Loss induced delocalization of topological boundary modes

Domain: physics
Published: 2025-11-24
Editorial outcome: accept

Abstract
The introduction of non-Hermiticity has inspired abundant intriguing phenomena to topological physics. As a significant part of non-Hermiticity, losses are generally inevitable, represent a natural tendency of open systems, and can be readily regulated by straightforward passive components. However, losses are originally considered obstructive to topological systems and usually enable the localization of states. Here, we present that reciprocal loss can induce delocalization of topological boundary modes, which is essentially distinct from non-reciprocal mechanisms. By precisely regulated imaginary hopping via loss, the end modes in one dimension and the corner modes in two and three dimensions of topological insulators are delocalized to the bulk, accompanied by the non-Hermitian skin effects. Our theory is implemented in passive electric circuits with resistance as non-Hermitian loss, and the topological delocalized modes are well visualized experimentally. The realized topological delocalized modes are still robust against disorders and possess a broadband property. Our work establishes an important missing link between topology and loss, and brings a unique vision for compact topological applications.

Introduction
Topological materials, a subject of intense research, are mainly known for their topological boundary modes (TBMs), where the bulk properties give rise to these unique modes exponentially localized at ends, boundaries or defects of lattice. These TBMs are characterized by their robustness against local disorders or immunity to backscattering, thereby exploring widely in condensed matters, photonic, acoustic, and electric circuit systems. With the presence of non-Hermitian interaction of gain, loss or non-reciprocity, the system usually hosts the complex spectrum, enabling innovative mechanisms to manipulate TBMs. For example, the gain medium can perform the TBMs to generate amplified topological whispering gallery modes as a sound laser. The non-reciprocal hoppings generally can give rise to the non-Hermitian skin effect (NHSE), which causes the eigenstates of open lattice to exponentially collapse toward its boundaries, making the TBMs further localized or even delocalized. Such topological delocalized modes (TDMs) remain topologically nontrivial and exhibit the same robustness as localized TBMs, enabling topological applications based on all sites regardless the bulk of lattice. However, the non-reciprocal hoppings enabled by active devices easily render the system complicated and unstable, thereby hindering device miniaturization and the advancement of high-dimensional systems.

As an important component of non-Hermiticity, loss mirrors the inevitable trend of open systems, which is originally thought of harmful and limited due to energy dissipation. However, recent studies discover that loss can play an important role in topological physics, which facilitate the re-formulation of Bloch band theory and phases can be transformed into nontrivial ones with TBMs, such as end mode and corner mode. On the other hand, the loss can contribute systems to form nonzero spectral winding numbers, resulting in NHSEs, including the geometry-dependent skin effect for the bulk modes and hybrid skin-topological effects for the TBMs. Distinguished from complicated gain and non-reciprocal interactions, the loss modulation necessitates only passive and simplest structures that afford greater flexibility and maintain system stability independent of dimensions, site numbers and non-Hermitian hopping strengths. It obviously promotes the development of high-dimensional topological physics and is applicable across diverse fields. Nevertheless, loss is inherently reciprocal and is generally considered leading to the localization of states in previous understanding. Considering the state propagation, loss naturally induces state decay in space and be localized. In the context of steady state, previous studies widely accept that loss can induce generalized reciprocal skin effects and localize eigenstates at boundaries. A counterintuitive question is that whether the reciprocal loss can be harnessed to delocalize the originally localized TBMs.

Here, we provide the reciprocal solution and demonstrate the delocalization of TBMs in non-Hermitian topological insulators with hopping loss. Our design is based on the bilayer strategy with pseudospin degree of freedom, and the loss plays the role to spin polarized non-reciprocal imaginary hoppings. They can achieve the NHSEs whose accumulated directions are locked to spins. Through the precisely regulated imaginary hopping via loss, the TDM can be still controlled via GBZ protected by a nonzero topological invariant. Meanwhile, the TDM originates from spin-dependent wavefunction topology, while its delocalization direction aligns with the spin polarized skin effect that relates to spin-dependent spectral topology. It is fundamentally distinguished from previous non-reciprocal studies that rely on spinless wavefunction and spectral topologies. We first introduce the lattice model with topological end modes in one-dimensional (1D) topological insulator. The spin polarized TDM and complex spectra are visualized in 1D passive electric circuit with resistances in experiments. Then, we extended the study to two-dimensional (2D) and three-dimensional (3D) topological corner modes (TCMs), where the TDMs are well observed in 2D and 3D passive electric circuits. The solid results in 1D, 2D and 3D all demonstrate that the reciprocal theory of loss utilization and the passive experimental method bring significant insights into non-Hermitian physics.

Results
Loss induced delocalization of topological end mode in 1D
We commence by a tight-binding model on a duplex non-Hermitian chain, pictorially shown in the top panel of Fig. 1a, which is stacked by two identical Su-Schrieffer-Heeger (SSH) single chains employing interlayer hoppings. Each unit cell consists of four sites 1–4, the intracell hopping is pure imaginary with iγ, where γ represents the amplitude of designed loss; the intercell hopping is t1 and ±t0 indicate the positive and negative interlayer hoppings, respectively. On the basis of sites 1–4, the non-Hermitian Hamiltonian in momentum space can be expressed as
H(k) = (t1 cosk + iγ)τxσ0 + t1 sinkτyσ0 + t0τyσy, (1)
where k represents the wave vector, τ and σ denote the Pauli matrices for sublattice and layer degrees of freedom, respectively. It can be separated into two independent blocks via the unitary transformation H0(k) = T H(k)T-1 with T = 1/sqrt(2) (τ0 + iτz) ⊗ σx, denoted as H0(k) = H01(k) 0 / 0 H01(k), where H01(k) = (t1 cosk + iγ)τx + (t1 sink + t0)τy. As shown in the bottom panel of Fig. 1a, the two blocks are named spin up and spin down respectively. As a result, through the designed loss regulated by γ, the spin-dependent non-reciprocal imaginary hoppings are obviously constructed with i(γ±t0) for spin up and i(γ∓t0) for spin down. Notably, the underlying physics of the bilayer model is essentially distinct from that in previous non-reciprocal studies. The entire system is inherently reciprocal and obeys spinful anomalous time-reversal symmetry, UTHT(k)UT-1 = H(-k) with UT2 = -1, which fundamentally differs from the non-reciprocal systems that maintain time-reversal symmetry, UT H*(k)UT-1 = H(-k) with UT2 = 1.

In the presence of loss, the open boundary condition (OBC) and periodic boundary condition (PBC) spectra are calculated in the top panel of Fig. 1b, which do not coincide with each other. In particular, the band closure points (|E|=0) of them are inconsistent, indicating the broken bulk-boundary correspondence. To better describe the topological properties of the spectrum, the Hamiltonian of either spin can be written in the form of H(β) = 0 Hba(β) / Hab(β) 0, where β is the Bloch phase factor β = eik. Under PBC, β is a unit circle on the complex plane, forming the Brillouin zone (BZ). Under OBC, since the introduction of loss, k is a complex value, and the trajectory formed by β does not coincide with BZ, forming the generalized Brillouin zone (GBZ). Based on the GBZ, we employ the non-Bloch winding number WGBZ to describe the topological properties of the OBC spectrum, WGBZ = 1/2πi ∫GBZ dβ/β arg Hab(β)Hba(β) (Supplementary Section 1). As shown in the bottom of Fig. 1b, the topological phase transition points are at γ = ± sqrt(t1^2 + t0^2), which are the same with the closure points of OBC spectrum. The spectra and topological invariants are the same for two spins. Hence, by setting |γ| < sqrt(t1^2 + t0^2), we can get a pair of spin polarized topological end modes at |E|=0, which are not affected by the closure of the PBC spectra.

The Bloch winding number WBZ defined in BZ is employed to describe the localization property of end mode, as shown in the bottom of Fig. 1b, where WBZ = 1/2πi ∫BZ dβ/β arg Hab(β)Hba(β) (Supplementary Section 1). It is also the same for two spins. When the Bloch winding number changes, one end mode is expected to reverse its localization direction and localize at the opposite end. Therefore, at the transition point γc = ± t0 ± |t1|, the end mode is naturally delocalized as a TDM. In this time, the PBC spectrum closes and coincides with the end mode of OBC spectrum under chiral symmetry, forming the Bloch points, as shown in Fig. 1c.

Although the spectra for two spins are identical on the complex plane, the PBC spectral winding directions are opposite, caused by the introduced loss and arising from spin-dependent spectral topology. This indicates the appearance of spin polarized NHSEs (left-localized and right-localized skin effects for spin up and spin down, respectively), which essentially represent the generalized reciprocal skin effects with zero current functionals (as shown in Fig. S1 of Supplementary Section 1). Accordingly, the delocalization directions of end modes align with the spin polarized skin NHSEs. For instance, when γ=γc, the right end mode for spin up and the left end mode for spin down are simultaneously delocalized in the left and right directions, respectively. To simplify the discussion, we cut off two sites at right end of the duplex chain and leave only end modes at left for two spins (Supplementary Section 2). In addition, the Bloch winding number cannot predict the appearance or disappearance of end mode in OBC. The nontrivial topological property of end mode should still be described by the non-Bloch winding number, while the critical parameter γc has no influence on it. Consequently, the loss induced TDM is always nontrivial yet delocalized rather than localized, characterizable by both Bloch and non-Bloch winding numbers, analogous to the winding tuple.

To intuitively illustrate the loss induced TDM, we employ the GBZ approach by calculating the Bloch phase factor. In fact, a point in the GBZ corresponds to an OBC eigenstate, whose wavefunction at l-th sublattice site satisfies ψl ∝ βl. For the Bloch point, owing to the same eigenvalues of PBC and OBC spectra, it is at βGBZ = βBZ. The Bloch phase factors of the left end modes for spin up and spin down can be derived as βE1"=0 = -i(γ-t0)/t1 and βE1#=0 = -i(γ+t0)/t1 at E=0, respectively, which represent the wavefunction decay ratios of end modes (Supplementary Section 1). Their wavefunctions at l-th sites A thus obey Al," ∝ (βE1"=0)l and Al,# ∝ (βE1#=0)l, while that at sites B are zero as Bl = 0. As observed, the loss can change the decay ratios of the end mode via the spin-dependent non-reciprocal imaginary hoppings, thereby counteracting the localization.

When γ=0, both |βE1"=0| < 1 and |βE1#=0| < 1 holding within the BZ indicate that the end modes are localized at left ends, as shown in Fig. 1d. When γ=γc, |βE1#=0| = 1 intersects with BZ and forms Bloch point in spectra, resulting a TDM rather than a localized mode for spin down, as shown in Fig. 1e. In this time, the Bloch winding number is changed (Supplementary Section 1). Meanwhile, the left end mode for spin up becomes more localized with |βE1"=0| < 1. Therefore, when taking γ=γc, the excellent TDM for spin down occupying the entire chain and the bound mode for spin up being more localized are plotted in Fig. 1f, which are reshaped from the topological end modes.

The reciprocal bilayer model, combining pseudospin degree of freedom with loss, consequently determines the formation of spin polarized TDM, where the designed loss is mainly responsible for the delocalization of topological end modes. It is fundamentally distinct from previous works on loss induced delocalization of TBMs, we introduce a 1D passive electric circuit that based on the duplex non-Hermitian chain, as depicted in Fig. 2a. The electric circuit systems are widely accepted as the critical platform for research and experimental demonstration of non-Hermitian topological physics. In matrix form of Kirchhoff’s law, the response of a circuit system is described by I=JV, where I and V denote the input current and voltage at all nodes, and J is the circuit Laplacian or admittance matrix similar to the Hamiltonian of system. Naturally, the eigenvalues j and eigenstates ϕ of Laplacian J take the similar roles to the eigenvalues and eigenstates of Hamiltonian. Supplementary Section 4 recapitulates how our Laplacian relates to the tight-binding model. For our 1D electric circuit with a given frequency ω=2πf, the Laplacian can be written as J(k) = iω H(k) + E, where E represents the diagonal matrix. Obviously, the real and imaginary parts of the Laplacian’s eigenvalues are inversely related to the counterparts of the Hamiltonian. The circuit unit cell corresponding to the one of lattice model is detailed shown in the right side of Fig. 2a, where the imaginary intracell hopping is indicated by the resistance R1 with iγ ∝ i/ωR1, which still represents loss in circuit; the positive and negative interlayer hoppings are constructed through the inductance L0 with t0 ∝ 1/ω2L0 and capacitance C0 with t0 ∝ -C0, respectively. In addition, C2 is capacitance to act on E. In the same way, the Laplacian can also be decoupled into two spin parts, which still possesses spin polarized topological end modes. To form the TDM for spin down, it is needed R1 = 2πf0/(C0 ∓ C2) to satisfy γ=γc.

Hence, by taking the resistance with loss, we can similarly delocalize the end mode of the Laplacian for spin down. With successfully obtaining the imaginary part, the PBC spectra of two spins are independently measured, as shown in Fig. 2b, c, respectively, which form the complex loops. While the opposite winding directions along the momentum k have also been observed, these results clearly indicate spin polarized NHSEs originating from spin-dependent spectral topology. The measured OBC bulk eigenstates of Laplacian shown in Fig. 2d demonstrate them with left and right skin directions. The measured OBC spectra for two spins are shown in Fig. 2e, f, respectively, which exhibit the end modes are still in the gap, while the TDM for spin down is delocalized as shown in Fig. 2g, while the bound mode of spin up is simultaneously obtained at operated frequency f=178 kHz.

Second-order delocalization of TCM via loss in 2D
Furthermore, we demonstrate the ability of loss can be extended to high-dimensional topological systems. A rhombic geometry of bilayer kagome lattice in 2D is introduced and schematically shown in Fig. 3a. A unit cell contains three sites 1–3 in each layer, and the intracell hopping is zero between sites 2 and 3, while the hoppings in m and n directions are the same with the above 1D chain. Hence, the Hamiltonian H2(k) can also be separated into two independent blocks with H0 = H2" 0 / 0 H2#, as shown in the bottom of Fig. 3a (Supplementary Section 5). Notably, the equivalent non-reciprocal imaginary hoppings of each spin are constructed in m and n directions via the deliberately introduced loss. For spin down case, the Hamiltonian H2# = 0 i(γ+t0)+h12 i(γ-t0)+h13 / i(γ-t0)+h12* 0 i(γ-t0)+h23 / i(γ-t0)+h13* i(γ-t0)+h23* 0, (2) where h12 = t1 e-ikx, h13 = t1 e-i(kx+sqrt(3)ky)/2 and h23 = t1 e-i(-kx+sqrt(3)ky)/2. The PBC and OBC spectra are shown in Fig. 3b. In this case, the spectrum of PBC has nonzero spectral area in the complex plane. And more importantly, a second-order TCM emerges in OBC spectrum, which can be similarly delocalized to form a TDM via loss. To demonstrate it, the wavefunction decay ratios of TCM are calculated by solving the OBC real-space eigen-equations, which are expressed as E2#C#(i,g) = t1 A#(i,g+1) + i(γ+t0)A#(i,g) + t1 B#(i-1,g+1), (3) where A#(i,g), B#(i,g) and C#(i,g) represent the wavefunction components on the i-th and g-th sublattice sites in m and n directions, respectively. Since the wavefunction of TCM has nonzero components only at sites A with a zero energy of E2#=0, the wavefunction decay ratio is γ+t0 = t1, which is the same as the 1D case. Therefore, the TCM is delocalized to form a TDM in the entire 2D spin down lattice. The TDM here is still topologically nontrivial, which is demonstrated by employing the non-Hermitian bulk polarization and shown in Fig. 3c (Supplementary Section 5). Notably, non-Hermitian bulk polarization is accurate when the non-Hermitian intensity is not strong. The nontrivial phase, colored green, exhibits Wannier centers positioned at the centers of downward-pointing triangles, marked by red stars in Fig. 3c. In the trivial phase, represented by blue color, the Wannier centers are located at the centers of upward-pointing triangles. The bulk polarization cannot be calculated when the PBC spectrum is closed, which is represented by gapless (gray color). Compared with the originally localized TCM, the bound mode and TDM have been clearly depicted in Fig. 3d.

The second-order delocalization of TCM is visualized by utilizing a 2D passive electric circuit, as shown in Fig. 4a, which is designed on the bilayer kagome topological lattice. The corresponding circuit unit cell is schematically shown in Fig. 4b, where the circuit elements connected in m and n directions are the same with the 1D circuit. Again, the circuit Laplacian related with H2(k) can be similarly derived at a given frequency (Supplementary Section 6). Figure 4c, d show the measured OBC spectra of Laplacian for two spins at f=173.2 kHz, respectively, where the blue backgrounds indicate the PBC spectra calculated in theory, separating two bulk bands. As observed, the measured TCMs are still in the gap. The measured eigenstate intensities of TCMs are shown in the right side of Fig. 4c, d, respectively, indicating the shape change of their wavefunctions. Owing to the passive bilayer structure combines pseudospin degree of freedom and loss, the originally localized TCM is more bound in the left-lower corner for spin up, and is delocalized forming a nearly flat TDM for spin down. The intensities of bulk and edge wavefunctions are also experimentally observed in Fig. S8 of Supplementary Section 6, which both exhibit the corner NHSEs.

To verify the topological feature of loss induced TDM, we provide the OBC real and imaginary spectra in the presence of resistance disorder δR1. The corresponding eigenvalue’s real part of the corner mode is shown in the right. The measured OBC imaginary spectrum of spin down in a certain frequency range. The intensity of the marked eigenstate is shown on the right. The red and gray points in (e, f) indicate the eigenvalues of corner and other modes, respectively. The calculated OBC imaginary part of spectrum as a function of the resistance disorder δR1, shown in Fig. 4e prove that the TDM is still embedded in the imaginary band gap, robust against such disorders. Meanwhile, the loss induced TDM is also robust against defects, as discussed in Supplementary Section 7. In particular, we show the measured imaginary spectra in a broad frequency range in Fig. 4f. Although the condition of forming a TDM needs a fixed frequency f0 to keep the interlayer hoppings equal, the TCMs are still well delocalized in a certain frequency range. The intensity distribution of eigenstate at f=168.4 kHz shown in the right panel of Fig. 4f is a concrete example to reveal the broadband property of TDM. These results denote the advantage of loss in inducing TBMs’ delocalization.

Third-order TDM delocalized from TCM through loss in 3D
In 3D case, a bilayer pyrochlore lattice with deliberately introduced loss is proposed and shown in Fig. 5a, where the unit cell contains four sites 1–4 in each layer, and the hoppings in m, n and l directions are the same with the 1D chain. The two independent spin models can also be derived via the unitary transformation from the Hamiltonian H3(k) (Supplementary Section 8). The Hamiltonian of spin down model can be expressed as H3# = 0 i(γ+t0)+h12 i(γ+t0)+h13 i(γ+t0)+h14 / i(γ+t0)+h12* 0 i(γ+t0)+h23 i(γ+t0)+h24 / i(γ+t0)+h13* i(γ+t0)+h23* 0 i(γ+t0)+h34 / i(γ+t0)+h14* i(γ+t0)+h24* i(γ+t0)+h34* 0, (4) where h12 = t1 e-i(kx+ky)/2, h13 = t1 e-i(ky+kz)/2, h14 = t1 e-i(kx+kz)/2, h23 = t1 e-i(kz-kx)/2, h24 = t1 e-i(kz-ky)/2 and h34 = t1 e-i(kx-ky)/2. The bottom sites of the OBC bilayer lattice are cut off, leaving only TCMs at top corners for two spins. By similarly employing the OBC real-space eigen-equations for spin down, the TCM’s wavefunction decay ratio at top corners is γ+t0 = t1. The TDM is still topologically nontrivial, which is characterized by the non-Hermitian bulk polarization, as shown in Fig. 5b (Supplementary Section 8). The nontrivial phase (green) and trivial phase (blue) indicate the Wannier centers (red stars) located at the centers of the downward-pointing and upward-pointing tetrahedrons, respectively. Consequently, the originally localized third-order TCM can be fully delocalized in the whole 3D spin down lattice via loss, as shown in Fig. 5c.

The third-order delocalization of TCM is achieved by employing a 3D passive electric circuit, as shown in Fig. 5d. Owing to the loss modulation necessitates only passive and simple components, the 3D passive electric circuit based on the bilayer pyrochlore lattice is easily designed on the 2D plane (Supplementary Section 9). Figure 5e schematically depicts the circuit unit cell, where the same circuit elements as those in the 1D circuit are employed in m, n and l directions. The simulated and measured OBC spectra for spin down. The blue backgrounds denote the PBC spectrum in theory. The corresponding measured eigenstate intensity of the TDM is plotted on the right. The measured OBC imaginary spectrum of spin down in a certain frequency range. The red and gray circles indicate the eigenvalues of corner and other modes, respectively. The intensity of the marked TCM is shown on the right. The simulated and measured OBC spectra for spin down at f=171.9 kHz. The TCM remains in the gap, while its wavefunction shows remarkable delocalization, giving rise to the TDM. The intensity of the topological bound mode for spin up is also experimentally observed (Supplementary Section 9). This TDM also possesses broadband property, as demonstrated in Fig. 5g. The intensity distribution of eigenstate at f=167.5 kHz indicates that the TDM is almost uniformly distributed even it deviates from the required frequency. Meanwhile, the topological properties of the TDMs remain unaffected across different dimensions within a certain frequency range (Supplementary Section 10). These excellent results in 1D, 2D and 3D all demonstrate that the proposed passive system with loss always maintains stability, featuring a simple design and compact size, regardless of both the number of sites and system dimensions, paving the way for high-dimensional or other systems.

Methods
Mapping between the eigenstates of the Laplacian J and the voltage distribution
For a given frequency ω=2πf, the canonical form of Laplacian is given by the measured Laplacian J0=TJT-1. The Keysight E5061B ENA vector network analyzer is employed to perform the measurements, where the input current is generated from the voltage source injected into each node through a resistance. It needs to be mentioned that, the theoretical frequency of forming a flat TDM is f0 = 1/2πsqrt(L0C0) as the critical frequency f0 to make sure this. Meanwhile, we connect certain grounding capacitances to different nodes to ensure that their respective on-site potentials of circuit Laplacian are equal. However, when the frequency deviates from the critical one, the on-site potential of each node is no longer strictly equal. At this time, it is not rigorous to transform the Laplacian into two spins via the unitary transformation. Therefore, in this paper, we primarily discuss the TDM of the spin Laplacians at the critical frequency. For the frequency variation near f0, the Laplacian operator is approximately regarded as being capable of being transformed into two spins. The experimental results exhibit that the TDM can remain uniform in a certain frequency near f0, which so called the broadband property. In addition, we have thoroughly discussed the uncertainties in experimental measurements, which can be found in Supplementary Section 13.

Sample fabrications and experimental measurements
The specific values of the circuit elements used for the passive non-Hermitian circuits are chosen with manufacturing tolerances as C0=220nF(±5%), C1=68nF(±5%), C2=369nF(±5%), L0=4.4uH(±10%) and R1=6.65Ω(±0.1%). The 1D, 2D and 3D OBC circuits include 26 nodes, 80 nodes and 100 nodes, respectively. In the 2D and 3D samples, we take the same values for the circuit elements to the 1D one. The experimental measurements for the eigenvalues and eigenstates of circuit Laplacian are performed by measuring the impedance matrix G = V/I. When we set a local input current I at node b, the voltage response V against ground at any node a is measured. The complete matrix G is the inverse of the circuit Laplacian J, which contains the full information of system, including eigenvalues and eigenstates of admittance. The 1D electric circuit shown in Fig. 2a possesses two three-port switches, where the circuit can be switched to PBC by connecting left end and right end; or switched to OBC by connecting the left end to the ground and the right end to the remaining two sites. Specifically, for the measurement of PBC, we place the input current on each node of a single unit cell in turn and measure the corresponding voltage responses of all the circuit nodes. By employing the Fourier transform, the Laplacian in the momentum space can be obtained. To measure the Laplacian of OBC, we place the input current on each node of the whole circuit in turn and measure the corresponding voltage response of all nodes. The measured Laplacian of spin model is obtained by the unitary transformation of the measured Laplacian J0=TJT-1. The Keysight E5061B ENA vector network analyzer is employed to perform the measurements, where the input current is generated from the voltage source injected into each node through a resistance. It needs to be mentioned that, the theoretical frequency of forming a flat TDM is f0=161.8 kHz due to the requirement of equal interlayer hoppings, the best TDMs formed in experiments are at 178 kHz, 173.2 kHz and 171.9 kHz for 1D, 2D and 3D samples, respectively. It is mainly owing to the actual values of the circuit components have certain deviations compared with the nominal values, which can be improved by screening circuit elements with smaller errors.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents a study on how loss can induce the delocalization of topological boundary modes in non-Hermitian systems. The authors demonstrate that imaginary hopping regulated by loss can drive boundary modes into the bulk, accompanied by spin-polarized non-Hermitian skin effects. Their findings are validated through passive electric circuit experiments, showing agreement with theoretical predictions. The work challenges conventional views on loss and contributes to the field of non-Hermitian topology.\n\nThe paper presents a significant contribution by demonstrating that loss can lead to delocalization rather than localization, challenging conventional understanding. The theoretical framework is well-developed and supported by non-Bloch band theory and winding numbers. Experimental validation using passive electric circuits strengthens the credibility of the study, and the work demonstrates robustness against disorder. The findings have potential applications in photonics, acoustics, and condensed matter systems.\n\nThere are some points that could improve the manuscript further. It would be helpful if the authors could provide a more intuitive explanation of why loss counteracts localization. A clearer description of how the voltage distribution in the circuit maps onto the eigenstate wavefunctions would also be beneficial. Additionally, the authors could consider discussing potential realizations in photonic or acoustic systems. A quantitative metric defining the transition from localized to delocalized modes would clarify the analysis. Further elaboration on how frequency variations affect the topological properties of the modes would strengthen the argument.\n\nThis study is a valuable contribution to non-Hermitian topological physics, with strong theoretical and experimental support. Addressing these points would enhance the clarity and impact of the manuscript. I support its publication in Nature Communications.", "reviewer_2": "Summary and Assessment:\nThis study examines how controlled losses can delocalize topological boundary modes (TBMs) in non-Hermitian systems. The theoretical framework builds on well-established concepts in non-Hermitian physics, particularly the non-Hermitian skin effects (NHSEs). The authors study a model that decomposes into two block-diagonal components (spin up and spin down), each equivalent to a classic non-Hermitian chain (Hatano-Nelson model). By tuning the loss parameter in each chain, they observe the stanard topological transition from modes localized on the left to modes delocalized on the right, with the critical case in between showing complete bulk delocalization.\n\nWhile the theoretical analysis is technically sound and correctly applied, it does not introduce fundamentally new insights, as the underlying mechanisms are already well-documented in the literature. The primary contribution lies in the experimental demonstration of these effects using passive electrical circuits. Verification through electric systems is relatively standard, as they offer excellent controllability for testing non-Hermitian phenomena. However, this specific demonstration of loss-induced TBM delocalization had not been achieved previously to my knowledge. The experiments are well-executed, as evidenced by the strong quantitative agreement between numerical simulations and experimental measurements across all key results. This close match demonstrates the system's precise control and reliable implementation. The measurements convincingly validate the theoretical predictions, including the robustness of the delocalized modes against disorder and their broadband characteristics.\n\nKey Observations:\nTheory: Competent but not groundbreaking. The analysis applies known principles of non-Hermitian topology without extending them significantly.\nExperiment: The standout aspect of the paper. While electric circuits are a common platform, the implementation is carefully done, and the results appear reliable and true. The excellent match between numerical and experimental plots confirms both the accuracy of the theoretical framework and the quality of the experimental realization.\n\nSignificance and Limitations:\nThe work provides useful experimental confirmation of theoretical predictions, which may interest researchers in topological physics.\nThe broader impact is somewhat limited, as the results align with existing theoretical expectations rather than opening new research directions.\nThe choice of electric circuits, while standard for such studies, is appropriate given the need for precise control over loss parameters. The high degree of agreement between theory and experiment validates this approach.\n\nOverall Evaluation:\nThis is a solid experimental study that fills a gap by demonstrating a predicted but previously unobserved phenomenon. The theoretical framework, while not novel, is correctly applied, and the experimental work is executed with care. The strong correspondence between numerical and experimental results speaks to the quality of both the theoretical predictions and their experimental realization. The paper makes a valid contribution to the field by providing clear experimental evidence for loss-induced delocalization of TBMs. While not groundbreaking, it represents a well-executed study that advances the experimental verification of non-Hermitian topological effects.", "reviewer_3": "In the manuscript by Tang et al, the authors employ loss to theoretically and experimentally demonstrate the delocalization of the topological states. They show this phenomenon for both first-order and second-order topological phases in one and two dimensions, respectively. The theoretical calculations as well as the experimental results seem to be very sound. However, at this stage, I can not justify publishing this work in a high-impact journal such as Nature Communications. This paper might be suitable for more subject-specific journals. Below, I provide a detailed report.\n\nThe authors have mentioned in their manuscript that such delocalization of the topological modes has been discussed for a non-Hermitian system, Refs. [22-28]. In many of these references, the delocalization of the topological states has been explicitly shown employing the SSH model. Thus, I am wondering how this current work stands out compared to the previous studies. In particular, the author should discuss the fundamental differences from the other works.\n\nThe authors extensively discuss the skin effects, the generalized Brillouin zone, and other things in the first half of the manuscript. I do not want to say that this is bad. However, the discussions related to the delocalized topological modes are short, given that this is their primary focus. Again, I liked how they built on their discussions, which led to the delocalized modes, but discussions related to this part should be extended.\n\nThey mention multiple times in the manuscript that loss can enable the localization of states. However, this statement is not very clear. Did they want to point out about the skin modes or something else?\n\nWhile explaining the delocalized modes, they make some statements: “At this time, the PBC spectrum closes at the Bloch points [44], which overlap with the eigenvalue of the end modes under OBC, as depicted in Fig. 1c. This provides the standing-wave solution for open boundary.” However, I feel that this discussion should be expanded a bit to explain the delocalized modes.\n\nI am curious about the location of the critical \\gamma_c in the topological phase diagram of this model. Does this point separate two different topological phases? They can study the winding number as a function of \\gamma and show what happens before and after the \\gamma_c point.\n\nI do not see any topological invariants for the corner modes in two dimensions.\n\nI am also curious about the applications of these delocalized topological modes. Can they suggest some examples where it is beneficial to have delocalized modes compared to localized modes?"}, "author_response": "RESPONSES TO REVIEWERS’ COMMENTS\nResponse to Referee #1:\nThis manuscript presents a study on how loss can induce the delocalization of\ntopological boundary modes in non-Hermitian systems. The authors demonstrate\nthat imaginary hopping regulated by loss can drive boundary modes into the bulk,\naccompanied by spin-polarized non-Hermitian skin effects. Their findings are\nvalidated through passive electric circuit experiments, showing agreement with\ntheoretical predictions. The work challenges conventional views on loss and\ncontributes to the field of non-Hermitian topology.\nThe paper presents a significant contribution by demonstrating that loss can lead\nto delocalization rather than localization, challenging conventional understanding.\nThe theoretical framework is well-developed and supported by non-Bloch band\ntheory and winding numbers. Experimental validation using passive electric\ncircuits strengthens the credibility of the study, and the work demonstrates\nrobustness against disorder. The findings have potential applications in photonics,\nacoustics, and condensed matter systems.\nThere are some points that could improve the manuscript further.\nResponse:\nWe would like to thank the referee for the careful reading and positive comments. More\nspecifically, we thank and are encouraged by your supportive evaluations “The work\nchallenges conventional views on loss and contributes to the field of non-Hermitian\ntopology.” and “The paper presents a significant contribution by demonstrating that\nloss can lead to delocalization rather than localization, challenging conventional\nunderstanding.” We also thank you for raising the following important comments and\nindicating explicitly the points to further improve the manuscript.\nIn the following, we have addressed your comments and suggestions point-by-point\nand incorporated appropriate changes in the revised manuscript. We believe the revised\nmanuscript has been improved accordingly.\n1. Comment:\nIt would be helpful if the authors could provide a more intuitive explanation\nof why loss counteracts localization.\nResponse:\nreciprocal imaginary hoppings based on the designed bilayer structure, which can\nchange the wavefunction decay ratio of the topological boundary mode (TBM),\nthereby counteracting the localization.\nWe provided a quantitative relation between the loss and the localization\n(wavefunction decay ratio) of TBM. According to Eq. (S1) in Supplementary\nMaterials, the amplitude of the left TBM’s wavefunction for spin down at 𝑙-th site\n𝑙−1\nA can be derived as |𝐴 | = 𝑐 |𝜖 | , where the wavefunction decay ratio is\n↓,𝑙 ↓ 𝐴,↓\n𝜖 = 𝛽\n𝐸\n1↓\n=0\n= −𝑖\n𝛾+𝑡0\nand 𝑐 is the normalized coefficient. It can be clearly\n𝐴,↓ 1 𝑡1 ↓\nobserved that the magnitude of the loss 𝛾 will affect the localization of TBM, as\nshown in Fig. R1a. When 𝜸 < 𝜸 there is |𝜖 | < 1, the TBM is localized at the\n𝑐 𝐴,↓\nleft end; when 𝜸 = −𝑡 ±|𝑡 | as the critical 𝜸 , there is |𝜖 | = 1, resulting in a\n0 1 𝑐 𝐴,↓\ntopological delocalized mode (TDM) instead of a localized mode; when 𝜸 > 𝜸\n𝑐\nand then |𝜖 | > 1, the TBM is more delocalized at the right end. Therefore, loss\n𝐴,↓\ncounteracts TBM’s localization by changing its wavefunction decay ratio. The\ncorresponding wavefunction intensities are shown in Fig. R1b to illustrate this.\nThe intuitive explanation has been added in the main text, and the quantitative\nrelation with Fig. R1 has been added in Supplementary Materials.\nFig. R1. Loss induced delocalization of TBM for spin down. a Magnitude of the\nleft TBM’s wavefunction decay ratio as a function of 𝜸. When |𝜖 | = 1, the TBM\n𝐴,↓\nis delocalized as a TDM. b Wavefunction intensities of the TBM as a function of\n𝜸.\n2. Comment:\nA clearer description of how the voltage distribution in the circuit maps onto\nthe eigenstate wavefunctions would also be beneficial.\nWe appreciate your important suggestion. The mapping between the eigenstates of\nthe circuit Laplacian 𝐽 and the voltage distribution can be understood by\nemploying the Green’s function. In matrix form of Kirchhoff’s law, the response of\na circuit system is described by 𝑰 = 𝐽𝑽, where 𝑰 denotes the input current and 𝑽\nis the voltage response at each node. For a given frequency 𝜔 = 2𝜋𝑓, the canonical\nform of circuit Laplacian is given by\n𝐽(𝜔) = 𝑃Λ𝑃−1 = ∑𝑁 𝑗 𝝓𝑅𝝓𝐿,\n𝑛=1 𝑛 𝑛 𝑛\nwhere\n𝑃 = [𝝓𝑅,𝝓𝑅,⋯,𝝓𝑅],\n1 2 𝑁\nΛ = diag(𝑗 ,𝑗 ,⋯,𝑗 ),\n1 2 𝑁\n𝑃−1 = [𝝓𝐿;𝝓𝐿;⋯;𝝓𝐿],\n1 2 𝑁\nand 𝑗 , 𝝓𝑅 and 𝝓𝐿 are the eigenvalue, right and left eigenstate of 𝐽(𝜔) ,\n𝑛 𝑛 𝑛\nrespectively. The Green’s function is obtained as\n𝐺(𝜔) = 𝐽−1(𝜔) = 𝑃Λ−1𝑃−1 = ∑𝑁 𝜙𝑛 𝑅𝜙𝑛 𝐿 .\n𝑛=1\n𝑗𝑛\nAs a result, we can describe the voltage distribution of the circuit system to the input\ncurrent via 𝑽 = ∑𝑁 𝜙𝑛 𝑅𝜙𝑛 𝐿 𝑰, which indicates the mapping between the eigenstates\n𝑛=1\n𝑗𝑛\nand voltage distribution. This mapping relation has also been discussed in Refs.\n[Nat. Phys. 16, 747-750 (2020); Nat. Commun. 12, 7201 (2021)], which have been\ncited as [35, 20] in the original manuscript. Therefore, for an input current at one or\nseveral nodes with frequency 𝜔 , the corresponding voltage response is the\ncombination of the eigenstates and the input current.\nThe above discussion has been added in the method of main text.\n3. Comment:\nAdditionally, the authors could consider discussing potential realizations in\nphotonic or acoustic systems.\nResponse:\nWe thank you for your constructive suggestion. According to this suggestion, we\nexpanded the discussion to acoustic systems, which demonstrates the universality\nof our work. As an intuitive example, the simulated realization of the loss induced\nTDM was performed by using the commercial finite element analysis software\nshown as follows:\nDesign of 1D phononic crystal (PC). To be concrete, the 1D PC corresponding\nto the tight-binding model is constructed in Fig. R2a, where the acoustic cavity and\ncoupling waveguide can be viewed as the site and hopping terms in the tight-binding\nmodel. The PC possesses 58 cavities and remains two topological end modes\nlocalized at the left end. All cavities are the same to ensure they possess the same\neigenfrequencies 𝑓 , which are designed as 𝑓 = 4314 Hz. The lattice constant is\n𝐸 𝐸\n𝑎 = 65.1 mm, the height and width of cavities are chosen as ℎ = 40 mm and\n𝑏 = 15 mm, respectively. The intracell hopping is 𝑡+𝑖𝜸 and the intercell one is\n𝑡 , where the designed loss is induced by the pink holes on waveguide, as shown in\n𝜌 𝑐 to simulate the absorption of sound waves, where 𝜌 = 1.29 kg/m3 and\n0 0 0\n𝑐 = 345 m/s are the density and velocity of sound, respectively. The magnitude\n0\nof the loss can be controlled by adjusting the area of holes. The positive and negative\ninterlayer hoppings ±𝑡 can be achieved by using the dipole mode of cavity, as\n0\nshown in Fig. R2b. Since the dipole modes in top and bottom cavities have the\nmirror symmetry for x-y plane, they have opposed phases for each other. When there\nis coupling waveguide among two cavities, the dipole mode splits into two modes\nwith eigenfrequencies 𝑓 = 𝑓 ±𝜅 [Phys. Rev. Lett. 124, 206601 (2020)]. For the\n𝐸 0\npositive hopping (𝜅 > 0), the frequency of the same phase mode is higher than\n0\nthat of opposed phase one, as shown in the top of Fig. R2b. On the contrary, there\nis a negative hopping (𝜅 < 0) shown in the bottom of Fig. R2b.\nloss is fixed. The critical parameters for forming the TDM correspond to the\nintersection points of the periodic boundary condition (PBC) and open boundary\ncondition (OBC) bands, which can be identified by scanning the intercell hopping\nparameters while fixing the remaining parameters with 𝑤 = 5 mm. Notably, we\nadjusted the hopping in PC by sweeping the side length 𝑤 of the waveguides,\n1\nwhile keeping the waveguide lengths fixed at half the cavity length to ensure the\ncenter frequency remained unaffected [Phys. Rev. Appl. 14, 024023 (2020)]. The\ncorresponding numerical energy and eigenfrequency of tight-binding model and PC\nare shown in Figs. R2c and R2d, respectively, where the critical parameters are\n4\nmarked by the red stars with 𝑤 = 8.5 mm. As observed, although the PBC band\n1\nis closed, the eigenfrequencies of TBMs are still in the gap of OBC band. The TBMs’\neigenstates are subsequently simulated and the sound pressure fields are obtained\nwithin 3ℎ/4 of each cavity. The corresponding sound pressure fields of the spin\n1\npolarized eigenstates can be obtained by the basis of 𝑃 = [𝑃 +(−)𝑖𝑃 ,𝑃 +\n↑(↓) 1 3 2\n√2\n(−)𝑖𝑃 ]𝑇. As shown in Fig. R2e, the full extended sound pressure distribution of\n4\nspin down chain indicates the well-formed TDM, while the more bound pressure\ndistribution of spin up chain represents the topological bound mode. The TBM in\nHermitian case is displayed by black dotted line.\nThe above discussions with Fig. R2 have been added in Supplementary\nMaterials.\nFig. R2. Realization of the TDM in an 1D acoustic system. a Illustration of the 1D\nduplex PC lattice. Each unit cell consists of four sites 1-4 and can be decoupled into\ntwo independent spin models. b Sound pressures of two eigenmodes generated by\nthe positive and negative hoppings, respectively. c, d Energy and eigenfrequency\nbands of OBC as a function of 𝑡 and 𝑤 , respectively. The parameters of tight-\n1 1\nbinding model are 𝑡 = 0.6, 𝑡 = 0.6 and 𝜸 = 0.4. e Normalized sound pressure\n0\nintensity distributions of TBMs. The TBM is fully delocalized for spin down and\nmore localized for spin up.\n5\n4. Comment:\nA quantitative metric defining the transition from localized to delocalized\nmodes would clarify the analysis.\nResponse:\nWe appreciate your insightful comment. We have written the wavefunction of the\nTBM in the form of spatial decay. By quantitatively analyzing the variation of the\nwavefunction decay ratio with 𝜸, as shown in Fig. R1, we defined whether the TBM\nis delocalized or localized (please see the response of Comment 1). Besides this, we\nquantitatively studied the localization property of the TBM by employing Bloch\nwinding number in 1D and defined the transition point in the topological phase\ndiagram, clearly showing the transition from localized to delocalized modes.\nThe topological phases of this model. Although the bulk-boundary\ncorrespondence is breakdown, the Bloch winding number defined in Brillouin zone\n(BZ) can still be employed to describe the localization property of end modes in 1D\n[Phys. Rev. A 97, 052115 (2018); Phys. Rev. B 103, 195414 (2021); Phys. Rev. B\n107, 035424 (2023)]. While the critical parameter 𝜸 can be understood as the\n𝒄\ntransition point of Bloch winding number, as shown in Fig. R3. Before and after\nthe transition point, the localization direction of the end mode will change; at the\ntransition point, the end mode naturally becomes delocalized. However, it needs to\nbe mentioned that, the Bloch winding number cannot predict the appearance or\ndisappearance of end mode in OBC. The presence of end mode under OBC should\nstill be described by the non-Bloch winding number defined in generalized Brillouin\nzone (GBZ), while the critical parameter 𝜸 has no influence on it. Therefore, to\n𝒄\nbetter describe the topological property of the end mode in 1D, we employ both\nBloch and non-Bloch winding numbers, similar to the winding tuple of Ref. [Phys.\nRev. B 111, L041406 (2025)].\nThe definition of Bloch and non-Bloch winding numbers. Considering an 1D\nSSH model, the Bloch and non-Bloch Hamiltonians with chiral symmetry have the\n0 𝐻 (𝜷) \nsame form as 𝐻(𝜷) = ( 𝒂𝒃 ) , but 𝜷 = 𝜷 and 𝜷 = 𝜷 ,\n𝐻 (𝜷) 0 BZ GBZ\n𝒃𝒂\nrespectively. The Bloch winding number is defined as 𝑊 = \nBZ\n1\n∮ arg\n𝐻𝑎𝑏 (𝜷BZ )\n𝑑𝜷 , while the non-Bloch one is expressed as 𝑊 = \nBZ GBZ\n2𝜋\nBZ\n𝐻𝑏𝑎 (𝜷BZ )\n6\n1\n∮ arg\n𝐻𝑎𝑏 (𝜷GBZ )\n𝑑𝜷 . Evidently, the ratio of two Laurent polynomials\nGBZ\n2𝜋\nGBZ\n𝐻𝑏𝑎 (𝜷GBZ )\n𝑀(𝜷) =\n𝐻𝑎𝑏 (𝜷)\ndefines a meromorphic function on the complex plane, while either\n𝐻𝑏𝑎 (𝜷)\n𝐻 (𝜷) = 0 or 𝐻 (𝜷) = 0 will make 𝑀(𝜷) become zero or infinite, indicating\n𝑎𝑏 𝒃𝒂\nthe zero or pole points. These winding numbers have physical meanings and can be\nunderstood by the difference between pole and zero points in BZ and GBZ, i.e.,\n𝑊 = 𝑍 −𝑃 and 𝑊 = 𝑍 −𝑃 , where 𝑍 (𝑃 ) denotes the total\nBZ BZ BZ GBZ GBZ GBZ\nnumber of zero (pole) points inside the BZ or GBZ [Phys. Rev. B 99, 201103(R)\n(2019); Phys. Rev. B 110, 035424 (2024); Phys. Rev. B 110, 214113 (2024)].\nMoreover, owing to the winding numbers derived from the zero-pole method are\nidentical to twice the results calculated by the “𝑄 matrix” [Phys. Rev. Lett. 121,\n086803 (2018)], we multiplied the topological invariants by 1/2.\nFigure R3 with related description has been added in the revised Fig. 1 of main\ntext, and the calculation of topological invariants has been added in Supplementary\nMaterials.\nFig. R3. Topological phase of spin down part of 1D model. Topological phase\ndiagram with 𝑡 = 0.6 and 𝑡 = −1 . It is the same for two spins. The blue ×\n0 1\nmarkers and black circles indicate the non-Bloch and Bloch winding numbers,\nrespectively.\n5. Comment:\nFurther elaboration on how frequency variations affect the topological\nproperties of the modes would strengthen the argument.\nResponse:\nThanks for your valuable comment. We calculated the topological invariants of 1D,\n2D and 3D TBMs as a function of frequency to illustrate the influence of frequency\n7\nvariations on the topological properties of TBMs. For our circuit with a given\nfrequency 𝜔 = 2𝜋𝑓 , the Laplacian can be written as 𝐽(𝑘) = 𝑖𝜔[𝐻(𝑘)+𝐸 ] ,\n0\nwhere the Hamiltonian has the same topological property with the Laplacian at any\nfrequency. Therefore, we mainly focus on the circuit’s Hamiltonian. Since the onsite\npart of each site is only equal at the critical frequency 𝑓 that satisfies equal\n0\n1\npositive and negative frequency hoppings, i.e., = 𝐶 ,\n𝜔\n0\n2𝐿0 0\nwe only consider the\ninfluence of the frequency variation near the 𝑓 on the topological properties of\n0\nthe TBMs. At this time, the spin models can still be approximately obtained by the\nunitary transform of 𝐽′(𝑘) = 𝑇𝐽(𝑘)𝑇−1.\nFor the spin down part of the 1D circuit, we can obtain its Hamiltonian as\n0 −𝐶 e−𝑖𝑘 +𝑖( 1 − 𝐶0 − 1 )\n𝐻 (𝑘) = [\n2 𝜔𝑅1 2 2𝜔2𝐿0\n].\n1↓ −𝐶 e𝑖𝑘 +𝑖( 1 + 𝐶0 + 1 ) 0\n2 𝜔𝑅1 2 2𝜔2𝐿0\nWhile its topological invariants represented by the Bloch and non-Bloch winding\nnumbers can also be calculated. The topological phase diagram of the 1D spin down\nmodel with frequency range 150-170 kHz is displayed in Fig. R4a. As observed, the\ntopological property of the TBM characterized by the non-Bloch winding number\nwill not be affected within a certain frequency range. The localization property of\nthe TBM characterized by the Bloch winding number will be influenced by the\nfrequency change. When the frequency reaches 𝑓 , it corresponds to a perfect TDM.\nFig. 4Ra are shown in Figs. R4b and R4c, respectively. Furthermore, we employed\nthe non-Hermitian bulk polarizations to describe the topological properties of the\ncorner modes in 2D and 3D.\nIn the 2D system, the non-Hermitian bulk polarization [Commun. Phys. 6, 347\n(2023)] is defined as 𝑃 = − 1 ∫ Tr[𝐴 ]𝑑2𝑘 , where 𝑖 = 𝑥,𝑦 , 𝐴 (𝑘) =\n𝑖 𝑆 𝐵𝑍 𝑖 𝑖\n𝑖⟨𝑢𝐿 |𝜕 |𝑢𝑅 ⟩ is the Berry connection of the lowest band, 〈𝑢𝐿 | and |𝑢𝑅 〉\n𝑚,𝑘 𝑘𝑖 𝑛,𝑘 𝑚,𝑘 𝑛,𝑘\nrepresent the 𝑚 -th left and 𝑛 -th right eigenvectors, respectively. It needs to be\nmentioned that the left eigenvectors are obtained via 𝐻†(𝑘)|𝑢𝐿 〉 = 𝐸∗ |𝑢𝐿 〉 to\n𝑚,𝑘 𝑚 𝑚,𝑘\nsatisfy the biorthogonal relations for non-Hermitian case [Phys. Rev. Lett. 121,\n136802 (2018)]. The bulk polarization can be subsequently calculated via Wilson\n1\nloop approach with 𝑃 = ∫ 𝑑𝑣 , where 𝑣 is the Berry phase along the closed\n𝑦 2𝜋 𝑘𝑦 2 2\nloop 𝑘 at fixed 𝑘 , while 𝑃 can be obtained via the similar way. We calculated\n𝑥 𝑦 𝑥\nthe bulk polarization of 2D spin down model along the 𝑘 and 𝑘 directions of\n𝑥 𝑦\nBZ (Fig. R4d) and draw the topological phase in Fig. R4e. Notably, the corner mode\nmaintains the nontrivial phase with 𝑃 = −1/2√3 unchanged within a certain\n𝑦\nfrequency range.\nIn the 3D system, we can define the bulk polarization of the lowest band for 3D\n1\nspin down model as 𝑃 = ∫ 𝑑𝑣 , where the Wannier band 𝑣 is calculated\n3 2𝜋 𝑏3 3 3\nalong the direction of the reciprocal lattice vector 𝑏 with (𝑘 ,𝑘 ) vary along\n3 1 2\n(𝑏 ,𝑏 ) in momentum space [Sci. Adv. 6, eaay4166 (2020)], as shown in Fig. R4d.\n1 2\nThe bulk polarizations 𝑃 and 𝑃 in the other two directions can be obtained via\n1 2\nthe similar way. The corresponding topological phase diagram is shown in Fig.\nR4f, indicating that the corner mode maintains the nontrivial phase with 𝑃 = −1/4\n3\nand is not affected by frequency variations.\nIn summary, within a certain frequency range, the topological properties\nof the 1D, 2D and 3D TBMs will not be affected by frequency variations even\nthey become the TDMs. The Bloch winding number calculated in 1D illustrates\nthat the frequency variation will affect the localization property of TBMs. However,\nthe experimental results confirm that within a certain frequency range, the\nlocalization properties of the TDMs change very little, reflecting their broadband\ncharacteristics.\nThe above discussions with Fig. R4 have been added in Supplementary\nMaterials.\nFig. R4. Topological phases of 1D, 2D and 3D models for spin down as a function\n9\nof frequency. All the element values of the circuit have been fixed except for the\nfrequency. a Topological phase of 1D model. The red and gray lines mark 154 kHz\nand 161.8 kHz, respectively. b, c Complex admittance spectra of OBC and PBC at\nthe marked frequencies of a, respectively. d First BZs of the kagome and pyrochlore\nmodels, respectively. e, f Topological phases of 2D and 3D models, respectively.\nThis study is a valuable contribution to non-Hermitian topological physics, with\nstrong theoretical and experimental support. Addressing these points would\nenhance the clarity and impact of the manuscript. I support its publication in\nNature Communications.\nResponse:\nWe appreciate the valuable suggestions and comments provided by the referee, which\nhave greatly help us to improve our manuscript. We also thank and are encouraged by\nyour support for the publication of our manuscript. We hope that our work will become\nmore accessible and meet the high standards of Nature Communications after careful\nrevision. We sincerely thank you for your help and guidance.\nSummary and Assessment:\nThis study examines how controlled losses can delocalize topological boundary\nmodes (TBMs) in non-Hermitian systems. The theoretical framework builds on\nwell-established concepts in non-Hermitian physics, particularly the non-\nHermitian skin effects (NHSEs). The authors study a model that decomposes into\ntwo block-diagonal components (spin up and spin down), each equivalent to a\nclassic non-Hermitian chain (Hatano-Nelson model). By tuning the loss parameter\nin each chain, they observe the standard topological transition from modes\nlocalized on the left to modes delocalized on the right, with the critical case in\nbetween showing complete bulk delocalization.\nWhile the theoretical analysis is technically sound and correctly applied, it does\nnot introduce fundamentally new insights, as the underlying mechanisms are\nalready well-documented in the literature. The primary contribution lies in the\nexperimental demonstration of these effects using passive electrical circuits.\nVerification through electric systems is relatively standard, as they offer excellent\ncontrollability for testing non-Hermitian phenomena. However, this specific\ndemonstration of loss-induced TBM delocalization had not been achieved\npreviously to my knowledge. The experiments are well-executed, as evidenced by\nthe strong quantitative agreement between numerical simulations and\nexperimental measurements across all key results. This close match demonstrates\nthe system's precise control and reliable implementation. The measurements\nconvincingly validate the theoretical predictions, including the robustness of the\ndelocalized modes against disorder and their broadband characteristics.\nResponse:\nWe would like to thank the referee for your careful reading of this manuscript and\nraising important comments, which promote us to further improve the manuscript. We\nalso appreciate your recognition of the experimental part of this manuscript, such as\n“However, this specific demonstration of loss-induced TBM delocalization had not been\nachieved previously to my knowledge.”\nAfter a careful comparison with previous studies, we clarify that our work is a\nfundamental innovation in both theory and experiment. Our work not only\nexperimentally paves the way for the exploration and verification of unique topological\n11\nstates in reciprocal non-Hermitian systems, but also theoretically proposes the novel\nreciprocal strategy for delocalizing localized topological boundary modes (TBMs)\nin high-dimensional or other systems and challenges the traditional understanding\nof loss. It is essentially different from the complex non-reciprocal ways.\nAccording to your comments, we have carried out a major revision of the\nmanuscript. The revised manuscript highlights the innovations in both theoretical and\nexperimental aspects of our work, and provides substantial supplements and\ndiscussions, which can be summarized as following:\n(1) We have added detailed discussions on the inherent distinct between our\nreciprocal work and the non-reciprocal studies in realizing topological delocalized\nmodes (TDMs), as well as the differences from traditional understanding of loss. The\nreciprocal theoretical model, combining pseudospin degree of freedom and loss, has\nbeen further emphasized, as it yields counterintuitive results in reciprocal systems and\nnew insights into non-Hermitian physics.\n(2) We have further theoretically and experimentally expanded the study of loss\ninduced TDM to 3D system, where the fully extended TDM is well visualized via the\nsimple passive circuit. The well-measured results show that the reciprocal theory to\nachieving the TDM has significant advantages, benefitting its simple configuration and\nexcellent stability.\n(3) Besides extending to higher-dimensional systems, we have demonstrated that\nour theories are universal by realizing loss induced TDMs in other fields, such as\nacoustics, supported by numerical simulations.\n(4) We have provided the detailed explanations about the formation of loss induced\nTDMs and demonstrated their nontrivial topological properties, including 1D, 2D and\n3D topological invariants.\nWe appreciate your patient comments and suggestions and have tried our best to\nincorporate appropriate changes in the revised manuscript. Based on our responses and\nrevisions, our manuscript has been greatly improved, and we hope the referee could\nrecommend it for publication in Nature Communications.\nIn the following, we address your comments point-by-point.\nKey Observations:\nTheory: Competent but not groundbreaking. The analysis applies known\nprinciples of non-Hermitian topology without extending them significantly.\nWe do appreciate your comment. The novelty and significance of our theory need\nto be clarified and extended. Indeed, the TDMs we obtained are similar to those\nrelying on known non-reciprocal principles in previous studies, but the\nreciprocal theoretical strategy and the topological origin of the loss induced\nTDMs are fundamentally different from the non-reciprocal one. The theoretical\nmodel is essentially based on a reciprocal system, and the method of loss utilization\nexhibits significance, thus enabling easier extension and implementation in high-\ndimensional or other systems compared with complex non-reciprocal interactions.\nAfter carefully considering the reported studies in delocalizing the TBMs and the\nprevious understanding of loss in non-Hermitian systems, we emphasize that our\nwork is indeed a valuable innovation, which has brought unique results both\ntheoretically and experimentally.\nThe significance of our theory:\n(1) Our theoretical models are essentially built on reciprocal systems with\npseudospin degrees of freedom, and fundamentally differ from the models in\nprevious studies on spinless non-reciprocal systems. Their symmetries and\ntopological origins are fundamentally distinct. As illustrated in Fig. R5a, our\ntheoretical model originally designs a bilayer structure and integrates pseudospin\ndegrees of freedom and loss, which is a spinful reciprocal system. It obeys spinful\nanomalous time-reversal symmetry, 𝑈 𝐻𝑇(𝑘)𝑈−1 = 𝐻(−𝑘) with 𝑈2 = −1 ,\n𝒯 𝒯 𝒯\nwhile the spinless non-reciprocal models in previous studies maintain time-reversal\nsymmetry, 𝑈 𝐻∗(𝑘)𝑈−1 = 𝐻(−𝑘) with 𝑈2 = 1. According to the different time-\n𝑇 𝑇 𝑇\nreversal symmetry, the reciprocal bilayer system is characterized by spin-dependent\nwavefunction and spectral topologies, which differ from the spinless topology in\nnon-reciprocal systems. The spectral topology can be classified by employing the\ncurrent functional [Nat. Commun. 13, 2496 (2022)], i.e., 𝐽 [𝑛] =\n𝜶\n∑ ∮ 𝑑𝑘𝑑𝑛(𝐸 ,𝐸∗)∇ 𝐸 (𝑘), where 𝑛(𝐸 ,𝐸∗) represents the distribution function\n𝑖 BZ 𝑖 𝑖 𝜶 𝑖 𝑖 𝑖\nwhen the system reaches to a steady state, 𝑖 indicates the energy band index, and\n∇ is the directional derivative along direction 𝜶 in d-dimension. The skin effects\n𝜶\nof the bilayer model’s eigenstate originate from the spin-dependent spectral\ntopology related to zero current functionals 𝐽 = 0, which distinguishes from the\nspinless spectral topology in non-reciprocal model with non-zero current\n13\nfunctionals 𝐽 ≠ 0 . Therefore, in our reciprocal model, the nontrivial TDMs\noriginate from spin-dependent wavefunction topology, while their delocalization\ndirections align with the spin polarized skin effects that relate to spin-dependent\nspectral topology. For instance, the TBMs are delocalized for one spin and are more\nlocalized for the other. Compared with non-reciprocal studies in TDMs, the\npresented reciprocal model here belongs to distinct topologies and exhibits\nabundant phenomena.\n(2) Our theoretical models proposed here are both ingenious and\nsignificant, which provide a new paradigm for non-Hermitian physics and\nendow reciprocal loss with the capability to delocalize TBMs from their\nlocalized state. By solely regulating the magnitude of loss, the in-gap TBM under\nopen boundary condition (OBC) can coincide with the periodic boundary condition\n(PBC) spectrum, as shown in Fig. R5a, thereby being delocalized into a TDM.\nWe\ncan precisely delocalize the spin polarized TBMs’ wavefunctions and even make\nthem delocalized at the other end, as shown in Fig. R5b. The amplitudes of the left\n𝑙−1\nTBMs’ wavefunctions at 𝑙 -th site A can be derived as |𝐴 | = 𝑐 |𝜖 | and\n↑,𝑙 ↑ 𝐴,↑\n𝑙−1\n|𝐴 | = 𝑐 |𝜖 | for spin up and spin down, respectively. The wavefunction\n↓,𝑙 ↓ 𝐴,↓\ndecay ratios are 𝜖 = 𝛽\n𝐸\n1↑\n=0\n= −𝑖\n𝜸−𝑡0\nand 𝜖 = 𝛽\n𝐸\n1↓\n=0\n= −𝑖\n𝜸+𝑡0\n, 𝑐 and\n𝐴,↑ 1 𝑡1 𝐴,↓ 1 𝑡1 ↑\n𝑐 are the normalized coefficients. When 𝜸 < 𝜸 there is |𝜖 | < 1, the TBM for\n↓ 𝑐 𝐴,↓\nspin down is localized at the left end; when 𝜸 = −𝑡 ±|𝑡 | as the critical 𝜸 , there\n0 1 𝑐\nis |𝜖 | = 1, resulting in a TDM instead of a localized mode; when 𝜸 > 𝜸 and\n𝐴,↓ 𝑐\nthen |𝜖 | > 1, the TBM is more delocalized at the right end. The corresponding\n𝐴,↓\nwavefunction decay ratios of the left TBMs are displayed in Fig. R5c. Our work\ntheoretically achieves the delocalization of originally localized TBMs through only\nloss, which is essentially different from previous researches.\n(3) Our theory represents the first proposal that reciprocal loss can\ndelocalize TBMs, offering a fundamentally distinct insight for understanding\nof loss in non-Hermitian topology. Previous studies of non-Hermitian topology\nwidely employ the non-reciprocal interactions to delocalize the TBMs. However, it\nis difficult to intuitively understand that single loss can delocalize the originally\nlocalized state, not only due to its inherent reciprocity but also because the\ntraditional understanding of loss is that it makes state localized. Among almost all\n14\nprevious studies, it is widely accepted that the loss can cause the localization of\neigenstate at boundary, i.e., the generalized reciprocal skin effect\n[Nat. Commun. 13, 2496 (2022); Nat. Commun. 14, 4569 (2023)]. Nevertheless, to the best of our\nknowledge, there is no theory and experiment to discuss how to delocalize the\noriginally localized states by employing reciprocal loss. The common theoretical\nproposals of loss utilization struggle to achieve this requirement. A concrete\nexample is that the conventional 1D non-Hermitian SSH single-layer model with\nloss at the same position as the bilayer model, specified as 𝑡 +𝑖𝜸, as depicted in\nthus, it cannot be delocalized as a TDM. Therefore, our theory challenges the\ntraditional understanding of non-Hermitian loss, and obtains the so-called\n“counterintuitive result”.\n(4) Our theoretical strategy by means of loss is more readily extensible and\nimplementable in research and experiment involving high-dimensional or\nother systems, compared with complicated non-reciprocal interactions.\nThe\nnon-reciprocal methods are usually used in realizing the non-Hermitian topological\neffects, however, they entail two widely recognized drawbacks in experiment: first,\nthey predominantly rely on active devices or bulky configurations that necessitate\nexternal intervention, significantly increasing system bulk and impeding device\nminiaturization and development in high-dimensional systems; second, once the\nnumber of active components increases or the non-reciprocal hopping strength\nsurpasses a certain magnitude, the system may become unstable accompanied by\nself-excited oscillations, which limit the number of sites and complicate the\nvalidation of models in high-dimension. Reciprocal loss modulation necessitates\nonly passive, simple structures that afford greater flexibility, suit compact\nsystems, and enable strong non-Hermitian hopping in high-dimensional\nsystem with stability. To prove this, we extended our theoretical model to 3D\nsystem by designing a bilayer pyrochlore lattice, as shown in Fig. R6a. Owing to\nthe introduction of loss, the originally localized topological corner mode becomes\nmore bound for spin up, while delocalizing to form a flat TDM for spin down, which\nremains topologically nontrivial (Figs. R6b and R6c). The corresponding\nexperimental results in 1D, 2D and 3D all demonstrate that our passive system\nalways maintains stability, featuring a simple design and compact size, regardless\nof both the number of sites and system dimensions. In addition, our theoretical\n15\nstrategy can be extended to acoustic systems, as shown in Fig. R2, which exhibits\nthe loss induced acoustic TDM. Therefore, our study is significant even in\ntheoretical aspect, which is universal and represents a fundamental innovation in\nnon-Hermitian topological physics.\nThe main discussions about significance have been added in the main text.\nFigure R5 with corresponding discussion has been added in Supplementary\nMaterials. The main component of Fig. R6 as the theoretical part of a new figure\n(Fig. 5), and its related discussion have been added in the main text.\nFig. R5. The proposed bilayer theoretical model. a Bilayer theoretical model with\npseudospin degree of freedom. The OBC and PBC spectra are not coincide with\neach other. b Wavefunction intensities of the TBMs for two spins as a function of\n𝜸. c Magnitudes of the left TBMs’ wavefunction decay ratios for two spins as a\nfunction of 𝜸. d Conventional 1D single-layer SSH model with non-Hermitian loss\nin the intracell hopping. The deviation between the spectrum closure points arises\nfrom the inadequate length of the OBC chain. The parameters are chosen as 𝑡 =\n0\n0.6, 𝑡 = −1 with 𝜸 = 0.4.\n1 𝑐\nConceptual illustration of the bilayer pyrochlore lattice. b Topological phase\ndiagram of spin down model with 𝑡 = 0.6 . The green, gray and blue colors\n0\nindicate the nontrivial phase, gapless and trivial phase, respectively. The blue star\nmarks the parameters of TDM. The red stars indicate the location of Wannier centers.\nc Wavefunction intensities of topological corner modes at different parameters for\nspin up and spin down, respectively. The parameters are chosen as 𝑡 = 0.6, 𝑡 =\n0 1\n−1 with 𝜸 = 0.4.\n𝑐\nExperiment: The standout aspect of the paper. While electric circuits are a\ncommon platform, the implementation is carefully done, and the results appear\nreliable and true. The excellent match between numerical and experimental\nplots confirms both the accuracy of the theoretical framework and the quality\nof the experimental realization.\nResponse:\nWe appreciate the referee for your recognition of our experiment. The non-\nHermitian loss in circuit only requires a resistor, which makes our experimental\nsystem simple and stable. The excellent experimental results indirectly reflect the\nfeasibility and advantages of our theoretical strategy. To further enhance the\nsignificance of our work, we have provided the 3D experimental demonstration, as\nshown in Fig. R7. Owing to the reciprocal loss modulation necessitates only passive\n17\nand simple components, the 3D passive electric circuit based on the bilayer\npyrochlore lattice is easily designed on the 2D plane (Figs. R7a and R7b). While\nthe well visualized 3D TDM (Fig. R7c) and the broadband property (Fig. R7d)\nverify the capability of our theoretical strategy once again.\nFigure R7 as the experimental part of the new figure (Fig. 5), and its related\ndiscussion have been added in the main text.\nFig. R7. Observation of the third-order delocalization effect in 3D via loss. a\nPhotograph of the 3D passive circuit sample, which is designed on the 2D plane. b\nSchematic diagram of the designed circuit unit cell for the dashed rectangle in a. c\nSimulated and measured OBC spectra for spin down. The blue backgrounds denote\nthe PBC spectra in theory. The corresponding measured eigenstate intensity of the\nTDM is plotted on the right. d Measured OBC imaginary spectrum of spin down in\na wide frequency range. The intensity of the marked eigenstate is shown on the right.\nThe red and gray circles indicate the eigenvalues of corner and other modes,\nrespectively.\nSignificance and Limitations:\nThe work provides useful experimental confirmation of theoretical predictions,\nwhich may interest researchers in topological physics.\nResponse:\nWe appreciate your positive comment, which encourages us to further improve our\nwork.\ntheoretical expectations rather than opening new research directions.\nResponse:\nWe appreciate your important feedback. Given the ubiquity of loss, the investigation\nof loss induced non-Hermitian physics has long garnered extensive attention.\nHowever, the existing theoretical expectations mainly hold that reciprocal loss\npromotes state localization, yet no theory and experiment have investigated\nhow loss could be utilized to achieve localized state delocalization. At this point,\nour work not only challenges the existing theoretical expectations of loss but also\nestablishes a foundation for future explorations in non-Hermitian physics.\nAs compared with the complex non-reciprocal approaches, our work presents a\nnovel reciprocal paradigm in both theoretical framework and experimental design,\nwhich can be readily generalized to explore and validate non-Hermitian topological\nphenomena in high-dimensional or other systems. The well visualized 3D TDMs in\nsimple passive circuit demonstrate our points (Fig. R7). Moreover, we broadened\nour investigation to acoustic systems to demonstrate the universality of our work.\nThe corresponding 1D simulated results shown in Fig. R2 illustrate this extension.\nFollowing careful integration of the referee’s constructive suggestions,\nwe\nbelieve that our work now holds broad interest and fundamental significance\nThe choice of electric circuits, while standard for such studies, is appropriate\ngiven the need for precise control over loss parameters. The high degree of\nagreement between theory and experiment validates this approach.\nResponse:\nWe appreciate your acknowledgment of the appropriateness of our experimental\napproach. We believe the revised manuscript based on your comments can further\nvalidate the accuracy and applicability of our work and contribute a broader impact.\nOverall Evaluation:\nThis is a solid experimental study that fills a gap by demonstrating a predicted\nbut previously unobserved phenomenon. The theoretical framework, while not\nnovel, is correctly applied, and the experimental work is executed with care.\nThe strong correspondence between numerical and experimental results\nspeaks to the quality of both the theoretical predictions and their experimental\n19\nrealization. The paper makes a valid contribution to the field by providing\nclear experimental evidence for loss-induced delocalization of TBMs. While\nnot groundbreaking, it represents a well-executed study that advances the\nexperimental verification of non-Hermitian topological effects.\nResponse:\nWe greatly appreciate the referee for your thorough evaluation and kind remarks on\nour manuscript, which have helped us refine the manuscript to better convey the\nsignificance of these findings. As highlighted in the response to Key Observations,\nthe reciprocal loss induced state delocalization has never been predicted either\ntheoretically or experimentally before. Thus, our work represents the first\ntheoretical proposal and experimental demonstration that loss can delocalize TBMs\nacross different dimensions, establishing a significant reciprocal paradigm and\noffering a fundamentally distinct insight for understanding loss in non-Hermitian\ntopology. In addition, your recognition of our study as “a solid experimental work\nthat fills a gap” and “a valid contribution to the field by providing clear\nexperimental evidence for loss-induced delocalization of TBMs” is particularly\nencouraging, for which we extend our sincere gratitude.\nAccording to your comments, we have made major revisions of the manuscript\nto further enhance the innovation of our work, including clarifying and extending\nthe significance of the theoretical part, adding theoretical and experimental\ndemonstrations of the 3D loss induced TDM, demonstrating the universality of our\nwork by providing acoustic simulations, and providing substantial discussions and\nexplanations on the loss induced TDMs.\nWe believe that our work now embodies both broad impact and fundamental\nsignificance. Following the careful integration of your constructive insights, we\nsincerely hope that the referee could recommend our work for publication in Nature\nCommunications, and we would greatly appreciate any further feedback you might\nprovide.\nWe extend our sincere gratitude for your invaluable help and professional\nguidance.\nIn the manuscript by Tang et al, the authors employ loss to theoretically and\nexperimentally demonstrate the delocalization of the topological states. They show\nthis phenomenon for both first-order and second-order topological phases in one\nand two dimensions, respectively. The theoretical calculations as well as the\nexperimental results seem to be very sound. However, at this stage, I can not justify\npublishing this work in a high-impact journal such as Nature Communications.\nThis paper might be suitable for more subject-specific journals. Below, I provide\na detailed report.\nResponse:\nWe would like to thank the referee for your careful reading of our manuscript and\nproviding valuable feedbacks. In particular, your comments on our discussions of the\nmanuscript’s key points, the fundamental distinctions from other works, and other\nsubstantive issues have been of great help to our study, for which we extend our sincere\ngratitude. We have thoroughly evaluated the innovativeness of our study. Our work\nrepresents the first theoretical proposal and experiment that loss can delocalize\ntopological boundary modes (TBMs) for different dimensions, offering a\nfundamentally distinct insight for understanding of loss in non-Hermitian physics.\nCompared to the previous works with unnatural non-reciprocal interactions, our work\npresents a novel paradigm in achieving the topological delocalized modes (TDMs) via\na reciprocal way.\nIn response to the referee’s comments, we have carried out a major revision of the\nmanuscript. The revised manuscript hosts enhanced innovations, which can be\nsummarized as following:\n(1) We have added detailed discussions on the fundamental differences between our\nloss study and non-reciprocal works in realizing TDMs, as well as the differences from\nthe traditional understanding of loss. The significance of our work has been enhanced.\n(2) We have provided the detailed explanations about the formation of loss induced\nTDMs and demonstrated their nontrivial topological properties, including 1D, 2D and\n3D topological invariants.\n(3) We have further theoretically and experimentally expanded the study of loss\ninduced TDM to 3D system, where the fully extended TDM is well visualized via the\nsimple passive circuit. The well-measured results show that the reciprocal theory to\nachieving the TDM has significant advantages, benefitting its simple configuration and\nexcellent stability.\n(4) Besides extending to higher-dimensional systems, we have demonstrated that\nour reciprocal theory is universal by realizing loss induced TDMs in other fields, such\nas acoustics. We then provided the simulated demonstrations.\nWe are committed to making improvements based on your guidance to ensure that\nthe manuscript meets the high standards of Nature Communications. We hope the\nreferee could re-evaluate our work based on our responses and revisions.\nIn the following, we address your comments point-by-point.\n1. Comment:\nThe authors have mentioned in their manuscript that such delocalization of\nthe topological modes has been discussed for a non-Hermitian system, Refs.\n[22-28]. In many of these references, the delocalization of the topological states\nhas been explicitly shown employing the SSH model. Thus, I am wondering\nhow this current work stands out compared to the previous studies. In\nparticular, the author should discuss the fundamental differences from the\nother works.\nResponse:\nWe are grateful to the referee for raising the insightful issues of our manuscript,\nwhich will greatly help us improve it. Indeed, the TDMs have been explicitly shown\nin previous studies via the non-reciprocal SSH model, but our work based on the\nreciprocal loss are fundamentally different. Our work stands for the first proposal\nand experiment that loss can delocalize TBMs for different dimensions, providing\nan essentially distinct insight for understanding of loss in non-Hermitian physics.\nThe reciprocal theoretical strategy, the topological origin and the experimental\nmethod of the loss induced TDMs are distinct from the non-reciprocal ones, leading\nto more conducive to extension and implementation for the research and\nexperimental validation involving high-dimensional or other systems. As a result,\nwe could clarify that our work is indeed a significant innovation, which has brought\ncounterintuitive results. The detailed reasons are shown as follows:\n(1) Our work represents the first theoretical proposal and experimental\ndemonstration that reciprocal loss can delocalize TBMs for different\ndimensions, offering a fundamentally distinct insight for understanding of loss\n22\nin non-Hermitian physics.\nThe previous understanding of loss. Loss often leads to the decay and\nlocalization of states because it makes the wavevector 𝑘 become complex 𝑘 =\n𝑘 +𝑖𝑘 . The corresponding eigenstate with space propagation can be written as\n𝑟 𝑖\n𝜓(𝑟) ∝ 𝑒𝑖𝑘𝑟 = 𝑒−𝑘𝑖𝑟𝑒𝑖𝑘𝑟𝑟, which decays in space and be localized. Therefore, loss\nis usually considered harmful and limited. On the other hand, on the perspective of\nsteady states, loss plays the same important role as nonreciprocity and gain, because\nit can help reshape the wavefunctions of eigenstates instead of making them decay.\nAmong almost all previous studies, it is widely accepted that the loss can make the\nenergy spectrum complex to form a non-zero winding number, causing the\nlocalization of eigenstate at boundary, i.e., the generalized reciprocal skin effect\n[Nat. Commun. 13, 2496 (2022); Nat. Commun. 14, 4569 (2023]. It appears that,\nthe presence of loss is typically employed to make the state localized. To the best\nof our knowledge, there is no theory and experiment to discuss how to\ndelocalize the originally localized states by employing loss.\nOur work utilizing loss to delocalize the TBMs is both ingenious and\nsignificant. Our reciprocal bilayer structure combines pseudospin degree of\nfreedom and loss, as illustrated in the left side of Fig. R8a. By solely regulating the\nmagnitude of loss, the in-gap TBM under OBC can coincide with the PBC spectrum,\nas shown in the middle of Fig. R8a, thereby delocalizing the TBM into a TDM.\nPlease see the response of Comment 4 for the detailed explanation of TDM’s\nformation mechanism. Interestingly, such a single-layer structure with the same loss\ninteraction cannot exhibit the TDM, and a concrete example is calculated in Fig.\nR5d. As a result, our study offers a fundamentally distinct insight (so called\n“counterintuitive result”) for understanding of non-Hermitian loss, and establishes\na new paradigm for manipulating TBMs via a reciprocal way. Meanwhile, the\nproposed reciprocal strategy with loss employs simple passive components, which\nundoubtedly contribute to making the non-Hermitian systems remain stable while\nmaintaining compact size. For instance, the non-Hermitian loss in the circuit only\nrequires a single resistor, as shown in the right side of Fig. R8a, which strengthens\nthe advantages of our work. Thus, the presented reciprocal theory and experiment\nbring new insights to the study of states in non-Hermitian physics.\n(2) Our work naturally based on spinful reciprocal systems, fundamentally\n23\ndiffers from previous studies on spinless non-reciprocal systems, and is more\nreadily extensible and implementable in research and experiment involving\nhigh-dimensional or other systems.\nTheir symmetries and topological origins are fundamentally distinct. Our\ntheoretical models obey spinful anomalous time-reversal symmetry,\n𝑈 𝐻𝑇(𝑘)𝑈−1 = 𝐻(−𝑘) with 𝑈2 = −1 , while the non-reciprocal models in\n𝒯 𝒯 𝒯\nprevious studies maintain time-reversal symmetry, 𝑈 𝐻∗(𝑘)𝑈−1 = 𝐻(−𝑘) with\n𝑇 𝑇\n𝑈2 = 1. According to the different time-reversal symmetry, the reciprocal bilayer\n𝑇\nsystem is characterized by spin-dependent wavefunction and spectral topologies,\nwhich differ from the spinless topology in non-reciprocal systems. The spectral\ntopology can be classified by employing the current functional [Nat. Commun. 13,\n2496 (2022)], i.e., 𝐽 [𝑛] = ∑ ∮ 𝑑𝑘𝑑𝑛(𝐸 ,𝐸∗)∇ 𝐸 (𝑘) , where 𝑛(𝐸 ,𝐸∗)\n𝜶 𝑖 BZ 𝑖 𝑖 𝜶 𝑖 𝑖 𝑖\nrepresents the distribution function when the system reaches to a steady state, 𝑖\nindicates the energy band index, and ∇ is the directional derivative along\n𝜶\ndirection 𝜶 in d-dimension. The skin effects of the bilayer model’s eigenstate\noriginate from the spin-dependent spectral topology related to zero current\nfunctionals 𝐽 = 0, which distinguishes from the spinless spectral topology in non-\nreciprocal model with non-zero current functionals 𝐽 ≠ 0 . Therefore, in our\nreciprocal model, the nontrivial TDMs originate from spin-dependent wavefunction\ntopology, while their delocalization directions align with the spin polarized skin\neffects that relate to spin-dependent spectral topology. For instance, the TBMs are\ndelocalized for one spin and are more localized for the other. Compared with non-\nreciprocal studies in TDMs, the presented reciprocal model here belongs to distinct\ntopologies and exhibits abundant phenomena.\nTheir experimental methods are essentially different. The non-reciprocal\nmethods are widely used in realizing the non-Hermitian topological effects.\nFor\ninstance, the conventional 1D non-reciprocal SSH model for realizing TDMs is\nshown in Fig. R8b, which is identical to that reported in Ref. [Nature 608, 50-55\n(2022)]. By tuning the strength of non-reciprocal interaction, the in-gap TBM under\nOBC can coincide with PBC spectrum, forming the TDM. However, even in circuit\nimplementations, a single non-reciprocal hopping usually requires a complicated\nimpedance converter circuit with external DC power supplies. The typical\nimpedance converter circuit of Refs. [Nat. Phys. 16, 747-750 (2020); Nat. Commun.\n24\n12, 7201 (2021)] is illustrated in the right side of Fig. R8b. Such non-reciprocal\ninteractions entail two widely recognized drawbacks in experiment: first,\nthey\npredominantly rely on active devices or bulky configurations that necessitate\nexternal intervention, significantly increasing system bulk and impeding device\nminiaturization and development in high-dimensional systems; second, once the\nnumber of active components increases or the non-reciprocal hopping strength\nsurpasses a certain magnitude, the system may become unstable accompanied by\nself-excited oscillations, which limit the number of sites and complicate the\nvalidation of models in high-dimension. Reciprocal loss modulation necessitates\nonly passive, simple structures that afford greater flexibility, suit compact\nsystems, and enable strong non-Hermitian hopping in high-dimensional\nsystem with stability.\nTo enhance the significance of our work, we further explored the\ndelocalization of third-order topological corner mode via loss in 3D systems.\nThe bilayer pyrochlore lattice is shown in Fig. R9a. The topological corner mode is\ndelocalized in the whole 3D spin down lattice via only loss, which is still\ntopologically nontrivial (Figs. R9b and R9c). Owing to the reciprocal loss\nmodulation necessitates only passive and simple components, the 3D passive\nelectric circuit based on the bilayer pyrochlore lattice is easily designed on the 2D\nplane, as shown in Figs. R9d and R9e. While the well visualized 3D TDM is shown\nin Fig. R9f and the demonstration of its broadband property is shown in Fig. R9g.\nThe solid experimental results in 1D, 2D and 3D all demonstrate that our passive\nsystem always maintains stability, featuring a simple design and compact size,\nregardless of both the number of sites and system dimensions. In addition, our\ntheoretical strategy can be extended to acoustic systems, as shown in Fig. R2, which\nexhibits the loss induced acoustic TDM.\nThe main discussions about fundamental differences from the previous works\nhave been added in the main text. Figure R8 with corresponding discussion has been\nadded in Supplementary Materials. Figure R9 as the new figure (Fig. 5), and its\nrelated discussion have been added in the main text.\ntheoretical model with pseudospin degree of freedom. The OBC and PBC spectra\nare not coincide with each other. The reciprocal hopping with loss is realized by\nutilizing only a resistor. b Conventional 1D non-reciprocal SSH model. The OBC\nand PBC spectra are plotted in the middle. The typical non-reciprocal hopping is\nrealized by employing the impedance converter circuit. The parameters are 𝑡 =\n0\n0.6, 𝑡 = −1 with 𝜸 = 0.4 and 𝜹 = 0.4.\n1 𝑐 𝑐\nFig. R9. Demonstration of the third-order delocalization effect in 3D via loss. a\nSchematic of the 3D bilayer pyrochlore lattice. b Topological phase diagram of spin\ndown model with 𝑡 = 0.6. The green, gray and blue colors indicate the nontrivial\n0\n26\nphase, gapless and trivial phase, respectively. The blue star marks the parameters\nof TDM. The red stars indicate the location of Wannier centers. c Wavefunction\nintensities of TCMs at different parameters for spin down, respectively. The\nparameters are chosen as 𝑡 = 0.6, 𝑡 = −1 with 𝜸 = 0.4. d Photograph of the\n0 1 𝑐\n3D passive circuit sample, which is designed on the 2D plane. e Schematic diagram\nof the designed circuit unit cell for the dashed rectangle in d. f Simulated and\nmeasured OBC spectra for spin down. The blue backgrounds denote the PBC\nspectra in theory. The corresponding measured eigenstate intensity of the TDM is\nplotted on the right. g Measured OBC imaginary spectrum of spin down in a certain\nfrequency range. The intensity of the marked eigenstate is shown on the right. The\nred and gray points indicate the eigenvalues of corner and bulk modes, respectively.\n2. Comment:\nThe authors extensively discuss the skin effects, the generalized Brillouin zone,\nand other things in the first half of the manuscript. I do not want to say that\nthis is bad. However, the discussions related to the delocalized topological\nmodes are short, given that this is their primary focus. Again, I liked how they\nbuilt on their discussions, which led to the delocalized modes, but discussions\nrelated to this part should be extended.\nResponse:\nWe appreciate your valuable feedback and have rearranged the whole manuscript\ncarefully. Specifically, we thank and are encouraged by your comment “Again, I\nliked how they built on their discussions, which led to the delocalized modes…”.\nAccording to your comments, the revised manuscript now focuses on the loss\ninduced delocalized topological modes and added substantial discussions related to\nthem. We believe the present manuscript more effectively captures the focal points\nand underscores the significance of our research. The main revisions can be\nsummarized as follows:\n(1) We provided the detailed explanations and topological phase diagrams\nto describe the TDMs. The Bloch winding number defined in Brillouin zone (BZ)\nhas been employed to describe the localization property of TBMs. While the\nnontrivial topological phases are still described by the non-Bloch winding number\ndefined in generalized Brillouin zone (GBZ). The non-Hermitian bulk polarizations\nhave been added to illustrate the topological properties of the 2D and 3D TDMs.\ndiscussions in the main text.\n(2) We extended the discussion on the loss induced delocalization of corner\nmodes in 3D systems, presenting the corresponding theoretical model of a\nbilayer pyrochlore lattice and the experimental validation (please see the\nresponse of Comment 1). At this point, the loss induced TDMs have been explored\nin theory and experiment for different dimensions, enhancing the significance of\nour manuscript. The revisions can be found in Fig. 5 as well as the corresponding\ndiscussions in the main text.\n(3) We have removed the discussions and related figures on skin effects and\nGBZ of bulk modes to Supplementary Materials. It should be noted that the GBZ\napproach is crucial for explaining the delocalization of TBM. We have added the\nBloch phase factors of TBMs in the revised Fig. 1, and related discussions in the\nmain text.\n(4) We demonstrated such loss induced TDMs can be realized in other fields,\nsuch as acoustics. We have implemented the simulations in an 1D acoustic system\nand shown the loss induced acoustic TDM by employing Comsol Multiphysics. The\ndetailed design and simulated results have been provided in Supplementary\nMaterials.\n3. Comment:\nThey mention multiple times in the manuscript that loss can enable the\nlocalization of states. However, this statement is not very clear. Did they want\nto point out about the skin modes or something else?\nResponse:\nWe are sorry for this confusion. As mentioned in the response of Comment 1, this\nstatement has two meanings. First, considering the state propagation with 𝜓(𝑟) ∝\n𝑒−𝑘𝑖𝑟𝑒𝑖𝑘𝑟𝑟 , loss inherently induces state decay in space and be localized. The\nsecond is exactly as you said. In the context of steady state, previous studies widely\naccept that loss can induce generalized reciprocal skin effects and localize\neigenstates at boundaries. Therefore, in previous understanding, loss has often\nplayed the role of state localization. We mention the common loss induced\nlocalization in order to emphasize that the loss induced delocalization is an\n28\ninnovative and counterintuitive phenomenon.\nWe have explained the loss induced localization in the main text.\n4. Comment:\nWhile explaining the delocalized modes, they make some statements: “At this\ntime, the PBC spectrum closes at the Bloch points [44], which overlap with the\neigenvalue of the end modes under OBC, as depicted in Fig. 1c. This provides\nthe standing-wave solution for open boundary.” However, I feel that this\ndiscussion should be expanded a bit to explain the delocalized modes.\nResponse:\nWe appreciate your helpful suggestion. The formation of the delocalized modes can\nbe understood by using the Bloch point and GBZ approach as follows:\nThe Bloch point and GBZ approach. The Bloch points are the intersection\npoints of PBC and OBC spectra, whose eigenstates are delocalized modes, which\nalso called flat standing-wave modes [Phys. Rev. Lett. 123, 246801 (2019); Phys.\nRev. Res. 1, 023013 (2019)]. As an intuitive illustration, we employ the GBZ\napproach by calculating the Bloch phase factor 𝜷 = 𝑒𝑖𝑘 to explain the delocalized\nmodes. Under PBC, because 𝑘 ∈ (−𝜋,𝜋] , 𝜷 is a unit circle on the complex\nBZ\nplane, forming the BZ. Under OBC, since 𝑘 is a complex value, the trajectory\nformed by 𝜷 typically does not coincide with BZ on the complex plane,\nGBZ\nforming the GBZ. In fact, a point in the GBZ corresponds to an OBC eigenstate,\nwhose wavefunction at 𝑙 -th sublattice site satisfies 𝜓 ∝ 𝜷𝑙 [Phys. Rev. Lett.\n𝑙 GBZ\n121, 086803 (2018)]. The |𝜷 | < 1 (|𝜷 | > 1 ) indicates that the OBC\nGBZ GBZ\neigenstate will be a left-localized (right-localized) skin mode. For the Bloch point,\nowing to the same eigenvalue of PBC and OBC spectra, there is 𝜷 = 𝜷 .\nGBZ BZ\nAs a\nresult, the Bloch point is visible in the intersection point of BZ and GBZ, i.e.,\n|𝜷 | = 1, giving rise to the delocalized mode for its eigenstate.\nGBZ\nIn our work, we use the Bloch point and GBZ approach to explain the\nformation of the delocalized modes. Here we take the left end mode for spin down\nas an example. Its Bloch phase factor can be derived as 𝜷\n𝐸\n1↓\n=0\n= −𝑖\n𝜸+𝑡0.\nAnd its\n1 𝑡1\nwavefunction at 𝑙-th site A thus obeys 𝐴 ∝ (𝜷 𝐸 1↓ =0 )𝑙, while that at site B is zero\n𝑙 1\nas 𝐵 = 0 . When 𝜸 = 𝜸 , the eigenvalue of end mode coincides with PBC\n𝑙 𝑐\n𝐸 =0\nspectrum, forming the Bloch point (Fig. R10a). In this time, |𝜷 1↓ | = 1\n1\n29\nintersects with BZ (Fig. R10b), resulting in a delocalized mode, rather than a\nlocalized mode. The case of 𝜸 = 0 is shown in Figs. R10c and R10d for\ncomparison. The eigenvalue of the end mode dose not coincide with PBC spectrum\n𝐸 =0\nand |𝜷 1↓ | < 1 is inside the BZ. So, the eigenstate of the end mode is localized\n1\nat the left boundary.\nThe main part of the above discussion has been added in the main text to explain\nthe delocalized modes. Figures R10b and R10d as the Bloch phase factors of end\nmode for spin down have been added in the revised Fig. 1 of main text.\nFig. R10. Explanation of delocalized modes for spin down. a Complex spectra of\nOBC and PBC at 𝜸 = 𝜸 . The zero energy of OBC intersects with PBC spectrum\n𝑐\n𝐸 =0\nforming Bloch point. b Relation between BZ and Bloch phase factor 𝜷 1↓ of left\n1\nend mode at 𝜸 = 𝜸 . c Complex spectra of OBC and PBC at 𝜸 = 0. d Relation\n𝑐\n𝐸 =0\nbetween BZ and Bloch phase factor 𝜷 1↓ at 𝜸 = 0. The parameters are chosen\n1\nas 𝑡 = 0.6, 𝑡 = −1 with 𝜸 = 0.4.\n0 1 𝑐\n5. Comment:\nI am curious about the location of the critical \\gamma_c in the topological\nphase diagram of this model. Does this point separate two different topological\nphases? They can study the winding number as a function of \\gamma and show\nwhat happens before and after the \\gamma_c point.\nResponse:\n𝜸 is the transition point of topological phase defined by the Bloch winding number.\n𝑐\nBelow, we study the topological phase diagram as a function of 𝜸 described by\nBloch and non-Bloch winding numbers in 1D, and discuss the physical meaning of\nthe transition point 𝜸 .\n𝑐\nThe topological phases of this model. Although the bulk-boundary\ncorrespondence is breakdown, the Bloch winding number defined in BZ can still be\nemployed to describe the localization property of end modes in 1D [Phys. Rev. A\n97, 052115 (2018); Phys. Rev. B 103, 195414 (2021); Phys. Rev. B 107, 035424\n(2023)]. While the critical parameter 𝜸 can be understood as the topological\n𝑐\ntransition point of Bloch winding number, as shown in Fig. R11a. Before and after\nthe transition point, the localization direction of the end mode will change; at the\ntransition point, the end mode naturally becomes delocalized. However, it needs to\nbe mentioned that, the Bloch winding number cannot predict the appearance or\ndisappearance of end mode in OBC. The presence of end mode under OBC should\nstill be described by the non-Bloch winding number defined in GBZ, while the\ncritical parameter 𝜸 has no influence on it. Therefore, to better describe the\n𝑐\ntopological property of the end mode in 1D, we employ both Bloch and non-Bloch\nwinding numbers, similar to the winding tuple of Ref. [Phys. Rev. B 111, L041406\n(2025)].\nThe definition of Bloch and non-Bloch winding numbers. Considering an 1D\nSSH model, the Bloch and non-Bloch Hamiltonians with chiral symmetry have the\n0 𝐻 (𝜷) \nsame form as 𝐻(𝜷) = ( 𝒂𝒃 ) , but 𝜷 = 𝜷 and 𝜷 = 𝜷 ,\n𝐻 (𝜷) 0 BZ GBZ\n𝒃𝒂\nrespectively. The Bloch winding number is defined as 𝑊 = \nBZ\n1\n∮ arg\n𝐻𝑎𝑏 (𝜷BZ )\n𝑑𝜷 , while the non-Bloch one is expressed as 𝑊 = \nBZ GBZ\n2𝜋\nBZ\n𝐻𝑏𝑎 (𝜷BZ )\n1\n∮ arg\n𝐻𝑎𝑏 (𝜷GBZ )\n𝑑𝜷 . Evidently, the ratio of two Laurent polynomials\nGBZ\n2𝜋\nGBZ\n𝐻𝑏𝑎 (𝜷GBZ )\n𝑀(𝜷) =\n𝐻𝑎𝑏 (𝜷)\ndefines a meromorphic function on the complex plane, while either\n𝐻𝑏𝑎 (𝜷)\n𝐻 (𝜷) = 0 or 𝐻 (𝜷) = 0 will make 𝑀(𝜷) become zero or infinite, indicating\n𝑎𝑏 𝒃𝒂\nthe zero or pole points. These winding numbers have physical meanings and can be\nunderstood by the difference between pole and zero points in BZ and GBZ, i.e.,\n𝑊 = 𝑍 −𝑃 and 𝑊 = 𝑍 −𝑃 , where 𝑍 (𝑃 ) denotes the total\nBZ BZ BZ GBZ GBZ GBZ\n31\nnumber of zero (pole) points inside the BZ or GBZ [Phys. Rev. B 99, 201103(R)\n(2019); Phys. Rev. B 110, 035424 (2024); Phys. Rev. B 110, 214113 (2024)].\nMoreover, owing to the winding numbers derived from the zero-pole method are\nidentical to twice the results calculated by the “𝑄 matrix” [Phys. Rev. Lett. 121,\n086803 (2018)], we multiplied the topological invariants by 1/2. The corresponding\ntopological phases of the spin down part of 1D model have been calculated as a\nfunction of 𝜸 and shown in Fig. R11a.\nThe physical meaning of the transition point 𝜸 . As an intuitive illustration,\n𝒄\nthe Bloch phase factor 𝜷𝐸=0 of end mode is exactly a zero (pole) point for making\n𝐻 (𝜷) = 0 or 𝐻 (𝜷) = 0, and 𝜷 = 0 is always a pole point for making 𝜷−1\n𝑎𝑏 𝒃𝒂\ninfinite. In the spin down part of 1D model, the 𝜷\n𝐸\n1↓\n=0\n= −𝑖\n𝜸+𝑡0\nof left end mode\n1 𝑡1\nand 𝜷 = 0 are the only two pole points in the GBZ before OBC gap close.\n𝐸 =0\nTherefore, when the 𝜷\n𝐸 =0\n1↓\nis inside the BZ, there are 𝑊 = 1 and 𝑊 = 1;\nwhen 𝜷\n𝐸 =0\n1↓\nintersects with BZ, i.e., 𝜸 = 𝜸 , the 𝑊 is thus changed to 𝑊 =\n1 𝑐 BZ BZ\n1/2 while the 𝑊 = 1 keeps unchanged; when 𝜷\n𝐸 =0\n1↓\nintersects with GBZ,\nGBZ 1\nthe 𝑊 is changed to 𝑊 = 0, indicating the disappearance of end mode.\nThe\nGBZ GBZ\narg(𝑀(𝜷)) at 𝜸 = 𝜸 displayed in Fig. R11b clearly shows the change of end\n𝑐\nmode’s localization property, where the × marker indicates the pole point.\nFigure R11a with related description has been added in the revised Fig. 1 of\nmain text. The calculation of topological invariants, the other discussions and Fig.\nR11b have been added in Supplementary Materials.\nFig. R11. Topological phase of the 1D model for spin down. a Topological phase\ndiagram for spin down with 𝑡 = 0.6 and 𝑡 = −1. It is the same for spin up. The\n0 1\nblue × markers and black circles indicate the non-Bloch and Bloch winding\nnumbers, respectively. b 𝜷-plane colored by arg(𝑀(𝜷)) at 𝜸 = 𝜸 with GBZ in\n𝑐\n32\nblack line, BZ in black dotted line, and poles of 𝑀(𝜷) as × marker. The\n𝐸 =0\n𝜷 1↓ = 𝑖 is a pole of 𝑀(𝜷) and intersects with BZ, indicating the change of\n6. Comment:\nI do not see any topological invariants for the corner modes in two dimensions.\nResponse:\nWe do appreciate your comment. The topological invariants of corner modes in 2D\nand 3D have been calculated by the non-Hermitian bulk polarizations as follows:\nIn the 2D system, the non-Hermitian bulk polarization [Commun. Phys. 6, 347\n(2023)] is defined as 𝑃 = − 1 ∫ Tr[𝐴 ]𝑑2𝑘 , where 𝑖 = 𝑥,𝑦 , 𝐴 (𝑘) =\n𝑖 𝑆 BZ 𝑖 𝑖\n𝑖⟨𝑢𝐿 |𝜕 |𝑢𝑅 ⟩ is the Berry connection of the lowest band, 〈𝑢𝐿 | and |𝑢𝑅 〉\n𝑚,𝑘 𝑘𝑖 𝑛,𝑘 𝑚,𝑘 𝑛,𝑘\nrepresent the 𝑚 -th left and 𝑛 -th right eigenvectors, respectively. It need to be\nmentioned that the left eigenvectors are obtained via 𝐻†(𝑘)|𝑢𝐿 〉 = 𝐸∗ |𝑢𝐿 〉 to\n𝑚,𝑘 𝑚 𝑚,𝑘\nsatisfy the biorthogonal relations for non-Hermitian case [Phys. Rev. Lett. 121,\n136802 (2018)]. The bulk polarization can be subsequently calculated via Wilson\n1\nloop approach with 𝑃 = ∫ 𝑑𝑣 , where 𝑣 is the Berry phase along the closed\n𝑦 2𝜋 𝑘𝑦 2 2\nloop 𝑘 at fixed 𝑘 , while 𝑃 can be obtained via the similar way. We calculated\n𝑥 𝑦 𝑥\nthe bulk polarization of 2D spin down model along the 𝑘 and 𝑘 directions of\n𝑥 𝑦\nBZ (Fig. R12a) and draw the topological phase diagram in Fig. R12b. The nontrivial\nphase with 𝑃 = −1/(2√3) is described by the green color, indicating the\n𝑦\nWannier centers located at the centers of the downward-pointing triangles, marked\nby red stars in Fig. R12b. The trivial phase with 𝑃 = 0 represented by blue color\n𝑦\nindicates the Wannier centers located at the centers of the upward-pointing triangles.\nThe Wannier band 𝑣 calculated along the closed loop 𝑘 for our critical\n2 𝑥\nparameters (marked by blue stars in Fig. R12b) is displayed in Fig. R12c. It’s worth\nnoting that, the bulk polarization cannot be calculated when the PBC band is closed,\nwhich is represented by gapless in Fig. R12b.\nIn the 3D system, we can define the bulk polarization of the lowest band for 3D\n1\n𝑷 = ∫ 𝑑𝑣 , where the Wannier band 𝑣 is calculated\n3 2𝜋 𝑏3 3 3\nalong the direction of the reciprocal lattice vector 𝑏 with (𝑘 ,𝑘 ) vary along\n3 1 2\n33\n(𝑏 ,𝑏 ) in momentum space [Sci. Adv. 6, eaay4166 (2020)], as shown in Fig. R12d.\n1 2\nThe bulk polarizations 𝑃 and 𝑃 in the other two directions can be obtained via\n1 2\nthe similar way. The corresponding topological phase diagram is shown in Fig.\nR12e, where the nontrivial phase with 𝑃 = −1/4 and trivial phase with 𝑃 = 0\n3 3\nindicate the Wannier centers located at the centers of the downward-pointing and\nupward-pointing tetrahedrons, respectively. The Wannier band 𝑣 for the critical\n3\nparameters is shown in Fig. R12f.\nThe topological phase diagrams have been added in Figs. 3 and 5 of main text.\nThe details of the calculation and discussion have been added in Supplementary\nMaterials.\nFig. R12. Topological phases of the 2D and 3D models for spin down. a First BZ\nof the kagome lattice. b Topological phase diagram of 2D model with 𝑡 = 0.6.\n0\nc\nWannier band 𝑣 vary with 𝑘 . d First BZ of the pyrochlore lattice. e Topological\n2 𝑥\nphase diagram of 3D model with 𝑡 = 0.6. f Wannier band 𝑣 varies with 𝑘 and\n0 3 𝑥\n𝑘 . The green, gray and blue colors indicate the nontrivial phase, gapless and trivial\n𝑦\nphase, respectively.\n7. Comment:\nI am also curious about the applications of these delocalized topological modes.\nCan they suggest some examples where it is beneficial to have delocalized\nmodes compared to localized modes?\nResponse:\nThanks for your comment. Owing to the delocalized feature, these TDMs have\n34\npotential applications, such as large-area single mode topological amplifier and\nhigh-performance energy harvesting in gap.\nWe mainly take the large-area single mode topological amplifier as an example.\n𝜕𝝍 𝑖𝑔𝑷\nThe time evolution of the field is expressed as 𝑖 = 𝐻𝝍−𝑖𝜸 𝝍+\n𝜕𝑡 0 1+|𝝍|2/𝐼sat\n[Science 359, eaar4003 (2018)], where 𝐻 is the Hamiltonian of system, 𝜸 is the\n0\nglobal loss in each site, the third term indicates gain 𝑔 via stimulated emission that\nis inherently saturable at 𝐼 and 𝑷 represents the spatial profile of the pump.\nWe\nsat\nbegin with the 1D spin down model shown in Fig. 1a of main text, setting 𝜸 =\n0\n0.01 on all sites, 𝑔 = 0.016 on all A sites and 𝐼 = 10 . The equivalent non-\nsat\nreciprocal imaginary hopping is constructed by loss 𝜸 . As shown in Fig. R13a,\nsufficient gain causes the imaginary part of the TBM’s energy to become positive,\ndriving it to become the single amplification mode, whereas bulk modes maintain\nnegative energies. The instantaneous total intensity of the amplification mode with\n𝐼 (𝑡) = ∑𝑁 |𝜓 (𝑡)|2 can be derived by solving the time evolution of the field. The\n𝑡 𝑖=1 𝑖\ninitial field of 𝝍 is set with random complex amplitudes of 0.005(𝑚+𝑛𝑖) and\n𝑚,𝑛 ∈ (−1,1), while the Hamiltonian is set with 𝜸 = 0 for TBM and 𝜸 = 𝜸\n𝑐\nfor\nTDM. The numerical results are shown in Fig. R13b, indicating that the\namplification modes of TBM and TDM finally become steady after experience a\ncertain period of evolution, respectively. As observed, even under the same gain\nconditions, the total field intensity of TDM is much larger than that of TBM,\ndisplaying the high energy conversion. Meanwhile, the corresponding intensity\ndistributions of the wavefunction evolving over time are shown in Figs. R13c and\nR13d, respectively. These results demonstrate that the extended TDM induced by\nloss can indeed be employed to realize energy amplification across the whole lattice.\nCompared with amplifier constructed by localized TBM, which can only\namplify energy at specific sites and often require huge bulk lattices, the\namplification mode constructed by TDM can amplify energy at all sites, regardless\nof the lattice size. This significantly improves the spatial utilization and\nflexibility of the amplifier, as multi-channel energy output can be achieved with\nonly a small lattice, greatly saving costs and footprints. The uniform intensity\ndistribution of TDM is also beneficial to avoiding excessive energy concentration\nat a single point, thereby preventing device damage. Meanwhile, it is demonstrated\nthat by employing the simple design with the loss and combining the gain, such an\n35\nextended amplification mode with nontrivial topological property can be achieved.\nCompared with the complex non-reciprocal regulation, this approach is\nundoubtedly conducive to the development of amplifier devices and promotes the\nexploration and development of applications based on topological insulators with\ndelocalized modes.\nThe discussions of TDMs’ potential applications have been added in the\ndiscussion of main text. Figure R13 with related discussions has been added in\nSupplementary Materials.\nFig. R13. Large-area single mode topological amplifier achieved by loss induced\nTDMs in 1D case. a Complex spectrum of OBC for spin down model with site\nnumber of 𝑁 = 13. The zero mode with positive imaginary energy is exactly the\nsingle amplification mode. b Time evolution of the instantaneous total intensity\n𝐼 (𝑡) for TBM and TDM with 𝜸 = 0 and 𝜸 = 𝜸 , respectively. c, d Wavefunction\n𝑡 𝑐\nintensities of the TBM and TDM at different times, respectively.\nIn the end, we are sincerely grateful for your insightful feedbacks, which have\nsignificantly enhanced the quality of our work. After incorporating the referee’s\nfeedback, we believe that our work is both of general interest and fundamental\nimportance, and deserves to be published in Nature Communications. It would be\ngreatly appreciated if the referee could re-evaluate the manuscript and provide\nfurther feedback that may assist in its improvement.\nResponse to Referee #1:\nI have carefully examined the authors’ rebuttal and the revised version of the\nmanuscript. I appreciate the considerable effort made to address the concerns\nraised in my initial review. The clarifications regarding the experimental setup\nand the expanded discussion of the results have notably improved the\nmanuscript’s overall clarity and readability. These changes strengthen the\npresentation and help the reader better understand the significance of the work.\nResponse:\nWe would like to express our appreciation to the referee for your careful reading of our\nrevised manuscript. Your recognition of our revised manuscript gives us great\nencouragement, and further motivates us to refine our work. We also thank you for\nproviding the following valuable comments and constructive suggestions.\nIn the following, we have addressed your comments and suggestions point-by-point\nand have carefully revised the manuscript accordingly based on your guidance.\n1. Comment:\nThat said, I believe a few minor revisions are still necessary to ensure the\nmanuscript fully meets the journal’s standards. Some parameter definitions\nremain insufficiently clear, particularly in the early sections of the text, where\nreaders would benefit from explicit definitions and units when such quantities\nare first introduced. In addition, while some figures have been improved,\nothers still suffer from small font sizes and inconsistent labeling, which can\nhinder interpretation. Enhancing the visual presentation of these figures by\nenlarging axis labels, improving contrast, and unifying notation would be\nhighly beneficial.\nResponse:\nThank you for your helpful comments. Based on your comments, we have carefully\nexamined the parameters’ definitions and the figures’ presentations, and made\nnecessary modifications to them. The main revisions are shown as follows:\nThe revisions on parameters’ definitions.\n(1) In the line 107 on page 4, we added the definitions of the wavevector:\n“where 𝑘 represents the wavevector”\n1\n(2) In the line 123 on page 4, we added the definitions of the non-Bloch winding\nnumbers and the corresponding descriptions:\n“To better describe the topological properties of the spectrum, the Hamiltonian of\n0 𝐻 (𝜷) \neither spin can be written in the form of 𝐻(𝜷) = ( 𝒂𝒃 ) , where 𝜷 is\n𝐻 (𝜷) 0\n𝒃𝒂\nthe Bloch phase factor 𝜷 = 𝑒𝑖𝑘. Under PBC, 𝜷 is a unit circle on the complex\nBZ\nplane, forming the Brillouin zone (BZ). Under OBC, since the introduction of loss,\n𝑘 is a complex value, and the trajectory formed by 𝜷 does not coincide with\nGBZ\nBZ, forming the generalized Brillouin zone (GBZ). Based on the GBZ, we employ\nthe non-Bloch winding number 𝑊 to describe the topological properties of the\nGBZ\nOBC spectrum, where 𝑊 = \n1\n∮ arg\n𝐻𝑎𝑏 (𝜷GBZ )\n𝑑𝜷 (Supplementary\nGBZ GBZ\n2𝜋\nGBZ\n𝐻𝑏𝑎 (𝜷GBZ )\nSection 1).”\n(3) In the line 136 on page 5, we added the definitions of the Bloch winding\nnumbers:\n“The Bloch winding number 𝑊 defined in BZ is employed to describe the\nBZ\nlocalization property of end mode, as shown in the bottom of Fig. 1b, where 𝑊 = \nBZ\n1\n∮ arg\n𝐻𝑎𝑏 (𝜷BZ )\n𝑑𝜷 (Supplementary Section 1).”\nBZ\n2𝜋\nBZ\n𝐻𝑏𝑎 (𝜷BZ )\nThe revisions on figures’ presentations.\nWe have enlarged and standardized the font size of text within each figure and\nthe font size of axis labels to ensure consistency. We have also adjusted the positions\nof axis labels for neat alignment, and modified the font weight and contrast of\ncertain text elements to enhance visibility. Additionally, we have appropriately\nresized some subfigures to achieve a more uniform overall appearance. We believe\nthese figures now have all been great improved based on your guidance.\n2. Comment:\nFurthermore, although the authors briefly mention certain limitations, a more\nthorough discussion would provide a balanced perspective. In particular,\nhighlighting possible uncertainties in experimental measurements and clearly\nstating the assumptions in the theoretical model would help strengthen the\nscientific rigor of the work. It would also be advisable to incorporate a few\nmore recent references, particularly from the last three years, that are directly\nrelevant to the measurement methodology used in this study. Such additions\nwould enhance the manuscript’s contextual background and demonstrate\nawareness of the latest developments in the field.\nResponse:\nWe do appreciate your constructive comments. According to your comments, we\nhave conducted a comprehensive discussion of the limitations. In addition, we have\ncarefully reviewed and incorporated the recent references that are directly relevant\nto the measurement methodology used in this study. The main discussions and\nrevisions are shown as follows:\nAssumptions in the theoretical model\nIt needs to be mentioned that, the positive and negative interlayer hoppings are\n1\nconstructed through the inductance 𝐿 with 𝑡 → and capacitance 𝐶 with\n0 0 𝜔2𝐿0 0\n1\n−𝑡 → −𝐶 , respectively, which requires 𝑓 = as the critical frequency 𝑓\n0 0 0\n2𝜋√𝐿0𝐶0\nto make sure this. Meanwhile, we connect certain grounding capacitances to\ndifferent nodes to ensure that their respective on-site potentials of circuit Laplacian\nare equal. However, when the frequency deviates from the critical one, the on-site\npotential of each node is no longer strictly equal. At this time, it is not rigorous to\ntransform the Laplacian into two spins via the unitary transformation. Therefore, in\nthis paper, we primarily discuss the topological delocalized mode (TDM) of the spin\nLaplacians at the critical frequency. For the frequency variation near 𝑓 , the\ninto two spins.\nUncertainties in experimental measurements.\nThe uncertainties in experimental measurements here are mainly originated\nfrom three aspects.\nThe first is the inevitable nominal errors between circuit components and their\ntrue values. The circuit components with manufacturing tolerances here are 𝐶 =\n0\n220 nF (±5%) , 𝐶 = 68 nF (±5%) , 𝐶 = 369 nF (±5%) , 𝐿 =\n1 2 0\n4.4 uH (±10%) and 𝑅 = 6.65 Ω (±0.1%) , which cause the actual circuit\ncause the measured eigenvalue spectra to fluctuate accordingly, while the\ncorresponding wavefunction distribution of the TDM is no longer uniformly\nextended. As a demonstration, we introduce the nominal error of each component\n3\ninto the numerical calculation as the disorder and calculate 30 times. The spectra\nunder open boundary condition (OBC) in 2D are presented in Figs. R1a and R1b\nfor two spins. The corner modes’ wavefunctions from a randomly selected single\ncalculation is shown on the right side. These calculated results are similar to the\nfluctuated experimental results, which demonstrate our point.\nIn addition, the deviations of components’ central values from their nominal\nvalues cause a corresponding shift in the critical frequency, which results in a\ndiscrepancy between the measured frequency of the TDM and its theoretical value.\nAs a simple example, we assume that the average value of the inductances for\ninterlayer hopping decreases by 8% compared to the previous value. The OBC\nimaginary spectrum of spin down in a certain frequency range is shown in Fig. R2a.\nIn this case, the critical frequency is no longer 𝑓 = 161.8 kHz but becomes 𝑓′ =\n0 0\n168.8 kHz , and the corner mode’s wavefunction at 161.8 kHz is no longer\ndistributed very uniformly compared with it at 168.8 kHz, as shown in Figs. R2b\nand R2c.\nThe third is the uncertainty introduced by experimental instrument and the\ncircuit PCB itself. Experimental instruments introduce noise (or preserve noise\nduring signal acquisition). This noise is superimposed on the useful voltage signal,\nresulting in a finite signal-to-noise ratio (SNR) in the measured responses and\ncausing them to exhibit fluctuations. For nodes with inherently small voltage\nresponses, the noise introduces a higher relative error in the measurements. To\nmitigate such errors, it is common practice to increase the amplitude of the input\nsignal and implement proper grounding between the instrument and the PCB circuit.\nDue to the integrated design of the PCB, there is mutual interference between\ncomponents, as well as between wires during the circuit’s response. For example,\ninductors in the PCB exhibit mutual inductance with other inductors, which affects\ntheir respective values; meanwhile, there are interactions between parallel wires,\nthe wires need to be disorganized to avoid parallel interference between them [Nat.\nCommun. 12, 7201 (2021)].\nTherefore, experimental measurement uncertainty can be minimized by\nselecting high-accuracy components, employing precision instruments, improving\nthe signal-to-noise ratio (SNR), and optimizing PCB layout to mitigate parasitic\neffects and crosstalk.\nWe have added recent references in the relevant field, including references [49-\n55]. When describing that circuit systems serve as the important platform for\nstudying and experimentally verifying non-Hermitian topological physics, we have\nincorporated references [49-55], which are directly relevant to the measurement\nmethodology used in this study. For instance, Observation of continuum Landau\nmodes in non-Hermitian electric circuits. Nat. Commun. 15, 1798 (2024), which is\ncited as the reference [49].\nThese above discussions have been added in the Method of the main text and\nthe Supplementary section 13. While the references have been added at the\ncorresponding positions in the main text.\nFig. R1. Calculated results that consider the manufacturing tolerances of circuit\ncomponents. a, b Calculated OBC spectra at critical frequency under disorders for\ntwo spins, respectively. Each spectrum is calculated 30 times. The corresponding\nwavefunction intensities of the corner modes for a randomly selected calculation\nare shown on the right.\ncomponents. a Calculated OBC imaginary spectra in a certain frequency range for\nspin down. b, c Wavefunction intensities of the corner mode at frequencies marked\nby black and blue stars in a, respectively. Owing to the deviation of circuit\ncomponent values from their central values, the wavefunctions deviate from the\nideal TDM.\nOverall, I am satisfied with the progress made in response to my earlier comments,\nand I believe the manuscript is much improved. With the minor adjustments\nsuggested above, it will be well-positioned for publication. Therefore, I recommend\nthe manuscript for Minor Revision.\nResponse:\nWe would like to express our sincere gratitude to the referee, not only for the valuable\ncomments you provided earlier but also for the constructive feedback you have offered\nnow. These comments and feedback are like a beacon, guiding us in the direction of\nimproving the manuscript. Meanwhile, we also thank your support for the publication\nof our manuscript. We believe that our work, which has undergone revisions based on\nyour guidance, is now more cogent and meets the high standards of Nature\nCommunications.\nIn the end, we sincerely thank you again for your help and guidance.\nI appreciate the authors’ hard work in improving the quality of the manuscript.\nThe authors have made some appropriate revisions and answered my queries\nsufficiently. I recommend the manuscript for publication in Nature\nCommunications. However, I have a small suggestion.\nResponse:\nWe appreciate the valuable suggestions and comments that you provided earlier, which\nhave greatly helped us to significantly improve the quality of our manuscript. We also\nthank and are encouraged by your recommendation for the publication. We believe that\nour work now become more accessible and meet the high standards of Nature\nCommunications based on your guidance.\n1. Comment:\nIt would be nice if the authors rephrase the sentence in the abstract: “By\nprecisely regulated imaginary hopping via loss, the 1D end mode, 2D and 3D\ncorner modes of topological insulators are delocalized to the bulk,\naccompanied by the non-Hermitian skin effects.” The end modes in one\ndimension and the corner modes in two and three dimensions are zero-\ndimensional.\nResponse:\nWe are grateful to the referee for your careful review of our manuscript and for also\npointing out the issues described in our abstract. Based on your suggestion, the\nsentence in the abstract have been changed as “By precisely regulated imaginary\nhopping via loss, the end modes in one dimension and the corner modes in two and\nthree dimensions of topological insulators are delocalized to the bulk, accompanied\nby the non-Hermitian skin effects.”\nFinally, we would like to reiterate our sincere gratitude to you for your important\nhelp.\n7"}]

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