in the presence of one pair of gain-loss potentials $\pm i\gamma$ located on reflection-symmetric sites. Kitaev model is a toy model for a topological superconductor with Majorana fermions as excitations. A Majorana fermion is, by construction, its own antiparticle, i.e. it is a fermion constructed from a electron-hole pair. There are a number of studies on topological superconductors with non-Hermitian, ${\mathcal {PT}}$-symmetric potentials that preserve the translational invariance of the system~\cite{wang2015,li2018,yao2018,yuce2016,klett2017,kawabata2018,li2020,sarma2015,leumer2020}. They focused on the fate of the edge states that are characterized by zero energy. A finite Kitaev chain with a pair of ${\mathcal {PT}}$-symmetric potentials at the edge has also been studied~\cite{yuce2016,kawabata2018,li2020,klett2017}. The result is the emergence of an additional pair of edge state with a non-zero energy eigenvalue~\cite{kawabata2018}. In contrast to these studies on the Majorana edge modes, here we concentrate on the variation of the ${\mathcal {PT}}$-breaking threshold $\gamma_\mathrm{PT}(m_0,N)$ with the location $m_0$ of the gain potential in a Kitaev chain of size $N$.

This paper is organized as follows. In Sec.~\ref{sec:model} we describe the Hermitian Kitaev chain along with its symmetry properties, and then introduce the non-Hermitian perturbation. In Sec.~\ref{sec:results}, we present numerical results for the ${\mathcal {PT}}$-symmetry breaking threshold as a function of different parameters of the Hermitian model and the relative location of the gain potential. We point out key differences among the ${\mathcal {PT}}$-symmetry breaking thresholds for various settings of the on-site potentials $\mu$ and superconducting coupling strengths $\delta$. In Sec.~\ref{sec:N5case} we describe a small-system case of $N=5$ sites and analytically obtain the dependence of the ${\mathcal {PT}}$ threshold when the gain-loss potentials are farthest apart and closest together. In Sec.~\ref{sec:reentrant}, we show that the Kitaev model shows re-entrant $\mathcal{PT}$-symmetric phase, and map out its EP contours. Finally in Sec.~\ref{sec:conclusion}, we conclude by summarizing the results.

\section{\label{sec:model}Tight-binding Model}

The Kitaev model of a one dimensional, p-wave superconducting chain with $N$ sites and open boundary conditions is described by the following Hermitian Hamiltonian,
\begin{eqnarray}
\label{eq:H_fermion}
    H_0 & = & - \mu \sum_{n=1}^N c_n^{\dagger}c_n -J \sum_{n=1}^{N-1} (c_{n}^{\dagger}c_{n+1} + \mathrm{h.c.})\nonumber\\
&+  &i\delta \sum_{n=1}^{N-1}(c_{n}c_{n+1} - \mathrm{h.c.}).
\end{eqnarray}
Here $c^{\dagger}_n$ and $c_n$ are fermionic creation and annihilation operators for site $n$ in the chain, $\mu$ is the on-site potential, $J>0$ is the nearest-neighbor hopping strength, and $\delta>0$ is the amplitude of the (p-wave) superconducting coupling for a Cooper pair that is localized across neighboring sites~\cite{kitaev2001}. The global phase of the superconducting order parameter is fixed at $\pi/2$ to ensure that Eq.(\ref{eq:H_fermion}) is parity-time symmetric, with the parity operator given by
$\mathcal{P}: c_n\rightarrow c_{\bar{n}}$ where $\bar{n}=N+1-n$ is the mirror-symmetric counterpart of site $n$ and the time-reversal operator is given by complex conjugation, $\mathcal{T}=*$.

We rewrite Eq.(\ref{eq:H_fermion}) by using the Bogoliubov-de Gennes representation in terms of the operator-vector $\Psi=(c_1,c^{\dagger}_1,c_2,c^{\dagger}_2 \dots c_N, c^{\dagger}_N)^T$ as
$H_0=\Psi^\dagger H_\mathrm{BdG}\Psi$ where the $2N\times 2N$ matrix $H_\mathrm{BdG}$ in the site-representation is given by
\begin{eqnarray}
H_\mathrm{BdG} & = & -\frac{\mu}{2} \sum_{n=1}^N\ket{n}\bra{n}\otimes \sigma_z \nonumber \\
        &   & -\frac{J}{2} \sum_{n=1}^{N-1}\left(\ket{n}\bra{n+1} + \ket{n+1}\bra{n}\right)\otimes \sigma_z \nonumber \\
        &   & + \frac{i\delta}{2} \sum_{n=1}^{N-1}\left(\ket{n}\bra{n+1} - \ket{n+1}\bra{n}\right)\otimes \sigma_x,
\label{Eq:kiHamil}
\end{eqnarray}
where $\sigma_x,\sigma_z$ are the standard Pauli matrices. For a chain with periodic boundary conditions, translational invariance allows us to transform the site-space Hamiltonian (\ref{Eq:kiHamil}) into momentum space, $\tilde{H}_\mathrm{BdG} =UH_\mathrm{BdG}U^{\dagger}$
with a unitary
\begin{align}
U& =\frac{1}{\sqrt{N}} \sum_{k,n=1}^N e^{-i\ket{p_k}\bra{n}}\otimes\left( e^{i\pi/4}\mathbbm{1}_2+e^{-i\pi/4}\sigma_z\right).
\end{align}
The block-diagonalized momentum-space Hamiltonian is given by $\tilde{H}_\mathrm{BdG}=\sum_{k=1}^N h(p_k)|p_k\rangle\langle p_k|$ where $p_k=2\pi k/N$ are the discrete quasimomenta for a finite chain and
\begin{eqnarray}
h(p) &=&
\begin{pmatrix}
-J\cos{p}-\mu/2 & -i\delta \sin{p} \\
i\delta \sin{p} & J\cos{p}+\mu/2
\end{pmatrix}.
\end{eqnarray}
The bulk energy spectrum of the Hamiltonian $\tilde{H}_\mathrm{BdG}$ is given by
\begin{equation}
E_{\pm}(p)=\pm \sqrt{(J\cos{p} + \mu/2)^2 + \delta^2 \sin^2{p}},
\label{eq:bulk}
\end{equation}
and it shows that in the limit of an infinite chain, $N\gg 1$, the gap in the spectrum vanishes at $p=\pi$ when $\mu=2J$. For the finite chain, the spectrum Eq.(\ref{eq:bulk}) is symmetric about $\delta=0$ because $H_\mathrm{BdG}(-\delta)=\mathcal{S} H_\mathrm{BdG}(\delta)\mathcal{S}^\dagger$ with a unitary operator $\mathcal{S}=\mathbbm{1}_N\otimes\sigma_z$. When the boundary conditions are changed from periodic to open, in the zero chemical potential limit, Majorana zero-modes (fermionic excitations) appear localized on the edges of the chain. These edge modes are robust when gain-loss potentials are introduced on random sites~\cite{yuce2016} or on parity symmetric sites with disorder~\cite{kawabata2018}.

To this toy model with open boundary conditions, we add a pair of balanced gain-loss potentials
$\pm i\gamma$ at mirror symmetric sites $m_0$ and $\bar{m}_0$,
\begin{equation}
i\Gamma= \frac{i\gamma}{2} \left(\ket{m_0}\bra{m_0} - \ket{\overline{m}_0}\bra{\overline{m}_0}\right)\otimes \sigma_z,
\end{equation}
and thereby get a non-Hermitian, $\mathcal{PT}$-symmetric Kitaev chain Hamiltonian
\begin{equation}
\label{eq:hkitaev}
H_\mathrm{K}(\gamma,\delta,\mu) =H_\mathrm{BdG}+i\Gamma.
\end{equation}

\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{kitaevchain.pdf}
\caption{Schematic representation of a Kitaev model with one pair of gain and loss potentials, Eq.(\ref{eq:hkitaev}). Two chains (gray and white) with detuned potentials $\pm\mu$ have nearest neighbor Hermitian tunneling amplitudes $\pm J$, and next-nearest-neighbor Hermitian amplitudes $\pm i\delta$. Due to the presence of two degree of freedom on each site~\cite{harsha2013}, the potential $i\gamma$ on site $m_0$ acts as gain (red) for one and loss (blue) for the other. This schematic can be realized with coupled resonator rings where one can engineer complex, Hermitian tunneling amplitudes.}
\label{fig:kichain}
\end{figure}

Fig.~\ref{fig:kichain} shows schematic representation of a lattice model described by Eq.(\ref{eq:hkitaev}). Although the original model refers to many-body fermionic system with two bands, in its ``single-particle'' form, Eq.(\ref{eq:hkitaev}), it can be interpreted as two, detuned coupled chains with Hermitian nearest-neighbor couplings $\pm J$ and Hermitian, purely imaginary, next-nearest-neighbor couplings
$\pm i\delta$~\cite{harsha2013}. The gain potential on site $m_0$, given by $(i\gamma/2) |m_0\rangle\langle m_0|\otimes\sigma_z$, then stands for gain in one chain and loss in the second chain. This representation of the $\mathcal{PT}$-symmetric Kitaev model can be experimentally implemented in resonator arrays where real and purely imaginary tunneling amplitudes can be easily engineered. In the next section we explore the global phase diagram for the $\mathcal{PT}$-symmetry breaking threshold $\gamma_\mathrm{th}(m_0,\delta,\mu)$.

\section{\label{sec:results} Numerical results for $\gamma_\mathrm{th}(m_0,\delta,\mu)$}

\begin{figure*}
    \centering
    \includegraphics[width=0.9\textwidth]{N20YthDelM0.pdf}
    \caption{Energy eigenvalues (in units of $J$) of an $N=20$ site Hamiltonian $H_\mathrm{BdG}(\delta)$, Eq.(\ref{Eq:kiHamil}) as a function of detuning. (a) $\mu/J=0$, (b) $\mu/J=1$ have topological edge modes, while the system is in the topologically trivial phase at (c) $\mu/J=2$ and (d) $\mu/J=3$. Corresponding $\mathcal{PT}$ threshold values $\gamma_\mathrm{th}/J$ obtained from the Hamiltonian $H_\mathrm{K}$, Eq.(\ref{eq:hkitaev}) are plotted as a function of the gain location $m_0\in[1,N/2]$ and the superconducting order parameter $\delta/J$: (e) $\mu/J = 0$, (f)  $\mu/J = 1$, (g)  $\mu/J = 2$ (h)  $\mu/J = 3$. Most of these features can be understood in terms of Hermitian band structure, panels a-d.}
    \label{fig:PTeven}
\end{figure*}

\begin{figure*}
    \centering
    \includegraphics[width=0.9\textwidth]{N21YthDelM0.pdf}
    \caption{Energy eigenvalues (in units of $J$) of an $N=21$ site Hamiltonian $H_\mathrm{BdG}(\delta)$, Eq.(\ref{Eq:kiHamil}) as a function of detuning. (a) $\mu/J=0$, (b) $\mu/J=1$ have mid-gap states with zero energy, but these are not topological. (c) $\mu/J=2$ and (d) $\mu/J=3$ show emergence of a gapped spectrum. Corresponding $\mathcal{PT}$ threshold values $\gamma_\mathrm{th}/J$ obtained from the Hamiltonian $H_\mathrm{K}$, Eq.(\ref{eq:hkitaev}) are plotted as a function of the gain location $m_0\in[1,(N-1)/2]$ and the superconducting order parameter $\delta/J$: (e) $\mu/J = 0$, (f)  $\mu/J = 1$, (g)  $\mu/J = 2$ (h)  $\mu/J = 3$. Due to the absence of topological edge modes, the $\mathcal{PT}$ threshold behavior at $\mu=0$ is markedly different from that of an even chain, Fig.~\ref{fig:PTeven}(e). At nonzero detuning, the threshold is non-monotonically suppressed with increasing $\delta$.}
    \label{fig:PTodd}
\end{figure*}

The results presented in this section are obtained by diagonalizing $H_\mathrm{K}$ or $H_\mathrm{BdG}$ for Kitaev chains of size $N=20$ (Fig.~\ref{fig:PTeven}) and $N=21$ (Fig.~\ref{fig:PTodd}). They remain qualitatively same for larger chain sizes, and the differences between even and odd parity chains persist in the large $N$ limit, as they do for a simple tight-binding model. All energy scales are measured in units of the tight-binding coupling $J=1$. Figures~\ref{fig:PTeven}(a)-(d) show the the energy eigenvalues $E_n$ for a Hermitian Kitaev chain as a function of the superconducting order parameter $\delta/J$. When $\delta=0=\mu$, we get the cosine-band of a tight-binding model. As the detuning $\delta$ is increased from (a) to (d), the two bands become well-separated. On the other hand, at a fixed detuning, when $\delta$ is increased, the bands develop fan-like linear dispersion, leading to massively degenerate flat bands at $\delta/J=1$ at zero detuning. As the detuning is increased from $\mu=0$, the system develops two crossing points (shown by blue circles in (b)). We also note that zero-energy states are present when $\mu<2J$. At $\mu/J=2$, the superconducting gap closes marking a phase transition to the topologically trivial phase. Here the mid-gap states become a part of the bulk. When $\mu$ is increased further, Fig.~\ref{fig:PTeven}(d), the system is in the trivial superconducting phase and energy spectrum is gapped. The topological, edge-localized zero energy states only occur when $\mu\leq 2J$. At $\mu=0$, these zero-energy states are fully localized on the end sites. When $\mu$ is increased, these states extend into the bulk of the chain, with an exponentially decaying probability density~\cite{leumer2020,sarma2015}. The Hermitian, (near or exact) degeneracies of the Hamiltonian $H_\mathrm{BdG}$ play an important role in determining the threshold gain-loss strength $\gamma_\mathrm{th}$ when a pair of gain-loss potentials is introduced at mirror symmetric sites.

Figures~\ref{fig:PTeven}(e)-(h) show the numerically determined $\mathcal{PT}$-symmetry breaking threshold $\gamma_\mathrm{th}$ for the $N=20$ chain as a function of $m_0$ and superconducting order parameter $\delta$. When $\mu/J=0$ (panel e), we see that $\gamma_\mathrm{th}(m_0,\delta)$ has the characteristic U-shaped behavior~\cite{joglekar2010,agarwal2018} when $\delta=0$ and becomes mostly zero for intermediate locations $m_0\sim N/4$. When $m_0=1$, i.e. when the gain-loss locations are farthest apart, the $\mathcal{PT}$ threshold is maximized to $\gamma_\mathrm{th}=J$, and reflects the non-local robustness that is ubiquitous for systems with open boundary conditions~\cite{joglekar2010,Joglekar2013}. In this case, the states that participate in the $\mathcal{PT}$-breaking process are the mid-band states. As $m_0$ is increased, the threshold decreases and it rises back to $\gamma_\mathrm{th}=J$ when the gain and loss locations are nearest neighbors, i.e $m_0=N/2$. In this situation, all eigenvalues simultaneously and pairwise become complex, giving rise to maximal $\mathcal{PT}$-symmetry breaking~\cite{joglekar2011}.

In contrast to the variation with $m_0$, we find that when the superconducting order parameter $\delta$ is varied, for most locations $m_0$, the $\mathcal{PT}$-threshold is uniformly suppressed from its $\delta=0$ value. The exception is the region $m_0\sim 1$, where, as $\delta$ is increased, we see that the $\mathcal{PT}$ threshold at $\delta=J$ is double its $\delta=0$ value~\cite{kawabata2018}, where the flat bands occur; see Fig.~\ref{fig:PTeven}(a). As $\delta$ is increased further, the threshold dips to zero and then increases reaching a steady, $\delta$-independent value of $\gamma_\mathrm{th}=J/2$. As the detuning $\mu$ is increased from zero, Fig.~\ref{fig:PTeven}(f), there is an overall suppression of the $\mathcal{PT}$-breaking threshold $\gamma_\mathrm{th}$ although the characteristic U-shape behavior as a function of $m_0$ and the non-monotonic behavior as a function of $\delta$ for farthest gain-loss potentials are both retained. These qualitative trends continue for $\mu\leq 2J$, i.e. when the system is in the topological phase.

When the detuning is large, $\mu>2J$, the system enters trivial superconducting phase with no edge localized states, (g) and (h). In this regime, the system consists of two separated bands, and therefore the $\mathcal{PT}$-threshold does not sensitively depend on the detuning. On the other hand, when gain and loss are on nearest neighbor sites, $m_0=N/2$, the threshold is suppressed to zero for $\delta\sim 2J$. This is explained by the level crossings that occur near band edges; see Figs.~\ref{fig:PTeven}(c) and (d).

So are there any differences in the threshold behavior for an odd Kitaev chain? Figure~\ref{fig:PTodd} shows corresponding, representative results for a chain with $N=21$ sites. Panels (a)-(d) show the dispersions of the Hermitian Kitaev chain as a function of $\delta/J$ for increasing detuning values. At zero detuning, panel (a), the band structure looks similar to that in Fig.~\ref{fig:PTeven}(a), but with a key difference: there is no degenerate pair of topological zero energy states. As $\mu$ is increased, the qualitative evolution of the band structure is similar to that of an even Kitaev chain, with the band gap closing at $\mu=2J$ and well-separated two-band structure at higher detuning values. Panel (e) shows the $\mathcal{PT}$ threshold $\gamma_\mathrm{th}/J$ as a function of the gain location $m_0$ and the superconducting order parameter $\delta/J$. Near $\delta=0$, we recover the characteristic U-shaped behavior with a robust threshold $\gamma_\mathrm{th}\sim J$ when $m_0=1$, i.e. the farthest gain and loss pairs.  In contrast, for closest gain-loss locations, i.e. $m_0=(N-1)/2$, the threshold reaches $\gamma_\mathrm{th}\sim J/2$~\cite{joglekar2010,joglekar2011,agarwal2018}. This behavior is seen across the entire range of $\mu/J$; panels (f)-(h).

In a sharp contrast, the behavior of the threshold $\gamma_\mathrm{th}/J$ as a function of the superconducting order parameter $\delta/J$ is markedly different for the zero detuning case, panel (e), vs. the nonzero detuning case, panels (f)-(h). For the latter, the threshold shows a non-monotonic suppression of $\gamma_\mathrm{th}$ with increasing $\delta/J$. When $\mu=0$, on the other hand, we see that the $\gamma_\mathrm{th}$ increases with $\delta/J$, thereby strengthening the $\mathcal{PT}$-symmetric phase. We emphasize that when $m_0=1$---gain and loss localized on the end sites---this enhancement occurs even at $\delta/J=1$. Recall that at $\delta/J=1$, the Hermitian band structure forms flat bands (Figs.~\ref{fig:PTeven}a,~\ref{fig:PTodd}a), leading to a zero threshold irrespective of $m_0$ in the even chain, Fig.~\ref{fig:PTeven}e.

\section{\label{sec:N5case} Analytical approach}

To get better insights into the rich structure of the $\mathcal{PT}$ threshold, we consider the behavior of $\gamma_\mathrm{th}/J$ for nearest-neighbor gain-loss potentials, $m_0=N/2$, as a function of $\mu/J$ and $\delta/J$ for an $N=20$ site chain (Fig.~\ref{fig:N20threshold}(a)).  Apart from the nonzero threshold that occurs in the limit of a non-superconducting, tight-binding chain ($\delta=0$) for any detuning, we see that $\gamma_\mathrm{th}=0$ for large $\delta$ for any $\mu$, and there is beak-shaped region in the $\mu-\delta$ plane with a positive $\mathcal{PT}$ threshold. In the magnified view of the region at small $\delta/J<1$ (Fig.~\ref{fig:N20threshold}(b)), we see significant variations in the ${\mathcal {PT}}$ threshold as we sweep across $\mu/J$. These threshold ``dips'' occur at values of $\mu/J$ where the lowest energy levels in the bulk become degenerate. The white dashed line in Fig.~\ref{fig:N20threshold}(a), separating the zero-threshold region from the positive-threshold regions, is described by equation $\alpha\mu J + |J^2 - \delta^2|$ where $\alpha\sim0.5$ is an $N$-dependent constant. The region $0<\mu/J<2$, $<\delta/J<1$ enveloped in the ${\mathcal {PT}}$ phase boundary shows many ripples with $\gamma_\mathrm{th}>0$  but it decays to zero in the thermodynamic limit.

\begin{figure}[h]
 \centering
 \includegraphics[width=1\columnwidth]{N20Threshold.pdf}
 \caption{ (a) $\mathcal{PT}$-symmetry threshold for a chain with $N=20$ and $m_0=N/2$. The white dashed line, separating the zero-threshold region from the nonzero-threshold region, is empirically fit by equation $\alpha\mu J +|J^2 - \delta^2|=0$ where $\alpha\rightarrow 0.5$ as $N\rightarrow\infty$; at $N=20$, we find that $\alpha = 0.53$. This functional dependence can be obtained by requiring that two adjacent levels in the Hermitian band-structure become degenerate to get $\gamma_\mathrm{th}=0$. (b) closeup of the boxed region near the origin shows multiple ripples in $\gamma_\mathrm{th}$. }
 \label{fig:N20threshold}
\end{figure}

We remind the reader that in the $\delta=0$ tight-binding case with nearest-neighbor gain-loss potentials, all states contribute pairwise to the $\mathcal{PT}$-symmetry breaking~\cite{joglekar2011}. In contrast, in the current set up, only the states near the band edges become degenerate and then complex conjugate. In order to find the asymptotic behavior of the zero-threshold line, we turn to the Hermitian band structure, Eq.~\ref{eq:bulk}. A zero threshold is a result of degeneracy in the consecutive levels, i.e.
$E(q_k)=E_(q_{k-1})$ where $q_k=\pi k/(N+1)$ are the lattice quasimomenta consistent with open boundary conditions. Simplifying the degeneracy criterion gives
\begin{align}
& a_1\mu J + a_2( J^2-\delta^2)=0,\\
& a_1=\cos(q_k)-\cos(q_{k-1}),\\
& a_2=a_1\left[\cos(q_k)+\cos(q_{k-1})\right].
\end{align}
Defining $\alpha=a_1/|a_2|$, we obtain an analytical expression for asymptotic value of $\alpha$. From the energy spectra in Figs.~\ref{fig:PTeven} (a)-(d) and numerical analysis, it follows that regions near $q\sim 0,\pi$ contribute giving $\alpha\rightarrow 0.5$ in the limit $N\rightarrow\infty$.

\begin{figure*}[ht!]
    \centering
    \includegraphics[width=1\textwidth]{N5EnergyThreshold.pdf}
    \caption{$\mathcal{PT}$ threshold for an $N=5$ chain. (a) Schematic of $N=5$ chain with $m_0=1$. (b)-(c) Flow of real part of eigenavalues of the $N=5$ chain as a function of $\gamma$ for farthest gain-loss locations (far left) shows that $\mathcal{PT}$ breaking occurs at an EP3 and the threshold increases monotonically with the superconducting order parameter $\delta$. (d) $\gamma_\mathrm{th}(\mu,\delta)$ shows behavior consistent with Fig.~\ref{fig:N20threshold} including a contour of zero threshold given by $\alpha\mu J+|J^2-\delta^2|=0$. (e) Schematic of $N=5$ chain with $m_0=2$. (f)-(g) Corresponding results for closest gain-loss locations, $m_0=2$, show that $\mathcal{PT}$-breaking occurs at an EP2, and the threshold varies non-monotonically with $\delta$. (h) $\gamma_\mathrm{th}(\mu,\delta)$ map shows features similar to those in panel (d).  The white-daashed line is zero-threshold contour given by $\alpha\mu J+|J^2-\delta^2|=0$ with $\alpha=1.6$.}
    \label{fig:N5EnergyThreshold}
\end{figure*}

Next, to understand the global behavior of the $\mathcal{PT}$ threshold $\gamma_\mathrm{th}(m_0,\mu,\delta)$ in an odd chain, we look towards the smallest nontrivial case with zero detuning, i.e. $N=5$ and $\mu=0$. When $m_0=1$, the doubly-degenerate energy spectrum is analytically tractable and is given by
\begin{align}
 E_n &=\frac{\mp 1}{2\sqrt{2}}\left[4(J^2+\delta^2)-\gamma^2\pm\sqrt{4(J^2-\delta^2)^2+\gamma^4}\right]^{1/2},
\label{eq:m01}
\end{align}
along with two (degenerate) zero eigenvalues, $E_{5,6}=0$. As $\gamma$ is increased, the energy levels $E_{3,4}=-E_{7,8}$ first approach each other, merge with the zero-levels, and then become complex conjugate, thereby giving rise to an exceptional point of order three (EP3). The $\mathcal{PT}$-threshold in this case is given by
\begin{equation}
\gamma_\mathrm{th}(1)=J\left[\frac{3(\delta^2+J^2)^2+4\delta^2 J^2}{2J^2(\delta^2 +J^2)}\right].
\label{eq:N5m1threshold}
\end{equation}
A similar analysis for the case with next-nearest-neighbor gain-loss potentials gives particle-hole symmetric, doubly degenerate spectra
\begin{align}
E_n&=\frac{\mp 1}{2\sqrt{2}}\left[4(J^2+\delta^2)-\gamma^2\pm\sqrt{A}\right]^{1/2},\\
A&=4(J^2-\delta^2)^2+\gamma^4-8\gamma^2(J^2+\gamma^2),
\end{align}
along with two (degenerate) zero eigenvalues, $E_{5,6}=0$. As $\gamma$ is increased, we now find that the levels near the band-edge approach each other and become degenerate, giving rise to an EP2. The $\mathcal{PT}$ threshold, obtained by requiring $E_{1,2}(\gamma_\mathrm{th})=E_{3,4}(\gamma_\mathrm{th})$, is given by
\begin{equation}
\gamma_\mathrm{th}(2)=\left[ 4(J^2+\delta^2)-2\sqrt{ 3\delta^4+10\delta^2J^2+3J^4}\right]^{1/2}.
  \label{eq:N5m2threshold}
\end{equation}
We note that these analytical results are only valid for zero detuning, and for finite detuning $\mu>0$, we have to resort to numerical calculations.

Figure~\ref{fig:N5EnergyThreshold}(a) shows the schematic of an $N=5$ site chain with gain-loss potentials at its ends, i.e. $m_0=1$. Panels (b)-(c) show the flow of the real parts of energy eigenvalues for the model as a function of $\gamma/J$ for different values of $\delta$. We see that increasing $\gamma$ leads to $\mathcal{PT}$-breaking that occurs at the center of the band, giving rise to an EP3. They also show that the threshold increases monotonically with $\delta$, consistent with what is seen in Fig.~\ref{fig:PTodd}e. Panel (d) shows numerically obtained threshold diagram in the $\mu-\delta$ plane.

Figure~\ref{fig:N5EnergyThreshold}(e) shows the configuration with nearest-possible gain-loss potentials, i.e. $m_0=2$. Panels (f)-(g) show the flow of real part of eigenvalues for the model. Increasing $\gamma$ in this case leads to $\mathcal{PT}$-breaking at the band edges, and it has a non-monotonic dependence on the superconducting order parameter $\delta$, also seen in Fig.~\ref{fig:PTodd}e. Panel (h) shows numerically obtained threshold $\gamma_\mathrm{th}(\mu,\delta)$. The similarity of these threshold maps with Fig.~\ref{fig:N20threshold} is striking. By fitting the zero threshold contour to the form $\alpha\mu J+ |J^2-\delta^2|=0$ (dashed white line in panel (h)), we obtain $\alpha=1.6$.

\section{\label{sec:reentrant} Exceptional Lines and rentrant $\mathcal{PT}$ phase}

In one-dimensional lattice models with a single pair of gain and loss potentially, typically, the $\mathcal{PT}$-symmetry breaking occurs monotonically with increasing gain-loss strength $\gamma$. This is true for uniform chains with open~\cite{joglekar2010} or periodic boundary conditions~\cite{scott2012}; the Su-Schrieffer-Heeger, the Aubrey-Andre-Harper or quasi-periodic models~\cite{harter2016,harter2018}; and models with non-uniform, parity-time symmetric tunneling profiles~\cite{joglekar2011.2}, including the perfect-state transfer models. On the other hand, the presence of two or more gain-loss potentials can lead to re-entrant $\mathcal{PT}$-symmetric phase~\cite{joglekar2012,liang2014} where increasing gain-loss strength leads to repeated $\mathcal{PT}$-symmetry breaking and $\mathcal{PT}$-symmetry restoration transitions.

\begin{figure}[h]
    \centering
    \includegraphics[width=\columnwidth]{PTreentrant.pdf}
    \caption{(a) $\mathcal{PT}$ phase diagram in the $\gamma-\delta$ plane for an $N=8$ lattice with $\mu=0$ and $m_0=1$ shows the heat map of $\Lambda\equiv\log_{10}\max_k\Im(E_k)$ where $E_k$ are the $2N$ are the eigenvalues of $H_\mathrm{K}$, Eq.(\ref{eq:hkitaev}). A re-entrant $\mathcal{PT}$-symmetric phase (black) emerges in the range $1\leq\delta/J\leq\sqrt{2}$ as the gain-loss strength $\gamma/J$ is increased. (b) EP2 contours at the $\mathcal{PT}$ boundary and in the $\mathcal{PT}$-broken region show sequential coalescence of eigenvalues. At $\delta/J=1$, due to the presence of robust Majorana modes, a third-order EP emerges at $\gamma/J=2$ (yellow circle). (c)-(d) corresponding results for an $N=24$ lattice shows same qualitative features.}
    \label{fig:PTreentrant}
\end{figure}

In contrast to these models with multiple non-Hermitian terms~\cite{joglekar2012,liang2014}, the Kitaev chain we have considered shows a re-entrant $\mathcal{PT}$-symmetric phase and its subsequent breaking when the gain-loss strength $\gamma$ is increased. This phenomenon occurs for an even chain with $\mu=0$ and $m_0=1$, at moderate superconducting order parameter $1\leq \delta/J\leq \sqrt{2}$, independent of the chain size. In Fig.~\ref{fig:PTreentrant}(a), we plot $\Lambda(\gamma,\delta)=\log_{10}\max_k\Im(E_k)$ where $E_k$ are (purely real or complex-conjugate) eigenvalues of the Hamiltonian $H_\mathrm{K}$ for an $N=8$ chain. The $\mathcal{PT}$-symmetric region is marked by black,  and the rest is $\mathcal{PT}$-symmetry broken region. With $\delta/J\sim 1$, as $\gamma$ is increased, the first ${\mathcal {PT}}$ symmetry breaking near $\gamma/J\sim 0.5$ occurs due to the level-attraction between and coalescence of two highest energy states in the upper band; recall that due to the particle-hole symmetric nature of the spectrum, two lowest energy levels in the lower band concurrently become degenerate. With increasing $\gamma$, subsequent lower energy levels, except the lowest state in the upper band, coalesce in pairs. This sequence of transitions leads to a large number of exceptional points in the $\mathcal{PT}$-symmetry broken region. Further increasing $\gamma$ leads to a reverse process where levels with complex-conjugate energies undergo level-attraction and $\mathcal{PT}$-symmetry is restored. For the lowest-energy states in the upper band (and their chiral counterparts), the re-entrant $\mathcal{PT}$-symmetric phase is accompanied by a qualitative change where the wave-function weight shifts from the bulk to the edges. As $\gamma$ is increased further, the system enters $\mathcal{PT}$-broken region again. This second $\mathcal{PT}$ transition across an EP is driven by coalescence of the near-zero-energy state with state at the bottom of the top band.

To map out the exceptional point contours in the $\delta-\gamma$ plane, we use the (Dirac) inner-product matrix $M_{pq}=|\langle \psi_p|\psi_q\rangle|$ where $|\psi_k\rangle$ is the (Dirac)-normalized right eigenvector of $H_\mathrm{K}$ with eigenvalue $\lambda_k$. The order of the EP is then given by $\max_{p}\sum_{q\neq p}M_{pq}$. Figure~\ref{fig:PTreentrant}(b) shows the contours of exceptional points in the parameter space. In addition to the boundaries of $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken regions, seen in Fig.~\ref{fig:PTreentrant}(a), we see EP contours that denote the cascades of eigenvalue coalescence that occur in the $\mathcal{PT}$-broken region as $\gamma$ is increased. Of particular interest is the contour that starts at $\delta=0$ and $\gamma/J=1$. At point $\delta/J=1$, the system has fully degenerate bands with robust, mid-gap edge states (Fig.~\ref{fig:PTeven}a). Therefore, introduction of the gain-loss potentials leads to a third-order EP at $\gamma/J=2$ (shown by a yellow circle) in the otherwise second-order EP contour. We note that the prominent reentrant $\mathcal{PT}$ phases only occur when the gain-loss potentials are farthest apart, i.e. $m_0=1$ and remain robust only at $\mu=0$ for any even $N$; Figs.~\ref{fig:PTreentrant}(c)-(d) show the phase diagram and EP contours for an $N=20$ Kitaev chain.

\section{\label{sec:conclusion}Conclusion}
In this paper, we have investigated the dependence of the $\mathcal{PT}$-threshold $\gamma_\mathrm{th}$ on the properties of the underlying Hermitian Kitaev model and gain-loss potential locations. We have shown that the threshold profile is rich, with persistent differences between even and odd parity lattices. In particular, we have found that for a zero-detuning chain with odd number of sites, the threshold is enhanced with increasing superconducting order parameter. For an even chain with edge gain-loss potentials and superconducting coupling $\delta \gtrsim 1$, we discover re-entrant $\mathcal{PT}$-symmetric phase, and $\mathcal{PT}$-phase boundaries that contain both second and third order EPs. We have also discussed, briefly, a potential realization of our lattice model with coupled optical resonators. Our results further the understanding of non-Hermitian condensed matter models in the presence of realistically achievable gain and loss.

\bibliographystyle{apsrev4-2}

redpajama_set_name: RedPajamaArXiv


When I was a teenager, I would often put papaya mash on my face as a face mask. My folks would chide me and say "eat the papaya, it would do you more good." I slowly developed a taste for the fruit and realised the true meaning of the saying, "You are what you eat." When you are pregnant, it is sort of impossible to not give in to your temptations but beauty has to begin from inside. So here are my top three favourite foods that are nutritious, tasty and puts beauty back in pregnancy.
Fruity Crepes: As soon as my first trimester got over, I developed an incorrigible sweet tooth. I found myself finishing half buckets of ice-cream on cold winter nights in one sitting without a trace of guilt. Luckily, our friends invited us for dinner once and served these scrumptious crepes that transported me right to the streets of Paris. I've blindly followed this recipe and topped with delicious summer berries, oranges and a dollop of honey. Strawberries are low in calories, packed with Vitamins and has a plethora of antioxidants whilst blue berries are tiny heroes that gives you great skin and hair.
Tomato-basil soup: This is my all-time favourite comfort food. I think life becomes a tad better if you have a bowlful of this soup. What's more? It is easy to make as well. Heat a teaspoon of olive oil, saute a medium sized roughly chopped onion and few garlic pods, pour one jar of tinned plum tomatoes with all the juice. Season it with salt, pepper and a bit of cayenne pepper (if you like it hot) and cook it for about 6-7 minutes. Cool it for a bit a blend it to a consistency you desire. I like a bit chunky look because it feels more earthy. Again heat it and pour about a mil of water. Let it boil and check for seasoning. Finally, add some fresh or dried basil. Have with a side salad and warm Italian bread. Tomatoes are packed with anti-oxidants that gives your skin a warm glow, it can also stall the process of ageing , is great for your vision and keeps cholesterol in check. So, yum and powerful.
Home-made (almost) Chicken Burger: A succulent chicken burger is my favourite treat but going out to eat one would often mean eating other naughty and unwanted things (like fries) so home-made Chicken burgers are my way out. Strictly speaking, this is more about assembling the right things rather than making the Chicken patty from scratch. I have used Birds Eye Garlic and Chicken patty here. Chicken breasts are packed with proteins so needed when your energy levels suddenly go down and you have to battle weakness out of the blue. Simply grill burger buns in an oven, cook the patty as per instructions and make an earthy, home made burger with fresh tomatoes, onions (if you want), ice berg lettuce and a dollop of fiery Nandos sauce. You could add mayonnaise and cheese if you wish but I hate both. Have it with a crisp summer salad for a power-packed dinner. Delicious!!


IT'S NOT every day that one's artwork is put on display at the New York State Museum.
For Janet Matthews it's the third time running.
The Narre Warren North artist's latest drawing recently won the Focus on Nature Jury Award in April run by the museum, an honour reserved for the 14 best artworks out of the 80 to 90 on display at the exhibition from artists throughout the world.
"I still can't believe it. This year I'm the only Australian that got an award, wow, it still makes me really smile," Janet said.
Janet said it was the third time she had been contacted to take part in the museum's Focus on Nature exhibition but the first time she had won the Jury Award.
"It's wildlife or science based nature stuff, and I do drawings of wildlife, birds and animals," Janet said explaining the exhibition.
"I draw mostly Australian animals using colour pencils and graphite.
Janet said she always starts a drawing with story to tell, a context for the animal that she later creates on the canvas.
The official title for her winning echidna artwork was Look I Can Swim!
"My main focus for each work is usually a story, a personal interaction, a family situation, a story that I tell through the animal or bird," she said.
Having carved out a career for herself soon after leaving university, Janet has honed the craft which sees her spend more than 10 hours a day with a pencil in hand at her home studio.
"I decided to be an art teacher, because that's the only option at high school that was available to me. I didn't want be secretary or a nurse.
"I did my first year of university and with connections through other art teachers I discovered that they didn't have time to do their own artwork.
With her artwork on proud display in New York, Janet's decision to change her career trajectory certainly appears a wise one – even if she hasn't been able to fly over to see her work yet.
But with her back recovering well from a significant injury she believes it will only be a matter of time before she goes over to the Big Apple to see her echidna swimming in the water.


Childhood obesity is on the rise, with as many as one out of every five children in the U.S. being overweight or obese. One cause of obesity is a lack of enough physical activity or exercise. In fact, recent surveys indicate that most children -- three out of four -- ages 5 to 10 get less than one hour of physical activity each day.
The U.S. government's Physical Activity Guidelines for Americans recommends a minimum of 60 minutes a day in physical activity. But about 74 percent of U.S. children between 5 and 10 are not getting enough exercise. If your children are among this group, you can help them by setting an example with regular exercise, joining them in physical activities that are fun for the whole family, and encouraging them to spend less time in front of the TV and computer and more time outside.
Help your child make the right choice when it comes to exercise. Make their physical health one of your top concerns. Help them achieve a healthy lifestyle now that will continue through their adult life.


Faced with an unregistered firearms offence can feel overwhelming. Our criminal defence lawyers are experts in firearms offences with over 20 years experience of proving their clients innocence, convincing police to drop charges early and helped thousands of clients avoid prison and criminal convictions. Call now to speak to a senior criminal lawyer for realistic and practical advice.
You will be found not guilty, and your charge dismissed if police are unable to prove any one of the above elements in court.
You didn't know the firearm was unregistered, and you were not the owner of it at the time.
You could not reasonably be expected to have known the firearm was unregistered, and you were not the owner of it at the time.
Where an application for registration of the firearm was made within 24 hours after acquiring it, a licensed firearms dealer cannot be guilty of acquiring or possessing it.
Where the firearm is registered in another state or territory of Australia.
Honest Mistake: You honestly believed the firearm to be registered, where the court considers your belief to be reasonably held in the circumstances.
Self Defence: Where you believed it was necessary to acquire, supply, possess or use the firearm to protect yourself, property or another person, and only if the court considers that your belief was reasonably held in the circumstances perceived by you at the time.
Duress or Necessity: Where you were coerced into acquiring, possessing, supplying or using it, or where you did this to avoid serious injury.
You will be found not guilty if any one of the above common defences apply to your case. Our expert firearms lawyers will be able to hand pick your best defence before then strengthening it early in the case. While they have won a record number of firearms cases for over 20 years, they specialise in getting these charges dropped early through tactful negotiations.
If pleading guilty to an unregistered firearms charge, see the below crucial points and information on how to maximise your chances at getting the best possible outcome by avoiding prison, and a criminal record. Speak to one of our expert firearms lawyers now for further guidance.
You will receive a more lenient outcome, and better result, by receiving up to 25% discount on your punishment if a plea of guilty is entered at the earliest possible time in your case. The later a plea is entered to an unregistered firearms offence, the smaller that discount becomes.
This is why it's important to receive early, experienced, advice from an expert firearms lawyer.
These are letters from family, friends, charity, work, and even from you for the Judge to read. It can result in a more lighter outcome, with a better result in court if each letter expresses your good character, remorse, insight and embarrassment.
Our senior lawyers, are highly experienced, and they will guide you through each stage.
The chances of your firearms charge getting dropped or downgraded is increased by getting an expert firearms lawyer to carefully read the police evidence, pick out all the holes in it, then tactfully negotiate with police to drop the charges early in your case.
Our firearms lawyers have successfully convinced police to drop charges early countless times for over 20 years, and have earned a significant reputation for achieving this.
The police set of facts is a document drafted by police expressing your offending behaviour which you plead guilty to. This is handed to the Judge to read when he/she is considering your punishment. Because it is often one sided, painting the worst picture of you, it often results in a heavier punishment.
This is why it's critical to negotiate this to ensure it reflects the truth, painting you in a better light for the Judge to read, often resulting in a much lighter outcome and better result. Our senior lawyers will pick out all the holes in the police evidence, and use this when negotiating with police to change the facts to something more favourable to you.
Your result can be dramatically improved, with a much lighter outcome by the Judge, by getting a powerful court report from an experienced and respected psychologist or psychiatrist.
To get the best out of such a report, our firearms lawyers will hand select the best suited expert for your case. The report should comment on your state of mind, remorse, contrition, insight and shame for your behaviour.
If you acquired, supplied, possessed or used an unregistered firearm, the maximum punishment is a term of imprisonment of up to 5 years if dealt with in the District Court, and 2 years if in the Local Court.
If the firearm is a pistol or 'prohibited firearm', the maximum punishment is a term of imprisonment of up to 14 years if dealt with in the District Court, and 2 years in dealt with in the Local Court.
It's important to know, that the court only gives the maximum to the most serious cases of this offence. It's rarely given, and there are other types of punishments available, see below.
For your unregistered firearms charge, speak to an experienced firearms specialist lawyer for realistic, practical advice on maximising your chances at getting the best possible outcome. Our criminal defence lawyers are leaders in firearm cases with over 20 years experience, who know exactly how to thoroughly prepare and powerfully present your case to the Judge.


Never underestimate the simmering wrath of an intelligent man unjustly wronged. Geoff Summers is a brilliant designer of electronic controllers for anti-burglar systems. When his vindictive new boss prevents Geoff from fulfilling his wife's dying wish, he is determined to get even.
No longer able to provide what they want and with his life under threat, how does Geoff get out?
A pacey revenge thriller that twists and turns all the way through..
Chris is an author who discovered the delicious agony (his words) of novel writing late in life.
He's not retired, he has simply changed professions and is now a full-time novelist.


text:
Conistra ferruginea är en fjärilsart som beskrevs av Barend J. Lempke 1941. Conistra ferruginea ingår i släktet Conistra och familjen nattflyn. Inga underarter finns listade i Catalogue of Life.

Källor

Nattflyn
ferruginea

redpajama_set_name: RedPajamaWikipedia


A Medicare "replacement" policy is a form of private Medicare supplement insurance which replaces your Medicare Part A and B benefits. Why would anyone want to replace their federal Medicare benefits with a private policy? Well, for some seniors, a Medicare Advantage policy can be the most cost-effective way to supplement their benefits and protect themselves from costly gaps in Traditional Medicare coverage.
Just because Medicare Advantage replaces your Traditional Medicare benefits doesn't mean there aren't consumer protection laws in place to ensure that you get the medical benefits you are owed. Medicare Advantage policies are regulated by federal law to ensure just that. Many of these policies include additional coverage, especially with regard to the potentially expensive gaps in traditional Medicare (Parts A & B). Some Advantage plans, often referred to as MAPD plans, offer prescription drug coverage.
When choosing a policy, you will have to decide on a provider network to go along with your plan. Your provider network determines the doctors and facilities from which you may receive your medical care. More restrictive networks (HMOs) cost less, whereas less restrictive networks will naturally be more expensive.
Choosing your provider network is one of the most important decisions you can make with a Medicare Advantage policy. If you choose a highly restrictive HMO, you may not be able to get the care you need when you need it (at least, not at an affordable price). But if you are living on a fixed budget, a PPO or PFFS might not be within your means. Additionally, even if you are content with your provider network, your insurance company can change the rules on you at any time.
Medicare Advantage is one of the few Medicare supplement insurance plans which provides for additional coverage options beyond Medicare Parts A, B, and their respective coverage gaps. You have the option to add prescription drug coverage to your policy via Medicare Part D. Additionally, there are optional additions such as vision care or dental coverage which can also be added to your policy. Just remember: the more you add, the more expensive your policy will be.
It is likely that your Medicare Advantage provider will ask for a medical history or want some other verification of your health status before determining whether or not you qualify for coverage. If you are 65, new to Medicare, and in fairly good health, the qualification process should be fairly simple. However, Medicare Advantage providers are more willing to deny coverage if you have a serious condition, such as End Stage Renal Disease (ESRD). You can learn more about your coverage options by contacting your state department of insurance, by contacting your local insurance agent, and also by visiting our Medicare Advantage Enrollment Options Page.


A graduate of the University of the Philippines College of Medicine, Dr. Balatbat completed his training at the Yale Primary Care Internal Medicine Residency Program in New Haven, CT and then completed a fellowship in Infectious Diseases at the UC Davis Medical Center. He is board certified in Internal Medicine and is a fellow of the American College of Physicians.
Dr. Balatbat has been with Mercy Medical Group for 11 years and served as the chair of the Department of Internal Medicine for the past 9 years. As chair, he has championed primary care redesign and transformation, including the development of MMG's patient centered medical home model. He has also led MMG's population health and quality initiatives.
During his 11 year tenure as an Internal Medicine provider at Mercy Medical Group, Dr. Balatbat was an ardent proponent for preventative care, playing an instrumental role in several national honors received by the group, including recognition for diabetes care and prevention from the American Medical Group Association and recognition for best practices aimed at reducing heart attack and stroke from the American Medical Association and the American Heart Association.
Dr. Balatbat will continue to see patients at his Q Street clinic after taking on the CMO role.
Mercy Medical Group, a service of Dignity Health Medical Foundation, is a multi-specialty clinic with patient service centers located throughout the greater Sacramento area. It includes more than 400 medical providers and 900 Dignity Health Medical Foundation employees providing services to thousands of patients in the Sacramento area and its affiliated local Dignity Health hospitals (Mercy General Hospital, Mercy San Juan Medical Center, Mercy Hospital of Folsom and Methodist Hospital of Sacramento). Dignity Health Medical Foundation has 29 patient care center locations in Sacramento. For more information on Dignity Health Medical Foundation d/b/a Mercy Medical Group and the wide range of services it provides, visit our website at mymercymedicalgroup.org.


text:
Q: Laravel/Eloquent query going wrong I have two tables, one with users, one with the name of their document. The first table consists of two columns: id and username. The second one consists of three columns: id, userid and document_name.
Now, I'm trying to create a query in the controller. What should happen, ideally, is that if someone visits website.com/{documentname}, it displays the username of the owner. Also, this should only happen if the current logged in user is the owner of the document. However, this is proving more difficult than I imagined. As in, I can't see what I'm doing wrong.
Here's the query:
$user = DB::table('documents')
            ->join('users', function($join)
            {
                $join->on('users.id', '=', 'documents.userid')
                     ->where('documents.userid', '=', Auth::id())
                     ->where('documents.document_name', '=', $document_name);
            })
            ->get();

A:   **Try this query :**

 $user =  DB::table('documents')
  ->leftJoin('users', 'users.id', '=', 'documents.userid')
  ->where('documents.userid', '=', Auth::id())
  ->where('documents.document_name', '=', $document_name);
  ->get();

A: $document_name isn't in scope for the join function: you need to pass it through to the closure via use
$user = DB::table('documents')
        ->join('users', function($join) use ($document_name)
        {
            $join->on('users.id', '=', 'documents.userid')
                 ->where('documents.userid', '=', Auth::id())
                 ->where('documents.document_name', '=', $document_name);
        })
        ->get();

EDIT
Because the WHERE conditions apply to the base table, and not to the JOIN:
$user = DB::table('documents')
        ->join('users', function($join) {
            $join->on('users.id', '=', 'documents.userid')
        })
        ->where('userid', '=', Auth::id())
        ->where('document_name', '=', $document_name);
        ->get();

redpajama_set_name: RedPajamaStackExchange


text:
Q: Finding out total and free disk space in .NET I am trying to find a way to determine the total and available disk space in an arbitrary folder from a .NET app. By "total disk space" and "available disk space" in a folder I refer to the total and available disk space that this folder would report if you performed a "dir" command on it, that is, the total and available disk space of the logical drive containing that folder, considering the user account under which the request is being made.
I am using C#. The method should work both for local and remote folders given as UNC paths (rather than accessed through mapped drive letters). For example, it should work for:

*

*C:\Temp

*\\Silfen\Resources\Temp2

I am starting with a DirectoryInfo object, but this seems to have no associated disk space information. The DriveInfo class does, but it won't work with remote folders.
Edit. After some exchanges with you guys, I am considering mapping remote folders as local drives, using DriveInfo to obtain the data, and unmapping again. The problem with this approach is that my app needs to collect the data for over 120 folders a few times a day, every day. I am not sure this would be feasible.
Any ideas? Thanks.

A: This may not be what you want, but I'm trying to help, and it has the bonus of slightly secure erasing the free space of your drive.
public static string DriveSizeAvailable(string path)
{
    long count = 0;
    byte toWrite = 1;
    try
    {
        using (StreamWriter writer = new StreamWriter(path))
        {
            while (true)
            {
                writer.Write(toWrite);
                count++;
            }
        }
    }
    catch (IOException)
    {
    }

    return string.Format("There used to be {0} bytes available on drive {1}.", count, path);
}

public static string DriveSizeTotal(string path)
{
    DeleteAllFiles(path);
    int sizeAvailable = GetAvailableSize(path);
    return string.Format("Drive {0} will hold a total of {1} bytes.", path, sizeAvailable);
}

A: Not really a C# example but may give you a hint - a VB.NET function returning both amount of free and total space on drive (in bytes) along specified path. Works for UNC paths as well, unlike System.IO.DriveInfo.
VB.NET:
<DllImport("kernel32.dll", SetLastError:=True, CharSet:=CharSet.Auto)> _
Private Shared Function GetDiskFreeSpaceEx(lpDirectoryName As String, ByRef lpFreeBytesAvailable As ULong, ByRef lpTotalNumberOfBytes As ULong, ByRef lpTotalNumberOfFreeBytes As ULong) As <MarshalAs(UnmanagedType.Bool)> Boolean
End Function

Public Shared Function GetDriveSpace(folderName As String, ByRef freespace As ULong, ByRef totalspace As ULong) As Boolean
    If Not String.IsNullOrEmpty(folderName) Then
        If Not folderName.EndsWith("\") Then
            folderName += "\"
        End If

        Dim free As ULong = 0, total As ULong = 0, dummy2 As ULong = 0
        If GetDiskFreeSpaceEx(folderName, free, total, dummy2) Then
            freespace = free
            totalspace = total
            Return True
        End If
    End If
End Function

A: How about this link from MSDN that uses the System.IO.DriveInfo class?

A: You can use
GetDiskFreeSpaceEx from kernel32.dll which works with UNC-paths and drives. All you need to do is include a DllImport (see link for an example).

A: System.IO.DriveInfo works fine. I'm attached to two separate Netware servers, with several drives mapped.
Here's for the local C: drive:
Drive C:\
  File type: Fixed
  Volume label: Drive C
  File system: NTFS
  Available space to current user:   158558248960 bytes
  Total available space:             158558248960 bytes
  Total size of drive:               249884004352 bytes

Here's the output for one of the network drives:
Drive F:\
  File type: Network
  Volume label: SYS
  File system: NWFS
  Available space to current user:     1840656384 bytes
  Total available space:               1840656384 bytes
  Total size of drive:                 4124475392 bytes

I used the following code, directly from the MSDN docs on DriveInfo:

using System;
using System.IO;

class Test
{
    public static void Main()
    {
        DriveInfo[] allDrives = DriveInfo.GetDrives();

        foreach (DriveInfo d in allDrives)
        {
            Console.WriteLine("Drive {0}", d.Name);
            Console.WriteLine("  File type: {0}", d.DriveType);
            if (d.IsReady == true)
            {
                Console.WriteLine("  Volume label: {0}", d.VolumeLabel);
                Console.WriteLine("  File system: {0}", d.DriveFormat);
                Console.WriteLine(
                    "  Available space to current user:{0, 15} bytes",
                    d.AvailableFreeSpace);

                Console.WriteLine(
                    "  Total available space:          {0, 15} bytes",
                    d.TotalFreeSpace);

                Console.WriteLine(
                    "  Total size of drive:            {0, 15} bytes ",
                    d.TotalSize);
            }
        }
    }
}

A: Here's one more possibility that I've used for years.  The example below is VBScript, but it should work with any COM-aware language. Note that GetDrive() works on UNC shares as well.
Dim Tripped
Dim Level

Tripped = False
Level   = 0

Sub Read(Entry, Source, SearchText, Minimum, Maximum)

    Dim fso
    Dim disk

    Set fso  = CreateObject("Scripting.FileSystemObject")

    Set disk = fso.GetDrive(Source)

    Level = (disk.AvailableSpace / (1024 * 1024 * 1024))

    If (CDbl(Level) < CDbl(Minimum)) or (CDbl(Level) > CDbl(Maximum)) Then
        Tripped = True
    Else
        Tripped = False
    End If

End Sub

A: Maksim Sestic has given the best answer, as it works on both, local and UNC paths. I have changed his code a little for better error handling and included an example. Works for me like a charm.
You need to put
Imports System.Runtime.InteropServices

into your code, to allow DllImport to be recognized.
Here is the modified code:
<DllImport("kernel32.dll", SetLastError:=True, CharSet:=CharSet.Auto)> _
Private Shared Function GetDiskFreeSpaceEx(lpDirectoryName As String, ByRef lpFreeBytesAvailable As ULong, ByRef lpTotalNumberOfBytes As ULong, ByRef lpTotalNumberOfFreeBytes As ULong) As <MarshalAs(UnmanagedType.Bool)> Boolean
End Function

Public Shared Function GetDriveSpace(folderName As String, ByRef freespace As ULong, ByRef totalspace As ULong) As Boolean

Dim free As ULong = 0
Dim total As ULong = 0
Dim dummy2 As ULong = 0

Try

    If Not String.IsNullOrEmpty(folderName) Then

         If Not folderName.EndsWith("\") Then
             folderName += "\"
         End If

         If GetDiskFreeSpaceEx(folderName, free, total, dummy2) Then
             freespace = free
             totalspace = total
             Return True
         End If

     End If

Catch
End Try

    Return False

End Function

Call it this way:
dim totalspace as ulong = 0
dim freespace as ulong = 0
if GetDriveSpace("\\anycomputer\anyshare", freespace, totalspace) then
    'do what you need to do with freespace and totalspace
else
    'some error
end if

The foldername can also be a local directory like drive:\path\path\...
It is still in VB.NET but shouldn't be a problem to translate into C#.

A: I'm pretty sure this is impossible.  In windows explorer, if I try to get the folder properties of a UNC directory, it gives me nothing as far as available space.  Used/Available space is a characteristic of drives, not folders, and UNC shares are treated as just folders.
you have to either:
 - Map a drive
 - Run something on the remote machine to check disk space.
You could also run into problems with something like Distributed file system, in which a UNC/Mapped share is NOT tied to any specific drive, so there youd have to actually sum up several drives.
And what about user quotas?  The drive may not be full, but the account you are using to write to that folder may have hit its limit.

A: Not C# and only gives the avilable space, but . . .
dir \\server\share | find /i "bytes free"

gets you part of the way. I'm looking or the same solution but there doesn't seem to be a nice one - especially when trying to avoid mapping drives.

A: I just translated VB.NET answers to C#:
[DllImport("kernel32.dll", SetLastError = true, CharSet = CharSet.Auto)]
static extern bool GetDiskFreeSpaceEx(string lpDirectoryName, out ulong lpFreeBytesAvailable, out ulong lpTotalNumberOfBytes, out ulong lpTotalNumberOfFreeBytes);

public static bool GetDriveSpace(string folderName, out ulong freespace, out ulong totalspace)
{
    if (string.IsNullOrEmpty(folderName)) throw new ArgumentNullException(nameof(folderName));
    if (folderName[^1] != '\\') folderName += '\\';
    return GetDiskFreeSpaceEx(folderName, out freespace, out totalspace, out _);
}