Ice rule breakdown and frustrated antiferrotoroidicity in an artificial colloidal Cairo ice

The Cairo geometry is a type of Euclidean plane tiling made of a sequence of connected pentagons which share vertices with two types of coordination numbers, namely z = 3 and z = 4 [1, 2]. Besides its aesthetic beauty, as testified by the presence of the Cairo geometry in numerous artistic paintings and pavements, especially in Egypt [3], such lattice is also important in frustrated spin systems. For example, it has been experimentally found in different magnetic compounds such as the Bi₂Fe₄O₉ [4] and the Bi₄Fe₅O₁₃F [5], apart from being the subject of theoretical studies on Ising-type models [6–9]. Recently, such geometry has been considered as an interesting way to organize interacting dipolar nanoislands on a plane, also known as artificial spin ice systems (ASIs) [10–12]. ASIs are lattice of ferromagnetic elements that interact via in-plane dipoles and are arranged to produce geometric frustration effects [13–20]. In the Cairo geometry, recent experimental works have found a rich behavior due to frustration [21, 22], while Monte Carlo simulations reported the presence of long-range order [23]. Even mechanical analogues of Cairo artificial spin ice were realized via 3D-printing [24].

Particle ice systems are soft matter analogues of ASIs but based on interacting colloids constrained to move within a lattice of double wells [25, 26]. In contrast to ASIs that feature in-plane dipoles, the colloidal particles present out-of-plane, induced dipoles and pair interactions that can be tuned by an external field. The microscopic size of the particles allows the use of optical microscopy to visualize their dynamics and thus, to extract all relevant degrees of freedom. The particle ice was originally proposed with a set of double wells generated optically [25, 27], while experiments were realized by using lithographically patterned substrates [28–30]. Moreover, it was shown theoretically that, for a lattice of single coordination number z, the colloidal ice is analogous to an ASI since the low energy states fulfill similar ice-rules, i.e. minimization of the associated topological charge [31]. Such similarity however, breaks down for lattices of mixed coordination such as decimated systems [32]. Indeed, in a colloidal ice, particles at a vertex tend to repel each other, and therefore the single vertex energy is different from ASI. If we consider the colloid as a token of a topological charge, then each vertex wants to push away as much charge as possible. In certain geometries this is impossible and the same trade-off, corresponding to the ice rule, is realized on each vertex. Thus, the colloidal ice for an extended, single coordinated lattice behaves as a spin ice [31, 33]. In other cases that is not true [32], as we show below for the Cairo geometry, and a transfer of topological charge takes place between vertices of different coordination, breaking the ice rule.

This fact underlines that particle based ice offers the possibility of investigating a rich set of physical phenomena, different than ASIs [30, 33–35]. And indeed many other recent realizations testify to the broader phenomenology of particle-based systems as models for geometric frustration. These include confined microgel particles [36–40], mechanical metamaterials [41–46], patterned superconductors [47, 48], skyrmions in liquid crystals [49, 50] and interacting macroscopic rods [51, 52].

In this article we experimentally realize a Cairo colloidal ice by confining repulsive paramagnetic colloidal particles within a lattice of lithographic double wells, as shown in figure 1(a). To place and move these particles within the topographic traps, we use a modified set of laser tweezers that generate an optical ring rather than a focalized spot. This strategy allows to easily trap and move paramagnetic particles, avoiding heating due to absorbed light. By investigating the low energy states, we find that topological charges can accumulate in different sublattices, breaking locally the ice rules. We complement our finding with Brownian dynamic simulations, which show good agreement with the experimental data in terms of fraction of vertices, topological charges and net chirality associated to each pentagonal plaquette of the Cairo lattice. Finally we perform simulations on an extended system to calculate chirality correlation functions and to elucidate the system frustration at high field strength.

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The schematic in figure 1(b) and the optical microscope image in figure 1(c) illustrate the basic features of a Cairo colloidal ice. The system presents a lattice of lithographic elliptical traps composed of two wells of lateral elevation $h \sim 3\,\mathrm{\mu m}$ that are connected by a small central hill. These wells are filled with paramagnetic colloidal particles of diameter $d = 10\,\mathrm{\mu m}$ and magnetic volume susceptibility κ = 0.025. A particle of volume $V = \pi d^3/6$ has to overcome a gravitational potential $U_g = \Delta \rho V g h \sim 2000 \, \mathrm{k_B T}$ to jump outside the double well due to thermal fluctuations. Here $\Delta \rho = \rho_p - \rho_w \sim 0.6 \,\mathrm{g\,cm^{-3}}$ is the difference between the mass density of the particle (ρ_p ) and the dispersing medium (ρ_w ). Thus, the particles are essentially confined within the elliptical traps and cannot change their location from one well to another unless subjected to an external force, such as the repulsion from a neighboring colloid. We induce such repulsion by applying an external magnetic field B perpendicular to the sample pane, figure 1(b). Once the field is applied, each particle acquires an induced dipole moment $\boldsymbol{m} = V \kappa \boldsymbol{B}/\mu_0$, being µ₀ the permeability of vacuum. Thus, a pair of particles (i, j) placed at a relative distance $r = |\boldsymbol{r}_i-\boldsymbol{r}_j|$ experience a repulsive dipolar interaction which is isotropic and inversely proportional to r³, $U_{dd} = \mu_0 m^2/(4\pi r^3)$. This interaction potential can be tuned via the applied field, B . For an amplitude of B = 10 mT, the potential strength is $U_{dd} = 122 \, k_B T$ for the closest distance between two particles in a z = 3 vertex being $r = 13\,\mathrm{\mu m}$, while is $U_{dd} = 4.7 \, k_B T$ for the farthest distance of $r = 38.4\,\mathrm{\mu m}$ in a z = 4 vertex in the Cairo geometry.

The mapping between the colloidal ice and an ASI [25] can be obtained by assigning an Ising-like spin to each double well, such that it points where the particle is located, figure 1(d). Using this mapping, one can distinguish between different type of vertices depending on the lattice coordination. For example, for the z = 4 (square lattice) there are 6 possible arrangements of the particles with different energetic weights, while for the z = 3 (honeycomb lattice) these reduce to 4. Moreover, in analogy to the ASI, one can assign a topological charge to each vertex defined here as $q = 2n-z$ being n the number of spins that point toward the vertex center. Note that we can talk of topological charges when considering the vertices within a lattice, not isolated ones. By this notion of charge, an extended lattice is overall charge-neutral, and thus charges appear in pairs and disappear only when annihilated by other defects in order to guarantee the charge conservation, $\sum q_{i} = 0$. For example, the vertex with four colloids pointing outwards ($q = -4$) is characterized by the lowest energetic weight of the z = 4, and thus, when considered alone it will be the natural state of repulsive colloids. However, within a lattice the $q = -4$ is topologically connected to the $q = +4$ due to particle conservation and thus they are unlikely to occur. One can prove that in a lattice, lower absolute values of the topological charges $|q| = 0,1$, corresponding to the ice-rule, are favored [33]. The ice rules, highlighted by the green boxes in figure 1(d) are a prescription of the minimization of $|q|$, given by vertices with q = 0 for z = 4 and $q = \pm 1$ for z = 3. Indeed, even though the ice rules were originally defined for a tetrahedral coordination system [53, 54], we find it useful to extend them to other coordination numbers as a the z = 3, figure 1(d).

Regarding the Cairo geometry, we have a mixture of z = 4 and z = 3 vertices, the latter characterized by unequal lengths of the double wells, as shown in figure 1(d). Upon analysis of the total magnetic interaction energy of a vertex U, we find that the z = 4 vertices have same energy hierarchy as the square colloidal ice investigated in previous works [28, 29]. Here $U = \sum_{i = 1}^{N_v-1}\sum_{j = i+1}^{N_v}U^{dd}_{ij}$, being N_v the number of particles in a vertex. However, the presence of the small double well in the z = 3 vertices, i.e. a length of $4.53\,\mathrm{\mu m}$ while in the z = 4 the length is $10\,\mathrm{\mu m}$, induces an energetic spitting of the vertices 1-in-2-out and 2-in-1-out depending on the location of the paramagnetic colloid. This energy difference between the z = 3 vertices, which does not affect the associated topological charge, is particular of the Cairo geometry, and it is not present in the z = 3 vertices of the classical honeycomb [55, 56] and triangular [57] colloidal ice, where all traps have the same length. Even if the energy difference is relatively small, as we will show later this will induce a disordered ground state.

The Cairo lattice is realized via soft-lithography using polydimethylsiloxane (PDMS). This substrate was first designed using a commercial software (AutoCad, Adobe) and transferred above a 5 inch soda-lime glass substrate covered with a 500 nm Cromium (Cr) layer. The chosen geometric parameters, as shown in figure 1(a), are $a = 19.54\,\mathrm{\mu m}$, $l = 26.7\,\mathrm{\mu m}$ and $\theta = 30^{\circ}$. We use laser lithography (DWL 66, Heidelberg Instruments Mikrotechnik GmbH) to write the double wells on the substrate with a 405 nm laser diode working at a speed of $5.7\,\mathrm{\, mm^2 \, min^{-1}}$. After that, we replicate the double wells of the Cr mask in the PDMS following two steps. In the first one we duplicate the structure using an epoxy-based negative photoresist (SU-8) on top of a silicon wafer. Then we covered the structure with liquid PDMS by spinning the sample at 4000 rpm during 1 min with an angular speed of 2000 rpm (Spinner Ws-650Sz, Laurell). With this process we obtain a layer of ≈20 $\mu m$ thickness. The PDMS is solidified by baking for 30 min at $95\,^{\circ}$C in a leveled hot plate. After solidification of the PDMS, we peel off the structure with the help of a cover glass (MENZEL-GLASER, Deckglaser). The resulting sample is ${\sim}170 \, \mu m$ thick, and transparent enough to visible light.

Once fabricated, the sample is placed on the stage of an inverted optical microscope (TiU, Nikon) which is connected to a complementary metal oxide semiconductor camera (MQ013CG-E2, Ximea) able to record videos of the particle dynamics at 30 Hz. The microscope is equipped with a 40 × oil immersion objective (Nikon, numerical aperture 1.3) which is used both for observation and optical trapping purpose.

One microscope port is modified in order to accommodate an incoming beam generated by a butterfly laser diode (wavelength λ = 976 nm, operated at a power of 70 mW, BL976-SAG300 Thorlabs). The optical path of the laser is composed of a series of lenses including a spatial light modulator (SLM, Hamamatsu X10468-03) which is commanded by a LCOS-SLM controller (Hamamatsu), figures 2(a) and (b). The holograms are generated with a custom made Labview program.

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The SLM is used in the holographic optical tweezers (HOTs) which are composed of 4 lenses. The first two constitute a telescope and have focal lengths $f = -75$ mm (Thorlabs LC 1582-B) and f = 175 mm (Thorlabs LA 1229-B), respectively. After the SLM there is another telescope with a focal length of f = 500 mm (Thorlabs LA1908-B) which focuses the beam after being deflected in order to filter the 0 mode via a diaphragm, reaching a final lens of f = 750 mm (Thorlabs LA 1727-C), as shown in the detailed diagram of figure 2(b). In this work, we use polystyrene paramagnetic colloids having a nominal diameter of $d = 10\,\mathrm{\mu m}$, standard deviation ${\lt}0.5\,\mathrm{\mu m}$ that are highly doped with nanoscale iron oxide grains ${\sim}20\%$ by (Sigma-Aldrich, Cat. number 496 64). Due to the optical absorption of the iron oxide grains, we have implemented a novel strategy to trap the colloids without damaging them due to the generated heat. Instead of using a single focalized spot, we program the SLM such that the deflected beam generates an optical ring as shown in figures 2(a) and (c).

The external magnetic field was generated via a custom made coil located below the sample. The magnetic coil is powered by an amplifier (BOP-20 10 M, KEPCO), that is computer controlled using a digital analogue card (NI 9269) with a custom made LabVIEW program. The field was applied via a ramp at a rate $0.05\,\mathrm{mT\ s^{-1}}$ until reaching a maximum value of 10 mT.

We complement the experiments performing Brownian dynamics simulations using as input parameters the experimental data. In particular, we use Euler's method to integrate the overdamped equations of motion for each colloidal particle i at position r _i :

$$\begin{align} \gamma\frac{\mathrm d \boldsymbol{r}_i}{\mathrm dt} = \boldsymbol{F}_i^{\mathrm{T}} + \boldsymbol{F}_i^{\mathrm{{\,}dd}} + \boldsymbol{\eta} \, \, \, , \end{align} \tag{ 1 }$$

being $\gamma = 0.033\,\mathrm{pN \, s \, m^{-1}}$ the friction coefficient. Further terms in equation (1) are the force from the topographic double well $\boldsymbol{F}_i^{\mathrm{T}}$ which is modeled as a piece-wise harmonic bistable potential,

$$\begin{align} \boldsymbol{F}_i^{\mathrm{T}} = -\boldsymbol{e}_{\perp}kR_{\perp}+\boldsymbol{e}_{\parallel}\delta \, \, \, . \end{align} \tag{ 2 }$$

Here $(\boldsymbol{e}_{\parallel},\boldsymbol{e}_{\perp})$ are unit vectors oriented parallel and perpendicular with respect to the line of length L that joins the two minima in the double well, whose associated vector is $\boldsymbol{R}\equiv (R_\parallel,R_\perp)$. Moreover $\delta = \xi_1 R_\parallel$ if $|R_\parallel| \unicode{x2A7D} \frac{L}{2}$ and $\delta = k (\frac{L}{2}-|R_\parallel|) \mathrm{sign} (R_\parallel)$ otherwise. As stiffness we use $k = 1 \cdot 10^{-4} \,\mathrm{pN\,nm}^{-1}$ which keeps the particle confined to the elongated region around the center of the trap, and $\xi_1 = 3 \,\mathrm{pN\,nm}^{-1}$ that creates a potential hill equivalent to the gravitational hill within the double wells.

The dipolar force on a particle i is given by,

$$\begin{align} \boldsymbol{F}_i^{\mathrm{{\,}dd}} = \frac{3\mu_0}{4\pi} \sum_{j\neq i} \frac{\boldsymbol{m}^2\hat{r}_{ij}}{|r_{ij}|^4} \, \, \, , \end{align} \tag{ 3 }$$

being $\mu_0 = 4\pi \times 10^{-7}\,\mathrm{H\,m}^{-1}$ and $\hat{r}_{ij} = (\mathbf{r}_i-\mathbf{r}_j)/|\mathbf{r}_i-\mathbf{r}_j|$. Dipolar interactions are calculated in an iterative form such that the global field B includes also that generated by all other dipoles. Moreover, we apply a large cut-off distance of $200 \,\mathrm{\mu m}$ to consider the effect of long range dipolar interactions.

Finally the last term in equation (1) is a random force characterized by a zero mean, $\langle \boldsymbol{\eta} \rangle = 0$ and delta correlated, $\langle \boldsymbol{\eta} \left(t\right) \boldsymbol{\eta} \left(t^{^{\prime}}\right) \rangle = 2k_\mathrm{B}T\gamma \delta(t-t^{^{\prime}})$, being $k_\mathrm{B}$ the Boltzmann constant and $T = 300\mathrm{\,K}$ the ambient temperature.

To reproduce the experimental results we start by solving equation (1) with $N_1 = 180$ particles which are arranged along $3\times 3$ unit cells and open boundary conditions. We avoid the effects of the boundaries (charge accumulation) by excluding a shell of vertices from the statistical analysis, as shown in figure 1(c), and by increasing the number of realizations. We have checked that the effect of topological charge transfer occurs both in small ($3 \times 3$ unit cells) and large ($10 \times 10$ unit cells) system size. Note also that the effect of including the outer region will be to have new vertices of coordination number z = 2 where most of the positive topological charges accumulate. To consider a larger system when measuring the chirality correlation functions, we extend the simulations also to $N_2 = 2000$ particles, where 800 are arranged along the z = 3 vertices and 400 in the z = 4. This situation corresponds to a colloidal ice of $10 \times 10$ unit cells also with open boundary conditions. To avoid that most of the particles in the z = 3 vertices localize on top of the topographic hills, due to strong dipolar forces, we also reduce the particle magnetic susceptibility to $\kappa_2 = 0.005$ and raise the spring constant of the central hill to $\xi_2 = 25\,\mathrm{pN\,nm}^{-1}$. In all cases, we numerically integrate the equation of motion using a time step of $\Delta t = 0.01$ s.

We start our experiments by first randomly placing the particles with the optical ring within the double wells according to a random number generator. In the initial, random configuration the highly charged monopoles $q = \pm 4$ are the $10\%$ of the total vertices, $q = \pm 3$ are the $15\%$, while the $25\%$ of vertices corresponds to low charged $q = \pm 2$, the rest is left to the ice rule vertices. Then, we slowly raise the applied field with a ramp of $0.05\,\mathrm{mT \, s^{-1}}$. Figure 3(a) shows the evolution of the fraction of topological charges as classified in figure 1(d), for the Cairo ice. By increasing B , we find that already after $B\sim 3$ mT, the fraction of high topological charges $q = \pm 4$ in the z = 4 and $q = \pm 3$ in the z = 3 vertices reduces almost to zero in favor of the low charged ones. In particular, vertices obeying the ice selection rules in the z = 4 rise up to the $50\%$, being only overcome by the $q = -1$ in the z = 3 (${\sim}60\%$), while the $q = +1$ reduces to ${\sim}30\%$. This reduction is accompanied by a slight increase of the charged monopoles $q = +2$ and a decrease of the $q = -2$.

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Crucially, figure 3(b) plots the average topological charge ($\bar{q} = \frac{1}{N_z}\sum q_{z_i}$, being N_z the number of vertices z) for vertices of coordination 3 and 4. It shows a net separation of topological charges between vertices of different coordination, and thus charge transfer between sublattices, breaking the ice rule. Thus, while the total topological charge is conserved, or $\sum q = 0$, at the sublattice level we observe a transfer of topological charges as the field reaches B = 8 mT. A net fraction of positive monopoles accumulate in the z = 4 vertices, while the z = 3 vertices are, on average, negatively charged. Note also the relatively good agreement between experimental data (open symbols) and numerical simulations (continuous lines), which are plot together in both graphs, while small deviations fall within the experimental/simulation error bars.

Importantly, charge transfer effect between sublattices does not occur in an ASI, whose ice rule is instead robust in most geometries as it is inscribed into the energetics of the vertices. It only occurs in a colloidal ice with mixed coordination geometry such as the Cairo. This effect results from the different nature of geometric frustration in ASI and in the colloidal system [31]. While both systems display similar behavior in terms of vertex fraction for single coordination lattices, in a mixed coordination geometry the difference at the single vertex level emerges: repulsive colloids cannot emulate in-plane ferromagnetic spins as in ASI, and topological charges tend to redistribute in order to minimize the system energy.

While charge transfer and ice-rule fragility had been predicted [31] and experimentally verified [32] in a decimated square ice, the sign of this effect in the Cairo lattice is inverted. In [32], negative monopoles form on the z = 4, breaking the ice rule. In that case the z = 3 vertices, which unlike the z = 4 vertices are charged even when obeying the ice rule, do not violate such rule but screen the negative charge of the z = 4 vertices. This was done by changing their relative admixture of ±1 charges, and thus assuming a net positive charge. In the Cairo system instead, the mechanism is similar but the sign of the charges is inverted. This is because Cairo is structurally different from a decimated square, with shorter and longer traps and thus a more complex energetics, as shown in figure 1(d). This allows also for structural transitions in the sign of the transferred charge as Cairo is deformed, and which we study elsewhere, while here we focus only on the Cairo geometry.

5.1. Frustrated antiferrotoroidicity in the Cairo system

To better characterize the disorder of the charge-unbalanced, low-energy state of strongly-interacting Cairo colloidal ice, we study the chirality value, or associated toroidal moment, to each pentagonal plaquette. As shown in the inset in figure 4(a), this moment acquires a maximum value of $\chi = -1$ (+1) for a full counter clockwise (clockwise) cell. Two natural questions are, firstly whether the plaquettes are naturally chiral, and secondly what is the mutual arrangement of these chiralities. To answer the first question, note that for a single pentagonal plaquette, a configuration that is fully chiral obeys the ice rule, that we know breaks down at high fields. At the same time, the long range interactions among colloids in further neighboring traps favors chirality. In the Cairo geometry the plaquette chirality is also favored by the asymmetry in the vertices of coordination z = 3. The particles within the shorter double wells require a smaller energetic cost to be displaced than that in the longer one. Thus, at high field strength the z = 4 vertices prefer to follow the ice rules, and parallel plaquettes separated by the same z = 3 vertex will acquire the same chirality that is opposite to that of plaquettes in nearest z = 3 vertices, see small inset in figure 4(d), left. Now, independently of how the colloidal particle in the z = 3 vertex is displaced, it will create always one fully chiral plaquette (χ = 1) and another with a chirality of $3/5$ of the same sign (a nearest z = 3 vertices will create a −1 and a $-3/5$). When normalized with respect to the number of plaquettes, we obtain a chirality value of 0.8.

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Thus we expect that at large field strengths, the average absolute chirality of the systems, $\bar{\chi} = \frac{1}{N_{pl}}\sum_{i}|\chi_i|$ being N_pl the number of plaquette, would tend towards $\bar{\chi} \rightarrow 0.8$. However we found that this was not the case, as shown in figure 4(c), where both experiments and simulations show that at the largest applied field $\bar{\chi} \sim 0.5$. This lower value was due to the fact that, in the Cairo geometry, the magnetic colloids were found to locate above the central hill at the largest field amplitude, B = 10 mT. Indeed this effect is shown in the first graph in figure 4(b), where histograms of the particle positions from the simulations are reported for both the long and short double wells in the Cairo geometry. In this situation, since the trap bistability is lost, it is difficult to extract an accurate determination of $\bar{\chi}$. We note that the particle localization in the central hill at high filed strength may be reminiscent of the symmetrisation of hydrogen bonds which is observed at high pressure ${\sim}70$ GPa. In the latter case, the hydrogen atoms tend to localize in the middle between the two oxygen ones and the molecular character of water in ice is lost [58, 59].

To keep the bistability, we have performed further simulations using a larger system size and stronger confinement, i.e. increasing the hill spring constant from $\xi_1 \to \xi_2 = 25\,\mathrm{pN \, nm^{-1}}$ and decreasing the magnetic volume susceptibility to $\kappa_1 \to \kappa_2 = 0.005$. The resulting histograms of the particle positions, shown in the right panel of figure 4(b), confirm that for these new parameters the bistability is recovered even for a larger field of B = 25 mT. Thus, for the latter system, we obtain the value of $\bar{\chi} \sim 0.7$ at the largest field which is in the ballpark of our estimate $\bar{\chi} = 0.8$ based on a single plaquette. Note that figure 4(a) shows an alternation of sign among many nearest neighboring plaquettes. However, the lattice of the pentagonal plaquettes is not bipartite, and thus frustrates the anti alignment of the plaquettes. A configuration in which all the neighboring plaquettes have opposite chirality (e.g. the check board pattern observed for the square spin ice system [60]) is geometrically impossible which suggests that the system has, at least, a disordered landscape of low energy states. As figure 4(a) suggests, the configuration corresponds to an antiferrotoroid. Moreover, we have checked that also with the new numerical conditions ($N_2,\xi_2,\kappa_2$) we observe the topological charge transfer similar to figure 3(b).

To answer the second question, concerning the mutual distribution of chiralities, we note that figure 4(a) shows a largely antiferrotoroidal arrangement. However, the lattice of the plaquettes is frustrated and therefore no full antiferrotoroidicity can exist. We explore this effect by calculating the following nearest neighbor correlations:

$$\begin{align} \bar{\psi} = \frac{1}{N_{nn}}\sum_{\langle ij \rangle}\left(1-\frac{\chi_i \chi_j}{|\chi_i \chi_j|}\right)\frac{1}{2}\, \, ; \, \, \, \, \, \, \bar{\sigma} = \frac{1}{N_{nn}}\sum_{\langle ij \rangle}\frac{\chi_i \chi_j}{\bar{\chi}} \, \, ; \end{align} \tag{ 4 }$$

where the sum is performed over the N_nn nearest neighboring plaquettes. Both correlations counts how many links among nearest neighboring plaquettes are antiferrotoroidal, regardless of the intensity of the chiralities of the plaquettes. If all the nearest neighboring plaquettes were antiferrotoroidal (which is impossible because of the frustration of the lattice of the plaquettes) then $\bar{\psi}$ would be equal to 1, and $\bar{\sigma} \to -1$. Moreover σ allows to distinguish between the ferrotoroidal and antiferrotoroidal behavior depending on whether positive or negative, respectively.

We plot $\bar{\psi}$ and $\bar{\sigma}$ in figures 4(d) and (e). In the left images, both correlations show similar trends between experiments and numerical simulations. For the large system size (right images) $\bar{\psi}$ increases steadily with the field reaching 0.68: almost the $68\%$ of the links are anti-ferromagnetic, namely 3.5 over 5 nearest neighbors. This results also reflects the arrangement of the toroidal moments shown in figure 4(a) for B = 25 mT, where the system organizes in a lattice of full chiral cells placed in an alternating order. These results allow to characterize the disorder of the strongly coupled state of the Cairo ice in terms of plaquette chirality as a frustrated antiferrotoroid. The charge transfer prevents all plaquettes from reaching maximum chirality, while the lattice of the plaquettes frustrates antiferrotoroidicity, preventing order.

We have investigated the arrangement of repulsive magnetic colloids confined in a lattice of double wells in the Cairo geometry. To experimentally realize such structure we have modified the lithographic process and developed a novel technique to trap absorbing magnetic particles using an optical ring. This strategy could be extended to other works aiming at trapping light-adsorbing magnetic particles to avoid undesired heating effects. We have observed that the Cairo ice breaks the ice rule as predicted for mixed coordination geometries, however, it does so with an inversion of the net charge transfer with respect to a previous experimental realization [32]. Moreover we have characterized this novel ensemble but looking at the effective toroidal moment associated to each pentagonal plaquette. We have found that the strongly coupled ensemble of the Cairo colloidal ice is a massively degenerate antiferroid. Our results could be potentially observed in magnetic compounds characterized by parallel lattice planes with spin lying on a Cairo lattice and having strong horizontal and weak vertical interactions. These materials are currently matter of research interest [4, 5, 61–65], although monitoring the spin dynamics remain a challenging task.

Further extensions of our work include investigating the transition from Shakti to Cairo (ongoing) or using different size of particles to investigate hysteresis and memory effects [27, 35] that could emerge when the particles localize above the double wells [66]. Finally, it will be also interesting to investigate how the topological charges freeze or move after long time and the presence of aging of topological defects in our system. This will require longer observation time, beyond our current experimental capabilities, and thus could be a challenge for future work.

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 811234). P T acknowledge support the Generalitat de Catalunya under Program 'ICREA Acadèmia', the Spanish Ministry of Science and Innovation (no. PID2022-137713NB-C22) and the Agència de Gestió d'Ajuts Universitaris i de Recerca (no. 2021 SGR 00450). The work of Cristiano Nisoli was carried out under the auspices of the U S DoE through the Los Alamos National Laboratory, operated by Triad National Security, LLC (Contract No. 892333218NCA000001).

All data that support the findings of this study are included within the article.