Chiral Hinge Transports in Disordered Non-Hermitian Second-Order Topological Insulators

Searching various topological phases has attracted tremendous attention because of their fundamental interests and exotic properties for potential applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. One such feature is the emergence of one-dimensional (1D) chiral channels along the edge of two-dimensional topological insulating systems at the Fermi level [11, 12, 13]. Chiral edge states result in quantized conductances for the well-known quantum anomalous Hall effect paradigm [5]. Recently, the study has been expanded to the higher-order topological insulators that follow the generalized bulk-boundary correspondence  [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. One widely investigated example is the strong three-dimensional second-order topological insulator (3DSOTI) that supports 1D chiral hinge states localized at the intersection of two surfaces  [24, 25, 26].

The robustness of these topological phases against disorders is universal [27, 28, 29]. Concerning chiral hinge states in 3DSOTIs, on the one hand, their robustness has been recognized in the study of Hermitian systems [30, 33, 32, 34, 31]. Consequently, disordered Hermitian 3DSOTIs are characterized by the quantized transmission coefficients (equal to the number of chiral channels) and the vanishingly-small fluctuations due to the absence of backward scattering. On the other hand, for non-Hermitian systems, the widely accepted paradigm asserts no quantized transport for chiral edge states in Chern (first-order topological) insulators [35, 36, 37]. The effective Hamiltonians of such chiral edge states are non-Hermitian, and the corresponding eigenenergies are complex numbers. Recently, we have shown that the chiral hinge states of a non-Hermitian 3DSOTI can be Hermitian under certain conditions [38]. So far, the fate and the transport of chiral hinge states in non-Hermitian 3DSOTIs under disorders are still open questions, especially for the Hermitian chiral hinge states.

Here, we numerically investigate the nature of chiral hinge states of disordered 3DSOTIs subject to constant non-Hermitian and random Hermitian on-site potentials. The main findings are as follows: Both Hermitian and non-Hermitian chiral hinge states can persist in the presence of finite disorders. As the disorder strength increases, the non-Hermitian 3DSOTI undergoes first a transition to the non-Hermitian three-dimensional first-order topological insulator (3DFOTIs) featured by the surface states and then follow by another transition to the topologically-trivial metal. The mean transmission coefficients of Hermitian chiral hinge states of non-Hermitian 3DSOTIs are still equal to the number of chiral hinge channels, similar to the chiral hinge states of the Hermitian 3DSOTIs [30], but the transmission coefficients of of the non-Hermitian chiral hinge states can be any number. In both cases, the fluctuations of transmission coefficients are finite and increase with the disorders and the system sizes. Furthermore, the fluctuations of transmission coefficients are not due to the broken chirality of hinge states but from the incoherent scattering by non-Hermitian potentials. The distributions of transmission coefficients of chiral hinge states follow universal forms that depend on the Hermiticity of chiral hinge states. Our findings highlight the importance of the incoherent processes in non-Hermitian topological insulators, which are often ignored in the literature [35, 36, 37], and should apply to other non-Hermitian topological insulators that support chiral boundary states.

The paper is organized as follows: A tight-binding model for the non-Hermitian 3DSOTI is introduced in Sec. II. Non-Hermiticities effect on chiral hinge states in the clean limit is discussed. The study of disorder-driven the quantum phase transitions of non-Hermitian 3DSOTIs is presented in Sec. III. Section IV gives a comprehensive investigation of quantum transports through chiral hinge states in non-Hermitian 3DSOTIs followed by the discussion in Sec. V. The conclusion is given in Sec. VI.

We consider following tight-binding model on the cubic lattices

which belongs to the class A of the Altland-Zirnbauer (AZ) classifications for non-Hermitian systems [39]. Here, $c^{\dagger}_{\bm{i}}$ and $c_{\bm{i}}$ are the single-particle creation and annihilation operators at site $\bm{i}=(n_{1},n_{2},n_{3})$. Particles governed by Eq. (1) can be either fermions like electrons or bosons like magnons. $\hat{x}_{\nu=1,2,3}$ are unit vectors along three Cartesian axes. $\Gamma^{\mu=1,2,3,4,5}=(s_{0}\otimes\sigma_{3},s_{1}\otimes\sigma_{1},s_{3}\otimes\sigma_{1},s_{2}\otimes\sigma_{1},s_{0}\otimes\sigma_{2})$ are four-by-four nonunique gamma matrices that satisfy $\{\Gamma^{\mu},\Gamma^{\nu}\}=2\delta_{\mu,\nu}\Gamma^{0}$ and $\Gamma^{\mu\nu}=[\Gamma^{\mu},\Gamma^{\nu}]/(2i)$, where $\Gamma^{0}$ and $s_{0},\sigma_{0}$ are four-by-four and two-by-two unit matrices, respectively, and $s_{\mu},\sigma_{\mu}$ are the Pauli matrices acting on some subspaces (e.g., spins or orbitals). $t$ is the hopping energy and is used as the energy unit ($t=1$) here. $M$ and $B$ are the masses for controlling the band inversions of bulk states and surface states, respectively. The non-Hermiticity is encoded in the on-site potential $\sum_{\bm{i}}c^{\dagger}_{\bm{i}}(i\sum_{\mu=0,1,2,3,4,5}\gamma_{\mu}\Gamma^{\mu})c_{\bm{i}}$ with $\gamma_{0,1,2,3,4,5}$ measure the strengths of non-Hermitian potentials. $v_{\bm{i}}$ is a white noise that distributes uniformly in the range of $[-W/2,W/2]$.

In the absence of the disorder, our model can be blocked diagonalized in the $\bm{k}$ space due to the Bloch’s theorem, $H=\sum_{\bm{k}}a^{\dagger}_{\bm{k}}h(\bm{k})a_{\bm{k}}$ with

Without the $\gamma$-terms, Hamiltonian (2) describes a Hermitian reflection-symmetric 3DSOTI with the reflection plane of $x=0$ for $0<B<1$ and $1+B<M<3-B$ [30]. Chiral hinge states appear at the boundaries when two surfaces meet at the reflection plane. Namely, if we apply Eq. (2) to a cubic lattice of size $L_{\parallel}/\sqrt{2}\times L_{\parallel}/\sqrt{2}\times L_{\perp}$ with open boundary conditions (OBCs) on surfaces perpendicular to $(110),(1\bar{1}0),(001)$, the in-gap chiral hinge modes are found at the two hinges $x=0,y=\pm L_{\parallel}$.

The chiral hinge states are visible in Fig. 1(a) which displays the energy spectrum of quasiparticles $\text{Re}[E](k_{3})$ of Eq. (1) of $M=2$, $B=0.2$, and $\gamma_{0,1,2,3,4,5}=0$ for a rectangle bar with OBCs on the surfaces perpendicular to $(110)$ and $(1\bar{1}0)$ and periodic boundary condition (PBC) in the $z-$direction. Colors in Fig. 1(a) encode the natural logarithm of the participation ratios $\ln[p_{2}]$. The participation ratio is defined as $p_{2}(E)=1/(\sum_{\bm{i}}\sum^{4}_{p=1}|\psi_{\bm{i},p}(E)|^{4})$ with $\psi_{\bm{i},p}(E)$ being the wave function of a right eigenstate of $H$ of a site $\bm{i}$. The participation ratio measures the numbers of occupied sites of a given state of the eigenvalue $E$ [40]. $\ln[p_{2}]$ for the in-gap hinge states is small (in comparison with the number of lattice sites), see Fig. 1(a).

Non-Hermiticities are introduced through the $\gamma$-terms in Eq. (2). $i\gamma_{0}\Gamma^{0}$ of $\gamma_{0}<0$ ($\gamma_{0}>0$) describes a non-local quantum incoherent loss (gain) of the chiral hinge states. $i\gamma_{j}\Gamma^{j}$ of $j=1,2,3$, acting on the eigenstates of the Hermitian part, adds a pre-factor $\exp[\gamma_{j}x_{j}/t]$ to the Bloch wave function of the 3DSOTI and leads to the non-Hermitian skin effect [41]. The remaining two terms, $i\gamma_{4}\Gamma^{4}$ and $i\gamma_{5}\Gamma^{5}$, make the Dirac masses of bulk and surface states complex such that the phase boundaries between the 3DSOTI and the second-order Weyl semimetals are tuned by the non-Hermitian potentials [30]. In what follows, we ignore the two terms $i\gamma_{4}\Gamma^{4}$ and $i\gamma_{5}\Gamma^{5}$.

Through the same approach used in Ref. [38], we derive the effective low-energy continuum Hamiltonian of chiral hinge states

in the basis of the zero-energy hinge states localized at the two hinges $x=0,y=\pm L_{\parallel}/2$. The Hermitian part of $tp_{3}\sigma_{3}$ gives two chiral hinge channels propagating in opposite directions. The $\gamma_{0}$-term is a non-local gain or loss if $\gamma_{0}$ is positive or negative, respectively. While eigenstates of $h_{\text{hinge}}(\gamma_{3}=0)$ are extended on the two hinges, $h_{\text{hinge}}(\gamma_{3}\neq 0)$’s eigenstates are exponentially localized at the end of the hinges, i.e., the corners. As we illustrated in Ref. [38], the hinge states of a non-Hermitian 3DSOTI will be Hermitian if neither the gain/loss nor the non-Hermitian skin effect is presented, say $\gamma_{0}=\gamma_{3}=0$ and $\gamma_{1,2}\neq 0$.

The above picture can be seen by the spatial distributions of wave functions of a right eigenstate of Eq. (1) on a lattice of size $L_{\parallel}=L_{\perp}=16$ for three cases: (i) $\gamma_{0}=-0.01,\gamma_{1,2,3}=0$, and $E=0-0.01i$; (ii) $\gamma_{1}=-0.01,\gamma_{0,2,3}=0$, and $E=0$; (iii) $\gamma_{3}=-0.01,\gamma_{0,1,2}=0$, and $E=0$. The distributions are shown in Figs. 1(b,c,d). While the wave functions of hinge states spread over the whole hinges for cases (i) and (ii), it is localized at the two corners $x=0,y=\pm L_{\parallel},z=L_{\perp}$ for the case (iii) where $\gamma_{3}=-0.01$ and $\gamma_{0,1,2}=0$. Such corner states are due to the non-Hermitian skin effect on hinge states and will localized at $x=0,y=\pm L_{\parallel},z=-L_{\perp}$ if we choose $\gamma_{3}=0.01$ and $\gamma_{0,1,2}=0$. Since the focus of this work is on the chiral hinge states, we set $\gamma_{3}=0$ and only consider cases (i) and (ii) in what follows.

Now, we concentrate on the fate of hinge states of non-Hermitian 3DSOTIs in the presence of finite disorders. To this end, we turn on the disorder potentials $W>0$ and consider a specific energy $E=0.02+i\gamma_{0}$ and $M=2,B=0.2$ in Eq. (1). As shown in Fig. 1(a), the $\text{Re}[E]=0.02$ state is a chiral hinge state in the clean limit. In the presence of disorders, if a state of a complex eigenvalue $E$ is still the hinge state, its wave function should be highly concentrated on the hinges. We thus use the following quantity to identify a hinge state:

with the first summation over all lattice sites on the two hinges ($x=0,y=\pm L_{\parallel}/2$) and $|\psi_{\bm{j},p}(E)|$ being the wave function amplitude of a right eigenstate $|\psi(E)\rangle$ of the eigenvalue $E$ on site $\bm{j}$. Here, we choose the normalization condition $\langle\psi(E)|\psi(E)\rangle=1$. The identification of a hinge state is guided by the following features [30, 31]: For hinge states, $\eta_{W,L}$ approaches a finite value for $L_{\parallel}=L_{\perp}=L\gg 1$, while $\eta_{W,L}$ decreases with $L$ for surface and bulk states. Numerically, we compute $\eta_{W,L}$ through the exact diagonalization by using the Scipy library [42] and the KWANT package [43].

Figures 2(a,b) display $\langle\eta_{W,L}\rangle$ of $E=0.02+i\gamma_{0}$ for various $W$ and $L$ of cases (i) $\gamma_{0}=-0.01,\gamma_{1,2,3}=0$; (ii) $\gamma_{1}=-0.01,\gamma_{0,2,3}=0$; where the hinge states are non-Hermitian (dissipative) and Hermitian for $W=0$, see Eq. (3). The symbol $\langle\cdot\rangle$ stands for the ensemble average. For both non-Hermitian and Hermitian hinge states, curves $\langle\eta_{W,L}\rangle$ of different $L$ merge at $W=W_{c,1}$ for $W\leq W_{c,1}$ and increases with the decreases of $L$ for $W>W_{c,1}$. For dissipative chiral hinge states, $W_{c,1}\simeq 1.8$, and for Hermitian chiral hinge states $W_{c,1}\simeq 1.4$. Hence, in each case, chiral hinge states can survive at finite disorders, which commensurate with the observation in Hermitian systems [30, 31].

The 3DSOTI-to-3DFOTI transitions can be understood as follows. In the absence of disorders, there are three types of states in a non-Hermitian 3DSOTI: topological hinge states, topological surface states, and trivial bulk states. In Ref. [38], we have shown that topological hinge states in the complex energy gap of topological surface states and topological surface states in the bulk energy gap are due to the band inversion of surface and bulk states, respectively. Disorders tend to close the energy gap of surface states first and then bulk states. Therefore, the disorder-induced quantum phase transitions occur first from non-Hermitian 3DSOTIs to 3DFOTIs, then from 3DSOTIs to 3D non-hermitian metals. The numerical verification of the assertion for case (i) is presented in Appendix A.

Naturally, we expect that a 3DFOTI-to-3D-metal transition occurs at a stronger disorder $W_{c,2}>W_{c,1}$ where the gap of bulk states closes. Since the 3DFOTIs are featured as bulk-gap states localized on the surfaces, the following quantity

which describes the occupation probability, or inverse participation ration, of a state on the surfaces or hinges of a sample, should be independent of $L$ for surface/hinge states and decrease with $L$ for the bulk states [30, 31]. Hence, if there exists a 3DFOTI-to-3D-metal transition, we expect that $\langle\zeta_{W,L}(E)\rangle$ is independent of $L$ for $W<W_{c,2}$. This is indeed what we found as shown in Figs. 2(c,d,e,f) where $W_{c,2}\simeq 3.0$ for cases (i) and (ii). Therefore, our numerical data confirm the existences of surface states for $W\in[W_{c,1},W_{c,2}]$.

For stronger disorders $W>W_{c,2}$, the states of $E=0.02+i\gamma_{0}$ in Fig. 2(c,d) are bulk states. Such states are extended states, and a third quantum phase transition from non-Hermitian diffusive metals to Anderson insulators, i.e., the Anderson localization transitions, happens at a higher critical disorder $W_{c,3}$. This can be seen from the finite-size scaling analysis of participation ratio $p_{2}$ for a given energy $E$, which satisfies a single-parameter scaling function near $W_{c,3}$ [44, 45]

In Eq. (6), $D$ is the fractal dimension of the critical wave function, $f(x=L/\xi)$ is the unknown scaling function, $C$ is a constant, and $y>0$ is the exponent of the irrelevant variable. $\xi$ is the correlation length and diverges as $\xi\sim|W-W_{c,3}|^{-\nu}$ with $\nu$ being the critical exponent characterized the universality class of Anderson localization transitions. The identification of an Anderson localization transition is guided by the following observations: (i) For extended (localized) states, the quantity $Y_{L}(W)=p_{2}L^{D}-CL^{-y}$ increases (decreases) with system size $L$. (ii) Near $W_{c,3}$, $Y_{L}$ collapses to a two-branch smooth function, i.e., $f(x)$.

Numerical evidence of Anderson localization transitions is displayed in Figs. 3(a,c) for (i) $\gamma_{0}=-0.01,\gamma_{1,2,3}=0$ and (ii) $\gamma_{1}=0.01,\gamma_{0,2,3}=0$, respectively. Clearly, there exist the crossing points at the critical disorders $W_{c,3}=18.73\pm 0.02$ for case (i) and $W_{c,3}=17.4\pm 0.1$ for case (ii), where $Y_{L}(W)$ are independent of system sizes. Data of $d\ln[Y_{L}]/dL$ is positive (negative) for $W<W_{c,3}$ ($W>W_{c,3}$), indicating the non-Hermitian system is a diffusive metal (Anderson insulator). $\ln[Y_{L}(x=\ln[L|W-W_{c,3}|^{\nu}])]$ near $W_{c,3}$ merge to two branches of one smooth scaling function, see Figs. 3(b,d).

The finite-size scaling analysis yields $\nu=1.47\pm 0.03$, $D=1.73\pm 0.06$, $C=1.2\pm 0.1$, $y=0.7\pm 0.1$ for case (i) and $\nu=1.2\pm 0.1$, $D=2.03\pm 0.05$, $C=0.8\pm 0.2$, $y=1.4\pm 0.2$ for case (ii). The critical exponent $\nu$ of case (i) is identical to that of Anderson localization transitions of its Hermitian counterpart [30], indicating that the appearance of the non-Hermitian potential $\sum_{\bm{i}}c^{\dagger}_{\bm{i}}(i\gamma_{0}\Gamma^{0})c_{\bm{i}}$ does not change the universality class. On the other hand, the critical exponent of case (ii), say $\nu=1.2\pm 0.1$, is very close to that of symmetry class AI^† in the non-Hermitian AZ classification of the non-Hermitian systems [46].

Through numerical analysis of $\langle\eta_{W,L}(E)\rangle$ and $\langle\xi_{W,L}(E)\rangle$, we have proven that a non-Hermitian 3DSOTI undergoes transitions to a 3DFOTI at $W_{c,1}$ then to a topological trivial system at $W_{c,2}$. We further substantiate that the topological trivial system is a diffusive metal in $[W_{c,2},W_{c,3}]$ and becomes an Anderson insulator for $W>W_{c,3}$ through the finite-size scaling analysis of participation ratios $p_{2}$. Our numerical results thus suggest a general route of quantum phase transitions of non-Hermitian 3DSOTIs, see Fig. 4, which is very similar to that of Hermitian 3DSOTIs [30].

Having determined the phase diagram of disordered non-Hermitian 3DSOTIs, we are now in the position to investigate the quantum transport of hinge states. We analyse the transmission coefficient of a Hall bar of size $L_{\parallel}/\sqrt{2}\times L_{\parallel}/\sqrt{2}\times L_{\perp}$ connected to two semi-infinite leads at two ends along the $z-$direction. Particles in the Hall bar are govern by the Hamiltonian of Eq. (1), and the leads are clean and Hermitian, i.e., Eq. (1) without the disorders and the $\gamma$-terms. Some previous works have studied the conductivities of non-Hermitian Chern insulators within the framework of Kubo-Greenwood formalism [35, 36, 37] by assuming conductor together with leads remains quantum mechanically coherent. This approach neglects the intrinsic loss of quantum coherence due to the non-Hermicities in the self-energy. Here, we use Büttiker’s approach combined with the non-equilibrium Green function method (NEGF) to calculate the transmission coefficients [47, 48, 49, 50], where a virtual lead of zero net particle flux is introduced to model the incoherent processes that are not in the usual Kubo-Greenwood formalism.

Figure 5 shows some typical results of mean transmission coefficients $\langle T\rangle$ for $E=0.02+i\gamma_{0}$. In general, transmission coefficients for the $\pm z$-moving fluxes are different in disordered non-Hermitian systems [51]. While, for the hinge channels governed by Eq. (3), the transpose symmetry is protected, and disorder-average transmission coefficients for the $\pm z$-moving fluxes are the same. Without the non-Hermitian potentials, the mean transmission coefficient is quantized at 1, i.e., $\langle T\rangle=1$, and the fluctuations of transmission coefficients, defined as $\Delta T=(\langle T^{2}\rangle-\langle T\rangle^{2})^{1/2}$, are vanishingly small ($\Delta T\sim 10^{-7}$ in our numerical calculations) until $W_{c,1}=2.6$, see Fig. 5(a,b). This is a standard feature of chiral hinge states in disordered Hermitian 3DSOTIs [30].

In the presence of a non-local loss $\gamma_{0}=-0.01,\gamma_{1,2,3}=0$, $\langle T\rangle$ is not quantized any more even for $W<W_{c,1}$ where the hinge states persist. This can be seen by Fig. 5(c) and its inset. As shown in Fig. 5(d), the fluctuations of transmission coefficients are also non-zero ($\Delta T\sim 10^{-3}$) and at least four order of magnitude larger than those of Hermitian cases. In addition, $\Delta T$ increases with $W$ and the system sizes $L$, indicating that the fluctuation of transmission coefficients is not due to numerical errors and cannot be eliminated in the thermodynamic limit where $L\to\infty$.

Differently, for Hermitian hinge states in non-Hermitian 3DSOTIs where $\gamma_{1}=-0.01$ and $\gamma_{0,2,3}=0$, the mean transmission coefficient $\langle T\rangle$ is still equal to the number of chiral hinge channel (here is 1). The values of mean transmission coefficients in Fig. 2(e) depend neither on the width of the Hall bar nor on disorders in $W<W_{c,1}$, a behavior which is strongly reminiscent of the Hermitian chiral hinge channels [see Fig. 2(a)]. The plateau of $\langle T\rangle=1$ can also be observed for $\gamma_{2}=-0.01,\gamma_{0,1,3}=0$ (not shown here). The akin results have been observed in non-Hermitian 3DSOTIs and other non-Hermitian topological systems like Chern insulators and topological Anderson insulators in our early work [38]. However, the fluctuations of transmission coefficients are always non-zero. Similar to the non-Hermitian chiral hinge states, the fluctuations of transmission coefficients are not due to numerical error and increase with the system sizes, see Fig. 5(f).

Similar behaviors of the mean transmission coefficient $\langle T\rangle$ and the fluctuation of transmission coefficient $\Delta T$ have been observed for non-Hermitian 3DSOTIs with different disorders. In Appendix B, we have considered two additional types of disorders: “non-Hermitian” on-site disorder and non-uniform in the subspace of interior degree of freedom. Again, we find that $\langle T\rangle$ equals 1 if the chiral hinge states are Hermitian; otherwise, $\langle T\rangle\neq 1$. In both cases, $\Delta T$ is finite.

A fairly simple theory, where the incoherent processes are artificially encoded in the particle density, can explain the mean transmission coefficient $\langle T\rangle$ displayed in Figs. 5(a,c,e). Within the framework of NEGF, the real parts of energy spectra $\text{Re}[E]$ are interpreted as the quasi-particle energies, and the imaginary parts $\text{Im}[E]$ associate with incoherent processes. Take electrons as examples. The second law of thermodynamics requires $\text{Im}[E]\leq 0$ such that one can treat $\hbar/(2|\text{Im}[E]|)$ as the average relaxation time of an electron. Without disorders, the current of a coherent hinge mode with momentum $k$ is given by $I_{k}=\rho_{k}v_{k}f(E_{k})$ with the electron density $\rho_{k}$, the Fermi-Dirac distribution $f(E_{k})$, and the group velocity $v_{k}=\partial_{k}E_{k}/\hbar$ [52]. In the presence of non-Hermitian potentials, we expect that the electron density is reduced by a factor of $\exp[-2|\text{Im}[E]|L_{\perp}/t]$ due to the relaxation of electrons.

Consider two reflectionless leads with chemical potentials $\mu_{1}$ and $\mu_{2}$, respectively, and $|\mu_{1,2}|<2\Delta$ with $2\Delta$ being the gap of the energies of hinge states. At the zero temperature, we obtain the net current flow of $I=(e/h)(\mu_{1}-\mu_{2})e^{-2|\text{Im}[E]|L_{\perp}/t}$ and the dimensionless conductance, which is equal to the transmission coefficient according to the Landauer formula [47],

see Appendix C for more details. Therefore, for a non-local loss $\gamma_{0}<0$ and $\gamma_{1,2,3}=0$ where $|\text{Im}[E]|=|\gamma_{0}|>0$, the coherent propagation of an electron in the hinge channel is destroyed when it interacts with the surroundings, which leads to $T<1$. Differently, for $\gamma_{0}=\gamma_{3}=0$ and $\gamma_{1,2}\neq 0$, the hinge channels are Hermitian with real energy spectra [see Eq. (3)] such that it retains its identity even at lead and $T=1$. Even though Eq. (7) is derived for $W=0$, we find it still works well for $\langle T\rangle$ at weak disorders [see the solid lines in Figs. 2(a,c,e) without any fitting parameter].

Still, there is one crucial question: Why transmission coefficients of hinge channels fluctuate in non-Hermitian 3DSOTIs? In Hermitian systems, the fluctuations come from the random opening and closing of a conducting channel as disorder configurations vary due to the inevitable diffusion of defects and impurities at finite temperatures. Each conducting channel contributes exactly one quantum conductance no matter what its dispersion is. Thus, there is no fluctuation of the transmission coefficients in one-dimensional chiral hinge channels as long as disorders do not destroy the channel. One may attribute the fluctuation in transmission coefficients to the broken chirality of hinge states due to the non-Hermitian potentials.

To test the correctness of this conjecture, we add a local loss $-i\delta\Gamma^{0}$ for $y>0$ ($y<0$) of a very large $\delta=t$, i.e., particles in one hinge $x=0,y=L_{\parallel}/2$ ($x=0,y=-L_{\parallel}/2$) completely decay, and calculate the corresponding mean transmission coefficient $\langle T_{+}\rangle$ ($\langle T_{-}\rangle$). The chirality of the hinge state can be defined as

If the chirality is preserved, say a up-moving current flux state propagating in one hinge $x=0,y=L_{\parallel}/2$ without backward scattering, we expect $\langle T_{+}\rangle=\langle T\rangle$, $\langle T_{-}\rangle=0$, and $C=1$. Otherwise, $C\neq 1$ if the chirality is broken.

Figure 6(a) shows $\langle T\rangle$, $\langle T_{+}\rangle$, and $\langle T_{-}\rangle$ as a function of disorder strengths $W$ for $\gamma_{0}=-0.01,\gamma_{1,2,3}=0$. Clearly, $\langle T_{-}\rangle$ vanishes (about $10^{-11}$) due to the strong loss, while $\langle T\rangle=\langle T_{+}\rangle$ for $W<W_{c,1}\simeq 1.8$ [the same critical disorder obtained by Fig. 2(a)], indicating no backward scattering of particles in such hinge channel. This can be also seen from Fig. 6(b): $C$ is always quantized as long as $W<W_{c,1}$, no matter what kind of non-Hermicities is applied. Therefore, the quantized $C$ should be convincing evidence of the preservation of chirality.

The quantized chirality $C=\pm 1$ (the sign depends on $B$) behaves as a fingerprint of chiral hinge states in non-Hermitian disordered 3DSOTIs, similar to the quantized quadrupole for the corner states in Hermitian topological insulators [14]. Specifically, we utilize the quantized $C=1$ to identify the non-Hermitian 3DSOTI phase in the $\log_{10}[|\gamma_{0}|]-W$ plane numerically, see Fig. 6(c). Our numerical data suggest the critical disorder $W_{c,1}$ decreases with $|\gamma_{0}|$.

Although the chirality of hinge states is preserved and the backward scattering is absent, the fluctuation of transmission coefficients can still arise from the scattering of the non-Hermitian potentials. This can be seen from a general formulation of a two-terminal multi-channels conductor in the beautiful book by Datta [47]:

with $M_{1}$ being the number of conducting channels of lead 1. $R_{11}$ is the reflection coefficient, and $T_{12}$ is the transmission coefficient from lead 2 to 1. $\tilde{\Gamma}_{1(2)}=i(\Sigma^{R}_{1(2)}-\Sigma^{A}_{1(2)})$, $G^{R}=[E-H-\Sigma^{R}]^{-1}$ and $G^{A}=[E-H^{\dagger}-\Sigma^{A}]^{-1}$ are the retarded and advanced Green’s functions with $H$ being the Hamiltonian of the conductor and $\Sigma^{R(A)}$ being the total retarded (advanced) self-energy, respectively. $\text{Tr}[\tilde{\Gamma}_{1}G^{R}[i(H-H^{\dagger})]G^{A}]$ is the transmission coefficient for a particle incident in the virtual lead to reach the lead 1. Therefore, one cannot self-consistently compute the transmission coefficient $T_{12}$ as if there were no incoherent processes (virtual lead) since it is affected by these events. Following Büttiker’s approach, we obtain the transmission coefficient of the chiral hinge states

see Appendix D for more details. Here, $M_{1}=1$ since we consider the chiral hinge transport. The contributions from the chiral hinge states and the scattering by non-Hermitian potentials are separated in the first and the second terms of Eq. (10). Therefore, even though no fluctuation is in the first term due to the preservation of chirality (no backward scattering means $R_{11}=0$), scattering by non-Hermicities always leads to fluctuations of transmission coefficients since it relates to the bulk Hamiltonian $H$ rather than the topological boundary states. Such fluctuations increase with $i(H-H^{\dagger})$ that is proportional to the strength of non-Hermicities.

As the fluctuations of transmission coefficients of chiral hinge states in non-Hermitian 3DSOTIs are inherent, the understanding of their distributions $P(T)$ should be very interesting. Numerically, we find $P(T)$ of metallic (bulk and surface) and localized states of Eq. (2) follow the Gaussian and the log-normal distributions, respectively, similar to its Hermitian counterparts [53]. While, for $P(T)$ of chiral hinge states, their Hermiticities play a significant role again. For Hermitian chiral hinge states, $P(T)$ is also Gaussian. However, it becomes very asymmetric with a long tail towards small $T$ ($T<\langle T\rangle$) if the chiral hinge states are dissipative, see Fig. 6(d). Exactly, the distributions always evolve to a one-sided exponential distribution $P(T)\sim\exp[-\lambda|T-\langle T\rangle|]$, where the coefficient $\lambda$ depends on the disorders and the non-Hermitian potentials.

We would like to discuss the experimental relevance of our theory before the conclusion. Loss can be introduced by the relaxation time of electrons such that results of $\gamma_{0}<0$ and $\gamma_{1,2,3}=0$ can be directly applied to the electronic 3DSOTIs, e.g., Bi_2-xSm_xSe₃ [54]. However, gains are not allowed for electronic systems in principle. We thus suggest the chiral hinge magnons as a potential platform for observing the reported Hermitian chiral hinge transport, e.g., $\gamma_{1}\neq 0$ and $\gamma_{0,2,3}=0$. The realization of chiral hinge magnons has been illustrated in Ref. [55]. The loss of magnons, known as the Gilbert damping, is the intrinsic property of spin systems [56], while, recently, it has been shown that the gain (the negative Gilbert damping) can be realized by the magneto-optical interaction and artificially tuned [57]. In real magnetic systems, disorders exist ubiquitously. In Ref. [58], it is shown that the forms of randomness used in this work are good approximations of disorders and can be used to explain the origin of the well-known spin Seebeck effect. Instead of magnonic chiral hinge states, all reported results are nevertheless appropriate for phononic [59, 60, 61] and photonic [62] topological insulators, where gains are allowed.

Besides, Hermitian chiral boundary (edge or hinge) states can appear in a large family of topological phases, including Chern insulators, 3DSOTIs, and topological Anderson insulators [38]. The physics discussed here is applicable to other non-Hermitian topological phases with Hermitian chiral edge states. Stimulated by this, we propose another system, the 2D laser cavity network on a honeycomb lattice that supports chiral edge states, as an easy-to-access platform to test our predictions. We show the flexibility of this proposal in Appendix E.

In conclusion, chiral hinge transport in disordered non-Hermitian 3DSOTIs is studied numerically. Chiral hinge states can be either Hermitian or non-Hermitian, depending on the form of the non-Hermitian potentials. Both can survive at finite disorders and undergo the transitions to surface states when increasing the disorder up to a critical value $W_{c,1}$. The mean transmission coefficients are equal and nonequal to the number of chiral channels for Hermitian and non-Hermitian chiral hinge states. The transmission coefficients always fluctuate whether the chiral hinge states are Hermitian or non-Hermitian. The fluctuations of transmission coefficients come from the scatterings of non-Hermitian potentials that describe incoherent processes involving the interactions between particles and their surroundings. Furthermore, the Hermiticity of chiral hinge states also affects the distribution of transmission coefficients. The distribution of Hermitian chiral hinge states is Gaussian; otherwise, it is a one-sided exponential distribution.