Monopole-mediated light control of half skyrmion topology in nematic liquid crystals

Abstract

Skyrmions, with their robust topologically protected properties, have demonstrated significant potential for applications in spintronic devices. Despite their promise, the manipulation of topological invariants within these protected structures has remained a complex challenge. In this work, we present a method to orchestrate the topological transformation of half skyrmions through monopoles, which are singular point defects endowed with nontrivial topological charges. Through experiments and simulations, we identified eight distinct types of emergent monopoles. The mutual transformation between different half skyrmions can be induced by manipulating the profiles of topologically protected monopoles with light irradiation. Furthermore, a pair of monopoles and antimonopoles exhibit both attractive and repulsive interactions, depending on the topological structure of the half skyrmion that separates them. Leveraging the dynamic characteristics of monopoles, we have effectively used them as carriers for colloidal particles. This study of topological transitions in nematic liquid crystals offers valuable insights into fundamental physical phenomena and enhances our grasp of the subtle dynamics of topological matter, potentially leading to advances in the design of smart materials and devices with novel functionalities.

Skyrmions, known for their robust topological protection, hold great promise for spintronic applications, but controlling their topological invariants has been a long-standing challenge. This work demonstrates light-induced manipulation of half skyrmions via eight identified monopole types, enabling topological transformations, tunable monopole interactions, and particle transport.

Introduction

Topological stability, often referred to as topological protection, is a characteristic of certain objects that maintain their properties under continuous deformations, which is a fundamental attribute of topological objects^1,2. This topological protection ensures that topologically stable structures cannot be easily created or destroyed, a feature that facilitates the robust transport of information carriers and opens up promising pathways for spintronics and memory device applications^3–5. As an exemplar of topological stability, skyrmions have been observed in various condensed matter systems, including magnetic solids^6–15, liquid crystals (LCs)^16–21 and Bose-Einstein condensates²². In three-dimensional (3D) systems, skyrmions manifest as string-like structures associated with a quantized flux, leading to the formation and annihilation of skyrmion strings^9,11,21,23. The sources or sinks of these emergent fluxes are recognized as emergent monopoles or antimonopoles, with skyrmion strings bridging them^23–25. Topologically, skyrmions are categorized as ${\pi }_2\left({\mathbb{S}}^2\right)$ entities^24,26–28. Distinct 2D skyrmions are identified by integers or fractions known as the “skyrmion number”, calculated by the integral $N_{sk}=\frac{1}{4\pi }\iint d^2\mathbf{rn}\cdot \left(\frac{\partial \mathbf{n}}{\partial \mathrm{x}}\times \frac{\partial \mathbf{n}}{\partial \mathrm{y}}\right)$, which measures the number of times the vector field $\mathbf{n}\left(\mathbf{r}\right)=\mathbf{n}\left(x,y\right)$ wraps the unit sphere³. Skyrmions with $N_{SK}=\pm 1$ are termed full skyrmions, while those with $\pm 1/2$ are known as half skyrmions^12,13,25,29. The topological protection of skyrmions stems from their invariant topological number, which remains unchanged even when the skyrmion undergoes deformation or movement. This property endows skyrmions with a resilience to destruction, making them stable configurations in their respective systems^25,30.

Emergent monopoles or antimonopoles, as described in the literature^{24,26,31–37}, are characterized by an integer topological charge $q=\frac{1}{4\pi }\iint d\theta d\psi \mathbf{n}\cdot \left({\partial }_{\theta }\mathbf{n}\times {\partial }_{\psi }\mathbf{n}\right)$, which, akin to the skyrmion number, determines the number of times the unit sphere is encircled by the director $\mathbf{n}\left(\mathbf{r}\right)=\mathbf{n}\left(\theta ,\psi \right)$ on any surface that encompasses the defect core^38,39. In the realm of topological spin orders, monopoles ($q=+1$) and antimonopoles ($q=-1$) serve as sources or sinks of the emergent field. Their presence in chiral ferromagnets and chiral LCs is pivotal to the transition between topologically distinct phases, often acting as nucleation points for initiating topological transitions^{23,24,40–42}. In the context of nematic LCs, the 3D topological defects^43–47, namely, monopoles^48–50 within the emergent field, are classified as ${\pi }_2\left({\mathbb{S}}^2/{\mathbb{Z}}_2\right)$ due to the property $\mathbf{n}\left(\mathbf{r}\right)\equiv -\mathbf{n}\left(\mathbf{r}\right)$²⁷. Theoretically, a hyperbolic hedgehog defect can evolve from a radial hedgehog through a series of continuous distortions of the director, passing through intermediate configurations such as the ‘circular’ configuration³⁸. This implies that radial, hyperbolic, and all intermediate hedgehogs are topologically equivalent, indicating that the charge sign of the hedgehog remains constant throughout the texture transformation process. The manipulation of these topologically protected monopoles to orchestrate the topological transitions in skyrmionics configurations presents a significant scientific and technological challenge.

In this work, we explore the manipulation of monopoles and antimonopoles within various fractional skyrmion strings²¹, such as half-anti-skyrmion (HAS), half-Néel-skyrmion (HNS), and half-Néel-bimeron (HNB) in a nematic LC with complex topological patterns. Both experiments and simulations have illustrated that an HNS can directly transform into an HAS via a monopole or antimonopole, resulting in a change in the skyrmion number. When these skyrmion strings are driven out-of-equilibrium by light actuation, the topological transitions between different skyrmion configurations can be mediated by the motion of monopoles or antimonopoles. Concurrently, the mutual transformation between distinct profiles of monopoles (or antimonopoles) can be achieved, with the topological charge remaining unchanged. Furthermore, we demonstrate tunable attraction and repulsion between monopole-antimonopole pairs by controlling the irradiation. When different geometries of the topological patterns are employed, monopole-antimonopole pairs are produced within the skyrmion loops, and their interactions are dictated by the topological nature of the skyrmion strings. Finally, by leveraging the dynamic characteristics of monopoles, we have developed various trajectories to govern their movement, which can be immediately applied to collect and transport colloidal particles. Thus, this work, based on a nematic environment with chirality induced by incompatible topological patterns, provides a route to exploring the rich interplay between topology, geometry, and material properties. It also opens opportunities to designing new tools to transport colloidal assembly.

Results

Creation of topological monopole-antimonopole pairs in half skyrmions

Topological defects, which are a necessary result of broken continuous symmetry, can be observed in the liquid crystal system employing a heating-cooling phase transition⁵¹. To create topologically protected skyrmion strings, a nematic LC slab with geometric frustration by different topological patterns is prepared. The bottom substrate is of alternating splay-bend distortions with director field designed as $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, where $\alpha =\pi x/L$, L = 100 μm is the period, and the top surface is of uniform alignment along the y-axis, (Fig. 1a and Supplementary Fig. 1, 2). As the sample thickness is $\ge$20 μm, after a nematic LC 4’-pentyl-4-cyanobiphenyl (5CB) is injected (Supplementary Fig. 1c), half skyrmion strings are formed in the splay regions. After the LC sample is heated to the isotropic phase at 38 °C, it is cooled down to the nematic phase at 25 °C at a cooling rate of 2 °C/min. Monopole ($q=+1$) and antimonopole ($q=-1$) pairs respectively are generated in the half skyrmion string (Fig. 1b and Supplementary Fig. 3a, b).

Fig. 1. Creation of topological monopole-antimonopole pairs in half skyrmions.

a Schematic of designed topological patterns. The bottom surface adopts alternating splay-bend distortions, and the top surface is along the y-axis. The half skyrmion strings are produced in the splay regions (pink). b Polarizing optical microscope (POM) image with an inserted red plate of two oppositely charged point defects in the half skyrmions. The red dashed line indicates the slow axis of a 530 nm retardation plate inserted between the polarizers. The red dotted circle denotes the monopole (q = +1), while the blue dotted circle indicates the antimonopole (q = −1). c Schematic vector field of a HAS (1/2, −1/2, 0) with far field F = 0; Red (blue) region shows that the y-component of the director is positive(negative), while the yellow region shows that the director is in-plane. d HNS (−1/2, 1/2, 0) with far field F = 0. e Simulated topological structures of a circular antihedgehog (q = −1) in the half skyrmion. f Simulated topological structures of a hyperbolic hedgehog (q = +1). g Vector field of the circular antihedgehog (q = −1). h Vector field of the hyperbolic hedgehog (q = +1). i Designed topological patterns. The bottom surface is alternating splay-bend distortions, and the top surface is along the x-axis. The half bimeron strings are produced in the bend regions (pink). j HNB (1/2, 0, −π/2) with far field F = −π/2. k HNB (−1/2, 0, −π/2) with far field F = −π/2. l Polarizing optical micrograph with an inserted red plate of two oppositely charged point defects in the half bimeron. m Simulated intermediate antihedgehog (q = −1) in the half bimeron. n Simulated intermediate hedgehog (q = +1) in the half bimeron. o Vector field of the intermediate antihedgehog (q = −1). p Vector field of the intermediate hedgehog (q = +1). All scale bars are 100 μm.

The topological configurations of the skyrmions (Supplementary Fig. 4) and monopoles (Supplementary Fig. 5–7) in our system are revealed through continuum simulations. A skyrmion (N_sk, m, F) in our model is characterized by three distinct parameters: the skyrmion number $N_{sk}$, vorticity $m$ and far field ($F$), which is defined as the angular orientation of the uniform surface relative to the y-axis. We have identified the monopole located on the left in Fig. 1b as a circular antihedgehog with a topological charge $q=-1$, as shown in Fig. 1e, g. Conversely, the monopole on the right in Fig. 1b is classified as a hyperbolic hedgehog with $q=+1$, as shown in Fig. 1f, h. The sign of the topological charge ($q=\pm 1$) is determined by applying the right-hand rule to the director field configuration. When the directors at three orthogonal surface points satisfy $\mathit{x}\times \mathit{y}=\mathit{z}$, the configuration represents a monopole ($q=+1$); conversely, $\mathit{x}\times \mathit{y}=-\mathit{z}$ indicates an antimonopole ($q=-1$)^38,52. The skyrmions associated with the monopole $q=-1$ are designated as HAS (1/2, −1/2, 0) on the left-hand-side (Fig. 1c) and HNS (−1/2, 1/2, 0) (Fig. 1d) on the right-hand-side, respectively. Mathematical definitions of these topological textures are described in the Supplementary text. On the other hand, the monopole $q=+1$ is linked to HNS (−1/2, 1/2, 0) on the left and HAS (1/2, −1/2, 0) on the right.

In our half skyrmion system, two half skyrmions are connected by monopoles, Fig. 1b–d. These half skyrmions exhibit identical polarities, denoted in blue, and share the same far field vector, represented in red. However, their vorticities, indicated in yellow, are vectors that point in opposite directions, leading to a discordance between them and the subsequent monopole formation. This opposition is a direct consequence of the boundary condition imposed by the top and bottom pattern designs. Details on skyrmion number, polarity, vorticity, and far field forms are elaborated in the Supplementary text. It is revealed that the formed monopoles in this experiment are circular (anti) hedgehog (q = ±1) and hyperbolic (anti) hedgehog (q = ±1) (Fig. 1g, h and Supplementary Fig. 6), also see Supplementary text. According to the simulation results, a circular antihedgehog with q = −1 and a hyperbolic hedgehog with q = +1 are shown in Fig. 1e, f, respectively, and Supplementary Fig. 7a–h.

Another type of skyrmion string, half bimerons, can also be created if the uniform alignment is along the x-axis (Fig. 1i–k). After the phase transition, monopole-antimonopole pairs are generated in the half bimeron (Fig. 1l and Supplementary Fig. 3c, d). The profile of the antimonopole (Fig. 1o) is different from both the circular and hyperbolic antimonopole shown in Fig. 1g and Supplementary Fig. 6c. This intermediate antihedgehog is obtained from the radial antihedgehog (q = −1) (Supplementary Fig. 5b) by rotating the director π/3 about a certain fixed axis at every point, see Fig. 1m and Supplementary Fig. 7i–l. Likewise, the intermediate monopole configuration (Fig. 1p) is obtained from the radial hedgehog (q = +1) (Supplementary Fig. 5a) by rotating the director 2π/3 about a certain fixed axis at every point, Fig. 1n and Supplementary Fig. 7m–p.

Topological transition of half skyrmions by monopole-antimonopole pair

When the far field is F = 0 (top uniform alignment along the y-axis), the skyrmion string is divided into three different half skyrmions by a monopole-antimonopole pair: HAS (1/2, 0) on both sides, and HNS (−1/2, 0) in the middle (Fig. 2a). Please note that, for convenience, the notation for skyrmions will be simplified to the form of skyrmion (N_sk, F) in the following text and figures. The monopole-antimonopole pair spontaneously move towards each other until they meet and annihilate (Fig. 2b, c and Supplementary Movie 1). During this process, the HNS (N_SK = −1/2) string with a length of 180 μm completely converts to an HAS (N_SK = 1/2) string within 41 s (Fig. 2d). This transition is driven by the pair annihilation of the circular antihedgehog (q = −1, Fig. 2b), and the hyperbolic hedgehog (q = +1, Fig. 2c), which act as the monopole and antimonopole of the emergent director field. However, when top uniform alignment along the −y-axis, F = π, the monopole-antimonopole pair repel each other and move out of the observation range (Supplementary Fig. 8a–d and Supplementary Movie 2). HNS (N_SK = ±1/2) can also be stabilized by manipulating the monopoles using optical tweezers (Supplementary Fig. 8e and Supplementary Movie 3).

Fig. 2. Topological transformation of half skyrmions by monopole-antimonopole pairs.

a Monopole-antimonopole pair formation, attraction and annihilation in a skyrmion string. The red dotted circle denotes the monopole (q = +1), while the blue dotted circle indicates the antimonopole (q = −1). b Vector field of the circular monopole with F = 0. c Hyperbolic antimonopole with F = 0. d Topological transition from HNS (−1/2,0) to HAS (1/2,0). e Monopole-antimonopole pair formation, attraction and annihilation in a bimeron string. f Intermediate antimonopole with F = −π/2. g Intermediate monopole with F = −π/2. h Topological transition from HNB (−1/2, −π/2) to HNB (1/2, −π/2). i Free energy comparison of different half skyrmions. The yellow (F = −π/2) and blue (F = π/2) frames are the enlarged parts of the energy comparison between HNBs in different far fields. j Under different far field conditions, the free energy difference ∆f between half skyrmions N_SK = +1/2 and N_SK = −1/2. Scale bars are 20 μm.

A similar phenomenon of mutual attraction and repulsion exists between the intermediate hedgehogs with opposite charges produced within the half bimeron (Fig. 2e and Supplementary Movie 4). When the uniform alignment on the top surface is along the −x-axis, F = −π/2, a half bimeron string is split into three segments by a monopole-antimonopole pair: HNB (1/2, −π/2) on both sides, and HNB (−1/2, −π/2) in the middle. The intermediate antihedgehog (q = −1) (Fig. 2f) and intermediate hedgehog (q = +1) (Fig. 2g) spontaneously annihilate, Fig. 2e. As a result, HNB (−1/2, −π/2) transforms into HNB (1/2, −π/2) (Fig. 2h). During this process, the 120 μm long HNB (N_SK = −1/2) string completely converts to an HNB (N_SK = 1/2) string within 373 s (Fig. 2e)

The simulations reveal that HNS (N_SK = ±1/2) possess higher free energy (Fig. 2i). As a result, HNS (N_SK = ±1/2) always transitions to HAS (N_SK = ±1/2) through either annihilation (Fig. 2a) or repulsion (Supplementary Fig. 8a) of monopoles and antimonopoles. Additionally, there is an energy difference between HNB (N_SK = −1/2) and HNB (N_SK = +1/2), as shown in the inset of Fig. 2i. When F = 0 or π (in the splay region), a significant energy difference between HNS and HAS, $∣\mathrm{\Delta }f∣~280k_{B}T/{\mathrm{\mu }\mathrm{m}}^3$, leads to a fast motion of the two defects (Fig. 2a, j). In contrast, when F = π/2 or −π/2 (in the bend region), the small free energy difference between the two HNB structures, $∣\mathrm{\Delta }f∣~5k_{B}T/{\mathrm{\mu }\mathrm{m}}^3$, results in a slower motion of the defects (Fig. 2e, j).

Tunable topological transformation of half skyrmions

The half skyrmion strings can be driven to translational out-of-equilibrium by irradiating the uniform alignment with linearly polarized blue light (wavelength = 455 nm, beam intensity = 510 mW/cm²)^21,53–55. Following the translated motion of half skyrmions, the mutual interaction of monopole-antimonopole can be tuned between attraction and repulsion, meanwhile inducing the topological transition of the half skyrmions. First, half bimerons are generated at locations of x = 0, 2L, …, if the top surface alignment is along the x-axis (Fig. 3a). At t = 0 s, the half bimeron is at x = 0 in the bend region. A pair of monopole and antimonopole is created in this half bimeron string, Fig. 3a, dividing a half bimeron string into three segments, with the middle part being an HNB(−1/2, −π/2), while the left and right parts are both HNB(1/2, −π/2).

Fig. 3. Tunable topological transformation of half skyrmions.

a Image sequence of the response of the monopole-antimonopole pair in the translation of half skyrmions. The white arrow indicates the translation direction. The colored trajectories of two defects are shown in the last frame at t = 383 s. Red and blue dotted circles mark the monopole (q = +1) and antimonopole (q = −1) configurations, respectively. b Topological profiles of the half skyrmion strings on the left and right sides change following the order: HNB(1/2, −π/2), HAS(1/2,0), HNB(1/2, π/2), HNS(1/2,π), and HNB(1/2, −π/2). c Topological profiles of the half skyrmion strings in the middle part follow the order: HNB(−1/2, −π/2), HNS(−1/2,0), HNB(−1/2, π/2), HAS(−1/2, π) and HNB(−1/2, −π/2). d Antimonopole configurations follow the order of an intermediate antimonopole(F = −π/2), a circular antimonopole, an intermediate antimonopole(F = π/2), a hyperbolic antimonopole and an intermediate antimonopole(F = −π/2). During this transformation process, topological charge (q = −1) remains unchanged. e Monopole configurations follow the order of an intermediate monopole(F = −π/2), a hyperbolic monopole, an intermediate monopole(F = π/2), a circular monopole and an intermediate monopole(F = −π/2). During this transformation process, topological charge (q = +1) remains unchanged. f The pair of oppositely charged defects is enabled to annihilate by controlling the irradiation. Scale bars are 100 μm.

Under light illumination, at t = 120 s, the half bimeron string moves to the splay region at x = L/2 (Fig. 3a and Supplementary Movie 5). The antimonopole moves towards the right and the monopole to the left, with the monopole-antimonopole pair exhibiting attraction-like interaction. The profiles of the antimonopole and monopole change from intermediate states to circular and hyperbolic, respectively, Fig. 3d, e. At the same time, the bimerons on the two sides are transitioned to HAS (1/2, 0) and the bimeron in between is transformed to HNS (−1/2, 0) (Fig. 3b, c). At t = 181 s, the string moves to the bend region at x = L. Half skyrmions continue to change into half bimerons (Fig. 3a–c). The monopole-antimonopole pair starts to move in opposite directions, exhibiting a repulsion-like interaction, Supplementary Movie 5. The bimerons on the two sides are transformed to HNB(1/2,π/2) while the middle part to HNB(−1/2, π/2), Fig. 3b, c. The profiles of the antimonopole and monopole continue to change to intermediate, respectively, Fig. 3d, e. At t = 248 s, the string moves to the splay region at x = 3L/2, the bimerons on the two sides are transitioned to HNS(1/2, π) while the middle part to HAS (−1/2, π), Fig. 3b, c. The profiles of the antimonopole and monopole change to hyperbolic and circular, respectively and the pair shows strong repulsion-like interaction, Fig. 3d, e. At t = 383 s, the skyrmion string completes the translation with a period of 2L = 100 μm, the monopole pair starts to move to each other, exhibiting attraction-like interaction again.

On the left and right sides, the topological structures of the half skyrmion strings are transformed in the following order: HNB(1/2, −π/2), HAS(1/2,0), HNB(1/2, π/2), HNS(1/2, π), and HNB(1/2, −π/2). The transformations in the middle part occur in the following order: HNB(−1/2, −π/2), HNS(−1/2, 0), HNB(−1/2, π/2), HAS(−1/2, π) and HNB(−1/2, −π/2). The configurations of the antimonopole and the monopole keep changing, but the charge remains unchanged. Single point defect in the half bimeron string can also be manipulated by light irradiation to achieve the topological transitions of half skyrmions (Supplementary Fig. 9 and Supplementary Movie 6).

During this process, since HAS (N_SK = ±1/2) possesses lower energy than HNS (N_SK = ±1/2), Fig. 2i, the monopoles move to HNS, leading to the growth of HAS. Due to the motion of the monopoles, the lengths of the skyrmion strings are tuned by light irradiation; meanwhile, the skyrmion number remains unchanged due to topological protection when the above four topological textures are mutually transformed. Likewise, the profiles of the monopoles are constantly changing, meanwhile the charges (q = ±1) are topological protected and remain constant. It is worth mentioning that the addition of an external optical field is crucial in enabling the interconversion among the four types of monopole structures and effectively prevents the annihilation between monopoles of opposite charges. For instance, in Fig. 3f, at t = 440 s, the monopole-antimonopole pair is very close. If illumination stops, they would inevitably meet and annihilate (Supplementary Movie 7). Afterwards, the whole skyrmion string is transitioned to an HAS(1/2,0) with N_SK = +1/2. Thus, the topological transition between different skyrmion strings can be tuned by controlling the mutual interaction of the monopole-antimonopole pair through programming the light irradiation.

A monopole-antimonopole pair in half skyrmion string loops

When the director field on the bottom surface is given by $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, with $\alpha =\frac{\pi \sqrt{x^2+y^2}}{L}$, and the top surface is aligned along the −x-axis, a half skyrmion loop containing 4 different half skyrmion types is formed²¹ (Fig. 4a, b and Supplementary Fig. 10a). After phase transition, a pair of opposite charged monopoles were created within the loop. The loop is determined to be composed of six alternating segments of half skyrmions and half bimerons (Supplementary Fig. 10b, c). Both the monopole and antimonopole adopt intermediate profiles, and they are positioned in the opposite bend regions (Fig. 4b–d). By calculating the free energy of the entire system, the system has only one stable state, achieved when the azimuthal angle of the monopole (q = +1) is φ₊₁ = 0° and that of the antimonopole (q = −1) is φ₋₁ = 180°, as shown in Fig. 4d, e (t = 0 s).

Fig. 4. A monopole-antimonopole pair in half skyrmion loops.

a Schematic of director fields of top and bottom surfaces. b Through phase transition, a monopole-antimonopole pair is generated in the loop and stably located in the bend regions. c Two intermediate hegehog structures with opposite charges. d The free energy calculation reveals that the monopole and antimonopole are stabilized at ${\phi }_{+1}=0^{\circ }$ and ${\phi }_{-1}=180^{\circ }$. e Image sequence of dynamics of monopole-antimonopole during loop shrinkage. Double arrow indicates the top uniform alignment. Red and blue dotted circles mark the monopole (q = +1) and antimonopole (q = −1) configurations, respectively. f The colored trajectories of two defects. g Schematic of director fields of top and bottom surfaces. h Director field of the bottom surface. i Circular hedgehog with a negative charge and hyperbolic hedgehog with a positive charge. j Image sequence showing the autonomous motion of a monopole-antimonopole pair in a pure half skyrmion loop. k Colored trajectories of the two defects. (Scale bar, 100 μm).

Next, the string loop is driven out-of-equilibrium to shrink by light irradiation (Fig. 4e, f and Supplementary Movie 8). During the shrinking process, the monopole-antimonopole pair rotates clockwise (CW) along the string. For example, at t = 0 s, the monopole and antimonopole are located at φ₊₁ = 0 and φ₋₁ = π, respectively, Fig. 4e. Upon irradiation, as the top surface alignment changes to the y-axis at t = 410 s, the monopole-antimonopole pair rotates 90° CW to φ₊₁ = −π/2 and φ₋₁ = π/2, Fig. 4e. Continuous rotation of the uniform alignment to the x-axis results in the monopole and antimonopole at φ₊₁ = π and φ₋₁ = 0 at t = 650 s (Fig. 4e). After one full period of string loop shrinkage, the motion trajectory of the monopole pair manifests two spiral shapes, as shown in Fig. 4f (t = 1090 s).

Note that the topological charges and profiles of the monopole-antimonopole pair remain unchanged throughout the loop shrinking process, as they are always drawn to the bend regions with the lowest energy within the system (Fig. 4d and Supplementary Fig. 10f). Since the pair moves in the same direction and at the same speed, they will never meet each other unless the loop is annihilated. During the expansion of the loop, the pair will rotate counterclockwise (CCW) along the loop in response to the light irradiation with unchanged topological charges (Supplementary Fig. 10d–f and Supplementary Movie 8).

When the director field on the bottom surface adopts $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, where $\alpha =\frac{\pi \sqrt{x^2+y^2}}{L}+\mathrm{a}\mathrm{rctan}\frac{y}{x}$ (Fig. 4h) and the top surface is designed as $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, with $\alpha \left(x,y\right)=m{\tan }^{-1}\frac{y}{x}+{\alpha }_0$, where m = +1 is an integer topological charge, and ${\alpha }_0=\frac{\pi }{2}$ sets the distortion with a pure bend (circular pattern) (Fig. 4g, h), a pure half skyrmion loop of HNS(−1/2,0) is generated. In Fig. 4j and Supplementary Movie 9, at t = 0 s, after phase transition, a pair of monopole and antimonopole is created in this pure half skyrmion loop (Fig. 4i), resulting in the loop being composed of two different types of half skyrmions, HNS(−1/2,0) and HAS(1/2,0). Afterwards, the circular antimonopole(q = −1) autonomously rotates CW along the loop, while the hyperbolic monopole(q = +1) rotates CCW concurrently until they meet and annihilate. Finally, at t = 1640 s, the loop becomes a pure HAS(1/2,0). The physical underpinning is consistent with the situation shown in Fig. 2. Besides, the annihilation of intermediate monopole-antimonopole pairs in the pure half bimeron loop is also demonstrated (Supplementary Fig. 10g–k and Supplementary Movie 9).

Colloidal transport by monopole-enabled topological transition of skyrmions

A spiral shaped pure half skyrimon HAS(1/2,0) is created, Fig. 5d, when the director field on the bottom surface follows $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, where $\alpha =\frac{\pi \sqrt{x^2+y^2}}{L}$ (Fig. 5a), and the top surface is designed as $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, with $\alpha \left(x,y\right)={\tan }^{-1}\frac{y}{x}+\frac{\pi }{2}$ (Fig. 5b, c).

Fig. 5. Colloidal transport by monopole-enabled topological transition of skyrmions.

a Director field of the bottom surface. b Director of the field of the top surface. c 3D schematic of the assembled cell. d Spiral-shaped half skyrmion HAS(1/2,0). e The hyperbolic monopole (q = +1) in the spiral-shaped string spontaneously moves from the center to the boundary along the string, the colored trajectory of the monopole is shown in the last image at t = 734 s and (f) The circular antimonopole (q = −1) in the spiral-shaped string spontaneously moves from the boundary to the center along the string, the trajectory is shown in the last image at t = 666 s. g The hyperbolic monopole (q = +1) in the string collects colloidal particles during its spontaneous movement within the string and transports them from the center to the boundary. A pure spiral-shaped skyrmion string is shown after the colloids are cleared from the skyrmion at t = 4044 s. h Hyperbolic hedgehog with a positive charge. i Circular antihedgehog with a negative charge. j Half skyrmion structures can continuously change from HNS (N_SK = −1/2) to HAS (N_SK = +1/2) by motion of monopoles. Red and blue dotted circles mark the monopole (q = +1) and antimonopole (q = −1) configurations, respectively. (Scale bar, 100 μm).

A monopole-antimonopole pair is generated after the phase transition. At t = 0 s, the circular antimonopole (q = −1) is moved by the optical tweezer to the boundary of the pattern and eliminated, while the hyperbolic monopole (q = +1) is brought closer to the center position (Fig. 5e, h, and Supplementary Movie 10). As such, the skyrmion string is of HNS(−1/2,0) (Fig. 5j). Due to the instability of HNS(−1/2,0) and the lower energy state of HAS(1/2,0), the hyperbolic monopole spontaneously moves from the center following the spiral trajectory at an average speed of 2 μm/s (Supplementary Fig. 11f) and HNS(−1/2,0) is transformed to HAS(1/2,0) (Fig. 5j). On the contrary, a circular antimonopole (q = −1) can be created close to the boundary and moves with an average speed of 2.32 μm/s (Supplementary Fig. 11g) towards the center along the half skyrmion (Fig. 5f, i and Supplementary Movie 10).

We have developed the transport function of the monopoles in the spiral half skyrmion string. Colloidal particles in different numbers are placed within the spiral string at different positions. The motion of the monopoles enables the transport of a colloidal chain (20 colloids) from the center to the boundary (Fig. 5g and Supplementary Movie 11) or from the boundary to the center (Supplementary Fig. 11e and Supplementary Movie 11). Note that the speed of point defects decreases as the number of colloidal particles increases. For example, a colloidal chain propelled by the hyperbolic monopole (q = +1) exhibits an average speed of 0.35 μm/s (Supplementary Fig. 11h). This corresponds to an Ericksen number E_r ≪ 1, which is orders of magnitude smaller than the flow speed (tens to 10³ μm/s with E_r = 2 ~ 100) required for director reorientation^56,57. Thus, hydrodynamic effects are negligible in this regime.

Discussion

The topological protection ensures the topologically stable skyrmions cannot be created or destroyed. Emergent monopoles, acting as sources or sinks of the emergent field, play a pivotal role in the transition between topologically distinct phases, enabling the initiation of topological transitions. However, how to manipulate the topologically protected monopoles to program the topological transitions of skyrmion configurations remains a grand challenge. Harnessing their properties could lead to advancements in the control and engineering of topological phases, with potential applications in data storage⁵⁸, quantum information processing⁵⁹, photonic texture manipulation^4,20 and other fields where topological stability is paramount.

In this work, we have meticulously manipulated monopoles to facilitate topological transitions between various types of skyrmion strings, each with distinct topological invariants, notably the skyrmion number. Our pivotal discovery is the transformation of half skyrmions through the intervention of monopoles, which are topological point defects. We generated eight diverse topological profiles of point singularities within half skyrmion strings. These include hyperbolic hedgehog defects with charge q = ±1 and circular hedgehog topological defects with charge q = ±1. Furthermore, four intermediate (anti)hedgehog configurations (q = ±1) were created in half bimeron strings. Subsequently, we demonstrate the mutual transitions between different skyrmions in the presence of monopoles or antimonopoles, leading to a change in the skyrmion number. This is reminiscent of the unwinding of the skyrmion phase in solid-state chiral magnets mediated by emergent magnetic monopoles²³. Here, the sources or sinks of emergent flux are determined by $\widehat{\mathbf{n}}$ configurations and quantized due to topology.

The demonstrated transformation in the skyrmion number signifies a pivotal leap in our comprehension of soliton matter structures. Our research delves into the fascinating dynamics of monopoles and antimonopoles, clarifying how their interactions are tuned by the perturbation of half skyrmions into a non-equilibrium state through the application of a light field. Throughout this process, the configurations of the monopoles or antimonopoles evolve, yet the topological charge remains invariant. Simultaneously, the monopoles (or antimonopoles) traverse the skyrmion string, synchronizing with the string’s translation. This movement modulates the configurational changes of the half skyrmions in a reversible manner, contingent upon the regions of the topological patterns. Furthermore, the interactions between monopoles and antimonopoles can be orchestrated by programming the geometry of the topological patterns. Besides, by harnessing the dynamic behavior of monopoles, we have proposed various trajectory schemes to govern their movement, effectively utilizing them as carriers for colloidal particles. The versatile nature of nematic liquid crystals offers a pliable platform for exploring the nuanced interplay between topology, geometry, and material order. The demonstrated effect opens opportunities in the design of micromachines for cargo transport.

Methods

Materials

Glass substrates were thoroughly cleaned in an ultrasonic bath containing dishwasher detergent, followed by rinsing with isopropyl alcohol and drying in an oven at 85 °C for 15 min. The substrates were further treated in a UV-ozone chamber for 10 min. A photosensitive material of azo dye SD1 (from Nanjing Leyao Technology Co., LTD) solution (Supplementary Fig. 1a), which is mixed with n,n-dimethylformamide (DMF) at 0.2 wt%, is spin-coated on the glass substrates at 3000 rpm for 30 s. Then, the substrates are baked at 100 °C for 1 h. Upon exposure to linearly polarized light, the photosensitive azo dye on the glass substrates will align perpendicularly to the direction of linear polarization⁶⁰. The orientation of the liquid crystal (LC) director is consistent with the alignment of the azo dye molecules.

Monomer RM257, as shown in Supplementary Fig. 1b (purchased from Nanjing Leyao) is mixed with toluene at a concentration of 10 wt%. The photoinitiator Irgacure 651 (from Nanjing Leyao) was added at a concentration of 5 wt% RM257. Subsequently, this solution was spin-coated onto the patterned SD1 substrates at 3000 rpm for 30 s. To polymerize the substrates, unpolarized ultraviolet light with an intensity of 1.4 mW/cm² was used for 30 min. The resulting polymer pattern replicates the pattern of SD1 alignment beneath it.

A nematic liquid crystal (LC) called 4′-pentyl-4-cyanobiphenyl (5CB), the chemical structure as shown in Supplementary Fig. 1c. The colloidal particles of radius 2.5 µm dispersed in the 5CB was injected into the photopatterned cell at 25 °C. The colloids were manipulated by using laser tweezers from JCOPTIX, China.

Patterned surface alignment

The topological pattern is achieved through a maskless photopatterning setup that utilizes a projector display^55,61,62, (Supplementary Fig. 2a). The topological pattern in Fig. 4g and Fig. 5b with $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, with $\alpha \left(x,y\right)=m{\tan }^{-1}\frac{y}{x}+{\alpha }_0$, where m = +1 is an integer topological charge, and ${\alpha }_0=\frac{\pi }{2}$ sets the distortion with a pure bend (circular pattern), (Supplementary Fig. 2b), is produced by using the pair triangle segments. At the beginning, the pair triangle segments aligned along the x-axis (Supplementary Fig. 2e). The rotation speed of the polarizer in the clockwise sense is set as R₁ = 10^o/5s. The segments rotate at a speed of R₂ = 10 μm/5s. Hence, after π rotation of the polarizer, the designed pattern is obtained, as shown in Supplementary Fig. 2b.

The topological pattern in Supplementary Fig. 10g with $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, with $\alpha \left(x,y\right)=m{\tan }^{-1}\frac{y}{x}$, where m = +1 is an integer topological charge, resulting in a distortion with a pure splay (radial pattern), (Supplementary Fig. 2c), is produced by using the pair triangle segments. At the beginning, the pair triangle segments aligned along the y-axis. The rotation speed of the polarizer in the clockwise sense is set as R₁ = 10°/5s. The segments rotate at a speed of R₂ = 10 μm/5s.

The topological pattern in Fig. 4h with $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, where $\alpha =\frac{\pi \sqrt{x^2+y^2}}{L}+\mathrm{a}\mathrm{rctan}\frac{y}{x}$, (Supplementary Fig. 2d), is produced by using the pair spiral-shaped segments. The rotation speed of the polarizer in the clockwise sense is set as R₁ = 10°/5s. The segments rotate at a speed of R₂ = 10 μm/5s. Thus, the period of this pattern is 100 μm, as shown in Supplementary Fig. 2d.

The alternating splay-bend pattern is designed with director field$\widehat{\mathbf{n}}=\left(n_x,n_y\right)=\left(\cos \alpha ,\sin \alpha \right)$, where α(x,y) = πx/L and L = 100 μm is the period (Supplementary Fig. 2f). The segment with a rectangular shape and the polarizer rotate starting along the x-axis. The rotation speed of the polarizer in the clockwise sense is set as R₁ = 10°/5s. The segments will be displaced along the −y-axis at a speed of R₂ = 10 μm/5s, Supplementary Fig. 2f.

The topological pattern in Fig. 4b, 5a with $\widehat{\mathbf{n}}=\left(n_x,n_y,n_z\right)=\left(\cos \alpha ,\sin \alpha ,0\right)$, where $\alpha =\frac{\pi \sqrt{x^2+y^2}}{L}$, (Supplementary Fig. 2g), is produced by using the concentric segments²¹. The rotation speed of the polarizer in the clockwise sense is set as R₁ = 10°/5s. The segments shrink at a speed of R₂ = 10 μm/5s. Thus, the period of this pattern is 100 μm, as shown in Supplementary Fig. 2g.

Sample preparation

The top substrate is uniformly aligned, while the bottom substrates adopt different topological patterns. The two substrates are assembled to form a sandwich cell with 20 μm glass spacers separating the two substrates. At 25 °C, nematic LC 5CB was injected into the cell. As the LC filled the cell, skyrmion strings were formed, and the samples were viewed using a polarizing optical microscope. The phase transition process was then carried out using a hot stage (from INSTEC), afterwards, monopole-antimonopole pair generated in the string. To drive the sample out-of-equilibrium, a collimated LED (from Thorlabs) emitting at a wavelength of 455 nm and with a light intensity of 510 mW/cm² at the focus point was used as the light source. Since the bottom substrate of the two-dimensional (2D) topological pattern is layered with a liquid crystal polymer, it remains unaffected by light.

Polarizing optical microscopy

We utilized a SOPTOP CX40P polarized optical microscope (POM) that included a 10x Plan, N.A. = 0.30 objective and a 50x Plan, N.A. = 0.55 objective to capture optical microscopy images. Using a 9 MP Sony Exmor CMOS Sensor Microscope Camera, we were able to capture images at a resolution of 4096 × 2160 pixels.

Simulation method for generating POM images

We have performed calculations using the open-source code: https://nemaktis.readthedocs.io/en/latest/intro/overview.html⁶³

Monopole-antimonopole motion distance measurement

Micrographs were processed by ImageJ, and the motion of monopole (antimonopole) was processed by tracking the displacement of their motion using the ImageJ MTrackJ Plugin. To calculate the moving distance of the monopole (antimonopole), each step of the trajectory is measured as (x_i, y_i), and the starting point is (x₀, y₀). The distance between each step and the starting point is $\sqrt{{\left(x_i-x_0\right)}^2+{\left(y_i-y_0\right)}^2}$.

Numerical simulations

The total free energy F_total of a nematic liquid crystal is given by

$$F_{\mathrm{total}}={\int }_V\left(f_{\mathrm{LdG}}+f_{\mathrm{el}}\right)\mathrm{d}V+{\int }_S{f}_{\mathrm{surf}}\mathrm{d}S,$$ 1

where f_LDG is the short-range Landau-de Gennes free energy, f_el is the long-range elastic energy, and f_surf is the surface anchoring-induced free energy. The Landau-de Gennes free energy density f_LdG is calculated as⁶⁴

$$f_{\mathrm{LdG}}=\frac{A_0}{2}\left(1-\frac{U}{3}\right)\mathrm{Tr}\left({\mathbf{Q}}^2\right)-\frac{A_{0}U}{3}\mathrm{Tr}\left({\mathbf{Q}}^3\right)+\frac{A_{0}U}{4}{\left(\mathrm{Tr}\left({\mathbf{Q}}^2\right)\right)}^2,$$ 2

where Q is the tensorial order parameter from an ensemble average over unit vector n (representing the molecular orientation), Q = 〈nn − I/3〉. Parameter U controls the magnitude of S₀ of a homogenous static system through

$$S_0=\frac{1}{4}+\frac{3}{4}\sqrt{1-\frac{8}{3U}}.$$ 3

The elastic energy density f_el is alternatively written as (Einstein summation rule assumed)

$$f_{\mathrm{el}}^{\mathrm{Q}}=\frac{1}{2}{L}_1\left({\partial }_k{Q}_{ij}\right)\left({\partial }_k{Q}_{ij}\right)+\frac{1}{2}{L}_2\left({\partial }_k{Q}_{jk}\right)\left({\partial }_l{Q}_{jl}\right)+\frac{1}{2}{L}_3{Q}_{ij}\left({\partial }_i{Q}_{kl}\right)\left({\partial }_j{Q}_{kl}\right)+\frac{1}{2}{L}_4\left({\partial }_l{Q}_{jk}\right)\left({\partial }_k{Q}_{jl}\right),$$ 4

which can be mapped into the Frank-Oseen elastic free energy density⁶⁵

$$f_{\mathrm{el}}^{\mathrm{FO}}=\frac{1}{2}{K}_1{\left(\nabla \cdot \mathbf{n}\right)}^2+\frac{1}{2}{K}_2{\left(\mathbf{n}\cdot \nabla \times \mathbf{n}\right)}^2+\frac{1}{2}{K}_3{\left(\mathbf{n}\times \left(\nabla \times \mathbf{n}\right)\right)}^2-\frac{1}{2}{K}_{24}\nabla \cdot \left[\mathbf{n}\left(\nabla \cdot \mathbf{n}\right)+\mathbf{n}\times \left(\nabla \times \mathbf{n}\right)\right],$$ 5

where K₁, K₂, K₃ and K₂₄ are the splay, twist, bend and saddle-slay elastic moduli, respectively.

The mapping between constant sets K₁, K₂, K₃ and K₂₄ and L₁, L₂, L₃, and L₄ is via

$$\begin{array}{c}\hfill L_1=\frac{1}{2S_0^2}\left[K_2+\frac{1}{3}\left(K_3-K_1\right)\right],\hfill \\ \hfill L_2=\frac{1}{S_0^2}\left(K_1-K_{24}\right),\hfill \\ \hfill \begin{array}{c}\hfill L_3=\frac{1}{2S_0^3}\left(K_3-K_1\right),\hfill \\ \hfill L_4=\frac{1}{S_0^2}\left(K_{24}-K_2\right).\hfill \end{array}\hfill \end{array}$$ 6

The anchoring energy f_surf is calculated by the nondegenerate formula, the so-called the Rapini-Papoular⁶⁶.

$$f_{\mathrm{surf}}=\frac{1}{2}W{\left(\mathbf{Q}-{\mathbf{Q}}_{\mathrm{surf}}\right)}^2,$$ 7

where Q_surf is the preferred field of the surface, ${\mathbf{Q}}_{\mathrm{surf}}=S_0\left({\mathbf{n}}_s{\mathbf{n}}_s-\mathbf{I}/3\right)$, ${\mathbf{n}}_{\mathrm{s}}$ is the surface-preferred molecular orientation, and W is the anchoring strength. In our simulations, we assumed that the anchoring conditions on the top and bottom surfaces are strong, and therefore, the directors are almost fixed at the substrates. To simulate the half skyrmions driven out of equilibrium if the top surface is irradiated with linear polarization, we assumed that the ${\mathbf{Q}}_{\mathrm{surf}}$ of the top surface (where there is no pattern) changes linearly from one preferred direction to another during the simulation. After equilibrating the initial state, all rotations are run for a total number of steps of 250000 to ensure that the systems evolve quasistatically. Equation (1) calculates a thermodynamic potential that determines the stable or metastable solutions of the system. We define a molecular field

$$\mathbf{H}={-\left[\frac{\delta F_{\mathrm{total}}}{\delta \mathbf{Q}}\right]}^{\mathrm{st}},$$ 8

where […]^st is a symmetric and traceless operator. Assuming that all transitions are quasistatic processes, the evolution of the Q-tensor for bulk points follows

$${\partial }_t\mathbf{Q}={\mathrm{\Gamma }}_{\mathrm{s}}\mathbf{H},$$ 9

where Γ_S is the relaxation constant. For surface points, the evolution of Q is governed by

$$\frac{\partial \mathbf{Q}}{\partial t}=-{\mathrm{\Gamma }}_s\left[-\mathit{\nu }\cdot \frac{\partial f}{\partial \nabla \mathbf{Q}}+{\left[\frac{\partial f_{\mathrm{surf}}}{\partial \mathbf{Q}}\right]}^{\mathrm{st}}\right],$$ 10

where the unit vector ν represents the surface normal.

To simulate 5CB, the elastic constants are K₁ = 6 pN, K₂ = 3.9 pN, K₃ = 8.2 pN and K₂₄ = 3.9 pN. The corresponding constant $L_1=6\times {10}^{-12}\mathrm{N}$, $L_2=6\times {10}^{-12}\mathrm{N}$, $L_3=4.27\times {10}^{-12}\mathrm{N}$, and $L_4=0$. The numerical parameters used here are $A_0=1.37\times {10}^5\mathrm{J}/{\mathrm{m}}^3$ and U = 3.5. The characteristic length scale is set to the nematic coherence length, the defect core size, with ${\xi }_N=\sqrt{L_1/A_0}\approx 6.63\mathrm{nm}.$

The initial configuration for the half bimeron under the topological pattern of alternating splay–bend distortions was generated as follows⁶⁷:

$$n_x=\cos \left\{\pi \left[1+\left(-\frac{x}{L}+\frac{1}{2}\right)\tanh \left[\eta \frac{\pi }{H}\left(x-x_w\right)\right]\right]\left(1-\frac{z}{H}\right)\right\}$$ 11

$$n_y=\sin \left\{\pi \left[1+\left(-\frac{x}{L}+\frac{1}{2}\right)\tanh \left[\eta \frac{\pi }{H}\left(x-x_w\right)\right]\right]\left(1-\frac{z}{H}\right)\right\}\tanh \left[\eta \frac{\pi }{H}\left(x-x_w\right)\right]$$ 12

$$n_z=\sin \left\{\pi \left[1+\left(-\frac{x}{L}+\frac{1}{2}\right)\tanh \left[\eta \frac{\pi }{H}\left(x-x_w\right)\right]\right]\left(1-\frac{z}{H}\right)\right\}\sec \mathrm{h}\left[\eta \frac{\pi }{H}\left(x-x_w\right)\right]$$ 13

where L represents the period of the surface pattern and H corresponds to the height of the cell thickness. The half bimeron is positioned at x = x_w, and the parameter η controls the string thickness depending on the elastic modulus K and the ratio between H/L. Different configurations can be obtained by multiplying a rotation matrix around the rotation axis and the z-axis.

To obtain the monopole structure, we link two half skyrmions as initial conditions. One half skyrmion is obtained from rotating the other half skyrmion π around z-axis. The monopole positions can be adjusted by the linking positions of two half skyrmions.

In this work, the forward Euler method is used for time integration. The time derivative $\frac{\mathrm{d}{Q}_{\mathrm{ij}}}{\mathrm{d}t}$ is approximated by a forward finite difference:

$$\frac{\mathrm{d}{Q}_{\mathrm{ij}}}{\mathrm{d}t}=\frac{Q_{\mathrm{ij}}^{t+\mathrm{\Delta }t}-Q_{\mathrm{ij}}^t}{\mathrm{\Delta }t}={\mathrm{\Gamma }}_{\mathrm{s}}{H}_{\mathrm{ij}}^t,$$ 14

where $H_{\mathrm{ij}}^t$ denotes the molecular field at time t (as defined in Eq. (8)). Spatial derivatives required for calculating the molecular field are approximated using second-order finite differences. In the bulk, second-order central finite differences are employed. On the surface, one-sided second-order differences are employed. For instance, along with the z-direction, second-order forward differences at the bottom surface and second-order backward differences at the top surface are used to implement the surface boundary conditions. Regarding the boundary conditions, for every simulation, the z direction is limited by the two patterned surfaces, while for the simulations involving the C-pattern, a periodic boundary condition is used along the x and y directions. For any other pattern, due to the lack of transitional symmetry, a Neumann boundary condition ($\frac{\partial f}{\partial \nabla Q_{\mathrm{ij}}}\cdot \mathit{\nu }=0$) is implemented along the x and y directions by imposing zero anchoring walls at the edges of the simulation box.

Author contributions

Z.A., J.J., and C.P. wrote the article. Z.A., R.W., J.Z., Q.S., and K.T. performed the experiments. W.T., X.Z., F.V., and R.Z. performed the numerical modeling and theoretical analysis. Z.A., X.Z., J.J., Z.C., and J.Z. analyzed the data. All authors participated in discussing and writing the manuscript.

Peer review

Peer review information

Nature Communications thanks Uroš Tkalec, Dong Ki Yoon, and the other, anonymous, reviewer for their contribution to the peer review of this work. A peer review file is available.

Data availability

All the data used to make the plots in main figures and Supplementary Figures are provided with this paper. Source data are provided with this paper.

Code availability

Codes used to make the plots in the main figures and Supplementary Figures are provided in the source data.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-64188-2.