Square-Lattice Artificial Spin Ice

Updated 10 July 2026

Square-lattice ASI is an engineered frustrated magnetic system where elongated magnetic islands form four-island vertices obeying ice-rule constraints.
It enables detailed studies of vertex energetics, phase transitions from antiferromagnetic order to square ice, and the emergence of monopole-like defects.
Experimental and simulation techniques uncover tunable spin-wave modes, thermal activation pathways, and field-driven defect kinetics in these versatile arrays.

Square-lattice artificial spin ice (ASI) is a class of engineered frustrated magnetic systems built from elongated magnetic elements arranged on a square lattice such that four elements meet at each vertex. In its canonical nanoscale realization, each island behaves approximately as an Ising-like macrospin constrained by shape anisotropy; in macroscopic analogues, the same vertex physics can be realized with centimeter-scale hinged bar magnets whose whole bodies rotate mechanically. Across these realizations, square-lattice ASI serves as a controlled platform for studying vertex frustration, ordered and disordered ice-rule manifolds, monopole-like defects, Dirac strings, thermally activated relaxation, and reconfigurable collective excitations (Perrin et al., 2016, Scafuri et al., 16 Jan 2025).

1. Vertex architecture and local state space

The standard square-lattice ASI geometry consists of two orthogonal sublattices of elongated magnetic islands, one horizontal and one vertical, arranged so that four islands meet around each square vertex. Because each island is strongly shape-anisotropic, its moment is constrained primarily along its long axis, which yields the familiar square-ice vertex problem with $2^4=16$ local configurations (Perrin et al., 2016).

Vertex class	Local rule	Charge / role
Type I	$2$-in/$2$-out	$q=0$ ; lowest-energy ground-state vertex in conventional square ASI
Type II	$2$-in/$2$-out	$q=0$ ; higher-energy ice-rule vertex in conventional planar square ASI
Type III	$3$-in/$1$-out or vice versa	$q=\pm 2$ ; monopole-like charged defect
Type IV	$2$0-in or $2$1-out	$2$2; highest-energy vertex

In the conventional planar square lattice, type-I and type-II vertices both satisfy the ice rule, but they are not degenerate. Type-I vertices form the familiar flux-closure arrangement and constitute the ordered antiferromagnetic ground state; type-II vertices are the competing $2$3-in/$2$4-out class that becomes important under field bias, in defect strings, and in square-derived geometries such as pinwheel ASI (Perrin et al., 2016, Morley et al., 2017).

Type-III vertices are the central topological excitations of square ASI. They are created when the ice rule is violated locally and are interpreted as effective magnetic monopoles. In both nanoscale and macroscopic square ASI, separating a pair of opposite type-III charges leaves a string of reversed moments connecting them; in the conventional defect language, this is the Dirac string (Scafuri et al., 16 Jan 2025).

2. Energetics, square-ice conditions, and competing ordered phases

The conventional planar square lattice does not automatically realize the macroscopically degenerate square-ice manifold. Its key energetic asymmetry is that the nearest-neighbor interaction between orthogonal islands, $2$5, differs from the interaction between collinear nearest neighbors, $2$6. In that regime, the system is in the ordered $2$7 or $2$8 sector, so type-I vertices are lower in energy than type II and the low-temperature state is an ordered antiferromagnetic array rather than a Coulomb phase (Perrin et al., 2016).

A direct route to true square ice is to lift one sublattice vertically by a height offset $2$9, thereby reducing $2$0 while leaving $2$1 essentially unchanged. In the shifted geometry, the square-ice condition is $2$2. For $2$3 islands with lattice parameter $2$4 and gap $2$5, the square-ice-like regime appears around $2$6–$2$7: Bragg peaks disappear, the structure factor develops pinch points, and the measured correlation length at $2$8 is $2$9 lattice spacings, compared with $q=0$ 0 for finite-size theoretical ice-rule states on the same system size (Perrin et al., 2016). In that regime, monopoles are embedded in a disordered divergence-free background rather than confined by an ordered type-I environment.

A different controlled deformation rotates every island continuously from the square-ASI orientation to the $q=0$ 1 pinwheel geometry. In this square-derived family, the long-range order changes from antiferromagnetic to ferromagnetic at a rotation angle $q=0$ 2 that depends on inter-island spacing. Experimentally, the onset of ferromagnetic order occurs around $q=0$ 3 for $q=0$ 4 and around $q=0$ 5 for $q=0$ 6. The common point-dipole prediction of a universal transition near $q=0$ 7 fails, whereas a dumbbell-dipole description reproduces both the larger transition angle and its spacing dependence, with calculated $q=0$ 8 values from $q=0$ 9 at $2$0 to $2$1 at $2$2 (Strømberg et al., 2024). This establishes that near-field end-charge structure, not just center-point dipole geometry, controls phase boundaries in square-derived ASI.

3. Thermal activation, field-driven relaxation, and monopole kinetics

Thermally active square ASI exposes a second layer of physics: the ordering pathway depends not only on the ground-state hierarchy but also on the interplay between anisotropy barriers and dipolar coupling. In FePd square ASI with $2$3 islands and spacings $2$4, $2$5, and $2$6, raising the alloy magnetization from Fe:Pd $2$7 to $2$8 shifts the room-temperature $2$9 from $2$0 to $2$1. The low-$2$2 arrays begin fluctuating at $2$3–$2$4 but reach only about $2$5 type-I vertices, whereas the high-$2$6 arrays begin changing only around $2$7–$2$8 and then collapse within $2$9 into states with more than $q=0$ 0 type-I vertices by $q=0$ 1 (Morley et al., 2017). The distinction is not merely onset temperature: stronger interactions change the defect-mediated relaxation pathway from $q=0$ 2 string motion to $q=0$ 3 string motion.

FePd $q=0$ 4 pushes this logic further by combining strong dipolar coupling with an experimentally accessible thermal window. For disconnected elliptical islands $q=0$ 5 on a $q=0$ 6 square lattice, thermal annealing above $q=0$ 7 followed by cooling at $q=0$ 8 yields perfect or nearly perfect type-I ordering, with observed defect-free domains as large as $q=0$ 9 (Drisko et al., 2015).

In sub-$3$0 square ASI made from $3$1 Permalloy islands, SQUID relaxation measurements show Arrhenius-type Néel-Brown behavior rather than glassy freezing. The relaxation follows

$3$2

with attempt times of order $3$3 to $3$4. The measured $3$5 values cluster around $3$6 in much of the thermally active regime, which the paper interprets as evidence for effectively one-dimensional chain-like or string-like dynamics inside the two-dimensional square lattice (Porro et al., 2017).

Field can tune not only ordering pathways but also the density and kinetics of magnetic charges. In thermally active conventional square ASI, the type-I and type-II vertices become degenerate on the diamond boundary

$3$7

so the Dirac-string tension vanishes. At that boundary, the system enters a field-induced high-density monopole plasma, in which simulations show a large increase in free type-III defects and the equilibrium noise spectrum is closest to Lorentzian, with $3$8 approaching the Brownian limit $3$9 in the fit

$1$0

Broadband spontaneous magnetization-noise measurements over $1$1 to $1$2 and field maps integrated over $1$3 to $1$4 directly reveal this plasma-like regime in the conventional square lattice (Goryca et al., 2020).

Field-driven reversal in dense square ASI is also strongly disorder-sensitive. Ensemble MOKE measurements on $1$5 permalloy islands found a non-monotonic coercive field with a local maximum near $1$6 in the $1$7 array, while micromagnetic simulations showed that SEM-derived edge roughness of $1$8 was essential to reproduce the observed angular coercivity (Kohli et al., 2011).

4. Collective excitations, FMR, and spin-wave transport

Square-lattice ASI is also a collective-wave system. In a nearest-neighbor macrospin treatment linearized about a square-ice ground state, the four-sublattice problem reduces to an $1$9 dynamical matrix with four positive-frequency branches, $q=\pm 2$ 0, $q=\pm 2$ 1, $q=\pm 2$ 2, and $q=\pm 2$ 3. The lowest antisymmetric branch $q=\pm 2$ 4 is acoustic-like in the zero-anisotropy limit, while finite shape anisotropy opens a gap

$q=\pm 2$ 5

For realistic $q=\pm 2$ 6 permalloy islands at $q=\pm 2$ 7, the anisotropy parameters are estimated as $q=\pm 2$ 8, $q=\pm 2$ 9, and $2$00, so the spectrum is strongly gapped and only weakly dispersive around a large anisotropy-controlled baseline (Lasnier et al., 2020).

Semi-analytical and micromagnetic work subsequently showed that square ASI is not limited to bulk-like macrospin modes. For $2$01 Permalloy stadiums on a $2$02 lattice, internal edge-bending states produce additional low-frequency bands that are explicitly compared to impurity bands in semiconductors. In that picture, vortex and remanent square-ASI states have different unit cells, and C-, S-, or onion-like end states reconfigure the collective band structure. Modest in-plane fields below $2$03 tune the $2$04-point modes and move bulk and edge-derived bands through crossings, so the same nanostructure acts as a reconfigurable magnonic crystal (Iacocca et al., 2015).

State-resolved FMR at the level of a single square vertex makes the local spectral fingerprints even more explicit. For four $2$05-thick permalloy islands with fixed short axis $2$06, variable long axis $2$07, and vertex gap $2$08, the remanent vertex-localized modes for the $2$09 geometry occur at distinct frequencies: type I near $2$10, type II near $2$11, type III$2$12 at $2$13, $2$14, and $2$15, and type IV near $2$16 with a higher-order mode near $2$17. By contrast, the bulk-like remanent resonances cluster near the isolated-island bulk frequency around $2$18, making them poorer identifiers of vertex populations in extended arrays (Gomez-Iriarte et al., 2024).

Material asymmetry adds a further control axis. In bicomponent square ASI, with Ni$2$19Fe$2$20 on one sublattice and Co$2$21Fe$2$22 on the other, VNA-FMR spectra display sublattice-specific branches NH, NV, CH, and CV, but the essential result is not simple superposition. At $2$23, the resonance of one sublattice exhibits discrete frequency steps when the complementary sublattice switches, demonstrating inter-lattice dynamical back-action and mode-profile conversion between central and higher-order excitations (Lendinez et al., 2020).

Charge configurations also guide propagation. In micromagnetic simulations of square ASI made from $2$24 Ni$2$25Fe$2$26 bars with $2$27 gaps, charged vertices produce an additional charged-vertex mode near $2$28–$2$29 and strongly anisotropic transport. In a $2$30 open-edge array, the bulk mode transmits to the nearest neighboring vertex with coefficient $2$31 in the preferred direction but only $2$32 in the orthogonal direction, whereas the charge-free $2$33 state yields nearly symmetric transmission of about $2$34 along both directions (Arora et al., 2022).

5. Macroscopic, three-dimensional, and out-of-plane extensions

Square-lattice ASI is not confined to planar nanomagnet arrays. In a macroscopic mechanical analogue, sixty bar magnets of length $2$35 are mounted on low-friction rotary supports on a square lattice with $2$36. Each rotor is modeled in the magnetic-monopole approximation by two charges $2$37, and its motion obeys

$2$38

In an extended $2$39 lattice-site system of $2$40 magnets, the ground-state dispersion has two bands spanning roughly $2$41 to $2$42, with degeneracy near $2$43 along $2$44 and $2$45. In the finite sixty-magnet array, waves rapidly lose coherence through intermodal scattering; a Dirac string does not act as a simple waveguide, but it does support a localized mode near $2$46, below the bulk band, concentrated on the string itself (Scafuri et al., 16 Jan 2025).

Three-dimensional deformations of square ASI modify the same underlying vertex problem in a different way. In a tilted square ASI built from stadium-shaped nanomagnets with $2$47, $2$48, $2$49, and planar reference gap $2$50, each island is rotated out of plane by an angle $2$51. Two constructions are distinguished: fixed center spacing $2$52, and fixed gap with

$2$53

Both semi-analytical calculations and MuMax3 simulations find tunable FMR shifts and splittings in vortex and remanent states as $2$54 varies from $2$55 to $2$56; however, the paper also emphasizes that the diagonal demagnetizing-tensor approximation limits the semi-analytical model, while finite-difference micromagnetics for tilted stadiums are grid-dependent and extremely slow (Alatteili et al., 2024).

Out-of-plane square ASI provides a different square-lattice limit. In Hotspice, OOP square ASI is modeled as an Ising system with two checkerboard ground states. For pure nearest-neighbor exchange, the simulator reproduces the exact square-Ising transition

$2$57

With both exchange and dipolar interactions, the equilibrium state evolves from checkerboard order to stripe phases as the ratio $2$58 increases. The same framework was then used to demonstrate clocking in a $2$59 OOP square lattice by addressing the two checkerboard sublattices alternately with opposite fields (Maes et al., 2024).

6. Experimental and computational toolchain, and recurrent interpretive lessons

The square-ASI literature is methodologically plural. Real-space state reconstruction has been carried out with XMCD-PEEM in rotated square-to-pinwheel arrays, where thermalization at $2$60 and Voronoi-cell analysis track the AF-to-FM transition (Strømberg et al., 2024). MFM on vertically shifted square ice reconstructs vertex populations, structure factors, and pinch points after a $2$61-hour oscillatory-field demagnetization (Perrin et al., 2016). Lorentz TEM on FePd$2$62 square ASI resolves type-I ground-state order after annealing above $2$63 and cooling at $2$64 (Drisko et al., 2015). SQUID magnetometry measures both thermally activated relaxation and collective ferromagnetic hysteresis in large-area arrays (Porro et al., 2017, Bingham et al., 2022). MOKE accesses full-array coercive response in situ (Kohli et al., 2011). VNA-FMR, micromagnetic mode maps, and time-domain FFT methods resolve state-dependent microwave spectra (Lendinez et al., 2020, Gomez-Iriarte et al., 2024). More recently, scanning single-NV magnetometry on square-lattice ASI with lattice constants $2$65 and $2$66 has extracted both axial and transverse stray-field components, weak-field-induced magnetization tilts below switching threshold, and effective saturation magnetization through iterative comparison with micromagnetic models (Spindler et al., 2 Sep 2025).

On the theory side, the same caution recurs across otherwise different models: square-lattice ASI is often more structurally detailed than a point-dipole Ising graph. For continuously rotating macrospin descriptions, one common starting point is a dipolar-plus-anisotropy Hamiltonian of the form

$2$67

which underlies small-amplitude dynamics in Heisenberg-like square-ASI models (Wysin et al., 2012, Lasnier et al., 2020). At the opposite end, Hotspice treats each element as an Ising-like macrospin with thermally activated switching, and then corrects the basic point-dipole picture by second-order finite-size terms, a dumbbell model for IP systems, and asymmetric switching barriers for chiral reversal pathways (Maes et al., 2024).

Three interpretive lessons recur. First, the square lattice by itself is not equivalent to the square-ice model: planar square ASI usually orders into type-I antiferromagnetic ground states unless $2$68 is deliberately rebalanced (Perrin et al., 2016). Second, center-point dipoles are often insufficient whenever end-field structure matters, as shown explicitly for the square-to-pinwheel transition and more generally for dense or strongly interacting arrays (Strømberg et al., 2024). Third, clean coherent pictures can break down in dynamics: small-spacing reversal develops nonuniform internal twist rather than pure Stoner-Wohlfarth rotation, while mechano-magnetic waves in macro-ASI scatter strongly enough that Dirac strings cannot be used as simple wave gates (Bingham et al., 2022, Scafuri et al., 16 Jan 2025).

Taken together, these results define square-lattice ASI as a family of experimentally and theoretically tractable frustrated lattices whose behavior is controlled by vertex energetics, defect topology, internal micromagnetic texture, and geometric reconfiguration. In the present literature, the square lattice supports ordered type-I antiferromagnets, Coulomb-phase-like square ice, field-induced monopole plasmas, state-selective FMR, charge-guided spin-wave transport, mechanically realized defect modes, and out-of-plane checkerboard clocking; the unifying theme is that all of these phenomena remain rooted in the four-island square vertex and the tunable competition among anisotropy, dipolar coupling, thermal activation, and external field.