Complex frequency detection in a subsystem

Abstract

Non-Hermitian physics, such as the non-Hermitian skin effect (NHSE), is well-established in classical platforms, but its emergence in intrinsically Hermitian or quantum systems remains a key challenge. Bridging this gap is crucial for connecting non-Hermitian concepts with foundational quantum many-body theory. Here, we systematically investigate this by studying a quantum subsystem with an effective non-Hermitian Hamiltonian arising from its exact frequency-dependent self-energy. We further employ complex-frequency detection, including excitation, synthesis, and fingerprint, to probe physical responses induced by complex driving frequencies. Our calculations reveal that both complex frequency excitation and synthesis are incompatible with the non-Hermitian approximation and cannot characterize the presence of the NHSE. In contrast, the complex-frequency fingerprint successfully detects the distinctive responses induced by the NHSE through the introduction of a double-frequency Green’s function. Our work provides a platform for studying non-Hermitian physics and its unconventional response in quantum systems rigorously without relying on any approximations.

Introduction

Non-Hermitian physics, particularly the non-Hermitian skin effect (NHSE), the phenomenon where an extensive number of non-orthogonal eigenstates become localized at the boundary^{1,2,3,4,5,6,7}, has recently attracted significant theoretical research interest^{5,6,7,8,9,10,11,12,13}. Many experimental developments have been made in classical wave platforms, including acoustic^14,15,16,17, optical^{18,19,20,21,22}, mechanical^23,24,25,26, and electrical circuit systems^{27,28,29,30,31}. These experiments have vividly demonstrated the existence of non-Hermitian skin modes in intrinsically dissipative systems. In many implementations, such as engineering asymmetric couplings in acoustic ring resonators¹⁵ and coupled optical fiber loops¹⁸, introducing gain and loss in photonic quantum walks¹⁹, the corresponding non-reciprocal or non-unitary dynamics associated with the NHSE have been further explored and corroborated.

However, these advances have predominantly involved phenomenological non-Hermitian terms, such as engineered gain and loss, implemented in classical wave platforms. In quantum systems, a key route to non-Hermitian descriptions originates from the self-energy within the Green’s function formalism^{13,32,33,34,35,36}. In real-space representation, the retarded Green’s function reads

$${G}^{{{{\rm{R}}}}}(\omega )=\frac{1}{\omega -{H}_{0}-\Sigma (\omega )},$$

(1)

where H₀ is the single-particle non-interacting Hamiltonian and Σ(ω) denotes the self-energy arising from, for example, electron-electron, electron-phonon, impurity, or substrate scattering. Since Σ(ω) is generally frequency-dependent, it is common practice to approximate it by a frequency-independent matrix, Σ(ω) ≃ Σ₀ = Σ(ω = 0), i.e., an approach termed the non-Hermitian approximation (NHA):

$${G}^{{{{\rm{R}}}}}(\omega )\simeq {G}_{{{{\rm{nH}}}}}^{{{{\rm{R}}}}}(\omega )=\frac{1}{\omega -{H}_{0}-{\Sigma }_{0}}.$$

(2)

This treatment often provides a good approximation to the spectral function and is widely employed in theoretical calculations. Under the NHA^{32,33,37,38,39,40,41,42,43,44}, the effective Hamiltonian

$${H}_{{{{\rm{nH}}}}}={H}_{0}+{\Sigma }_{0}$$

(3)

becomes non-Hermitian and can exhibit point-gap topology along with the corresponding non-Hermitian skin effect.

Recently, complex frequency detection techniques, including complex frequency excitation (CFE)^{45,46,47,48,49,50,51,52,53,54}, synthesis (CFS)^45,46,55, and fingerprint (CFF)⁵⁶, have garnered significant attention. As the names suggest, these methods enable the experimental detection of the corresponding Green’s function in the complex frequency domain (see Supplementary Note 2). This naturally raises a fundamental question: Given that the non-Hermitian approximation works well on the real-frequency axis, how does its validity extend over the entire complex plane? Moreover, if we apply complex frequency detection to such systems, what physical outcomes would emerge?

To address this question, we propose a subsystem model as illustrated in Fig. 1(a). When both excitations and responses are confined within a subsystem S, a frequency-dependent self-energy emerges, i.e., Σ_S(ω). Compared to dealing with the interacting system, the self-energy in this subsystem approach is often more tractable, both analytically and numerically, making it a practical and well-motivated route to exploring the corresponding complex frequency detection.

Fig. 1: Non-Bloch response in subsystem model.

a Schematic of subsystem S under excitation and detection, coupled to another subsystem ${S}^{{\prime} }$. b Spectra of H_S,nH under periodic boundary condition (PBC) and open boundary condition (OBC), denoted by blue and red circles, respectively. The nontrivial spectral topology is obtained via Eq. (12). Here, ω₁ = − 0.6 − 0.09i, ω₂ = − 0.6 − 0.128i, ω₃ = − 0.6 − 0.14i, ω₄ = − 0.6 − 0.16i, ω₅ = − 0.6 − 0.18i and ω₆ = − 0.6 − 0.21i. c Comparative analysis of the complex frequency Green’s function (red curve) obtained with and without the non-Hermitian approximation (NHA). The complex frequency Green’s function (CFGF) is calculated via Eq. (11) and Eq. (8), respectively, under OBC. The pink and blue shading represents regions where x ≤ x₀ and x > x₀ for $| {G}_{{{{\rm{S}}}}}^{{{{\rm{R}}}}}(\omega ){| }_{x{x}_{0}}$ (x₀ = 40), respectively. Detailed model is illustrated in Fig. 2(a) and defined via Eq. (4)–Eq. (6). Parameters are ${t}_{1}=1.2,{t}_{2}=-1,\lambda =-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{t}_{y}=3.5,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,{t}_{{{{\rm{A}}}}}=0.6,\gamma =0.1$, N_S = 80 and N_y = 300.

Our conclusion can be summarized as follows: As illustrated in Fig. 1(b), in our subsystem model H_S, under the NHA, H_S,nH = H_S + Σ_S(ω = 0) exhibits non-zero spectral winding under the periodic boundary condition (PBC), and the corresponding non-Hermitian skin effect under open boundary condition (OBC)^10,12. Now we consider the complex frequency Green’s function (CFGF) of H_S,nH under OBC. As illustrated in Fig. 1(c), frequencies from ω₃ to ω₅ (which possess non-zero spectral winding) exhibit rightward-dominant divergence in the CFGF upon excitation at x₀ = 40. Such a divergent behavior is a hallmark of the non-Bloch response^56,57,58 and can be used to characterize the presence of point gap topology as well as the NHSE^10,12,59,60 (See Supplementary Note 1).

However, if we consider the exact frequency-dependent self-energy without any approximation, the corresponding complex frequency Green’s function demonstrates that the unidirectional divergence becomes absent, as illustrated in Fig. 1(c) (right panel). This result raises three fundamental issues: (i) How to interpret this result? (ii) Does our subsystem truly exhibit the non-Hermitian skin effect and non-Bloch responses? (iii) What kind of result does the complex frequency detection method measure? Notably, as illustrated in Fig. 1(c), when ω lies near the real axis (e.g., ω₁), the exact and NHA Green’s functions behave similarly. However, when ω lies near the PBC spectral regions, the differences between them become dominant.

As summarized in Table 1, the long-time asymptotic behaviors of CFE and CFS yield the exact results with no non-Bloch response, whereas the CFF in the steady-state limit detects the approximated ones with the presence of non-Bloch response. Our work not only establishes a foundation for understanding the non-Hermitian skin effect and non-Bloch dynamics in quantum systems but also provides critical guidance for experimental verification.

Table 1 Results of CFE, CFS, and CFF for t → ∞

Results and Discussion

Model and subsystem Green’s function

Our Hamiltonian adopts the following Hermitian form:

$${\widehat{H}}_{{{{\rm{tot}}}}}={\widehat{H}}_{{{{\rm{S}}}}}+{\widehat{H}}_{{{{{\rm{S}}}}}^{{\prime} }}+{\widehat{H}}_{{{{{\rm{SS}}}}}^{{\prime} }},$$

(4)

where the hopping parameters are illustrated in Fig. 2(a). The subsystem S corresponds to a one-dimensional spinless Rice-Mele model with broken time-reversal symmetry¹³, positioned in the y = 1 region, with sublattices A and B represented by red and blue points, respectively. Applying the Fourier transformation, we obtain the Bloch Hamiltonian:

$${H}_{{{{\rm{S}}}}}(k)=({t}_{1}+{t}_{2}\cos k){\sigma }_{{{{\rm{x}}}}}+{t}_{2}\sin k{\sigma }_{{{{\rm{y}}}}}+(\lambda \sin k-{\mu }_{{{{\rm{S}}}}}){\sigma }_{{{{\rm{z}}}}},$$

(5)

where the Pauli matrix acts on the sublattice degree of freedom. We denote the number of unit cells in S by N_S.

Fig. 2: Model schematic and flat self-energy of the subsystem.

a Schematic representation of the tight-binding model for the composite system, with subsystem S explicitly demarcated by the red rectangular enclosure. Here, red and blue dots correspond to atoms in sublattices A and B in S, respectively; the black dot represents the atom in ${S}^{{\prime} }$; black lines indicate chemical bonds; and a double-headed arrow marks the next-nearest hopping. b and c A comparison between the slowly varying self-energy and the bare bands E_±(k) (marked by black lines) of the subsystem S. Here, the self-energy exhibits negligible variation (two orders of magnitude smaller than the bandwidth of S) across the band, which ensures the validity of the non-Hermitian approximation. The color bar in (b) and (c) ranges from −0.094 to 0.094 and 0.001 to 0.1, respectively. The parameters include ${t}_{1}=1.2,{t}_{2}=-1,\lambda =-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{t}_{y}=3.5,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,{t}_{{{{\rm{A}}}}}=0.6,\gamma =0.1$, N_S = 80 and N_y = 300.

The subsystem ${S}^{{\prime} }$ comprises a two-dimensional single-band model inhabiting the y ≥ 2 region, featuring anisotropic hoppings with t_x ≠ t_y. Its x-direction Bloch Hamiltonian adopts the layer-resolved form:

$${H}_{{{{{\rm{S}}}}}^{{\prime} }}(k)=\left(\begin{array}{cccccc}{h}_{{{{{\rm{S}}}}}^{{\prime} }}(k) & {t}_{y}{\sigma }_{0} & 0 & 0 & 0 & \ldots \\ {t}_{y}{\sigma }_{0} & {h}_{{{{{\rm{S}}}}}^{{\prime} }}(k) & {t}_{y}{\sigma }_{0} & 0 & 0 & \ldots \\ 0 & {t}_{y}{\sigma }_{0} & {h}_{{{{{\rm{S}}}}}^{{\prime} }}(k) & {t}_{y}{\sigma }_{0} & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \end{array}\right),$$

(6)

where ${h}_{{{{{\rm{S}}}}}^{{\prime} }}(k)=({t}_{x}+{t}_{x}\cos k){\sigma }_{{{{\rm{x}}}}}+{t}_{x}\sin k{\sigma }_{{{{\rm{y}}}}}-{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}{\sigma }_{0}$ describes the Hamiltonian at each y-layer using a two-site unit cell. The interlayer hopping in the subsystem $S^{\prime}$ is parameterized by t_yσ₀, and the total number of y-layers in the subsystem $S^{\prime}$ is denoted as N_y. Notably, although ${\widehat{H}}_{{{{{\rm{S}}}}}^{{\prime} }}$ is intrinsically a single-band system along the x direction, we adopt a two-band representation for conceptual and notational convenience. This choice circumvents the band-folding issue that would otherwise arise in Σ_S(k, ω), thereby preserving momentum consistency between H_S(k) and the self-energy. Importantly, the physical results, particularly the subsystem Green’s function of S, remain unaffected by this representational choice, as they are determined solely by projecting the full Green’s function onto the S subspace within the fixed total Hilbert space.

For conceptual clarity, we restrict inter-subsystem coupling to the A-sublattice (denoted as t_A; qualitative results hold for any t_B ≠ t_A, see Supplementary Note 8 and Fig. S3). In this case, as derived in Supplementary Note 3, the exact self-energy for the subsystem S becomes:

$${\Sigma }_{{{{\rm{S}}}}}(k,\omega )=\left(\begin{array}{cc}{\Sigma }_{{{{\rm{S}}}}}^{{{{\rm{AA}}}}}(k,\omega ) & 0\\ 0 & 0\\ \end{array}\right),$$

(7)

where ${\Sigma }_{{{{\rm{S}}}}}^{{{{\rm{AA}}}}}(k,\omega )={t}_{{{{\rm{A}}}}}^{2}{[1/(\omega -{H}_{{{{{\rm{S}}}}}^{{\prime} }}(k))]}_{11}$. Here, the subscript 11 denotes the (1, 1) matrix element of the Green’s function for ${H}_{{{{{\rm{S}}}}}^{{\prime} }}(k)$ within its two-band representation, corresponding to the A-sublattice. Fig. 2(b, c) presents the corresponding density plots of the real and imaginary components of ${\Sigma }_{{{{\rm{S}}}}}^{{{{\rm{AA}}}}}(k,\omega )$, respectively. Based on this exact self-energy, the exact Green’s function for subsystem S under PBC can be written as:

$${G}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{R}}}}}(k,\omega )=\frac{1}{\omega -{H}_{{{{\rm{S}}}},{{{\rm{eff}}}}}(k,\omega )},$$

(8)

where the frequency-dependent non-Hermitian Hamiltonian is explicitly expressed as

$${H}_{{\rm S},{{{\rm{eff}}}}}(k,\omega )={H}_{\rm S}(k)+{\Sigma }_{\rm S}(k,\omega +{{{\rm{i}}}}\gamma )-{{{\rm{i}}}}\gamma .$$

(9)

Here, ω spans the entire complex energy plane, while the parameter iγ serves as a trivial, constant dissipation, equivalent to the iη in conventional retarded Green’s function formalism of a Hermitian system. In practical numerical calculations, γ is chosen as a small value, which does not affect the qualitative results (See Supplementary Note 9 and Fig. S4–S6). Thus, the coupling to ${S}^{{\prime} }$, or the self-energy, is necessary to induce nontrivial non-Hermitian physics, such as NHSE and non-Bloch phenomenon. It is precisely the detection of non-Bloch responses in a physically Hermitian system that demonstrates the significance of complex frequency detection, as discussed later. For notational simplicity, we retain the explicit iγ in the self-energy but treat it as implicit in the frequency argument of H_S,eff(k, ω).

Non-Hermitian approximation

We now check the validity of the non-Hermitian approximation (NHA). As shown in Fig. 2(b) and (c), the two energy bands of H_S(k), defined as E_±(k), are plotted against the real and imaginary components of the self-energy, respectively. Notably, within the band dispersion region, the self-energy can be well approximated by a constant due to its negligible variation across the band (two orders of magnitude smaller than the bandwidth). Consequently, the expansion point k = 0, ω = 0 (the band center of S) is chosen merely for convenience:

$${\Sigma }_{{{{\rm{S}}}}}^{{{{\rm{AA}}}}}(k,\omega +{{{\rm{i}}}}\gamma )\approx {\Sigma }_{{{{\rm{S}}}}}^{{{{\rm{AA}}}}}(0,{{{\rm{i}}}}\gamma )+{{{\mathcal{O}}}}(k)+{{{\mathcal{O}}}}(\omega ).$$

(10)

As further demonstrated in the Methods section, this negligible variation of the self-energy makes the choice of the approximation point arbitrary. This NHA yields the following approximated Green’s function

$${G}_{{{{\rm{S}}}},{{{\rm{nH}}}}}^{{{{\rm{R}}}}}(k,\omega )=\frac{1}{\omega -{H}_{{{{\rm{S}}}},{{{\rm{nH}}}}}(k)},$$

(11)

where the frequency-independent non-Hermitian Hamiltonian is given by:

$${H}_{{{{\rm{S}}}},{{{\rm{nH}}}}}(k)={H}_{{{{\rm{S}}}}}(k)+\left(\begin{array}{cc}{\Sigma }_{{{{\rm{S}}}}}^{{{{\rm{AA}}}}}(0,{{{\rm{i}}}}\gamma ) & 0\\ 0 & 0\end{array}\right)-{{{\rm{i}}}}\gamma .$$

(12)

To further verify the non-Hermitian approximation, we present a comparative analysis of exact versus approximated spectral functions, i.e., ${A}_{{{{\rm{S}}}},{{{\rm{eff}}}}/{{{\rm{nH}}}}}(k,\omega )=-(1/\pi ){{{\rm{Im}}}}\,{{{\rm{Tr}}}}\,{G}_{{{{\rm{S}}}},{{{\rm{eff}}}}/{{{\rm{nH}}}}}^{{{{\rm{R}}}}}(k,\omega \in {\mathbb{R}})$. As illustrated in Fig. 3(a) and (b), the precise agreement between these spectral functions confirms the validity of the NHA. This conclusion is further corroborated by the density of states (DOS) at real frequencies shown in Fig. 3(c), which follows the conventional definition, i.e., ${\rho }_{{{{\rm{S}}}},{{{\rm{eff}}}}/{{{\rm{nH}}}}}(\omega )=(1/{N}_{{{{\rm{S}}}}}){\sum }_{k}{A}_{{{{\rm{S}}}},{{{\rm{eff}}}}/{{{\rm{nH}}}}}(k,\omega )$. It should be noted, however, that this excellent agreement is confined to the real-frequency axis and deteriorates as the imaginary part of the frequency approaches the dissipation scale γ (see the validity domain analysis below).

Fig. 3: Verification of the non-Hermitian approximation at real and complex frequencies.

The spectral functions under non-Hermitian approximation (NHA), A_S,nH(k, ω), and from the exact case, A_S,eff(k, ω), respectively. The color bar ranges are [0.002, 2.98] for (a) and [0.003, 2.99] for (b). c A comparison between the real frequency density of states (DOS) with and without the NHA, i.e., ρ_S,nH(ω) (blue circles) and ρ_S,eff(ω) (green curve), respectively. d, e The complex frequency DOS with and without NHA, defined as ${D}_{{{{\rm{S}}}},{{{\rm{nH}}}}}(\omega \in {\mathbb{C}})$ and ${D}_{{{{\rm{S}}}},{{{\rm{eff}}}}}(\omega \in {\mathbb{C}})$ respectively, where the black circles indicate the poles of the complex frequency Green’s function. The poles in (d) form the spectra with a nonzero winding number. The non-Hermitian approximation remains accurate near the real axis but breaks down in the complex plane as $| {{{\rm{Im}}}}\,\omega |$ increases. For (d) and (e), we choose ω = − 1 − ω_ii, with ω_i ∈ [0.08, 0.16] to compute the winding number shown in Fig. 8 in the “Methods” section. Frequencies from Fig. 1c's non-Bloch region, i.e., ω₃, ω₄, ω₅ = − 0.6 − 0.14i, − 0.6 − 0.16i, − 0.6 − 0.18i are marked in (d) as red, green, and pink dots, respectively. The color bar ranges are [−10, 10] for (d) and [−2.8, 2.8] for (e). Other parameters are ${t}_{1}=1.2,{t}_{2}=-1,\lambda =-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{t}_{y}=3.5,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,{t}_{{{{\rm{A}}}}}=0.6,\gamma =0.1$, N_S = 80 and N_y = 300.

Importantly, as illustrated later in the “Methods” section and in Supplementary Note 4, the validity of NHA requires that the coupled environment ${S}^{{\prime} }$ possesses a sufficiently large number of degrees of freedom and a broad enough bandwidth compared to S. This explains our choice of a 2D anisotropic environment with t_y > t_x (t_x sets the bandwidth scale for S) and a sufficiently large N_y.

However, significant discrepancies emerge when considering the complex frequency domain. Figure 1(c) demonstrates a representative example of these differences. For comprehensive analysis, we introduce the complex-frequency density of states⁵⁶ across the entire complex plane, i.e.,

$${D}_{{{{\rm{S}}}},{{{\rm{eff}}}}/{{{\rm{nH}}}}}(\omega )=-\frac{1}{{N}_{{{{\rm{S}}}}}\pi }{\sum }_{k}{{{\rm{Im}}}}\,{{{\rm{Tr}}}}\,{G}_{{{{\rm{S}}}},{{{\rm{eff}}}}/{{{\rm{nH}}}}}^{{{{\rm{R}}}}}(k,\omega \in {\mathbb{C}}),$$

(13)

which serves not only as a tool for analytic continuation, revealing the pole structures of the complex frequency Green’s function⁵⁶, but also as an indirect observable that faithfully captures the system’s underlying quasiparticle picture in the complex plane. As shown in Figs. 3(d, e), the approximated CFGF displays two closed-loop poles, contrasting with the exact case that features a branch cut positioned at ${{{\rm{Im}}}}\,\omega =-\gamma$ (See Supplementary Note 5 for demonstration). Due to the vanishing of the spectral winding in the exact poles under PBC, it is expected that there is no non-Bloch response in the complex frequency Green’s function, which explains the paradox observed in Fig. 1(c). However, despite their distinct distributions, a profound connection exists: both the NHA loop-poles and the exact branch cut poles can be linked through an electrostatic mapping, as they generate an identical “electric field” along the real frequency axis. See Supplementary Note 7 for details.

As an aside, the poles of the Matsubara Green’s function are related to those of the complex frequency Green’s function by a Wick rotation and a constant γ-shift, detailed in Supplementary Note 6.

In summary, although the non-Hermitian approximation demonstrates excellent performance in simulating real-frequency Green’s function, it fundamentally modifies the global pole structures. The very success of the NHA in simulating real-frequency observables masks the fundamental disparity in the complex plane, thereby highlighting the critical need for our advanced approach to uncover genuine non-Bloch physics. Therefore, it is necessary to examine the nontrivial roles of the non-Hermitian approximation in variant complex-frequency detection methodologies.

Driven-dissipative equation

We now focus on the complex frequency detection of the subsystem. The theoretical framework originates from the following driving dissipative equation⁵⁶:

$${{{\rm{i}}}}\frac{{{{\rm{d}}}}\langle \widehat{{{{\bf{a}}}}}(t)\rangle }{{{{\rm{d}}}}t}=({H}_{{{{\rm{tot}}}}}-{{{\rm{i}}}}\gamma )\langle \widehat{{{{\bf{a}}}}}(t)\rangle +{{{\bf{F}}}}(t),$$

(14)

where H_tot represents the first quantized Hamiltonian of ${\widehat{H}}_{{{{\rm{tot}}}}}$ in real space. The parameter γ characterizes the uniform dissipation within the Green’s function. As detailed in Supplementary Note 2, $\langle \widehat{{{{\bf{a}}}}}(t)\rangle ={{{\rm{Tr}}}}\left[\widehat{{{{\bf{a}}}}}\widehat{\rho }(t)\right]={\{\langle {\widehat{{{{\bf{a}}}}}}_{{{{\rm{S}}}}}(t)\rangle ,\langle {\widehat{{{{\bf{a}}}}}}_{{{{{\rm{S}}}}}^{{\prime} }}(t)\rangle \}}^{T}$ describes the mean values of single-particle bosonic operators at each lattice site of the total system within the open quantum system framework. Here, $\widehat{{{{\bf{a}}}}}={({\widehat{a}}_{1},{\widehat{a}}_{2},\ldots ,{\widehat{a}}_{N})}^{T}$ denotes the bosonic annihilation operator over the total system, N is the matrix dimension of H_tot, and $\widehat{\rho }(t)$ is the density matrix operator. F(t) denotes the external drive. Experimentally, when F(t) is applied to the subsystem S, the corresponding response $\langle {\widehat{{{{\bf{a}}}}}}_{{{{\rm{S}}}}}(t)\rangle$ is detected. For example, consider an external driving defined as

$${{{\bf{F}}}}(t)=\theta (t){{{{\rm{e}}}}}^{-{{{\rm{i}}}}{\omega }_{{{{\rm{r}}}}}t}{\{0,\ldots ,0,{F}_{{{{\rm{S}}}},i},0,\ldots ,0\}}^{T},\,{\omega }_{{{{\rm{r}}}}}\in {\mathbb{R}},$$

(15)

which acts exclusively on the ith-site of the subsystem S. Following Supplementary Note 2, the induced response at the j-th site, i.e., $\langle {\widehat{a}}_{{{{\rm{S}}}},j}(t)\rangle$, which is the direct experimental observable, demonstrates proportionality to the real-frequency Green’s function in the long-time limit. This relationship can be expressed as:

$${\rm lim}_{t\to \infty }\langle {\widehat{a}}_{{{{\rm{S}}}},j}(t)\rangle ={\left[{G}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{R}}}}}({\omega }_{{{{\rm{r}}}}})\right]}_{ji}{F}_{{{{\rm{S}}}},i}{{{{\rm{e}}}}}^{-{{{\rm{i}}}}{\omega }_{{{{\rm{r}}}}}t},$$

(16)

where ${G}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{R}}}}}({\omega }_{{{{\rm{r}}}}})$ denotes the exact Green’s function of the subsystem defined in Eq. (8) (real space representation), and ω_r corresponds to the driving frequency. Given the predetermined values of F_S,i and ω_r, the real-frequency Green’s function becomes experimentally accessible through $\langle {\widehat{a}}_{{{{\rm{S}}}},j}(t)\rangle$ detection. This methodology maintains alignment with classical wave system protocols for real-frequency Green’s function measurement^{61,62,63,64,65}.

Complex frequency excitation

When extending the driving frequency from real (ω_r) to complex values (ω_c) in Eq. (15), the concept of complex frequency excitation (CFE) emerges. As demonstrated in Supplementary Note 2, The experimental procedure of the CFE is largely identical to that of the CFF, except that the harmonic driving field is replaced by one with temporal attenuation. Specifically, the solution to Eq. (14) under complex frequency driving

$${{{\bf{F}}}}(t)=\theta (t){{{{\rm{e}}}}}^{-{{{\rm{i}}}}{\omega }_{{{{\rm{c}}}}}t}{\{0,..,0,{F}_{{{{\rm{S}}}},i},0,\ldots ,0\}}^{T},\,{\omega }_{{{{\rm{c}}}}}\in {\mathbb{C}},$$

(17)

takes the form

$$\langle {\widehat{a}}_{{{{\rm{S}}}},j}(t)\rangle ={\left[{\chi }_{{{{\rm{S}}}}}({\omega }_{{{{\rm{c}}}}})\right]}_{ji}{F}_{{{{\rm{S}}}},i}{{{{\rm{e}}}}}^{-{{{\rm{i}}}}{\omega }_{{{{\rm{c}}}}}t},$$

(18)

where the response function χ_S(ω_c), serving as the key CFE observable, is expressed as

$${\chi }_{{{{\rm{S}}}}}({\omega }_{{{{\rm{c}}}}})={G}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{R}}}}}({\omega }_{{{{\rm{c}}}}})-{\left[{G}_{{{{\rm{tot}}}}}^{{{{\rm{R}}}}}({\omega }_{{{{\rm{c}}}}}){{{{\rm{e}}}}}^{-{{{\rm{i}}}}({H}_{{{{\rm{tot}}}}}-{{{\rm{i}}}}\gamma -{\omega }_{{{{\rm{c}}}}})t}\right]}_{{{{\rm{S}}}}}.$$

(19)

Here, the subscript S indicates the subsystem component, and ${G}_{{{{\rm{tot}}}}}^{{{{\rm{R}}}}}({\omega }_{{{{\rm{c}}}}})$ denotes the complex frequency Green’s function for the total system. We note that this solution assumes the initial condition $\langle \widehat{{{{\bf{a}}}}}(0)\rangle =0$, whose value will not affect the subsequent conclusions, as proved in Supplementary Note 2.

A critical observation arises from the above formulation: when the complex frequency ω_c resides above the branch cut determined by iγ (light pink region in Fig. 4(a)), the corresponding response function will converge to the exact CFGF of the subsystem in the t → ∞ limit, i.e.,

$${\rm lim}_{t\to \infty }{\chi }_{{{{\rm{S}}}}}({\omega }_{c})={G}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{R}}}}}({\omega }_{{{{\rm{c}}}}}),\,{{{\rm{Im}}}}\,{\omega }_{{{{\rm{c}}}}} > -\gamma .$$

(20)

Conversely, frequencies below the iγ line will yield divergent responses due to the second term’s divergence. A compact demonstration can be made by expanding the response function as

$${\chi }_{{{{\rm{S}}}}}({\omega }_{{{{\rm{c}}}}})=P\mathop{\sum }_{n=1}^{N}\frac{\left|{\psi }_{n}\right\rangle \left\langle {\psi }_{n}\right|}{{\omega }_{{{{\rm{c}}}}}+{{{\rm{i}}}}\gamma -{\epsilon }_{n}}({{{\bf{I}}}}-{{{{\rm{e}}}}}^{-{{{\rm{i}}}}({\epsilon }_{n}-{{{\rm{i}}}}\gamma -{\omega }_{{{{\rm{c}}}}})t}){P}^{{{\dagger}} },$$

(21)

where $P=({I}_{2{N}_{{{{\rm{S}}}}}\times 2{N}_{{{{\rm{S}}}}}},{0}_{2{N}_{{{{\rm{S}}}}}\times (N-2{N}_{{{{\rm{S}}}}})})$ denotes the projection matrix, and $\left|{\psi }_{n}\right\rangle$ and ϵ_n denote the eigenstates and eigenvalues of the Hermitian matrix H_tot under OBC. Notably, by Taylor expansion of the exponential function, one can see that Eq. (21) has no poles; thus, ${{{\rm{Im}}}}\,{\omega }_{{{{\rm{c}}}}}=-\gamma$ becomes the convergence boundary. As numerically verified in Fig. 4(b), only the ω₇ case converges to the exact CFGF (blue dashed lines).

Fig. 4: Comparison of complex frequency excitation and synthesis.

a The poles of the subsystem complex frequency Green’s function with and without non-Hermitian approximation under (NHA) open boundary conditions (OBC). In the complex plane, the light pink shading represents the region within the branch cut, and the blue shading represents the area beyond the non-Hermitian spectra. The area between them is shaded light yellow. b A comparison between the time-dependent response function χ_S(ω_c) for the complex frequency excitation (CFE, gray curve), and the complex frequency Green’s function with (red dashed line) and without NHA (blue dashed line). The result is obtained under OBC, exhibiting the failure of NHA in the complex plane. Here ω₇ = − 1 − 0.05i, ω₈ = − 1 − 0.115i, and ω₃ ~ ω₅ in Fig. 1c are marked as red, green, and pink dots, respectively. j₀ = 30, and i₀ = 36. c A comparison of the observable $\langle {\widehat{a}}_{{{{\rm{S}}}},{j}_{0}}(t)\rangle$ between the CFE (gray curve) and complex frequency synthesis (CFS) at ω₇ with different synthesis domains and spectral resolution (green, blue, and orange curves, respectively). Here i₀ = 36. Crucially, the line ${{{\rm{Im}}}}\,\omega =-\gamma$ serves as a critical boundary: CFE (and equivalently CFS) responses undergo a qualitative change from steady-state convergence to divergence when crossing it. d Error analysis of the CFS. The error residuals, defined as the relative difference of the complex frequency synthesis (CFS) results from the complex frequency excitation (CFE) results in (c) (green, blue and orange curves, respectively). Other parameters are ${t}_{1}=1.2,{t}_{2}=-1,\lambda =-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{t}_{y}=3.5,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,{t}_{{{{\rm{A}}}}}=0.6,\gamma =0.1$, N_S = 80 and N_y = 300.

For comparative analysis, we show the NHA-predicted complex frequency Green’s function as red dashed lines, which exhibit clear deviations from our calculated response functions. This discrepancy stems from the modification of CFGF pole structures under the non-Hermitian approximation (Fig. 3(d, e)), rendering it invalid for frequencies distant from the real axis. The failure originates near the system’s exact poles, where the approximation error becomes singular, rendering the perturbative expansion invalid. Fig. 4(b) explicitly demonstrates this limitation: for ω₇, converged results deviate from NHA predictions; ω₈ exhibits response divergence despite NHA suggesting convergence.

Key conclusions are summarized in Table 1: (i) The CFE exclusively detects the exact complex frequency Green’s function, i.e., Eq. (8), above the total Hamiltonian’s imaginary gap (light pink region); (ii) NHA fails in regions far from the real axis, an incompatibility that holds in the case of a frequency-dependent self-energy; (iii) Absence of non-Bloch response in long-time dynamics due to the vanishing of the spectral winding number.

Complex frequency synthesis

We now examine the complex frequency synthesis (CFS), a method based on the following observation: within the framework of CFE, the nonzero component of F(t) in Eq. (17) can be transformed through Fourier analysis as

$$\theta (t){{{{\rm{e}}}}}^{-{{{\rm{i}}}}{\omega }_{{{{\rm{c}}}}}t}=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\frac{{{{\rm{i}}}}}{\omega -{\omega }_{{{{\rm{c}}}}}}{{{{\rm{e}}}}}^{-{{{\rm{i}}}}\omega t}{{{\rm{d}}}}\omega ,$$

(22)

where ${{{\rm{Im}}}}\,{\omega }_{{{{\rm{c}}}}} < 0$ and t > 0. Experimentally measured real frequency responses enable comprehensive reconstruction of the CFE solution through signal synthesis (see Supplementary Note 2):

$$\langle {\widehat{{{{\bf{a}}}}}}_{{{{\rm{S}}}},{\omega }_{{{{\rm{c}}}}}}(t)\rangle =\frac{{{{\rm{i}}}}}{2\pi }\mathop{\sum }_{\omega ={\omega }_{{{{\rm{a}}}}}}^{{\omega }_{{{{\rm{b}}}}}}\frac{\Delta \omega }{\omega -{\omega }_{{{{\rm{c}}}}}}\langle {\widehat{{{{\bf{a}}}}}}_{{{{\rm{S}}}},\omega }(t)\rangle ,$$

(23)

where [ω_a, ω_b] defines the synthesis domain and Δω specifies the spectral resolution defined as the sampling interval between adjacent discrete frequencies. Following the measurement steps detailed in supplementary Note 2, the key observables of the CFS are the series of $\langle {\widehat{{{{\bf{a}}}}}}_{{{{\rm{S}}}},\omega }(t)\rangle$ within the synthesis domain. The quantity $\langle {\widehat{{{{\bf{a}}}}}}_{{{{\rm{S}}}},{\omega }_{{{{\rm{c}}}}}}(t)\rangle$ is the numerically synthesized result.

As summarized in Table 1, CFS demonstrates theoretical equivalence with CFE. Specifically, as detailed in Supplementary Note 2, the established exact identity:

$$\langle {\widehat{{{{\bf{a}}}}}}_{{{{\rm{S}}}},{\omega }_{{{{\rm{c}}}}}}(t)\rangle =\frac{{{{\rm{i}}}}}{2\pi }{\int }_{-\infty }^{\infty }\frac{1}{\omega -{\omega }_{{{{\rm{c}}}}}}\langle {\widehat{{{{\bf{a}}}}}}_{{{{\rm{S}}}},\omega }(t)\rangle {{{\rm{d}}}}\omega$$

(24)

serves as rigorous proof of this equivalence. A quantitative validation is presented in Fig. 4(c), where the result derived from CFE is compared with synthesized responses across progressively expanding synthesis domains and refined spectral resolutions Δω. The convergence toward the CFE result confirms the validity of the CFS method, which is further verified by the error residuals in Fig. 4(d).

Complex frequency fingerprint

To capture the non-Bloch response, the recently developed complex frequency fingerprint (CFF) method under real-frequency driving is necessary. Its experimental implementation comprises three concise steps (details in Supplementary Note 2 and Fig. S1): (i) Apply a harmonic drive ${F}_{i}(t)=\theta (t){{{{\rm{e}}}}}^{-{{{\rm{i}}}}{\omega }_{{{{\rm{r}}}}}t}{F}_{{{{\rm{S}}}},i}$ at a single site i and measure the induced response ${\langle {\widehat{a}}_{j}(t)\rangle }_{{F}_{i}}$ at all sites j within subsystem S; (ii) repeat step (i) sequentially for every site i = 1, …, 2N_S to construct the full response matrix χ_S(ω_r, t), with elements ${[{\chi }_{{{{\rm{S}}}}}({\omega }_{{{{\rm{r}}}}},t)]}_{ji}={\langle {\widehat{a}}_{j}(t)\rangle }_{{F}_{i}}/({F}_{{{{\rm{S}}}},i}{{{{\rm{e}}}}}^{-{{{\rm{i}}}}{\omega }_{{{{\rm{r}}}}}t})$; (iii) define the CFF as

$${{{{\mathscr{G}}}}}_{{{{\rm{S}}}},{\omega }_{{{{\rm{r}}}}}}({\omega }_{{{{\rm{c}}}}}\in {\mathbb{C}};t)=\frac{1}{({\omega }_{{{{\rm{c}}}}}-{\omega }_{{{{\rm{r}}}}})+{[{\chi }_{{{{\rm{S}}}}}({\omega }_{{{{\rm{r}}}}},t)]}^{-1}},$$

(25)

with an auxiliary parameter ω_c. As demonstrated in Supplementary Note 2, as t → ∞, the CFF specifically measures the following double-frequency Green’s function,

$${G}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{CFF}}}}}(k,{\omega }_{{{{\rm{c}}}}},{\omega }_{{{{\rm{r}}}}})=\frac{1}{{\omega }_{{{{\rm{c}}}}}-{H}_{{{{\rm{S}}}},{{{\rm{eff}}}}}(k,{\omega }_{{{{\rm{r}}}}})},$$

(26)

where ${\omega }_{{{{\rm{c}}}}}\in {\mathbb{C}}$ and ${\omega }_{{{{\rm{r}}}}}\in {\mathbb{R}}$. Notably, here we clarify that as detailed in Supplementary Note 2, ω_c and ω_r in the CFF method are independent parameters: ω_c is an artificially introduced complex variable, while ω_r is a controlled parameter corresponding to the frequency of the actual driving field defined in Eq. (15). Crucially, for any fixed driving parameter ω_r, this double-frequency Green’s function precisely corresponds to the CFGF associated with the non-Hermitian Hamiltonian H_S,eff(k, ω_r). As ω_r evolves, the corresponding double-frequency Green’s function changes accordingly. Figure 5(a) and (b) exemplify this behavior through the following complex-frequency DOS

$${D}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{CFF}}}}}({\omega }_{{{{\rm{c}}}}},{\omega }_{{{{\rm{r}}}}})=-\frac{1}{{N}_{{{{\rm{S}}}}}\pi }\mathop{\sum }_{k}{{{\rm{Im}}}}\,{{{\rm{Tr}}}}\,{G}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{CFF}}}}}(k,{\omega }_{{{{\rm{c}}}}},{\omega }_{{{{\rm{r}}}}})$$

(27)

at frequencies ω_r = 4 and ω_r = 5, respectively.

Fig. 5: The results of complex frequency fingerprint.

a, b The complex frequency density of states for the complex frequency fingerprint (CFF) defined as ${D}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{CFF}}}}}({\omega }_{{{{\rm{c}}}}},{\omega }_{{{{\rm{r}}}}})$ at ω_r = 4 and ω_r = 5, respectively. Both results in (a) and (b) exhibit nontrivial spectral topology, with the color bar ranging from −10 to 10. The poles of the double-frequency Green’s function are marked by black circles. Other parameters are ${t}_{1}=1.2,{t}_{2}=-1,\lambda =-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{t}_{y}=3.5,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,{t}_{{{{\rm{A}}}}}=0.6,\gamma =0.1$, N_S = 80 and N_y = 300.

For any given ω_r, once H_S,eff(k, ω_r) exhibits the NHSE, the corresponding non-Bloch response can be detected via the double-frequency Green’s function by setting ω_c with a nonzero spectral winding. An example is illustrated on the left side of Fig. 1(c) with ω_r = 0. Moreover, for the non-Hermitian skin effect in H_S,eff(k, ω_r) to be pronounced, ω_r must lie within the spectral range of ${S}^{{\prime} }$. If ω_r lies far outside this region, the self-energy becomes vanishingly small, causing the NHSE, even if theoretically present, to be imperceptible. Furthermore, the nonzero spectral winding originates primarily from two intrinsic properties of subsystem S: the breaking of time-reversal symmetry and its asymmetric coupling to ${S}^{{\prime} }$ (t_A ≠ t_B)¹³. It is not explicitly tied to the topological properties of the full system ${\widehat{H}}_{{{{\rm{tot}}}}}$.

In summary, as illustrated in Table 1: (i) The CFF methodology effectively identifies double-frequency Green’s function governed by Eq. (26); (ii) For each driving frequency ω_r, the CFF is described by the non-Hermitian Hamiltonian at ω_r, i.e., H_S,eff(k, ω_r); (iii) This framework establishes a unified platform for the systematic exploration of NHSE and non-Bloch responses in a Hermitian quantum system.

Notably, the inequivalence between CFF and CFE primarily stems from the frequency dependence of the self-energy, reflecting the breakdown of the Markovian approximation. However, even when the Markovian approximation is valid, if the subsystem Hamiltonian ${\widehat{H}}_{{{{\rm{S}}}}}$ is intrinsically non-Hermitian and hosts nontrivial spectral topology, CFE and CFF remain inequivalent due to virtual gain (divergent response from CFE, see Supplementary Note 2). Meanwhile, the CFF approach continues to correctly capture the response across the full complex-frequency plane under real-frequency excitation⁵⁶.

Conclusion

In this work, we have established a framework for detecting complex frequency responses in a frequency-dependent Hamiltonian of the subsystem. Our comparative study reveals that while both CFE and CFS methods yield identical results characterizing the exact complex frequency Green’s function without non-Hermitian skin effect signatures, the CFF approach uniquely manifests non-Bloch responses through its detection of double-frequency Green’s function. Our work elucidates the physical origin of non-Hermitian Hamiltonians in quantum systems through exact theoretical formalisms, bypassing traditional approximation constraints. These advancements provide foundational insights for decoding non-Hermitian phenomena in quantum systems and offer concrete guidance for experimental implementations.

Methods

Validity domain analysis of the NHA

We perform an error analysis to determine the validity domain of the self-energy under the non-Hermitian approximation. For this purpose, we define a relative error function, e.g., ${{{\rm{Err}}}}({{{\rm{Im}}}}\,{\Sigma }_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{AA}}}}}(k,\omega +{{{\rm{i}}}}\gamma ))=1-{{{\rm{Im}}}}\,{\Sigma }_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{AA}}}}}(0,0+{{{\rm{i}}}}\gamma )/{{{\rm{Im}}}}\,{\Sigma }_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{AA}}}}}(k,\omega +{{{\rm{i}}}}\gamma )$, Err(ρ_S,eff(ω)) = 1 − ρ_S,nH(ω)/ρ_S,eff(ω) and Err(A_S,eff(k, ω)) = 1 − A_S,nH(k, ω)/A_S,eff(k, ω), which quantifies the deviation of the NHA result from the exact counterpart. Notably, here the subscript “eff" emphasizes the case of exact frequency-dependent effective Hamiltonian in Eq. (8). As shown in Fig. 6(a), the relative error in the imaginary part of the self-energy remains negligible (below 5%) within the subsystem bandwidth, demonstrating the validity of the NHA in this region, as well as the free choice of the approximation point beyond k = 0, ω = 0. Moreover, as previously indicated in Fig. 2(b), the real part of the self-energy varies only slightly over the same bandwidth, further supporting the accuracy of the approximation. This validity domain is also corroborated by Fig. 6(b), where the relative error of the DOS stays sufficiently small between the band edges (marked by red dashed lines).

Fig. 6: Error analysis of the NHA: deviation from the exact result.

a Relative error of the imaginary part of the self-energy. The color bar ranges from −0.17 to 0.04. b Relative error of the density of states (DOS, green curve); the red dashed lines mark band edges at ω = ± 2.21, and the gray shaded areas indicate the band gaps ( ≈ 0.5). c Relative error of the spectral function at ${{{\rm{Im}}}}\,\omega =-0.08$. The color bar ranges from −0.65 to 0.4. d Validity breakdown diagram of the non-Hermitian approximation (NHA) versus ${{{\rm{Im}}}}\,\omega ,{t}_{A}$ and t_y/t_x. Boundary lines are drawn as yellow, blue, green, and red curves, with their corresponding validity regions filled with the respective colors. Here, ${t}_{1}=1.2,{t}_{2}=-1,\lambda =-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,\gamma =0.1$, N_S = 80 and N_y = 300, ${E}_{\min }=-2.21,{E}_{\max }=2.21$. In (a)–(c) t_y = 3.5, t_A = 0.6.

Outside the bandwidth, the relative error of the imaginary part of the self-energy grows significantly, exceeding 10%, indicating the breakdown of the non-Hermitian approximation. Nevertheless, since the density of states decays rapidly in this region, the NHA remains applicable on the real frequency axis.

In contrast, the non-Hermitian approximation fails in the complex plane. As illustrated in Fig. 6(c), the spectral function at ${{{\rm{Im}}}}\,\omega =-0.08$ exhibits substantial errors under the NHA, even within the bandwidth, highlighting the limitation of the method for complex frequencies.

To illustrate the validity domain of non-Hermitian approximation with respect to ${{{\rm{Im}}}}\,\omega$, t_A and t_y/t_x, we again employ the error function for the density of states, Err(ρ_S,eff(ω)). The condition ${{{\rm{Im}}}}\,\omega > -\gamma$ ensures that the DOS ρ_S,eff/nH(ω) is well-defined along ${{{\rm{Re}}}}\,\omega$. Specifically, we scan the relative error across ${{{\rm{Re}}}}\,\omega \in [{E}_{\min },{E}_{\max }]$, where ${E}_{\min /\max }$ denote the lower/upper band edge of H_S(k). The NHA is considered to fail whenever the relative error exceeds 5% at any ${{{\rm{Re}}}}\,\omega$ within this interval. As shown in Fig. 6(d), the shaded regions below the curves, each corresponding to a different ${{{\rm{Im}}}}\,\omega$, indicate where the non-Hermitian approximation remains valid. Above these curves, the approximation fails, indicating the necessity of an anisotropic environment ${S}^{{\prime} }$ with a bandwidth sufficiently broad relative to S. Notably, as ${{{\rm{Im}}}}\,\omega$ approaches the branch cut, the validity region shrinks and the boundaries become more unstable (oscillatory), suggesting that a larger N_y is required to converge the self-energy at higher t_y/t_x ratios.

Furthermore, we consider the separability of ω_c and ω_r in the double-frequency density of states ${D}_{{{{\rm{S}}}},{{{\rm{eff}}}}}^{{{{\rm{CFF}}}}}({\omega }_{{{{\rm{c}}}}},{\omega }_{{{{\rm{r}}}}})$, which relates to the validity of the non-Hermitian approximation. The essential information for the complex-frequency DOS is contained in the eigenvalues of the effective non-Hermitian Hamiltonian H_S,eff(k, ω_r). The distribution of its eigenvalues thus serves to characterize the validity of the NHA.

Specifically, under the NHA at ω_r = 0, the eigenvalue trajectories form loops enclosing a fixed area. As ω_r varies, the point gap topology persists, but the shape and area of these loops change. Thus, we can define Ar_eff(ω_r) as the area enclosed by the eigenvalue loops of H_S,eff(k, ω_r). Then the validity of the non-Hermitian approximation can be quantified by the relative error function Err(Ar_eff(ω_r)) = 1 − Ar_eff(0)/Ar_eff(ω_r). As shown in Fig. 7, using a 5% relative error as the demarcation standard, the validity domain of the NHA is indicated by the light blue shaded area, where the relative error remains within 5%, exhibiting the separability of ω_c and ω_r within this region. Beyond this threshold, in the light pink shaded region, the non-Hermitian approximation begins to fail, and the exact treatment should be considered. This result clearly demonstrates the dependence of the complex DOS on ω_r, underscoring its inherent double-frequency nature.

Fig. 7: Evolution of the ω_r-dependence from negligible to significant.

The relative error in the area enclosed by the eigenvalues of H_S,eff(k, ω_r) as a function of ω_r, using a 5% threshold (the black dashed line) to indicate the validity of non-Hermitian approximation (light blue and light pink regions represent validity and failure, respectively). Here, ${t}_{1}=1.2,{t}_{2}=-1,\lambda =-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{t}_{y}=3.5,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,{t}_{{{{\rm{A}}}}}=0.6,\gamma =0.1$, N_S = 80 and N_y = 300.

The loop-cut tracking method and the winding number

Here, we provide a quantitative diagnostic tool for verifying the point gap topology and the branch cut beyond mere visual plotting.

For the NHA case, we consider E₁(k) and E₂(k) as the two sets of eigenvalues of H_S,nH(k). As shown in Fig. 3(d), these eigenvalues form the two distinct loops on the complex plane. In our numerical calculation, we discretize the momentum as ${k}_{i}=-\pi +(i-1)\frac{2\pi }{{N}_{{{{\rm{S}}}}}}$ with i = 1, 2, …, N_S. Focusing on E₁(k) as an example (the left loop in Fig. 3(d)), we define the relative phase angle between consecutive eigenvalues as $\theta ({k}_{i})={{{\rm{Arg}}}}({E}_{1}({k}_{i+1})-{E}_{1}({k}_{i}))$, where ${{{\rm{Arg}}}}$ returns the phase within [−π, π].

As shown in Fig. 8(a), the blue circles indicate the evolution of this relative phase for N_S = 80. The complete 2π transition clearly tracks the winding of E₁(k) and demonstrates the existence of a loop. The result is further confirmed by the results for N_S = 300 (marked by crosses), which show convergence. This outcome can be verified by the winding number defined in Supplementary Note 1:

$${\nu }_{{{{\rm{nH}}}}}(\omega )=\frac{1}{2\pi {{{\rm{i}}}}}{\int }_{-\pi }^{\pi }{{{\rm{d}}}}k{\partial }_{k}{{{\rm{ln}}}}\,\det [{H}_{{{{\rm{nH}}}}}(k)-\omega ].$$

(28)

Fig. 8: Diagnostic verification of the loop and the branch cut.

a, b The tracked relative phase of eigenvalues (poles) for the non-Hermitian approximation (NHA) and the exact case, respectively. Blue circles and brown crosses represent the case of N_S = 80 and N_S = 300, respectively. c, d The corresponding winding number (red circles) for the NHA and exact case, with ω = − 1 − ω_ii, where ω_i ∈ [0.08, 0.16]. These results confirm the presence of loop and branch cut characteristics.

In this case, we choose ω = − 1 − ω_ii with ω_i ∈ [0.08, 0.16]. As ω_i varies, this frequency trajectory crosses the loop in Fig. 3(d) and the branch cut in Fig. 3(e). As shown in Fig. 8(c), the corresponding winding number jumps from zero to one when the complex frequency encounters the loop, demonstrating the point gap topology.

For the exact case, it can be proved (see Supplementary Note 5) that the poles are given by the eigenvalues of

$${H}_{{{{\rm{tot}}}}}(k)-{{{\rm{i}}}}\gamma =\left(\begin{array}{cc}{H}_{{{{\rm{S}}}}}(k) & {H}_{{{{{\rm{SS}}}}}^{{\prime} }}\\ {H}_{{{{{\rm{SS}}}}}^{{\prime} }}^{{{\dagger}} } & {H}_{{{{{\rm{S}}}}}^{{\prime} }}(k)\end{array}\right)-{{{\rm{i}}}}\gamma$$

(29)

where ${H}_{{{{{\rm{S}}}}}^{{\prime} }}(k)$ is given by Eq. (6). Under PBC, the coupling between S and ${S}^{{\prime} }$ is

$${H}_{{{{{\rm{SS}}}}}^{{\prime} }}=\left(\begin{array}{ccccc}{t}_{{{{\rm{A}}}}} & 0 & 0 & \ldots & 0\\ 0 & {t}_{{{{\rm{B}}}}} & 0 & \ldots & 0\end{array}\right),$$

(30)

which is a 2 × 2N_y matrix. Similarly, we denote the eigenvalues of H_tot(k) − iγ as ${\epsilon }_{1}(k),{\epsilon }_{2}(k),\ldots ,{\epsilon }_{2{N}_{{{{\rm{y}}}}}+2}(k)$, and analyze ϵ₁(k) to examine its “cut" property. The result in Fig. 8(b) shows numerically negligible variation (~10⁻⁷, which originates from the inherent truncation error of the numerical diagonalization process), indicating that these poles share the same imaginary part, thus forming a cut. This conclusion is further confirmed by the winding number

$${\nu }_{{{{\rm{ext}}}}}(\omega )={\int }_{-\pi }^{\pi }{{{\rm{d}}}}k{\partial }_{k}{{{\rm{ln}}}}\,\det [{H}_{{{{\rm{tot}}}}}(k)-\omega -{{{\rm{i}}}}\gamma ].$$

(31)

As shown in Fig. 8(d), the vanishing winding number demonstrates the trivial spectral topology of the branch cut.

Quantitative verification of the non-Bloch region

This section provides a quantitative demonstration of the non-Bloch region in Fig. 1, going beyond the visual representation. For clarity, we define the spatial distribution of the Green’s function under NHA as

$${\psi }_{{{{\rm{nH}}}},i}(\omega )=\sqrt{| {[{G}_{{{{\rm{S}}}},{{{\rm{nH}}}}}^{{{{\rm{R}}}}}(\omega )]}_{2i-1,{j}_{0}}{| }^{2}+| {[{G}^{\rm R}_{{{{\rm{S}}}},{{{\rm{nH}}}}}(\omega )]}_{2i,{j}_{0}}{| }^{2}}{| }_{{{{\rm{Nor}}}}},$$

(32)

where i = 1, 2, …, N_S and ∣_Nor denotes normalization. To characterize the skin effect, we introduce the skin length ∣ξ_l(ω)∣ through the relation ${{{\rm{ln}}}}\,{\psi }_{{{{\rm{nH}}}},i}(\omega ) \sim {{i}}/{\xi }_{{{{\rm{l}}}}}(\omega )$, with ω belonging to the non-Bloch region. Since the exponential scaling requires using bulk sites away from boundaries to avoid edge effects, we numerically fit ${{{\rm{ln}}}}\,{\psi }_{{{{\rm{nH}}}}}(\omega )$ over the sites i ∈ [N_S/4, 3N_S/4]. This choice is arbitrary but does not affect the steady-state results as N_S → ∞.

For ω in the non-Bloch region and λ < 0 (the sign of λ does not affect the complex spectrum), we find ξ_l(ω) > 0. With a left-side excitation at j₀ = 40, ψ_nH(ω) localizes to the right, so we denote ξ_l,L(ω) = ξ_l(ω) for λ < 0. Conversely, for λ > 0 and a right-side excitation at j₀ = 2N_S − 39, ψ_nH(ω) localizes to the left, and we define ξ_l,R(ω) = − ξ_l(ω) for λ > 0.

Figure 9(a) shows that the skin length converges to definite values as N_S increases. The coincidence of ξ_l,L and ξ_l,R at large N_S demonstrates left/right excitation symmetry. The steady-state skin lengths for ω₃, ω₄, ω₅ are obtained as 107.2, 48.4, and 187.7, respectively.

Fig. 9: Quantitative evidence of the non-Bloch response.

a The skin length ξ_l,L(ω) (lines) and ξ_l,R(ω) (circles) for ω₃, ω₄ and ω₅ (red, blue and green colors, respectively) in the non-Bloch region of Fig. 1(c), which converges as N_S increases. Excitation sites are j₀ = 40 (λ = − 1) and j₀ = 2N_S − 39 (λ = 1). b The relative center location x_0,L (lines) and x_0,R (circles) of ψ_nH(ω) approaches 0.5 for ω₃, ω₄ and ω₅ (red, blue and green colors, respectively) as N_S increases, which demonstrates the non-reciprocal localization. Here ω₃, ω₄, ω₅ = − 0.6 − 0.14i, − 0.6 − 0.16i, − 0.6 − 0.18i. All other parameters are ${t}_{1}=1.2,{t}_{2}=-1,{\mu }_{{{{\rm{S}}}}}=0.2,{t}_{x}=1,{t}_{y}=3.5,{\mu }_{{{{{\rm{S}}}}}^{{\prime} }}=0.3,{t}_{{{{\rm{A}}}}}=0.6,\gamma =0.1$ and N_y = 300.

Next, to quantify left-right asymmetry in the non-Bloch region, we define the distribution center:

$${x}_{0}(\omega )=\frac{1}{{N}_{\rm S}}\mathop{\sum }_{i=1}^{{N}_{\rm S}}i| {\psi }_{{{{\rm{nH}}}},i}(\omega ){| }^{2},$$

(33)

where x₀(ω) ∈ (0, 1). Values x₀(ω) > 0.5 (λ < 0) indicate rightward localization, while x₀(ω) < 0.5 (λ > 0) indicate leftward localization. For convenience, we define the relative center location as x_0,L(ω) = x₀(ω) − 0.5 (λ < 0) and x_0,R(ω) = 0.5 − x₀(ω) (λ > 0).

As shown in Fig. 9(b), the relative center location approaches 0.5 with increasing N_S, confirming the asymmetry of non-Bloch response. The convergence of x_0,L(ω) and x_0,R(ω) further exhibits left-right excitation symmetry.