Experimental probe of point gap topology from non-Hermitian Fermi-arcs

Abstract

The gap in spectra of a physical system is fundamental in physics, while gap topology further restricts possible occurrent gaps of topological boundary states. The emergence of non-Hermiticity unveils a unique gap type known as the point gap, which forecasts the wavefunction localization, known as the non-Hermitian skin effect. Therefore, experimentally identifying the point gap in the complex frequency plane through a real operating frequency can become a tool for the systematic investigation of skin effects. Here, we utilize a Weyl phononic crystal to demonstrate that the point gap constituted by bulk and Fermi-arc surface states can be observed experimentally by a real-space field mapping technique. The identified point gaps forecast various skin effects and their evolutions. We further experimentally demonstrate the hinge skin effect in a parallelogram structure. Our work provides a feasible recipe to explore point gap topology experimentally in a variety of systems and certainly stimulates the research on skin effects in three-dimensional systems.

Introduction

Non-Hermiticity emerges instead of being introduced when a real-world system becomes open in the physical sense, for instance, exchanging energy or momentum with the surroundings, thus leading to non-Hermiticity being omnipresent regardless of the details. A natural consequence is that the eigenvalue spectra expand from real values in the Hermitian case to complex ones, generalizing all the concepts defined previously on the real spectra and enriching our knowledge of Hermitian physics^1,2,3. As the fundamentals, both degeneracy and gap types have been imparted to non-Hermitian unique ingredients known as exceptional points (EPs) and point gaps (PGs)^3,4,5,6. The EP possesses identical eigenvalues but defective eigenvectors, and the fruitful EP configurations, dubbed as exceptional geometry, sizably fertilize the Hermitian nodal structures^{7,8,9,10,11,12,13,14,15,16}. Viewed from the spectra, the EP only differs from the Hermitian degeneracy by the critical exponents nearby, but the PGs have no Hermitian counterparts, as exhibited in Fig. 1a, b. The Hermitian eigenvalue spectra being always line segments only possess the ${\mbox{Re}}\left(E\right)$ line gaps (${L}_{r}$), but the complex eigenvalue spectra give rise to the PGs, which are declared to open nontrivially at a particular energy $E$ indicating that the winding numbers of the eigenvalues with respect to $E$ are nonzero, as dictated by ${{{\rm{PG}}}}_{\pm }$ in Fig. 1a⁴.

Fig. 1: Point gaps from Weyl points and the ensuing Fermi arcs.

a Line and point gaps in one-dimensional (1D) systems. b Point gaps (PGs) formed by bulk and surface states in 2D or 3D systems. ${L}_{r}$ and ${L}_{i}$ respectively denote the ${\mbox{Re}}\left(E\right)$ and ${\mbox{Im}}\left(E\right)$ line gaps, while ${{{\rm{PG}}}}_{\pm }$ stands for the PG with its subscript labeling the eigenvalue winding number herein. c Schematics of the unit cell and Brillouin zone of a representative tight-binding model supporting Weyl points and Fermi arcs (FAs). The lattice constants in all ${xyz}$ directions are set to 1 for simplicity. Two pairs of Weyl points at the ${{\rm{K}}}$ (${{{\rm{K}}}}^{{\prime} }$) and ${{\rm{H}}}$ (${{{\rm{H}}}}^{{\prime} }$) points with oppositely topological charges when γ =0  labeled in the Brillouin zone by blue and red spheres. d A rectangular block setup with the blue and yellow (gray) surfaces being the zigzag (armchair) boundaries, wherein the blue and yellow solid lines sketch the typical surface states on the FAs. e Schematics of the bulk and surface bands when $\gamma \ne 0$. The yellow and blue lines represent the FAs, which connect the exceptional point (EP) ring with distinct ${\mbox{Re}}\left(E\right)$ and have the distinct ${\mbox{Im}}\left(E\right)$ as differentiated by color. f PG formed by bulk states and FA surface states at ${k}_{z}=\pm 0.5\pi$. g The equal frequency contour in the ${k}_{x}$-${k}_{z}$ plane at ${\mbox{Re}}\left(E\right)=0$ with its color showing Im (E). h The spectral function $A\left(\omega =0,{{\boldsymbol{k}}}\right)$ in the ${k}_{x}$-${k}_{z}$ plane. i Diagrammatical representation of the hinge skin effect under different lifetime layouts at a fixed ${k}_{z}=-0.5\pi$. The orientation and color of the arrows denote the propagation direction and lifetime of the corresponding surface waves. j The corresponding calculated eigenstate. The parameters used are t₀ = −1, ${t}_{z}=-0.5$, and $\gamma =$0.15.

The significance for both manifests in the eigenvectors (wavefunctions). The EP also highlights its eigenvectors by the critical exponents³, while the nontrivial PG forecasts the wavefunction localization, known as the non-Hermitian skin effect (NHSE)^{17,18,19,20,21,22,23}. Concisely, the nontrivial PG formed by the bulk states under the periodic boundary conditions (PBCs) indicates that all the states will localize to the surface under the open boundary conditions (OBCs)^17,18,19. Such dramatic differences in both wavefunction and spectra beg for non-Bloch band theory, in which the generalized Brillouin zone (GBZ) replacing the Brillouin zone (BZ) plays the central role^{24,25,26,27,28}. Using GBZ, the OBC spectra within the nontrivial PGs are then resolved. Recent efforts in d-dimensional (dD, $d\ge 2$) or continuum systems reveal that the occurrence of NHSEs relies on boundary conditions (BCs) and geometric shapes^{29,30,31,32,33,34}, wherein the central task is still to resolve the OBC spectra within the nontrivial PGs. Moreover, the higher dimension affords more types of states, thus letting the PGs not be limited to the bulk states and can be formed solely by surface states or mixed surface and bulk states, as depicted in Fig. 1b^29,30. The ensuing NHSE is then termed the nth-order NHSE in a dD system of a size ${L}^{d}$, implying the number of skin states is $O({L}^{d-n+1})$^{35,36,37,38,39}$.$

Since the PGs are the bedrock in the non-Hermitian systems, it is not overemphasized their experimental probing. However, the experiment is always performed using the real operating energy/frequency (wiggles in Fig. 1a, b), which generally does not overlay with the nontrivial PGs. Note that we only discuss the nontrivial PGs afterward because the trivial PGs imply the zero density of states⁴, and thus, we use the PG for simplicity. Hence, how to directly identify a PG lying in the complex plane through the real-frequency measurement requires investigation. Aiming to reveal the validity of the experimental probe of the PGs, together with the fact that the higher dimensional system lifts the stringent constraints for the PGs being formed in 1D, we construct a non-Hermitian Weyl phononic crystal (PC) by introducing inhomogeneous losses as the prototype. The PC is known to host the Hermitian Weyl points (WPs) and the associated open Fermi arcs (FAs)^40,41. The non-Hermiticity introduced results in the PGs formed by the bulk states and FAs. We then measure the pressure fields excited by sources at different positions and perform Fourier transformation on the measured fields. The magnitudes of the Fourier component correspond to imaginary parts of the eigenfrequency, and thus, the PG is experimentally probed and verified. The confirmed PGs give rise to the hybrid NHSE, geometry-dependent skin effect (GDSE), and a hinge NHSE. As evidence, the hinge skin states, which localize at the hinge formed by two zigzag boundaries, have been further observed experimentally.

Results

Tight-binding model of point gap

First, to showcase the PGs originating from the non-Hermitian FAs and the feasibility of their experimental probe, we introduce a 3D tight-binding model (TBM)

$$H\left({k}_{x},{k}_{y},{k}_{z}\right)=\left(\begin{array}{cc}-i\gamma +{t}_{z}{f}_{z}\left({k}_{x},{k}_{y},{k}_{z}\right) & {t}_{0}{f}_{0}\left({k}_{x},{k}_{y}\right)\\ {t}_{0}{f}_{0}\left({k}_{x},-{k}_{y}\right) & {t}_{z}{f}_{z}\left({k}_{x},{k}_{y},{-k}_{z}\right)\end{array}\right),$$

(1)

where ${f}_{z}({k}_{x},{k}_{y},{k}_{z})=2\cos ({k}_{x}-{k}_{z})+4\cos (\sqrt{3}{k}_{y}/2)\cos ({k}_{x}/2+{k}_{z})$ and ${f}_{0}({k}_{x},{k}_{y})={e}^{-i{k}_{y}/\sqrt{3}}+2\cos ({k}_{x}/2){e}^{i{k}_{y}/2\sqrt{3}}$. With the hexagonal prism unit cell shown in Fig. 1c, ${t}_{0}$ (${t}_{z}$) is the hopping in the ${xy}$ plane (along the $z$ direction), $\gamma$ is the typical onsite loss that determines the non-Hermiticity, and all parameters in Eq. (1) are assumed to be real. The above TBM hosts WPs and FAs in the Hermitian scenario (Fig. 1c, d). When non-Hermiticity is present, Fig. 1e schematically draws the bulk and surface bands, wherein an EP ring replaces each corresponding WP but with the distinct $E$. It is then natural to ask the ensuing FAs, which can be investigated in the projected band structure scheme with the calculation setup depicted by the black dashed box in Fig. 1d. The OBCs (PBCs) are applied in the $y$-direction ($x$- and $z$-directions), which are labeled as the $y$-OBC configuration. Two topological surface bands are clearly seen within the bulk ${L}_{r}$ gap (See Methods), and their typical state profiles are located at the zigzag boundaries in yellow and blue (Fig. 1d), indicating the presence of FAs. Such survival of FAs is because of the still-existing ${L}_{r}$ in the OBC setup, which is assured by the 3D GBZ defined by the ameba theory being the same as its BZ for our reciprocal TBM [$H\left({{\boldsymbol{k}}}\right)={H}^{T}\left({{\boldsymbol{-}}}{{\boldsymbol{k}}}\right)$]²⁷ (see Methods for details). However, the dramatic difference in ${\mbox{Im}}\left(E\right)$ results in two FAs having their individual lifetime $\tau$ [$\propto {\mbox{Im}}{\left(E\right)}^{-1}$], as differentiated by the yellow and blue lines in Fig. 1e. Intuitively, the yellow (blue) FA, which has a large (small) lifetime, indicates that the states are concentrated at the yellow (blue) zigzag boundary, with its outmost site being the lossless (lossy) one. These two distinct FAs, together with the bulk states, enclose a nonzero area, i.e., PGs, for different ${k}_{z}$ (${k}_{z}\ne 0,\pi$). For ${k}_{z}=\pm 0.5\pi$, the clockwise and anticlockwise loops correspond to ${{{\rm{PG}}}}_{+}$ and ${{{\rm{PG}}}}_{-}$, respectively, as illustrated in Fig. 1f.

Aiming to quantify the role of $\tau$ in the surface bands, we generalize the equal frequency contour (EFC) defined previously in non-Hermitian bulk bands ${{\mathbb{K}}}_{{\mbox{B}}}$ to the surface bands as ${{\mathbb{K}}}_{{\mbox{S}}}=\left\{{{{\boldsymbol{k}}}}_{{{\boldsymbol{\parallel }}}}\in {\mbox{Surface}}{{\rm{BZ}}}{{\boldsymbol{|}}}{\mathrm{Re}}\left[{E}_{i}\left({{{\boldsymbol{k}}}}_{{{\boldsymbol{\parallel }}}}\right)\right]{{\boldsymbol{=}}}E\right\}$, where ${{{\boldsymbol{k}}}}_{{{\boldsymbol{\parallel }}}}$ belongs to a surface BZ, $i={\mathrm{1,2}},\ldots$ is the band index, and $E{\mathbb{\in }}{\mathbb{R}}$ is a given excitation energy. Unlike the Hermitian scenario, each state on the EFC, whatever the bulk or surface, now has a unique lifetime $\tau \propto {\mbox{Im}}{\left(E\right)}^{-1}$. Figure 1g shows the EFC of the zigzag boundary at ${\mbox{Re}}\left(E\right)=0$, and the two topological surface states clearly have distinct lifetimes. This discrimination results in various skin effects for different boundary configurations, including hybrid SE, GDSE, hinge SE, and their evolutions. When ${k}_{z}$ scans, both the surface states on the FAs and bulk states can become localized with the mode profile dependent on ${k}_{z}$, under different 2D geometries, the hybrid SE and GDSE can be found at two selected ${k}_{z}$ (See Methods). As a representative, when two zigzag boundaries are connected to form a parallelogrammic shape in Fig. 1i, their FAs have two types, as shown in yellow and blue. The hinge SE only occurs when the surface waves with a longer lifetime (yellow arrows) impinge on the surface possessing a shorter lifetime surface wave (blue arrows). Any waves on the yellow surfaces can propagate to the hinge highlighted in red, while any waves on the blue surfaces decay voluntarily, giving rise to the hinge SE, which has been validated by the calculated eigenstates shown in Fig. 1j. Due to the prophecy capability, to probe the PGs, which are usually excited by a given real energy, the EFC of the bulk and surface plays a crucial role in non-Hermitian systems, and thus, experimentally acquiring the knowledge of EFC is pivotal.

Since ${{\mathbb{K}}}_{{\mbox{S}}}$ and ${{\mathbb{K}}}_{{\mbox{B}}}$ are a set of ${{\boldsymbol{k}}}$ points on which the states dominate under the excitation of $E$, we employ the spectral function $A\left(E,{{\boldsymbol{k}}}\right)=-{\mbox{ImTr}}\left[1/\left(E-H\left({{\boldsymbol{k}}}\right)\right)\right]$, i.e., the density of the state (DOS), as the experimental indicator. The $A\left(E,{{\boldsymbol{k}}}\right)$ results are shown in Fig. 1h and directly reflect the features of EFC in Fig. 1g, with the magnitude of $A\left(E,{{\boldsymbol{k}}}\right)$ signifying the state lifetime, that is, the EFC color (See Methods). Therefore, in the experiment, an excitation is placed at different positions to stimulate the eigenstates, and the associated spatial field distributions are then measured and Fourier transformed. The amplitudes of the Fourier components correspond to $A\left(E,{{\boldsymbol{k}}}\right)$, indicating the state lifetime and the subsequent EFC.

Experimental probe of point gap in a Weyl phononic crystal

For implementing the TBM in acoustic waves, a layer-stacking PC with chiral interlayer couplings is designed and fabricated by 3D printing of plastic stereolithography material (DSM IMAGE8000). Figure 2a exhibits a cuboid sample, where each layer consists of two non-equivalent cavities at the nearest distance of $a=2{{\rm{cm}}}$ arranged in a hexagonal lattice. The corresponding unit cell is shown in Fig. 2b. The radius and height of the cavity are ${r}_{c}=0.4a$ and ${h}_{c}=0.3\sqrt{3}a$, respectively. The block with a square cross-section of a side length $w=0.21\sqrt{3}a$ connecting two nearest-neighbor cavities, offers the intralayer coupling. The lattice constant along the z direction is $L=0.5\sqrt{3}a$. The chiral tube of a diameter $d=0.24a$ provides the interlayer coupling. The vertical cylinders with a radius ${r}_{v}=0.05\sqrt{3}a$ much smaller than that of the cavity are for the convenience of experimental measurements. The topological properties of this Weyl PC have been investigated in previous works^40,41. To reveal the non-Hermitian physics in this PC, we introduce losses by inserting sponges (black materials in Fig. 2a, b) into the cavities. Here, the grid meshes with a thickness of $1{{\rm{mm}}}$ embedded in each cavity are deliberately designed to hold the sponges to facilitate experimental measurements. The simulated bulk band structure of the non-Hermitian Weyl PC is given in Supplementary Note 1. It is worth mentioning that as all the phenomena in this work do not depend on the specific value of the losses, i.e., the imaginary part of the sound velocity of air, the sound velocity in simulation is not fitted to the experiment. Thus, in all simulations, the mass density and sound velocity of air are set to $\rho =1.18{{\rm{kg}}}/{{{\rm{m}}}}^{3}$ and $v=(343+30i){{\rm{m}}}/{{\rm{s}}}$, respectively.

Fig. 2: Experimental probe of non-Hermitian Fermi-arc surface states.

a Cuboid sample. The cyan rectangles and hexagons represent the headphone, microphone, and unit cell, respectively. The black materials are the sponges inserted into one of the two non-equivalent cavities for introducing loss. b Schematic and enlarged view of the unit cell. Two non-equivalent cavities are arranged in a hexagonal lattice with the nearest distance $a=2\,{{\rm{cm}}}$. The radius and height of the cavity are ${r}_{c}=0.4a$ and ${h}_{c}=0.3\sqrt{3}a$, respectively. The block connecting two nearest-neighbor cavities with a square cross-section of a side length $w=0.21\sqrt{3}a$. The height of the unit cell is $L=0.5\sqrt{3}a$. The chiral and vertical tubes connect the two layers with diameters $d=0.24a$ and ${r}_{v}=0.05\sqrt{3}a$, respectively. c Surface state dispersions along the ${k}_{x}$ direction for different ${k}_{z}$. Color maps (circles) are the measured (simulated) results. d Fermi-arc surface states of the two non-equivalent boundaries at the real part of the frequency $4.75\,{{\rm{kHz}}}$. The radii of the circles are proportional to imaginary parts of the eigenfrequency of the surface states on opposite zigzag boundaries. e, f Corresponding results of the Hermitian Weyl phononic crystal.

The measured surface state dispersions of the real part of frequency for different ${k}_{z}$ are demonstrated in Fig. 2c, as denoted by the color maps. Here, we only investigate the surface states on the zigzag boundaries. The gray areas and white circles represent the simulated bulk and surface states, respectively. The two non-equivalent zigzag boundaries corresponding to whether the outermost cavity is lossy or not result in the surface states presenting different lifetimes during propagation, that is, they have different imaginary parts of eigenfrequencies. The correlation between the eigenproblem with complex eigenfrequencies and the response of surface waves in experiments at real frequencies has been established through the spectral function. The relevant discussion can be conducted based on the TBM and has been given in Methods. In the experiment, a headphone with a diameter of 5 mm placed at multiple positions in the sample is used to excite the FA surface states. A microphone with a diameter of $3\,{{\rm{mm}}}$ is fixed on a 3D automatic scanning stage to record the amplitude and phase information of the pressure fields through the network analyzer (Keysight 5061B). Then, all the measured pressure fields are Fourier transformed and summed together. The amplitudes of the Fourier components characterize the response strength between the surface wave and point source. In Fig. 2c, the brighter color indicates the stronger response with a larger DOS, while the darker color corresponds to the weaker response. This is consistent well with the simulated results, where the imaginary parts of eigenfrequencies are proportional to the radii of the white circles. The simulated spectral property of the surface states and the corresponding eigenfield distributions are illustrated in Supplementary Note 2. In contrast, in the Hermitian case, the difference between the surface state dispersions entirely disappears, as measured in Fig. 2e.

Second-order hinge skin effect

Different lifetimes of the surface states on the two boundaries predict abnormal wave propagation. Figure 2d shows the measured and simulated FA surface states on the two boundaries at the real frequency $4.75\,{{\rm{kHz}}}$. To demonstrate the wave propagation behavior, a right parallelogrammic sample with four zigzag boundaries is fabricated, as shown in Fig. 3a. Boundaries 1 (in yellow) and 2 (in blue) correspond to the outmost cavities without and with the sponges inside, respectively. To excite the surface wave, two point sources with a phase difference of $0.5\pi$ are placed in two adjacent layers as marked by the red stars in Fig. 3a. The wave propagates on boundary 1, concentrates at the hinge, and cannot bend into boundary 2, as measured in Fig. 3b. This is because of the large difference of the imaginary parts of eigenfrequencies of the two surface states on boundaries 1 and 2. For ${k}_{z}=-0.5\pi /L$, the imaginary parts of the eigenfrequencies of the surface states on boundary 1 are much smaller than that on boundary 2. Therefore, the incident wave propagates and slowly attenuates on boundary 1, and when it encounters the hinge, it is concentrated at the hinge without transmitting to boundary 2. For comparison, in the Hermitian case, the wave can propagate from boundary 1 to boundary 2, as shown in Fig. 3c, which is also interpreted by the FA surface states in Fig. 2f. The attenuation of the wave amplitude during the propagation is due to the global air loss in the real structure. Similarly, becaused of the reciprocity of the system, if the two point sources with a phase difference of $-0.5\pi$ are placed on boundary 4, the propagating wave can also be concentrated at the hinge intersecting by boundaries 3 and 4 (See Supplementary Note 3).

Fig. 3: Second-order hinge skin effect.

a A parallelogrammic sample. b, c Measured pressure field distributions of the non-Hermitian and Hermitian cases at the frequency of $4.75\,{{\rm{kHz}}}$. Two point sources with a phase difference of $0.5\pi$ are placed at the positions marked as red stars in (a). d, e Measured pressure fields in the outmost unit cell of the hinge states in the non-Hermitian and Hermitian cases at the frequency of $4.75\,{{\rm{kHz}}}$ for ${k}_{z}=-0.5\pi /L$.

To confirm the hinge states experimentally, a headphone is placed randomly at multiple positions on the bottom (or top) of the sample for excitation, and a microphone is applied to probe the pressure fields on the boundaries. Through Fourier transforming a series of the measured fields and summing them together, the field distributions for a fixed ${k}_{z}$ can be obtained. The detailed experimental measurement is described in Methods. Here, only the pressure fields in the outmost unit cell on the boundary are detected. The pressure field for ${k}_{z}=-0.5\pi /L$ is shown in Fig. 3d, concentrated in the lower left corner, and almost no fields are distributed in the other three corners. Note that the hinge states in the upper right corner exist at ${k}_{z}=0.5\pi /L$. For comparison, we also measure the system in the Hermitian case, as illustrated in Fig. 3e. The pressure field is localized at both the upper right and lower left corners, and no fields are distributed in the other two corners. All the measured results agree well with the simulated ones (See Supplementary Note 3).

Conclusions

In conclusion, we have experimentally probed the PGs from the FA surface states with distinct lifetimes on two non-equivalent zigzag boundaries of a non-Hermitian Weyl PC. Various SEs emerge in different boundary configurations. For instance, by changing ${k}_{z}$, the hybrid SE appears in both the rectangular prism and parallelogram prism structures, while GDSE only exists in the rectangular prism structure and not in the parallelogram one (see Supplementary Note 4). Furthermore, at the hinge formed by two zigzag boundaries, the second-order hinge SE is observed unambiguously. Our work provides a flexible platform to explore non-Hermitian physics in higher dimensions, especially the probe recipe of PGs, and certainly can be generalized to other waves and even electron systems. Due to the crucial role of GBZ in investigating non-Hermitian topology, our system will promote the exploration and even identification of GBZ in 3D systems together with other experimental techniques. In addition, the skin effect in Weyl semimetal will inspire research on skin effects in other semimetals, such as Dirac semimetal and nodal ring semimetal, and trigger studies into the evolution of various skin effects.

Methods

Non-Bloch properties of the non-Hermitian TBM

To reveal the non-Bloch properties of the non-Hermitian TBM defined by Eq. 1 in the main text, we first demonstrate that it supports topological surface states only rather than skin effects when the OBCs (PBCs) are applied in the $y$-direction ($x$- and $z$-directions). In other words, the zigzag boundary only supports non-Hermitian FAs but not the bulk NHSEs. We then finalize the non-Bloch property by proving that the 3D GBZs for the TBM in Eq. 1 are the same as its BZ.

As shown in Fig. 4a, with ${k}_{x}=0.5\pi$ and ${k}_{z}=0.5\pi$, the eigenfrequencies under the 3D-PBC configuration ${E}^{{{\rm{P}}}}({k}_{x},{k}_{y},{k}_{z})$ exhibiting two blue arcs enclose zero areas and overlap the eigenfrequencies under $y$-OBC configuration (the black markers), denoted ${E}^{y-{\mbox{O}}}\left({k}_{x},{k}_{z}\right)$. The blue spectral arc, ensuring a zero eigenvalue winding number, directly implies the absence of NHSE. Meanwhile, two surface states labeled by cyan and red markers in Fig. 4a survive with their mode profiles shown in Fig. 4b. Figure 4c further confirms the conclusion that ${E}^{{{\rm{P}}}}$ overlaps ${E}^{y-{\mbox{O}}}$ except for the surface states. Considering different ${k}_{z}$, two topological surface bands are clearly seen within the bulk ${L}_{r}$ gap (Fig. 4d). It indicates that the hinge state shown in Fig. 1i, supported by the system with a parallelogram geometry formed by zigzag boundaries under the $z$-PBC and ${xy}$-OBC configuration, arises from the surface states shown in Fig. 4d.

Fig. 4: Non-Hermitian bulk and surface bands of the zigzag boundary.

a–c Calculated eigenfrequencies under the configuration with the 3D periodic boundary conditions (PBCs) (${E}^{{{\rm{P}}}}$) and the $y$-open boundary condition (OBC) and ${xz}$-PBCs (${E}^{y-{\mbox{O}}}$) by fixing ${k}_{z}=0.5\pi$. The gray area in (a) and (c) represents ${E}^{{{\rm{P}}}}$ for ${k}_{y}\in \left[-\pi ,\pi \right]$ and ${k}_{x}\in \left[-\pi ,\pi \right]$, while the solid blue arcs in (a) highlight ${E}^{{{\rm{P}}}}$ by further fixing ${k}_{x}=0.5\pi$. The black (cyan and red) markers in (a) are the bulk (surface) states in ${E}^{y-{\mbox{O}}}$ with ${k}_{x}=0.5\pi$. The spatial distributions of two surface states are shown in (b). The blue markers in (c) represent ${E}^{y-{\mbox{O}}}$ for ${k}_{x}\in \left[-\pi ,\pi \right]$. d ${\mbox{Re}}\left({E}^{y-{\mbox{O}}}\right)$ as functions of ${k}_{x}$ and ${k}_{z}$ with the color displaying ${{\rm{Im}}}\left({E}^{y-{\mbox{O}}}\right)$. A bulk ${L}_{r}$ gap is seen when ${k}_{z}\in \left[-0.85\pi ,-0.15\pi \right]\cup \left[0.15\pi ,0.85\pi \right]$, wherein two topological surface states survive but with separate lifetimes $\tau$.

Figure 1g shows the EFC of the zigzag boundary at Re $E=0$ (the purple plane in Fig. 4d). We have generalized EFC defined in non-Hermitian bulk bands ${{\mathbb{K}}}_{{\mbox{B}}}=\left\{{{\boldsymbol{k}}}\in {{\rm{bulk}}}{{\rm{BZ}}}{{\boldsymbol{|}}}{\mathrm{Re}}[{E}_{i}({{\boldsymbol{k}}})]{{\boldsymbol{=}}}E\right\}$ to the surface bands ${{\mathbb{K}}}_{{\mbox{S}}}$ in the main text. When ${k}_{z}$ scans, both the surface states on the FAs and bulk states can become localized with the mode profile dependent on ${k}_{z}$. The calculated eigenmodes at two selected ${k}_{z}$ exhibit hybrid SE (second row) and GDSE (third row), respectively, under different 2D geometry, as shown in Fig. 5. This is because the states on the EFC ${{\mathbb{K}}}_{{\mbox{S}}}$ at ${k}_{z}=0.1{{\rm{\pi }}}$ and ${k}_{z}=0.25{{\rm{\pi }}}$ are bulk and surface states. Figure 5 thus verifies the prescription based on the generalized EFC.

Fig. 5: Various skin effects under different boundary configurations.

a–c Calculated mode profiles (second and third rows, shown in magenta) under different boundary configurations (first row) at two selected ${k}_{z}$ (labeled on the left) illustrate hybrid skin effect (SE) (second row) and geometry-dependent skin effect (GDSE) (third row), respectively. In the first row, the yellow and blue lines represent two types of zigzag boundaries, the orange line represents the armchair boundary, and the gray line represents an arbitrary boundary.

We now handle the 3D GBZ. Since GBZ needs to perform an analytic continuation to the Hamiltonian, we first define the complex wavenumber ${k}^{{\prime} }=k-i\mu$ with $k$ and $\mu$ being real numbers. We then prove that the 3D GBZ of the TBM model here is the same as its BZ. There are two equivalent statements to this conclusion: (1) ${\sigma }^{U}={\sigma }^{P}$, where ${\sigma }^{U}$ is the uniform OBC spectrum for any regular lattice geometry and ${\sigma }^{P}$ is the PBC spectrum; (2) The location of the minimum of the Ronkin function, which precisely determines the decay factor of an OBC eigenstate, is fixed at $({\mu }_{x},{\mu }_{y},{\mu }_{z})=({\mathrm{0,0,0}})$, where ${\mu }_{x,y,z}=-{{\rm{Im}}}({k^{\prime} }_{x,y,z})$^27,42$.$ Physically speaking, it indicates the extended characteristics of all OBC bulk states. It is worth pointing out that the ameba theory defines the higher-dimensional GBZ as the thermal dynamical limit of the non-Hermitian system by averaging out all the finite-sized details, including geometric shapes and boundaries. Therefore, the GBZ by the ameba theory can only tell the OBC spectra and the ensuing line gaps, and the other details, such as the FAs concerned here, shall be investigated individually.

To illustrate the 3D uniform OBC spectrum ${\sigma }^{U}$ of the model, which shall be independent of any given boundary shapes, we consider the parallelogram geometry (Fig. 6a) as the experimental sample shown in Fig. 3a and add imaginary random local perturbations at the boundary sites to eliminate any favorable wavenumbers existing (Fig. 6b). It clearly shows that as the perturbation $w$ increases, ${\sigma }^{O}(w)$ extends to fill the PBC spectrum area ${\sigma }^{P}$. This is a preliminary numerical verification of the statement (1).

Fig. 6: Confirmation of 3D generalized Brillouin zone by spectra, ameba, and Ronkin function.

a Schematics of the geometry of the tight-binding model identical to the experimental sample shown in Fig. 3a. b The gray area represents the 3D periodic boundary condition spectra ${\sigma }^{P}$. The discrete dots are the calculated eigenfrequencies from diagonalizing of the real-space Hamiltonian shown in (a) with an imaginary on-site random potential distributed uniformly in $[-w,w]$ added at each boundary site. The orange, blue, and green markers correspond to $w=0$, $w=0.2$, and $w=0.3$, respectively. c, e Ronkin functions ${R}_{f}\left({{\boldsymbol{\mu }}}\right)$ on different slices of the 3D parameter space (${\mu }_{x},{\mu }_{y},{\mu }_{z}$) taken at ${E}_{1}\notin \,{\sigma }^{U}$ (c) and ${E}_{2}\in \,{\sigma }^{U}$ (e) labeled in (b). The solid lines (dashed lines) are the outer (inner) boundaries of the corresponding amebas ${A}_{f}$, and the blue single dot in (e) denotes ${{{\boldsymbol{\mu }}}}_{\min }({E}_{2})$. d, f The 3D ameba taken at ${E}_{1}$ and ${E}_{2}$.

An analytical proof is proposed as follows. We first give a brief introduction to the ameba method and the Ronkin function herein. Consider a characteristic polynomial as

$$f\left(E,{{\boldsymbol{\beta }}}\right)=\det \left[E-H({{\boldsymbol{\beta }}})\right],$$

(2)

where ${{\boldsymbol{\beta }}}{{\boldsymbol{=}}}\left({\beta }_{1},{\beta }_{2},\ldots ,{\beta }_{d}\right)$ and $d$ is the spatial dimension with ${\beta }_{i}={e}^{i{k}_{i}^{{\prime} }}={e}^{{\mu }_{i}}{e}^{i{k}_{i}}$ and $i={\mathrm{1,2}},\ldots ,d$. For a given energy $E$, the log-moduli of the zero locus of $f$ form a shape known as ameba

$${A}_{f}\left(E\right)=\left\{{{\boldsymbol{\mu }}}=\log \left|{{\boldsymbol{\beta }}}\right|:f\left(E,{{\boldsymbol{\beta }}}\right)=0\right\}\subset {{\mathbb{R}}}^{d},$$

(3)

in which ${{\boldsymbol{\mu }}}{{\boldsymbol{=}}}\left({\mu }_{1},{\mu }_{2},\ldots ,{\mu }_{d}\right)=\log \left|{{\boldsymbol{\beta }}}\right|{{\boldsymbol{=}}}\left(\log \left|{\beta }_{1}\right|,\log |{\beta }_{2}|,\ldots ,\log |{\beta }_{d}|\right)$. The associated analytic tool in the study of ameba, namely the Ronkin function, is introduced

$${R}_{f} \left({{\boldsymbol{\mu }}}\right)= \int {\left(\frac{{dk}}{2 \pi }\right)}^{d} \log \left|f \left(E,{{\boldsymbol{\beta }}} \right)\right|= \int {\left(\frac{{dk}}{2\pi }\right)}^{d} \log |f(E,{e}^{{{\boldsymbol{\mu }}}+i{{\boldsymbol{k}}}})|,$$

(4)

where $f\left(E,{e}^{{{\boldsymbol{\mu }}}+i{{\boldsymbol{k}}}}\right){{\boldsymbol{=}}}f(E,{e}^{{\mu }_{1}+i{k}_{1}},{e}^{{\mu }_{2}+i{k}_{2}},{\ldots },{e}^{{\mu }_{d}+i{k}_{d}})$ and ${\left(\frac{{dk}}{2\pi }\right)}^{d}=\frac{d{k}_{1}}{2\pi }\ldots \frac{d{k}_{d}}{2\pi }$ indicating that the domain of integration is the d-dimensional torus ${T}^{d}={\left[0,2\pi \right]}^{d}$. Reference ²⁷ proves that the absence (presence) of a hole in the ameba of the characteristic polynomial is an indicator of the given energy $E$ being (being not) in the uniform OBC spectrum ${\sigma }^{U}$. The statement can be rephrased in terms of the Ronkin function: If and only if the minimum of the Ronkin function is reached at a single point ${{{\boldsymbol{\mu }}}}_{\min }(E)$ instead of in a hole with nonzero size, will $E$ be in the ${\sigma }^{U}$. The GBZ condition in higher dimensions becomes $\log \left|{{\boldsymbol{\beta }}}\right|={{{\boldsymbol{\mu }}}}_{\min }(E)$, which reverts to $\left|{\beta }_{M}\right|=\left|{\beta }_{M+1}\right|$ in 1D.

In our case, we rewrite the Hamiltonian defined in Eq. 1 as

$$H\left({\beta }_{x},{\beta }_{y},{\beta }_{z}\right)=\left(\begin{array}{cc}-i\gamma +{t}_{z}{f}_{z}\left({\beta }_{x},{\beta }_{y},{\beta }_{z}\right) & {t}_{0}{f}_{0}\left({\beta }_{x},{\beta }_{y}\right)\\ {t}_{0}{f}_{0}\left({\beta }_{x},{\beta }_{y}^{-1}\right) & {t}_{z}{f}_{z}\left({\beta }_{x},{\beta }_{y},{\beta }_{z}^{-1}\right)\end{array}\right),$$

(5)

where ${f}_{z}({\beta }_{x},{\beta }_{y},{\beta }_{z})=({\beta }_{y}^{3}+{\beta }_{y}^{-3})({\beta }_{y}{\beta }_{z}+{\beta }_{y}^{-1}{\beta }_{z}^{-1})+{\beta }_{x}^{-2}{\beta }_{z}+{\beta }_{x}^{2}{\beta }_{z}^{-1}$ and ${f}_{0}({\beta }_{x},{\beta }_{y})={\beta }_{y}^{-2}+{\beta }_{y}({\beta }_{x}+{\beta }_{x}^{-1})$ with ${\beta }_{x}={{e}^{{\mu }_{x}/2}e}^{i{k}_{x}/2}$, ${\beta }_{y}={{e}^{{\mu }_{y}/2\sqrt{3}}e}^{i{k}_{y}/2\sqrt{3}}$, and ${\beta }_{z}={{e}^{{\mu }_{z}}e}^{i{k}_{z}}$. For given ${E}_{1}$ and ${E}_{2}$ shown in Fig. 6b, the corresponding Ronkin functions ${R}_{f}\left({{\boldsymbol{\mu }}}\right)$ and amebas ${A}_{f}\left(E\right)$ are shown in Fig. 6c, e, respectively. For ${E}_{1}\notin {\sigma }^{U}$, a hole (blue dashed line in Fig. 6c) exists, as shown in Fig. 6d. Conversely, for ${E}_{2}\in {\sigma }^{U}$, the hole disappears (Fig. 6f), and the Ronkin function reaches its minimum at a single point ${{{\boldsymbol{\mu }}}}_{\min }(E)$ marked by the blue dots in Fig. 6e. We can get ${{{\boldsymbol{\mu }}}}_{\min }{{\boldsymbol{=}}}(0,0,0)$, indicating that the GBZ of this model is the same as its BZ.

A concise and intuitive proof can be stated as follows. The Ronkin function is always convex, guaranteeing that when the central hole exists, the minimum is reached on the central hole; otherwise, the minimum is reached at a single point in the ameba. Thus, the symmetry analysis of the model can simplify the task. As the system is reciprocal, i.e., $H\left({{\boldsymbol{\beta }}}\right)={H}^{T}\left({{{\boldsymbol{\beta }}}}^{-1}\right)$, leading to the ${C}_{2}$-symmetry of Ronkin function, i.e., ${R}_{f}\left({{\boldsymbol{\mu }}}\right)={R}_{f}\left({{\boldsymbol{-}}}{{\boldsymbol{\mu }}}\right)$. This symmetry, together with the convex property, guarantees that if the minimum is reached at a single point, then it must locus on the origin $({\mathrm{0,0,0}})$, i.e., ${{\rm{GBZ}}}={{\rm{BZ}}}$.

Probe of point gaps from the Green’s function perspective

We illustrate here that the spectral function $A\left(E,{{\boldsymbol{k}}}\right)=-{\mbox{ImTr}}\left[1/\left(E-H\left({{\boldsymbol{k}}}\right)\right)\right]$ can differentiate the spectral arcs and loops, namely, identify the PGs. For simplicity, we deliberately design a spectral arc and loop below the real frequency axis shown in Fig. 7a–d, respectively. We use ${E}_{B}{{\boldsymbol{=}}}0{\mathbb{\in }}{\mathbb{R}}$ as the given excitation frequency, and $A\left(E={E}_{B},{{\boldsymbol{k}}}\right)$ reaches its maximum at $\{{k}_{1},{k}_{2}\}$, which lies on the EFC of bulk bands ${{\mathbb{K}}}_{{\mbox{B}}}$ for $E={E}_{B}{{\boldsymbol{=}}}0$. The full width at half maximum of $A\left(E={E}_{B},{{\boldsymbol{k}}}\right)$ reflects the corresponding lifetime $\tau \propto {\mbox{Im}}{\left({E}^{P}\right)}^{-1}$. Since the spectral loops enclose a nonzero area as shown in Fig. 7b, the wavevectors $\left\{{k}_{1},{k}_{2}\right\}$ are mapped to different complex frequencies possessing different lifetimes, leading to the asymmetry of the spectral function shown in Fig. 7d. Thus, we can utilize this method to identify the spectral loops, i.e., PGs formed by two surface states of the non-Hermitian Weyl TBM, as shown in Fig. 7e–g.

Fig. 7: Spectral function A(E,k) identifying point gaps in the complex frequency plane.

The 1D periodic boundary condition spectrum with the model Hamiltonian $H\left(k\right)=-i+{e}^{{ik}}+t{e}^{-{ik}}$ for $t = 1.0$ and $t = 1.2$, respectively in (a, b). c, d The corresponding spectral function $A\left(E={E}_{B},{{\boldsymbol{k}}}\right)$ as a function of $k$ for the Hamiltonicans in (a, b), respectively. The filled stars, open circles, and open triangles denote the position of ${E}_{B}$, ${k}_{1}$, and ${k}_{2}$. e The $y$-open boundary condition band structure for the Hamiltonian defined in Eq. 1 with ${k}_{z}=0.5\pi$. f The corresponding spectral function $A\left(E,{{\boldsymbol{k}}}\right)$ in the $E-{k}_{x}$ plane. g The spectral function $A\left(E={E}_{S},{{\boldsymbol{k}}}\right)$ at the white dashed line as a function of ${k}_{x}$. The red and cyan dashed lines (circle and triangle) highlight the position of ${k}_{x1}$ and ${k}_{x2}$.

Experimtanal measurements

In this section, we will describe in detail the experimental measurements of Figs. 2, 3 in the main text. The experimental setup is schematically plotted in Fig. 8a. A headphone of a diameter $5\,{{\rm{mm}}}$ is applied to excite the bulk and surface states. A microphone of a diameter $3\,{{\rm{mm}}}$ is fixed on a 3D automatic scanning stage to collect the pressure fields. To measure the surface state dispersions and the Fermi-arc surface states in Fig. 2, a cuboid sample with a size of $15\times 6\times 11$ unit cells was fabricated, as shown in Fig. 8b. The headphone is placed at multiple positions in the next-nearest outmost unit cell of the middle layer to excite the surface states, as marked by the red star in the red area in Fig. 8b. The pressure fields on the front boundary are probed by the microphone, as illustrated in Fig. 8c. The amplitude and phase information of the pressure fields are recorded by the network analyzer (Keysight 5061B). Then Fourier transform is performed on the measured fields to obtain the surface state dispersions and FA surface states. The surface state dispersions of the back boundary are obtained by the same process. To observe the abnormal wave propagation of the surface states, a right parallelogrammic sample with a size of $10\times 10\times 11$ unit cells was fabricated, as shown in Fig. 8d. Two point sources, marked by two red stars in Fig. 8e, with a phase difference $0.5\pi$ is put in two adjacent layers along the $z$ direction. The pressure fields are also collected by the microphone as shown in Fig. 3b, c of the main text. To confirm the hinge states for a certain ${k}_{z}$, the right parallelogrammic sample is also used. The headphone is placed randomly at multiple positions in the top area of the sample (20 red circles in Fig. 8f), the microphone is applied to probe the pressure fields on the boundaries. Taking 16 measurements, then Fourier transforming these measured data and summing them together, the field distributions for ${k}_{z} < 0$ can be obtained, as show in Fig. 3d, e of the main text. The field distributions for ${k}_{z} > 0$ can be obtained by placing the headphone in the bottom area of the sample.

Fig. 8: Experimental measurement.

a Schematic of the experimental setup. The acoustic bulk and surface states are excited by a headphone. A microphone is fixed on a 3D automatic scanning stage to collect the pressure fields by the network analyzer. b A cuboid sample for measuring the surface state dispersions and Fermi-arc surface states. c The pressure field on the front boundary probed by the microphone. d A right parallelogrammic sample for measuring the abnormal wave propagation and hinge states. e, f The schematics of the experimental setup corresponding to (d).