Introduction

The hallmark of topological physics1,2,3,4,5,6,7,8 is the robust boundary mode protected by the non-trivial bulk gap—known as the bulk-boundary correspondence. As more and more such topological modes have been discovered at various locations in diverse lattice types—such as kinks, edges, surfaces, corners, hinges, vortices, disclinations, and dislocations, there is a growing consensus that a general correspondence exists between the bulk and the topological defects9,10,11,12, where certain order parameters of the lattice change discontinuously. However, among all possible topological defect states up to three dimensions (Fig. 1a), the monopole state remains undiscovered13.

Fig. 1: Bulk-topological-defect correspondence and the zero-mode trilogy.
figure 1

a Summary of the topological modes at topological defects (marked as red points, lines, and plane), where the Dirac mass (with components mx, my, mz) has an undefined singularity in various dimensions. The blue arrows represent the phase (or sign) of the mass terms in the bulk, on the surface, or on the edge. b Zero-mode trilogy. σi are the Pauli matrices, γij = σi⊗σj are the gamma matrices, and Γijk = σi ⊗ σj ⊗ σk, where ⊗ is the Kronecker product. SSH Su–Schrieffer–Heeger18, HCM Hou–Chamon–Mudry28, DFB distributed feedback56, VCSEL vertical-cavity surface-emitting laser57, TCSEL topological-cavity surface-emitting laser40. The scaling behavior between the mode spacing (FSR) and mode volume (V) is listed.

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In topological band theory, the order parameters defining the topological defects can be understood as the mass terms14,15, the scalar or vector parameters that gap out the band degeneracies known as the Dirac points, whose low-energy theory is described by the celebrated Dirac equation. The topological defects of Dirac masses, in one, two, and three dimensions (3D), were first studied by Jackiw, Rebbi, and Rossi16,17, who derived the zero-energy modes in kink, vortex, and monopole mass settings (Fig. 1b). The kink solution16, of one mass term, has been realized in polyacetylene as in the Su–Schrieffer–Heeger model18 in 1D, and on the edge and surface of 2D and 3D topological insulators. The vortex solution17, of two mass terms, has found connections to the Majorana modes in 2D superconductors19, the dislocation line states20,21 in 3D, and has been recently demonstrated in phononic22,23,24 and photonic crystals25,26,27 based on the Kekulé-distorted graphene model28.

However, regarding the monopole solution, of full three mass terms in the dimension of our real world where the Dirac equation was originally written, no progress has been made towards its physical discovery since the analytical result was obtained in 197616. In this work, we theoretically propose and experimentally demonstrate this monopole topological mode in a Dirac crystal with hedgehog real-space modulations. We further show that the monopole mode has the optimal mode spacing in the limit of large mode volume.

Results

Jackiw–Rebbi monopole mode

After the proposals of the Dirac monopole in Maxwell’s equations29 and the ’t Hooft–Polyakov monopole in Yang–Mills equations30,31, Jackiw and Rebbi found a zero-mode solution bound to the magnetic monopole in the Dirac equation16. The monopole zero mode arises at a 3D topological point defect of three independent mass terms in an 8-by-8 extended Dirac equation shown in Fig. 1b. There has to be a minimal of eight bands to allow three mass terms for the 3D hedgehog construction without breaking the time-reversal symmetry.

We first generalize the Jackiw–Rebbi single-monopole mass profile into the multi-monopole case of arbitrary charges. Although the mass distribution can be chosen to be spherically symmetric for a single monopole, it cannot be for monopoles of charge >132. Our choice of the spatial mapping of the Dirac-mass vectors \(\hat{{{\bf{m}}}}({{\bf{r}}})=(\hat{{m}_{x}}({{\bf{r}}}),\hat{{m}_{y}}({{\bf{r}}}),\hat{{m}_{z}}({{\bf{r}}}))\) is expressed as

$$\hat{{m}_{x}}({{\bf{r}}}) =\sin ({W}_{\theta }\theta )\cos ({W}_{\phi }\phi )\\ \hat{{m}_{y}}({{\bf{r}}}) =| \sin ({W}_{\theta }\theta )| \sin ({W}_{\phi }\phi )\\ \hat{{m}_{z}}({{\bf{r}}}) =\cos ({W}_{\theta }\theta )\\ W =\frac{1}{8\pi }{\int}_{A}{\rm {d}}A{\epsilon }^{\theta \phi }{\epsilon }^{xyz}\hat{{m}_{x}}{\partial }_{\theta }\hat{{m}_{y}}{\partial }_{\phi }\hat{{m}_{z}}={W}_{\theta }{W}_{\phi }$$
(1)

where θ and ϕ are the polar and azimuthal angles in the spherical coordinate; Wθ and Wϕ are the corresponding winding numbers. W is the monopole charge, measuring how many times the three-component Dirac-mass vectors wrap around a closed surface (A) enclosing the monopole. The absolute value in \(\hat{{m}_{y}}\) ensured that the 3D wrapping number (W) is expressed by the product of the two angular winding numbers. If we use the standard parametrization without the absolute value33, \(W=\frac{1-{(-1)}^{{W}_{\theta }}}{2}{W}_{\phi }\) (see Supplementary Note I for comparison). The reason is that the polar angle θ ∈ [0, π] generates alternating signs of mz that unwind each other, while the absolute sign on either mx or my reverses the sign of the in-plane winding wherever mz unwinds in the polar direction.

In the band-theory language, the above Dirac theory works at the vicinities of the Dirac points in the band structures. The 8-by-8 Dirac equation can be constructed by coupling the two 4-by-4 3D Dirac points. The three mass terms mean three independent ways or perturbations to open the bandgap. In condensed matter, the same low-energy Hamiltonian describes the 3D Majorana zero modes34,35, although there has been no clue where to find them. In topological-defect classifications10, the kink, vortex, and monopole zero modes in 1D, 2D, and 3D all belong to the same Altland–Zirnbauer symmetry class BDI with integer invariant (\({\mathbb{Z}}\)) under time-reversal and chiral symmetries. Of course, the chiral symmetry is not rigorously present in our acoustic system, since the spectrum is not exactly up-down symmetric with respect to the Dirac frequency.

Double 3D Dirac point

The starting point of our design is a crystal hosting two 3D Dirac points to form an 8-by-8 Dirac theory. Our Dirac acoustic crystal is defined by the 3D periodic implicit functions

$$\begin{array}{rlr}{f}_{{\rm {D}}}({{\bf{r}}})&=\sum\limits_{{{\rm{cyc}}}}\sin (\frac{4\pi }{a}x)\cos (\frac{2\pi }{a}y)\sin (\frac{2\pi }{a}z)&\\ &+3\sum\limits_{{{\rm{cyc}}}}\cos (\frac{4\pi }{a}x)\cos (\frac{4\pi }{a}y),\end{array}$$
(2)

where a is the lattice constant and \({\sum}_{{{\rm{cyc}}}}\) is the sum over the cyclic permutation of (x, y, z). The volume of fD(r) ≥ −1.56 is filled with sound-hard material like plastic, and the rest of the volume is air. This lattice geometry of space group \(Ia\bar{3}d\) (#230) is improved upon a previous blue-phase-I (BPI) structure36,37,38 to have two fully frequency-isolated Dirac points, locating at the high-symmetry momenta (±P) in the Brillouin zone of the body-centered-cubic (BCC) primitive cell (Supplementary Note II). By taking the simple-cubic super-cell as shown in Fig. 2, two Dirac points fold to the momentum R point of the simple-cubic Brillouin zone, forming an 8-by-8 double Dirac point.

Fig. 2: Acoustic realization of the double Dirac point and the three mass terms for the monopole design.
figure 2

a The ideal Dirac acoustic crystal with the 8-band Dirac point in the super-cell. b, d, f The perturbed geometries defined by fD(r) + δfi(r), (i = x, y, z), whose band structures are plotted in (c) and (e). The rainbow-colored arrows represent the hedgehog form of the Dirac mass vector (mx, my, mz).

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Here we explain a general approach to construct photonic and phononic crystals of a specific space group G using triply periodic functions FG(r), like the one in Eq. (2). This is done through a generic Fourier expansion39\({F}_{G}({{\bf{r}}})={\sum }_{i}[{A}_{i}\cos ({{{\bf{k}}}}_{i}\cdot {{\bf{r}}})+{B}_{i}\sin ({{{\bf{k}}}}_{i}\cdot {{\bf{r}}})]\) with the real coefficients Ai and Bi, satisfying FG(r) = FG(gr) for each space-group element g ∈ G. The reciprocal lattice vectors \({{\bf{k}}}=(l\frac{2\pi }{a},m\frac{2\pi }{b},n\frac{2\pi }{c})\) are defined by the lattice constant a, b, c, and integers l, m, n. One usually takes the lowest l, m, and n values for the simplest structures. We use this method to obtain the Dirac structure of Eq. (2) (detailed in Supplementary Note III), as well as the symmetry-breaking perturbations in Eq. (3) to engineer the Dirac masses.

Dirac-mass engineering

In this section, we present the physical realizations of the mass terms for the double 3D Dirac point through symmetry arguments and space group relations. Each mass term gaps the Dirac degeneracy, by breaking the specific symmetries and reducing the #230 space group to its subgroup. By identifying the corresponding subgroup for each mass term, we can construct the structural perturbations of each subgroup using the periodic-function method described in the previous section.

The total number of four mass terms (mx, my, mz, \(m^{\prime}\)) and their symmetry properties are derived from the low-energy (k ⋅ p) theory of the eight-band Dirac point, given in Fig. 1b and detailed in Supplementary Note IV. All four masses are invariant under time-reversal symmetry, while only mx is invariant under inversion symmetry (centrosymmetric). The first two masses (mx, my) break the BCC primitive translations and expand the BCC primitive cell to the cubic supercell; they couple the two four-band Dirac points to open a gap and are thus called the “super-cell masses”. The latter two masses (mz, \(m^{\prime}\)) break the inversion symmetry to gap the two Dirac points in the BCC primitive cell; they are thus called the “primitive-cell masses”.

The two primitive-cell masses (mz, \(m^{\prime}\)) can be realized in the subgroups #220 and #214, respectively. These are the only two BCC subgroups that remain when inversion is removed from #230. They represent two independent ways to break inversion and correspond to the two different masses. Further details on the mass-subgroup relations can be found in Supplementary Note V.

The two super-cell masses (mx, my) can both be realized in the same subgroup #205, the only centrosymmetric subgroup of #230 that remains when the primitive translations are removed. Physically, this implies that the two distinct super-cell masses can be realized by identical structures that are symmetry-related. To understand this, we note that though both masses have inversions in #205, they do not share the same inversion center. This is consistent with the symmetry analysis, which shows that only one super-cell mass, mx, is invariant under inversion. These two inversion centers, (0,0,0) and \((\frac{1}{4},\frac{1}{4},\frac{1}{4})\), are originally related by the symmetry operations in #230 but absent in #205. One such symmetry is the glide translation \(\{{M}_{01\bar{1}}| (\frac{1}{4},\frac{1}{4},\frac{1}{4})\}\), where \({M}_{01\bar{1}}\) is a mirror, and squaring the glide is the BCC primitive translation. Correspondingly, mx and my can also be transformed into each other under the glide that translates between the inversion centers. The fact that super-cell masses can be related under fractional lattice translations also appears in the previous low-dimensional examples, such as the SSH model18, Dirac-vortex cavity27, and one-way fiber21.

After identifying the subgroups (#205, #205, #220, #214) for the four Dirac masses (mx, my, mz, \(m^{\prime}\)), we construct the geometric perturbations (δfx, δfy, δfz, \(\delta f^{\prime}\)) within each subgroup. When these perturbations are added to the original #230 structure fD, they gap the Dirac points. The method for finding these perturbation functions within specific subgroups is identical to that used in finding fD(r) in Eq. (2).

The functional forms of the first three perturbations are listed in Eq. (3), in which the coefficients (amplitudes) are chosen to maximize the size of the common gap as plotted in Fig. 2. It is easy to check that δfx(r) and δfy(r) are related by the glide symmetry and share the same band structure in Fig. 2e, ensuring the identical bandgap frequencies and size. Since \(\delta f^{\prime}\) (in Supplementary Note VI) opens the smallest bandgap (shown in Supplementary Note VII) compared with the other three masses, we do not include it for the cavity construction.

$$\begin{array}{rlr}\delta {f}_{x}({{\bf{r}}})&=0.60\sum\limits_{{{\rm{cyc}}}}\cos (\frac{4\pi }{a}x)\sin (\frac{4\pi }{a}y)\sin (\frac{2\pi }{a}z)&\\ \delta {f}_{y}({{\bf{r}}})&=0.60\sum\limits_{{{\rm{cyc}}}}\sin (\frac{4\pi }{a}x)\cos (\frac{4\pi }{a}y)\cos (\frac{2\pi }{a}z)\\ \delta {f}_{z}({{\bf{r}}})&=0.36\sum\limits_{{{\rm{cyc}}}}\cos (\frac{4\pi }{a}x)\sin (\frac{2\pi }{a}y)\cos (\frac{2\pi }{a}z)\end{array}$$
(3)

Monopole-cavity construction

With three independent mass realizations at hand, we construct the monopole cavity by applying the direction-dependent perturbations to the otherwise uniform 3D Dirac crystal, where (δfx, δfy, δfz) are aligned to the three spatial axes. The geometry of the monopole cavity is defined by

$${f}_{{\rm {D}}}({{\bf{r}}})+\sum\limits_{i=x,y,z}\hat{{m}_{i}}({{\bf{r}}},{W}_{\theta },{W}_{\phi })\delta {f}_{i}({{\bf{r}}})\ge -1.56$$
(4)

where the hedgehog mass vector \(\hat{{m}_{i}}({{\bf{r}}},{W}_{\theta },{W}_{\phi })\) is given in Eq. (1) with the monopole charge W = WθWϕ.

The numerical results of the single- and multi-monopole modes are presented in Fig. 3. To design the single-monopole cavity, we take Wθ = Wϕ = +1, whose mass field is illustrated in Fig. 3a. A single mid-gap mode appears in the numerical spectrum in Fig. 3b, whose frequency deviation from that of the original Dirac point is less than 1%. Plotted in Fig. 3c is the pressure field of the topological mode localized at the cavity center, with four cubic cells cladding in each direction. To design the multi-monopole cavity, we take Wθ = Wϕ = +2 with W = +4 as an example. The high-charge hedgehog field is illustrated in Fig. 3d. The numerical results in Fig. 3e, f show four nearly degenerate mid-gap modes, whose frequency differences are within 1%.

Fig. 3: Single and multi-monopole cavities.
figure 3

a–c Dirac-mass field, eigenvalue spectrum, and the modal field distribution of the single-monopole cavity. d–f Dirac-mass field, eigenvalue spectrum, and the modal field distributions of the multi-monopole cavity with charge four.

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Acoustic experiments

Encouraged by the simulation results, we 3D-print the acoustic monopole cavity with stereolithography using photo-curable resin. As shown in Fig. 4a, the sample has a total volume of (36 cm)3 and a lattice constant of a = 3 cm. This bi-continuous structure has two types of through holes, separated by 15 mm (\(\frac{a}{2}\)), whose diameters are about 1.9 and 2.8 mm allowing stainless-steel tubes to get inside the sample for spectral and spatial measurements. The sound source is an earphone whose output tube tip, a diameter of 2.4 mm, is placed near the cavity center. A micro-tube probe (B&K-4182) of diameter 1.2 mm is used to detect the sound pressure at any point in the sample. The acoustic signal is a broadband pulse of a 12.0 kHz central frequency and a 6.4 kHz bandwidth, generated and recorded by a signal analysis module (B&K-3160-A-042), the spectrum recorded in each measurement is averaged a hundred times.

Fig. 4: Experimental results on the monopole cavity.
figure 4

a Photograph of the experiment setup and a zoom-in of the sample surface. The 3D-printed cavity has a lattice constant a = 3 cm and a sample size of (12a)3. b The frequency response of the cavity shows a single mid-gap resonance. c–e Probe scans of the pressure field distributions inside the sample at three resonant peaks across the bandgap.

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The two spectra in Fig. 4b demonstrate the existence of the bandgap and the localized mid-gap mode. When the probe is off-center, the low signal response indicates a bandgap region roughly from 11 to 12.5 kHz. When the probe is centered, a resonate mode peaks in the middle of the gap at 11.76 kHz, whose quality factor is  ~73. The field profiles of the resonant modes are mapped out by scanning the probe tube inside the through holes of the sample using a motorized stage. The pressure fields are recorded, with the step size of 7.5 mm (= \(\frac{a}{4}\)), on the three orthogonal surfaces crossing the cavity center. In Fig. 4c–e, we plot the signal profiles at the frequencies of the three central resonate modes. It is obvious that only the mid-gap mode is localized as a defect state. The robustness of the topological mode is tested in Supplementary Note VIII, with over a hundred of steel sticks penetrating the cavity.

Optimal scaling behavior

The free spectral range (FSR), of a resonator cavity, is the frequency spacing between adjacent modes that shrinks as the mode volume (V) expands. A wider FSR improves the wavelength-multiplexing bandwidth, the spontaneous emission rate, and the stability of single-mode operation. Although conventional cavities have an inverse proportionality between modal spacing and modal volume (FSR ∝ V−1), the recent Dirac-vortex cavity27 exhibits FSR \(\propto {V}^{-\frac{1}{2}}\) that drops much slower than the common scaling rule and is beneficial for broad-area single-mode lasers40. Here, we show the monopole cavity offers the optimal scaling of FSR \(\propto {V}^{-\frac{1}{3}}\) (Fig. 1b). A general theoretical argument is given in Supplementary Note IX to derive the above scaling laws using the concept of density-of-states.

We demonstrate numerically, in Fig. 5, the scaling law of the monopole cavity. There are two methods to enlarge the volume of the monopole topological mode. One method is to lower the bandgap size40, which is twice the FSR for the mid-gap mode. The other method is to expand the central singular point27 (R = 0) into a volume of non-modulated Dirac lattice with a finite radius R, shown in Fig. 5b inset. As R increases, the volume of the mid-gap mode increases and the high-order cavity modes drop into the bandgap. We take this finite-R method because it requires smaller computational domains and resources. For the same mode volume, the finite-R cavity is more spatially confined than the R = 0 version, due to the non-decaying wavefunction in the gapless core and the fastest decaying tails in the cladding region by choosing the largest bandgap. The FSR \(\propto {V}^{-\frac{1}{3}}\) relation is clearly identified in the simulation results from R = 2a to 10a in Fig. 5b. Due to the size limitation of 3D-printing and the high acoustic absorption, the experimental verification of the scaling law and the multi-monopole resonances are challenging, as shown in Supplementary Note X. These challenges might be alleviated in photonic platforms.

Fig. 5: Numerical demonstration of FSR \(\propto {V}^{-\frac{1}{3}}\) scaling law of the monopole topological mode at large volumes.
figure 5

a Resonance spectra from the cavities of core radius (R) from 0 to 10a. The frequency of the monopole topological mode ω0 remains almost constant in the middle of the bandgaps. b Free spectral range versus mode volume (normalized by λ0 = 2πc/ω0) relation fitted with the various scaling powers. The inset illustrates the cavity construction with the monopole core radius R.

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The unique advantage of the monopole cavity is the robust mechanism in pinning the defect modes at the middle of the bandgap as the FSR is being tuned to zero in a well-controlled fashion. The non-topological defect modes lack the mechanism to stabilize a single mode at the center of the gap in the first place. Comparing with the topological corner modes of third-order 3D topological crystals41,42,43,44,45, the monopole design avoids sharp interfaces as well as the extra defect modes associated with the additional surfaces and hinges.

Discussion

We experimentally realize the Jackiw–Rebbi monopole mode proposed in their seminar paper from 1976, by observing the topological mode of a 3D topological point defect in a lattice. The method for constructing mass hedgehogs is applicable to other 3D Dirac and Weyl systems46,47,48,49,50. Our work completes the zero-mode trilogy, of kink-vortex-monopole, as well as the bulk-topological-defect correspondence within three spatial dimensions and within Hermitian theory. Extensions to the time dimension (such as instantons51 and time crystals), synthetic spaces52,53, as well as non-Hermitian systems can be explored in the future.

Methods

Eigen-frequency solutions

The band structures and eigen-mode solutions are calculated by COMSOL Multiphysics. The solid-air interface is set as hard-wall boundaries. The largest resonator size in computation is (8a)3, taking up the memory size of 380 GB (2666 MHz). The unwanted modes localized at the scattering boundaries can be identified from their low-quality factors.

The mesh generation for large complex structures is challenging. First, we generate the iso-surface mesh from the implicit function using the marching cubes algorithm. Second, we improve the surface meshes with MeshLab54 using a quadric-based edge collapse strategy. Finally, the improved surface mesh is imported into COMSOL for meshing the bulk with first-order free tetrahedrons. It takes a day to mesh the cavity and another day to compute the eigen-spectrum, containing 200 eigenmodes around the Dirac frequency, using two Intel Xeon Gold 6140 CPUs of 36 cores at 2.30 GHz.

Time-domain simulations

To further increase the simulation size, for obtaining the scaling law of the monopole resonator, we switch to an open-source time-domain package k-Wave55. It is based on pseudo-spectral methods and the geometries are discretized in simple cuboids (\(\Delta x=\frac{a}{32}\), a = 3 cm). The computation domain includes the cavity volume of [2(R + 4a)]3 with R from 0 to 10a, one more cell of air surrounding the cavity, and the perfectly matched layers as the boundary condition.

We have to set the solid material parameters very carefully, in order to make the large-scale computation feasible. The two material parameters involved are the mass density (\(\rho ^{\prime}\)) and the sound velocity (\(c^{\prime}\)), whose product is the sound impedance (\(\rho ^{\prime} c^{\prime}\)). First, \(\rho ^{\prime} c^{\prime} \, \gg \, {\rho }_{{\rm {air}}}{c}_{{\rm {air}}}\) approximates the hard-wall boundary condition lacked in k-Wave. Second, \(c^{\prime} \, < \, {c}_{{\rm {air}}}\) maximizes the time step due to the Courant condition (\(\Delta t < \frac{\Delta x}{\sqrt{3}{c}_{{\rm {air}}}} < \frac{\Delta x}{\sqrt{3}c^{\prime} }\)). Third, we set the bulk modulus \(\rho^{\prime} c^{\prime 2} \, \ll \, {\rho }_{{\rm {air}}}{c}_{{\rm {air}}}^{2}\) to ensure the numerical stability that we observe in k-wave. The above three conditions lead to our choices of \(c^{\prime}=1{0}^{-14}\) m/s and \(\rho ^{\prime}=1{0}^{32}\) kg/m3 for the hard material in simulation with Δt = 1.2 × 10−6 s.

To obtain the spectral response of the cavity, as in Fig. 5a, we excite the modes with a broadband pulsed source near the cavity center and hundreds of point detectors evenly distributed inside the cavity. After 105 time steps, the recorded spectra are averaged and normalized by the source spectrum. To obtain the modal volumes of the monopole topological modes, as in Fig. 5b, we excite the cavity with a narrow-band signal at the resonate frequency and apply a Blackman window with 40,000 time steps. The resulting fields are used to compute the modal volume as \(V=\frac{\iint \!\!\!\int {\rm {d}}x{\rm {d}}y{\rm {d}}z \,| p({{\bf{r}}}){| }^{2}}{{\rm {max}}[| p({{\bf{r}}}){| }^{2}]}\), where p(r) is the pressure. The time consumption for the largest cavity of R = 10a is about three weeks, including both the spectral and modal calculations, and the maximum memory in use is about 90 GB.