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\begin{document}
\title{Linking DC transport properties and entanglement negativity in non-Hermitian superlattices}
%\title{Non-Hermitian linear response formalism for optical conductivity in non-Hermitian Dirac Hamiltonians}
\author{L. S. Lima*}
\affiliation{Department of Physics, Federal Center for Technological Education of Minas Gerais, 30510-000, Belo Horizonte, MG, Brazil. E-mail: lslima@cefetmg.br}
\date{\today}
\begin{abstract}
\hspace{6cm}{\bf{Abstract}}\\
We investigate how non-Hermitian effects in one-dimensional superlattices incorporating gain and loss imbalance, asymmetric hopping, and complex potentials modify both electrical transport  and quantum correlations. Using a family of generalized Su–Schrieffer–Heeger (SSH) superlattice Hamiltonians with tunable sublattice asymmetry and non-reciprocity, we compute, linear response transport coefficients, and mixed-state entanglement measures across parameter regimes (\emph{$\emph{PT}$-symmetric broken regime}). We find robust signatures linking the onset of the skin effect and spectral winding to enhanced local currents and strongly inhomogeneous entanglement profiles, and topological edge modes to finite entanglement negativity in otherwise mixed steady states. Our results clarify when transport and entanglement provide complementary probes of non-Hermitian phase structure, and suggest experimental observables in photonic and cold-atom realizations.
\end{abstract}
\maketitle
%
\section{Introduction}{\label{1}}

\begin{figure}
 \centering
\includegraphics[width=11.0cm]{Fig_super_lattice.eps} \\
\caption{Schematic representation of tetramerized superlattice with four sites (labeled by $(a_1,a_2,a_3,a_4)$ per unit cell. $t_1$, $t_2$ and $t_3$ are the intracell couplings, whereas $t_4$ is the intercell hopping. There
are three different choices of balanced onsite gain and loss: case I with $(i\alpha,-i\alpha,i\alpha,-i\alpha)$, case II with $(i\alpha,i\alpha,-i\alpha,-i\alpha)$, and case III
with $(i\alpha,-i\alpha,-i\alpha,i\alpha)$  The solid-green circles (dashed-blue circles) indicate positive (negative) imaginary parts, $\pm\alpha$.}\label{fig_9}
\end{figure}
\noindent
In the past decade, non-Hermitian quantum systems have attracted growing attention as fertile grounds for realizing unconventional topological phases and transport phenomena that have no direct counterparts in Hermitian settings \cite{Bergholtz,Ashida,Gong,Wang,Gardiner,HP,Lei,N,AH,kevin,prl,LJ,yan,leonardoslima,lslimasci2,lslimapl,physlett,physe}. In contrast to Hermitian Hamiltonians, whose spectra are strictly real and whose topological properties are well captured by conventional bulk–boundary correspondence, non-Hermitian systems generally exhibit complex energy spectra and host distinctive effects such as the non-Hermitian skin effect (NHSE) \cite{Yao,Kunst,Helbig,Weidemann}. In the presence of open boundary conditions (OBC), the NHSE leads to an extensive accumulation of eigenstates near the system boundaries, profoundly modifying both spectral and transport properties. These features call for a reexamination of the foundations of topology and transport in quantum matter.

A paradigmatic model in the study of topological phases is the Su–Schrieffer–Heeger (SSH) model, which describes a one-dimensional dimerized lattice supporting topologically protected edge states \cite{Su}. Generalizations of the SSH model to higher dimensions, such as two-dimensional arrays of coupled SSH chains, provide a versatile framework for exploring richer band structures, including Dirac-like dispersions and higher-order topological phases \cite{Benalcazar,Lin}. When nonreciprocal hopping processes and on-site gain and loss are introduced, these systems naturally enter the non-Hermitian regime, giving rise to novel spectral and topological characteristics that strongly influence their dynamical and transport behavior \cite{Lee,Kiong,Longhi}.

A key issue in this context is the role played by critical points associated with the closing and reopening of complex energy gaps, particularly under OBC, and how these spectral transitions manifest in experimentally accessible transport quantities such as electrical conductivity. While in Hermitian systems bulk gap closings typically coincide with topological phase transitions and leave clear signatures in transport, non-Hermitian systems often display boundary-sensitive spectral features that are absent under periodic boundary conditions (PBC), reflecting the breakdown of conventional bulk–boundary correspondence \cite{Kunst,Okuma,Zhang,Kawabata}.

%In this work, we study electrical transport in non-Hermitian superlattices by explicitly evaluating transport coefficients within the Kubo–Greenwood formalism. We consider three gain-loss configurations as described in Fig.~\ref{fig_9}, where each one revealing unique behaviors in multigap topological states: $\mathcal{PT}$-symmetric pattern $(i\alpha,-i\alpha,i\alpha,-i\alpha)$: The gain and loss do not affect the multigap topological phases of the superlattice. $\mathcal{PT}$-symmetric pattern $(i\alpha,i\alpha,-i\alpha,-i\alpha)$: The non-Hermiticity in this case disrupts the midgap topological phase. Anti-$\mathcal{PT}$-symmetric pattern $(i\alpha,-i\alpha,-i\alpha,i\alpha)$: The gain and loss in this configuration drives a topological phase transition from trivial to nontrivial in the midgap. In Ref.~\cite{Gaoyu}, the influence of different gain-loss configurations on the multigap topological edge states has been analyzed. 
%We investigated how non-Hermitian parameters $\alpha$  including nonreciprocity and gain/loss strength, reshape both AC and DC conductivities \cite{Kawabata,Yokomizo,Ahang,Lee2}. Furthermore, we explore the relationship between the entanglement negativity $E_N$ and the Drude weight $D_S$, which serves as a quantitative indicator of DC transport, thereby shedding light on the interplay between quantum correlations and transport in non-Hermitian systems.
In this work, we study electrical transport in non-Hermitian superlattices using the Kubo–Greenwood formalism. We analyze three gain–loss configurations (Fig.~\ref{fig_9}) with distinct impacts on multigap topology. For the $\mathcal{PT}$-symmetric pattern $(i\alpha,-i\alpha,i\alpha,-i\alpha)$, the multigap phases remain robust. In contrast, the configuration $(i\alpha,i\alpha,-i\alpha,-i\alpha)$ suppresses the midgap topological phase. For the anti-$\mathcal{PT}$-symmetric case $(i\alpha,-i\alpha,-i\alpha,i\alpha)$, a topological transition emerges in the midgap from trivial to nontrivial. The role of such gain–loss profiles on edge states was discussed in Ref.~\cite{Gaoyu}.
 We further show that non-Hermitian parameters $\alpha$, incorporating nonreciprocity and gain–loss strength, strongly reshape both AC and DC conductivities~\cite{Kawabata,Yokomizo,Ahang,Lee2}. Finally, we relate the entanglement negativity $E_N$ to the Drude weight $D_S$, linking quantum correlations to coherent DC transport in non-Hermitian systems.
\noindent


The remainder of this paper is organized as follows. In Section \ref{2}, we introduce the model and analyzed the different phases, spectrum properties and topological edge states. In Section \ref{3}, we compute the transport coefficients and quantifier of entanglement quantum correlation, analysing the interplay between DC conductivity and entanglement negativity.  Finally, in Section \ref{5}, we discuss the physical implications of our findings and outline directions for future work.


\section{Model and Formalism}{\label{2}}
%\subsection{Hamiltonian in momentum space}

The real-space Hamiltonian is given by
\begin{eqnarray}{\label{model}}
\hspace{-0.0cm}\mathcal{H}=\sum_{j=1}^{N}\sum_{l=1}^{3}\left[t_na^{\dag}_{j,l}a_{j,l+1}+t_4a^{\dag}_{j,4}a_{j+1,1}+\hbox{H.c.}\right]\nonumber\\
\hspace{-0.0cm}+i\alpha\sum_{j=1}^{N}\left[\sum_{l=1}^{4}(-1)^{l+1}a^{\dag}_{j,1}a_{j,1}+\sum_{l=1}^{2}(a^{\dag}_{j,l}a_{j,l})-\sum_{l=3}^{4}(a^{\dag}_{j,l}a_{j,l})\right]\nonumber\\
\hspace{-0.0cm}-i\alpha\sum_{j=1}^{N}\left[\sum_{l=1}^{2}(-1)^{l+1}a^{\dag}_{j,l}a_{j,l}-\sum_{l=3}^{4}(-1)^{l+1}a^{\dag}_{j,l}a_{j,l}\right],\nonumber\\
\end{eqnarray}
with $\alpha$ representing the gain or loss rate and $t=t_1^2+t_2^2+t_3^2+t_4^2$. In the Hermitian limit, performing the Fourier transformation, the Block Hamiltonian is obtained as
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
\hspace{-1.0cm}  \mathcal{H}_1(k)=t_1\sigma_x\otimes\sigma^{+}+t_2\sigma_0\otimes\sigma_0+(t_4e^{ik}-t_2)\sigma_0\otimes\sigma_{-}\nonumber\\
\hspace{-1.0cm}  +(t_4e^{-ik}-t_2)\sigma_0\otimes\sigma_{+}%+t_2\sigma_0\otimes\sigma_{+}+t_2\sigma_0\otimes\sigma_{-}
+t_3\sigma_x\otimes\sigma^{-} +i\alpha\sigma_0\otimes\sigma_z\\
\hspace{-1.0cm}  \mathcal{H}_2(k)=t_1\sigma_x\otimes\sigma^{+}+t_2\sigma_0\otimes\sigma_0+(t_4e^{ik}-t_2)\sigma_0\otimes\sigma_{-}\nonumber\\
\hspace{-1.0cm}  +(t_4e^{-ik}-t_2)\sigma_0\otimes\sigma_{+}%+t_2\sigma_0\otimes\sigma_{+}+t_2\sigma_0\otimes\sigma_{-}
+t_3\sigma_x\otimes\sigma^{-} +i\alpha\sigma_z\otimes\sigma_0
%\hspace{-1.0cm}  \mathcal{H}_3(k)=t_1\sigma_x\otimes\sigma^{+}+t_2\sigma_0\otimes\sigma_0+(t_4e^{ik}-t_2)\sigma_0\otimes\sigma_{-}\nonumber\\
%+(t_4e^{-ik}-t_2)\sigma_0\otimes\sigma_{+}%+t_2\sigma_0\otimes\sigma_{+}+t_2\sigma_0\otimes\sigma_{-}
%+t_3\sigma_x\otimes\sigma^{-} +i\alpha\sigma_z\otimes\sigma_z\nonumber\\
\end{eqnarray}
where
\begin{eqnarray*}
% \nonumber % Remove numbering (before each equation)
  \sigma_{+}=\left(
               \begin{array}{cc}
                 0 & 1 \\
                 0 & 0 \\
               \end{array}
             \right), \hspace{0.5cm}
             \sigma_{-}=\left(
               \begin{array}{cc}
                 0 & 0 \\
                 1 & 0 \\
               \end{array}
             \right)\nonumber\\
               \sigma^{+}=\left(
               \begin{array}{cc}
                 1 & 0 \\
                 0 & 0 \\
               \end{array}
             \right), \hspace{0.5cm}
             \sigma^{-}=\left(
               \begin{array}{cc}
                 0 & 0 \\
                 0 & 1 \\
               \end{array}
             \right).
\end{eqnarray*}


%The presence of non-Hermitian terms generates symmetries such as time-reversal $\mathcal{T}_{+}\mathcal{H}_{1,2,3}(k)\mathcal{T}^{-1}_{+}=\mathcal{H}_{1,2,3}(-k)$ , with $\mathcal{T}_{+}=\sigma_0\otimes\sigma_0$, particle-hole symmetry $\eta\mathcal{H}^{*}_{1,2,3}(k)\eta^{-1}=-\mathcal{H}_{1,2,3}(-k)$ with $\eta=\sigma_0\otimes\sigma_z$ and pseudo-anti-Hermiticity symmetry $\Gamma\mathcal{H}_{1,2,3}(k)\Gamma^{-1}=-\mathcal{H}_{1,2,3}(-k)$. Moreover, for $t_1=t_3$, $\mathcal{H}_{1,2}(k)$ presents parity-time symmetry $(\mathcal{PT})\mathcal{H}_{1,2}(k)(\mathcal{PT})^{-1}=\mathcal{H}_{1,2}(k)$, with $\mathcal{P}=\sigma_x\otimes\sigma_x$, while $\mathcal{H}_{3}(k)$ is anti-$\mathcal{PT}$ symmetric  $(\mathcal{PT})\mathcal{H}_{3}(k)(\mathcal{PT})^{-1}=-\mathcal{H}_{3}(k)$ with $\mathcal{P}=\sigma_x\otimes\sigma_y$. In the following, we have that the multigap topological edge states of the superlattice exhibit distinct responses under different gain-loss patterns. Hereafter we set $t_1=t_3=1$ as the energy unit for simplicity.
The presence of non-Hermitian terms generates symmetries such as time-reversal $\mathcal{T}_{+}\mathcal{H}_{1,2}(k)\mathcal{T}^{-1}_{+}=\mathcal{H}_{1,2}(-k)$, with $\mathcal{T}_{+}=\sigma_0\otimes\sigma_0$, particle-hole symmetry $\eta\mathcal{H}^{*}_{1,2}(k)\eta^{-1}=-\mathcal{H}_{1,2}(-k)$ with $\eta=\sigma_0\otimes\sigma_z$. %and pseudo-anti-Hermiticity symmetry $\Gamma\mathcal{H}_{1,2,3}(k)\Gamma^{-1}=-\mathcal{H}_{1,2,3}(-k)$.
 Moreover, for $t_1=t_3$, $\mathcal{H}_{1,2}(k)$ presents parity-time symmetry $(\mathcal{PT})\mathcal{H}_{1,2}(k)(\mathcal{PT})^{-1}=\mathcal{H}_{1,2}(k)$, with $\mathcal{P}=\sigma_x\otimes\sigma_x$. %while $\mathcal{H}_{3}(k)$ is anti-$\mathcal{PT}$ symmetric  $(\mathcal{PT})\mathcal{H}_{3}(k)(\mathcal{PT})^{-1}=-\mathcal{H}_{3}(k)$ with $\mathcal{P}=\sigma_x\otimes\sigma_y$.
 In the following, we have that the multigap topological edge states of the superlattice exhibit distinct responses under different gain-loss patterns. Hereafter we set $t_1=t_3=1$ as the energy unit for simplicity.

%\emph{Phase diagram and topological edge states:}
\subsection{Phase diagram, spectral properties and topological edge states}

The eigenvalues of Hamiltonian $\mathcal{H}_1$ are given by
\begin{eqnarray}{\label{gap1}}
% \nonumber % Remove numbering (before each equation)
  E_{\pm,\pm}(k) =\pm\sqrt{t/2-\alpha^2\pm (1/2)\sqrt{t^2-4F(k)}},
\end{eqnarray}
where $F(k)=t_1^2t_3^2+t_2^2t_4^2-2t_1t_2t_3t_4\cos k$. The band gaps between $j$th and $(j+1)$th bands are given  at $k=0$ and $k=\pi$ respectively. As $\alpha$ is increased, five distinct regions can be defined: a $\mathcal{PT}$-symmetric phase for $\alpha<\sqrt{t/2-\sqrt{t^2/4-F(0)}}$ where all eigenvalues are real. a $\mathcal{PT\mathcal}$-broken phase when
$\sqrt{t/2-\sqrt{t^2/4-F(0)}}<\alpha<\sqrt{t/2-\sqrt{t^2/4-F(\pi)}}$, which the central two bands have real and imaginary eigenvalues separated by exceptional points. A partial-$\mathcal{APT}$-symmetric phase for $\sqrt{t/2-\sqrt{t^2/4-F(\pi)}}<\alpha<\sqrt{t/2+\sqrt{t^2/4-F(\pi)}}$
 where the eigenvalues are real for two bands and imaginary for other bands. $\mathcal{PT}$-broken phase when $\sqrt{t/2+\sqrt{t^2/4-F(\pi)}}<\alpha<\sqrt{t/2+\sqrt{t^2/4-F(0)}}$, of which two bands have real and imaginary eigenvalues separated by exceptional points. $\mathcal{APT}$-symmetric phase for $\alpha>\sqrt{t/2+\sqrt{t^2/4-F(0)}}$ where the eigenvalues are purely imaginary.

\noindent
\begin{figure}
 \centering
\includegraphics[width=8.0cm]{phase-diag.eps} \\
\caption{Plot (a) Phase diagram of the Hermitian topological superlattice: region I is the topological phase of the first and third band gap with Zak phases $\mathcal{Z}_1=\pi,\mathcal{Z}_2=0$. Region II represents the trivial phase with $\mathcal{Z}_{1,2}=0$; region III is the topological phase of the middle band gap $\mathcal{Z}_1=0,\mathcal{Z}_2=\pi$; region IV is the completely topological phase with $\mathcal{Z}_{1,2}=\pi$. Plot (b) Phase diagram of the non-Hermitian Hamiltonian $\mathcal{H}_1$ with $t_2=1$, which contains a $\mathcal{PT}$-symmetric (PTS) phase where all eigenvalues are real, two $\mathcal{PT}$-broken (PTB I and PTB II) phases with real and imaginary eigenvalues separated by exceptional points, a partial $\mathcal{APT}$-symmetric (APT) phase with purely imaginary energies.}\label{fig_8}
\end{figure}
\noindent

In the Eq.~(\ref{gap1}), when $t_1=t_3$, $t_2=t_4$, the first and third gaps close, while the second gap closes for $t_1t_3=t_2t_4$, where at these gaps close signify the presence of topological phase transitions. Moreover, the topological properties can be characterized by the sum of Zak phases of all the isolated bands below the corresponding gap: topological phase of the first and third band gap with Zak phases $\mathcal{Z}_1=\pi,\mathcal{Z}_2=0$. Region II represents the trivial phase with $\mathcal{Z}_{1,2}=0$; region III is the topological phase of the middle band gap $\mathcal{Z}_1=0,\mathcal{Z}_2=\pi$; region IV is the completely topological phase with $\mathcal{Z}_{1,2}=\pi$. The boundaries coincide with the locations of gap-closing points. The nontrivial Zak phase implies that a pair of topologically protected edge states will emerge at the boundaries of the finite lattice according to bulk-edge correspondence.
\begin{eqnarray*}
% \nonumber % Remove numbering (before each equation)
  \mathcal{Z}_l=i\int_{-\pi}^{\pi}(\langle\psi_1(k)|\partial_k|\psi_1(k)\rangle+\langle\psi_2(k)|\partial_k|\psi_2(k)\rangle\nonumber\\
  +\ldots +\langle\psi_l(k)|\partial_k|\psi_l(k)\rangle)dk\hspace{0.2cm} \hbox{mod} 2\pi,
\end{eqnarray*}
where $|\psi_l(k)\rangle$ is the Bloch wave functions with eigenvalue $E_l$. Furthermore, the Zak phase can take only the values zero or $\pi$, denoting the trivial and nontrivial topological phases, respectively.

In Fig.~\ref{fig_8} (a), we show the phase diagram $t_4$ vs $t_2$ of the Hermitian topological superlattice ($\alpha=0$). The four distinct phases are represented: region I which is the topological phase of the first and third band gap with Zak phases $\mathcal{Z}_1=\pi,\mathcal{Z}_2=0$. Region II which represents the trivial phase with $\mathcal{Z}_{1,2}=0$; region III which is the topological phase of the middle band gap $\mathcal{Z}_1=0,\mathcal{Z}_2=\pi$; region IV which is the completely topological phase with $\mathcal{Z}_{1,2}=\pi$.

 In Fig.~\ref{fig_8} (b), we present the phase diagram of the non-Hermitian $\mathcal{H}_1$ with $t_2=1$.  The phase diagram contains five distinct phases: $\mathcal{PT}$-symmetric (PTS) phase where all eigenvalues are real, two $\mathcal{PT}$-broken (PTB I and PTB II) phases with real and imaginary eigenvalues separated by exceptional points, a partial $\mathcal{APT}$-symmetric (APT) phase with purely imaginary energies.

For the non-Hermitian superlattice governed by the Hamiltonian $\mathcal{H}_2(k)$, its eigenvalues for $t_1=t_3$ are given by
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  E_{\pm,\pm}(k) =\pm\sqrt{(t-2\alpha^2)\pm (1/2)\sqrt{G(k)-16\alpha^2t_1^2}},\nonumber\\
\end{eqnarray}
 where $G(k)=(t_2^2-t_4^2)^2+4t_1^2t_2^2+4t_1^2t_4^2+8t^2_1t_2t_4\cos k$. Supposing the midgap $\Delta E=0$, we have
\begin{eqnarray*}
\hspace{-0.25cm}\alpha_c^{\pm}=\sqrt{\pm(1/2)\left[\sqrt{G(0)-t^2+(2t_1^2-t_2^2-t_4^2)^2}-t_1^2+t_2^2/2+t_4^2/2\right]},
\end{eqnarray*}
where the mid-gap is gapless when $\alpha^{-}_c<\alpha<\alpha^{+}_c$.

%For the case with a smaller $\alpha$, the bands are real-valued and gapped, indicating the $\mathcal{PT}$ symmetry is not broken. As the non-Hermiticity increases, the top and botton bands touch their neighboring bands, accompanied by the emergence of exceptional points. In this case, the $\mathcal{PT}$ symmetry is broken for the top and bottom bands, while it is retained for the middle gaps. Moreover, the band gap of the two bands in the middle closes gradually. Furthermore, increasing the gain-loss rate, the $\mathcal{PT}$ symmetry is spontaneously broken for the middle two bands. Interestingly, the middle bands turn into real-valued and the midgap reopens with a further enlarged gain-loss rage $\alpha$.
For small values of $\alpha$, the energy bands remain entirely real and gapped, indicating that $\mathcal{PT}$ symmetry is preserved. As the degree of non-Hermiticity increases, the upper and lower bands approach and eventually touch their adjacent bands, accompanied by the exceptional points. In this regime,
 $\mathcal{PT}$ symmetry is broken for the top and bottom bands, while it remains intact within the intermediate gaps. Concurrently, the gap between the two bands gradually closes. With a further increase in the gain–loss rate $\alpha$, $\mathcal{PT}$ symmetry becomes spontaneously broken for the middle bands as well. Interestingly, upon increasing $\alpha$ beyond this regime, the middle bands recover real-valued spectra, and the corresponding midgap reopens with an enlarged separation.

%\emph{Topological Invariants:}
\subsection{Topological Invariants}

The one-dimensional non-Hermitian chiral-symmetric system: $\Gamma\mathcal{H}^{\dag}(k)\Gamma^{-1}=-\mathcal{H}(k)$ has its topological properties characterized by Chern number \cite{Hidden}. The Chern number is obtained for a $2D$ effective Hermitian Hamiltonian $\mathcal{H}_e(k,\varepsilon)=\Gamma[\varepsilon-i\mathcal{H}(k)]$, with $\Gamma=\sigma_0\otimes\sigma_z$ and $\varepsilon$ denotes the imaginary part of the energy. To overcome the periodicity of $\varepsilon$, we define $\mathcal{H'}_{e}(k,\varepsilon)=\mathcal{R}_{\delta}\mathcal{H}_{e}(k,\varepsilon)\mathcal{R}_{\varepsilon}^{\dag}$
 with $\mathcal{R}_{\varepsilon}=\exp[i\frac{\pi}{4}(1+\tanh\varepsilon)\mathcal{G}]$, where $\mathcal{G}=\sigma_x\otimes\sigma_0$. Thus, the Chern number is obtained as
 \begin{eqnarray}
 % \nonumber % Remove numbering (before each equation)
   C=\lim_{\varepsilon\rightarrow\infty}\frac{\varepsilon}{2\pi} \int_{-\pi}^{\pi}\Omega(k)dk,
 \end{eqnarray}
where
\begin{eqnarray*}
\lim_{\varepsilon\rightarrow\infty}\mathcal{H'}_{e}(k,\varepsilon)= \lim_{\varepsilon\rightarrow-\infty}\mathcal{H'}_{e}(k,\varepsilon)=|\varepsilon|\Gamma
\end{eqnarray*}
 due to $\mathcal{R}_{-\infty}=\sigma_0\otimes\sigma_0$, $\mathcal{R}_{-\infty}=i\mathcal{G}$
 and
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
\hspace{-0.75cm} \Omega(k)=\sum_{n\leq n_F,m>n_F}\hbox{Im}\bigg\{2\langle\psi^{nm}(k)|\mathcal{H'}_e(k)|\psi^{mn}(k)\rangle\nonumber\\ \times\frac{\langle\psi^{mn}(k)|\mathcal{H'}_e(k)|\psi^{nm}(k)\rangle}{\left(E_{nm}(k)-E_{mn}(k)\right)^2}\bigg\},
\end{eqnarray}
where $n_F$ is the number of occupied bands, $|\psi^{mn}(k)\rangle$ is eigenstate of $\mathcal{H'}(k)$ with eigenvalues $E_{mn}(k)$. The Chern number $C$ is an integer, and its value remains strictly constant under smooth perturbations that preserver the band gap separating the occupied and empty bands.




\section{Results and Discussion}{\label{3}}
\subsection{Transport coefficients}%{\label{3}}

We get the current operator continuity Equation $\mathcal{J}$ from $\partial\rho/\partial t+\nabla\cdot\mathcal{J}=\mathcal{J}^{1,2}_{N}$ where $\rho=a^{\dag}(\mathbf{x},t)a(\mathbf{x},t)$ and $\mathcal{J}^{1,2}_{N}$ is the contribution of the dissipative non-Hermitian term. We found
\begin{eqnarray}
%\partial_{t}\rho=\sum_{j}\mathcal{J}_{ij}$,\nonumber\\
\mathcal{J}_{il}=2i \sum_{j=1}^{N}\sum_{l=1}^{3}[t_n(a^{\dag}_{j,l}a_{j,l+1}-a^{\dag}_{j,l+1}a_{j,l})\nonumber\\+t_4(a^{\dag}_{j,l}a_{j+1,l}-a^{\dag}_{j+1,l}a_{j,l})]+\mathcal{J}^{1,2}_{N}%\nonumber\\
%\hspace{-0.0cm}+i\alpha\sum_{j=1}^{N}\left[\sum_{l=1}^{4}(-1)^{l+1}a^{\dag}_{j,1}a_{j,1}+\sum_{l=1}^{2}(a^{\dag}_{j,l}a_{j,l})-\sum_{l=3}^{4}(a^{\dag}_{j,l}a_{j,l})\right]\nonumber\\
%\hspace{-0.0cm}-i\alpha\sum_{j=1}^{N}\left[\sum_{l=1}^{2}(-1)^{l+1}a^{\dag}_{j,l}a_{j,l}-\sum_{l=3}^{4}(-1)^{l+1}a^{\dag}_{j,l}a_{j,l}\right],\nonumber\\
\end{eqnarray}
The longitudinal electric conductivity at ${q}=0$ in $x$ direction is given by \cite{kubo}
\begin{eqnarray}
\sigma(\omega)=-\frac{i}{N}\sum_{m,n=\pm}\sum_{{k}}\left[\frac{p_{nm}({k})-p_{mn}({k})}{E_{nm}({k})-E_{mn}({k})}\right]\nonumber\\ \times\frac{\langle{k},nm|\mathcal{J}({k})|{k},mn\rangle\langle{k},m|\mathcal{J}({k})|{k},nm\rangle}{E_{nm}({k})-E_{mn}({k})+\omega+i0^{+}},
\end{eqnarray}
where $E_{nm}({k})$ $n,m=\pm$ are the energies of the $n,m\hbox{th}$ bands and $p(x)=(e^{x}+1)^{-1}$, with $x=E_{nm}({k})/k_BT$ is the Fermi-Dirac distribution function. $\mathcal{J}({k})$, is the electric current operator, at point ${k}$ into the first Brillouin zone given by \cite{Han}. From above formula, we get the real part of the dynamical conductivity given by \cite{Mahan,Wen}:
\begin{equation}
\hbox{Re} \sigma(\omega)=D_S(T)\delta(\omega)+\sigma^{reg}(\omega),
\end{equation}
where
\begin{eqnarray}\label{sigmareg}
%\langle\mathcal{J}_{\alpha}({k},\omega)\rangle=\sum_{\beta}\sigma_{\alpha\beta}({k},\omega)ik_{\alpha} h_{\beta}({k},\omega),\nonumber\\
%\sigma_{\alpha\beta}({k},\omega)=\hbox{Re}[\sigma_{\alpha\beta}({k},\omega)]+i\hbox{Im}[\sigma_{\alpha\beta}({k},\omega)]\nonumber\\
%\mathcal{J}({k},\omega)=\sigma({k},\omega) E({k},\omega),\nonumber\\
\hspace{-1.9cm}{D_S(T)}=\frac{\pi}{N}\sum_{n,m=\pm}\sum_{{k}}\frac{\partial p_{nm}({k})}{\partial E_{nm}({k})}\langle{k},nm|\mathcal{J}({k})|{k},nm\rangle\langle{k},nm|\mathcal{J}({k})|{k},nm\rangle,\nonumber\\
\hspace{-1.9cm}\sigma^{reg}(\omega)=-\frac{1}{N}\sum_{n,m=\pm}\sum_{{k}}\left[\frac{p_{nm}({k})-p_{mn}({k})}{E_{nm}({k})-E_{mn}({k})}\right]\nonumber\\ \hspace{-0.8cm}\times\langle{k},nm|\mathcal{J}({k})|{k},mn\rangle\langle{k},mn|\mathcal{J}({k})|{k},n\rangle%\nonumber\\ \times
\delta\left(\omega-E_{nm}({k})-E_{mn}({k})\right),\nonumber\\
\end{eqnarray}
where
\begin{eqnarray}\label{sigmareg2}
\hspace{-1.0cm} D_S(T)=\frac{\pi}{N}\sum_{n,m=\pm}\sum_{{k}}\bigg\{\left(t_1+t_2+t_3-t_4\right)^2\sin(k)\nonumber\\
\times\tanh\left(\frac{E_{nm}({k})-E_{mn}({k})}{2k_BT}\right)\nonumber\\
\hspace{-1.0cm}-\left(t_1+t_2+t_3-t_4\right)^2\cos^2(k)\left[\frac{p_{nm}({k})-p_{mn}({k})}{E_{nm}({k})-E_{mn}({k})}\right]\nonumber\\
\hspace{-1.0cm}\times\left[1-e^{(E_{nm}({k})-E_{mn}({k}))/k_BT}\right]\mathcal{P}\left(\frac{1}{E_{nm}({k})-E_{mn}({k})+\omega}\right),\nonumber\\
\hspace{-1.0cm}\sigma^{reg}(\omega)=\sum_{n,m=\pm}\sum_{{{k}}}\left(t_1+t_2+t_3-t_4\right)^2\cos^2(k)\nonumber\\ \hspace{-1cm}\times\left[\frac{p_{nm}({k})-p_{mn}({k})}{E_{nm}({k})-E_{mn}({k})}\right] \left[1-e^{(E_{nm}({k})-E_{mn}({k})))/k_BT}\right]\nonumber\\ \times\delta\left(E_{nm}({k})-E_{mn}({k})+\omega\right)\nonumber\\
\end{eqnarray}
%\begin{eqnarray}\label{sigma}
%\hspace{-1cm}\hbox{Re} \sigma(\omega)=\sum_{n,m=\pm}\sum_{{{k}}}\left(t_1+t_2+t_3-t_4\right)^2\cos^2(k)\nonumber\\ \hspace{-1cm}\times\left[\frac{p_{nm}({k})-p_{mn}({k})}{E_{nm}({k})-E_{mn}({k})}\right] \left[1-e^{(E_{nm}({k})-E_{mn}({k})))/k_BT}\right]\nonumber\\ \times\delta\left(\omega-E_{nm}({k})-E_{mn}({k})\right)\nonumber\\
%\hspace{-1cm}+\frac{\pi}{N}\sum_{n,m=\pm}\sum_{{k}}\bigg\{\left(t_1+t_2+t_3-t_4\right)^2\sin(k)\nonumber\\
%\times\tanh\left(\frac{E_{nm}({k})-E_{mn}({k})}{2k_BT}\right)\nonumber\\
%\hspace{-1cm}-\left(t_1+t_2+t_3-t_4\right)^2\cos^2(k)\left[p_{nm}({k})-p_{mn}({k})\right]\nonumber\\ \times\left[1-e^{(E_{nm}({k})-E_{mn}({k}))/k_BT}\right]\nonumber\\ \times
%\mathcal{P}\left(\frac{1}{\omega-E_{nm}({k})-E_{mn}({k})}\right)\bigg\}\delta(\omega),\nonumber\\
%\end{eqnarray}
where $\mathcal{P}(\ldots)$ means principal value.


The Kubo formula is extended to non-Hermitian systems,
with the conductivity $\sigma(\omega)$ depending on the current operator $\mathcal{J}({k})$ with the modifications made to account for the complex energy spectrum and
biorthogonal basis, and provide a validation against the known Hermitian case to ensure accuracy. The differences are: Biorthogonal matrix elements, where both numerators involve left–right matrix elements; no complex conjugation appears by default. The Complex poles, where denominators carry complex transition energies; their imaginary parts regularize the $\omega\rightarrow 0$ behavior and encode gain/loss-induced broadening. The causality, where the retarded prescription $+i0^{+}$ is kept, ensuring the same analyticity and Kramers–Kronig structure for the retarded correlator and stability required.
%

{The key reasons the Kubo formula survives in many non-Hermitian settings are as following: In first, if the non-Hermitian Hamiltonian $\mathcal{H}_{\hbox{eff}}$ is used to describe the system, and a pseudo-equilibrium or steady state is reached via coupling to a reservoir, the system may still exhibit equilibrium-like features. The steady state can resemble a thermal Gibbs-like state in a biorthogonal ensemble. Moreover, in open quantum systems, fluctuation and response functions can be defined using Keldysh field theory. Even in the presence of non-Hermiticity, Kubo-type linear response functions can be formulated, and the fluctuation–dissipation relation can be preserved if the dynamics equilibrate and satisfy stationarity and causality in the effective description. In some non-Hermitian systems, effective unitarity can emerge in the unbroken-symmetric phase, allowing fluctuation–dissipation relation to hold within certain parameter regimes \cite{Lei,kevin}.}
\noindent
\begin{figure}
 \centering
\includegraphics[width=8.5cm]{non-herm_cond_super2.eps} \\
\caption{Plots (a) and (b) $D_S$ vs. $T$ for $\alpha=0.5$ and $D_S$ vs. $\alpha$ for $T=0.01$ for the Hamiltonian $\mathcal{H}_1$. Plots (c) and (d) $D_S$ vs. $T$ for $\alpha=0.5$ and $D_S$ vs. $\alpha$ for $T=0.01$ for the Hamiltonian $\mathcal{H}_2$.}\label{fig_5}
\end{figure}
\noindent

%In Fig.~\ref{fig_5}, we show the behavior of $D_S(T)$ as a function of $T$ and $\alpha$. As $T \rightarrow 0$, $D_{S}$ is finite, exhibiting a pronounced divergence of DC conductivity at $T=0$. However, the presence of disorder or at finite $T$, the delta peak $\delta(\omega)$ in $\hbox{Re} \sigma(\omega)$ is broadened into a Lorentzian, resulting in a finite conductivity. As discussed in Ref.~\cite{scalapino}, the Drude weight $D_S(T)$ is defined as $D_S(T)=\langle -K\rangle-\Lambda({k}=0,\omega)$, where  $\langle -K\rangle$ is the kinetic energy of the particles and $\Lambda({k},\omega)$ denotes the current–current correlation function. In the limit $\omega\rightarrow 0$, $\hbox{Re}\sigma(\omega)$ contains a delta-function contribution $D_S\delta(\omega)$, implying a zero-resistance state. On the other hand, the Meissner effect corresponds to the current response to a static $\omega=0$ and transverse gauge vector potential $\vec{A}({k},\omega=0)=0$. In the small-${k}$ limit, one finds  $D_S/\pi=D/\pi=\langle -K\rangle-\Lambda(k\rightarrow 0,i\omega_m=0)$, where $i\omega_m=\omega+i0^{+}$ and $D$ is the superfluid stiffness, which is finite in a superfluid state. Physically, $D_S$ measures the ratio of the density of mobile charge carriers to their mass, while $D$ measures the ratio of the superfluid density to the mass. The difference between $D_S$ and $D$ lies in the order in which the limits $k\rightarrow 0$ and $i\omega_n\rightarrow 0$ are taken. At zero temperature and in the absence of disorder, the nature of the ground state is determined by the asymptotic behavior of $D_S$ and $D$ as the system size approaches the continuum limit. In this limit, both $D_S$ and $D$ are finite for a superconductor, $D_S$ remains finite while $D=0$  for a metal, and both $D_S=D =0$ for an insulator.
In Fig.~\ref{fig_5}, we show the Drude weight $D_S(T)$ as a function of temperature $T$ and non-Hermiticity $\alpha$. In the limit $T\rightarrow 0$, $D_S$ remains finite, implying a divergent DC conductivity. At finite temperature or in the presence of disorder, the delta peak $\delta(\omega)$ in $\hbox{Re}\sigma(\omega)$ broadens into a Lorentzian, yielding a finite conductivity. Following Ref.~\cite{scalapino}, the Drude weight is defined as $D_S(T)-\langle K\rangle-\Lambda(k=0,\omega)$, where $\langle-K\rangle$ is the kinetic energy and $\Lambda(k,\omega)$ the current–current correlation function. In the limit $\omega\rightarrow 0$, $\hbox{Re}\sigma(\omega)$ contains a singular term $D_S\delta(\omega)$, signaling dissipationless transport. The Meissner response instead probes the static $(\omega=0)$  transverse limit. For $k\rightarrow 0$, the superfluid stiffness is $D=\pi\left[\langle-K\rangle-\Lambda(k\rightarrow 0, i\omega_n=0)\right]$, with $i\omega_n=\omega+i0^{+}$. The distinction between $D_S$ and $D$ arises from the non-commutativity of the limits $k\rightarrow 0$ and $\omega\rightarrow 0$.  Physically, $D_S$  measures the charge stiffness, while $D$ quantifies the superfluid density.
 In the thermodynamic limit at $T=0$  and without disorder, these quantities characterize the ground state:  $D_S,D\ne 0$ for a superconductor; $D_S\ne 0,D=0$ for a metal; and
$D_S=D=0$ for an insulator.


\subsection{Quantum correlations and entanglement}%{\label{6}}

 Entanglement negativity $E_N$  is defined as the logarithm of the trace norm of the partial transpose of the density matrix of the system $\rho_A$. Overall, the entanglement negativity is a powerful tool for characterizing the entanglement properties of quantum systems, and its behavior can provide important insights into the fundamental properties of quantum mechanics and potential applications of quantum information science.
However, $E_N$ presents a large deficiency i.e. a failure in satisfying the discriminant property, either that the entanglement $E_N(\rho)=0$ if and only if $\rho$ is separable \cite{K}. It is used much often as a measure of thermal entanglement for disjoint intervals. %Consequently, the negativity has been proven to be useful to detect topological order,\cite{C,Y}.
 $E_N$ is given for a mixed state $\rho_{GE}$ by \cite{Plenio,Giles}
\begin{eqnarray}{\label{en1}}
% \nonumber % Remove numbering (before each equation)
  {N}(\rho) = \frac{\|\rho_A^{T}\|_1-1}{2},
\end{eqnarray}
where  $\|X\|_1=\hbox{Tr}|X|=\hbox{Tr}\sqrt{X^{\dag}X}$ is the trace norm and $\rho_A^{T}$ is the partial transpose of $\rho_A$ with respect to the subsystem $A$. The logarithmic negativity is \cite{Plenio}
\begin{equation}{\label{en2}}
E_{N}(\rho)=\log_2\|\rho_A^{T}\|_1.
\end{equation}
%


 In general, the system will thermalise when the Gibbs distribution $\rho$, is given by $\rho\propto e^{-\mathcal{H}/k_BT}$, where the statistical ensemble describing the system for long time is expected to be the canonical ensemble, being the density matrix of the canonical ensemble given by \cite{Calabrense,Davide,Calabrense2}.

The quantum state evolves in time under the non-Hermitian Hamiltonian $\mathcal{H}(t)$ as the model Eq.~(\ref{model}), according the von Neumann equation
\begin{equation}\label{von}
  i\hbar\frac{\partial \rho(t)}{\partial t}=\left[\mathcal{H}(0),\rho\right]+\left\{\mathcal{H}_1(t),\rho\right\}%\mathcal{H}(t)\rho(t)-\rho(t)
\end{equation}
where $\rho(0)=\rho_0$, $\mathcal{H}_1(t)=-if(t)A(t)$, with $A(t)=e^{i\mathcal{H}_0t/\hbar}Ae^{-i\mathcal{H}_0t/\hbar}$. $f(t)$ is a non-negative time-dependent function for instance, a rectangular pulse of strength $s$ and duration $\delta t$, $f(t)=\hbar s\left[\theta(t-t_w)-\theta(t-t_w-\delta t)\right]/\delta t$ ($\theta(x)$ is the step function) at fixed waiting time $t_w$. To linear order in the perturbation theory, we can write
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \rho(t) =\rho_0-\hbar^{-1}\int_{0}^{t}dt'\left\{A(t),\rho_0\right\}f(t')
\end{eqnarray}
where one gets
\begin{equation}
\hbox{Tr}\left[\rho(t)\right]=1-2\hbar^{-1}\int_{0}^{t}dt'\langle 0|A(t')|0\rangle f(t').
\end{equation}

We say that the system relaxes locally if the limit $\lim_{t\rightarrow\infty}\lim_{N\rightarrow\infty}\rho_A(t)=\rho_A(\infty)$ exists for any finite subsystem $A$. Its stationary state is defined as $\lim_{N\rightarrow\infty}\hbox{Tr}_B(\rho)=\rho_A(\infty)$, where $B$ is the complement of $A$. In the absence of other conserved quantities isolated systems thermalize to thermal equilibrium
\begin{eqnarray*}
\rho=\rho^{Gibbs}=[\hbox{Tr}(e^{-\beta\mathcal{H}})]^{-1}e^{-\beta\mathcal{H}},
\end{eqnarray*}
 where $\beta$ is fixed by the initial value of the energy density
\begin{eqnarray*}
h\equiv \lim_{N\rightarrow\infty}\frac{1}{N}\hbox{Tr}(\rho^{Gibbs}\mathcal{H}).
\end{eqnarray*}

Thus, we get $\rho$ as
\begin{eqnarray}
 \nonumber % Remove numbering (before each equation)
\rho=Z^{-1}e^{-\frac{1}{k_BT}\sum_{n=\pm,m=\pm}\sum_{{k}}E_{nm}({k})a^{\dag}_{{k}}a_{{k}}},
\end{eqnarray}
where ${Z}$ is the partition function. The entanglement negativity is given by
%Thus, we obtain the entanglement negativity given by
\begin{eqnarray}{\label{EN}}
E_{{N}}=-\log_2\left(Z^{-1}e^{-\frac{1}{k_BT}\sum_{n=\pm,m=\pm}\sum_{{k}}E_{nm}({k}){p}(E_{nm}({k}))}\right).\nonumber\\
\end{eqnarray}
%where $Z$ is the canonical partition function. %, $p(x)$ is the Fermi-Dirac distribution function $p(x)=(e^{x}+1)^{-1}$, and $x=E_{\pm,\pm}({k})/k_BT$, where we consider $k_B=1$ in all calculations.
We assume a thermal state associated with an effective Hamiltonian, for which the resulting density matrix is mathematically well defined and physically motivated through a connection to an effective equilibrium description. An alternative and more general approach for a non-Hermitian Hamiltonian such as Eq.~(\ref{model}) is to explicitly determine the steady-state density matrix by solving a Lindblad master equation, thereby modeling the coupling to the environment. In this framework, the steady state $\rho_{\mathrm{ss}}$ can be obtained either from the long-time limit of the non-unitary dynamics or within a biorthogonal formalism, allowing the entanglement negativity to be evaluated for a genuinely non-thermal state.

The bipartition of the system plays a crucial role in the interpretation of the entanglement negativity. A common choice is to divide the chain at symmetry-related locations or at regions of particular physical relevance, such as the center of the system, where topological edge or interface states may emerge in certain phases. Alternatively, one can systematically shift the cut along the chain and monitor the spatial dependence of $E_N$, which provides insight into how quantum correlations are distributed across the lattice.

More generally, the choice of bipartition is not merely a geometric consideration but a diagnostic tool tailored to the specific topological features under investigation. Depending on the location of the partition, $E_N$ can probe distinct aspects of the underlying topological phase, including edge states, bulk entanglement patterns, and the influence of non-Hermitian effects. By judiciously selecting the bipartition, one can therefore access information about topological invariants and uncover how non-Hermiticity reshapes the entanglement structure in comparison to the Hermitian limit.

Typically, one might choose the bipartition of the chain at symmetry points or regions of interest, such as at the center of the chain, where topological edge states are expected to reside in certain phases. Alternatively, one might study how $E_N$ changes as the cut is moved across different sites, which can provide insight into how entanglement is distributed throughout the system.

{We analyzed quantum correlations via entanglement negativity in a subsystem of ${N}$ spins block, considering the partition with ${N}$ spins for which the local entropy is calculated and in following making ${N} \rightarrow\infty$ in the partition, for the continuum theory is valid.} Moreover, {We assume a thermal state for a effective Hermitian Hamiltonian correspondent to the non-Hermitian model Eq.~(\ref{model}), the result thus is mathematically defined and physically justified by a connection to an effective equilibrium. However, an another approach for a non-Hermitian Hamiltonian as Eq.~(\ref{model}), is to compute the true steady-state density matrix from the Lindblad equation (if we model the open system environment explicitly), either from the long-time limit of non-unitary evolution
or from a biorthogonal formalism, where we compute the entanglement negativity from the non-thermal $\rho_{ss}$}.

%
\begin{figure}
 \centering
\includegraphics[width=8.5cm]{non-herm_ent_super2.eps} \\
%\includegraphics[width=7.0cm]{non-herm_ent_superENvsa2.eps} \\
\caption{Plots (a) and (b) $E_N$ vs. $T$ for $\alpha=0.5$ and $E_N$ vs. $\alpha$ for $T=0.01$ for the Hamiltonian $\mathcal{H}_1$. Plots (c) and (d) $E_N$ vs. $T$ for $\alpha=0.5$ and $E_N$ vs. $\alpha$ for $T=0.01$ for the Hamiltonian $\mathcal{H}_2$.}\label{fig_7}
\end{figure}
\noindent
%

In Fig.~\ref{fig_7}, we present the temperature dependence of the entanglement negativity $E_{N}$. As $T \rightarrow 0$, $E_{N}$ exhibits a pronounced divergence, which can be attributed to the strong enhancement of quantum fluctuations in the zero-temperature limit. In addition, we observe that increasing the non-Hermitian parameter leads to only minor quantitative changes in the $E_{N}$ curves. This weak sensitivity reflects the fact that the overall symmetry of the energy spectrum undergoes an abrupt transition from being predominantly real to predominantly imaginary, without substantially altering the low-temperature scaling behavior of the entanglement.


% \section{Results and discussion}{\label{4}}
 \subsection{Discussion}%{\label{4}}


\emph{Observable signatures of spectral splitting:} When spectral splitting occurs along the real-energy axis, the system exhibits two distinct resonant frequencies, resulting in a characteristic double-peak structure in the transmission or optical response as a function of frequency. In contrast, splitting along the imaginary-energy axis leads to a single resonance, accompanied by asymmetric linewidths and pronounced variations in amplitude or effective gain. This behavior can be quantitatively identified through linewidth analysis or by examining growth–decay dynamics. Concomitantly, the corresponding eigenstate profiles undergo a qualitative transformation, evolving from spatially extended modes to boundary-localized states. Such edge localization can be directly probed via near-field imaging techniques in photonic platforms or through site-resolved occupation measurements in cold-atom systems.
 Importantly, the emergence of non-Bloch Dirac points is reflected in abrupt variations of the Drude weight, while discontinuities or sharp features in the regular conductivity  provide clear signatures of these spectral transitions.

\emph{Identification of conductivity resonances:} The position of non-Bloch Dirac points relative to the surrounding energy gaps plays a decisive role in determining the resonant frequencies associated with interband optical transitions. When a non-Bloch Dirac point is situated deep within a real line-gapped phase, the optical gap remains well defined, giving rise to sharp and well-resolved peaks in the optical conductivity. Conversely, as the Dirac point approaches the boundary of a complex-energy region, the effective gap narrows or closes, leading to significant broadening of the conductivity peaks due to the finite imaginary components of the eigenvalues. This regime is typically accompanied by an enhanced low-frequency response.
 Within this framework, the frequency window ($\omega-\omega_0$) serves as a direct probe of transitions relative to the lowest excitation threshold set by the non-Bloch gap. As the Dirac points move closer to the gap edges, the threshold frequency decreases, and the corresponding conductivity features become increasingly pronounced.

\emph{Numerical assessment of robustness and signal-to-noise:} Although both $D_S$ and $\sigma^{reg}(\omega)$ are derived analytically, their robustness can be quantitatively validated through numerical simulations. To this end, we introduce weak random perturbations of small amplitude to the Hamiltonian matrix elements and repeat the transport calculations over a large ensemble of disorder realizations. The resulting noise level is quantified by the standard deviation
 of the corresponding observable. In addition, we examine the sensitivity of the results to the choice of the infinitesimal imaginary regulator ($i0_{+}$) entering the response functions. Finally, by comparing results obtained using thermal occupation factors at low temperature with those based on step-function occupations, we assess the stability of
 $D_S$  and $\sigma^{reg}(\omega)$ with respect to both numerical and physical regularization schemes.


\subsection{Linking $D_S$ and  $E_N$}


\begin{figure}
    \centering
\includegraphics[width=5.5cm]{herm_ent_super_DsvsEN.eps} \\
\caption{$D_S$ vs $E_N$, at low temperature: $T=0.01J$ held fixed.}\label{fig_38}
\end{figure}


From Eqs.~(\ref{sigmareg2}) and (\ref{EN}), we found the link between $D_S$ and $E_N$  in the following way
\begin{eqnarray}{\label{inter}}
\hspace{-1.0cm}D_S(T)=\frac{\pi}{N}\sum_{{k}}\left(t_1+t_2+t_3-t_4\right)\sin k\left(\frac{e^{E_N/2}-Ze^{-E_N/2}}{e^{E_N/2}+Ze^{-E_N/2}}\right)\nonumber\\
\hspace{-1.0cm}-\left(t_1+t_2+t_3-t_4\right)^2\left[\frac{\cos^2(k)}{k_BT(\log_2Z-E_N)}\right]\left[1-\frac{Ze^{-E_N}-1}{Z^{-1}e^{E_N}-1}\right]\nonumber\\ \times
\mathcal{P}\left(\frac{1}{k_BT(\log_2Z-E_N)+\omega}\right),\nonumber\\
\end{eqnarray}
{In Fig.~\ref{fig_38}, we plot the Drude weight $D_S$ vs. $E_N$ as a way to clarify the relationship between entanglement negativity and DC conductivity. We found that the Drude weight of the electric conductivity remains finite even in the limit where $E_N\rightarrow 0$, i.e., when quantum correlations are absent. As $E_N$ increases, the Drude weight exhibits a slight decrease, demonstrating that there is slight dependence of the DC conductivity vs $E_N$. Physically, this behavior indicates that stronger quantum correlations tend to inhibit coherent charge transport, while regimes with reduced entanglement are more favorable to metallic conduction.}


%\section{Conclusions and Outlook}{\label{5}}
\section{Summary and Outlook}{\label{5}}

We analyze the link between DC transport and quantum correlations given by entanglement negativity in one-dimensional superlattices.

The Skin effect further amplifies boundary disorder, making robust observation challenging. Moreover, measurement of transport are sensitive to complex-frequency conductivity e.g., THz spectroscopy in photonics, or AC impedance in topolectrical circuits. Direct observation of sixth-order scaling in condensed matter physics is likely impractical. Synthetic platforms are more realistic.

From a broader perspective, studying the AC and DC conductivities of the SSH model is essential for understanding the low-temperature transport properties of topological insulators, especially when non-Hermitian effects are present. Our results highlight how non-Hermiticity fundamentally alters the transport response, revealing new routes to engineer and control electronic properties in low-dimensional topological systems.

\appendix{\bf Corresponding author:}\\
 *E-mail: lslima@cefetmg.br.

\appendix{\bf Funding Declaration:}\\
 This research has received none funding. 
 
\appendix{\bf Data availability:}\\ 
All data generated or analysed during this study are included in this manuscript.

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\end{thebibliography}
%
\end{document}
















Due to quadratic form of model Eq.~(\ref{model}) and the Gaussian form of the ground state, the reduced density matrix can be written as \cite{Chang,Fossati}
\begin{equation}{\label{dens}}
  \rho_A=\frac{1}{Z_A}\prod_{i,j\in A}e^{\left(c^{\dag}_{i}K^A_{ij}c_j\right)},
\end{equation}
where $K^{A}_{ij}$ is the kernel entanglement Hamiltonian that in certain integrable models can be written in terms of the density of the lattice Hamiltonian $h_{ij}$ with a linearly increasing local temperature \cite{Pasquale}
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  K^A\propto\sum_{i\ne j}K^A_{ij},\hspace{0.75cm}K^A_{ij}={ij}h_{{ij}}
\end{eqnarray}
with a non-trivial proportionality constant.













\emph{Topological Invariants:} The one-dimensional non-Hermitian chiral-symmetric system: $\Gamma\mathcal{H}^{\dag}(k)\Gamma^{-1}=-\mathcal{H}(k)$ has its topological properties characterized by Chern number \cite{Hidden}. The Chern number is obtained for a $2D$ effective Hermitian Hamiltonian $\mathcal{H}_e(k,\delta)=\Gamma[\delta,i\mathcal{H}(k)]$, where $\delta$ denotes the imaginary part of the energy. To overcome the periodicity of $\delta$, we define $\mathcal{H'}_{e}(k,\delta)=\mathcal{R}_{\delta}\mathcal{H}_{e}(k,\delta)\mathcal{R}_{\delta}^{\dag}$
 with $\mathcal{R}_{\delta}=\exp[i\frac{\pi}{4}(1+\tanh\delta)\mathcal{G}]$, where $\mathcal{G}=\sigma_x\otimes\sigma_0$. Thus, the Chern number is obtained as
 \begin{eqnarray}
 % \nonumber % Remove numbering (before each equation)
   C=\lim_{\delta\rightarrow\infty}\frac{\delta}{2\pi} \int_{-\pi}^{\pi}\Omega(k)dk,
 \end{eqnarray}
where
\begin{eqnarray*}
\lim_{\delta\rightarrow\infty}\mathcal{H'}_{e}(k,\delta)= \lim_{\delta\rightarrow-\infty}\mathcal{H'}_{e}(k,\delta)=|\delta|\Gamma
\end{eqnarray*}
 due to $\mathcal{R}_{-\infty}=\sigma_0\otimes\sigma_0$, $\mathcal{R}_{-\infty}=i\mathcal{G}$
 and
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
\hspace{-0.75cm} \Omega(k)=\sum_{n\leq n_F,m>n_F}\hbox{Im}\bigg\{2\langle\psi^{nm}(k)|\mathcal{H'}_e(k)|\psi^{mn}(k)\rangle\nonumber\\ \times\frac{\langle\psi^{mn}(k)|\mathcal{H'}_e(k)|\psi^{nm}(k)\rangle}{\left(E_{nm}(k)-E_{mn}(k)\right)^2}\bigg\},\nonumber\\
\end{eqnarray}
where $n_F$ is the number of occupied bands, $|\psi^{mn}(k)\rangle$ is eigenstate of $\mathcal{H'}(k)$ with eigenvalues $E_{mn}(k)$. The Chern number $C$ is an integer, and its value remains strictly constant under smooth perturbations that preserver the band gap separating the occupied and empty bands.




Experimental challenges: To suppress first–fifth order terms and isolate the sixth-order crossing requires sub-percent precision in hopping amplitude ratios $t_x, t_y, t_z$, which is very hard in solid-state but plausible in synthetic lattices  such as photonic crystals, cold atoms, topolectrical circuits.  The Skin effect further amplifies boundary disorder, making robust observation challenging. Moreover, measurement of transport are sensitive to complex-frequency conductivity e.g., THz spectroscopy in photonics, or AC impedance in topolectrical circuits. Direct observation of sixth-order scaling in condensed matter physics is likely impractical. Synthetic platforms are more realistic.

From a broader perspective, studying the AC and DC conductivities of the SSH model is essential for understanding the low-temperature transport properties of topological insulators, especially when non-Hermitian effects are present. Our results highlight how non-Hermiticity fundamentally alters the transport response, revealing new routes to engineer and control electronic properties in low-dimensional topological systems.










In recent years, non-Hermitian quantum systems have emerged as a rich platform for exploring novel topological phases and unconventional transport phenomena beyond the traditional Hermitian paradigm \cite{Bergholtz,Ashida,Gong,Wang,Gardiner,HP,Lei,N,AH,kevin,prl,LJ,yan,leonardoslima,lslimasci2,lslimapl,physlett,physe} Unlike Hermitian systems, where energy eigenvalues are guaranteed to be real and the bulk-boundary correspondence directly connects bulk invariants to boundary modes, non-Hermitian systems often feature complex spectra and exhibit striking effects such as the non-Hermitian skin effect (NHSE) \cite{Yao,Kunst,Helbig,Weidemann}, where an extensive number of eigenstates become localized at system boundaries under open boundary conditions (OBC). These features fundamentally challenge and enrich our understanding of band topology and transport in quantum materials.

Among the most studied models in topological condensed matter physics is the Su–Schrieffer–Heeger (SSH) model, which describes a one-dimensional dimerized chain that supports topologically protected edge states under suitable conditions \cite{Su}. Extending this model to two dimensions by stacking SSH chains introduces additional degrees of freedom, enabling exploration of higher-dimensional topological phases and Dirac-like features in the band structure \cite{Benalcazar,Lin}. Incorporating nonreciprocal hopping (i.e., asymmetric hopping amplitudes between sites) and on-site gain and loss further drives the system into the realm of non-Hermitian physics, where unique topological and transport properties arise \cite{Lee,Kiong,Longhi}.

A central question concerns how the critical properties of the system, particularly those associated with the closing and reopening of complex energy gaps under OBC influence measurable transport coefficients such as the electrical conductivity. In Hermitian systems, gap closings in the bulk spectrum typically signal topological phase transitions and can strongly affect transport. In non-Hermitian systems, however, the relevant spectral features may appear only in the open-boundary spectrum and disappear under periodic boundary conditions (PBC), owing to the breakdown of conventional bulk-boundary correspondence \cite{Kunst,Okuma,Zhang,Kawabata}.

In this work, we investigate electrical transport in on-Hermitian superlattices. To this end, we compute the transport coefficients using  the Kubo-Greenwood formula  and examine how the non-Hermitian parameters, such as the strength of nonreciprocity and gain/loss, modify the AC and DC conductivities \cite{Kawabata,Yokomizo,Ahang,Lee2}. The behavior of $E_N$ vs Drude weight $D_S$ which gives a measure of DC transport is analyzed.

















Our findings reveal that the emergence of non-Bloch Dirac points under OBC leads to substantial enhancement or suppression of conductivity, depending on system parameters. This highlights the intricate interplay between topology, non-Hermiticity, and boundary effects in determining the transport behavior. Beyond fundamental interest, these results suggest potential avenues for engineering quantum devices where electrical transport can be tuned via non-Hermitian effects, with possible realizations in photonic lattices \cite{Song,Pan}, electric circuits \cite{Hofmann,Helbig}, and cold atomic systems \cite{Ki}.




  In Fig.~\ref{fig_6}, we display the behavior of $D_S(T)$ as a function of $T$ for different values of non-Hermitian coupling $\eta$, into the range $\eta=\sqrt{1+\xi}$, with $-1<\xi<0$, where the system presents the non-Bloch Dirac points in the band structure. We consider $(r,c,\gamma,w,\eta)=(\frac{7}{8},\frac{1}{4},\frac{1}{4},1,\eta)$ such that the system goes from phase purely real $(R3)$ to real-line-gapped $(R4)$.

 \emph{Although we obtain a finite $D_S(T)$ for all temperatures, in the presence of disorder or at finite $T$, the delta peak $\delta(\omega)$ in $\hbox{Re} \sigma(\omega)$ is broadened into a Lorentzian, resulting in a finite conductivity. As discussed in Ref.~\cite{scalapino}, the Drude weight $D_S(T)$ is defined as $D_S(T)=\langle -K\rangle-\Lambda(\mathbf{k}=0,\omega)$, where  $\langle -K\rangle$ is the kinetic energy of the particles and $\Lambda(\mathbf{k},\omega)$ denotes the current–current correlation function. In the limit $\omega\rightarrow 0$, $\hbox{Re}\sigma(\omega)$ contains a delta-function contribution $D_S\delta(\omega)$, implying a zero-resistance state. On the other hand, the Meissner effect corresponds to the current response to a static $\omega=0$ and transverse gauge vector potential $\vec{A}(\mathbf{k},\omega=0)=0$. In the small-$\mathbf{k}$ limit, one finds  $D_S/\pi=D/\pi=\langle -K\rangle-\Lambda(k_x=0,k_y\rightarrow 0,i\omega_m=0)$, where $i\omega_m=\omega+i0^{+}$ and $D$ is the superfluid stiffness, which is finite in a superfluid state. Physically, $D_S$ measures the ratio of the density of mobile charge carriers to their mass, while $D$ measures the ratio of the superfluid density to the mass. The difference between $D_S$ and $D$ lies in the order in which the limits $k_y\rightarrow 0$ and $i\omega_n\rightarrow 0$ are taken. At zero temperature and in the absence of disorder, the nature of the ground state is determined by the asymptotic behavior of $D_S$ and $D$ as the system size approaches the continuum limit. In this limit, both $D_S$ and $D$ are finite for a superconductor, $D_S$ remains finite while $D=0$  for a metal, and both $D_S=D =0$ for an insulator.}

 We get a growing in the damping of the curves of $D_S$  with increasing of the strength of $\eta>0$, which tends to zero at high-temperature limit. In addition, we have a suave changing in the conductivity of the system with $\sin\phi$, where $\zeta_n^2$ reaches a maximum and minimum in $\sin\phi=1$ and $\sin\phi=-1$, respectively and $\eta=\sqrt{1+\xi}$ with $-1<\xi<0$.

 \emph{In Fig.~\ref{fig_5}, we show the behavior of $\sigma^{reg}(\omega)$ as a function of $\omega$ at range $\eta=\sqrt{1+\xi}$ with $-1<\xi<0$, where we consider a low-temperature value in the calculations such as $T=0.01J$. We take $(r,c,\gamma,w,\eta)=(\frac{7}{8},\frac{1}{4},\frac{1}{4},1,\eta)$, such that the system goes from phase purely real ($R3$) to real-line-gapped ($R4$). We varies the $\eta$ parameter into the region $\eta=\sqrt{1+\xi}$ with $-1<\xi<0$, where the system presents the non-Bloch Dirac points in the band structure. We get a divergence in $\sigma^{reg}(\omega)$  at $\omega\rightarrow 0$ (DC limit) for all cases analyzed however, this divergence of  $\sigma^{reg}(\omega)$ and a positive Drude weight obtained inf Fig.~\ref{fig_6} for all $T$ values cannot be associated to a superconducting state since the system does not present neither a pairing term nor interactions. Moreover, it is necessary the system to exhibit the Meissner effect as well.  The interaction among electrons and electron with impurities of the lattice make the free mean path becomes finite, smearing out the peak of the Dirac's delta function of the conductivity where it must stay finite when these effects are included}.  Moreover, as obtained for the Drude peak, we get a variation in $\sigma^{reg}(\omega)$ with changing of the parameter $\eta$, where a second peak emerges in $\sigma^{reg}(\omega)$ for $\xi=\sqrt{0.9}\simeq 0.94(9)$, close to the $\xi=0$ point (when $(1-\sqrt{\xi})^2-\eta^2=0$), where a phase transition from imaginary-line-gapped ($R1$) to gapless phase ($R5$) takes place, being the phase diagram symmetric at point $(\xi,\eta)=(0.0,1.0)$ and the entire spectrum changing abruptly from real to imaginary.

 In Figs.~\ref{fig_6} and \ref{fig_5}, the low-frequency transport is dominated by states near non-Bloch Dirac points (NBDPs) in the open-boundary spectrum, proximity to gap closing. When $\eta$ drives the system through the gapless point (here $\eta=0$ is the Hermitian/non-Hermitian gapless condition), more low-energy channels and Dirac crossings appear near the Fermi reference energy and the lifetime/linewidth ($\hbox{Im}\zeta=\Delta$) of the NBDP modes. Smaller lifetime or linewidth $\Delta$ (modes closer to real axis) produce sharper, longer-lived resonances and therefore much larger low-frequency conductivity (the denominators in Kubo formulas contain $(\zeta_{n}(\mathbf{k})-\zeta_{m}(\mathbf{k}))$ and pick up factors $\sim 1/\Delta$ when $\omega$ is small). The non-Hermitian skin effect concentrates current-carrying weight to boundaries; when NBDPs are boundary-localized, their matrix elements of the current operator are large and enhance both the Drude stiffness $D_S$ and $\sigma^{reg}$. The Conditions for sixth-order non-Bloch Dirac points means that the effective low-energy dispersion scales as $\zeta(\mathbf{k})\sim v{k}^6$, where $v$ is the velocity of excitations, rather than linear $({k})$, quadratic $(k^2)$, or quartic $(k^4)$ as in ordinary or higher-order band crossings.










The conductivity tensor is given by \cite{kubo}
\begin{eqnarray}{\label{conductivity}}
\sigma_{\alpha\beta}(\omega)=-\frac{i}{\mathcal{V}}\sum_{m,n}\sum_{\mathbf{k}}\left[\frac{p_n(\mathbf{k})-p_m(\mathbf{k}+\mathbf{q})}{\zeta_{n}(\mathbf{k})-\zeta_{m}(\mathbf{k}+\mathbf{q})}\right]\nonumber\\ \times\frac{\langle\mathbf{k},n|\mathcal{J}_{\alpha}(\mathbf{k})|\mathbf{k}+\mathbf{q},m\rangle\langle\mathbf{k}+\mathbf{q},m|\mathcal{J}_{\beta}(\mathbf{k})|\mathbf{k},n\rangle}{\zeta_n(\mathbf{k})-\zeta_m(\mathbf{k}+\mathbf{q})+\omega+i0^{+}}.
\end{eqnarray}
where $\zeta_n(\mathbf{k})$ is the energy of the $n\hbox{th}$ band, $\mathcal{V}$ is the volume of the system, $p(x)=(e^{x}+1)^{-1}$, with $x=\zeta_n({k})/k_BT$ is the Fermi-Dirac distribution function. $\mathcal{J}_{\alpha}(\mathbf{k})$, $\alpha=x$ is the electric current operator in $x$ direction, at point $\mathbf{k}$ into the first Brillouin zone given by \cite{Han}%which is obtained from continuity equation as % $\partial_tn_j =\sum_{j}\mathcal{J}_{ij}$, with $n_j=c^{\dag}_jc_j$ as
\begin{eqnarray}
\mathcal{J}_{\alpha}(\mathbf{k})=g\mu_B\frac{\partial H(\mathbf{k})}{\partial k_{\alpha}},\nonumber\\
%\mathcal{J}_x(\mathbf{k})=i\eta\sum_{\mathbf{k}}\psi^{\dag}_{\mathbf{k}}\sigma_z\psi_{\mathbf{k}}.
\mathcal{J}_x(\mathbf{k})=-g\mu_Bw\sum_{\mathbf{k}}\left(\sin k_x \sigma_x+\cos k_x\sigma_y\right).
\end{eqnarray}




\appendix{\bf APPENDIX A: NON-HERMITIAN LINEAR-RESPONSE FORMALISM}

The general form of non-Hermitian dissipation term $\mathcal{H}'$ added to a Hermitian system with Hamiltonian $\mathcal{H}_0$:
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \mathcal{H}' = -i\delta\sum_{j}\hat{\mathcal{O}}^{\dag}_j\hat{\mathcal{O}}_j+\hat{\mathcal{O}}^{\dag}_j\xi_{j}+\xi^{\dag}_{j}\hat{\mathcal{O}}_j
\end{eqnarray}
where $\xi$ and $\xi^{\dag}$ are white noise operators, that obeys $\langle\hat{\xi}_j(t)\hat{\xi}^{\dag}_i(t')\rangle=2\delta_{ji}\delta(t-t')$, the non-Hermitian response theory $\Delta\mathcal{W}(t)\equiv\mathcal{W}(t)-\mathcal{W}(0)$ is given as
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
\hspace{-1.25cm}  \Delta\mathcal{W} = -\delta\sum_{j}\int_{0}^{t}\left\langle\left\{\hat{W}(t),\hat{\mathcal{O}}^{\dag}_j(t')\hat{\mathcal{O}}_j(t')\right\}
  -2\mathcal{O}^{\dag}_j(t')\hat{W}(t)\hat{\mathcal{O}}_j(t')\right\rangle dt',\nonumber\\
%  \delta\mathcal{W}(t)\equiv\mathcal{W}(t)-\mathcal{W}(0),\nonumber\\
\end{eqnarray}
where $\{\ldots\}$ means anticomutator and $\mathcal{W}$ is the physical observable given as
\begin{eqnarray*}
\mathcal{W}=\langle\hbox{Tr}(\rho\hat{W}(t))\rangle_{\hbox{white noise}},
\end{eqnarray*}
 $\hat{W}$ is a Hermitian operator and $\rho=e^{\beta\mathcal{H}}/{Z}$ is the initial equilibrium density matrix of the non-perturbed Hermitian Hamiltonian with $\beta=1/k_BT$, $Z$ is the partition function of the equilibrium system. We consider $k_B=\hbar=1$ in all formulas. From stochastic integral equation for density operator $\rho(t)$
\begin{eqnarray}
%\rho(t)=\rho(0)+\int_0^t\mathcal{L}_0(s)\rho ds+\int_0^t\mathcal{L}_1(s)\rho dW(s),\nonumber\\
%d\rho(t)=-\frac{i}{\hbar}\left[\mathcal{H}(t),\rho\right]dt-i\sqrt{2\epsilon}\left[c^{\dag}c,\rho\right]dW(t),
\rho(t)=\rho(0)-\frac{i}{\hbar}\int_0^t\left[\mathcal{H}(t),\rho\right]dt-i\int_0^t\sqrt{2\varepsilon}\left[c^{\dag}c,\rho\right]dW(t),\nonumber\\
\end{eqnarray}
where
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
% -i\int_0^t\sqrt{2\varepsilon}\left[c^{\dag}c,\rho\right]dW(t)
% \int_0^t\sqrt{2\varepsilon}\left[c^{\dag}c,\rho\right]dW(t)
 \int_0^t\mathcal{L}_1(t)\rho dW(t)=\hbox{ms}-\lim_{t\rightarrow 0}\sum_{n=0}^{N}\frac{[\mathcal{L}_1(t_n)+\mathcal{L}_1(t_{n+1})]\rho}{2}\Delta W(t_n),\nonumber\\
\end{eqnarray}
 where $\hbox{ms}-\lim$ is the mean-square limit \cite{Gardiner},\\  $\mathcal{L}_1(t)\rho=-i\sqrt{2\varepsilon}\left[c^{\dag}c,\rho\right]$ is solved in Stratonovich interpretation and $-i\sqrt{2\varepsilon}\left[c^{\dag}c,\rho\right]:[0,\infty]\times\Omega\rightarrow\mathbb{R}$. Being the probability measure $P$ on a measurable space $\left(\Omega,\mathcal{F}\right)$, a function $P:\mathcal{F}\rightarrow[0,1]$, such that $P(\emptyset)=0$, $P(\Omega)=1$.  Furthermore,
%\begin{eqnarray}
%\mathcal{L}_0(t)\rho=-i\left[\mathcal{H}(t),\rho\right]/\hbar \hspace{0.25cm} \hbox{and} \hspace{0.25cm} \mathcal{L}_1(t)\rho=-i\sqrt{2\varepsilon}\left[c^{\dag}c,\rho\right]\nonumber\\
%\end{eqnarray}
 %are Liouville superoperators. % and $\epsilon$ is dephasing rate $\epsilon>0$.
the Wiener increment $dW(t)=\xi(t)dt$, where $\xi(t)$ is a white noise that presents the following properties: $\langle W(t)\rangle=0$ and $\langle W(t)W(t')\rangle=\min(t-t')$. The state of the system is described by the noise-averaged density operator %$\sigma\equiv \langle\rho \rangle$
which evolves in time according to the Lindblad master equation
\begin{eqnarray}
%\frac{d\sigma}{dt}=-\frac{i}{\hbar}\left[\mathcal{H}',\sigma\right]-\varepsilon\left(\left\{L^{\dag}L,\sigma\right\}-2L\sigma L^{\dag}\right),
\frac{d\langle\rho \rangle}{dt}=-\frac{i}{\hbar}\left[\mathcal{H}',\langle\rho \rangle\right]-\varepsilon\left(\left\{L^{\dag}L,\langle\rho \rangle\right\}-2L\langle\rho \rangle L^{\dag}\right),\nonumber\\
\end{eqnarray}
where $\varepsilon>0$ is dephasing rate,  $\mathcal{H}$ is the Hamiltonian of the system and the Hermitian Lindblad operator is\\ $L=c^{\dag}_jc_j$, which describes dissipation and decoherence \cite{lindblad}. $\rho$ satisfies the stochastic von Neumann equation \cite{HP,Lei,AH,kevin}.

The non-Hermitian response function is given by \cite{kevin}
\begin{eqnarray}
\Phi^{NH}(t,t')=-\frac{1}{\hbar}\Theta(t-t')\bigg[\langle 0|[A(t),B(t')]|0\rangle\nonumber\\-2\langle 0|A(t)|0\rangle\langle 0|B(t')|0\rangle\bigg],
\end{eqnarray}
where $|0\rangle$ is the ground state, $\Theta$ is the Heaviside step function. The non-Hermitian dynamic susceptibility is given by the Fourier transform
\begin{eqnarray}{\label{suc}}
\chi^{NH}(\tau,\omega)=\int_{-2\tau}^{2\tau}d\Delta t\Phi^{NH}\left(\tau+\frac{\Delta t}{2}, \tau-\frac{\Delta t}{2}\right)e^{i\omega\Delta t},\nonumber\\
\end{eqnarray}
where $\tau=it$. We can write  $\chi^{NH}(\omega)=\hbox{Re}\chi^{NH}(\omega)+i\hbox{Im}\chi^{NH}(\omega)$, where for conciseness, we remove out the $\tau$ argument from the following formulas
\begin{equation}
\chi^{NH}(\omega)=2\left[\hbox{Re}\chi^{NH}(\omega)+i\hbox{Im}\chi^{NH}(\omega)\right].
\end{equation}
From Eq.~(\ref{suc}), we get correlation spectrum
\begin{eqnarray}
%\hspace{-0.5cm}
S(\tau,\omega)=-\frac{\hbar}{2}\int_{-2\tau}^{2\tau}d\Delta t\Phi^{NH}(t,\Delta t)e^{i\omega\Delta t}\nonumber\\
-\frac{\hbar}{2}\int_{-2\tau}^{2\tau}d\Delta t\Phi^{NH}(t,\Delta t)e^{-i\omega\Delta t}\nonumber\\
=-\frac{\hbar}{2}\left[\chi^{NH}(\tau,\omega)+\chi^{NH}(\tau,-\omega)\right]%\nonumber\\
=-\hbar\hbox{Re}\chi^{NH}(\tau,\omega).\nonumber\\
\end{eqnarray}
Thus, from fluctuation-dissipation relation in thermal equilibrium, we get $S(\tau,\omega)=-\hbar\hbox{Re}\chi^{NH}(\tau,\omega)$, where $S(\tau,\omega)$ is the relaxation function of the system
\begin{equation}{\label{susc}}
\hbox{Re}\chi^{NH}(\omega)=-\coth\left(\frac{\hbar\omega}{2k_BT}\right)\hbox{Im}\chi^{NH}(\omega).
\end{equation}
The deviations from Equation above are quantified by the absolute error
\begin{equation}
\Delta_{abs}=\left\| -\hbox{Re}\chi^{NH}(\omega)\tanh\left(\frac{\hbar\omega}{2k_B\Theta}\right)\nonumber\\-\hbox{Im}\chi(\omega)\right\|_p,
\end{equation}
where $\|\ldots\|_p$ is a $L_p$ $(p=2)$ norm:
\begin{eqnarray*}
%\|g\|_2=\left[\frac{1}{\mathcal{D}}\int_{\mathcal{D}}d\omega|g(\omega)|^2\right]^{1/2},
\|h\|_p=\left[\frac{1}{\mathcal{D}}\int_{\mathcal{D}}d\omega|h(\omega)|\right]^{1/p},
\end{eqnarray*}
choosing a fixed integration domain $\mathcal{D}$.
 Thus, the effective temperature that best relates with the susceptibilities via the fluctuation-dissipation relation for a fixed waiting time $t_s$, is obtained from the least-squares method as
\begin{eqnarray}{\label{temper}}
\hspace{-0.75cm}T=\arg\min_{\Theta}\int d\omega\bigg[-\hbox{Re}\chi^{NH}(t_s,\omega)\tanh\left(\frac{\hbar\omega}{2k_B\Theta}\right)\nonumber\\-\hbox{Im}\chi(t_s,\omega)\bigg]^2.\nonumber\\
\end{eqnarray}



\section{Interplay of $E_N$ and conductivity}

 We get the influence of the mixed state measure given by $E_N$ on Drude weight given by
\begin{eqnarray}
D_S(T)\simeq\frac{\pi w}{N}\sum_{\mathbf{k}}\sin k_x\left(\sin k_x\sigma_x+\cos k_x\sigma_y\right)\left(\frac{1-Ze^{-E_N}}{1+Ze^{-E_N}}\right).\nonumber\\
%\hspace{-0.6cm}-\left(\sin k_x\sigma_x+\cos k_x\sigma_y\right)^2\cos^2(k_x)\left[p_n(\mathbf{k})-p_m(\mathbf{k})\right]\nonumber\\ \times\left[1-e^{(\zeta_n(\mathbf{k})-\zeta_m(\mathbf{k}))/k_BT}\right]
%\mathcal{P}\left(\frac{1}{\omega-\zeta_n(\mathbf{k})+\zeta_m(\mathbf{k})}\right)\bigg\}\delta(\omega),\nonumber\\
\end{eqnarray}
%where $\log_2Z=\sum_{\mathbf{k}}\log_2\left[1+e^{\zeta_{+}(\mathbf{k})}+e^{\zeta_{-}(\mathbf{k})}]$.

In Fig.~\ref{fig_38}, we plot the Drude weight $D_S$ as a function of the entanglement negativity $E_N$ to clarify the interplay between quantum entanglement and transport phenomena. Our analysis shows that the Drude weight of the electric conductivity remains finite even in the limit where $E_N\rightarrow 0$, i.e., when quantum correlations are absent. As $E_N$ increases, the Drude weight exhibits a slight but systematic decrease. These results demonstrate that there is slight dependence (change) of the conductivity vs entanglement negativity. Physically, this behavior indicates that stronger quantum correlations tend to inhibit coherent charge transport, while regimes with reduced entanglement are more favorable to metallic conduction.
\begin{figure}
    \centering
%\includegraphics[width=5.5cm]{non-herm_ent_ssh_DsvsEN.eps} \\
\caption{$D_S$ vs $E_N$, at low temperature: $T=0.01J$ and $D=J/10$, $B=0$ held fixed.}\label{fig_38}
\end{figure}



, $\psi_{\mathbf{k}}=\left(c_{A}(\mathbf{k}),c_{B}(\mathbf{k})\right)$, $c_{A}(\mathbf{k})$ and $c_{B}(\mathbf{k})$ are annihilation fermion operators of $A$ and $B$ sublattices. We write Eq.~(\ref{model1}) as
\begin{eqnarray}{\label{model}}
% \nonumber % Remove numbering (before each equation)
%  H(\mathbf{k})=\left(
%                  \begin{array}{cc}
%                    i\eta & (r+2c\cos k_y+w\cos k_x+\gamma)+iw\sin k_x \\
%                    (r+2c\cos k_y+w\cos k_x-\gamma)-iw\sin k_x & -i\eta \\
%                  \end{array}
%                \right)
\hspace{-0.5cm}H(k_x,k_y)=i\eta\sigma_z-w\sin k_x\sigma_y+(r+2c\cos k_y+w\cos k_x)\sigma_x\nonumber\\+i\gamma\sigma_y, \nonumber\\
\end{eqnarray}
where $i=\sqrt{-1}$, $\sigma=(\sigma_x,\sigma_y,\sigma_z)$ are Pauli matrices, $r,\gamma,c,\eta\in\mathbb{R}$, $r+\gamma$, $r-\gamma$ are asymmetric intracell hopping, $w$ is the inter-cell hopping, $c$ is the off-diagonal coupling along $y$ and $\eta$ is the intensity of on-site gain and loss \cite{Megan}. Introducing of the non-Bloch variable $\beta_x:=e^{ik_x}$, where $\beta_x$ corresponds to the complex-valued momentum $k_x$, the set of all permissible values of $\beta_x$ corresponds to the complex-valued momentum $k_x$ determining so, the generalized Brillouin zone. For each $\beta_x$ fixed  the Hamiltonian (\ref{model}) reduces to the one-dimensional lossy subsystem without non-Hermitian skin effect. Thus, imposing PBC or OBC in the $y$ direction results in an identical continuum spectrum in the thermodynamic limit, where one adopts PBC along the $y$ direction and OBC along the $x$ direction, forming a cylinder geometry.  Making $t\equiv r+2c\cos k_y$, we obtain Eq.~(\ref{model}) as
\begin{eqnarray}
\tilde{{H}}(\beta_x,k_y)=(t+w\beta_x)\sigma_x+i\gamma\sigma_y+i\eta\sigma_z.
\end{eqnarray}
Solving the eigenvalue problem $\det({\mathcal{H}}-\mathbf{I}\omega)=0$, we found
\begin{equation}{\label{dispersion}}
% \nonumber % Remove numbering (before each equation)
  \zeta_n({k}_y)=\pm\sqrt{t^2+w^2-\gamma^2-\eta^2+2w\sin\phi\sqrt{t^2-\gamma^2}},
\end{equation}
where $n=\pm$, $\phi\in[-\pi,\pi)$, $t=t(k_y)=r+2c\cos k_y$ is $k_y$-dependent, $v=i\sin\phi$ and
\begin{eqnarray}{\label{beta}}
% \nonumber % Remove numbering (before each equation)
  \beta_{x,\pm}=\frac{\sqrt{\gamma^2-t^2}}{t-\gamma}\left(v\pm\sqrt{1+v^2}\right), \\
  v\equiv\frac{\omega^2-t^2-w^2+\gamma^2+\eta^2}{2w\sqrt{\gamma^2-t^2}}.
\end{eqnarray}
The two $\beta_x$ solutions should have the same amplitude to allow standing waves to form in the $x$ direction under OBCs and $|\beta_{x,+}|=|\beta_{x,-}|$ and $|v+\sqrt{1+v^2}|=|v-\sqrt{1+v^2}|$. Since $\left(v+\sqrt{1+v^2}\right)\left(v-\sqrt{1+v^2}\right)=-1$, we set $v+\sqrt{1+v^2}=e^{i\phi}$.

For the general model, the solution for this coupled SSH model can be obtained using the generalized Brillouin zone (GBZ) where the model is represented as a 2D surface embedded in the 3D space of $\left(\hbox{Re}\beta_x, \hbox{Im}\beta_x,k_y\right)$ for $k_y\in[-\pi,\pi)$, where the intersection between the generalized Brillouin zone and the $k_y$ constant plane is a circle with radius \cite{Megan} $R={R}(k_y):=|\beta_x(k_y)|$,\\
%\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  ${R}(k_y)=\sqrt{|(t+\gamma)/(t-\gamma)|}$,
%\end{eqnarray}
where we can make $\beta_x={R}(k_y)e^{i\phi}$ and get the energy spectrum Eq.~(\ref{dispersion}), which can be entirely real, imaginary or a mix of real and imaginary components, determined by Eq.~(\ref{dispersion}). The generalized band structure is obtained taking $k_y\in[-\pi,\pi)$, where we get phase boundaries in the band structure. Thus, the phase diagram $\eta$ vs $\xi$ depends on $k_y$ and presents different phase boundaries, where the numerous crossings between them exhibit a rich variety of behaviors, comprising the separation of real and imaginary line-gapped-bands and transitions between real line gaps and imaginary line gaps presents two notable types of special $k_y$ points in the band structure: exceptional points (EPs) and non-Bloch Dirac points.

The phase diagram of the model Eq.~(\ref{model1}), showing the evolution of non-Bloch Dirac points (DPs) is derived analytically in terms of the energy line gaps in \cite{Megan}. The topological properties are characterized by mapping of the non-Hermitian  model to a Hermitian problem, showing the non-Bloch DPs  protected by a $\mathbb{Z}$ index. In robustness of the non-Bloch Dirac points, we generalize the model by introducing the next-nearest neighbor hopping to the model and studying the symmetry-protected stability of the Dirac points.

%\subsection{Non-Hermitian linear-response theory}%{\label{4}}


 %
\begin{figure}
 \centering
\includegraphics[width=7.0cm]{non-herm_cond_ssh_nb.eps} \\
\caption{Drude weight $D_S$ as a function of effective temperature at range $\eta=\sqrt{1+\xi}$ with $-1<\xi<0$. We take $(r,c,\gamma,w,\eta)=(\frac{7}{8},\frac{1}{4},\frac{1}{4},1,\eta)$, where we vary the non-Hermitian parameter $\eta$ as: $\eta=0,\sqrt{0.5},\sqrt{0.9}$, which corresponds to $\xi=-0.5,-0.1$. For all cases analyzed, we get $D_S$ finite and a decreasing of $D_S$ in the high-temperature limit.}\label{fig_6}
\end{figure}
\noindent
%




% Generalization to 3D stacked SSH: In two dimensions (2D): stacked SSH chains give Dirac-like crossings when nonreciprocity, gain/loss, and dimerization balance correctly. In three dimensions (3D), stack such 2D layers along a third axis ($k_z$), and introduce interlayer couplings that respect or break chiral symmetry. The non-Hermiticity enters via nonreciprocal hoppings ($t\pm\gamma$) along one or more axes, balancing gain/loss $\pm i\eta$ on sublattices. Complex interlayer couplings can tune the effective dimensionality of the Dirac spectrum. 