Non-Hermitian topology from edge transport in Hermitian quantum anomalous Hall systems

Abstract

Non-Hermitian physics, known for phenomena like exceptional points and the skin effect, has been most prominently realized in engineered systems relying on controlled gain and loss. Here we show that it can also arise naturally as an intrinsic transport response of a globally Hermitian quantum anomalous Hall system, without the need for external non-Hermitian engineering. We show that the interplay between unidirectional chiral edge modes and diffusive normal edge modes induces intrinsic non-reciprocal transport described by a continuum Hatano-Nelson model. Consequently, the non-Hermitian skin effect is encoded directly in experimentally accessible Hall-bar observables: the electrochemical potential and local heat dissipation acquire chirality-dependent exponential spatial profiles, while the longitudinal conductance decays exponentially with system size and the Hall conductance remains quantized. Using Landauer–Büttiker simulations, we confirm these transport signatures and identify magnetic topological insulators as a realistic platform for an intrinsic non-Hermitian transport response. Our results bridge non-Hermitian topology with mesoscopic transport, opening a pathway toward non-Hermitian topological devices in solid-state systems.

Here, the authors show that the interplay between chiral and normal edge transport in a quantum anomalous Hall system can generate non-Hermitian topology, producing measurable exponential patterns in voltage, heat dissipation, and conductance.

Introduction

Non-Hermitian (NH) physics has emerged as a fundamental paradigm for open systems coupled to external environments^1–3. The discovery of exceptional points, exotic non-Hermitian topological phases, and the non-Hermitian skin effect (NHSE) has broadened the landscape of quantum mechanics beyond the conventional Hermitian framework^4–21. To date, however, experimental realizations of these non-Hermitian phenomena have predominantly relied on engineered platforms with externally imposed gain and loss^22–32. Meanwhile, recent studies have shown that NH topology can also be encoded in Hermitian topological matter^33–36, suggesting that NH physics may also be explored directly through transport phenomena in Hermitian solid-state systems.

Magnetic topological insulators (TIs) hosting the quantum anomalous Hall (QAH) state with chiral edge modes and the axion insulator state^37–57 provide a particularly promising solid-state platform for exploring NH phenomena. Previous studies have approached NH physics in Hermitian topological electronic systems in several ways^{33,33,34,34,35,35,58–61}. In some cases, disorder or interactions generate effective self-energies associated with finite quasiparticle lifetimes, leading to NH band structures with momentum-space topology^62,63. In others, NH topology is encoded in reflection matrices or in effective boundary Hamiltonians obtained by integrating out the bulk^33–35. More recently, quantum Hall and QAH devices have provided a route to NH topology through multi-terminal conductance matrices, in which contact arms act as effective sites of a discrete Hatano-Nelson chain^58–61. Despite these important developments, directly isolating transport signatures of NH topology in a globally Hermitian electronic system, particularly the chirality-dependent exponential transport response associated with the NHSE, remains challenging. This motivates our central inquiry: can NH topology arise directly from the intrinsic edge dynamics of a globally Hermitian QAH system, with signatures appearing naturally in the spatially continuous evolution of local, experimentally measurable transport observables?

In this work, we show that realistic QAH insulators provide a solid-state transport setting in which non-Hermitian topology emerges directly from intrinsic edge dynamics. While ideal QAH insulators host only chiral edge modes, realistic samples often harbor non-topological normal edge modes due to imperfections^37,43,44. We show that the interplay between a chiral mode and a normal edge mode realizes the continuum Hatano-Nelson model. By mapping the edge transport equations to the HN Hamiltonian, we demonstrate that a Hermitian QAH system exhibits characteristic signatures of non-Hermitian topology, including the directional exponential behavior of the Green’s function and the NHSE. Here, the Hermitian QAH system refers to the QAH sample described by a Hermitian microscopic Hamiltonian. We further show that these features manifest in the experimentally measurable transport quantities of a Hall-bar device, where heat dissipation and the longitudinal conductivity acquire an exponential behavior while the Hall response remains robustly quantized. Our results uncover a condensed-matter realization of non-Hermiticity rooted in microscopic chiral dynamics, bridging non-Hermitian topology with mesoscopic transport.

Results

Quantum kinetic theory of QAH edge states

We begin by investigating the microscopic theory of edge transport in a 2D QAH insulator. As illustrated in Fig. 1, we consider the QAH insulator that hosts both a chiral edge mode and a normal edge mode. The coexistence of these modes has been confirmed by transport experiments on magnetic TI samples^42–44. The total Hamiltonian of the edge system is given by:

$$H=H_0+U\left(x\right)+U_D\left(x\right),$$ 1

where H₀ = diag{ε_1k, ε_2k} describes the normal and chiral modes with dispersions ε_1,k = ε_1,−k and ε_2k = cv_cℏk, respectively. The parameter v_c denotes the chiral mode velocity, and c = ± 1 indicates the propagation direction. The term U(x) represents a slowly varying electrostatic potential, while U_D(x) = ∑_α,iu_ασ_αδ(x − R_i) denotes the delta-function impurity potential with the randomly distributed impurity position R_i. Here, the Pauli matrices σ_α (α = 0, x) act on the band indices. The parameters u₀ and u_x represent the intra-band and inter-band scattering strengths, respectively.

Fig. 1. Nonreciprocal transport and NHSE in QAH edge states.

a Schematic plot of unidirectional chiral (blue) and bidirectional normal (red) edge modes. The nonreciprocal transport is highlighted by the unequal amplitude splitting of edge-injected wave packets. This imbalance serves as a dynamic signature of the NHSE. b Corresponding energy-momentum dispersion, showing the chiral (blue) and normal (red) edge modes relative to the Fermi level E_F.

To model the transport, we utilize the quantum Liouville equation for the density operator ${\widehat{\rho }}_t$:

$$\frac{\partial {\widehat{\rho }}_t}{\partial t}+\frac{\mathrm{i}}{\hslash }\left[\widehat{H},{\widehat{\rho }}_t\right]=0.$$ 2

After performing disorder averaging under the first-order Born approximation and applying the Wigner transformation(see Supplemental Material for details), we obtain the semiclassical Boltzmann transport equation:

$$\begin{array}{c}\frac{\partial f_{nk}}{\partial t}+v_{nk}{\partial }_x{f}_{nk}-{\partial }_{x}U\frac{\partial f_{nk}}{\hslash \partial k}\hfill \\ =-{\sum }_{n^{\prime }}\int \frac{\mathrm{d}{k}^{\prime }}{2\pi }{W}_{nk,n^{\prime }{k}^{\prime }}\left(f_{nk}-f_{n^{\prime }{k}^{\prime }}\right)\hfill \end{array}$$ 3

Here, f_nk is the distribution function for state k in band n and v_nk = ℏ⁻¹∂_kε_nk is the group velocity. The scattering rate is defined as $W_{nk,n^{\prime }{k}^{\prime }}=2\pi {\hslash }^{-1}{\sum }_{\alpha }{n}_{\alpha }{u}_{\alpha }^2{\left({\sigma }_{\alpha }\right)}_{n{n}^{\prime }}{\left({\sigma }_{\alpha }\right)}_{n^{\prime }n}\delta \left({\epsilon }_{nk}-{\epsilon }_{n^{\prime }{k}^{\prime }}\right)$, where n_α is the impurity density.

We consider a weakly nonequilibrium state induced by a slowly varying electrochemical potential $\tilde{\mu }\left(x\right)$. The small statistical force ${\partial }_x\tilde{\mu }$ produces only weak deviations of f_nk from the equilibrium state. Thus, we linearize the non-equilibrium distribution as $f_{nk}\simeq f_0\left({\epsilon }_{nk}-\mu \right)+f_{nk}^{\left(1\right)}$. Here f₀(ε_nk − μ) denotes the Fermi–Dirac distribution, and $\mu \left(x\right)=\tilde{\mu }\left(x\right)-U\left(x\right)$ is the local chemical potential that determines the local equilibrium occupation. The first-order corrections are derived as: $f_{1k}^{\left(1\right)}=\left[\mathrm{sign}\left(k\right)v_{1F}{\tau }_1-c{v}_{2F}{\tau }_2{g}_2/g_1\right]{\partial }_{\epsilon }{f}_0\left({\epsilon }_{1k}-\mu \right){\partial }_x\tilde{\mu }$, and $f_{2k}^{\left(1\right)}=c{v}_{2F}{\tau }_2{\partial }_{\epsilon }{f}_0\left({\epsilon }_{2k}-\mu \right){\partial }_x\tilde{\mu }.$ Here, τ₁ and τ₂ are the relaxation time for the distributions of normal edge channel (f_1k) and the chiral edge channel (f_2k) to reach equilibrium, while $v_{1F}=\mid {\hslash }^{-1}{\partial }_k{\epsilon }_{1k}{\mid }_{{\epsilon }_{1k}=E_F}$ and v_2F = v_c correspond to Fermi velocities. E_F is the Fermi level. The explicit forms of the relaxation time are $\hslash /{\tau }_1=2\pi \left(n_0{u}_0^2{g}_1+n_x{u}_x^2{g}_2\right)$ and $\hslash /{\tau }_2=2\pi n_x{u}_x^2{\left(g_1+g_2\right)}^2/g_1$, where $g_n=\int \frac{\mathrm{d}k}{2\pi }\delta \left(E_F-{\epsilon }_{nk}\right)$ is the density of states (DoS) of band n at Fermi energy. Here, the transport lifetime for the normal edge channel (τ₁) is governed by both scattering strengths u₀ and u_x. Conversely, the relaxation time of the chiral edge channel (τ₂) is determined solely by the inter-band scattering strength u_x. These results demonstrate that, while the chiral edge state is immune to intra-band backscattering (thereby preserving its unidirectional nature in isolation), the coexistence of a normal mode introduces dissipation via inter-band scattering.

The edge current is given by $j=-e{\sum }_n\int \frac{\mathrm{d}k}{2\pi }{v}_{nk}{f}_{nk}$, and takes the form

$$j\left(x\right)=\frac{1}{e}{\sigma }_D{\partial }_x\tilde{\mu }\left(x\right)-c\frac{e}{h}\left(\tilde{\mu }\left(x\right)-{\tilde{\mu }}_0\right)+j_0.$$ 4

Here, σ_D = e²(g₁D₁ + g₂D₂) is the one-dimensional conductivity with diffusion constant $D_n={\left(v_{nF}\right)}^2{\tau }_n$, which arises from backscattering due to the normal edge modes. ${\tilde{\mu }}_0$ denotes the constant electrochemical potential at equilibrium. The first term in Eq. (4) corresponds to the diffusive current driven by the statistical force ${\partial }_x\tilde{\mu }$. The second term describes the chiral current carried by the chiral edge mode. The final term, j₀, represents a circulating equilibrium current, which does not contribute to charge transport. Eq. (4) remains unchanged even when inelastic scattering is included, whose main effect is to renormalize σ_D.

The charge continuity equation is given by ∂ρ/∂t + ∂_x(j − j₀) = S, where S is the source term and the charge density is given by $\rho =-e{\sum }_n\int \frac{\mathrm{d}k}{2\pi }{f}_{nk}={\rho }_0-e{g}_0\left(\tilde{\mu }-{\tilde{\mu }}_0\right)+\mathcal{O}\left({\left(\tilde{\mu }-{\tilde{\mu }}_0\right)}^2\right)$ at low temperature. $-e{g}_0=\partial \rho /\partial \tilde{\mu }$ and ρ₀ is the charge density at equilibrium. In the absence of the source term, we obtain the following effective Schrödinger-like equation of motion:

$$\mathrm{i}\hslash \frac{\partial \tilde{\mu }}{\partial t}=-\mathrm{i}\left[\frac{{\widehat{p}}^2}{2m}+\mathrm{i}cv\widehat{p}\right]\tilde{\mu }\equiv -\mathrm{i}{H}_{\mathrm{HN}}\left(\widehat{p}\right)\tilde{\mu }.$$ 5

Here, $\widehat{p}=-\mathrm{i}\hslash {\partial }_x$ is the momentum operator, m = e²g₀ℏ/(2σ_D) is the effective mass, and v = 1/(hg₀) denotes the drift velocity. Equation (5) demonstrates that edge transport in a QAH insulator is governed by the non-Hermitian Hamiltonian H_HN, which represents the continuum limit of the HN model⁶⁴ derived in the long-wavelength limit. Notably, the non-Hermitian topology of this system stems directly from the inherently nonreciprocal propagation of the chiral edge modes. Unlike the wavefunction in the standard non-Hermitian Schrödinger equation, the electrochemical potential $\tilde{\mu }$ is a directly measurable quantity in experiments. This distinction provides an experimentally accessible route for studying non-Hermitian phenomena.

NHSE of non-Hermitian edge dynamics

To explore the NHSE in the non-Hermitian Hamiltonian, we investigate the Green’s function of the system under a periodic boundary condition (PBC), defined as $G\left(E,x,x^{\prime }\right)=⟨x∣{\left[E-H_{\mathrm{HN}}\right]}^{-1}∣x^{\prime }⟩$. It takes the form:

$$G\left(E;x,x^{\prime }\right)=-\frac{\mathrm{i}m}{{\hslash }^2\kappa }\left(\frac{{\mathrm{e}}^{\mathrm{i}\kappa \left(x-x^{\prime }\right)}}{1-{\mathrm{e}}^{2\mathrm{i}L{k}_{+}}}-\frac{{\mathrm{e}}^{-\mathrm{i}\kappa \left(x-x^{\prime }\right)}}{1-{\mathrm{e}}^{2\mathrm{i}L{k}_{-}}}\right){\mathrm{e}}^{\frac{c\left(x-x^{\prime }\right)}{\lambda }}$$ 6

which is valid for the interval $0\le x-x^{\prime }<2L$. For other coordinate differences, the Green’s function satisfies the periodicity condition $G\left(E;x,x^{\prime }\pm 2L\right)=G\left(E;x,x^{\prime }\right)$. L is the length of the QAH sample. Here, $\kappa =\sqrt{2mE-{\left(cmv\right)}^2}/\hslash$ and k_± = − ic/λ ± κ with H_HN(ℏk_±) = E. The parameter λ = ℏ/(mv) = 2σ_Dh/e² serves as a characteristic length scale governing the exponential behavior of the non-Hermitian system. λ is determined solely by the normal-mode diffusive conductivity σ_D. This implies that stronger impurity scattering (or a reduced number of normal modes) yields a smaller σ_D resulting in a shorter λ and thus a faster exponential decay.

Notably, the Green’s function exhibits a distinct directional exponential amplification that becomes fully visible under open boundary conditions (OBC). In this case, the right eigenstates of H_HN, ${\psi }_{n,R}\left(x\right)=L^{-1/2}{\mathrm{e}}^{cx/\lambda }\sin \left(n\pi x/2L\right)$ with $n\in \mathbb{Z}$, are exponentially localized at the boundary for c = ± 1, directly signaling the NHSE. The NHSE is the manifestation of the non-Hermitian topology^65,66. The non-Hermitian topological winding number is defined as $w\left(E\right)=\frac{1}{2\pi \mathrm{i}}\int \mathrm{d}k\frac{\mathrm{d}}{\mathrm{d}k}\ln \left[H_{\mathrm{HN}}\left(\hslash k\right)-E\right]$. For our model, $w\left(E\right)=\frac{1}{2}\left[\mathrm{sign}\left(\mathrm{Im}\left(k_{+}\right)\right)+\mathrm{sign}\left(\mathrm{Im}\left(k_{-}\right)\right)\right]$, which can take nonzero values for c = ± 1, leading to the emergence of the NHSE^65,66. In contrast, w(E) is always zero for c = 0, corresponding to a topologically trivial phase without the NHSE. The correspondence between the exponential spatial asymmetry of the Green’s function under PBC and the eigenstate localization under OBC constitutes a hallmark of non-Hermitian topology. This result explicitly demonstrates the presence of the NHSE in our model.

This exponential behavior can be observed on the edge transport of a QAH insulator. We consider a setup with a source at x = 0 and a drain at x = L [see Fig. 2a], described by the source term S(x) = Iδ(x) − Iδ(x − L). Solving the steady-state charge continuity equation yields the following relation for the electrochemical potential:

$$\tilde{\mu }\left(x\right)=\frac{\hslash I}{e{g}_0}\left[G\left(0;x,0\right)-G\left(0;x,L\right)\right]$$ 7

This equation directly links the experimentally measurable electrochemical potential (or voltage profile, $V\left(x\right)=-\tilde{\mu }\left(x\right)/e$) to the Green’s function of the non-Hermitian Hamiltonian. In addition, the local heat dissipation P_D(x) provides another experimentally accessible quantity^67–72 as a probe of NHSE. In our model, $P_D\left(x\right)={\sigma }_D/e^2{\left({\partial }_x\tilde{\mu }\right)}^2$, which reduces to $P_D\left(x\right)=\left(I^2/{\sigma }_D\right){\mathrm{e}}^{4cx/\lambda }/{\left(1+{\mathrm{e}}^{2cL/\lambda }\right)}^2$ for 0 ≤ x ≤ L. Consequently, when c = ± 1, $\tilde{\mu }$ and P_D exhibit a distinct spatial exponential profile along the lower edge [see Fig. 2b, c whereas for c = 0, this exponential behavior disappears. For the upper edge (−L ≤ x ≤ 0), the exponential spatial profile also appears, but the localization occurs at the opposite end of the Hall bar. Thus, observing these features in the transport of a QAH insulator can provide clear experimental evidence of the robust NHSE.

Fig. 2. Spatial exponential profiles along a QAH edge.

a Schematic of a Hall-bar device hosting a chiral mode and a normal mode. The positive directions of the local one-dimensional boundary coordinates along the upper and lower edges are chosen to follow their respective chiral propagation directions. Spatial profiles of the electrochemical potential (b) and heat dissipation (c) along the lower edge of the QAH sample for c = 0, ± 1, with L = 5λ. The sign of c = ± 1 specifies the propagation direction of the chiral mode, whereas c = 0 indicates that no chiral edge mode is present. d Longitudinal conductivity σ_xx as a function of the sample length L for c = 0 and 1, evaluated at x = 0.4L, with L₀ = 0.3L and W = 3λ.

Experimentally, the longitudinal and Hall resistances are typically measured using the Hall-bar device of size L × W (see Fig. 2(a)). The longitudinal and Hall resistances are obtained from the voltage measurements as ρ_xx = [V(x) − V(x + L₀)]W/(IL₀) and ρ_xy = [V(x) − V( − x)]/I. The corresponding longitudinal and Hall conductance are obtained from the tensor relations: ${\sigma }_{xx}={\rho }_{xx}/\left({\rho }_{xx}^2+{\rho }_{xy}^2\right)$, and ${\sigma }_{xy}={\rho }_{xy}/\left({\rho }_{xx}^2+{\rho }_{xy}^2\right)$. Based on this, the longitudinal conductance is expressed as:

$${\sigma }_{xx}=\left\{\begin{array}{c}\frac{2{\sigma }_D}{W}=\frac{e^2}{h}\frac{\lambda }{W},\mathrm{for}L\ll \lambda \mathrm{or}c=0\hfill \\ \begin{array}{c}\mid c\mid \frac{e^2}{h}\frac{W}{L_0}\left[\mathrm{\Theta }\left(c\right){\mathrm{e}}^{2c\left(x-L_0-L\right)/\lambda }\right)\hfill \\ \left(+\mathrm{\Theta }\left(-c\right){\mathrm{e}}^{2cx/\lambda }\right],\hfill \end{array}\mathrm{for}x,L\gg \lambda .\hfill \end{array}\right)$$ 8

where Θ(c) is the Heaviside step function. This equation describes σ_xx in two distinct regimes as shown in Fig. 2(d). For L ≪ λ or c = 0, transport is dominated by the normal edge modes, and σ_xx exhibits a classical ohmic law. In contrast, for L ≫ λ, the chiral edge mode begins to influence electric transport, leading to an exponential dependence of σ_xx either on x at fixed L or on L at fixed x/L. Therefore, σ_xx(c = 1) ≪ σ_xx(c = 0) for L ≫ λ, providing direct experimental evidence for the exponential decay of conductance in QAH state. Such a strong contrast in longitudinal conductance between QAH state (c = 1) and the axion insulator state (c = 0) has already been observed in magnetic topological insulators (TIs) experimentally^50,56,73. Previously, exponential decay of conductance was observed mostly in systems with Anderson localization^74,75 or quantum tunneling⁷⁶. Our results show that non-Hermitian topology offers an alternative mechanism for such exponential behavior in condensed-matter systems.

Experimental realization in magnetic TIs

To demonstrate the experimental feasibility of our proposal, we investigate the transport properties of edge states in magnetic TIs using the Landauer-Büttiker formalism^77–78. We consider a 3D TI with independent magnetization on the top and bottom surfaces (see the insets of Fig. 3a, b). The system is described by the four-band effective Hamiltonian H_MTI = H_TI + H_M, where $H_{\mathrm{TI}}\left(\mathbf{k}\right)={\sum }_{i=x,y,z}{A}_i{k}_i{\sigma }_x\otimes s_i+\left(M_0-{\sum }_{i=x,y,z}{B}_i{k}_i^2\right){\sigma }_z\otimes s_0$ represents a 3D TI⁷⁹ with model parameters A_i, B_i, and M₀. Here, σ_i and s_i denote Pauli matrices acting on the orbital and spin degrees of freedom, respectively. The Zeeman splitting term is H_M = M(z)σ₀ ⊗ s_z, where M(z) = M_t on the top surface, M(z) = M_b on the bottom surface, and M(z) = 0 elsewhere.

Fig. 3. Energy spectra and size-dependent transport in magnetic topological insulators.

a, b Energy spectra of the surface states for parallel (c = 1,QAH) and antiparallel (c = 0, Axion insulator) magnetizations, respectively. Chiral edge states, normal edge states, and bulk states are shown in blue, red, and gray, respectively. The resulting dispersions reproduce those reported in Fig. 4o of Chang et al.⁴³. Insets show schematics of the corresponding magnetization and edge-state configurations. Longitudinal conductance σ_xx and Hall conductance σ_xy as functions of the system size L at several Fermi energies E_F for c c = 1 and d c = 0. e Spatial profile of the edge voltage for c = 0, ± 1 at E_F = 0.2, calculated using the multiterminal Hall-bar geometry shown in (f). f Schematic of the multiterminal Hall-bar geometry used to extract the spatial voltage profile. g, h Spatial distributions of the local heat dissipation P_D for g c = 1 and h c = 0 at E_F = 0.2. The model parameters are A_x = A_y = 1.0, A_z = 0.4, B_x = B_y = 0.4, B_z = 0.5, M₀ = 1.0, M_t = 0.4, and M_b = 0.4 for the QAH state and M_b = − 0.4 for the axion-insulator state. In (e–h), L = 240. The remaining parameters are x = L/3, L₀ = L/3, W = 100, and L_z = 6. The lattice constant is set to unity.

Figure 3a, b illustrate the band structures of TI under different magnetization configurations. For parallel magnetization, the magnetic TI realizes a QAH state with Chern number c = ± 1, characterized by the emergence of chiral edge states traversing the bulk energy gap and 1D normal edge states residing on the side surfaces. Upon magnetic reversal to an antiparallel alignment induced by an external magnetic field, the system transitions into an axion insulator with c = 0. In this regime, the Hall conductance vanishes, and only the 1D normal states persist. Crucially, the transport in these edge states is governed by the non-Hermitian HN Hamiltonian H_HN, as derived in Eq. (5).

Figure 3c, d show σ_xy and σ_xx as functions of the system length L for different magnetization configurations. For the QAH state (c = 1), σ_xy rapidly approaches the quantized value e²/h, while σ_xx decreases toward zero with increasing L. This confirms that the quantization of σ_xy remains robust even in the presence of normal edge modes. In contrast, for the axion insulator state (c = 0), σ_xy vanishes and σ_xx is independent of the system size, consistent with transport through only normal edge modes. Notably, the NHSE is directly manifested in the exponential voltage distribution V(x)/V(0) observed in the QAH phase (c = ± 1) [Fig. 3e], which stands in stark contrast to the linear distribution found in the axion insulator (c = 0) regime. This exponential voltage profile can be experimentally measured using a multi-terminal Hall bar device depicted in Fig. 3f. Furthermore, local heat dissipation P_D [Fig. 3g, h] reveals an exponential localization at the sample edges for the QAH state, in sharp contrast to the uniform distribution observed in the axion insulator state. A detailed quantitative comparison between theory and numerical results, presented in the Supplemental Material, shows good agreement across different system parameters.

The good agreement between our Landauer-Büttiker simulations (in Fig. 3) and analytic theory (in Fig. 2) underscores the reliability of our findings and the experimental feasibility of observing the NHSE in magnetic TIs. Recent experiments have already demonstrated magnetic switching between QAH and axion insulator phases via external fields in both magnetically doped (Bi, Sb)₂Te₃ TI sandwich structures and MnBi₂Te₄^50,56,73. Moreover, the interplay between chiral and normal edge states has been shown to be a primary dissipation mechanism in these magnetic TI heterostructures^37,43,44 (see, e.g., Fig. 4o in the experimental work of Chang et al.⁴³). This suggests that the proposed intrinsic non-Hermitian topology is well within the reach of current experimental techniques.

Fig. 4. Schematic illustration of the lattice model obtained from discretizing the QAH Hamiltonian.

Unidirectional blue arrows indicate the chiral channel originating from the chiral edge state, while bidirectional red arrows denote the normal channel arising from the normal edge state. The curved lines represent virtual leads attached to the corresponding lattice sites.

Discussion

We emphasize that the exponential behavior discussed here is fundamentally different from Anderson localization. The latter is characterized by the Anderson localization length ξ, which describes exponential decay of single-particle wavefunctions due to phase-coherent quantum interference. In contrast, our mechanism originates from nonreciprocal transport of the edge modes governed by the continuum Hatano–Nelson equation, introducing a characteristic length λ that governs the spatial evolution of the electrochemical potential. The proposed non-Hermitian mechanism can be experimentally distinguished from Anderson localization by two criteria. First, exponential decay of conductance in our model appears only in the QAH state but not in the axion-insulator state of the same device in the classical transport regime. This exponential behavior persists while Anderson localization is suppressed by dephasing when increasing temperature. Second, reversing magnetization reverses edge chirality and thus the direction of exponential voltage and dissipation profiles, providing a chirality-dependent signature absent in conventional Anderson localization. Notably, in the ideal QAH limit with only a chiral edge mode (σ_D = 0), the exponential voltage and heat-dissipation profiles disappear.

In summary, we have shown that a Hermitian quantum anomalous Hall system can intrinsically host non-Hermitian topology through the interplay between chiral and diffusive edge modes. Without introducing externally imposed dissipation or contact engineering, the edge transport dynamics naturally reduce to a continuum Hatano-Nelson model, leading to an experimentally measurable non-Hermitian skin effect manifested in voltage distribution, heat dissipation, and exponential longitudinal conductance. Our findings reveal that non-Hermitian topology need not rely on externally engineered gain/loss or reservoir-induced non-Hermitian band structures, but can instead emerge from microscopic chiral transport processes within a globally Hermitian quantum platform. This intrinsic mechanism provides a realistic and scalable route toward observing non-Hermitian phenomena in solid-state devices. Given the rapid experimental progress in quantum anomalous Hall heterostructures, the predicted exponential transport signatures should be directly accessible with existing measurement techniques. More broadly, this work establishes a bridge between mesoscopic transport theory and non-Hermitian topology, and opens new avenues for designing topological nonreciprocal electronic functionalities based on quantum materials.

Methods

Numerical method for the voltage and heat dissipation in magnetic topological insulators

We discretize the Hamiltonian H_MTI in the main text on cubic lattices:

$$H_{\mathrm{MTI}}=\left[{\sum }_{\mathbf{i}}{\mathbf{c}}_{\mathbf{i}}^{†}{\mathcal{M}}_0{\mathbf{c}}_{\mathbf{i}}+\left({\sum }_{\mathbf{i},j=x,y,z}{\mathbf{c}}_{\mathbf{i}}^{†}{\mathcal{T}}_j{\mathbf{c}}_{\mathbf{i}+{\widehat{\mathbf{e}}}_j}+\text{H}.\text{c}.\right)\right]+{\sum }_{\mathbf{i}}{\mathbf{c}}_{\mathbf{i}}^{†}{\mathcal{M}}_z{\mathbf{c}}_{\mathbf{i}}$$ 9

where ${\mathcal{M}}_0=\left(M_0-{\sum }_{j=x,y,z}2B_j/a^2\right){\sigma }_z\otimes s_0$, ${\mathcal{M}}_z=M_z\left(z\right){\sigma }_0\otimes s_z$, and ${\mathcal{T}}_j=B_j/a^2{\sigma }_z\otimes s_0-i{A}_j/\left(2a\right){\sigma }_x\otimes s_j$. Here, a = 1 is the cubic lattice constant. c_i (${\mathbf{c}}_{\mathbf{i}}^{†}$) is the annihilation (creation) operator at site i, and ${\widehat{\mathbf{e}}}_j$ is a unit vector in the j direction for j = x, y, z. The sample width, length, and thickness are given by W = N_ya, L = N_xa, and L_z = N_za, respectively.

We study the edge transport behavior in a six-terminal Hall-bar device. Within the Landauer–Büttiker formalism^80,81, the electric current and the heat current are given by

$$\begin{array}{c}{I}_p=\frac{e^2}{h}{\sum }_{q\ne p}\left(T_{qp}{V}_p-T_{pq}{V}_q\right),\hfill \\ P_D\left(p\right)=\frac{e^2}{2h}{\sum }_{q\ne p}{T}_{qp}{\left(V_p-V_q\right)}^2,\hfill \end{array}$$ 10

where V_p is the voltage at lead p. I_p denotes the current flowing out of lead p, while P_D(p) denotes the heat dissipation at lead p. The transmission probability from lead q to lead p is expressed as $T_{pq}\left(E_F\right)=\mathrm{Tr}\left[{\mathit{\Gamma }}_p{\mathit{G}}^r{\mathit{\Gamma }}_q{\mathit{G}}^a\right]$, with the linewidth function ${\mathit{\Gamma }}_p=i\left({\mathit{\Sigma }}_p^r-{\mathit{\Sigma }}_p^{r†}\right)$ and the retarded Green’s function ${\mathit{G}}^r={\left[E_F-H_{\mathrm{MTI}}-{\sum }_p{\mathit{\Sigma }}_p^r\right]}^{-1}$. Here ${\mathit{\Sigma }}_p^r$ is the self-energy due to the coupling between the central region and lead p. For system sizes exceeding the dephasing length, the transport behavior is consistent with semiclassical Boltzmann transport theory. The dephasing effect and heat dissipation are modeled by attaching Büttiker virtual leads to the side surfaces^69,77,78,80 (see Fig. 4). Note that we use the wide-band approximation for the real and virtual leads with Γ_p = Γ_pI_p, where Γ_p is the dephasing strength⁷⁸. I_p is 4n_p × 4n_p unit matrix. In our model, Γ_p = 2.0 for the source and drain, and Γ_p = 0.4 for the other leads. n_p is the number of sites coupled to lead p, where n_p = N_yN_z for the source and drain leads and n_p = N_z for other leads.

In our calculations, the source and drain leads act as current electrodes with fixed voltage ±V₀ and other leads act as voltage electrodes. Combining the Landauer-Büttiker formula with those boundary conditions in the leads, one can calculate the voltage of each lead and the longitudinal current I. V(x) denotes the voltage of the virtual lead located at position x along the one-dimensional coordinate. The longitudinal and Hall resistances are obtained from the measured voltages as ρ_xx = [V(x) − V(x + L₀)]W/(IL₀) and ρ_xy = [V(x) − V( − x)]/I. The corresponding longitudinal and Hall conductance are obtained from the tensor relation: ${\sigma }_{xx}={\rho }_{xx}/\left({\rho }_{xx}^2+{\rho }_{xy}^2\right)$, and ${\sigma }_{xy}={\rho }_{xy}/\left({\rho }_{xx}^2+{\rho }_{xy}^2\right)$. Only electrons in the edge states participate in the electrical transport when E_F is located in the bulk gap at zero or low temperature. Therefore, we just need to consider the dephasing process on the edge. We simulate the dephasing process by using 2n_x uniformly distributed Büttiker’s virtual leads on the top and bottom when we calculate the voltage and conductance. The heat dissipation distribution is calculated by using n_x × n_y uniformly distributed Büttiker’s virtual leads on the sample. Throughout our calculations, we set n_i/N_i = 0.2. N_i and n_i are the number of total sites and selected sites in the i direction for i = x, y, respectively.

Author contributions

X.-C. X. and C.-Z. C. conceived the idea and supervised the research. H. Z. performed the analytical and numerical calculations with the help of X.-C. X., C.-Z. C., and M. L. H. Z. and C.-Z. C wrote the manuscript with input from all other authors. All authors discussed the results and commented on the manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

This work was financially supported by the National Key R and D Program of China (Grant no. 2025YFA1412100), the National Natural Science Foundation of China (Grants nos. 123B2058, Nos.12574043), and the Natural Science Foundation of Jiangsu Province Grant (BK20230066). X.C.X. acknowledges support from the Innovation Program for Quantum Science and Technology (Grant no. 2021ZD0302400). This work was supported by High-performance Computing Platform of Peking University.

Data availability

The source data underlying the main and Supplementary Figs. are provided with this paper as a Source Data file. Source data are provided with this paper.

Code availability

The computation code for getting the theoretical predictions is available from the corresponding authors upon reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-77208-6.