⇦Chinese Physics Letters, 2022, Vol. 39, No. 1, Article code 010301Express Letter Anomalous Transport Induced by Non-Hermitian Anomalous Berry Connection in Non-Hermitian Systems Jiong-Hao Wang (王炅昊)1, Yu-Liang Tao (陶禹良)1, and Yong Xu (徐勇)1,2* Affiliations 1Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, China 2Shanghai Qi Zhi Institute, Shanghai 200030, China Received 12 November 2021; accepted 12 December 2021; published online 16 December 2021 *Corresponding authors. Email: yongxuphy@tsinghua.edu.cn Citation Text: Wang J H, Tao Y L, and Xu Y 2022 Chin. Phys. Lett. 39 010301    Abstract Non-Hermitian materials can exhibit not only exotic energy band structures but also an anomalous velocity induced by non-Hermitian anomalous Berry connection as predicted by the semiclassical equations of motion for Bloch electrons. However, it is unclear how the modified semiclassical dynamics modifies transport phenomena. Here, we theoretically demonstrate the emergence of anomalous oscillations driven by either an external dc or ac electric field, which arise from non-Hermitian anomalous Berry connection. Moreover, it is a well-known fact that geometric structures of electric wave functions can only affect the Hall conductivity. However, we are surprised to find a non-Hermitian anomalous Berry connection induced anomalous linear longitudinal conductivity independent of the scattering time. We also show the emergence of a second-order nonlinear longitudinal conductivity induced by non-Hermitian anomalous Berry connection, violating a well-known fact of its absence in a Hermitian system with symmetric energy spectra. These anomalous phenomena are illustrated in a pseudo-Hermitian system with large non-Hermitian anomalous Berry connection. Finally, we propose a practical scheme to realize the anomalous oscillations in an optical system.
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 DOI:10.1088/0256-307X/39/1/010301 © 2022 Chinese Physics Society Article Text Non-Hermitian physics has recently been one of active subjects intensively studied in various branches of physics ranging from optical and acoustic systems, cold atomic systems to condensed matter materials.[1–5] Existence of non-Hermitian terms in a Hamiltonian, such as gain or loss, can lead to exotic energy band structures that have no counterparts in Hermitian systems, such as band structures with exceptional points or rings.[6–19] Recently, it has been shown that non-Hermitian physics may also arise in disordered or strongly correlated systems due to finite lifetimes of quasiparticles.[20–27] This motivates us to ask how transport phenomena are modified in non-Hermitian systems. The semiclassical dynamics of Bloch electrons in external fields has proven to be a powerful theoretical framework to account for various transport properties.[28–33] For instance, the semiclassical equations of motion predict an anomalous transverse velocity arising from the geometric structures of electric wave functions.[31] The geometric structures are involved in the semiclassical equations of motion in terms of Berry curvature rather than Berry connection, which is gauge dependent. Such an anomalous transverse velocity can only induce a Hall current instead of a longitudinal current. Given that non-Hermitian physics can exist in various systems, it is important to ask how the semiclassical dynamics should be modified in a non-Hermitian system. In fact, one of the authors has derived the following semiclassical equations of motion for Bloch electrons in an external electric force $\boldsymbol {F}=-e{E}_\mu {\boldsymbol e}_\mu$ in Ref. [7] (also see Refs. [2,34,35]): $$\begin{alignat}{1} \dot { {r}}_\lambda={}&\partial_\lambda \bar{\varepsilon}_n- e\epsilon_{\lambda\mu\nu}\varOmega_{n,\mu}{E}_\nu, \label{SME1a}~~ \tag {1a}\\ \dot {{k}}_\lambda={}&-e{E}_\lambda,~~ \tag {1b} \end{alignat} $$ where $\boldsymbol {r}=r_\mu{\boldsymbol e}_\mu$ and $\boldsymbol {k}=k_\mu{\boldsymbol e}_\mu$ denote the center of mass of a wave packet in real and momentum space, respectively, $\epsilon_{\lambda\mu\nu}$ is the Levi–Civita symbol, and ${\boldsymbol e}_\mu$ is the unit vector along the $\mu$ direction ($\mu=x,y,z$). Here, to simplify notations, we have set $\hbar=1$, defined $\partial_\lambda=\partial_{k_\lambda}$, adopted the Einstein summation convention, and will set the lattice constant to one henceforth. $\varOmega_{n,\lambda}=i\varepsilon_{\lambda\mu\nu} \langle{\partial_\mu u_{n}^R}| {\partial_\nu u_{n}^R}\rangle$ is the Berry curvature in the $n$th band, which accounts for the intrinsic anomalous Hall effects (note that only the Berry curvature defined by the right eigenstates is relevant to the velocity[35]). Here, $|{u_{n}^R}\rangle$ is the normalized right eigenstate of a generic non-Hermitian Hamiltonian $H({\boldsymbol k})$ in momentum space in the $n$th band, i.e., $H({\boldsymbol k})|{u_{n}^R(\boldsymbol k)}\rangle=\varepsilon_n({\boldsymbol k}) |{u_{n}^R(\boldsymbol k)}\rangle$ with $\langle{u_{n}^R(\boldsymbol k)}|u_{n}^R(\boldsymbol k)\rangle=1$. In fact, for a non-Hermitian Hamiltonian, there appears a normalized left eigenstate $\langle{ {u}_{n}^L(\boldsymbol k)}|$ satisfying $\langle{ {u}_{n}^L(\boldsymbol k)}|H(\boldsymbol {k})=\langle{ {u}_{n}^L(\boldsymbol k)}| \varepsilon_n({\boldsymbol k})$ and $\langle {u}_n^L(\boldsymbol {k})|u_n^R(\boldsymbol {k})\rangle=1$, which coincides with the Hermitian conjugation of the corresponding right eigenstate in the Hermitian case. The emergence of the different left eigenstate leads to an effective energy spectra (note that the second part does not contribute to the distribution function): $$ \bar{\varepsilon}_n({\boldsymbol {k}})={\rm Re}[\varepsilon_n({\boldsymbol {k}})] -e\bar{{A}}_{n,\mu}(\boldsymbol {k}){E}_\mu,~~ \tag {2} $$ where a non-Hermitian anomalous Berry connection (NHABC) arises, $$ \bar{{A}}_{n,\mu}(\boldsymbol {k})\equiv{\rm Re}[{A}_{n,\mu}(\boldsymbol {k})-\tilde{{A}}_{n,\mu}(\boldsymbol {k})].~~ \tag {3} $$ Clearly, the Berry connection is involved in the equation through the difference of the right-right Berry connection ${A}_{n,\mu}(\boldsymbol {k})=i\langle{u}_n^R(\boldsymbol {k})|\partial_\mu u_n^R(\boldsymbol {k})\rangle$ and the left-right Berry connection $\tilde{{A}}_n(\boldsymbol {k})=i\langle{u}_n^L(\boldsymbol {k})|\partial_{\mu}u_n^R(\boldsymbol {k})\rangle$, showing the fact that this term can only appear in non-Hermitian systems. Such a term is nonzero in a generic non-Hermitian system [except in a $\mathcal{PT}$ (product of inversion and time-reversal symmetry) or $C_2\mathcal{T}$ (product of two-fold rotational and time-reversal symmetry) symmetric system].[35] Based on Eq. (1a), this term leads to a non-Hermitian anomalous velocity $$ {v}_{\scriptscriptstyle{\rm NA},\lambda}=-e{E}_\mu \partial_\lambda \bar{{A}}_{n,\mu}.~~ \tag {4} $$ Despite the fact that the semiclassical equations of motion have been derived in Refs. [7,34], it remains an important open question of whether such a modified dynamics will result in anomalous transport. In this Letter, we study two classes of transport phenomena: coherent dynamics of one electron, and linear and nonlinear conductivities of many electrons. We find that the existence of the non-Hermitian anomalous velocity results in anomalous features in oscillations driven by either dc or ac electric fields. For the linear longitudinal conductivity, we are surprised to find an NHABC induced anomalous longitudinal conductivity that is independent of the scattering time. In addition, it is a well-known fact that in a Hermitian system with symmetric energy spectra, a second-order nonlinear longitudinal conductivity is forced to vanish. Remarkably, we find a second-order nonlinear longitudinal conductivity induced by the NHABC. These results suggest that the geometric structures of wave functions can induce not only a Hall current but also a longitudinal current in a non-Hermitian system. We demonstrate these anomalous phenomena in a pseudo-Hermitian system with large NHABC. Model. To demonstrate the anomalous transport properties, we start by studying the NHABC in a one-dimensional (1D) two-band non-Hermitian system described by the following Hamiltonian in momentum space: $$ H(k)={\boldsymbol d}\cdot{\boldsymbol {\bar\sigma}}+d_0,~~ \tag {5} $$ where $$ \bar{\sigma}_x=\left( \begin{array}{cc} 0 & a \\ b & 0 \\ \end{array} \right), ~\bar{\sigma}_y=\left( \begin{array}{cc} 0 & -ia \\ ib & 0 \\ \end{array} \right), ~\bar{\sigma}_z=\left( \begin{array}{cc} q^{-1} & 0 \\ 0 & -q \\ \end{array} \right)~~ \tag {6} $$ are $q-$deformed Pauli matrices with $a=\sqrt{(1+q^2)/2}$, $b=\sqrt{(1+q^{-2})/2}$ and $q>0$.[36,37] Note that such matrices have also been used to construct non-Hermitian Chern insulators, Weyl semimetals and chiral topological insulators.[37] Let us first consider a system with time-reversal symmetry with $$ d_x=t_0+t_1\cos k,~ d_y=t_2\sin k, ~d_z=m, ~d_0=0,~~ \tag {7} $$ where $t_0$, $t_1$, $t_2$ and $m$ are real parameters. Although the Hamiltonian is non-Hermitian when $q\neq 1$, it is pseudo-Hermitian,[38] and its energies $\varepsilon_{\pm}$ are real.[35] Such a real energy spectrum indicates the absence of the skin effects due to the absence of the winding number even though the Hamiltonian in real space has asymmetric hopping.[39–41] Although the system is topologically equivalent to its Hermitian counterpart, we find that the NHABC emerges, that is, $$ \bar{A}_\pm(k)=\frac{b(a-b)(t_2 d_x\cos k+t_1 d_y \sin k)(\xi_\pm+mc_1)}{2\xi_\pm[(\xi_\pm+mc_1)^2+b^2(d_x^2+d_y^2)]},~~ \tag {8} $$ where $\xi_{\pm}=\pm \sqrt{ab(d_x^2+d_y^2)+m^2c_1^2}$ and $c_1=(1+q^2)/(2q)$. For simplicity, we consider $m=0$ so that the expression reduces to $$ \bar{A}_\pm(k)=c_2\frac{t_2(t_1+t_0\cos k)}{(t_0+t_1\cos k)^2+t_2^2 \sin^2 k},~~ \tag {9} $$ where $c_2=-(1-q)/[2(1+q)]$ ($|c_2| < = 1/2$). In this case, $\bar{A}_{\pm}$ does not depend on the band index, and we thus drop the band index henceforth. Clearly, when $q=1$, the Hamiltonian is Hermitian so that the term vanishes. Specifically, consider the NHABC at $k=0$ or $\pi$, which reads $\bar{A}(0)=c_2 t_2/(t_0+t_1)$ and $\bar{A}(\pi)=-c_2 t_2/(t_0-t_1)$. Remarkably, one of the terms diverges when either $t_0+t_1=0$ or $t_0-t_1=0$. When $|t_0/t_1| < 1$, we find that $\bar{A}_{\pm}=c_2t_1/t_2$ at $k={\arccos}(-t_0/t_1)$, which diverges at $t_2=0$. We therefore conclude that the NHABC can be large in the model. We remark that when $\bar{A}$ diverges, the energy gap also closes at the corresponding momentum. With an energy gap, $\bar{A}$ is distributed around the momenta associated with minima of direct gaps, as shown in Fig. 1(a). Anomalous Oscillations in an External dc Electric Field. We now study the dynamics of a wave packet in a periodic potential subject to a dc electric field so that the wave packet undergoes a Bloch oscillation. Without loss of generality, we consider a 1D case. Based on the semiclassical equations of motion (1)a), the position and quasimomentum of a wave packet evolves as $$\begin{alignat}{1} x(t)={}&-\frac{1}{e{E}}[\bar{\varepsilon}[k(t)]-\bar{\varepsilon}(k_0)] =x_{_{\scriptstyle \mathrm{H}}}(t)+x_{\scriptscriptstyle{\rm NA}}(t),~~~~ \tag {10a}\\ {k}(t)={}&-e{E} t+{k}_0.~~ \tag {10b} \end{alignat} $$ Here, we have set $x(0)=0$, and $k_0$ is the initial quasimomentum of the wave packet. The position is determined by two parts: the weighted decrease in the real part of the energy spectrum, $x_{_{\scriptstyle \mathrm{H}}}(t)=-{\rm Re}[\varepsilon({{k(t)}})-\varepsilon({{k_0}})]/(e{E})$, and the increase in the NHABC, $x_{\scriptscriptstyle{\rm NA}}(t)=\bar{{A}}[{k}(t)]-\bar{{A}}({k_0})$. In a Hermitian system, $x_{\scriptscriptstyle{\rm NA}}$ vanishes so that the position is entirely determined by $x_{_{\scriptstyle \mathrm{H}}}(t)$, which exhibits an oscillation with a period of $T_{\mathrm{B}}=2\pi/(e{E})$. Another feature in the Hermitian case is that besides at the time of integer multiples of the period at which the wave packet returns to the initial position, this also occurs at other return times $t_{\rm r}$ with ${\rm Re}[\varepsilon({{k(t_{\rm r})}})]={\rm Re}[\varepsilon({{k_0}})]$. For example, consider the energy spectrum in Fig. 1(b). The energies at $k_0$ and $k_{\rm r}$ are equal so that the wave packet must move back to the original position at time $t_{\rm r}$ when $k(t_{\rm r})=k_{\rm r}$. Similarly, the energies at $k_0^\prime$, $k_{r1}$, $k_{r2}$ and $k_{r3}$ are equal, leading to the fact that the return happens at times $t_{r1}$, $t_{r2}$ and $t_{r3}$ corresponding to $k(t_{r1})=k_{r1}$, $k(t_{r2})=k_{r2}$ and $k(t_{r3})=k_{r3}$. However, for a non-Hermitian system, the rule is generically violated due to the contribution of $x_{\scriptscriptstyle{\rm NA}}(t)$, in the sense that even though $x_{_{\scriptstyle \mathrm{H}}}$ vanishes, $\bar{{A}}({k_{\rm r}})$ is not necessarily equal to $\bar{{A}}({k_0})$. In fact, the return times are shifted or lifted depending on $x(t)=0$. Indeed, Fig. 1(c) illustrates that at time $t_{\rm r}=T_{\mathrm{B}}/2$, $x(t_{\rm r})$ is lifted so that $x(t)>0$ when $t\in(0,T_{\mathrm{B}})$, in stark contrast to the Hermitian case shown by the blue line.
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Fig. 1. (a) The computed NHABC and (b) energy spectra versus $k$. In (b), the circles denote the states with the same energy in a certain band. The time evolution of the center of mass of a wave packet in real space (c) under a dc electric field and (d1)–(e1) under a sinusoidal ac electric field with the corresponding $g$ function plotted in (d2) and (e2), respectively. The electric field is ${E}=0.2$. In (c), $k(0)=0$. In (d1) and (d2), $k(0)=0$ and $k(T/2)=-\pi$. In (e1) and (e2), $k(0)=-0.2$ and $k(T/2)=-1.3$. The blue and red lines and solid circles show the results obtained by numerically solving the Schrödinger equation, while the grey ones show the results obtained by numerically solving the semiclassical equations of motion (1a). The results imply that the semiclassical dynamics agrees well with the dynamics of a wave packet governed directly by the Schrödinger equation. The blue (red) lines or solid circles correspond to the results for the Hamiltonian (5) with parameters (7) and $q=1$ (Hermitian) [$q=0.1$ (non-Hermitian)], respectively. Here $t_0=1$, $t_1=0$ and $t_2=2$.
Anomalous Oscillations in an External ac Electric Field. To exhibit anomalous oscillations in an ac electric field, we require that the field should be either positive or negative in each half cycle and is antisymmetric with respect to $T/2$ ($T$ is the time period), i.e., $E(t)=-E(T-t)$. The requirements are naturally satisfied by commonly used ac electric fields, such as sinusoidal, triangular and square waveforms. With the electric field, the quasimomentum of a wave packet center first moves from $k_0=k(0)$ to $k_m=k(T/2)$ in the first half cycle and then returns to $k_0$ in the second half cycle. One can also prove that the displacements of the wave packet over the first and second half cycles in a Hermitian system are equal, i.e., $x_H(T/2)-x_H(0)=x_H(T)-x_H(T/2)$ based on the result $k(t)=k(T-t)$ for $0\le t\le T/2$. Thus, we can define the displacement as a discrete function $$ g(n)=x[(n+1)T/2]-x(nT/2),~~ \tag {11} $$ which is a constant function in the Hermitian case. For example, for a dynamics of a wave packet in a Hermitian system shown by the blue lines in Figs. 1(d1)–1(e1), the associated $g$ functions shown in Figs. 1(d2)–1(e2) are constant functions. In a specific case with ${\rm Re}[\varepsilon(k_0)]={\rm Re}[\varepsilon(k_m)$], $g(n)=0$, showing that a wave packet returns to the initial position over each half cycle as shown in Fig. 1(d2). However, in the non-Hermitian case, $$ g(n)=C+(-1)^n [\bar{{A}}({k_m})-\bar{{A}}({k_0})].~~ \tag {12} $$ Here $C=x_{_{\scriptstyle \mathrm{H}}}[(n+1)T/2]-x_{_{\scriptstyle \mathrm{H}}}(nT/2)$, which is constant contributed by the energy dispersion. $C$ vanishes when ${\rm Re}[\varepsilon(k_0)]={\rm Re}[\varepsilon(k_m)$]. It is clear to see that $g$ is no longer a constant function when $\bar{{A}}({k_m})\neq \bar{{A}}({k_0})$, and it varies with respect to $n$ with a period of $2$. The period change can be clearly seen in the $g$ functions (red solid circles) shown in Figs. 1(d2)–1(e2) for a dynamics of a wave packet in a non-Hermitian system shown by the red lines in Figs. 1(d1)–1(e1), Such a period change of the function can be directly measured in experiments. Anomalous Linear and Nonlinear Longitudinal Conductivities. To investigate the electric response to an electric field for many electrons in a system with impurities, we employ the semiclassical equations of motion together with the Boltzmann equation. In the relaxation time approximation, the Boltzmann equation for the distribution function $f$ of electrons is $$ -e\tau E_\mu \partial_\mu f+\tau \partial_t f=f_0-f,~~ \tag {13} $$ where $f_0$ is the equilibrium distribution function without external fields, and $\tau$ is the scattering time. In a non-Hermitian case, $f_0[\varepsilon({\boldsymbol k})]=\{\exp [({\rm Re}[\varepsilon({\boldsymbol k})]-\mu)/(k_{\scriptscriptstyle{\rm B}} T)]+1\}^{-1}$ corresponds to the Fermi–Dirac distribution for the real part of the eigenenergy with $\mu$ being the chemical potential and $T$ being the temperature; the imaginary part of the eigenenergy plays the role of the scattering time. For generality, we consider an ac electric field, i.e., $E_\mu={\rm Re}\{\mathcal{E}_{\mu} e^{i\omega t}\}$, with $\omega$ being the angular frequency. To see the effects of the NHABC, we expand the distribution function up to the second order: $f={\rm Re}\{f_0+f_1+f_2 \}$ with $f_1=g_{1}^{\omega}e^{i\omega t}$ and $f_2=g_2^{0} +g_2^{2\omega} e^{i2\omega t}$. Based on the above Boltzmann equation, one obtains[33] $$\begin{align} g_1^\omega ={}&\frac{e\tau \mathcal{E}_\mu \partial_\mu f_0}{1+i\omega \tau},~~ g_2^0=\frac{(e\tau)^2 \mathcal{E}_\mu^* \mathcal{E}_\nu \partial_{\mu\nu}f_0}{2(1+i\omega \tau)}, \\ g_2^{2\omega}={}&\frac{(e\tau)^2 \mathcal{E}_\mu \mathcal{E}_\nu \partial_{\mu \nu}f_0}{2(1+2i\omega\tau)(1+i\omega\tau)}.~~ \tag {14} \end{align} $$ Combined with the semiclassical equation, we derive the current $$j_\lambda=-e\int_k fv_\lambda={\rm Re}\{j_\lambda^0+j_\lambda^{\omega}e^{i\omega t}+j_\lambda^{2\omega} e^{i2\omega t}\}, $$ with $$\begin{alignat}{1} j_\lambda^0 &={e}\int_k (-g_2^0 \partial_\lambda {\rm Re}(\varepsilon)+\frac{e}{2} g_1^\omega \bar{\varOmega}_{\lambda\nu} \mathcal{E}_\nu^*),~~ \tag {15a}\\ j_\lambda^{\omega} &= e\int_k (-g_1^{\omega} \partial_\lambda {\rm Re}(\varepsilon) + ef_0\bar{\varOmega}_{\lambda\nu}\mathcal{E}_\nu),~~ \tag {15b}\\ j_\lambda^{2\omega} &= {e}\int_k (-g_2^{2\omega}\partial_\lambda {\rm Re}(\varepsilon)+\frac{e}{2} g_1^{\omega}\bar{\varOmega}_{\lambda\nu}\mathcal{E}_\nu),~~ \tag {15c} \end{alignat} $$ where the terms $j_\lambda^0$, $j_\lambda^{\omega}$ and $j_\lambda^{2\omega}$ describe the rectified, first harmonic and second harmonic currents, respectively, and $\int_k\equiv \int_{_{\scriptstyle \mathrm{BZ}}} d^d k/(2\pi)^d$, an integral over the first Brillouin zone with $d$ being the dimension of a system. Here we have introduced a new quantity $$ \bar{\varOmega}_{\lambda\nu}=\varepsilon_{\lambda\mu\nu}\varOmega_\mu +\partial_\lambda\bar{A}_\nu,~~ \tag {16} $$ where the first term is associated with a Berry curvature dipole resulting in nonlinear Hall effects,[33] and the second term results from the NHABC. When $\lambda=\nu$, the quantity is completely determined by the NHABC, i.e., $\bar{\varOmega}_{\nu\nu}=\partial_\nu\bar{A}_\nu$. For simplicity, we utilize the constant relaxation time approximation. Consider a system with ${\rm Re}[\varepsilon(-{\boldsymbol k})]={\rm Re}[\varepsilon({\boldsymbol k})]$, such as a system with either time-reversal symmetry or inversion symmetry. Then, $\partial_{\mu\nu}f_0 \partial_\lambda {\rm Re}(\varepsilon)$ is an odd function with respect to $\boldsymbol k$, forcing the corresponding integrals to vanish. We hence obtain $j_\lambda^0=\chi_{\scriptscriptstyle\lambda\mu\nu}\mathcal{E}_\mu\mathcal{E}_\nu^*$, $j_\lambda^{\omega}=\sigma_{\lambda\mu}\mathcal{E}_\mu$, $j_\lambda^{2\omega}=\chi_{\scriptscriptstyle\lambda\mu\nu}\mathcal{E}_\mu\mathcal{E}_\nu$, where $\sigma_{\lambda\mu}$ is the linear conductivity tensor with $$ \sigma_{\lambda\mu}=e\int_k \Big[-\frac{e\tau \partial_\mu f_0}{1+i\omega \tau} \partial_\lambda {\rm Re}(\varepsilon) + ef_0\bar{\varOmega}_{\lambda\mu}\Big],~~ \tag {17} $$ and $\chi_{\scriptscriptstyle\lambda\mu\nu}$ is the second-order nonlinear conductivity tensor with $$\begin{align} \chi_{\scriptscriptstyle\lambda\mu\nu}={}&\frac{e^3\tau}{2(1+i\omega \tau)}\int_k \partial_\mu f_0 \bar{\varOmega}_{\lambda\nu}\\ ={}&-\frac{e^3\tau}{ 2(1+i\omega \tau) }\int_k f_0 \bar{D}_{\lambda\mu\nu},~~ \tag {18} \end{align} $$ where $\bar{D}_{\lambda\mu\nu}=\partial_\mu\bar{\varOmega}_{\lambda\nu}= \epsilon_{\lambda \alpha \nu}\partial_\mu \varOmega_\alpha+\partial_{\lambda\mu}\bar{A}_\nu$ with the first term being the local Berry curvature dipole[33] and the second term induced by the NHABC.
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Fig. 2. (a) The second-order nonlinear longitudinal conductivity $\chi$ versus the chemical potential $\mu$ for the 1D Hamiltonian (5) in terms of parameters in (7) with time-reversal symmetry at zero temperature. Here, $t_0=1$, $t_1=6$ and $t_2=0.7$. (b) The linear conductivity $\sigma^{\mathrm{NA}}=\sigma_{xx}^{\mathrm{NA}}$ resulted from the NHABC versus the chemical potential $\mu$ for the 1D Hamiltonian (5) with $d_0=t_3\cos(k)$, $d_x=t_0+t_1\cos(k)$, $d_y=t_2\cos(k)$ and $d_z=0$ at zero temperature. Here, $t_0=1$, $t_1=6$, $t_2=2$ and $t_3=5$. Inset in (b): $t_3=-5$ so that the conductivity becomes negative.
We now focus on the longitudinal conductivity. It is a well-known fact that for a Hermitian system with symmetric energy spectra with $\varepsilon(-{\boldsymbol k})=\varepsilon({\boldsymbol k})$, there does not exist a second-order nonlinear longitudinal conductivity. However, Eq. (15a) remarkably shows that a second-order nonlinear longitudinal conductivity arises in a non-Hermitian system due to the geometric structures of wave functions, i.e., $$ \chi_{\scriptscriptstyle{\lambda\lambda\lambda}}=-\frac{e^3\tau }{ 2(1+i\omega \tau) }\int_k f_0 \partial^2_\lambda\bar{A}_\lambda.~~ \tag {19} $$ In the dc limit, $\chi$ scales as $\tau$ instead of $\tau^2$. At high frequencies $\omega\tau \gg 1$ but below the interband transition threshold, the prefactor in $\chi$ is independent of the scattering time so that the nonlinear longitudinal conductivity directly measures the geometric structure in the NHABC. With time-reversal symmetry, the nonlinear longitudinal conductivity can be nonzero due to the fact that $\bar{A}_{\lambda}(-{\boldsymbol k})=\bar{A}_{\lambda}({\boldsymbol k})$ enforced by time-reversal symmetry.[35] However, it is forced to vanish in an inversion symmetric system because inversion symmetry imposes constraints that $\bar{A}_{\lambda}(-{\boldsymbol k})=-\bar{A}_{\lambda}({\boldsymbol k})$,[35] making $f_0 \partial^2_\lambda\bar{A}_\lambda$ an odd function. In Fig. 2(a), we plot $\chi=\chi_{xxx}$ for a 1D Hamiltonian with time-reversal symmetry, showing that a significant nonlinear longitudinal conductivity arises when the Fermi energy is close to the band edge. Besides the nonlinear longitudinal conductivity, we are surprised to find that the NHABC can induce a linear longitudinal conductivity, $$ \sigma_{\lambda\lambda}^{\mathrm{NA}}=e^2\int_k f_0 \partial_\lambda\bar{A}_\lambda,~~ \tag {20} $$ which is independent of frequencies and the scattering time. Due to the constraint imposed by time-reversal symmetry, this conductivity is forced to vanish in a time-reversal invariant system. Instead, we consider a Hamiltonian that breaks time-reversal symmetry and exhibits antisymmetric $\bar{A}$; the NHABC induced linear conductivity reaches maxima when the Fermi surface is near the band edges as illustrated in Fig. 2(b). There, the conductivity becomes negative when either $t_3 < 0$ or $q < 1$. Before closing this section, we wish to briefly discuss the NHABC induced Hall effects in two dimensions. To have nonzero linear Hall effects, one has to break time-reversal symmetry. With time-reversal symmetry, the linear Hall effects are forced to vanish, and the nonlinear Hall effects are attributed to $\bar{\varOmega}_{\lambda\nu}$. To characterize the Hall current, we define three Hall pseudovectors as $d_\lambda^{(1)}=\epsilon_{\mu\nu}\chi_{\mu\nu\lambda}/2$ with $\epsilon_{\mu\nu}$ being the 2D Levi–Civita symbol, $d_\lambda^{(2)}=\epsilon_{\mu\nu}\chi_{\mu\lambda\nu}/2$, and $d_\lambda^{(3)}=\epsilon_{\mu\nu}\chi_{\lambda\mu\nu}/2$, which contribute to the Hall current as ${\boldsymbol j}_{1}^0=({\boldsymbol{ \mathcal{E} }} \times {\boldsymbol e}_z) (\boldsymbol{ d}^{(1)}\cdot{\boldsymbol {\mathcal{E}} }^*)/2$, ${\boldsymbol j}_1^{2\omega}=({\boldsymbol{ \mathcal{E} }} \times {\boldsymbol e}_z) (\boldsymbol{ d}^{(1)}\cdot{\boldsymbol {\mathcal{E}} })/2$, ${\boldsymbol j}_2^0=({\boldsymbol{ \mathcal{E} }^*} \times {\boldsymbol e}_z) (\boldsymbol{ d}^{(2)}\cdot{\boldsymbol {\mathcal{E}} })/2$, ${\boldsymbol j}_2^{2\omega}=({\boldsymbol{ \mathcal{E} }} \times {\boldsymbol e}_z) (\boldsymbol{ d}^{(2)}\cdot{\boldsymbol {\mathcal{E}} })/2$, ${\boldsymbol j}_3^{0}={\boldsymbol d}^{(3)} [(\boldsymbol{\mathcal{E}\times \boldsymbol{\mathcal{E}}^* })\cdot {\boldsymbol e}_z]/2$ and ${\boldsymbol j}_3^{2\omega}=0$. The NHABC cannot yield nonzero ${\boldsymbol d}^{(1)}$ since $\partial_{\mu\nu}\bar{A}_\lambda=\partial_{\nu\mu}\bar{A}_\lambda$. In summary, we have found anomalous coherent oscillations of a wave packet induced by the NHABC. While we demonstrate our prediction in a pseudo-Hermitian Hamiltonian, the anomalous oscillations in an ac electric field may also be observed in other non-Hermitian systems,[35] such as a system with skin effects,[42,43] given that the dynamics of a wave packet in a non-Hermitian Hamiltonian is independent of boundary conditions.[44] In the Supplementary Material, we also propose a practical scheme with coupled resonator optical waveguides[45–47] to observe the anomalous oscillations. We further provide a generic theory, showing the existence of an NHABC induced anomalous linear longitudinal conductivity independent of the scattering time in a time-reversal symmetry broken system and a second-order anomalous nonlinear longitudinal conductivity in a time-reversal invariant system. Given that non-Hermitian physics can widely exist in disordered or strongly correlated systems (the conductivity may be insensitive to boundary conditions even for a system with skin effects[26]), the anomalous longitudinal conductivities may be observed in these materials. Our work thus opens a new direction for studying anomalous transport phenomena induced by NHABC in non-Hermitian systems. Acknowledgments. We thank T. Qin and Y.-B. Yang for helpful discussions. The work was supported by the National Natural Science Foundation of China (Grant No. 11974201), and the Start-up Fund from Tsinghua University. 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