Introduction

Braiding, one of the oldest crafting techniques in human history, has evolved into a key concept across a broad spectrum of scientific disciplines. In biology, it manifests in the twisting and coiling of DNA strands and protein folding, essential for genetic information storage and cellular functions1. In physics, braiding serves as a fundamental framework for understanding various phenomena2,3,4,5,6,7,8,9,10,11, such as the exchange statistics of quasiparticles in topological phases of matter. Recently, the study of non-Hermitian spectral topology has introduced the concept of topological braiding in the complex-energy plane, revealing the intricate behavior of non-Hermitian bands12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28. Unlike braiding operations in topologically ordered phases of matter, which involves real-space exchange of quasiparticles, braiding in non-Hermitian spectral topology occurs in momentum space and involves the intertwining of complex energy bands, enabling braiding patterns that do not appear in Hermitian systems with real eigen-energies. Such braiding not only deepens our understanding of symmetry and topology but also offers tools for exploring non-Hermitian phases, thus expanding the boundaries of topological physics.

On the other hand, fractals lie at the intriguing intersection of geometry and complexity, defined by intricate patterns that repeat across multiple scales. These self-similar structures consist of smaller replicas of the overall shape, maintaining their characteristic patterns regardless of the scale of observation29. Beyond their mathematical elegance, fractal patterns appear ubiquitously in nature, from the delicate symmetry of snowflakes to nonlinear dynamics and nano-assembly of a polymer30,31,32,33,34,35,36. In condensed matter physics, fractality has also led to the unveiling of various quantum phases37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55. A typical example is the fractal eigen-spectrum37,38,39,40,41, such as Hofstadter’s butterfly, which describes the fractal energy spectrum of electrons in a two-dimensional lattice under a perpendicular magnetic field. This quantum fractal vividly illustrates the intriguing interplay between electrons and magnetic fluxes. Moreover, recent investigations have shown that fractal lattices can support topological boundary states and higher-order topological corner states even in the absence of insulating bulk modes42,43,44,45,46,47,48,49,50, providing deeper insights into the bulk-boundary correspondence and the nuanced relationship between topology and fractal dimensions. However, despite these advancements, the currently unveiled topological physics and fractal geometry have primarily focused on Hermitian systems. Investigations combining fractal geometry with non-Hermitian physics have been limited to non-Hermitian fractal lattice systems56 or non-Hermitian fractal exceptional points (eigenvalue degeneracies)57. The interplay between fractal geometry and non-Hermitian spectral topology remains largely unexplored. Motivated by this opportunity, it is interesting to ask whether fractal geometry can be realized in non-Hermitian band braiding? How such fractal braids impact non-Hermitian phenomena like skin effects? What experimental approaches can realize these structures. Resolving these questions not only bridge a gap between non-Hermitian spectral topology and fractal physics, but also unlocks possibilities for the design and control of non-Hermitian topological matter.

In this work, we first introduce the concept of non-Hermitian fractal braiding, which merges the fractal physics with non-Hermitian spectral topology. By employing a Lissajous construction approach, we establish the fractal braids where each braid segment recursively replicates at progressively smaller scales, generating an infinitely extending fractal pattern. This self-similar braiding behavior leads to the corresponding topological winding matrix exhibiting fractal characteristics. Through tight-binding lattice realizations of non-Hermitian fractal braiding, we further uncover the bulk-boundary correspondence for non-Hermitian fractal braiding system, revealing a scale-invariant relationship between fractal winding matrices and the emergence of non-Hermitian skin modes. To validate our theoretical results, we fabricate reconfigurable non-Hermitian topoelectrical circuits to construct the fractal braiding of non-Hermitian energy bands. Our work only bridges a gap between non-Hermitian spectral topology and fractal physics, but also unlocks possibilities for the design and control of non-Hermitian topological matter.

Results and Discussion

Theoretical methods for constructing fractal braiding using Lissajous method

We begin by introducing the concept of fractal braiding and outlining the general method for its mathematical realization. Fractal braiding intertwines the mathematical fractals with the topological pattern of braiding, creating intricate self-similar patterns. Figure 1a, b illustrates schematic representations of two fractal braids, where each braid segment undergoes recursive replication at progressively smaller scales. These self-similar structures can be described using Artin’s braid words58, specifically \({({{\sigma }_{2}}^{-1}{{\sigma }_{1}}^{-1})}^{3}\) and \({({{\sigma }_{2}}^{-1}{\sigma }_{1})}^{3}\), respectively. In this notation, the braid operation \({\sigma }_{i}\) represents the interchange of the ith and (i + 1)th strands, with the ith strand crossing beneath the (i + 1)th strand, while \({\sigma }_{i}^{-1}\) indicates the ith strand crossing above the (i + 1)th strand. As shown in Fig. 1a, the fractal braid consists of three distinct sections, highlighted in orange, green, and blue. Each of these sections functions as an effective strand, forming the first-generation braid—also known as the fundamental braid (Fig. 1c). As the fractal pattern iterates, each effective strand of the first-generation braid evolves into a second-generation braid (Fig. 1d). Subsequently, each effective strand of the second-generation braid is further divided into a third-generation braid shown in Fig. 1e. This recursive self-replication creates multiple layers of intricately interwoven segments, giving rise to the fractal braiding. Similarly, the braid in Fig. 1b also exhibits fractal characteristics, as illustrated in Fig. 1f–h. The key distinction between these two types of fractal braids lies in the braiding behavior of their fundamental braids, which follow circular and lemniscate trajectories in the transverse plane, as highlighted in the enlarged views of Fig. 1c–h. Here, all braid structures presented in this work exhibit non-Abelian characteristics with the number of strands N ≥ 3, as their braiding operations are inherently non-commutative. This non-Abelian property disappears in the case of N = 2, where the braiding reduces to a simpler Abelian scenario.

Fig. 1: The theoretical framework for constructing the fractal braiding using the Lissajous construction.
figure 1

This figure introduces the concept of fractal braiding, which intertwines mathematical fractals with the topological patterns of braiding. a, b illustrate schematic diagrams of two representative fractal braids. Both types of fractal braiding showcase the recursive self-replication of each braid segment at progressively smaller scales, leading to self-similar patterns defined by Artin’s braid words \({({{\sigma }_{2}}^{-1}{{\sigma }_{1}}^{-1})}^{3}\) and \({({{\sigma }_{2}}^{-1}{\sigma }_{1})}^{3}\), respectively. For these two braids, the first-generation braid comprises three sections (marked by orange, green, and blue lines). c–e and f–h illustrate the iterative processes of fractal patterns in (a) and (b). Each section is subdivided into three second-generation braids, which in turn are further divided into three third-generation braids with each strand returning to itself after traversing the Brillouin zone, creating an intricate fractal structure. The difference between the two fractal braids lies in their distinct braiding behaviors. The gray curve depicts the Lissajous trajectory of the first-generation braid in the xy-plane, while the colored curves depict the Lissajous trajectories generating higher-generation braids. The colored dots indicate the starting or ending points of these fractal braid.

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To further illustrate the mathematical structure of fractal braiding beyond its visual representation, we develop a general method to construct its spatial architecture. We characterize the fractal braid in a 3D parameter space (x, y, z), where the xy-plane serves as the transverse plane, and the z-axis represents the braiding direction ranging from 0 to 2π. We consider a fractal braid with a total of l generations. The braiding behavior and the number of effective strands (labeled by N) in each generation braid remain consistent across all generations. In this case, there are totally \({N}^{l}\) stands of the fractal braid, and the jth strand (with \(j\in [1,{N}^{l}]\)) is labeled by a vector \({{\bf{M}}}_{{\bf{j}}}=({m}_{j,1},{m}_{j,2},\ldots ,{m}_{j,l})\), where \({m}_{j,i}\) denotes the \({m}_{j,i}\)-th effective strand in the ith-generation braid with \(i\in [1,l]\). The value of \({m}_{j,i}\) across all generations is in the range of \([1,N]\). Consequently, there are \({N}^{l}\) distinct vectors \({{\bf{M}}}_{{\bf{j}}}(j\in [1,{N}^{l}])\) used to label all strands within the fractal braiding.

It has been shown that a conventional braid without fractal characteristics can be constructed using the Lissajous method59, in which each strand traces a Lissajous curve in the xy-plane as the braiding parameter varies. Here, we extend this approach to fractal braiding, expressing the transverse coordinate of the strand \({{\bf{M}}}_{{\bf{j}}}\) within the fractal braid as follows:

$$x\left({{\bf{M}}}_{{\bf{j}}},z\right)=\mathop{\sum }\limits_{i=1}^{l}{r}_{i}\cos [({\alpha }_{{m}_{j,i}}z-{\varphi }_{{{\boldsymbol{m}}}_{j,i}})],y\left({{\bf{M}}}_{{\bf{j}}},z\right)=\mathop{\sum }\limits_{i=1}^{l}{r}_{i}\sin [{\rm{\beta }}({\alpha }_{{m}_{j,i}}z-{\varphi }_{{{\boldsymbol{m}}}_{j,i}})]$$
(1)

Here, \({r}_{i}\) determines the transversal length scale of the ith-generation braid, while \({\alpha }_{{m}_{j,i}}\) and \({\varphi }_{{m}_{j,i}}\) represent the twisting parameter and initial phase of the \({m}_{j,i}\)-th strand in the ith generation braid. To achieve a self-similar fractal braiding structure, the twisting parameters and initial phases across all generations must satisfy \({\alpha }_{{{\boldsymbol{m}}}_{j,i}}=\alpha \) with \(i\in [1,l]\) and \({\varphi }_{{{\boldsymbol{m}}}_{j,i}}={\varphi }_{n}=\frac{2\pi }{N}(n-1)\) with \(n\in [1,N]\). The parameter \({\rm{\beta }}\) determines the transverse Lissajous trajectory in the xy-plane with z changing from 0 to 2π. Utilizing Eq. (1), the spatial structure of the fractal braid in Fig. 1a can be constructed with parameters \(N=l=3,\left({r}_{1},{r}_{2},{r}_{3}\right)=\left(\mathrm{1,0.4,0.05}\right),\alpha =1\) and \(\beta =1\). When \(\beta =1\), as \(z\) varies from 0 to 2π, the phases of both \(x\) and \(y\) change by 2π, resulting in a circle Lissajous trajectory in the xy-plane. Moreover, the fractal braid in Fig. 1b can be obtained with parameters \(N=l=3\), \(\left({r}_{1},{r}_{2},{r}_{3}\right)=(\mathrm{1,0.3,0.01})\), \(\alpha =1\) and \(\beta =2\). Here, for \(\beta =2\), with the variation of z, the phase of x changes by 2π while the phase of y changes by 4π, resulting in a 2-lemniscate trajectory in the xy-plane. It is worth noting that the fractal braids with higher-order lemniscates in the transverse plane can also be constructed by the Lissajous construction with \(\beta \ge 3\), where \(\beta =3\) corresponds to the case with a 3-lemniscate, as shown in Supplementary Note 1. Once the trajectory shape is determined by the parameter β, increasing the fractal generation l introduces additional terms to the summation in Eq. (1). As a result, compared to the first-generation braid, multi-generation Lissajous trajectories exhibit abrupt transitions, manifested as an increased number of Lissajous patterns. In contrast, changing the number of base braids N only shifts the positions of the band endpoints along the existing Lissajous trajectory, without introducing abrupt transitions of the spectral graphs (see Supplementary Note 2 for detailed examples and analysis).

It is worth noting that the main text focuses on link-type fractal braids, where each strand returns to itself after completing a cycle over the first Brillouin zone. However, our Lissajous-based construction is more general and allows for the generation of diverse braid types by tuning the parameters N and \({\alpha }_{{m}_{j,i}}\). Specifically, when the twisting parameter \({\alpha }_{{m}_{j,i}}\) takes fractional values of the form \(\frac{\gamma }{N}\) with \(\gamma \in Z\), the resulting braid endpoints alternate and reconnect with those of other strands, forming knot-type braids (see details in Supplementary note 3).

Fractal braiding of non-Hermitian complex-energy bands

In this section, we demonstrate how fractal braids can be embedded within the braiding of one-dimensional (1D) non-Hermitian bands. The 1D non-Hermitian lattice Hamiltonian in momentum space is denoted by a \(n\times n\) matrix \(H(k)\), where all energy bands \(E\left(k\right)=[{E}_{1}\left(k\right),{E}_{1}\left(k\right),\ldots ,{E}_{n}\left(k\right)]\) considered here are spectrally separated from one another. The number of energy bands is given by \(n={N}^{l}\), where N represents the number of effective strands within the fundamental braid, and l indicates the total number of generations for the fractal braid. Since the momentum k is defined within the first Brillouin zone of [0, 2π], each energy band can be viewed as a strand of a periodic braid in the (Re[\(E\left(k\right)\)], Im[\(E\left(k\right)\)], \(k\)) space. To construct the fractal braiding in non-Hermitian bands, we directly replace (x, y, z) in Eq. (1) with (Re[\(E\left(k\right)\)], Im[\(E\left(k\right)\)], k), resulting in:

$${\mathrm{Re}}[{E}_{j}\left(k\right)]=\mathop{\sum }\limits_{i=1}^{l}{r}_{i}\cos [({\alpha }_{{m}_{j,i}}k-{\varphi }_{{{\boldsymbol{m}}}_{j,i}})],{\rm{Im}}[{E}_{j}\left(k\right)]=\mathop{\sum }\limits_{i=1}^{l}{r}_{i}\sin [{\rm{\beta }}({\alpha }_{{m}_{j,i}}k-{\varphi }_{{{\boldsymbol{m}}}_{j,i}})]$$
(2)

where each k-space energy band labeled by \(j\in [1,{N}^{l}]\) is mapped to a strand \({{\bf{M}}}_{{\bf{j}}}=({m}_{j,1},{m}_{j,2},\ldots ,{m}_{j,l})\) in the fractal braid. Thus, the complex-energy bands E(k) are able to exhibit the behavior of fractal braiding.

To further illustrate the fractal braiding of non-Hermitian bands, we design the non-Hermitian tight-binding lattice models that exhibit fractal braiding in their complex energy bands. It is well-established that the complex energy bands \(E\left(k\right)\) of a non-Hermitian Hamiltonian should satisfy the characteristic polynomial given by

$$q \, \left(\varepsilon ,k\right)=\mathop{\prod }\limits_{j=1}^{{N}^{l}}\left\{\varepsilon -{{Re}}\left[{E}_{j}\left(k\right)\right]-{iIm}[{E}_{j}\left(k\right)]\right\}$$
(3)

where \(\mathrm{Re}[{E}_{j}\left(k\right)]\) and \({Im}[{E}_{j}\left(k\right)]\) are determined using Eq. (2), and \(\varepsilon \) corresponds to the eigenenergy. The characteristic polynomial can be expanded in the series of \(\varepsilon \) as \(q\left(\varepsilon ,k\right)={\varepsilon }^{{N}^{l}}-{\sum }_{j=0}^{{N}^{l}-1}{t}_{j}(k){\varepsilon }^{j}\). The characteristic polynomial of this form corresponds to a companion matrix representation of the Hamiltonian as follows17, whose spectrum exhibits fractal braiding features:

$$H(k)={\left(\begin{array}{ccccccc}{t}_{n-1}(k) & {t}_{n-2}(k) & {t}_{n-3}(k) & \cdots & {t}_{2}(k) & {t}_{1}(k) & {t}_{0}(k)\\ 1 & 0 & 0 & \cdots & 0 & 0 & 0\\ 0 & 1 & 0 & \cdots & 0 & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & 0 & 0\\ 0 & 0 & 0 & \cdots & 1 & 0 & 0\\ 0 & 0 & 0 & \cdots & 0 & 1 & 0\end{array}\right)}_{n\times n}$$
(4)

This structure ensures that only the first row of the Hamiltonian contains the full spectral information via the coefficients \({t}_{j}\left(k\right)(j\in [0,n-1])\), while the unit entries on the sub-diagonal enforce nearest-neighbor hopping. Supplementary Tables 1 and 2 in Supplementary Note 4 display the matrix elements \({t}_{m}(k)\) for two fractal braids in Fig. 1a, b, respectively. Based on the k-space Hamiltonian of Eq. (4), the tight-binding lattice models featuring non-Hermitian fractal braids can be directly derived (see Supplementary Note 5 for details). In contrast to previously proposed lattice models that support conventional complex-energy braids, our designed non-Hermitian lattice models with fractal braids display more intricate non-local couplings, allowing their complex-energy bands to exhibit richer fractal braiding behaviors.

In addition to demonstrating that non-Hermitian band structures with fractal characteristics can be constructed by merging the theory of fractal braid with non-Hermitian band theory, we now focus on examining the spectral topology of fractal braiding within these non-Hermitian bands. Previous studies have established that the braiding topology of non-Hermitian N-band model is characterized by the winding of complex energy bands around one another. This winding behavior can be described using a N×N quantized winding matrix \(w\), where the complex-energy bands are represented in a braid diagram55. The extraction of the braid diagram is achieved through a projection process, as illustrated at the top chart of Fig. 2a. In this projection, the start and end points of the complex-energy bands are mapped onto a common line. Specifically, the quantized matrix element related to the ith and jth bands is given by

$${w}_{{ij}}=\frac{1}{2\pi i}\int {\partial }_{k}{ln}[{E}_{i}\left(k\right)-{E}_{j}\left(k\right)]{dk}$$
(5)

which is related to an integer \({v}_{{ij}}{\mathbb{\in }}{\mathbb{Z}}\) with \({w}_{{ij}}=\frac{{v}_{{ij}}}{2}\), and represents the difference between the number of clockwise and counterclockwise windings of the ith band around the jth band.

Fig. 2: Fractal characteristics of the winding matrix for the fractal braiding of non-Hermitian bands.
figure 2

a Winding matrix for non-Hermitian fractal-braiding energy bands with N = 3 and β = 2, showcasing the winding behavior of complex energy bands. The top subpanel illustrates the braid diagram, obtained by projecting the starting and ending points of the complex energy bands onto a plane perpendicular to the complex energy plane. The bottom subpanel displays the winding matrix for the fundamental braid, where the highlighted red matrix element indicates the counterclockwise braiding of the 2nd and 3rd bands. b Winding matrix for two-generation fractal braiding, derived from the fundamental braid. The right-hand subpanel corresponds to the band structure and braid diagram, emphasizing the fractal nature of the 9×9 winding matrix, where each 3×3 diagonal submatrix replicates the fundamental winding matrix from (a). The highlighted red elements reflect uniform winding behavior among the second-generation braids. c Winding number matrix for the three-generation fractal braids, demonstrating that the diagonal block matrix of the three-generation 27×27 matrix is identical to the two-generation winding matrix in (b), and the values of elements in each non-diagonal block are equal to the corresponding fundamental matrix elements shown in (a). The highlighted red elements reflect uniform winding behavior among the third-generation braids.

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Furthermore, the winding matrix for non-Hermitian fractal bands also exhibits fractal characteristics. To illustrate this feature, we consider the non-Hermitian fractal braids with β = 2 and N = 3, and the total generation number from l = 1 to 3. It is important to note that the braiding between two energy bands can involve both clockwise and counterclockwise contributions. This may lead to cancellations in the winding matrix elements, resulting in all elements of the winding matrix becoming zero, even when non-trivial braiding is present. To identify such non-trivial braids, we instead examine the winding matrix computed over a subinterval of the Brillouin zone. For non-Abelian braids with N ≥ 3, the presence of any non-zero winding matrix over a subinterval is sufficient to indicate non-trivial braiding topology28. Within this subinterval, the fractal structure of the winding matrix becomes apparent, and the underlying topological features are preserved.

For the complete band structure with β = 2 and N = 3, the corresponding quantized winding matrix consists entirely of zero elements. This outcome arises from the reversal of winding between each pair of bands across the full Brillouin zone, where the winding switches between [0, π] and [π, 2π]. Therefore, we focus on the winding matrix over the subinterval [0, π] to highlight the fractal characteristics of the winding matrix, as shown in Fig. 2. The bottom subpanel of Fig. 2a displays the calculated winding matrix with l = 1, aligning with that of the projected braid diagram. The highlighted matrix element (in red) is equal to 1, corresponding to the counterclockwise braiding of the 2nd and 3rd bands. In Fig. 2b, we calculate the winding matrix for the fractal braiding with two generations (l = 2). The right-hand subpanel shows the associated braiding structure and the projected braid diagram. The 9×9 winding matrix depicted in Fig. 2b can be derived from the 3×3 matrix shown in Fig. 2a. Each 3×3 diagonal submatrix is identical to the winding matrix with l = 1. This uniformity arises because each diagonal submatrix reflects the winding behavior of two strands within a single second-generation braid, mirroring the same winding pattern as the fundamental braid with l = 1. Furthermore, each 3 × 3 non-diagonal submatrix captures the winding behavior between two strands belonging to different second-generation braids, where the winding number remains consistent across any pair of these strands. This consistency arises because the winding number between strands from different second-generation braids is equivalent to the winding number between two strands of the first-generation braid formed by these two second-generation braids. Specifically, as illustrated by the red-highlighted submatrix in Fig. 2b, all winding elements equal to 1, originating from the red element in the winding matrix of Fig. 2a. To further illustrate this fractal characteristics, we calculate the winding matrix for the fractal braid with three generations, as displayed in Fig. 2c. Each 9 × 9 diagonal submatrix matches the two-generation winding matrix in Fig. 2b. Moreover, all elements in the red-highlighted non-diagonal submatrix are identical, corresponding consistently to the red-highlighted matrix element in Fig. 2a.

Based on the formation pattern of the winding matrix, we conclude that it exhibits the following self-similar block structure: \({w}^{(l)}={\mathscr{W}}{\mathscr{\otimes }}{J}_{N}^{\otimes ({\boldsymbol{l}}-{\boldsymbol{1}})}+{I}_{N}\otimes {w}^{(l-1)}\), establishing the fractal property into non-Hermitian spectral topology. Here \({\mathscr{W}}\) is the winding matrix of the base braid, \({J}_{N}\) is an all-ones matrix, and \({I}_{N}\) is the identity matrix, with N denoting the number of base braids. The term \({\mathscr{W}}{\mathscr{\otimes }}{J}_{N}^{\otimes (l-1)}\) signifies that to obtain the l-th generation winding matrix, the base matrix \({\mathscr{W}}\) is tensor-multiplied with \({J}_{N}\) repeated (l-1) times. Unlike structures defined in fractal geometry that possess non-integer fractal dimensions, our fractal energy band structure does not have a conventional fractal dimension. Instead, its fractal characteristics manifest in the topological braiding structure of complex energy bands, which is characterized by the fractal winding matrix.

Additionally, the winding matrix not only characterizes the spectral topology of non-Hermitian energy bands but also establishes a bulk-edge correspondence for non-Hermitian skin effects13,14,60,61,62,63. Specifically, in non-Hermitian systems, bulk-boundary correspondence links the signs of spectral winding numbers under periodic boundary conditions to the localization direction of skin eigenstates under open boundary conditions. This is in stark contrast to Hermitian systems, where bulk topological invariants defined on eigenstates typically quantify the number of topological edge states. Based on the non-Hermitian winding matrix, the bulk-edge correspondence of non-Hermitian systems falls into two types28. The first case involves two-band models, where the braiding behavior of two bands is Abelian. In this case, the presence of skin modes under open boundary conditions (OBCs) exhibits a direct correspondence with the winding matrix. Specifically, the sign of each matrix element \({w}_{{ij}}\) indicates the localization direction: \({w}_{{ij}} < 0\) suggests right-localized skin modes, while \({w}_{{ij}} > 0\) indicates left-localized skin modes. If all matrix elements are zero, the system consists solely of extended states. The second case applies to multi-band systems with \(N\ge 3\), where the winding behavior exhibits non-Abelian characteristics, meaning the band topology depends on the order of braiding operations. In this case, the relationship between the winding matrix and skin modes differs from the Abelian scenario. For instance, in multi-band non-Hermitian systems, zero values across all \({w}_{{ij}}\) do not necessarily imply the absence of skin effects; multiple reverse windings between the ith and jth bands may yield opposing signs among the winding elements, resulting in a net sum of \({w}_{{ij}}=0\). This scenario enables the coexistence of left- and right-localized modes with equal numbers in this multi-band model. Additionally, the positive and negative elements within the winding matrix indicate the existence of left- and right-localized skin modes, respectively. Considering the fractal properties of the winding matrix in non-Hermitian fractal bands, we observe that the proportion of specific-valued matrix elements remains constant across different fractal generations. This implies that the ratio of left- and right-localized skin modes remains unchanged with increasing fractal generation. In other words, the bulk-edge correspondence in non-Hermitian energy bands with multi-generational fractal structures can be characterized by the winding matrix of the fundamental braid.

To verify this unique property, we analyze a non-Hermitian lattice model featuring fractal-braiding energy bands, where the fundamental braid arises from a non-Abelian three-band braid with parameters β = 2 and N = 3 (the same to Fig. 2). Following this setup, we first construct the lattice model with l = 1 and compute the complex energy spectrum of a finite chain (100 units) under periodic boundary conditions (PBCs), as shown in Fig. 3a. It is worth noting that the minimal terms in the long-range coupling of the lattice model have been omitted. Under this approximation, although each braid trajectory may exhibit slight deviations, the overall braiding behavior of the system remains unchanged. In other words, the topological winding matrix retains the same form, thereby confirming the robustness of the scheme. It is observed that a 2-lemniscate structure emerges in the complex energy plane. The red points in Fig. 3a represent the energy spectrum under OBCs and lie within the PBC spectrum, signaling the presence of non-Hermitian skin effects. The certain points in the spectrum, identified as Bloch points, remain unchanged under OBCs, indicating the presence of extended states. For the band structure depicted in Fig. 2a, the corresponding quantized winding matrix consists entirely of zero elements. This outcome arises from the reversal of winding between any two bands, switching from 0 to π and from π to 2π. Consequently, in a finite system, non-Hermitian skin modes are anticipated to localize at two distinct endpoints in equal numbers, as previously discussed. To substantiate this, we compute the eigenvectors for this finite model under OBCs, as shown in Fig. 3b. The results reveal that eigenstates with energies in the range -1.0 to -0.06 exhibit left-localized skin modes, while those with energies from 0.06 to 1.0 display right-localized skin modes, aligning with predictions from the winding matrix. Additionally, certain states within the energy range [−0.06, 0.06] are Bloch states, demonstrating extended behavior.

Fig. 3: Verification of scalable non-Hermitian skin modes in lattice models with complex-energy fractal braids.
figure 3

a The calculated complex energy spectrum for the finite lattice model (100 units) with first-generation (l = 1) under PBCs and OBCs. The red points indicate the energy spectrum under OBCs, collapsing within the PBC spectrum. Other parameters are set as β = 2 and N = 3. b The calculated eigenvectors for a finite system with first-generation (l = 1) under OBCs, illustrating left-localized skin modes for energies within the range of [-1, -0.06] and right-localized skin modes for energies in the range of [0.06, 1.0], with Bloch states in the range of [-0.06, 0.06]. c. Energy spectra for systems with second-generation (l = 2) fractals under periodic and open boundary conditions. (d) The calculated eigenvectors for a finite system with l = 2 under OBCs.

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Furthermore, we calculate the energy spectra of the lattice models with two-generational (l = 2) fractal bands. It should be emphasized here that due to the presence of long-range couplings, the chosen lattice length must be sufficiently large to ensure the existence of sufficient amount of bulk lattice sites under OBCs. Grey and red dots in Fig. 3c display the corresponding energy spectra under PBCs and OBCs with the lattice length being 400 units. Consistently, the OBC spectrum collapses into the interior of the PBC spectrum, with Bloch points appearing at the center of the complex energy plane. Compared to Fig. 3a, the number of Lissajous trajectories increases abruptly with higher generation l, consistent with our theoretical predictions. Moreover, the calculated eigenvectors for the lattice with OBC reveal two oppositely localized non-Hermitian skin modes, as shown in Fig. 3d. It is worth noting that, due to the fractal characteristics of winding matrix discussed above, the ratio between left- and right-localized skin modes remains consistent with the case of l = 1. Additionally, for the systems corresponding to other fractal energy bands under the parameters \((\beta =3,N=4)\), we also calculate the spatial distributions of eigenmodes in finite lattice models, as detailed in Supplementary Note 6, which similarly reveal the characteristic skin modes of fractal braids. The left/right skin mode ratio remains strictly conserved throughout the fractal generation (Fig. 3). This preservation enables unprecedented multiscale engineering of non-Hermitian skin effects, a feature fundamentally unattainable in non-fractal topological systems.

Experimental observation of fractal braiding in non-Hermitian bands by reconfigurable electric circuits

In this section, we experimentally observe the fractal braiding of non-Hermitian bands. To implement the proposed non-Hermitian lattice model, we must realize non-reciprocal, long-range, and tunable couplings—a challenging feat in most quantum and classical systems. Recent advancements leveraging the equivalence between circuit networks and tight-binding lattice models have established topoelectrical circuits as a versatile platform for exploring topological physics64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90, where non-Hermitian, tunable, and long-range couplings can be implemented with high fidelity. Building on these unique advantages, we design and fabricate two types of non-Hermitian electric circuits with tunable node couplings. Each circuit configuration allows for the experimental observation of a distinct type of non-Hermitian fractal braiding.

First, we focus on the realization of the non-Hermitian model, which exhibits the three-generational fractal braiding with a circular trajectory. The corresponding k-space lattice, featuring a total of 27 bands, is shown in Fig. 4a. This lattice includes the constant unidirectional couplings (light green lines) between adjacent lattice sites, as well as the k-dependent unidirectional couplings \({t}_{3n}(k)\) with \(n\in \left[\mathrm{1,9}\right]\) (marked by dark green lines) connecting the first site and the 3n-th site. The specific form of the k-dependent unidirectional couplings can be found in Supplementary Table 1. To implement this k-space lattice structure, we design an electric circuit with totally 27 nodes, as shown in Fig. 4b. The constant nonreciprocal coupling between adjacent nodes is implemented by connecting a voltage follower in series with a capacitor, as illustrated in Fig. 4c. For the tunable nonreciprocal couplings between the 3n-th node with \(n\in [\mathrm{1,9}]\) to the first node, we use a specially designed multiplier module comprising two multipliers connected in parallel, with a capacitor (\({C}_{a,3n}\)) and resistor (\({R}_{b,3n}\)) serving as loads, as shown in Fig. 4d. By adjusting the control voltages in each multiplier module, denoted as \({V}_{a,3n}\) and \({V}_{b,3n}\), we can achieve tunable nonreciprocal coupling. In addition, the inductor \({L}_{g}\) is grounded at the first node, while resistors \({R}_{g}\) are grounded at other nodes. Using Kirchhoff’s equation, we find that the circuit Laplacian mirrors the tight-binding Hamiltonian of the mapped non-Hermitian lattice model by carefully designing the values of circuit parameters and the voltage frequency \(\omega \) (see Methods section for the detailed derivation of the circuit Laplacian). Given the consistency between the circuit Laplacian and k-space lattice Hamiltonian, we can observe the non-Hermitian fractal braiding by the designed electric circuit.

Fig. 4: Experimental results of the three-generation non-Hermitian fractal braiding with a circular Lissajous trajectory.
figure 4

a The k-space lattice model for the three-generation fractal braiding with a circular Lissajous trajectory, containing a total of 27 lattice sites within a unit cell. The k-dependent unidirectional couplings \({t}_{2n}(k)\) are established between the first node and the 2n-th node. b Circuit design for simulating a three-generation fractal braiding structure based on a circular Lissajous trajectory, containing a total of 27 nodes. c Schematic and photograph of the constant nonreciprocal coupling between adjacent nodes, achieved by connecting a voltage follower in series with a capacitor. The scale bar represents 5 mm. d Configuration and photograph for the circuit realization of the tunable k-dependent nonreciprocal coupling. This tunable coupling is implemented using two multipliers connected in parallel, with a capacitor and a resistor as loads. External input voltages control the real and imaginary parts of the coupling, respectively. The scale bar represents 5 mm. e–g Experimental reconstruction of the admittance eigen-spectrum sections for the circuit under various values of \({V}_{a,3n}\) and \({V}_{b,3n}\), implementing the k-dependent couplings. The solid lines represent the simulation results by the spice software LTspice (https://www.analog.com/cn/resources/design-tools-and-calculators/ltspice-simulator.html).

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The fabricated circuit sample consisting of 27 nodes is presented in Supplementary Note 7. Enlarged views of the voltage follower module and the multiplier module are shown in Fig. 4c, d. We then reconstruct the admittance band structures of the circuit (labeled by \({E}_{L}\)) by measuring the global voltage responses across all nodes when an alternating voltage excitation at \(\omega =100{kHz}\) is applied to each circuit node. Further details on the experimental setup are provided in Methods. The discrete points in Fig. 4e–g correspond to the experimentally reconstructed admittance eigen-spectra of the circuit with varied voltages of \({V}_{a,3n}\) and \({V}_{b,3n}\), which are applied to implement the lattice Hamiltonian at six different k values. Solid lines show the corresponding simulation results. As evident, the experimentally reconstructed admittance eigen-spectra align well with the simulations, successfully capturing the three-generation non-Hermitian fractal braid with a circular Lissajous trajectory. Based on the measured admittance eigen-spectra, we can further obtain the winding matrix of the three-generation non-Hermitian fractal braid. The measured 27 × 27 winding matrix exhibits three 3 × 3 diagonal matrices \(\left(\begin{array}{ccc}0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0\end{array}\right)\) and six 3 × 3 off-diagonal matrices \(\left(\begin{array}{ccc}1 & 1 & 1\\ 1 & 1 & 1\\ 1 & 1 & 1\end{array}\right)\), showcasing the topological braiding behavior of the non-Hermitian fractal bands.

Then, we explore the two-generation non-Hermitian fractal braid, which follows a lemniscate-shaped Lissajous trajectory, with its fundamental braid comprising four strands. The corresponding k-space lattice diagram, containing 16 sites within a unit cell, is shown in Supplementary Note 8. In this structure, the k-dependent unidirectional couplings \({t}_{2n}(k)\) (\(n\in [\mathrm{1,8}]\)) are established between the first site and the 2n-th node. The specific form of the k-dependent unidirectional couplings can be found in Supplementary Table 3. To realize this two-generation fractal braid, we design an electric circuit with sixteen circuit nodes. Similarly, the voltage follower module provides the constant nonreciprocal coupling, while the multiplier module enables tunable complex couplings. The detailed derivation of the circuit Laplacian, which corresponds to the tight-binding Hamiltonian of the mapped non-Hermitian lattice model, is provided in Methods. Figure 5a–d display the experimentally reconstructed admittance eigen-spectrum for the circuit, with each section corresponds to a second-generation braid. These measured results are consistent with simulation results presented by solid lines, demonstrating the successful implementation of the designed two-generation fractal braid with lemniscate-shaped Lissajous trajectory. The experimentally measured winding matrix of the fractal braid exhibits fractal characteristics, identical to the matrix obtained by iterating the winding matrix of the fundamental fractal braid described in Supplementary Note 6.

Fig. 5: Experimental results of the two-generation non-Hermitian fractal braiding with lemniscate-shaped Lissajous trajectory.
figure 5

a–d Experimentally measured admittance eigen-spectra of the circuit, segmented to illustrate each of the four basic braids composing the second-generation braid. Solid lines represent the simulation results.

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Conclusions

In conclusion, we have theoretically introduced and experimentally realized the topological fractal braiding of non-Hermitian bands, bridging a gap at the intersection of fractal physics and non-Hermitian spectral topology. By developing a theoretical framework based on Lissajous construction, we have demonstrated how self-similar braiding patterns can be engineered in non-Hermitian systems, with braid segments recursively replicating across scales to form fractal topological structures in complex-energy space. These fractal braids exhibit exactly self-similar winding matrices that give rise to a scale-invariant non-Hermitian skin effect - a phenomenon enabling multiscale control unattainable in conventional systems. Furthermore, the experimental realization of 27-band fractal braiding demonstrates scalable control of complex spectral topologies. This provides a template for engineering hierarchical non-Hermitian phenomena in synthetic materials. It should be noted that while we have focused on a specific form of the fractal winding matrix, other classes of fractal matrices and their corresponding fractal braids may potentially exist (see Supplementary Note 9 for a detailed discussion of three recursive forms of winding matrices). The exploration of such possibilities would constitute an interesting direction for future research.

The implications of non-Hermitian fractal braiding extend beyond our current implementation. This paradigm could be realized in diverse platforms including optical systems 38,39, cavity optomechanics 41, acoustics 44, and magnonics 49, offering opportunities for controlling classical and quantum waves with non-Hermitian fractal topology. More broadly, our findings unlock possibilities for the design and control of non-Hermitian topological matter, and enrich the exploration at the nexus of fractal physics and non-Hermitian topology.

Methods

The derivation of the circuit Laplacian

At first, we focus on deriving the circuit Laplacian \({L}_{27}(\omega )\) of the structure shown in Fig. 4b. Applying Kirchhoff’s law to twenty-seven circuit nodes, we get the following equations as

$${I}_{1}=\mathop{\sum }\limits_{n=1}^{9}\left(\frac{{V}_{a,3n}{V}_{3n}}{10V}-{V}_{1}\right)i\omega {C}_{a,3n}+\mathop{\sum }\limits_{n=1}^{9}\left(\frac{{V}_{b,3n}{V}_{3n}}{10V}{V}_{1}\right)\frac{1}{{R}_{b,3n}}+\frac{\left(0-{V}_{1}\right)}{i\omega {L}_{g}}$$
(6a)
$${I}_{n}=\left({V}_{n-1}-{V}_{n}\right)i\omega {C}_{0}+\frac{\left(0-{V}_{n}\right)}{{R}_{g}},n\in \left[2,27\right]$$
(6b)

Here, the transfer function of analog multipliers AD633 is \(U=\frac{({X}_{1}-{X}_{2})({Y}_{1}-{Y}_{2})}{10V}+Z\). Resistors (\({R}_{b,3n}\)) and capacitors (\({C}_{a,3n}\)) with \(n\in \left[\mathrm{1,9}\right]\) in each multiplier module satisfy the relationship of \({R}_{b,3n}=\frac{1}{\omega {C}_{a,3n}}\). In addition, the capacitor connected to the output of the voltage follower module is selected as \({C}_{0}=0.01C\). Choosing \({C}_{a,3n}=0.1C\), (\(n=\mathrm{1,8,9}\)) and \({C}_{a,3n}=C\) (\(n\in [\mathrm{2,7}]\)), Eq. (6) can be rewritten in the matrix form as

$$\left(\begin{array}{c}{I}_{1}\\ {I}_{2}\\ {I}_{3}\\ {I}_{4}\\ {I}_{5}\\ {I}_{6}\\ \vdots \\ {I}_{27}\end{array}\right)=i\omega C\left(\begin{array}{cccccccc}{D}_{1} & 0 & {a}_{3}+i{b}_{3} & 0 & 0 & {a}_{6}+i{b}_{6} & \cdots & {a}_{27}+i{b}_{27}\\ 0.01 & {D}_{2} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0.01 & {D}_{3} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0.01 & {D}_{4} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0.01 & {D}_{5} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0.01 & {D}_{6} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0.01 & {D}_{27}\end{array}\right)\left(\begin{array}{c}{V}_{1}\\ {V}_{2}\\ {V}_{3}\\ {V}_{4}\\ {V}_{5}\\ {V}_{6}\\ \vdots \\ {V}_{27}\end{array}\right)$$
(7)

with \({V}_{a,3n}=100{a}_{3n}\), \({V}_{b,3n}=-100{b}_{3n}\)(\(n=1,8,9\)) and \({V}_{a,3n}=10{a}_{3n}\), \({V}_{b,3n}=-10{b}_{3n}\)\((n\in \left[2,7\right])\). In this case, the circuit Laplacian \({L}_{27}(\omega )\) can be written as

$${L}_{27}\left(\omega \right)=i\omega C\left(\begin{array}{cccccccc}{D}_{1} & 0 & {a}_{3}+i{b}_{3} & 0 & 0 & {a}_{6}+i{b}_{6} & \cdots & {a}_{27}+i{b}_{27}\\ 0.01 & {D}_{2} & 0 & 0 & 0 & 0 & \vdots & 0\\ 0 & 0.01 & {D}_{3} & 0 & 0 & 0 & \vdots & 0\\ 0 & 0 & 0.01 & {D}_{4} & 0 & 0 & \vdots & 0\\ 0 & 0 & 0 & 0.01 & {D}_{5} & 0 & \vdots & 0\\ 0 & 0 & 0 & 0 & 0.01 & {D}_{6} & \vdots & 0\\ 0 & 0 & 0 & 0 & 0 & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0.01 & {D}_{27}\end{array}\right)$$
(8)

When the circuit parameters are set as \({C}_{a,3n}=1{nF}\), \({R}_{b,3n}=10k\varOmega (n=\mathrm{1,8,9})\), \({C}_{0}=100{pF}\), \({C}_{a,3n}=10{nF}\), \({R}_{b,3n}=1k\varOmega (n=[\mathrm{2,7}])\), the matrix elements of Eq. (8) are \({D}_{1}=-6.3+6.3i+\frac{1}{{\omega }^{2}{L}_{g}C}\), \({D}_{n}=-0.01+\frac{1}{\omega C{R}_{g}}\) for \(n\in \left[\mathrm{2,27}\right]\), \({a}_{3n}+i{b}_{3n}=\frac{{V}_{a,3n}}{100}-i\frac{{V}_{b,3n}}{100}\) for \(n=\mathrm{1,8,9}\) and \({a}_{3n}+i{b}_{3n}=\frac{{V}_{a,3n}}{10}-i\frac{{V}_{b,3n}}{10}\) for \(n\in \left[\mathrm{2,7}\right]\). We can see that when the resistance \({R}_{g}\) and the inductance \({L}_{g}\) satisfy the specified relationship \({L}_{g}=\frac{1}{6.29{\omega }^{2}C}\) and \({R}_{g}=\frac{1}{6.3\omega C}\) (with other circuit parameters being \({L}_{g}=1.59{mH}\), \({R}_{g}=158.7\varOmega \) and \(\omega =100{kHz}\)), the circuit Laplacian mirrors the tight-binding Hamiltonian of the mapped non-Hermitian lattice model. By adjusting the external voltages \({V}_{a,3n}\) and \({V}_{b,3n}\), the tunable couplings \({a}_{3n}+i{b}_{3n}\) can be implemented.

Next, we derive the Laplacian \({L}_{16}(\omega )\) of the other circuit shown in Supplementary Fig. 7b. Applying Kirchhoff’s law to the sixteen circuit nodes, we get the following equations as

$${I}_{1}=\mathop{\sum }\limits_{n=1}^{n=8}\left(\frac{{V}_{a,2n}{V}_{2n}}{10}-{V}_{1}\right)i\omega {C}_{a,2n}+\mathop{\sum }\limits_{n=1}^{n=8}\left(\frac{{V}_{b,2n}{V}_{2n}}{10}-{V}_{1}\right)\frac{1}{{R}_{b,2n}}+\frac{\left(0-{V}_{1}\right)}{i\omega {L}_{g}}$$
(9a)
$${I}_{n}=\left({V}_{n-1}-{V}_{n}\right)i\omega C+\frac{\left(0-{V}_{n}\right)}{{R}_{g}},n\in \left[2,16\right]$$
(9b)

Resistors \({R}_{b,n}\) and capacitors \({C}_{a,n}\) in each multiplier module satisfy \({R}_{b,n}=\frac{1}{\omega {C}_{a,n}}\left(n=[\mathrm{1,16}]\right)\). Meanwhile, other circuit elements are set as \({C}_{a,n}=10C\left(n=\mathrm{2,6,12,14}\right)\), \({C}_{a,n}=100C\left(n=\mathrm{4,8,10}\right)\), \({C}_{a,n}=C\left(n= 16\right)\), \({R}_{b,n}=\frac{0.1}{\omega C}\left(n=\mathrm{2,4,12}\right)\), \({R}_{b,n}=\frac{0.01}{\omega C}\left(n=\mathrm{6,8,10}\right)\) and \({R}_{b,n}= \frac{1}{\omega C}\left(n=\mathrm{14,16}\right)\). Eq. (9) can be written in the matrix form as

$$\left(\begin{array}{c}{I}_{1}\\ {I}_{2}\\ {I}_{3}\\ {I}_{4}\\ {I}_{5}\\ {I}_{6}\\ \vdots \\ {I}_{16}\end{array}\right)=i\omega C\left(\begin{array}{cccccccc}{D}_{1} & {a}_{2}+i{b}_{2} & 0 & {a}_{4}+i{b}_{4} & 0 & {a}_{6}+i{b}_{6} & \cdots & {a}_{16}+i{b}_{16}\\ 1 & {D}_{2} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & {D}_{3} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & {D}_{4} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & {D}_{5} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & {D}_{6} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & 0 & \cdots & 1 & {D}_{16}\end{array}\right)\left(\begin{array}{c}{V}_{1}\\ {V}_{2}\\ {V}_{3}\\ {V}_{4}\\ {V}_{5}\\ {V}_{6}\\ \vdots \\ {V}_{16}\end{array}\right)$$
(10)

with \({V}_{a,n}={a}_{n}\), \({V}_{b,n}=-{b}_{n}\left(n=2,12\right)\); \({V}_{a,n}=0.1{a}_{n}\), \({V}_{b,n}= -{b}_{n} \left(n=4\right)\); \({V}_{a,n}={a}_{n}\), \({V}_{b,n}=-0.1{b}_{n}\left(n=6\right)\); \({V}_{a,n}=0.1{a}_{n}\), \({V}_{b,n}=-0.1{b}_{n} \left(n=8,10\right)\); \({V}_{a,n}={a}_{n}\), \({V}_{b,n}=-10{b}_{n}\left(n=14\right)\) and \({V}_{a,n}=10{a}_{n}\), \({V}_{b,n}=-10{b}_{n}\left(n=16\right)\). In this case, the circuit Laplacian \({L}_{16}(\omega )\) can be written as

$${L}_{16}\left(\omega \right)=i\omega C\left(\begin{array}{cccccccc}{D}_{1} & {a}_{2}+i{b}_{2} & 0 & {a}_{4}+i{b}_{4} & 0 & {a}_{6}+i{b}_{6} & \cdots & {a}_{16}+i{b}_{16}\\ 1 & {D}_{2} & 0 & 0 & 0 & 0 & \vdots & 0\\ 0 & 1 & {D}_{3} & 0 & 0 & 0 & \vdots & 0\\ 0 & 0 & 1 & {D}_{4} & 0 & 0 & \vdots & 0\\ 0 & 0 & 0 & 1 & {D}_{5} & 0 & \vdots & 0\\ 0 & 0 & 0 & 0 & 1 & {D}_{6} & \vdots & 0\\ 0 & 0 & 0 & 0 & 0 & \ddots & \ddots & \vdots \\ 0 & 0 & 0 & 0 & 0 & \cdots & 1 & {D}_{16}\end{array}\right)$$
(11)

When the circuit parameters are set as \(C=100{pF}\), \({C}_{a,n}=1{nF}(n=2,6,12,14)\), \({C}_{a,n}=10{nF}(n=4,8,10)\), \({C}_{{a}_{3}}=100{pF}(n=16)\), \({R}_{b,n}=10k\varOmega (j=2,4,12)\), \({R}_{b,n}=1k\varOmega (n=6,8,10)\), \({R}_{b,n}=100k\varOmega (n=14,16)\), the matrix elements of Eq. (11) are \({D}_{1}=-341+332i+\frac{1}{{\omega }^{2}{L}_{g}C}\), \({D}_{n}=-1+\frac{1}{\omega C{R}_{g}}\left(n=[\mathrm{2,16}]\right)\),\({a}_{n}+i{b}_{n}={V}_{a,n}-i{V}_{b,n}\left(n=2,12\right)\), \({a}_{n}+i{b}_{n}={10V}_{a,n}-i{V}_{b,n}\left(n=4\right)\),\({a}_{n}+i{b}_{n}={V}_{a,n}-{i10V}_{b,n}\left(n=6\right)\),\({a}_{n}+i{b}_{n}={10V}_{a,n}-{i10V}_{b,n}\left(n=8,10\right)\),\({a}_{n}+i{b}_{n}={V}_{a,n}-{i0.1V}_{b,n}\left(n=14\right)\) and \({a}_{n}+i{b}_{n}={0.1V}_{a,n}-{i0.1V}_{b,n}\left(n=16\right)\). When the grounded resistance \({R}_{g}\) and the inductance \({L}_{g}\) satisfy the relationship \({L}_{g}=\frac{1}{340{\omega }^{2}C}\) and \({R}_{g}=\frac{1}{332\omega C}\) (with other circuit parameters being \({L}_{g}=2.94{mH}\), \({R}_{g}=301.2\varOmega \)), the circuit Laplacian mirrors the tight-binding Hamiltonian of the mapped non-Hermitian lattice model. Similarly, the tunable couplings can be realized by adjusting the external voltages \({V}_{a,n}\) and \({V}_{b,n}\).

Sample fabrications

We fabricate electronic circuits using LCEDA program software, where PCB stacking, internal layers, and the wiring method are properly designed. Here, sample PCBs have six-layer board structure, where the top and bottom layers are used to wire and place components. The two nonadjacent inner layers are used as the power supply layers for active devices and the remaining two inner layers are grounding layers, which not only synchronizes the power supply, but also facilitate heat dissipation. Furthermore, we made the wiring as short as possible while also selecting a relatively large width (\(0.6{mm}\)) to minimize parasitic effects. The voltage follower module utilizes an LM6171 OpAmp from Texas Instruments, with a parallel resistor \({R}_{f}\) and capacitor \({C}_{f}\) placed between the inverting input and output of the OpAmp to maintain stability. Additionally, a capacitor \({C}_{0}\) also is connected to the output to enhance stability. The analog multiplier module uses two AD633 analog multipliers from Analog Devices Inc. Each multiplier output is connected to a capacitor \({C}_{a,3n}\) and a resistor \({R}_{b,3n}\), with external voltages \({V}_{a,3n}\) and \({V}_{b,3n}\) applied through dedicated pins to tune the coupling term via voltage inputs. It is worth noting that the filtering capacitors are placed at each active device supply pin, which can filter AC noise and high-frequency interference signals to provide a stable DC operating voltage for the multipliers (\(0.01\mu F\)) and amplifiers (\(0.1\mu F\)). The amplifier driving the capacitive load introduces a resistor \({R}_{0}=1\varOmega \) close to the OpAmp to ensure output stability. The pins or SMP connectors are soldered to both the PCB nodes and the multiplier inputs for external voltage input. To ensure high accuracy and low losses of the circuit components (the disorder strength is only 1%), we use a WK6500B impedance analyzer to select circuit elements.

Experimental details

The experiment is based on discrete measurements of the circuit Green’s function matrix G for each momentum value k, from which the circuit Laplacian J is reconstructed to obtain the system’s band structure. Firstly, we successively set the ac input signal at each circuit node. At the same time, we choose \(\pm 15V\) dc voltage source to supply the active device and vary external voltages of the multipliers to achieve tunable couplings. To recover the circuit Laplacian eigenspectra, the voltage responses of all circuit nodes should be measured by the oscilloscope when an ac voltage with the form of \({V}_{0}{e}^{i\omega t}\) is applied to each circuit node. It is noted that the voltage source is connected in series with a precision resistor to the circuit node. Here, \({V}_{i}(j)\) corresponds to the measured voltage at node \(i\) under the voltage excitation at node \(j\) with the current at node \(j\) being \({I}_{j}\). The ac current \({I}_{j}\) is calculated by Ohm’s law \({I}_{j}=U/R\), where \(U\) is the voltage difference across the precision resistor \(R\). Based on the measured voltages and the ac current currents, we can get the matrix for the inverse of circuit Laplacian G with \({G}_{{ij}}={V}_{i}(j)/{I}_{j}\), after which the circuit Laplacian \(J\) can be recovered from \({\bf{J}}={{\bf{G}}}^{-1}\). Based on the recovered circuit Laplacian, the k-space non-Hermitian Hamiltonian can be obtained as \({\bf{H}}=\frac{1}{i\omega C}{\bf{J}}-D{\bf{I}}\) with I being an identity matrix of the same size as H and \(D\) being the diagonal matrix of J. To obtain the band structure, we diagonalize the reconstructed momentum-space Hamiltonian H(k), and the resulting eigenvalues correspond to the energy levels of the circuit at each momentum point \(k\). In particular, each momentum k is mapped to a distinct AD633 input voltage configuration. These experimentally obtained spectra can be directly compared with theoretical band predictions. In practice, we perform this measurement for a sufficiently dense set of \(k\)-points along high-symmetry paths in the Brillouin zone. This dense sampling ensures that the discrete eigenvalues of H(k) form a continuous band structure that closely matches the theoretical model.