Memory of topologically constrained disorder in Shakti artificial spin ice

TL;DR

This work reveals that topological constraints in Shakti artificial spin ice enable memory effects and complex, protocol-dependent dynamics that are absent in square ice. By comparing unidirectional and bidirectional field protocols, the authors show that a critical coupling ratio $oldsymbol{alpha}=J_{ m perp}/J_{ m parallel}$ governs a transition between degenerate topological order ($oldsymbol{alpha}<2$) and conventional antiferromagnetic order ($oldsymbol{alpha}>2$), with memory emerging as sequence-dependent bifurcations or persistent final states under various protocols. The minimal 4×4 Shakti supercell already captures the richness, displaying deterministic memory, stochastic branching, and path-dependent return states; large systems retain memory only up to certain field thresholds. These findings highlight how topology and correlated disorder can be harnessed for programmable, memory-bearing responses in frustrated magnetic metamaterials, with potential avenues to relate to dimer mappings, loop dynamics, and other topologically ordered systems in spin ice genera.

Abstract

Complex behaviors often sit at a critical threshold between order and disorder. But not all disorder is created equal. Disorder can be trivial or constrained, and correlated disorder can even be topological. Crucially, constrained disorder can harbor memory, leading to non-trivial, sequence-dependent responses to external manipulations. And yet the fascinating subject of "memory of disorder" remains poorly explored, as memory is often associated to the retention of metastable order. In recent years artificial frustrated materials -- in particular arrays of frustrated nanomagnets known as artificial spin ices -- have been employed to study complex disorders and its wealth of exotic behaviors, yet their memory properties have received much less attention. Here, we investigate both analytically and numerically the sequence-dependent responses of two somehow opposite yet related artificial spin ices: the Landau-ordered square spin ice and the disordered but topologically-ordered Shakti spin ice. We find that Shakti exhibits a pronounced sequence-dependent response, whereas in the square lattice, such path dependence is absent. Within Shakti, even the minimal periodic supercell demonstrates both deterministic and stochastic forms of sequence memory, depending on the interaction strength. Extending our study to cyclic driving, we find that retracing the same input path leads to enhanced memory retention. These results open new perspectives on how topological constraints and correlated disorder generate robust memory effects in frustrated artificial materials, hitherto examined mainly in terms of their ground-state kinetics and thermodynamics.

Figures (19)

• Figure 1: a) Dilution of the square lattice (top) yields the Shakti lattice (bottom). Representative perpendicular (blue) and parallel (red) interacting spin pairs are marked. b) Spin configurations of ascending energy, in four-, three- and two-coordinated vertices. c) Energy spectra of vertex types (top) and ground state configurations (bottom) for the Shakti lattice. Energy is plotted with respect to the minimal energy level for each coordination number. Colors correspond to vertex colors in b. The lattice does not permit all vertices to be in their lowest energy state, and the first excitation switches from $II_3$ to $II_2$ as the ratio of perpendicular to parallel interaction $\alpha = J_{\perp} / J_{\parallel}$ changes, leading to a degenerate ground state for $\alpha<2$ (left) and to an antiferromagnetic ground state for $\alpha>2$ (right).
• Figure 2: The protocols used to drive the artificial spin ice by an in-plane external magnetic field $\vec{h}=(h_x,h_y)$. Unidirectional protocols: a) One way (1D1W) X path; b) Two way (1D2W) XX path; c) Pulse (1DPulse) XX path. Bidirectional protocols: d) One way (2D1W) XY path; e) Two way (2D2W) XYYX path; f) Loop (2DLoop) XYXY path. All six protocols have a complementary variant in which X and Y are exchanged. Red dots indicate the maximal external fields applied during the process.
• Figure 3: Response of square artificial spin ice to one-way protocols. a) Magnetization at the end of the 1D1W or 2D1W protocols as a function of the final field. b) Left: Ground state configuration at zero field. Center: Representative configuration after 1D1W, namely at field $\vec{h}=(H,0)$, where $H$ is greater than the critical field $H_c=4(\alpha+1)$; all $x$-spins point along the field, while $y$-spins are arranged such that each column points in an arbitrary direction. Right: Configuration at the end of the 2D1W protocol, namely at the final field $\vec{h}=(H,H)$. The flipped spins with respect to the initial configuration are colored in green, and excited vertices are colored according to the scheme introduced in Fig \ref{['lattice']}b.
• Figure 4: Degenerate ground state configurations of the Shakti supercell. Excited 3-coordinated vertices are colored according to the scheme introduced in Fig. \ref{['lattice']}b. Configurations (i)-(iii) all have zero magnetization, while configurations (iv.a) and (iv.b) have positive $x$-magnetization and configurations (iv.c) and (iv.d) have positive $y$-magnetization.
• Figure 5: Response starting from the degenerate state. Minimum energy states obtained at the end of the one way unidirectional protocol 1D1W, starting at configuration (iii.a). The $\alpha<2$ case (top) shows bifurcations in the outcomes for $H>\alpha$ whereas for $\alpha>2$ (bottom), the response is deterministic. The degenerate ground state configuration (iv.a) is highlighted in yellow.