Topology by Design in Magnetic Nano-Materials: Artificial Spin

TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE

CRISTIANO NISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

Contents 1. Introduction2 2. Frustration, Topology, Ice, and Spin Ice4 3. Simple Artiﬁcial Spin Ices9 3.1. Kagome Spin Ice9 3.2. Square Ice 13 4. Exotic States Through Vertex-Frustration 15 5. Emergent Ice Rule, Charge Screening, and Topological Protection: Shakti Ice 17 6. Dimensionality Reduction: Tetris Ice 21 7. Polymers of Topologically Protected Excitations: Santa Fe Ice 22 8. Conclusions 24 References 24

Abstract Artificial Spin Ices are two dimensional arrays of magnetic, interacting nano-structures whose geometry can be chosen at will, and whose elementary degrees of freedom can be characterized directly. They were introduced at first to study frustration in a controllable setting, to mimic the behavior of spin ice rare earth pyrochlores, but at more useful temperature and field ranges and with direct characterization, and to provide practical implementation to celebrated, exactly solvable models of statistical mechanics previously devised to gain an understanding of degenerate ensembles with residual entropy. With the evolution of nano–fabrication and of experimental protocols it is now possible to characterize the material in real-time, real-space, and to realize virtually any geometry, for direct control over the collective dynamics. This has recently opened a path toward the deliberate design of novel, exotic states, not found in natural materials, and often characterized by topological properties. Without any pretense of exhaustiveness, we will provide an introduction to the material, the early works, and then, by reporting on more recent results, we arXiv:1711.00921v1 [cond-mat.mes-hall] 2 Nov 2017 will proceed to describe the new direction, which includes the design of desired topological states and their implications to kinetics.

Date: March 12, 2018. 1 2CRISTIANONISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

1. Introduction From quasi-particles, fractionalization, pattern formation, to the nano–machinery of life in DNA replication and transcription, or to the coherent behavior of a flock, an ant colony, or a human group, emergent phenomena are generated by the collective dynamics of surprisingly simple interacting building blocks. Indeed, much of the more resent research in condensed matter pertains to the modeling of unusual emergent behaviors, typically from correlated building blocks in natural materials, either at the quantum or classical level [1]. A few years ago [2] we proposed a different approach: design, rather than simply deduce, collective behaviors, through the interaction of simple, artificial building blocks whose interaction could lead to exotic states not seen in natural materials. Arrays of elongated, mutually interacting, single-domain, magnetic nano-islands ar- ranged along a variety of different geometries, (Fig. 1) were ideal candidates. The magnetic state of each island could be described by a classical Ising spin, and advances in lithography allowed their nano–fabrication in virtually any geometry. The advantage of such approach is twofold: (1) the low energy dynamics, which underlies possible exotic states, is dictated by geometry, which here is open to design; (2) characterization methods—Magnetic Force Microscopy (MFM), PhotoElectron Emission Microscopy (PEEM), Transmission Electron Microscopy (TEM), Surface Magneto-Optic Kerr Effect (MOKE), Lorentz Microscopy— allow direct visualization of the magnetic degrees of freedom for unprecedented validation. Nano-scale is an good choice choice: the size of the building blocks, which are shape- anisotropic, elongated nano–islands (typically, NiFe alloys 200 × 80 × 5 − 30 nm3 patterned by nano-lithography on a non-magnetic Si substrate), has to be inferior to the typical magnetic domain, to provide single domains with magnetization directed along the principal axis, that can be interpreted as switchable spins. These Artificial Spin Ices (ASI) were employed at first to study frustration in a controllable setting, to mimic the behavior of spin ice rare earth pyrochlores, but at more useful temperature and field ranges and with direct characterization, and to provide practical implementation to celebrated, exactly solvable models of statistical mechanics previously devised to gain an understanding of degenerate ensembles. Soon, a growing number of groups has extended the use of ASI [3], to investigate topological defects and dynamics of magnetic charges [4,5,6,7,8,9, 10], information encoding [11, 12], in and out of equilibrium thermodynamics [13, 14, 15, 16, 17, 18, 19, 20, 21, 22], avalanches [23, 24], direct realizations of the Ising system [25, 26, 27], magnetoresistance and the Hall effect [28, 29], critical slowing down [30], dislocations [31], spin wave excitations [32], and memory effects [33, 34]. Meanwhile similar strategies [35, 36, 37, 38, 39, 40] have found realization in trapped colloids [41, 42, 43], vortices in nano-patterned superconductors [44, 45] and even at the macroscale [46]. With the evolution of nano–fabrication and of experimental protocols it is now possible to characterize the material in real-time, real-space [47, 48, 49, 50, 51], and to realize virtually any geometry, for direct control over the collective dynamics. This has recently opened a path toward the deliberate design of novel, exotic states [52, 53, 54, 55, 56] not found in natural materials [57, 58]. TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 3

Figure 1. Artificial spin ice in its most common geometries. Left: Atomic force microscopy image of square ice showing its structure (figures from [2]). Center: a magnetic force microscopy image of square ice, showing the orien- tation of the islands’ magnetic moments (north poles in black, south poles in white); Type-I (pink) Type-II (blue) and Type-III (green) vertices are highlighted (see the text and Fig. 7 for a definition). Right: schematics of honeycomb spin ice.

Frustration is a fundamental ingredient in design: it controls the interplay of length and energy scales, dictating the emergent dynamical properties that lie at the boundaries between order and disorder, and leading to a lively, quasi-disordered ensemble called ice manifold, to be exploited in the design of exotic behaviors. Correlated spin systems have of course a long history in Physics. In classical statistical mechanics, the Ising model [59] paved the way to our understanding of long-range order from symmetry breaking as a second order phase transition, universality classes and scaling [60], and finally the renormalization group [61] with implications reaching well beyond condensed mater systems [62]. However, frustrated spin systems often do not order, generally resulting in quasi-disordered manifolds governed by some geometric or topological rule. Often their collective dynamics lends itself to emergent descriptions that are only partially reminiscent of the constitutive spin structure. The situation is somehow similar to everyday life, where frustration results from a set of constraints that cannot be all satisfied at the same time, leading to a manifold of compromises among which the choice is most often equivalent and can be influenced by a small bias. Thus, obstructed optimization provides high susceptibility that can generate the complex social dynamics we witness everyday. These analogies between social settings and frustrated materials are not merely philosophical: ideas borrowed from the frustrated spin ice physics have been exploited in social networks to describe wealth allocation [63]. Much as in life, frustration is understood in Physics as a set of constraints that cannot be all satisfied. Typically the constraint is the optimization of an energy, usually the pairwise interaction between elementary degrees of freedom. This too leads to a degenerate manifold 4CRISTIANONISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA which preserves non-zero entropy density at low temperatures, in apparent violation of the third law of thermodynamics. We will see how frustration is exploited in the design of artificial spin ices. Initially, the aim was pure exploratory science, with the goal to understand frustration and disorder in a controllable environment that could be characterized directly. These materials could mimic the frustrated ice rule that defines the exotic manifold of rare earth pyrochlores (see below), yet at room temperature rather than at the Kelvin scale. They could also provide the first realization of the celebrated exactly solvable models of statistical mechanics, such as the antiferromagnetic Ising system on a triangular lattice described above, or the various vertex models introduced and/or solved by Lieb, Wu and Baxter between the late 60s and early 80s [64, 65, 66, 67]. As both experimental protocols and theoretical understanding evolved, however, it became clear how the material could open new paths in a material-by-design effort: instead of finding, more or less serendipitously, natural materials of interesting or novel behavior, one could think about a bottom-up approach, where a suitable design could produce desired exotic properties. In this chapter we will start with a brief description of fundamental concepts and then earlier realizations, fleshing out the basics pertaining to their experimental protocols, nano- fabrication, and characterization (for details we refer to references or to the following reviews [3, 68, 69]. Geometric frustration should really be called topological, as it is essentially a topological property, yet it is based on interaction, which is instead geometric, and generally not topologically invariant: we will discuss the two issues in parallel, and see how new approaches and new designs, based on a different level of frustration, so called vertex-frustration, can indeed allow access to bona fide topological states.

2. Frustration, Topology, Ice, and Spin Ice The concept of geometric frustration in its broader mathematical form involves a geometric system, a manifold of degrees of freedom and a set of prescriptions on how they should arrange with respect to each other. The system is frustrated if there are loops along which not all these prescriptions can be satisfied (Fig. 1). Clearly the concept is very general and extends beyond Physics. One recognizes topology immediately in the nature of such definition: any homotopy, that is any continuous transformation that does not tear those loops, will lead to a system of the same frustration. In Physics, in general (a) these “prescriptions” correspond to the optimization of a certain energy, and (b) that energy is usually a pairwise interaction between binary degrees of freedom. An early example is the famous antiferromagnetic Ising model on a triangular lattice [70], a system of binary spins interacting antiferromagnetically on a triangular latice (Fig. 2). There the interaction among nearest neighbor spins cannot be satisfied simultaneously on a triangular plaquette, leading to a disordered manifold. The disorder is, however, non- trivial, and its entropy per spin is not merely s = kB ln(2) ' 0.6931kB, because rules apply, due to frustration: of all the energy links, only one per plaquette is frustrated, and it relieves the frustration on adjacent plaquettes. Indeed its entropy per spin at T = 0 is TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 5

Figure 2. Geometric frustration can be understood schematically as a set of prescriptions that cannot be satisfied simultaneously around certain loops. The red link on the figure on the left represents an “unhappy link” in a generally frustrated system. More specifically, for an Ising antiferro- magnet (right) the loop in question is a loop of interactions among nearest neighbors. On a triangular lattice, triangular loops are frustrated, as one of the three links (red) must be unhappy.

s ' 0.3383kB, different from zero, in violation of the third law of thermodynamics, and about half of the entropy of a completely random configuration. However, as most realistic interactions in Physics are geometric (for instance, the dipolar interaction between magnets is anisotropic) they immediately break the topological structure of frustration in a real system. We will discuss later how renouncing the point (b) opens the way to great freedom of design in artificial spin ices. Perhaps the first famous occurrence of frustration in the history of Physics pertained to water ice. In the 1930s Giaque and Ashley [71, 72] performed a series of carefully conducted calorimetric experiments, deduced the entropy of water ice at very low temperature, and found that it was not zero. The answer to this mystery would be provided by Linus Pauling a few years later [73]. Ice comes in many crystalline forms, but all imply oxygen atoms residing at the center of tetrahedra, sharing four hydrogen atoms with four nearest neighbor oxigen atoms (Fig. 3). Two of such hydrogens will be covalently bond, and two will realize an hydrogen bond: two are “in”, two are “out” of the tetrahedron. This is the so called ice-rule previously introduced by Bernal and Fowler [74]. Each tetrahedron has 6 admissible configurations out of the 24 = 16 ideally possible, and the collective degeneracy grows exponentially in the number of tetrahedra N as W N , leading to a non-zero entropy per tetrahedron s = kB ln W for this disordered manifold. In what can be considered as one of the most precise and felicitous back-of-the envelope estimate in the history of statistical 6CRISTIANONISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

Figure 3. The ice rule. Left: In water ice oxygen atoms sit at the center of tetrahedra, connected to each other by a hydrogen atom. Two of such protons are close (covalently bonded) to the oxygen at the center, two are further away, close to two of the four neighboring oxygens. Right: One might replace this picture with spins pointing in or out depending on whether the proton is close or far away. Then two spins point in, two point out. This corresponds to the disposition of magnetic moments on pyrochlore spin ices, rare earth titanates whose magnetic ensemble does not order at low temperature, because of frustration, and, much like water ice, provides non-zero low temperature entropy density. (Figures from ref [76]).

mechanics, Pauling counted such degeneracy as W = 3/2, remarkably close to both the experimental value and to the numerical value (W = 1.50685 ± 0.00015 [75]). These works pointed to the reality of exotic disordered states in the most common and vital substance on earth. Decades later, they also motivated the introduction by Lieb, Wu, and Rys of simplified models of mathematical physics, known as vertex models, which could in many cases be solved exactly [64, 65, 66, 77, 78]. Those are two dimensional models of in-plane spins impinging on vertices, where different energies are associated to different vertex-configurations, and whose statistical mechanics is usually solved via transfer matrix methods. Ice-like systems have then received renewed interest in the 1990s, when unusual behaviors were discovered in the low temperature regime of rare earth titanates such as Ho2Ti2O7 whose magnetic moments exhibit a net ferromagnetic interaction between nearest neighbor spins, yet no ordering at low temperature, suggesting strong frustration. Similar to protons in water ice, the magnetic moments of these materials reside on a lattice of corner-sharing tetrahedra, and they are constrained to point either directly toward or away from the center of a tetrahedron (Fig. 3). The resulting ferromagnetic interaction favors a 2-in/2out TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 7

Figure 4. The ice manifold (top left), an ensemble of spins obeying the ice rule (in each vertex two arrows point in, two out). One can obtain another realization of the ensemble only by flipping a proper loop of spins. Flipping a single spin creates a couple of magnetic monopoles of opposite charge (positive is red, negative is blue). The monopoles can be separated by further spin flips (creating a “Dirac string”, shown in red), interact via Coulomb interaction, but are topologically protected, as they can only be created and annihilated in pairs. ice-rule. The similarity noted by Harris et al [79] was confirmed experimentally by Ramirez et al. [80]. Besides providing important model systems with novel field-induced phase transitions and unusual forms of glassiness, and an early and practical example of a classical topological state, spin ices harbor a new fractionalization phenomenon in their low energy dynamics: emergent magnetic monopoles [81, 76]. To facilitate understanding consider the two-dimensional schematics of Fig. 4, which represent a disordered ice manifold, an ensemble of spins where all the vertices obey the ice rule. The reader will notice that it is impossible to explore the manifold by single spin flips, without breaking the ice-rule. Only by flipping proper loops of spins we can obtain a new configuration within the ice-manifold. This is already a hint of the topological nature of the state. If we flip one spin only, we create two defects (3-in/1-out and vice versa). We can separate those defects by further flips, and we have two deconfined magnetic monopoles, and one can prove through multipole expansion that their interaction is Coulomb [76]. 8CRISTIANONISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

Of course these monopoles are in effect simply the opposite ends of a long, floppy dipole, in red in figure, called the Dirac string; however, owing to the disorder of the manifold, the system is no longer reminiscent of the Dirac string connecting the monopoles. Thus, excitations over the ice manifold can be described by a fractionalization of the spins into individual, separable magnetic charges which interact via a Coulomb law. In view of the more complex geometries that we will discuss later, let us generalize the notion of ice manifold and ice rule for a general lattice, or graph, or network [63], whose edges are spins impinging in vertices of various coordination z. Then we say that a vertex of coordination z with n spins pointing toward it has topological charge q = 2n − z, corresponding to the difference between spins pointing in and out. In general, we call spin ice systems those in which |q| is minimized locally at each vertex (typically, but not necessarily, by nearest neighbor spin-spin interaction). For a lattice of even coordination, such as the square ice or pyrochlore ice introduced before, the ice manifold is characterized by zero charge on each vertex. However, for lattices of odd coordination there cannot be any charge cancellation, and thus in the ice manifold each vertex will have charge q = ±1 in equal fraction, as the total charge of a system of dipoles must always be zero. That is the case of Kagome ice, which we will discuss in the next section. These magnetic charges are topologically protected monopoles in square or pyrochlore ice: their magnetic charge is indeed also a topological charge and one can see from Fig. 3 that charges can only be created and annihilated in opposite pairs. One could indeed create a single monopole in an open system but that would simply imply pushing the second one at the boundaries. If we, however, placed the ensemble of Fig. 3 on a torus, thus without boundaries, then clearly there would not be a net monopole charge in the bulk. Even in an open system, the total net charge will be proportional to the flux of the magnetic moment through the boundaries, and as the latter is bound by the net magnetization of the spins, one finds that the density of net charge must scale with the reciprocal length of the system. These two-dimensional considerations extend to the three-dimensional spin ice. In more theoretical terms, the topological state of spin ice is an example of a so called Coulomb phase [82], a phase described not by an order parameter or a symmetry breaking, such as the ordered phases that fall within the Landau paradigm, but rather by a solenoidal emergent field (the coarse grained magnetization M~ ). One can say that the ice rule and its charge cancellation corresponds to a divergence-free condition ∇~ · M~ = 0 on each vertex. Then, a monopole in ~x0 is a source of divergence ∇~ · M~ = qmδ(~x − ~x0) of charge qm = Qq where Q = M/L (the magnetic moment M of the spin divided by its length L), and q is the topological charge of the vertex defined above. This is considered as an example of classical topological order [83]. Indeed, while quantum topological order has provided a valuable framework to conceptualize disordered states of spin liquids that escape a Landau symmetry breaking paradigm and cannot be obviously characterized by local correlations [84, 85], the importance of topological states had been recognized even earlier in classical physics [86]: in the theory of dislocations [87], liquid crystals [88], or topological transitions [89]. Recently, whether in direct analogy with quantum physics [90], in purely abstract terms [91, 92], or motivated TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 9 by real systems such as pyrochlore spin ices [83, 93, 82], a consistent notion of classical topological order in discrete systems has been proposed, to conceptualize (i) a degenerate, locally disordered manifold (ii) described by a topologically non-trivial, emergent field (iii) whose topological defects (in spin ice, magnetic monopoles [81, 76]) coincide with excitations above the manifold. Topological protection implies that states within the manifold can be linked only via collective changes of entire loops of a discrete degree of freedom. Thus any realistic low- energy dynamics happens necessarily above the manifold, through creation, motion, and annihilation of pairs of protected topological excitations. Typically, their constrained and discrete kinetics leads to ergodicity breaking, fractionalization and thus various forms of glassy behaviors [91, 94]. We will see later that such order can also be found in novel, non- trivial geometries of artificial spin ice characterized by vertex-frustration, such as Shakti spin ice.

3. Simple Artificial Spin Ices After introducing the main concepts, we warm up to the field of artificial spin ice by summarizing briefly the early work on classical geometries, based on the square and honeycomb lattices. Further, more general details about fabrication anc characterization will be discussed in the context of these early realizations. 3.1. Kagome Spin Ice. Even before artificial spin ice realizations [95, 96, 14] (Fig. 5) honeycomb structures have been extensively studied theoretically as they describe the two- dimensional behavior of the three dimensional spin ice pyrochlores under a magnetic field aligned along a particular crystalline axis. A honeycomb ice is often called the Kagome spin ice, as the spins reside on the edges of a honeycomb lattice, which is a Kagome lattice, the honeycomb dual lattice. In the context of artificial spin ice, Kagome represented the only simple geometry with a degenerate ice manifold. Indeed, as we will see in the next section, square ice has a frustrated yet perfectly ordered, antiferromagnetic ground state. Before proceeding with Kagome ice, some more general details on artificial spin ice materials are in order, starting with the energetics involved. In general, magnetic, elongated nano-islands can be described as nano–spins, binary degrees of freedom describing their magnetization along their principal axis. This is, however, already an approximation of the magnetic texture of the nano-structure: indeed both direct characterization and mi- cromagnetic simulations show potentially significant relaxation of the magnetization field at the tips of the islands, due to the local field of the surrounding islands. A further approximation, which seems to work surprisingly well, implies describing the inter-island interaction via a vertex model. There we assign energies to the various vertex configurations as in Fig. 5. The nano-islands being magnetic dipoles, one expects this approach to eventually break down. It does indeed in enticing ways, revealing inner, low entropy phases within the ice manifold. The equilibrium phases of the system have been investigated numerically [97, 98] via Metropolis Monte-Carlo simulations with full dipolar interaction. Figure 5 shows that at high temperature the system is paramagnetic. As temperature is reduced, we cross over week ending PRL 105, 047205 (2010) PHYSICAL REVIEW LETTERS 23 JULY 2010

the N vertices of a given lattice, each allocated among the a four degenerate vertex types according to a distribution N of degeneracy q , I; ...; IV. Calling D=N and ¼ ¼ N =D, we consider S lnM and maximize it under a vertex-energy¼ constraint¼ on the ensemble of defected vertices, or ln 1 ln 1 IV À Àð ÀIV Þ ð vÀ ÞÀ e I E E , where I lnq is the 2a 1 µm ð ¼ À Þ ¼À ¼ ‘‘entropy’’P of the defected ensemble. WeP obtain a canoni- cal distribution for the defects q exp E ðÀ e Þ (1) ¼ Z ð eÞ Type I Type II Type III Type IV [Z e is deﬁned by normalization of ] as well as an 10CRISTIANO(12.5%) NISOLI,(25%) THEORETICAL DIVISION,(50%) LOS ALAMOS(12.5%) NATIONALexpressionð LABORATORY,Þ for the auxiliary LOS ALAMOS, quantity NM, 87545, USA week ending PRL 106, 207202 (2011) PHYSICAL REVIEW LETTERS 20 MAY 2011 1 ; (2) ð eÞ¼exp 1 b ½À ð eÞ þ collapse with the critical exponents of the 2D Ising uni- where e is obtained by substitution of Eq. (1) into the versality class. Figures 3(c) and 3(d) show the scaling expressionð forÞ . Equations (1) and (2) provide the relative behavior of the speciﬁc heat c and charge susceptibility vertex population densities as 1Q. In obtaining the scaled curves, we have subtracted

nI I;nIII III;nIV IV; a size-dependent background contribution from the spe- ¼ ¼ ¼ (3) cific heat. n 1 : II ¼ð À Þþ II The spin order emerging on top of magnetic charge 2a 1 µm We compute the vertex energies by using a ‘‘dumbbell’’ order is expected to be that of the p3 p3 type shown model (as in Ref. [29]), in which the magnetic dipole is in Fig. 1(c), the same as in the short-rangeÂ model with an treated as a finite-size dumbbell of monopoles, and we ffiffiffi ffiffiffi antiferromagnetic second-neighbor exchange [12]. This Type I Type II consider only interactions between monopoles converging 1 can be understood as follows. In a charge-ordered state, (75%) (25%) in each vertex: Energies then scale as a l À (where a is the lattice constant and l the lengthð ofÀ theÞ islands). By every triangle has two majority spins pointing into (or out FIG. 1 (color online). Square and hexagonal artificial spin ice. imposing EI 0 and EIII 1, one finds EII of) the triangle and a minority spin pointing the other way. FIG. 2 (color online).¼ Temperature¼ dependence of¼ the (a) Schematics (top left) and MFM (top right) of the square p2 1 = p2 1=2 and E 4p2= 2p2 1 . In this specific heat c T and entropy perIV spin s T of the dipolar spin Such states can be represented by dimer coverings of a arrays and the 16 vertices of the square artificial ice (bottom). simpleð À dumbbellÞ ð À modelÞ of the¼ energetics,ð theÀ ratiosÞ be- ice (1)ffiffiffi withð (a)Þffiffiffi ferromagnetic exchangeffiffiffið Þ ffiffiffi J 0:5D and honeycomb lattice [Fig. 1(d)]; the dimers indicate loca- (b) Schematics (top left) and MFM (top right) of the hexagonal tween different vertex energies are independent¼ of the arrays with the 8 vertices of the hexagonal. White arrows show(b) antiferromagnetic exchange J 2:67D. The linear size tions of minority spins. The energy of such a state is array lattice constant. ¼À the vertex ground states, and the percentages indicate the vertexof the systemAs a simple is L test12. of The the dashed basic assumptions lines show levels of the of model entropy determined by the interactions between minority spins multiplicity. s 0:693 (Ising¼ paramagnet), 0.501 (spin ice), and 0.108 alone. To see that, picture a minority spin as a super- ¼ above [1], we consider the quantities ln 5nI=2nII and (charge-orderedln 8n =2n spinas deduced ice) per fromspin. the measuredð n , n Þ, and position of a majority spin and a minorityÀ spin of ð I IIIÞ I II þ differently under ac demagnetization: Square ice never nIII. These quantities should be proportional to the recip- double strength 2; in this representation, majority spins À finds (or closely approaches) the ground state, whereas rocal effective temperatures EII e and EIII e, since our form an inert background. We thus arrive at a model of Figure 5. Top, from left to right: Schematicscharge-ordering and MFM transition image of is the evidenced by Fig. 3(b), where demagnetized hexagonal ice returns the vertex-model predictions for the vertex populations [Eq. (3)] at high dimers with point dipoles of strength 2" directed along the groundhexagonal state with arrays at most with sparse the excitations. 8 vertices of the honeycomb/KagomeBinder’stemperatures fourth-order are well artificial cumulants approximated for by different a purely canonicalL cross at dimers and towards triangles with positive charge. The spinWe first ice. consider White the arrows case of show the square the vertex ice arrays, Ice I with state,Tc anddistribution0:267 theD percentages. that assigns indi- an anomalous multiplicity of 5 lattice constant a 400, 440, 480, 560, 680, and 880 nm. rather than 4 to type-II vertices—a fact that can be checked interaction energy of two dimers depends on their mutual cate the vertex¼ multiplicity. Type-I vertices haveThe lowerZ2 energysymmetry than of Type-II the order parameter Q suggests that Theand external correspond field in to our the rotational generalized demagnetization ice rule. Temperature isthe charge-orderingby direct dependencecalculation transition but (top which is in also the seems universality reasonable class as of position. It is minimized by increasing the number of initially strong enough to coerce every island into follow- there are four different multiplicities in the defected sam- right) of the specific heat c and entropy perthe spin Isings of model. the Kagome To verify spin this conjecture, we performed a second neighbors (distance between centers p3rnn) and ing the external field, but, as its magnitude decreases, ple and one in the background. In Fig. 2, we plot these two reducing the number of third neighbors (2r ). The dimer successiveice obtained islands by presumably ref [53]. begin The to dashed ‘‘fall away’’ lines from showfinite-size valuesquantities of scaling entropy against each analysis per other. spin A and linear found fit returns excellentE =E data nn ffiffiffi II III ¼ thes field,= 0. locked693 (Ising in by paramagnet),favorable magnetostatic 0.501 (Ice interactions I), and 0.1080:441 (charge-ordered, very close tospin the expected theoretical value configuration shown in Fig. 1(d) optimizes both. It is one of three states related to each other by lattice translations. The withice, their or Ice neighbors. II). Bottom: The accumulation the four phases of these of distinct KagomeE iceII=E orderedIII p2 by increasing1 = p2 1=2 0:453 obtained from ¼ð À Þ ð À Þ¼ corresponding spin order is shown in Fig. 1(c). There are a ‘‘defects’’temperature. carved in Figures the initial are uniform adapted set of from aligned ref type- [14, 53,the 30 dumbbell]. ffiffiffi approximation.ffiffiffi II vertices generates a well-defined statistical system. In an In Fig. 3(a), we plot the experimentally observed pop- total of 6 magnetic ground states related to each other by isotropic, vertex-gas approximation, where each vertex is ulations of each vertex type vs the effective reciprocal lattice translations and time reversal. 4nI treated as an independent entity, there are M temperature extracted from eEIII lnn and the theoreti- toward a disorderedN ice-manifold, called Ice I, where¼ each vertex has charge ±1. This is¼ as III The symmetry-breaking pattern of the magnetic order N! q ways to choose D defected vertices among cal curves for the vertex populations as a function of the N D ! N ! described above suggests that the magnetic transition is in muchð asÀ aÞ vertex model approximation would explain, as the ice-rule minimizes the energy of the vertices.Q the universality class of the 2D three-state Potts model. However, at lower temperature, we see a transition047205-2 toward charge ordering: the disor- However, Monte Carlo simulations of the dipolar ice dered plasma of magnetic charges residing on the vertices orders within an ionic crystal. model on systems up to L 36 fail to turn up any evidence ¼ The transition is of the Ising class and is due to the Coulomb interaction among magnetic of the Potts critical behavior. The lack of a singularity in charges. It can be replicated within a vertex-model approximation only by adding further the specific heat [Fig. 4(a)] is consistent with a Kosterlitz- interaction via Coulomb coupling between the charges of the vertices. Note that such state, Thouless (KT) transition. This unexpected result can be called Ice II, while being charge ordered, is still disordered in the spin structure, as there are

FIG. 3 (color online). Monte Carlo simulation of the charge- ordering transition in the spin-ice model (1). (a) and (b) show the temperature dependence of the staggered-charge order parameter FIG. 4 (color online). Specific heat c as a function of tempera- 4 2 2 Q and Binder’s fourth-order cumulant B4Q 1 Q =3 Q . ture in Monte Carlo simulations of (a) the dipolar spin-ice model À h i h i (c) and (d) show the scaling of specific heat c and charge-order Eq. (1) on the kagome and (b) the dimer model with attractive v2 susceptibility 1 Q2 Q 2 =NT using critical exponents on a honeycomb lattice. The peak of the specific-heat curve Q ¼ðh i À h i Þ 0, 7=4, and # 1 from the 2D Ising universality class. corresponds to p3 p3 magnetic and dimer ordering. ¼ ¼ ¼ Â 207202-3 ffiffiffi ffiffiffi

FIG. 3. (a) Lorentz contrast TEM image and (b) corresponding spin map of a square geometry sample that was heated above its TC and cooled back to room temperature showing perfect ground state ordering of type-I vertices. Such behavior is robust among the samples studied herein.

TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 11

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FIG. 4. Charge domain maps overlaid on top of Lorentz TEM images of kagome geometry f) samples of varying element lengtha g)a from 500 nm to 300 nm. Shorter h)belement b lengths tend to lead to greater degrees of charge-ordering and larger charge domains. We note that a magnetically saturated state would also be highly charge-ordering, however our samples are necessarily

demagnetized, with the net magnetization M x 0.2 for all data. Also see Fig. 1 for a close up of a Lorentz image without charge domains.

FigureFigure S8. Static S8. Staticstructure structure factors factors for magnetic for magnetic charge charge crystallites crystallites in artificial in artificial kagome kagome spin ice. spin ice. Figure 6. Magnetic(a) and charge(a) (b) and are (b) orderingthe are static the structurestatic in structure Kagome factors factors for magnetic ice. for magnetic (a)-(e) charge charge crystallites Charge crystallites for domain the for 260 the and 260 490 and nm 490 nm maps obtained via Lorenzlatticeslattice shown TEMs shown in Fig. relativein 4. Fig. 4. to the annealing of Fe-Pd alloy artificial Kagome icesThe of magnetic differentThe magnetic charge edge charge-charge- lengthcharge correlation correlation (from functions functions 500 at varying nm at varying to temperatures 300 temperatures nm) are showshown are shown in Fig. in S 9Fig.. S9. increasing size of theThe ionic differentThe different crystallites monopolar monopolar-charge of-charge pairs charges arepairs defined are as defined thein the lattice ininset the ofinset Fig. constant of (a) Fig. of (a)the of de-main the text.main The text. The correlationscorrelations clearly clearly increases increases as temperature as temperature is lowered is lowered toward toward the transition the transition temperature temperature Tc ~1.8 T c ~1.8 creases, and thus theJ. Fig.mutualJ . S Fig.10(a) S interaction10 shows(a) shows a snapshot a among snapshot of the of magneticmagnetic the magnetic charge charges charge configurations configurations increases. at the at effective the effective temperaturetemperature T=3.1 TJ,=3.1 corresponding J, corresponding to the tospin the-ice spin array-ice witharray lattice with latticeconstant constant 260 nm. 260 Both nm. theBoth the C is the charge-chargesnapshotsnapshot correlation configuration configuration and parameter the and corresponding the corresponding (C Fourier= Fourier transform 1 for transforma [Fig. fully S [F10ig.(b)] charge-S 10are(b)] similar are similar to those to those ordered state). Imagesfrom MFMfrom adapted MFM data shown data from shown in Figures ref in Figures [ 499(a,b)]. 4 and(a (f),b) S8(a) and Charge . S8(a) Fig. .S Fig.10 map(c )S 10 shows(c obtained) shows the structure the structure factor factor via MFM after annealingobtainedobtained by of averaging permalloyby averaging over ~3000 over Kagome ~3000different different charge ice charge configurations. of lattice configurations. constant 260 nm showing incipient domains of charge-ordering and (g) its static structure factor showing incipient peaks corresponding to crystalline order. (h) Static structure factor for lattice constant 490 nm, showing no incipient peaks. an exponentially growing (in the number of spins) number of possible spin configurations that correspond to the charge ordered state. Finally, further lowering the temperature, another transition leads to an ordered state, where order is brought in by the long range effects of the dipolar interaction. These states were variously investigated experimentally. Ice I proved easy to reach. In- deed, even non-thermal methods were able to reach it [95, 96, 14]. Those methods pertains FigureFigure S9. Magnetic S9. Magnetic charge charge-charge-charge correlation correlation obtained obtained from Montefrom Monte Carlo Carlo simulations simulations at at to thicker islands that are thusvarious notvarious temperatures. superparamagnetic temperatures. The temperature The temperature is measured at is room measured in units temperature in of units J. of J. (that is, do not flip their magnetization under thermal fluctuations). These islands are therefore coercive enough that MFM would provide a non-destructive characterization at room temperature. The AC demagnetization [15] of such samples is sufficient to reach the ice manifold, a fact WWW.NATURE.COM/NATURE | 9 which already points to its lack of topological protection. WWW.NATURE.COM/NATURE | 9 12CRISTIANO NISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

The facility with which such state could be reached is telling. Indeed, while magnetic charges are topologically protected in pyrochlore ices [83], as we saw above, they are not bona fide topological numbers in the ice manifold of Kagome ice. There, vertices of odd coordination can gain and lose charge freely from the surrounding, disordered, and overall neutral plasma of magnetic charges. Consequently, the ice-manifold can be explored from within by consecutive single-spin flips, without any need for collective loop-moves such as those shown in Fig. 4. Instead, the charge-ordered state, or Ice II, cannot be explored by individual spin flips. A glimpse of the Ice II phase shown in Fig. 5 should convince that any spin flip within the manifold will lead out of the manifold, as it will locally destroy the charge order. The Ice II state has been explored via thermal methods capable of providing a bona fide thermal spin ensemble. These are of three kinds: annealing at higher than room temperature followed by characterization at room temperature [50, 100, 99]; thermalization with real-time, real-space characterization [51, 49, 55]; and thermalization without real- space characterization [47, 30]. In the first, the material is not superparamagnetic at room temperature, but it is heated slightly above the Curie temperature of the nano-islands (which can vary, depending on the size and chemical composition of the nano-structure, from about 600 Celsius for permalloy down to about 100 Celsius for Fe-Pd alloys) and then annealed down into a frozen state, usually characterized via MFM. In the second method the nano-islands are chosen to be thin enough (usually thickness of 2-3 nm) to be superparamagnetic at room temperature or below, and thus need to be characterized via PEEM, at a proper beam source. In the third, various averaged quantities are extracted, such as the average flip rate of spins, through muon spectroscopy [30]—while spin noise spectroscopy [101, 102] should also be a viable method. Figure 6 shows the results of thermal annealing on artificial hexagonal ice made of permalloy [100], which demonstrate formation of crystallites of magnetic charges, due to the Coulomb interaction between the charges themselves. Control over the size of those ionic crystallites has also been obtained by employing an alloy of iron and palladium, rather than permalloy, which has a much lower Curie temperature [99]. However, nobody has yet reported any direct evidence of complete long range charge order in such a material, nor of the zero entropy phase of spin order (Fig. 5). Indirect indications that such low entropy phases within the Kagome ice manifold—or at least some kind of phases—can be reached were obtained via muon spectroscopy studies, involving islands that were too small and therefore too active to be imaged directly, but whose rate of magnetic flipping could be deduced from the relaxation time of muons im- planted on a gold cap over the two-dimensional array. There, the critical slowing down of the spins was measured and found to correspond to that of the numerically predicted transitions, where parameters for the numerical simulations were taken from the material [30]. These results, the first to probe deep inside the ice manifold of Kagome lattice, represent a strong corroboration of the existence of a complex phase diagram, most likely the theoretical predicted one. However, we should not forget that topological or ordered states can be hard to reach via spin dynamics of the Glauber kind [103], and that indeed the actual spin dynamics might be rather more complex than a Glauber model. Indeed these “spins” TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 13 are nanoscopic objects with their own magnetic reversal dynamics. Such specificity might bias certain kinetic pathways, leading to non-equilibration or ergodicity breaking even in Ising models that are not susceptible to these phenomena: thus the phases whose critical slowing down was experimentally revealed could be only reminiscent of the one predicted at equilibrium, which of course adds to their potential interest.

3.2. Square Ice. With the exception of the work of Tanaka et al [95], early works con- centrated on the square geometry (Fig. 1) [2, 13, 14]. Square ice also represented the benchmark on which to test demagnetization and annealing methods which lead to experimental protocols for thermal ensembles [18, 47, 50]. It is important to understand that square artificial spin ice does not resemble the square ice of Lieb [64], or the degenerate square ice described in Fig. 4, firstly because it admits topological defects in the form of magnetic monopoles absent in the six-vertex model, but most importantly because it is not degenerate. In this sense it shares similarities with the Rys F-model [78] but those should not be overstated, as the (physically unnatural) absence of monopoles in the latter leads to an infinitely continuous transition to antiferromagnetic ordering [65], whereas in the former the transition is of second order. Figure 7 shows the energetic hierarchy of vertices with 90◦ angles (including those of coordination z = 3, 2 to be discussed later). Because of the anisotropy of the dipolar interaction, nearest neighbor perpendicular islands interact more strongly than collinear ones, leading to lifting of degeneracy within the ice manifold. The system, if modeled at the vertex level, can be described as a J1,J2 antiferromagnetic Ising model on a square lattice, with a transition to antiferromagnetic ordering, which indeed has been obtained experimentally via thermal annealing, as shown in Fig. 7. Within the ordered state of square ice potentially interesting transitions have been proposed [104, 105, 106]: when the system is not degenerate, creating and separating a couple of monopoles requires energy proportional to the number of Type-II in the Dirac string (see Fig. 7). Much like quarks or Nambu monopoles [107] these pairs are linearly confined, and the tensile strength of their Dirac string drives the ordering as the temperature is reduced. However, one can imagine that a topological transition corresponding to monopole deconfinement might take place under proper conditions when the energy of the Dirac string is offset by its fluctuating entropy. Square ice, however, can be made properly degenerate, which means described by a spin ensemble such as the one of Fig. 4, revealing an emergent topological Coulomb phase. One way is to raise the vertical islands with respect to the horizontal ones [97], a method that has recently been pursued experimentally [108] demonstrating a degenerate manifold whose static structure factor coincides with the numerically computed one for a six-vertex model, thus providing the first artificial realization of a two dimensional Coulomb phase. We have also proposed to iterate such design on the axis perpendicular to the array, and we have designed layered structures that are geometrically different but topologically equivalent to three dimensional spin ice pyrochlores [109]. Those have not found realization yet. The only three dimensional realization of artificial spin ice was obtained by filling the voids of an artificial opal film with Cobalt [110, 111], a promising approach to bring to room RESEARCH LETTER RESEARCH LETTER

only begun to be explored experimentally18, and have not been visua- Supplementary Figs 1–3. Our expectation was that the island moments lizedonly directly begun as to experimental be explored evidenceexperimentally for the18, collective and have behaviour not been visua- of wouldSupplementary align into a Figsminimum-energy 1–3. Our expectation state on was cooling that thethrough island the moments Curie magneticlized directly charges as and experimental their usefulness evidence as degrees for the collective of freedom behaviour in low- of temperature,would align and into that a minimum-energy this state would state be on determined cooling through by the the lattice Curie energymagnetic collective charges physics. and We their use usefulness here a thermal as degrees annealing of freedom technique in low- geometry,temperature, that is, and the that artificial this spinstate ice would would be be determined thermalized. by the lattice to achieveenergy thecollective first experimental physics. We use validation here a thermal of the predicted annealing charge- technique Aftergeometry, annealing, that is, thewe artificialcollected spin MFM ice wouldimages be of thermalized. each array, which orderedto achieve phase: theincipient first experimental magnetic charge validation ordering of in the artificial predicted kagome charge- allowedAfter us annealing,to examine we the collected thermalized MFM state images of the of moment each array, arrange- which spinordered ice. phase: incipient magnetic charge ordering in artificial kagome ments.allowed Images us to of examine the annealed the thermalized artificial square state spin-ice of the moment arrays revealed arrange- Thespin current ice. gap between theory and experiment is in part a con- extensivements. Imagesdomains of of the ground-state annealed artificial ordering square (Fig. spin-ice 2). The arrays ground revealed state sequenceThe of current the present gap between limits on theory our and ability experiment to placeis the in magnetic part a con- consistsextensive of an domains antiferromagnetic of ground-state ordering ordering on (Fig. the 2). vertical The ground and hori- state RESEARCH LETTER momentssequence of artificial of the present spin ice limits in the on desired our abilitycollective to placethermodynamic the magnetic zontalconsists sublattices. of an antiferromagnetic The large size of the ordering domains on (in the many vertical cases and larger hori- state.moments Although of artificial a.c. demagnetization spin ice in the can desired produce collective well thermodynamic defined and thanzontal thesublattices. 10–20-mm-wide The large MFM size images) of the domains and the (in paucity many of cases isolated larger predictablestate. Although out-of-equilibrium a.c. demagnetization statistical can ensembles produce well of defined different and monopolarthan thecharge 10–20- excitations,mm-wide MFM especially images) in the and samples the paucity with theof isolated smal- monopolar charge excitations, especially in the samples with the smal- only begun to be explored experimentallyentropy,predictable it cannot18, and out-of-equilibrium approach have not thebeen two-fold visua- statistical degenerateSupplementary ensembles ground Figs of state 1–3. different ofOur expectationlest latticewas constants, that the underscore island moments the effectiveness of our annealing lest lattice constants, underscore the effectiveness of our annealing lized directly as experimentalsquare evidenceentropy, ice15 for. In it the cannot the collective case approach of artificial behaviour the kagome two-fold of would spin degenerate ice, align a.c. into demagnetiza- ground a minimum-energy state of process state at minimizingon cooling through the magnetostatic the Curie energy of the arrays. In the 15 process at minimizing the magnetostatic energy of the arrays. In the magnetic charges and theirtion usefulnesssquare couldasice easily degrees. In access the of case thefreedom of kagome artificial in low- ice kagome I phasetemperature, spin (in ice, which a.c. and the demagnetiza- that islands this statearrays would with be smallest determined lattice by constant, the lattice only the handful of islands located arrays with smallest lattice constant, only the handful of islands located energy collective physics. Weobey usetion the here could two-in/one-out a thermal easily annealing access orone-in/two-out the technique kagome ice pseudo-iceIgeometry, phase (in that rule),which is, probablythe the artificial islands on spin the ice isolated would be domain thermalized. walls are not in the ground state (Supplemen- on the isolated domain walls are not in the ground state (Supplemen- to achieve the first experimentalbecauseobey validation of the that two-in/one-out state’s of the massive predicted or entropy. one-in/two-out charge- Yet a.c.After pseudo-icedemagnetization annealing, rule), we probably failed collectedtary MFM Fig. 4). images Approximating of each array, the islands’ which magnetization as proportional because of that state’s massive entropy. Yet a.c. demagnetization failed tary Fig. 4). Approximating the islands’ magnetization as proportional ordered phase: incipient magneticto produce charge any ordering of the in predicted artificial kagome ordered phasesallowed governed us to examine by long- the thermalizedto the square state root of the of momentTC – T (where arrange-T is the physical sample temper- to the square root of T – T (where T is the physical sample temper- spin ice. rangeto interactions. produce any of the predicted orderedments. phases Images governed of the byannealed long- ature), artificial we square found spin-ice the excitations arraysC revealed of the 400-nm square lattice fit well to a range interactions. ature), we found the excitations of the 400-nm square lattice fit well to a The current gap between theoryRecently and a experiment few alternative is in ways part ato con- thermalizeextensive artificial domains spin ice of ground-state have Boltzmann ordering distribution (Fig. 2). The corresponding ground state to a temperature of T < 540 uC, Recently a few alternative ways to thermalize artificial spin ice have Boltzmann distribution corresponding to a temperature of T < 540 uC, sequence of the present limitsbeen on developed. our ability Morgan to place and the co-workers magnetic examinedconsists of3,19 anthe antiferromagnetic as-grown as shown ordering in Supplementary on the vertical Fig. and 6. hori- been developed. Morgan and co-workers examined3,19 the as-grown as shown in Supplementary Fig. 6. moments of artificial spin icestate in the of desired artificial collective squarethermodynamic spin ice and discoveredzontal thatsublattices. thermalization The large sizeTo of the gain domains further (in insight many into cases the larger development of these large ground- state of artificial square spin ice and discovered that thermalization To gain further insight into the development of these large ground- state. Although a.c. demagnetizationduring the can first produce stages of well growth defined is preserved, and than leaving the 10–20- the samplemm-wide in a MFMstate images) domains and in the annealed paucity artificial of isolated square spin ice, we extracted the during the first stages of growth is preserved, leaving the sample in a state domains in annealed artificial square spin ice, we extracted the predictable out-of-equilibriumvery low statistical energy ensembles state. This ofsingle-shot different approachmonopolar represents charge theexcitations, best pairwise especially correlations in the samples between with the islands smal- from our MFM images. The very low energy state. This single-shot approach represents the best pairwise correlations between islands from our MFM images. The entropy, it cannot approachenergy the two-fold minimization degenerate available ground to date state for of artificiallest lattice spin ice, constants, although underscore its correlation the effectiveness between two of our islands annealing is 11(21) if the islands’ magnetic 15 energy minimization available to date for artificial spin ice, although its correlation between two islands is 11(21) if the islands’ magnetic square ice . In the case of artificialreliance kagome on the spininitial ice, conditions a.c. demagnetiza- of growth limitsprocess the at degree minimizing of control the magnetostaticmoments are energy aligned of theto minimize arrays. In (maximize) the the dipolar interaction reliance on the initial conditions of growth limits the degree of control moments1 are aligned to minimize (maximize) the dipolar interaction tion could easily access the kagome ice I phase (in which the islands arrays with smallest lattice20 constant,energy only. A1 plotthe handful of correlation of islands as a located function of island separation is shown which can be obtained by this method. Kapaklis et al. fabricated 20 energy . A plot of correlation as a function of island separation is shown obey the two-in/one-out or one-in/two-outwhich can pseudo-ice be obtained rule), by thisprobably method.on Kapaklis the isolatedet al. domainfabricated walls arein notFig. in 3. the The ground distance state at (Supplemen- which the correlation falls to 1/e < 0.368 artificial spin ice in a material that has a Curie temperature (TC)of in Fig. 3. The distance at which the correlation falls to 1/e < 0.368 artificial spin ice in14CRISTIANO a material that NISOLI, has a Curie THEORETICAL temperature (TC)of DIVISION, LOS ALAMOS NATIONALNATURE PHYSICS LABORATORY,24 DOI: 10.1038/NPHYS3037 LOS ALAMOS, NM, 87545, USA because of that state’s massive entropy. Yet a.c. demagnetizationARTICLES failed tary Fig. 4). Approximating thecorresponds islands’ magnetization to a rough as estimate proportional ofNATURE the domain PHYSICS size .24 BecauseDOI: the10.1038/NPHYS3037 230230 K, below K, below room room temperature. temperature.ARTICLES Magneto-optical Magneto-optical Kerr Kerr effect effect (MOKE) (MOKE) corresponds to a rough estimate of the domain size . Because the to produce any of the predicted ordered phases governed by long- to the square root of TC – T (wheredipolarT interactionis the physical between sample islands temper- is largest for the arrays with the measurementsmeasurements of remanent of remanent states states revealed revealed evidence evidence for for the the creation creation of of dipolar interaction between islands is largest for the arrays with the range interactions. ature),a we found the excitationssmallest of the 400-nm lattice constants, square lattice the fitcorrelations well to a in these arrays decay the most thermalthermal excitations excitations slightly slightly below below the the Curie Curiea temperature. temperature. Although Although smallest lattice constants, the correlations in these arrays decay the most Recently a few alternativethis ways work to thermalize in some artificial sense represents spin ice have the successfulBoltzmann thermalization distribution corresponding of slowly. This to a temperature can be qualitatively of T < 540 confirmeduC, by examining the MFM this work in some sense3,19 represents the successful thermalization of slowly. This can be qualitatively confirmed by examining the MFM been developed. Morgan and co-workers examined the as-grown as shown in Supplementary Fig.images 6. of Fig. 2: smaller lattice constants produce larger ground-state artificialartificial spin spin ice, ice,it focused it focused on the on the remanent remanent state state (rather (rather than than the the true true images of Fig. 2: smaller lattice constants produce larger ground-state state of artificial square spin ice and discovered that thermalization To gain further insight intodomains. the development The correlation of these largebetween ground- islands at the largest lattice spacings groundground state), state), and and the MOKEthe MOKE measurements measurements were were sensitive sensitive only only to the to the domains. The correlation between islands at the largest lattice spacings during the first stages of growth is preserved, leaving the sample in a state domains in annealed artificialis insignificant square spin beyond ice, we one extracted or two lattice the constants. globalglobal properties properties of the of samples,the samples, so the so theexact exact microstates microstates of the of the samples samples is insignificant beyond one or two lattice constants. very low energy state. This single-shot approach represents the best21–23 pairwise correlations between islands from our MFM images. The were not determined. Several groups 21–23, inspired by these experi- The ground-state ordering observed in our artificial square spin ice were not determined. Several groups correlation, inspired between by these two experi- islands isThe11( ground-state21) if the islands’ ordering magnetic observed in our artificial square spin ice energy minimization availablements, to date have for artificial carried outspin simulations ice, although of its thermalized artificial spin-ice highlightshighlights a significant a significant difference difference between between artificial artificial square square spin spin ice ice and and reliance on the initial conditionsments, of growth have limits carried the degreeout simulations of control of thermalizedmoments are artificial aligned spin-iceto minimize (maximize) the dipolar interaction arrays, but post-growth thermalization of artificial1 Type spin I4ice (2) into its Type II4 (4) Type III4 (8) Type IV4 (2) which can be obtained by thisarrays, method. but Kapaklis post-growthet al. thermalizationfabricated20 energy of artificial. A plotType spin of I4 correlation(2) ice into its as a function Type II4 of(4) island separation Type is III shown4 (8) Type IV4 (2) zero-fieldzero-field ground ground state state has has not not yet yetbeen been experimentally experimentally demonstrated. demonstrated. a a320 nm b b360 nm artificial spin ice in a material that has a Curie temperature (TC)of in Fig. 3. The distance at which the320 correlation nm falls to 1/e < 0.368 360 nm The results of Morgan and co-workers demonstrated that therma- 16 μm × 16 μm 24 20 μm × 20 μm 230 K, below room temperature. Magneto-opticalThe results ofMorgan Kerr effect and (MOKE) co-workerscorresponds demonstrated to a that rough therma- estimate of16 the μm domain× 16 μm size . Because the 20 μm × 20 μm lization of artificial spin ice is possible, inspiring the present work in measurements of remanent stateslization revealed of evidence artificial for spin the ice creation is possible, of inspiringdipolar interaction the present between work in islands is largest for the arrays with the which we achieve such thermalization on demand and then use it to thermal excitations slightly belowwhich the we Curie achieve temperature. such thermalization Although onsmallest demand lattice and constants, then use theit to correlations in these arrays decay the most explore the long-range collective physics. We fabricated square and this work in some sense representsexplore the the successful long-range thermalization collective physics. of slowly. We fabricated This can be square qualitatively and confirmed by examining the MFM artificial spin ice, it focused onkagome thekagome remanent arrays arrays of state permalloy of (rather permalloy (Nithan80 (Ni theFe8020 trueFe) nanoislands20) nanoislandsimages of on Fig. ona silicon a 2: silicon smaller nitride nitride lattice constants produce larger ground-state substratesubstrate using using an electron an electron beam beam lithography lithographydomains. lift-off lift-off process The process correlation described described between islands at the largest lattice spacings ground state), and the MOKE measurements1 1 were sensitive only to the global properties of the samples,elsewhere soelsewhere the exact.Theislandsarerectangularwithsemicircularends(‘stadium-.Theislandsarerectangularwithsemicircularends(‘stadium- microstates of the samples is insignificant beyond one or two lattice constants. were not determined. Severalshaped’), groupsshaped’), nominally21–23 nominally, inspired 220220 by nm these nm long long byexperi- 80 by nm 80 nm wideThe wide and ground-state and 25 nm25 nm high. high. ordering The The observed in our artificial square spin ice single-domainsingle-domain character character of the of the islands islands was was confirmed confirmedType I3 (2) by byMFM MFM (right (right Type II3 (4) Type III3 (2)Type I2 (2) Type II2 (2) ments, have carried out simulations of thermalized artificial spin-ice highlightsType I a3 (2) significant difference Type II3 between (4) artificial Type square III3 (2) spin ice and Type I2 (2) Type II2 (2) arrays, but post-growth thermalizationpanelspanels of Fig. of of Fig. 1). artificial We 1). We studied spin studied ice arrays intoarrays with its with a wide a wide range range of latticeof lattice con- con- zero-field ground state has notstants yetstantsbeen (320–880 (320–880 experimentally nm nm for square, for demonstrated. square, and and 260–1,050 260–1,050a nm nm for for kagome) kagome) so as so to as to c c b d d have a range of magnetostatic interaction strengths.320 nm The arrays were 400400 nm 360nm nm 440440 nm nm The results of Morgan andhave co-workers a range ofdemonstrated magnetostatic that interaction therma- strengths.16 μm × The 16 μm arrays were 20 20μm μ 20×m 20 ×μ m 20μ m× μ 20m μm 2020 μm μ m× 20× 20 μm μm thenthen polarized polarized in a in strong a strong magnetic magnetic field field at room at room temperature, temperature, heated heated lization of artificial spin ice is possible, inspiring the present work in bcbc which we achieve such thermalizationin vacuumin vacuum on in nominallydemand in nominally and zero then zero magnetic use magnetic it to field field to an to anannealing annealing temper- temper- explore the long-range collectiveatureature abovephysics. above the We theislands’ fabricated islands’ Curie Curie square temperature temperature and (540–545 (540–545uCu asC measuredas measured by vibratingby vibrating sample sample magnetometry), magnetometry), and and then then allowed allowed to cool to cool slowly slowly to to kagome arrays of permalloy (Ni80Fe20) nanoislands on a silicon nitride substrate using an electron beamroomroom lithography temperature. temperature. lift-off Square processSquare and and described kagome kagome arrays arrays of all of alllattice lattice constants constants elsewhere1.Theislandsarerectangularwithsemicircularends(‘stadium-werewere fabricated fabricated together together on chips on chips so that so that all the all the arrays arrays on ona chip a chip shared shared shaped’), nominally 220 nmthe long samethe by same 80 annealing nm annealing wide protocol. and protocol. 25 nm Results high. Results The are are shown shown for for an anannealing annealing tem- tem- single-domain character of theperature islandsperature of was 545 of confirmed545uC, butuC, butqualitatively by qualitatively MFM (right consistent consistent results results were were obtained obtained on on panels of Fig. 1). We studiedmultiple arraysmultiple withsamples samples a wide annealed annealed range just ofjust latticeabove above the con- theCurie Curie temperature. temperature. Importantly, Importantly, by judiciousby judicious choice choice of substrate of substrate materials materials and and careful careful scanning scanning of theof the stants (320–880 nm for square, and 260–1,050 nm for kagome) so as to c FigureFigured 2 | 2MFM| MFM images images of annealedof annealed artificial artificial square square spin spin ice. ice. a– ad–,d The, The lattice lattice relevantrelevant temperature temperature range range in annealing in annealing experiments, experiments,400 nm we we were were able able to to 440 nm have a range of magnetostatic interaction strengths. The arrays were 20 μm × 20 μm constantsconstants are20 areμ givenm given× 20 at μ topatm top left, left, and and are are 320 320 nm nm (a), (a 360), 360 nm nm (b (),b), 400 400 nm nm (c ()c and) and then polarized in a strong magneticidentifyidentify field a temperature aat temperature room temperature, window window in heated which in which effective effective annealing annealing can can be be 440 nm (d). The image size is also indicated at top left. Notice the decrease in performed while avoiding film/substrate interdiffusion and lateral 440 nm (d). The image size is also indicated at top left. Notice the decrease in in vacuum in nominally zeroperformed magnetic fieldwhile to avoiding an annealing film/substrate temper- interdiffusion and lateral domaindomain size size with with increasing increasing lattice lattice constant, constant, a trend a trend also also revealed revealed in in Fig. Fig. 3 3by by ature above the islands’ Curiedegradation temperaturedegradation of the (540–545 of islands.the islands. DetailsC as Details measured of sample of sample fabrication fabrication and and this this anneal- anneal- correlations between islands. The ordering of moments is in stark contrast with u correlations between islands. The ordering of moments is in1,2 stark contrast with by vibrating sample magnetometry),ing processing process and can then can be allowed found be foundto in cool Methods, in Methods, slowly Supplementary to Supplementary Information Information and and thethe disordered disordered state state resulting resulting from from a.c. a.c. demagnetization demagnetization1,2. . room temperature. Square and kagome arrays of all lattice constants 554 | NATURE | VOL 500 | 29 AUGUST 2013 554 | NATURE | VOL 500 | 29 AUGUST 2013 were fabricated together on chips so that all the arrays on a chip shared ©2013 Macmillan Publishers Limited. All rights reserved the same annealing protocol. Results are shown for an annealing tem- ©2013 Macmillan Publishers Limited. All rights reserved perature of 545 uC, but qualitatively consistent results were obtained on FigureFigure 3 | 3 Mapping| Mapping the the shakti shakti lattice lattice to to the the six-vertex six-vertex model. model.a,a The, The vertex vertex types types found found in in the the shakti shakti lattice, lattice, showing showing only only the the short-island short-island shakti shakti lattice. lattice. multiple samples annealed just above the Curie temperature.Consistent Importantly, with the standard literature nomenclature for the vertices in artificial square spin ice, we number the vertices with Roman numerals in order of by judicious choice of substrate materials and careful scanningConsistent of the with the standard literature nomenclature for the vertices in artificial square spin ice, we number the vertices with Roman numerals in order of increasingincreasing energy. energy. WeFigure We distinguish distinguish 2 | MFM vertices vertices images with with of di annealed dierenterent numbers numbersartificial of of islandssquare islands byspin by a subscript aice. subscript a–d, indicatingThe indicating lattice the the number number of of islands. islands.b,b A, Amap map of of the the three-island three-island relevant temperature range in annealing experiments, we were able to vertexvertex configurations configurationsconstants in inthe the magnetic magnetic are given force force at microscopetop microscope left, and image are image 320 in innmFig. Fig.2 (ae.2),e. The 360 The solid nm solid ( linesb), lines 400 indicate indicate nm (c the) andthe boundaries boundaries of of the the plaquettes, plaquettes, the the dashed dashed lines lines indicate indicate identify a temperature window in which effective annealingthethe two-island two-island can be vertices vertices440 in nm in the the ( middled). middle The of image of each each plaquette, size plaquette, is also and indicated and the the directions directions at top left. of of the Notice the island island the moments moments decrease are are in reproduced reproduced as as red red arrows arrows in in the the upper-leftmost upper-leftmost performed while avoiding film/substrate interdiffusionplaquette and lateral for illustrative purposes. The circles indicate the location of the three-moment vertices; the Type I vertices are denoted by open circles, and the plaquette for illustrativedomain purposes. size with The increasing circles indicate lattice the constant, location a oftrend the three-momentalso revealed in vertices; Fig. 3 by the Type3 I3 vertices are denoted by open circles, and the higher energy Type II vertices are denoted by filled circles. c, The six degenerate ground-state configurations for arrangement of the three-moment vertex degradation of the islands. Details of sample fabrication andhigher this anneal- energy Typecorrelations3 II3 vertices arebetween denoted islands. by filled The circles. orderingc, The of moments six degenerate is in stark ground-state contrast◦ withconfigurations for arrangement of the three-moment vertex Figure 7. Top: Vertex-configurations1,2 for 90 angles of coordination z = ing process can be found in Methods, Supplementary Informationtypestypes on on a plaquette. a and plaquette.the disordered state resulting from a.c. demagnetization . 4, 3, 2 (degeneracy in brackets) listed in order of increasing energy. Middle: 554 | NATURE | VOL 500 | 29 AUGUST 2013 minimizeminimize the the global global energy energy of of the the system, system, the the lattice lattice topology topology re- re- individualindividual plaquettes plaquettes and and vertices vertices of of the the six-vertex six-vertex model model is is shown shown ©2013 Macmillan PublishersMFM Limited. images All rights of reserved thermally annealed square ice at different lattice constants quiresquires that that half half the the three-moment three-moment vertices vertices are are in in the the local local magneto- magneto- inin Fig. Fig.3c.3c. Note Note that that the the observed observed emergent emergent frustration frustration and and high high 31,32 staticstatic ground groundshowing state state and and half an half are areordered in in a higher-energy a higher-energy domain defect defect state crossed state31.,32. degeneracydegeneracy by a Dirac that that arises arises string from from the the topology (for topology the of of the the specimen shakti shakti lattice lattice is is a a AsAs a resulta result of of the the mixture mixture of of states states on on the the three-moment three-moment vertices, vertices, directdirect consequence consequence of of its its design design and and has has no no obvious obvious analogue analogue in in any any thethe shakti shaktiat ground ground 320 state state nm) has has intrinsic and intrinsic degeneracya degeneracy multi-domain arising arising from from the the ensemblenaturallynaturally occurring occurring separated lattice. lattice. by domain walls of freedomfreedommonopoles in in allocating allocating the the twoand two higher-energy higher-energy diract strings vertices vertices among among (at the 400 the nnByBy plotting and plotting the440 the vertex vertex nm); population population note as as a also functiona function the of of lattice frozen lattice spacing spacing fourfour possibilities possibilities on on each each plaquette; plaquette; the the two two defect defect vertices vertices on on each each (Fig.(Fig.4),4), we we demonstrate demonstrate explicitly explicitly that that we we are are obtaining obtaining the the ground ground plaquetteplaquettein can can monopole be be equivalently equivalently located pairs located at at (figures four four di dierenterent adapted sites sites for for a a state fromstate of of the refthe shakti shakti [100 lattice lattice]). by by tuningBottom: tuning the the strength strength the of of lowest the the moment moment totaltotal of of sixenergy six possible possible energeticallystate energetically of equivalent equivalentsquare configurations configurations ice as an (Fig. (Fig. antiferromagnetic3c).3c). interactions.interactions. Although Although tiling the the vertex vertex of distribution Type-I distribution is is closevertices; close to to random random Therefore,Therefore, each each plaquette plaquette can can be be mapped mapped precisely precisely onto onto a vertexa vertex of of atat large large lattice lattice spacing, spacing, when when the the spacing spacing is is small,4 small, we we obtain obtain the the 31,32 a classica classic two-dimensionalcreating two-dimensional and six-vertex six-vertex separating model model obeying obeying a monopole the the ice ice rule rule31.,32. pairpreciselyprecisely entails expected expected a vertex Dirac vertex distribution distribution string for(red) for the the short-island short-island of Type- lattice. lattice. ThisThis correspondence correspondence is is illustrated illustrated in in Fig. Fig.3b,3b, where where we we take take the the TheThe long-island long-island shakti shakti lattice lattice does does not not quite quite achieve achieve this this degenerate degenerate momentmomentII configuration configuration4 vertices, from from that Fig. Fig.2—the2 are—the first energeticallyfirst physical physical realization realization moresix-vertexsix-vertex costly state state within than within the the range range Type-I of of our our experiments, experiments,4, leading presumably presumably to ofof the the six-vertex six-vertexthe linear model model ground ground confinement state. state. The The correspondence correspondence of the pair.between between owingowing to to the the more-constrained more-constrained dynamics dynamics of of the the longer longer islands. islands.

© ©2014 2014 Macmillan Macmillan Publishers Publishers Limited. Limited. All All rights rights reserved reserved TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 15 temperature some of the features of spin ice pyrochlores. Of course, as always with three dimensional realizations, the challenge there lies not only in nano-fabrication, but also in characterization, as real-space methods are generally surface methods. Finally, another way to produce a Coulomb phase in square ice has been presented recently, and involves “rectangular ice” where vertical and horizontal islands diﬀer in length, and degeneracy is obtained for a proper critical value of their ratio [105, 112].

4. Exotic States Through Vertex-Frustration Until 2014, the only degenerate artificial spin ice was Kagome. As both nano–fabrication and characterization protocols evolved, it became clear that the initial inspiration of the entire project—to design exotic behaviors in the geometry of interacting, binary degrees of freedom—could become viable, if not for one problem: in real systems, the frustration of the pairwise interaction is wedded to the geometry. What this means is explained in Fig. 8 where brickwork spin ice and Kagome spin ice are shown to lead to completely different ground states, one disordered, the other ordered, despite the two geometries being topologically equivalent. Indeed, the dipolar interaction is not topologically invariant, but instead depends very much on the mutual arrangements of the dipoles. To overcome this limitation and gain freedom in the design of new materials capable of various states and unusual behaviors, the first step is to decouple frustration from geometry. As the pairwise interaction is anisotropic, something else will have to be frustrated. A possibility is the vertex itself. Consider a geometry made of 90◦ vertices of coordination z = 4, 3, 2 (Fig. 7). Each vertex has a unique configuration of minimal energy (up to a flip of all the spins). Imagine now arranging them in such a way that, however, not all vertices can be assigned to the lowest energy configuration [52]. This will lead to “unhappy vertices” (UV), that is, topologically protected local excitations (Fig. 8). In proper geometries, the degeneracy of the allocation of such vertices grows exponentially with the size of the system, leading to a degenerate low energy manifold [52, 56]. Crucial here is that within this manifold the system is usually captured by an emergent description that considers the allocation of these protected local excitations, rather than by its spin ensemble. As a consequence, other emergent properties appear that are in general not obvious nor indeed apparent in the local spin structure. While vertex models [77] were introduced to describe frustrated systems, they were themselves not frustrated. They simply subsumed the degeneracy of a frustrated system within a degenerate energetics. Vertex-frustrated geometries can thus be considered the first frustrated vertex models. Vertex-frustration is of course a nearest-neighbor level concept, although it induces topological states that are collective. However, the real materials being made of dipoles, other phases are present within their vertex-frustrated low energy manifold, much like inner phases are present in the diagram of Kagome above. Let us now see how this comes about in three such geometries: Shakti, Tetris, and Santa Fe. 16CRISTIANO NISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

Figure 8. Geometry vs. Topology. Topologically equivalent geometries leads to completely diﬀerent spin ensembles, due to the anisotropy of the dipolar interaction. The honeycomb spin ice (top right) is topologically equivalent to the ladder spin ice (middle, right) yet the nearest neighbor interactions lead to an ordered ground state in the latter (see also Fig. 7 for the energetic hierarchy of the vertices) and a disordered manifold in the former. Pairwise interactions are frustrated in both systems, however in the honeycomb lattice all the spins interacting in the vertex have the same mutual angle (top left) and thus any of the three interactions can be frustrated, whereas it is energetically favorable to frustrate the interaction between parallel spins in the ladder lattice (middle left). At the bottom is an example of vertex-frustration, where the allocation of vertices of lowest energy is frustrated, leading to “unhappy vertices” (blue circles) on certain loops, instead of unhappy energy links (red lines above). TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 17

5. Emergent Ice Rule, Charge Screening, and Topological Protection: Shakti Ice Consider the Shakti geometry in Fig. 9 [53]. Each minimal, rectangular loop of Shakti is frustrated. What it means is that it must be affected by an odd number of unhappy vertices [52, 53]. Because each unhappy vertex aways affects two nearby loops and costs energy, the lowest energy configuration is realized when nearby loops are dimerized by a single unhappy vertex [52]. If one considers the geometry, one finds (Fig. 9-c) that each plaquette made by two rectangular loops will host two unhappy vertices in 4 possible locations, much like the ice rule in water ice prescribes that 2 hydrogenatoms are within the tetrahedron containing each oxygen atom (Fig. 3), in 2 of the 4 possible allocations. In both cases the same ice-rule applies, but here in emergent form: not in terms of the original spins, but in terms of allocation of unhappy vertices. Thus, the lowest energy manifold, at the nearest neighbor vertex description employed here, corresponds then to an emergent six-vertex model. This has been shown experimentally [54]. Nonetheless, as we had cautioned before, this nearest neighbor description defines the ice-manifold, within which intervene other non-trivial phenomena, due to the long range nature of the interaction. A particularly interesting one regards the screening of magnetic charge. Shakti has multiple coordination, therefore while in its low energy state all the vertices of coordination z = 4 are in the ice rule, they are surrounded by vertices of coordination z = 3 which always have a magnetic charge ±1 (in natural units, previously defined), and are disordered. When a vertex of coordination z = 4 hosts a magnetic monopole, the overall neutral plasma of charge around it rearranges to screen it, as shown in Fig. 10 [54]. It is important to understand that magnetic monopoles are not proper topological charges for Shakti, as they are not protected. Each z = 4 vertex being surrounded by a sea of charges, it can gain or lose charge to and from it. However, the Shakti state is a bona fide topological phase, which means that some other topological charge should be identified in it. That the manifold has topological protection can be immediately suspected by noting that a single spin flip takes out of the manifold, and only a proper loop of collective spin flips realizes change within the manifold. This can be understood easily from Fig. 9, as all spins impinging in a z = 3 vertex also impinge into z = 4, 2 vertices, which are in their lowest energy in the manifold. Flipping such spin will thus necessarily cause excitations. To identify the topological structure, we go back to the properties of the low energy state. We saw that because each UV affects two nearby plaquettes (Fig. 9) and costs energy, the lowest energy configuration is realized when nearby plaquettes are “dimerized” by a single UV [52]. The ice manifold of Shakti is thus described by a dimer cover model on the lattice connecting the rectangular plaquettes, which is topologically equivalent to a square lattice (Fig. 11) (from now on called “dimer lattice”), and which can be solved exactly [113]. The following is then standard: a discrete, emergent vector field E~ can be introduced, perpendicular to each edge, of length 1 (o 3) if the edge is unoccupied (or occupied) by a 18CRISTIANO NISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

Figure 9. Theory of Shakti spin ice. The structure of the system (a) is such that its lowest energy spin ensemble (b) is disordered. A look at the spin structure in (b) does not seem particularly insightful. However, if we translate that spin map unto the allocation of locally excited vertices, denoted by circles in (c) we then see that each plaquette will host two and only two unhappy vertices in four possible positions. This is equivalent to a six vertex model (d) where pseudo-spins are assigned to each plaquettes and point toward (away from) the unhappy vertices in plaquette of vertical (horizontal) long island. Figures adapted from ref [53]. dimer, and direction entering (exiting) a gray square of Fig. 11 from top or bottom, and R ~ ~ exiting (entering) it from the sides. The “line integral” γ E · dl for such a discrete vector field along a directed line γ crossing the edges is the sum of the vectors along the line with sign taken along the line’s direction. For a complete cover the emergent field is irrotational H ~ ~ ~ ~ ( γ E · dl = 0) leading to the definition of a “height function” [91] h such that E = ∇h and thus demonstrating the topological state. TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 19

Figure 10. Realizations of Shakti Ice. On the top left, an MFM image of Shakti ice after annealing. On the top right, the experimental data of the left part is translated in terms of allocation of the unhappy vertices, on plaquettes (black dots). One can see how an emergent ice rule describes the system, as each plaquette can have only two of four slots for unhappy vertices occupied. Bottom: screening of monopoles from magnetic charges hQnni denotes the average magnetic charge surrounding a magnetic monopole on a z = 4 vertex, at the nearest neighbor level. Figures adapted from ref [54].

Beyond the standard dimer model, this picture can incorporate the low-energy excitations of Shakti ice as scramblings of the cover. As Fig. 11 shows, above the ground state a frustrated plaquette (i.e. a node of the dimer lattice) can be dimerized three times instead of one (over-dimerization) by UVs, or also diagonally by a Type-II4 or a Type-II2 vertex. In the presence of such scramblings the emergent vector ﬁeld E~ is not irrotational anymore. 20CRISTIANO NISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

ℇ 1 µm

Figure 11. Top: Shakti manifold as a dimer cover model. From left to right: Disordered spin ensemble for the ground state of Shakti ice manifold. The manifold is completely described by the allocation of the UVs (circles) which affect two nearby rectangular plaquettes (connected by the blue segments). Thus, an unhappy vertex is a dimer (blue segments) connecting frustrated plaquettes, and the ground state is a complete dimer-cover model on the (Ochre color) lattice with nodes in the center of rectangular plaquettes, topologically equivalent to a square lattice. There we introduce the emergent vector field E~ , as in the text. The circulation of the vector field along any closed loop is zero. Middle and Bottom: The Shakti’s low- energy manifold. XMCD image of Shakti spin ice, for a spin ensemble with one excitation (red and blue dots) and the corresponding emergent dimer cover representation. Now excitations appear as multiple occupancy and/or diagonal dimers (Type II2s). E~ is no longer irrotational and its circulation 1 H ~ ~ defines the topological charge as q = 4 γ E · dl. (Image copyright: Yuyang Lao.) TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS: ARTIFICIAL SPIN ICE 21

Figure 12. Tetris Ice. (a) XMCD-PEEM image of a 600 nm Tetris lattice. The black/white contrast indicates whether the magnetization of an island has a component parallel or antiparallel to the polarization of the incident X-ray, which is indicated by the yellow arrow. (b) Map of the moment conﬁgurations showing ordered backbones (blue) and disordered staircases (red). (Images from ref [55].)

Indeed its circulation around any topologically equivalent loop encircling a scrambling de- 1 H ~ ~ fines the quantized topological charge of the defect as q = 4 γ E · dl (Fig 11). Thus, the excitations of the Shakti ice manifold are topological charges, turning the discrete scalar field h that defines its order into a multivalued phase. We have now the full picture: a topological phase, which cannot be explored from within, but only via a discrete kinetics of excitations whose topological charge is conserved. This picture is emergent, and not at all evident from, or indeed reminiscent of, the original spin structure. It also has consequences for the kinetics, in terms of ergodicity breaking, non-equilibration, and glassiness, as it is typical of a topological state with topologically protected excitations, that cannot be reabsorbed into the manifold individually, and which evolves via a discrete kinetics. All these issues are still to be investigated in full, as Shakti ice might provide the first artificial, controllable, modifiable and fully characterizable magnetic system which provides non-topographic vistas of ergodicity breaking and non-equilibration as consequences of a classical topological order.

6. Dimensionality Reduction: Tetris Ice While Shakti spin ice provides a topologically protected low-energy manifold, no such protection is present in the ground state of Tetris ice (Fig. 12), which can be explored by consecutive spin ﬂips. As Fig. 12 shows, the lattice can be decomposed into T-shaped “tetris” pieces and it has a principal axis of symmetry. 22CRISTIANO NISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

The geometry can be considered as layered one-dimensional systems. On the blue islands in Fig. 12 there cannot be any Type-II3 unhappy vertex [52], and therefore the blue portion of the lattice, which we call backbone, must be ordered at the lowest energy. The unhappy vertices must reside on the red portions, which we call staircase, and which therefore remain disordered at low temperature. As temperature is lowered, we have thus a dimensional reduction of an alternating ordered-disordered one-dimensional system, which was indeed confirmed experimentally [55]. This dimensional reduction is also apparent in the kinetics. Tetris was the first of the new geometries to be characterized in real-time, real-space and from the supplementary information of ref [55] it is possible to watch clips of its kinetics as the temperature is lowered or raised. Starting at high temperature, all the spins flip at about the same rate. As the temperature is lowered, ordered domains begin to form in correspondence with the backbones, where eventually the spins become static, while the spins on the staircases continue to fluctuate. While Tetris spin ice’ lowest energy state described above has been confirmed experimentally, it follows from a nearest neighbor approximation. The profile of low energy excitations, however, has not been yet studied in any systematic way, and promises interesting new effects. For instance, as one-dimensional systems, the backbones can never order completely, and will always host excitations above the low-energy manifold. Of course Tetris is in fact a two-dimensional system, which decomposes into one-dimensional ones only in the lowest energy configuration. Slightly above such a manifold, one expects correlations among excitations that belong to different backbones. Such correlations must be controlled both by the magnetic interaction between these defects—as Tetris is, after all, a system of dipoles that can interact at long-range—but also through entropic interactions, since the backbones are separated by disordered staircases of non-zero density of entropy. None of the above issues has yet been studied theoretically, and they might indeed provide a useful setting to explore the onset of phase decoupling into lower-dimensional states, a broader problem relevant to liquid crystal phases [114] or weakly coupled sliding phases [115, 116].

7. Polymers of Topologically Protected Excitations: Santa Fe Ice We end this vista on how novel and unusual spin ice geometries influence topology with Santa Fe of Fig. 13, which was inspired by a terra cotta floor in the homonymous New Mexican capital—incidentally, the oldest in the United States. While Shakti and Tetris are maximally frustrated, which means that any minimal loop inside the geometry needs to be affected by an unhappy vertex, in Santa Fe only the dashed loops in the figure are frustrated and they are surrounded by unfrustrated ones. It is an inviolable topological constraint that at any energy frustrated loops can be affected by only an odd number of excitations, and unfrustrated ones by only an even number (or none). An unhappy vertex on a frustrated loop of Santa Fe lattice affects a nearby unfrustrated one. However, an unfrustrated loop can only be affected by an even number of defects, and thus there will be a second unhappy vertex on it, affecting in turn a nearby unfrustrated �� TOPOLOGY BY DESIGN IN MAGNETIC NANO–MATERIALS:�� ARTIFICIAL SPIN ICE 23

Figure 13. The Santa Fe Ice can support both frustrated (shaded, green) and unfrustrated loops. There, “polymers” of unhappy vertices (blue dots) thread through unfrustrated loops to connect frustrated ones. On the right, the brick ﬂoor at the convention center in Santa Fe, New Mexico, USA.

loop, et cetera. It follows that magnetic “polymers”, whose “monomers” are local protected excitations, must begin from and end into frustrated loops. As each monomer costs energy, the lowest energy configuration of the magnetic ensemble will correspond to the shortest possible polymers, which are made of three monomers, each connecting nearby frustrated loops as in Fig. 13. The entropy of such a state can easily be computed exactly. As each polymer dimerizes two frustrated loops, the degeneracy is given by the dimer cover model on the square lattice whose node is made of nearby frustrated loops, times the number of ways in which polymers can be chosen once their pinned ends are fixed. Thus the ice manifold decomposes into the direct product of two states: the dimer-cover manifold, which selects which loops are joined by the polymers, and the degeneracy of the polymers themselves. At low temperature the kinetics will reduce to the fluctuations of the magnetic polymers without changing the pinning location of their ends, and thus without changing the dimer cover picture. Thus, the low energy manifold can be explored from within, but only in part. The kinetics within the manifold remains local and the polymer’s fluctuations are uncorrelated. As the temperature rises the polymers lengthen to include more than three monomers. At that point they can bump into each other, fuse in a cross, and then separate in different ways. This transition can lead to a different dimerization, as the new polymers, emerging from “collisions” of the old ones, are now pinned to different ending point. Thus the dimer-cover ensemble is explored via this mechanism of polymer colliding, fusing together and then breaking again into different ones. This of course involves excitations over the ice manifold, further demonstrating the partial topological protection that pertains only to the dimer-cover sector of the low-energy manifold. 24CRISTIANO NISOLI, THEORETICAL DIVISION, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM, 87545, USA

8. Conclusions We have argued that by assembling together interacting, elementary building blocks, here Ising spins in the form of single domain, magnetic nano-islands, we can invert a tendency that has dominated condensed matter physics for half a century. Instead of finding serendipitously exotic states and behavior in nature, and then model them via higher level, emergent hamiltonians, one can devise materials—magnetic materials in this case–that do not exist in nature by developing first the model of their collective dynamics. Then, advances in nano–lithography and chemical synthesis can allow for their realization, while new thermal protocols afford characterization, often in real-time and real-space, for unprecedented direct validation of the theoretical expectations. We have shown that while the field began with simple geometries, reminiscent of natural materials, current advances make it possible to realize dedicated geometries of completely different properties and behaviors. These allow now to access rather sophisticated topological states, as their emergent description loses reminiscence of the original degrees of freedom, the underlying spin structure. This approach opens a new path in the material- by-design effort, on which unusual topological states can be deliberately designed.

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