Introduction

It is well known that the non-Hermitian quantum mechanics can provide a description of dissipative systems in a minimalist fashion1,2,3,4. It has gained much attention recently due to the rapid progresses in experimental implementations of non-Hermiticity as photonic experiments5,6,7,8,9,10,11,12 and ultra-cold atoms13, where has been revealed that the non-Hermiticity changes the properties of a large number of well-known quantum phenomena well explained in the Hermitian physics, ranging from quantum and topological phase transitions, to quantum magnetism14,15,16,17,18,19,20,21,22,23,24,25,26,27. The purpose of the present paper is to propose the spin transport and electric transport theories for the one-dimensional non-Hermitian XXZ model and the one-dimensional non-Hermitian Hubbard models respectively, using the non-Hermitian linear-response theory. The effect of splitting magnon bands, induced by the non-Hermitian parameters in properties as quantum correlation has been analyzed by us in the Refs.28,29,30. The paper is organized as follows. In Sect. Non-Hermitian linear-response theory for spin currents, we discuss about the non-Hermitian spin transport theory of quantum spin systems. In Sect. Results and discussion, we analyzed the case of the half-integer spin and integer spin one-dimensional non-Hermitian XXZ model and the case of strongly correlated electrons system modelled by the one-dimensional non-Hermitian Hubbard model, where we presented analytical and numerical results for the transport coefficients. In the last Sect. Outlook, we present our conclusions and final remarks.

Non-Hermitian linear-response theory for spin currents

In the linear response theory for non-Hermitian systems with the Hamiltonian in the form \(\mathcal {H}=\mathcal {H}_0+\mathcal {H}_{dis}\), where the non-Hermitian dissipation term corresponding to the Itô’s stochastic differential equation \(dx(t)=\gamma a(x(t),t) dt+b(x(t),t)\circ dW(t)\), with the Wiener increment \(dW(t)=\xi (t)dt\) and white noise term \(\xi (t)\)

$$\begin{aligned} {} \mathcal {H}_{dis}=\sum _{j}\left( -i\gamma \mathcal {B}^{\dag }_j\mathcal {B}+\mathcal {B}_j^{\dag }\xi _j+\xi _j^{\dag }\mathcal {B}_j\right) \end{aligned}$$
(1)

where \(\Gamma =2\gamma \), \(\gamma \in \mathbb {R}\) and \(\xi (t)\) is a stochastic noise that satisfies \(\langle \xi _i(t)\xi ^{\dag }_j(t')\rangle =\Gamma \delta _{ij}\delta (t-t')\), \(\mathcal {B}_j\) is a Hermitian operator. The linear response to the Eq. (1) is given by4

$$\begin{aligned} \delta \mathcal {H}(t)=-\Gamma \sum _{j}\int _{0}^{t}\langle \{\mathcal {W}(t),\mathcal {B}^{\dag }_j(t')\mathcal {B}_j(t')\}-2\mathcal {B}^{\dag }_j(t')\mathcal {W}(t)\mathcal {B}_j(t')\rangle dt', \end{aligned}$$
(2)

where the physical observable \(\mathcal {W}(t)\) is given by

$$\begin{aligned} \mathcal {W}=\langle \hbox {Tr}\left( \rho \mathcal {W}_{\mathcal {H}}(t)\right) \rangle _{\hbox {noise}},\end{aligned}$$
(3)
$$\begin{aligned} \rho =\frac{e^{-\mathcal {H}_0/T}}{\hbox {Tr}\left\{ e^{-\mathcal {H}_0/T}\right\} } \end{aligned}$$
(4)

and the operator \(\mathcal {W}(t)\) in the Heisenberg picture is given by \(\mathcal {W}(t)=e^{i\mathcal {H}^{\dag }t}\mathcal {W}e^{-i\mathcal {H}^{\dag }t}\). Here, we consider the linear response theory for a proper spin system described by the one-dimensional XXZ model, where we discuss the linear response theory for a one-dimensional electron system (electrical transport) as well. For the case of spins (spin transport), we have the response of the system to the frequency-dependent gradient of an external magnetic field \(\textbf{H}\) generates a spin current given by \(\mathcal {J}=\sigma \nabla \textbf{H}\), where the response linear to the external field in x direction is

$$\begin{aligned} \langle \mathcal {J}_{x}(l,t)\rangle =\sum _{j}\int _{-\infty }^{\infty }dt' \chi _{jS}(l,j,t-t')h^{z}(j,t'), \end{aligned}$$
(5)

being the response function defined as

$$\begin{aligned} \chi _{jS}(l,j,t-t')\equiv i\Theta (t-t')\langle 0|[\mathcal {J}_x(l,t),S^z(j,t')]|0\rangle , \end{aligned}$$
(6)

where \(\Theta \) is the Heaviside step function. On the other hand, the non-Hermitian response function is given by3

$$\begin{aligned} \chi _{jS}^{NH}(l,j,t-t')=-\frac{1}{\hbar }\Theta (t-t')\bigg [\langle 0|\{\mathcal {J}_x(l,t),S^z(j,t')\}|0\rangle \nonumber \\ -2\langle 0|\mathcal {J}_x(l,t)|0\rangle \langle 0|S^z(j,t')|0\rangle \bigg ], \end{aligned}$$
(7)

where \(\{\cdot \cdot \cdot \}\) is the unequal-time anti-commutator to establish the link between the response function and the correlation function. We have the non-Hermitian dynamic susceptibility as the Fourier transform

$$\begin{aligned} \chi _{jS}^{NH}(\tau ,\omega )=\int _{-2\tau }^{2\tau }d\Delta t\chi _{jS}^{NH}\left( \tau +\frac{\Delta t}{2},\tau -\frac{\Delta t}{2}\right) e^{i\omega \Delta t}, \end{aligned}$$
(8)

where \(\tau =it\). \(\chi _{jS}\) is the non-Hermitian response function

$$\begin{aligned} \chi _{jS}^{NH}(t,t')=\frac{i}{\hbar } \Theta (t-t')\langle 0|\{\mathcal {J}_x(t),S^z(t')\}|0\rangle . \end{aligned}$$
(9)

Performing the Fourier transform, we get the current response function in x direction

$$\begin{aligned} \langle \mathcal {J}_x(k,\omega )\rangle =\chi _{jS}^{NH}(k,\omega ){H}^z({k},\omega ), \end{aligned}$$
(10)

where the frequency and wave-vector-dependent susceptibility is given by

$$\begin{aligned} \chi _{jS}^{NH}({k},\omega )\equiv \frac{i}{N}\int _{0}^{\infty }dte^{i(\omega +i0^{+})t} \langle 0|\{\mathcal {J}_x({k},t),S^z(-{k},0)\}|0\rangle . \end{aligned}$$
(11)

In order to derive a formula for the spin conductivity, we need the equation of continuity for the spin current: \(\dot{S}^z({k},t)+i{k}\cdot \mathcal {J}_x({k},t)=0\). \(\chi _{jS}^{NH}\) can be transformed as follows:

$$\begin{aligned} \chi _{jS}^{NH}({k},\omega )=\frac{i}{N}\frac{1}{i(\omega +i0^{+})}\bigg [ik_x\int _{0}^{\infty }dte^{i(\omega +i0^{+})t}\nonumber \\ \times \langle 0|\{\mathcal {J}_x({k},t),\mathcal {J}_x(-{k},0)\}|0\rangle -\langle 0|\{\mathcal {J}_x({k},0),S^z(-{k},0)\}|0\rangle \bigg ]. \end{aligned}$$
(12)

In following, using the representation of the spin current operator in terms of spin operators

$$\begin{aligned} \mathcal {J}_x(j)=\frac{iJ}{2}\sum _x(S_{j}^{+}S_{j+x}^{-}-S_{j}^{-}S_{j+x}^{+}), \end{aligned}$$
(13)

where \(j+x\) is the nearest-neighbor site of the site j in the positive x direction, we can transform the second term as

$$\begin{aligned} \langle 0|\{\mathcal {J}_x({k},0),S^z(-{k},0)\}|0\rangle =\frac{i}{2}\sum _{l,x}\mathcal {J}_{l,l+x}\left( e^{ik_x}-1\right) \nonumber \\ \times \langle S_{j}^{+}S_{j+x}^{-}+S_{j}^{-}S_{j+x}^{+}\rangle . \end{aligned}$$
(14)

In the long-wavelength \(k_x\rightarrow 0\) limit, the susceptibility \(\chi _{jS}^{NH}({k},\omega )\) is proportional to \(ik_x\) and we can write

$$\begin{aligned} {} \langle \mathcal {J}_{x}({k},\omega )\rangle =-\frac{\langle -\mathfrak {K}_x\rangle -\mathfrak {F}_{xx}({k},\omega )}{i(\omega +i0^{+})}ik_xh^z({k},\omega ), \end{aligned}$$
(15)

where \(\langle -\mathfrak {K}_x\rangle \) is the kinetic energy

$$\begin{aligned} \langle \mathfrak {K}_x \rangle =-\frac{J}{N}\sum _{j}\left( S_j^{+}S_{j+x}^{-}+S_j^{-}S_{j+x}^{+}\right) , \end{aligned}$$
(16)

and \(\mathfrak {F}_{xx}\) is the retarded Green’s function defined in \(T=0\) by32

$$\begin{aligned} \mathfrak {F}_{xx}({{k}},\omega )=\frac{i}{ N}\int _{0}^{\infty }dte^{i(\omega +i0^{+}) t}\langle 0| [\mathcal {J}_x({{k}},t),\mathcal {J}_x(-{{k}},0)]|0\rangle , \end{aligned}$$
(17)

where the Eq. (15) exhibits the desired structure \(\langle \mathcal {J}\rangle =\sigma \nabla H^z\). Thus, the formula for the spin conductivity defined as the linear spin-current response to a uniform \(k=0\) , frequency-dependent gradient of the magnetic field

$$\begin{aligned} \sigma _{xx}(\omega )=-\frac{\langle -\mathfrak {K}_x\rangle -\mathfrak {F}_{xx}(k=0,\omega )}{i(\omega +i0^{+})}, \end{aligned}$$
(18)

where this complex expression can be decomposed into its real and imaginary parts as \(\sigma _{xx}(\omega )=\hbox {Re}\{\sigma _{xx}(\omega )\}+i\hbox {Im}\{\sigma _{xx}(\omega )\}\), where

$$\begin{aligned} \hbox {Re}\{\sigma _{xx}(\omega )\}=\pi \left[ \langle -\mathfrak {K}_{xx}\rangle -\hbox {Re}\{\mathfrak {F}_{xx}(k=0,\omega \rightarrow 0)\}\right] \delta (\omega )\nonumber \\ +\frac{\hbox {Im}\{\mathfrak {F}_{xx}(k=0,\omega )\}}{\omega }, \nonumber \\ \hbox {Im}\{\sigma _{xx}(\omega )\}=\pi \left[ -\hbox {Im}\{\mathfrak {F}_{xx}(k=0,\omega \rightarrow 0)\}\right] \delta (\omega )\nonumber \\ +\frac{\langle -\mathfrak {K}_x\rangle -\hbox {Re}\{\mathfrak {F}_{xx}(k=0,\omega )\}}{\omega }. \end{aligned}$$
(19)

Since \(\mathfrak {F}_{xx}(k=0,\omega \rightarrow 0)\) is the retarded susceptibility and the Fourier transform in time of a real function, we have that its real part is an even function of \(\omega \) and its imaginary part is an odd function of \(\omega \). As long as the odd function \(\hbox {Im}\{\mathfrak {F}_{xx}\}\) has no singularity at \(\omega =0\), we know that \(\hbox {Im}\{\mathfrak {F}_{xx}(k=0,\omega \rightarrow 0)\}=0\) and the first term contribution to \(\hbox {Im}\{\sigma _{xx}\}\) vanishes.

The regular part of the conductivity \(\sigma \) (continuum conductivity) in the context of Hermitian quantum mechanics is given by31,32,33,34,35,36: \(\hbox {Re}\left[ \sigma _{xx}(\omega )\right] =D_S(T)\delta (\omega )+\sigma ^{reg}(\omega )\), where

$$\begin{aligned} \langle \mathcal {J}_{\alpha }({k},\omega )\rangle =\sum _{\beta }\sigma _{\alpha \beta }({k},\omega )ik_{\alpha } h_{\beta }({k},\omega ),\nonumber \\ \sigma _{\alpha \beta }({k},\omega )=\hbox {Re}[\sigma _{\alpha \beta }({k},\omega )]+i\hbox {Im}[\sigma _{\alpha \beta }({k},\omega )]\nonumber \\ \sigma ^{reg}(\omega )=\frac{\hbox {Im} \{\mathfrak {F}({{k}}=0,\omega )\}}{\omega } \end{aligned}$$
(20)

and \(\alpha ,\beta =x,y,z\). \(D_S(T)\) is the spin Drude’s weight. The effective T that best relates the susceptibilities via fluctuation dissipation relation for a fixed waiting time \(t_w\) is given by3

$$\begin{aligned} T=\arg \min _{\Theta }\int d\omega \left[ -\chi '^{NH}(t_w,\omega )\tanh \left( \frac{\hbar \omega }{2k_{B}\Theta }\right) -\chi ''(t_{w},\omega )\right] ^{2},\nonumber \\ \end{aligned}$$
(21)

where \(\chi ^{NH}=\chi '^{NH}+i\chi ''^{NH}\), \(\chi =\chi '+i\chi ''\).

Figure 1
figure 1

\(\sigma ^{reg}(\omega )\) at \(T=0.0J\) for different values of non-Hermitian coupling \(\delta \), for the spin-1/2 non-Hermitian XXZ model. For \(\delta =0\) we have that the model is Hermitian and \(\delta \ne 0\) the model is non-Hermitian. We find that conductivity tends to zero at DC limit.

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Results and discussion

As example of application of the method, we consider the non-Hermitian XXZ model given by the Hamiltonian

$$\begin{aligned} \mathcal {H}=-\sum _{i,j\ne i}\left( S_i^{+}S_{j}^{-}+S_i^{-}S_{j}^{+}\right) +\Delta \sum _{i,j\ne i} S_{i}^{z}S_{j}^z+\sum _{j}\textbf{H}\cdot S_j, \end{aligned}$$
(22)

where the non-Hermiticity originates from complex magnetic field \(\textbf{H}=\left( 1,-i\delta ,0\right) \), \(i=\sqrt{-1}\), which can be understood as the spin-dependent losses and is within the reach of ultracold atom experiments. The degeneracy of the system breaks down when the external complex field presents the form \(\mathfrak {T}\textbf{H}\cdot S_j\mathfrak {T}^{-1}=-\textbf{H}^{*}\cdot S_j\). The external field also spoils the commutation relation \(\left[ \sum _{j}S_j^{z},\mathcal {H}\right] =0\).

We can recast the Hamiltonian above as

$$\begin{aligned} \tilde{\mathcal {H}}=-\sum _{i,j\ne i}\left( \tau _i^{+}\tau _{j}^{-}+\tau _i^{-}\tau _{j}^{+}+\Delta \tau _{i}^{z}\tau _{j}^z\right) +\sum _{i}\sqrt{1-\delta ^2}\tau ^x_j,\end{aligned}$$
(23)

which is the XXZ model in a transverse field. The transformation hold if and only if \(|\delta |\ne 1\). Considering a local complex field \(\textbf{H}\cdot S_N\), the Hamiltonian shares two eigenstates \(|\psi \rangle =\pm \sqrt{1-\delta }|\uparrow \rangle +\sqrt{1+\delta }|\downarrow \rangle \), that satisfies \(\mathcal {H}|\psi \rangle =(-N\Delta /4\pm \sqrt{1-\delta ^2}|\psi \rangle \), where \(|\psi \rangle \) coalesces at \(|\delta |=1\), where the ground state is dramatically changed from degeneracy to coalescence by the local complex field. Under a homogeneous global complex field, the Hamiltonian of the free spins in the absence of interaction describes the \(\mathcal{P}\mathcal{T}\)-symmetric hypercube graph of N dimension, and the system can be projected onto several invariant subspaces denoted by S. For \(\delta =1\), the \(n=2S+1\) of n-eigenstate coalescence in each subspace.

Half-integer spin S: Performing the Jordan-Wigner transformation

$$\begin{aligned} \tau _j^x =\frac{1}{2}-\bar{f}_jf_j, \nonumber \\ \tau _j^y =\frac{i}{2}\sum _{j<l}(1-2\bar{f}_jf_j)(\bar{f}_j-f_j), \nonumber \\ \tau _j^{z}=-\frac{1}{2}\sum _{j<l}(1-2\bar{f}_jf_j)(\bar{f}_j+f_j) \end{aligned}$$
(24)

to replace the quasi-spin operators by the new non-Hermitian operators \(f_j\) and \(\bar{f}_j\), where \(\bar{f}_j=\mathfrak {A}_jc_j^{\dag }\mathfrak {A}_j^{-1}\), \(f_j=\mathfrak {A}_jc_j\mathfrak {A}_j^{-1}\), \(c_j^{\dag }\), \(c_j\) are the creation and annihilation operators of spinless fermions. The new operators satisfy the fermionic anticommutation relation \(\{\bar{f}_j,f_j'\}=\delta _{j,j'}\). The parity of the number of fermions is a conservative quantity such that the Hamiltonian can be expressed as \(\tilde{\mathcal {H}}=\tilde{\mathcal {H}}_{+}\textbf{I}=\tilde{\mathcal {H}}_{-}\textbf{I}\), where \(\tilde{\mathcal {H}}_{+}=\tilde{\mathcal {H}}_{-}=-2(\bar{f}_{N}\bar{f}_1+\bar{f}_{N}f_1+\bar{f}_1f_{N}+f_1f_{N})\) and the Hamiltonian can be rewritten as

$$\begin{aligned} \tilde{\mathcal {H}}=\frac{1}{4}\sum _{j=1}^{N}[2g\sqrt{1-\delta ^2}(1-2\bar{f}_jf_j)\nonumber \\ +\Delta (\bar{f}_j\bar{f}_{j+1}+\bar{f}_jf_{j+1}+\bar{f}_{j+1}f_{j}+f_{j+1}f_{j})\nonumber \\ -(\bar{f}_j\bar{f}_{j+1}-\bar{f}_jf_{j+1}-\bar{f}_{j+1}f_{j}+f_{j+1}f_{j})\nonumber \\ +(1-2\bar{f}_{j+1}f_{j+1}-2\bar{f}_{j}f_j+4\bar{f}_{j+1}f_{j+1}\bar{f}_{j}f_j)]. \end{aligned}$$
(25)
Figure 2
figure 2

Behavior of the Drude’s weight \(D_S(T)\) as a function of T for the spin-1/2 non-Hermitian XXZ model. For \(\delta =0\) we have that the model is Hermitian and \(\delta \ne 0\), the model is non-Hermitian. The Drude’s weight is finite at \(T=0\) indicating thus, an ideal spin conductor at this limit.

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Taking the Fourier transform

$$\begin{aligned} f_j=\frac{1}{\sqrt{N}}\sum _{\textbf{k}}f_{\textbf{k}}e^{i{\textbf{k}\cdot \textbf{r}_j}}, \hspace{0.5cm}\bar{f}_j=\frac{1}{\sqrt{N}}\sum _{\textbf{k}}\bar{f}_{\textbf{k}}e^{-i{\textbf{k}\cdot \textbf{r}_j}}, \end{aligned}$$
(26)
$$\begin{aligned} \tilde{\mathcal {H}}=\frac{1}{4}\sum _{j=1}^{N}[2g\sqrt{1-\delta ^2}(1-2\bar{f}_{\textbf{k}}f_{\textbf{k}})\nonumber \\ +\Delta (e^{-i{\textbf{k}}}\bar{f}_{\textbf{k}}\bar{f}_{-{\textbf{k}}}+2\cos ({\textbf{k}})\bar{f}_{\textbf{k}}f_{{\textbf{k}}}+e^{i{\textbf{k}}}f_{{\textbf{k}}}f_{-{\textbf{k}}})\nonumber \\ -(e^{-i{\textbf{k}}}\bar{f}_{\textbf{k}}\bar{f}_{-{\textbf{k}}}-2\cos ({\textbf{k}})\bar{f}_{\textbf{k}}f_{{\textbf{k}}}+e^{i{\textbf{k}}}f_{{\textbf{k}}}f_{-{\textbf{k}}})\nonumber \\ +(1-4\bar{f}_{{\textbf{k}}}f_{{\textbf{k}}}+4\bar{f}_{{\textbf{k}}}f_{{\textbf{k}}}\bar{f}_{{\textbf{k}}}f_{\textbf{k}})]. \end{aligned}$$
(27)

where \(g=1\), \(|\textbf{k}|=k=2\pi (n+1/2)/N\), \(n=0,1,2,...,N-1\), the Hamiltonian can be written as

$$\begin{aligned} \tilde{\mathcal {H}}_{+}=\bar{\mathcal {H}}_{-}=\sum _{0<k<\pi }\bar{\chi }_{\textbf{k}}\bar{\mathcal {H}}_{+}(\textbf{k})\chi _{\textbf{k}}, \end{aligned}$$
(28)

with \(\bar{\chi }_{\textbf{k}}=(\bar{f}_{\textbf{k}},f_{-\textbf{k}})\), \(\chi _{\textbf{k}}=(f_{\textbf{k}},\bar{f}_{-\textbf{k}})^{T}\), and

$$\begin{aligned} \hspace{-1.0cm} \bar{\mathcal {H}}_{+}({\textbf{k}})=\left( \begin{array}{cc} (\Delta +1)\cos (k) -\lambda -1 &{} i(\Delta -1)\sin (k) \\ -i(\Delta -1)\sin (k) &{} -((\Delta +1)\cos (k) -\lambda -1) \\ \end{array} \right) ,\end{aligned}$$
(29)

where \(\lambda =2g\sqrt{1-\delta ^2}\) and we make \(g=1\) to simplify the notation. In following, making the non-Hermitian Bogoliubov transformation

$$\begin{aligned} \bar{\psi }_{\textbf{k}}=\cos \frac{\varphi _{\textbf{k}}}{2}\bar{f}_{\textbf{k}}+i\sin \frac{\varphi _{\textbf{k}}}{2}f_{-\textbf{k}}, \end{aligned}$$
(30)
$$\begin{aligned} \psi _{\textbf{k}}=\cos \frac{\varphi _{\textbf{k}}}{2}f_{\textbf{k}}-i\sin \frac{\varphi _{\textbf{k}}}{2}\bar{f}_{-\textbf{k}}, \end{aligned}$$
(31)

where \([\bar{\psi }_{\textbf{k}},\psi _{\mathbf {k'}}]=\delta _{\textbf{k},\mathbf {k'}}\) and \(\varphi _{\textbf{k}}=\tan ^{-1}[(\Delta +1)\sin ({k})/(\lambda +1-(\Delta +1)\cos ({k}))]\), the Hamiltonian is recast in the diagonal form \(\tilde{\mathcal {H}}_{+}=\sum _{k}\left( \bar{\psi }_{\textbf{k}}{\psi }_{\textbf{k}}-1/2\right) \), with the dispersion relation of quasi-particles given by

$$\begin{aligned} \omega _{\textbf{k}}=\sqrt{\left[ (\Delta +1)\cos ({k})-\lambda -1\right] ^2-(\Delta -1)^2\sin ^2({k})}, \end{aligned}$$
(32)

If \(|\delta |< 1\), the single-particle spectrum is real and \(|\delta |> 1\), the system presents a complex single-particle spectrum regardless of \(\textbf{k}\).

The regular part of the spin conductivity \(\sigma _{xx}\), (continuum conductivity) in the context of Hermitian quantum mechanics is given by31,32: \(\hbox {Re}\left[ \sigma _{xx}(\omega )\right] =D_S(T)\delta (\omega )+\sigma ^{reg}(\omega )\), where \(D_S(T)\) is the spin Drude’s weight, being given by

$$\begin{aligned} \hspace{-0.0cm}D_S(T)\approx -\frac{1}{N}\sum _{{k}}\frac{\cos (k_{x})}{\omega _{\textbf{k}}}[1+n(\omega _{\textbf{k}})], \end{aligned}$$
(33)

where \(n(\omega _{\textbf{k}})=1/(e^{\beta \omega _{\textbf{k}}}- 1)\) is the boson occupation number and \(\beta =1/T\). In the case of noninteracting magnons, an explicit expression of the kinetic energy can be obtained at the same level of approximation as for the current-correlation function. The unphysical singularities such as the Dirac’s delta peak in the conductivity formula, are removed once magnon interaction are included, where the Kramers-Kronig relations are fulfilled for the current-correlation functions calculated in the ladder approximation within some numerical error arising from the finite energy resolution, especially at the upper band edge where the correlation functions diverge. These calculations should be used to calculate the Drude weight by means of the sum rule. However, the \(k=0\) current correlations alone do not seen to provide enough information for this task35.

In Fig. 1, we present the behavior of \(\sigma ^{reg}(\omega )\) for different values of non-Hermitian coupling \(\delta \). We get the AC conductivity tending to zero at \(\omega \rightarrow 0\) however, as we have \(\sigma (0)=D_S\delta (\omega )\) and since that we obtain \(D_S\) finite, we must have a divergence in the DC current. However, the scattering among quasi-particles must introduce a spreading in the conductivity where in a real system the conductivity must to stay finite. The large peaks obtained for the conductivity are due to the behavior of the dispersion relation at range \((0.0<\omega /J<0.3)\), generating so, resonance effects on conductivity. In Figs. 5 and 6, we analyze the AC conductivity for the non-Hermitian XXZ model. In this case, we get the continuum conductivity tending to zero at DC limit, \(\omega \rightarrow 0\).

The behavior of the Drude’s weight \(D_S(T)\) as a function of T is displayed in Fig. 2 for the one-dimensional spin-1/2 non-Hermitian XXZ model. Since \(D_S(T)\) is finite for all T values, we get that the system is an ideal spin conductor. For \(\delta =0\) we have that the model is Hermitian and \(\delta \ne 0\) the model is non-Hermitian. Moreover, we get a significant difference among the behavior of the curves for the two models (Hermitian and non-Hermitian), due to transformation of the non-Hermitian Hamiltonian in Hermitian. For non-zero temperature, \(D_S(T)\) rises with the temperature due to spinons thermally excited for T non-zero.

Figure 3
figure 3

\(\sigma ^{reg}(\omega )\) at \(T=0.0J\) for different values of non-Hermitian coupling \(\delta \) for the spin-1 non-Hermitian XXZ model. For \(\delta =0\) we have the Hermitian model and \(\delta \ne 0\), the model is non-Hermitian. We find that conductivity tends to zero at DC limit, indicating so.

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Integer spin S: Considering the ferromagnet case, we perform the Holstein-Primakoff transformation linearized

$$\begin{aligned} \hspace{-0cm} S^{z}_j\approx \frac{\sqrt{2S}}{2}({a}^{\dag }_j+a_j),\hspace{0.3cm} S^{y}_j\approx \frac{\sqrt{2S}}{2i}({a}^{\dag }_j-a_j),\hspace{0.3cm}S^{x}_j=S-{a}^{\dag }_ja_j, \end{aligned}$$
(34)

where the operators \(a_j\) and \({a}^{\dag }_j\) and \({a}^{\dag }_j=\mathfrak {U}_jb_j^{\dag }\mathfrak {U}_j^{-1}\), \(a_j=\mathfrak {U}_jb_j\mathfrak {U}_j^{-1}\). \(b_j^{\dag }\), \(b_j\) are the creation and annihilation operators in finite dimensional space. The new operators satisfy the bosonic commutation relation \(\left[ {a}^{\dag }_j,a_j'\right] =\delta _{j,j'}\). In following, we introduced the Fourier transformed operators

$$\begin{aligned} a_{\textbf{k}}=\frac{1}{\sqrt{N}}\sum _{j=1}^{N} a_je^{i\textbf{k}\cdot r_j},\hspace{0.5cm}{a}^{\dag }_{\textbf{k}}=\frac{1}{\sqrt{N}}\sum _{j=1}^{N} {a}^{\dag }_je^{-i\textbf{k}\cdot r_j}. \end{aligned}$$
(35)

We obtain

$$\begin{aligned} \hspace{-1.5cm}{\mathcal {H}}=\left[ (\Delta +1)\cos (k)-\frac{S}{2}(\Delta -1)+S(S+\lambda )\right] \sigma _3 -(\Delta +1)\sin (k)\sigma _2, \end{aligned}$$
(36)

where \(\sigma _1\), \(\sigma _2\) and \(\sigma _3\) are the (spin-1/2) Pauli’s Matrices. We make the non-Hermitian Bogoliubov transformation

$$\begin{aligned} {\alpha }^{\dag }_{\textbf{k}}=\cos \frac{\theta _{\textbf{k}}}{2}{a}^{\dag }_{\textbf{k}}+i\sin \frac{\theta _{\textbf{k}}}{2}a_{-\textbf{k}}, \end{aligned}$$
(37)
$$\begin{aligned} \alpha _{\textbf{k}}=\cos \frac{\theta _{\textbf{k}}}{2}a_{\textbf{k}}-i\sin \frac{\theta _{\textbf{k}}}{2}{a}^{\dag }_{-\textbf{k}}, \end{aligned}$$
(38)

where \([{\alpha }^{\dag }_{\textbf{k}},\alpha _{\mathbf {k'}}]=\delta _{\textbf{k},\mathbf {k'}}\) and \(\theta _{\textbf{k}}=\tan ^{-1}[(\Delta +1)\sin ({k})/(\frac{S}{2}(\Delta -1)-S(S+\lambda )-(\Delta +1)\cos ({k}))]\). We get the diagonal Hamiltonian

$$\begin{aligned} \mathcal {H}=\sum _{{k}}\varepsilon _{\textbf{k}}\alpha ^{\dag }_{\textbf{k}}\alpha _{\textbf{k}} \end{aligned}$$
(39)

with the dispersion relation of quasi-particles

$$\begin{aligned} \hspace{-1.5cm} \varepsilon _{\textbf{k}}=\sqrt{\left( (\Delta +1)\cos (k)-\frac{S}{2}(\Delta -1)+S(S+\lambda )\right) ^2-(\Delta +1)^2\sin ^2(k)}. \end{aligned}$$
(40)

For \(k=\pi /2\), we have that the dispersion relation is given by \(\varepsilon _{\textbf{k}}=S\sqrt{(\Delta -1)^2-4(\Delta -1)(S+\lambda )+4(S+\lambda )^2-(\Delta +1)^2}\), where \(\varepsilon _{\textbf{k}}=2\sqrt{(S+\lambda )^2-1}\) in the isotropic point \(\Delta =1\). The unphysical situation will occur for \(\Delta _c>1\), where \((\Delta -1)^2-4(\Delta -1)(S+\lambda )+4(S+\lambda )^2-(\Delta +1)^2<0\). For \(k=\pi \), we have \(\varepsilon _{\textbf{k}}=\sqrt{\left( (\Delta +1)+\frac{S}{2}(\Delta -1)-S(S+\lambda )\right) ^2}=|(\Delta +1)+\frac{S}{2}(\Delta -1)-S(S+\lambda )|\in \mathbb {R}\).

Figure 4
figure 4

Behavior of the Drude’s weight \(D_S(T)\) as a function of T for the spin-1 non-Hermitian XXZ model. For \(\delta =0\) we have the Hermitian model and \(\delta \ne 0\), the model is non-Hermitian. The Drude weight is finite at \(T=0\) indicating thus an ideal spin conductor at this limit.

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In Fig. 3, we analyze the AC conductivity for the non-Hermitian spin-1 XXZ model. In this case, we get the continuum conductivity tending to zero at DC limit, \(\omega \rightarrow 0\). We display the behavior of \(\sigma ^{reg}(\omega )\) for different values of non-Hermitian coupling \(\delta \). As \(\sigma (0)=D_S\delta (\omega )\) and since that we obtain \(D_S\) finite for all T values, we must have a divergence in the spin current. However, the scattering among magnons must introduce a spreading in the conductivity where in a real system the conductivity must to stay finite. The large peaks obtained for the conductivity are due to the behavior of the dispersion relation at range \(0.0<\omega /J<0.5\) generating so, resonance effects on conductivity into this range of \(\omega \).

The behavior of the Drude’s weight \(D_S(T)\), as a function of T is displayed in Fig. 4 for the spin-1 one-dimensional non-Hermitian XXZ model. Since \(D_S(T)\) stays finite for all T values, we get an ideal spin conductor behavior for the system for all T values. Moreover, we get a significant difference among the behavior of the curves for both models (Hermitian and non-Hermitian), due to transformation of the non-Hermitian Hamiltonian in Hermitian. For non-zero temperatures values, \(D_S(T)\) rises with the temperature due to magnons thermally excited at this range of T. However, this description for T non-zero is only qualitative due to limitations of the spin wave approach used.

Non-Hermitian Hubbard model: The model is given by the Hamiltonian

$$\begin{aligned} {} \mathcal {H}=i\sum _{i,j}\sum _{\sigma =\uparrow ,\downarrow }t_{ij}\left( c_{i,\sigma }^{\dag }c_{j,\sigma }+c_{j,\sigma }^{\dag }c_{i,\sigma }\right) +U\sum _{j}n_{j,\uparrow }n_{j,\downarrow }, \end{aligned}$$
(41)

where \(c_{j,\sigma }\) \((c_{j,\sigma }^{\dag })\) are the usual annihilation (creation) operators of fermions with spin \(\sigma \in \{\uparrow ,\downarrow \}\) at site j and \(n_{i,\sigma }=c_{i,\sigma }^{\dag }c_{i,\sigma }\) is the number operator for a particle of spin \(\sigma \) on site j. In addition, U, \(t_{ij}\in \mathbb {R}\) are the interaction and kinetic energies respectively. The number of sites and particles are \(\mathcal {N}\) and \(\mathcal {M}\), respectively. The Hamiltonian above can describe a 1D homogeneous ring system in which \(it_{ij}=it\delta _{i,j+1}\). Each subspace is labelled by \({k}=2n\pi /N\) which is the momentum indexing the subspace. Each subspace labeled by k can be decomposed into four subspaces with \((S, S^z)=(0,0), (1,0)\) and \((1,\pm 1)\) in term of the spin symmetry. In each invariant subspace we have the two-particle solution, where the two-particles state is calculated in Ref.2. Based on the symmetry of the system, we have the paring mechanism through the exact solution with the two-particle subspace. The bound pair emerges in the (0, 0) subspace with energy of the bound pair given by \(\xi _{{\textbf{k}}}=\pm \sqrt{U^2-16t^2\cos ^2({k}/2)}\). The details about the bound pair are given in Ref.2. For the case of U negative, the lowest energy of a bound pair is \(\omega _{\pi }=-U\) locating on the subspace with \({k}=\pi \). Furthermore, the bound pair state is \(|\varphi ^{b}_{{\textbf{k}}}\rangle =\sum _rg_{{\textbf{k}}}|\phi _r^{-}({{k}})\rangle \) with \(g_{{\textbf{k}}}^{-}(j)=1/\sqrt{2}\) if \(j=0\) and \(e^{-\gamma j}\), if \(j\ne 0\), where \(\gamma =\ln \left[ \left( -U\pm \sqrt{U^2+4\lambda _{{\textbf{k}}}^2}\right) /4it\cos ({k}/2)\right] \). In the absence of on-site interaction \((U=0)\), only the scattering eigenstate with imaginary eigenvalue is present and the system does not accommodate a bound pair state. A nonzero interaction \((U\ne 0)\) leads thus, the emergence of a bound pair. When \(|U|>|4t|\), the system possesses the full real bound pair spectrum however, a small U results in the appearance of the imaginary bound pair energy.

Figure 5
figure 5

\(\sigma ^{reg}(\omega )\) at \(T=0.0\) for different values of non-Hermitian couplings U, for the one-dimensional non-Hermitian Hubbard model. We find \(\sigma ^{reg}(\omega \rightarrow 0)\) diverging at DC limit indicating thus, a super-current behavior.

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The continuum part of the spin conductivity \(\sigma ^{reg}(\omega )\), is defined in terms of the Green’s function \(\mathfrak {F}(k,\omega )\). We obtain the electrical current operator in terms of operators Fourier transformed \(c_{\textbf{k}}^{\dag }\) and \(c_{\textbf{k}}\) given by

$$\begin{aligned} \hspace{-1.0cm}\mathcal {J}_x(t)=\sum _{k}\mathcal {J}_x(k,t)= -\sum _{{k}}\frac{\cos (k_{x})}{\xi _{{\textbf{k}}}}c_{{\textbf{k}}}^{\dag }{\Lambda }_{{\textbf{k}}}(t)c_{{\textbf{k}}}. \end{aligned}$$
(42)

The non-Hermiticity of \(\mathcal {H}\) here, comes from the imaginary hopping \(it_{ij}\) where this imaginary hopping competes with the interaction leading to the unique properties of the considered system. However, the effective spin Hamiltonian of Eq. (41) is Hermitian in large U limit, where the introduction of such non-Hermiticity will significantly alter the magnetic correlation of the parent Hermitian system. Thus, the spin current operator obtained above Eq. (42), is related to the parent Hermitian system in same way the ground state with \(\eta \)-pairing superconductivity is obtained in this non-Hermitian setting2.

The current response function \(\mathfrak {F}({{k}},\omega )\) at non-zero T is given by32

$$\begin{aligned} \mathfrak {F}({{k}},\omega )=\frac{i}{ N}\int _{0}^{\infty }dte^{i\omega t}\langle 0| [\mathcal {J}_x({{k}},t),\mathcal {J}_x(-{{k}},0)]|0\rangle , \end{aligned}$$
(43)

where \(\mathfrak {F}({{k}}=0,\omega \rightarrow 0)\) is the susceptibility or retarded Green’s function32. The retarded Green’s function or dynamical correlation function is obtained after performing an analytical calculation, where we get the result

$$\begin{aligned} {\Lambda }_{{\textbf{k}}}(\omega )=\frac{1}{\pi ^2}\int _{0}^{2\pi }d\omega _1\mathfrak {F}_0({k},\omega _1)\tilde{\mathfrak {F}}_0({k},\omega -\omega _1), \end{aligned}$$
(44)

where \(\mathfrak {F}_0\), \(\tilde{\mathfrak {F}}_0\) are the bare propagators.

$$\begin{aligned} \mathfrak {F}_0({k},\omega )= \frac{1}{\omega -\xi _{{\textbf{k}}}+i0^{+}},\hspace{0.5cm}\tilde{\mathfrak {F}}_0({k},\omega )= -\frac{1}{\omega +\xi _{{\textbf{k}}}-i0^{+}}. \end{aligned}$$
(45)

\({\Lambda }_{{\textbf{k}}}(\omega )\) is the Fourier transform of \({\Lambda }_{{\textbf{k}}}(t)\), which is the dynamical correlation function. Thus, we get the regular part of the longitudinal spin conductivity \(\sigma ^{reg}(\omega )\) as

$$\begin{aligned} \hspace{-0.0cm}\sigma ^{reg}(\omega )=\sum _{{k}}\frac{\cos ^2(k_x)}{\xi _{{\textbf{k}}}} n(\xi _{{\textbf{k}}})\left( 1-e^{\xi _{{\textbf{k}}}/T}\right) \delta (\omega -2\xi _{{\textbf{k}}}),\nonumber \\ \hspace{-1.0cm}\sigma ^{reg}(\omega )=\sum _{{k}}\frac{\cos ^2(k_x)}{\xi _{{\textbf{k}}}}\tanh \left( \frac{\xi _{{\textbf{k}}}}{2T}\right) \delta (\omega -2\xi _{{\textbf{k}}}). \end{aligned}$$
(46)

The Drude’s weight is given by

$$\begin{aligned} \hspace{-0.0cm}D_S(T)\approx -\frac{1}{N}\sum _{k}\sin (k_x)\tanh \left( \frac{\xi _{{\textbf{k}}}}{2T}\right) , \end{aligned}$$
(47)

where \(n(\xi _{{\textbf{k}}})=1/(e^{\beta \xi _{{\textbf{k}}}}+ 1)\) is the fermions occupation number. In Figs. 5 and 6, we present the behavior of the Drude’s weight and \(\sigma ^{reg}(\omega )\) for different values of non-Hermitian coupling \(\delta \). We obtain the AC conductivity tending to zero at \(\omega \rightarrow 0\) however, as we have \(\sigma (0)=D_S\delta (\omega )\) and since that we obtain a \(D_S\) finite, we must have a divergence for the DC current. However, the scattering among particles must introduce a spreading in the conductivity where in a real system the conductivity must to stay finite. In Figs. 5 and 6, we analyzed the conductivity for the non-Hermitian model Eq. (41). In this case, we get a divergence in the continuum conductivity at DC limit, \(\omega \rightarrow 0\). The behavior obtained for the AC conductivity is due to the form of Eq. (46), which is a very complicated expression of k, involving thus, many processes that depends on k. Moreover, as we obtain a finite Drude’s weight for all values of T, we have a Dirac’s delta peak for the conductivity at \(\omega =0\) and consequently, we get that the transport is ideal in this point (\(\omega =0\)) for all values of T. For values nonzero of \(\omega \) (\(\omega \ne 0\)), we get a decreasing in the conductivity for higher values of T and \(\omega \), although this behavior is only qualitative due to approach used.

In all cases analyzed, the influence of dispersionless flat modes on longitudinal spin conductivity is only to give rise to a Dirac’s delta-like peak at frequency \(\omega =\xi _{{\textbf{k}}}\), where \(\xi _{{\textbf{k}}}\) is a plane mode in each case. Moreover, the presence of large peaks in the AC spin conductivity and a finite Drude’s weight \(D_S(T)\), indicate a supercurrent behavior for the system although, for we have a superconductor behavior is necessary that the system exhibits the Meissner effect as well37.

Figure 6
figure 6

Drude’s weight \(D_S(T)\) as a function of T for different values of non-Hermitian coupling U, for the one-dimensional non-Hermitian Hubbard model. The Drude’s weight is finite for all T/J indicating thus, an ideal conductor for all T values.

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Derivation of non-Hermitian Hamiltonian from Lindblad equation

To derive a non-Hermitian Hamiltonian from a Hermitian system interacting with a bath, we can start from the Lindblad master equation \(d\rho /dt=-i[\mathcal {H},\rho ]+\sum _{k}\left( L_k\rho L^{\dag }_k-1/2\{L^{\dag }_kL_k\rho \}\right) \), where \(\rho \) is the density matrix, \(\mathcal {H}\) is the Hermitian Hamiltonian, and \(L_k\) are the Lindblad operators representing the interaction with the bath. By focusing on the coherent part of the evolution and neglecting the quantum jumps which correspond to the stochastic terms \(\mathcal {H}_{eff}=\mathcal {H}-\frac{i}{2}\sum _{k}L^{\dag }_kL_k\). This non-Hermitian Hamiltonian captures the dissipation through the imaginary part, which accounts for the loss or decay rates due to the interaction with the environment. In general the exceptional points in non-Hermitian systems play a crucial role due to their unique mathematical and physical properties, where close to them, the system’s response to external perturbations is significantly amplified. This sensitivity can be harnessed in precision measurement and sensing applications, where small changes in the environment can induce large changes in the system’s observable properties. Moreover, exceptional points exhibits topological characteristics, such as phase transitions and non-trivial winding numbers in parameter space. These features can lead to robust and exotic phenomena like unidirectional invisibility and chiral transport. Furthermore, they enable novel ways to control quantum systems, such as adiabatic encircling around exceptional points, which can lead to non-reciprocal state transfer and enhanced state manipulation capabilities.

Outlook

In brief, we present the spin transport theory for the spin-1/2 and spin-1 one-dimensional non-Hermitian XXZ model and the electrical transport theory of strongly correlated electrons systems modelled by the one-dimensional non-Hermitian Hubbard model. We get the variation of the transport coefficients with the non-Hermitian coupling parameters of the system. Comparisons with existing theoretical predictions or experimental data for the spin transport in the half-integer spin and integer spin non-Hermitian one-dimensional XXZ models as well as for the electrical transport in the one-dimensional non-Hermitian Hubbard model can be performed. As far as I know, there are none experimental result for the spin transport and electrical transport for the non-Hermitian models studied here. In a general way, in quantum spin systems either real fields or complex fields generate a splitting of degenerate ground states, where the spins are aligned along the direction of the external magnetic field38,39,40,41,42,43,44. However, the eigenvalues and eigenvectors of the system with real spectrum will not suffer many changes with the external magnetic field and the initial state display a oscillating behavior and periodic among all possible spin configurations.

By end, non-Hermitian systems offer a complementary perspective by incorporating dissipation and gain directly into the system’s Hamiltonian. In general, a non-Hermitian Hamiltonian can be derived from a Hermitian system interacting with a bath by focusing on the system’s effective dynamics and neglecting the stochastic nature of quantum jumps. Thus, the approach proved here can provide new insights into the behavior of open quantum systems, such as spectral properties, where non-Hermitian Hamiltonians can have complex eigenvalues, where the real part corresponds to the energy levels and the imaginary parts represent decay rates or lifetimes of states. This dual nature allows for a more detailed understanding of the system’s stability and transient behavior. Dynamical Features, where non-Hermitian systems can exhibit unique dynamical phenomena, such as non-Hermitian skin effects, where eigenstates are exponentially localized at the system’s boundaries, and non-reciprocal transport, where the flow of particles or energy differs depending on the direction and finally, exceptional points, which are degeneracies where both the eigenvalues and eigenvectors of the Hamiltonian coalesce. Near these points, the system shows highly sensitive responses to perturbations, that can be exploited for enhanced sensing and control of quantum states. Dealing with formally non-Hermitian systems derived from Lindblad equations by ignoring quantum jumps provides new information about the spectral properties, dynamical features, and sensitivity of open quantum systems. In particular, exceptional points introduce unique opportunities for enhanced sensing, control, and exploration of topological effects in quantum dynamics.