Out-of-equilibrium microstates but effective thermodynamics in artificial kagome ice networks Breno Malvezzi Cecchi,1, 2, ∗ Sébastien Lacaze,2 Ondřej Brunn,2, 3, 4 Stanislav Krátký,3 Petr Meluźın,3 Johann Coraux,2 Kleber Roberto Pirota,1 and Nicolas Rougemaille2 1Gleb Wataghin Institute of Physics, State University of Campinas, 13083-859 Campinas, SP, Brazil 2Université Grenoble Alpes, CNRS, Grenoble INP, Institut NEEL, 38000 Grenoble, France 3Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic 4Institute of Physical Engineering, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic (Dated: May 5, 2025) Using magnetic force microscopy and Monte Carlo simulations, we investigate the low-energy properties of two artificial kagome ice structures. The two systems differ in that the first series of lattices consists of an assembly of physically disconnected nanomagnets coupled via magnetostatics, whereas the second series is made of fully connected honeycomb networks. Imaging the microstates resulting from a field demagnetization protocol, and analyzing their magnetic correlations in real and reciprocal space, we observe distinct behaviors between the two lattice types. While the former exhibits properties well-described by the dipolar kagome ice model equilibrated at a finite fictional temperature, the latter instead is found systematically out-of-equilibrium. Remarkably, this out- of-equilibrium physics can be reformulated into an at-equilibrium one by strengthening specific coupling terms in the spin Hamiltonian. We interpret this property as a result of the field-induced domain wall propagation that arises when demagnetizing a connected network, i.e., a field driven kinetic process that competes with the formation of local flux closure configurations that minimize the magnetostatic energy. Our findings highlight how micromagnetic effects bias the selection of spin liquid microstates during a field demagnetization protocol. I. MOTIVATION Spin liquids are disordered, yet correlated, low- temperature states of matter that generally develop in highly frustrated magnets [1–6]. When the spin vector field satisfies a set of conditions, their physics can be described by an emergent electrostatics through the con- cept of Coulomb phase [7, 8]. A large variety of classical (and quantum) spin liquids has been identified in the last years, and important effort is currently devoted to clas- sifying and characterizing these correlated states of mat- ter [9–12]. Although most of this research is explored in bulk magnetic compounds and spin models in three dimensions (notably on the pyrochlore lattice) [13–15], spin liquid physics can be also studied experimentally in two dimensions, for example in nanomagnetic arrays [16– 18], colloidal systems [19–21], qubit systems [22–24], or macroscopic lattices [25–27]. In this context, arrays of interacting nanomagnets al- low the physics of classical spin liquids to be investigated under the prism of nanoscience and magnetic imaging [28, 29]. Their main advantage lies in the experimen- tal ability to visualize spin liquid microstates directly in real space, at the scale of the individual degree of free- dom, and as a function of time [30–34]. The square ice [35, 36] and the fragmented (dipolar) kagome ice [37–39] are emblematic examples of two-dimensional Coulombic spin liquids that have been studied using lithographi- ∗ bmcecchi@ifi.unicamp.br cally patterned arrays of magnetic nanostructures [40– 49]. However, probing the ground state or low-energy configurations of these artificial Coulomb phases is chal- lenging [45–47]. In particular, overcoming the freezing of the spin dynamics, whether it is intrinsic (e.g., criti- cal slowing down when approaching a phase transition, change of the spin dynamics) or extrinsic (e.g., quenched disorder, low flipping rate associated to the thermaliza- tion protocol) is a delicate task. In the kagome ice, several strategies have been followed to reach the fragmented Coulomb regime [40, 42, 45– 47, 50, 51]. Besides the two main classes of methods, ei- ther based on the fabrication of thermally active systems [30, 31, 52] or arrays subjected to a field demagnetiza- tion protocol [53–55], two geometrical designs have been explored. In the first case, the arrays consist of N nomi- nally identical nanostructures coupled through magneto- statics, the intensity of which is adjusted by changing the separation distance between the elements [17]. In the sec- ond case, these nanostructures are physically connected at the lattice vertices, and the array actually consists of a single micromagnetic network [16, 56, 57]. To better understand how the fragmented spin liquid phase may be reached in an artificial kagome ice, we have fabricated these two lattice designs and we have subjected them to a field demagnetization protocol. Al- though it turns out to be impossible to observe spin frag- mentation in any of our lattices, the physics differ sub- stantially in the two cases. Whereas assemblies of discon- nected nanomagnets are found in an equilibrated spin liq- uid microstate, connected lattices appear systematically frozen in a non-equilibrated state. In these connected lat- mailto:bmcecchi@ifi.unicamp.br 2 tices, certain magnetic correlations in particular strongly deviate from those predicted numerically. As we shall see below, this out-of-equilibrium physics can be interpreted as a manifestation of the field-induced domain-wall prop- agation that occurs upon demagnetization. Surprisingly however, this kinetic process of pure micromagnetic ori- gin can be accounted for by a kagome spin ice Hamilto- nian through an ad hoc strengthening of a set of magnetic couplings. In this framework, the connected lattices be- have as if they were at thermodynamic equilibrium. Be- yond kagome ice structures, these results suggest that the kinetic process associated to the field demagnetiza- tion protocol may be exploited to effectively adjust the magnetic couplings. II. EXPERIMENTAL AND NUMERICAL DETAILS A. Artificial kagome ice structures Two types of sample were patterned by electron-beam lithography: one with physically disconnected nanomag- nets [Figs. 1(a-d)], and another with connected nano- magnets [Figs. 1(e-h)]. While all lattices studied in this work contain approximately 1,000 permalloy nano- magnets having a thickness of 25 nm, their dimensions slightly vary. The elements in the connected (discon- nected) lattices have a width of 200 nm (150 nm), a length of 1000 nm (750 nm), and a center-to-center separation of 960 nm between nearest-neighbors. In both cases, a 5- nm-thick Ti bottom layer was used to promote adherence to the silicon substrate. Since our nanomagnets are not thermally active at room temperature, the lattices were demagnetized through a standard field protocols known to bring many artificial spin systems to equilibrium [41, 44, 50, 53, 54, 58–61]. To do so, the samples were rotated at about 10Hz in an in-plane, sinusoidal magnetic field of fre- quency 0.25Hz with a slowly decaying amplitude, which is linearly ramped from ∼ 100mT to zero over several days. The magnetic configurations resulting from the demag- netization protocol were imaged using magnetic force mi- croscopy (MFM) [Fig. 1(c,g)]. The magnetic microstates of the disconnected lattices are obtained rather conve- niently: each nanomagnet exhibits bright and dark con- trasts at its two extremities [Fig. 1(c)], revealing its Ising character [Fig. 1(d)]. In contrast, the determination of the magnetic microstates of the connected lattices is more subtle [16, 57, 62], as it relates to the orientation of an heart-like feature [45] [Fig. 1(g)]. B. Monte Carlo simulations We compared the properties of our structures to those obtained by simulating spin models with an Hamiltonian (h) (g) (f) (e) (d) (c) (b) (a) FIG. 1. Disconnected (left panels) and connected (right pan- els) artificial kagome ice networks. (a, e) Scanning electron microscopy images of the whole lattice (scale bars are 10 µm). (b, f) Atomic force microscopy images of smaller regions of the lattices (scale bars are 2µm), and (c, g) corresponding magnetic images after demagnetization. (d, h) Magnetic con- figurations extracted from the magnetic images. of the form: H = − ∑ i̸=j Jij S⃗i · S⃗i, (1) where Jij is the coupling strength between the spins S⃗i and S⃗j . For dipolar coupling strengths, Jij are given by Jdip ij = D [ 3(S⃗i · r⃗ij)(S⃗j · r⃗ij) r5ij(S⃗i · S⃗j) − 1 r3ij ] , (2) with D denoting the dipolar constant, and r⃗ij the sep- aration vector between site i and site j. The values of the dipolar coupling strengths for the first seven neigh- bors are reported in Table I [neighbors are defined in the inset of Fig. 2(a)]. Monte Carlo simulations of these models were per- formed on a kagome lattice with 18 × 18 × 3 sites, i.e., 3 Neighbor 1st 2nd 3rd 4th 5th 6th 7th j index β γ ν δ τ η ϕ Jdip αj 1 -0.137 0.045 -0.036 0.014 0.037 0.014 TABLE I. Dipolar coupling terms Jdip αj for the first seven neighbors, given in units of Jdip αβ . having approximately the same size as the experimen- tal arrays, and assuming periodic boundary conditions. For the dipolar model discussed in Sec. III A, interactions were considered up to a cutoff radius slightly smaller than half the lattice size, and the dynamics includes both sin- gle spin and loop flips to ensure ergodicity at low tem- peratures [37, 63]. For the modified models discussed in Sec. III B, interactions were restricted to the first seven neighbors, and only single spin flips were implemented. In either case, the system was cooled from T/Jαβ = 100, using 103 modified Monte Carlo steps (MMCS) for equi- libration, followed by 103 MMCS for measurements at each probed temperature [64]. C. Correlation analysis For each imaged microstate, we computed the pair- wise spin correlations Cαj = ⟨S⃗α · S⃗j⟩ for the first seven neighbors, and the nearest-neighbor charge correlation Q = ⟨QiQi+1⟩, where the ⟨⟩ averages are taken over all pairs of neighbors. In addition, six different lattices of each type were imaged to improve statistics. The experimental set of spin correlators { Cexp αj } are compared to the numerical values CMC αj (T ) obtained with Monte Carlo simulations. Following previous works [60], we define the best fitting temperature of a given model as the temperature that minimizes the sum of squared residuals (SSR), SSR(T ) = 7∑ j=1 [ Cexp αj − CMC αj (T ) ]2 , (3) considering neighbors from j = 1 (nearest-neighbors) to j = 7 (seventh neighbors). III. RESULTS A. Out-of-equilibrium microstates The values of the nearest-neighbor charge correlation and the first seven spin-spin correlations, averaged over the six connected and six disconnected kagome lattices, are reported in Table II. We first compare these re- sults with the ground state correlations predicted by the nearest-neighbor kagome ice model [50, 65], also shown in Table II. At first sight, experimental and theoretical values agree reasonably well, both in sign and amplitude. The charge-charge correlation, however, differs substan- tially in the three cases. In particular, the charge corre- lator in the nearest-neighbor kagome ice model reaches a -0.116 plateau in the ice manifold, whereas it is twice and three times smaller in the disconnected and con- nected lattices, respectively. These differences cannot be accounted for by the theoretical standard deviations. Be- sides, careful inspection of the spin correlations also re- veals deviations between the two lattice types and the nearest-neighbor kagome ice model. The values mea- sured for the disconnected lattices slightly deviate from the predictions – especially for the second (Cαγ), third (Cαν) and fourth (Cαδ) correlators – but these differences are well accounted for by the dipolar kagome ice model, as we will see below. However, for the connected lat- tices, the values of Cαν and Cαη correlators are several times larger than expected, and here as well the differ- ences cannot be interpreted as statistical fluctuations. In this case, we stress that the Cexp αν and Cexp αη reach val- ues that are not even in the range of possible values of both the nearest-neighbor and dipolar kagome ice models [Fig. 2(a)], suggesting that another mechanism is at play. Disconnected lattice Connected lattice Nearest-neighbor model Cαβ 0.154 0.167 0.167 Cαγ -0.071 -0.049 -0.062 Cαν 0.085 0.210 0.101 Cαδ -0.104 -0.054 -0.075 Cατ 0.009 0.012 0.012 Cαη 0.019 0.079 0.019 Cαϕ 0.024 0.062 0.023 Q -0.214 -0.322 -0.116 TABLE II. Spin and charge correlators of the disconnected and connected artificial kagome ice networks (mean values computed by averaging over six distinct lattices in each case), and of the nearest-neighbor kagome ice ground state (obtained from Monte Carlo simulations). To be more quantitative, these experimental values are compared to the temperature dependence of the correla- tions obtained in the dipolar kagome ice [Fig. 2(a)]. Min- imizing the sum of squared residuals, we find that the model describes well the average measurements of the disconnected lattices at a fictional temperature T/Jαβ = 0.37. These arrays are equilibrated in the first spin liq- uid regime, and the fictional temperature is low enough to make the distinction between the nearest-neighbor model and its dipolar counterpart, consistent with previ- ous works [50, 57]. This is further confirmed by compar- ing the magnetic structure factors (MSFs) deduced from the Monte Carlo simulations at T/Jαβ = 0.37 [Fig. 2(b)] and the measurements [Fig. 2(c)]. This fictional temper- ature remains too high though to observe signatures of a magnetic fragmentation process [39, 42], as also revealed by the moderate value of the charge correlation (see Ta- ble II). The same correlation analysis performed on the con- 4 (a) Paramagnetic Spin liq uid Ground state Fragmented spin liq uid α β γ δ ν τ η φ FIG. 2. (a) Temperature dependence of the pairwise spin correlations predicted by the dipolar kagome ice model (solid lines), shown for the first six neighbors (see color code in the bottom right corner). Experimental correlators for disconnected (square markers) and connected (circle markers) artificial kagome ice are superimposed at their best fitting temperatures, T/Jαβ = 0.37 and T/Jαβ = 0.45, respectively. The inset shows the temperature dependence of the nearest-neighbor charge correlation. Magnetic structure factors of the: (b) dipolar kagome ice model at T/Jαβ = 0.37; (c) disconnected artificial kagome ice, averaged over six distinct lattices; and (d) connected artificial kagome ice, averaged over six distinct lattices. nected lattices shows that they are left out-of-equilibrium by the field demagnetization protocol. As mentioned above, this is unambiguously evidenced by the large val- ues of the Cexp αν and Cexp αη correlators, both falling out of the range predicted by the dipolar model [Fig. 2(a)]. We emphasize that these important differences are not due to a single lattice in the series that would artificially increase the average values of the correlators. Instead, all six measured lattices show the very same behavior, indicating that an ingredient is missing in our descrip- tion. Minimizing the SSR(T ) function, we find that a fictional temperature of T/Jαβ = 0.45 best describes the measurements [Fig. 2(a)]. With no surprise, the discrep- ancies in the correlations are clearly visible in reciprocal space, where the experimental MSF averaged over the six connected lattices [Fig. 2(d)] exhibits distinctive fea- tures compared to the one predicted by the dipolar model [Fig. 2(b)]. Two questions naturally arise at this point. Can we identify another spin Hamiltonian than reliably describes the physics of connected kagome lattices? What is the origin of the observed out-of-equilibrium physics? We address these two questions in the following sections. B. Identifying an alternative spin Hamiltonian The correlation analysis shows that the Cexp αν and Cexp αη correlators of the connected lattices strongly deviate from those expected in the dipolar kagome ice model. It is legitimate to assume that modifying the Jαν and Jαη coupling strengths could improve the agreement between numerical and experimental data. Of course, this is a crude argument as modifying a given coupling strength in the spin Hamiltonian will likely impact all correla- tors. Nevertheless, we expect to have the strongest ef- fect on the correlator that is directly associated to the coupling strength. The fact that Cexp αν and Cexp αη have 5 larger positive values than predicted [Fig. 2(a)] suggests to strengthen the ferromagnetic character of both Jαν and Jαη. To test this idea, we first increased Jαν , recomputed the temperature dependence of the spin correlators, and we recalculated the minimized sum of squared residu- als. Interestingly, we do observe that the agreement is improved by setting Jαν larger than its dipolar value Jdip αν . As shown in Fig. 3(a), SSR decreases substantially as Jαν increases, and reaches a minimum values when Jαν is about 2.25 times larger than Jdip αν . We emphasize that SSR takes into account the first seven spin corre- lators described above. Increasing Jαν thus leads to an overall improved agreement between the measurements and the predictions. This can be represented by plotting the single residuals ∆Cαj = Cexp αj − CMC αj , normalized to the standard deviation σαj of the predicted correlator CMC αj at the best fitting temperature. When this quan- tity ranges between −1 and +1, the difference between Cexp αj and CMC αj lies within one standard deviation. Set- ting Jαν = 2.25×Jdip αν , the agreement is clearly improved [compare the points from the dipolar (black dots) and modified (blue dots) spin Hamiltonians in Fig. 3(b)]. In fact, all correlators are well described by this modified model and only Cexp αη remains above one standard devia- tion. Following the same reasoning, we now fix Jαν = 2.25× Jdip αν and use Jαη as a free parameter. Performing the same analysis leads to a similar conclusion: increasing Jαη improves the agreement, which is optimum when Jαη = 2.20 × Jdip αη . All correlators now fit within one standard deviation [see orange dots in Fig. 3(b)]. To get better insights into the behavior of the modified model, we performed new Monte Carlo simulations with Jαν = 2.25 × Jdip αν and Jαη = 2.20 × Jdip αη , and we have recomputed the temperature dependence of the spin and charge correlations [Fig. 4(a)]. As revealed by the low- temperature values of the spin correlations, the ground state is a long-range ordered polarized configuration with all the spins of a given direction being ferromagnetically aligned. This is consistent with the large values of the Jαν and Jαη coupling strengths that favor the develop- ment of magnetic correlations along the main directions of the kagome lattice. We stress that magnetic fragmen- tation is absent in this spin Hamiltonian, and theQ = −1 value of the nearest-neighbor charge correlator, indicative of an antiferromagnetic charge order [Fig. 4(a)], is here associated to a long-range spin order. The spin liquid regime settles in at T/Jαβ ∼ 1.0 where Cαβ = 1/6, as expected when the ice rule is strictly obeyed. However, this spin liquid differs from the one ob- served in the dipolar kagome ice, as revealed by the tem- perature dependence of the spin-spin correlations [com- pare Fig. 2(a) and Fig. 4(a)]. This is also reflected in the magnetic structure factor, which exhibits distinct fea- tures [compare Fig. 4(b-d) and Fig. 2(b)]. Overall, the configurations observed experimentally are well described by the modified Hamiltonian when (a) S S R (b) FIG. 3. Results from fitting the spin correlators of connected artificial kagome ice networks using different spin models. (a) Sum of squared residuals SSR evaluated at the best fit- ting temperature, for models with increasing couplings Jαj between j neighbors. The solid blue curve is associated to the strengthening of Jαν (i.e., j = ν), with other couplings fixed at their dipolar values. The lowest SSR value occurs at Jαν = 2.25Jdip αν , marked by the dashed blue line. The solid orange curve shows the effect of strengthening Jαη (i.e., j = η), with Jαν = 2.25Jdip αν and the other couplings fixed at their dipolar values. The lowest SSR value is achieved at Jαη = 2.20Jdip αη , indicated by the dashed orange line. (b) Nor- malized residuals, ∆Cαj/σαj , of the first seven neighbors for different models, where ∆Cαj = Cexp αj − CMC αj and σαj is the standard deviation of CMC αj at the best fitting temperature. setting T/Jαβ = 0.12. This is illustrated in Fig. 4(a), where the average experimental correlators (large dots) are superimposed with those of the model. Remarkably, the model’s MSF computed at the same temperature [Fig.4(d)] reproduces all the features observed in the ex- perimental MSF [Fig. 2(d)], including those not captured by the dipolar kagome ice model. It is worth noticing that good agreement is also found for the correlators of each of the six connected lattices separately [small dots in Fig. 4(a)]. The model then captures the physics of all our measurements and not only of the average behavior. 6 ParamagneticSpin liquid Ground state α β γ δ ν τ η φ (a) (b) (c) (d) FIG. 4. (a) Temperature dependence of the pairwise spin correlations predicted by the spin model with strengthened couplings Jαν = 2.25Jdip αν and Jαη = 2.20Jdip αη (solid line), shown for the first seven neighbors (see color code in the bottom right corner). Experimental correlators of connected artificial kagome ice averaged over six lattices (circle markers) are superimposed at the best fitting temperature T/Jαβ = 0.12. The correlators of each connected lattice (small dots) are also reported. The inset shows the results for nearest-neighbor charge correlation. (b,d) Computed MSFs at (b) T/Jαβ = 0.050, (c) T/Jαβ = 0.12, and (d) T/Jαβ = 0.40. Interestingly, the six connected lattices show substan- tially different fictional temperatures. We also note that the agreement is better when the measurements reach the low temperature regime, whereas slight deviations from the predictions are observed for the lattices exhibiting the highest fictional temperature (T/Jαβ = 0.27). This effect is not only seen in the spin correlations, but on the charge correlation as well [Fig. 4(a)]. In fact, we ex- pect to observe stronger deviations with the model as temperature increases. The reason is that the demag- netization protocol we use competes with the polarized ground state, which is reached at the very beginning of the protocol when the applied field is sufficiently large to saturate the kagome lattice. Long demagnetization times strongly reduces, by definition, the residual magne- tization, thus implying a higher temperature when com- pared to the Monte Carlo simulations. Ultimately, an efficient demagnetization protocol should bring the lat- tice in a demagnetized configuration satisfying the ice rule. In other words, a perfectly demagnetized sample will be brought in a conventional spin liquid microstate such as the ones observed in disconnected lattice, and not in a high-temperature paramagnetic state as predicted by the model. We thus expect deviations with the model as the associated temperature is large, consistent with our experimental findings. C. Kinetic effects hindering equilibration Our results naturally lead to the question of why the connected artificial kagome ice fails to reach equilibrium, contrary to what is observed in disconnected lattices. We interpret this difference as a micromagnetic effect related to the field-driven domain wall propagation occuring in a two-dimensional network. First, we recall that in assemblies of disconnected nanomagnets, the (single spin flip) dynamics is associ- ated to the magnetization reversal of individual elements. This generally occurs through the nucleation of a reversed volume that subsequently expands in the entire element. An energy barrier must then be overcome. Once it hap- pens, the global stray field emitted by all nanomagnets is modified, potentially triggering the reversal of a neigh- boring element. In a connected lattice however, the spin dynamics is governed by the propagation of magnetic do- 7 FIG. 5. Length distribution of zigzag chains comprising ferro- magnetically aligned elements in disconnected and connected lattices. The length is defined as the number of elements in the chain, with counting done over all six samples of each lat- tice type. main walls within a single, fully connected network. The energy barrier to overcome is now mainly given by the pinning field at the vertex sites. Such a domain wall propagation has been already investigated in different contexts involving a honeycomb network [66–71]. Second, we note that the β, ν, and η neighbors corre- spond to successive spins aligned along the main direc- tions of the kagome lattice. The large positive values of Cexp αν and Cexp αη in connected lattices indicate the presence of ferromagnetic zigzag chains longer than what is ex- pected in the conventional spin liquid manifold. This can be quantified by plotting the length distribution of zigzag chains for the two lattice types, which suggests that a field-induced domain wall propagation favors longer head-to-tail lines in connected lattices (Fig. 5). The pres- ence of these polarized lines explains why our measure- ments are better described once coupling terms linking spins along a zigzag chain are strengthened. This being said, we must point out that zigzag chains are also observed in disconnected artificial kagome ice under an applied field [72–74], and are not specific to connected lattices. Our interpretation is that domain wall zigzag trajectories are kinetically favored over large distances in connected lattices, a process that competes with the influence of the magnetostatic interaction that energetically favors flux closure local configurations. In connected lattices, washing out the zigzag trajectories by a field protocol is challenging, whereas this effect can be strongly reduced in disconnected lattices. IV. SUMMARY AND PROSPECTS To summarize, we have studied two types of artificial kagome ice structures, the first consisting of an assembly of nanomagnets coupled by magnetostatics, the second being a single micromagnetic object having the form of a honeycomb network. When these two systems are sub- jected to a field demagnetization protocol, different be- haviors are observed. In the first case, we find spin con- figurations that are representative of a physics in ther- modynamic equilibrium, well described by the dipolar kagome ice model. In the second case, however, the spin configurations show strong signatures of non-equilibrated physics that we attribute to the field-induced domain wall propagation taking place during the demagnetiza- tion protocol. These lattices are characterized by a high density of ferromagnetically polarized zigzag lines, which induce unusually strong spin correlations along the main directions of the kagome lattice. Remarkably, this non- equilibrium physics can be reformulated into an effective physics at thermodynamic equilibrium by strengthening a set of well-chosen couplings. An alternative physics then emerges, in which a spin liquid, distinct from the kagome ice, sets in before experiencing a transition to a ferromagnetic long range order at low temperature. Our results suggest that one can now envision to in- duce non-equilibrium phenomena on purpose to explore a physics that differs from the one generally studied in nanomagnetic arrays. Carefully choosing the design of the lattice might permit a fine tuning of the mag- netic coupling strengths, at least in specific directions, thus allowing us to modify the effective phase diagram of the considered system in a controlled manner. In other words, model engineering might be potentially ap- proached by design. This strategy is in line with re- cent attempts to circumvent the limitations present in assemblies of nanomagnets in order to reach magnetic states that are difficult to access otherwise, specially in the kagome geometry [45–47]. It also extends to two- dimensional lattices the non-equilibrium phenomena ob- served in field-demagnetized artificial spin chains [61, 75], which are absent when similar systems are thermally an- nealed [76]. ACKNOWLEDGMENTS This work was supported by the São Paulo Research Foundation (FAPESP) through grants 2019/23317-6 and 2023/00132-6, by INCT of Spintronics and Ad- vanced Magnetic Nanostructures (INCT-SpinNanoMag) through CNPq 406836/2022-1, and by Agence Nationale de la Recherche through Projects No. ANR-22-CE30- 0041-01 “ArtMat”. 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B 111, L020408 (2025). https://doi.org/10.1088/1367-2630/17/1/013054 https://doi.org/10.1088/1367-2630/17/1/013054 https://doi.org/https://doi.org/10.1002/adma.202008135 https://doi.org/10.1038/nphys2669 https://doi.org/10.1038/nphys1794 https://doi.org/10.1038/nphys1794 https://doi.org/10.1088/1367-2630/15/3/035026 https://doi.org/10.1088/1367-2630/15/3/035026 https://doi.org/10.1103/PhysRevB.102.224420 https://doi.org/10.1103/PhysRevB.102.224420 https://doi.org/10.1103/PhysRevB.109.054425 https://doi.org/10.1103/PhysRevB.109.054425 https://doi.org/10.1103/PhysRevB.111.L020408 Out-of-equilibrium microstates but effective thermodynamics in artificial kagome ice networks Abstract Motivation Experimental and numerical details Artificial kagome ice structures Monte Carlo simulations Correlation analysis Results Out-of-equilibrium microstates Identifying an alternative spin Hamiltonian Kinetic effects hindering equilibration Summary and prospects Acknowledgments References