Nambu monopoles interacting with lattice defects in a two-dimensional artificial square spin ice

The interactions between an excitation (similar to a pair of Nambu monopoles connected by their associated string) and a lattice defect are studied in an artificial two-dimensional square spin ice. This is done by considering a square array of islands containing only one island different from all others. This difference is incorporated in the magnetic moment (spin) of the “imperfect” island and several cases are studied, including the special situation in which this distinct spin is zero (vacancy). We show that the two extreme points of a defective island behave like two opposite magnetic charges. Then, the effective interaction between a pair of Nambu monopoles with the defective island is a problem involving four magnetic charges (two pairs of opposite poles) and a string. We also sketch the configuration of the field lines of these four charges to confirm this picture. The influence of the string on this interaction decays rapidly with the string distance from the defect.

Figure 1

The four topologies for the vertices in an artificial square spin ice. The energy of these topologies increases from left to right. Topologies $T_1$ and $T_2$ obey the ice rule (two in/two out) but, they are not degenerate. Topology $T_3$ exhibits the configurations with three in/one out or one in/three out while in the topology $T_4$, one has four in or four out. Topology $T_1$ gives the ground state. Topology $T_2$ is associated with the stringlike excitations and $T_{3,4}$ are associated with magnetic monopolelike excitations.

Figure 2

Particular configuration of excitations in a lattice with a defective island [yellow (light gray) arrow between numbers 3 and 4]. We use two basic shortest strings in the separation process of the magnetic charges, which are indicated by the marked white arrows. Pictures (a) and (b) exhibit two kinds of strings, $I$ and $\mathit{II}$, respectively. The black circle is the positive charge while the red (dark gray) circle is the negative charge.

Figure 3

Data for $\Delta$ as a function of $r=R/a$, considering string type I (a) and II (b) (the string shapes I and II are shown in Fig. 2). In these figures the smallest distance between position 1 and the defect $\delta$ is larger than one spacing lattice. The magnitude of dipole moment is $|{\vec{S}}_l|=0$. The simulation data $\Delta$ are the points and represent the case with $\delta =5a$. The dashed lines are the fits to expression (2).

Figure 4

Data for $K_1$ as a function of the size of the defective island. Observe the linear behavior of $K_1$ as $S_l$ is increased.

Figure 5

Ground-state configuration of the system around the defective island for (a) $S_l$ smaller (b) $S_l$ greater than other islands. The white circle represents the negative charge induced by the lattice defect while, the gray circle represents the positive charge.

Figure 6

Data for $\Delta$ as a function of $r=R/a$, considering strings I (a) and II (b), for $\delta =1a$ and $S_l=0$. The simulation data for $\Delta$ are the black points and the red dashed line is the fit to expression (2).

Figure 7

The interaction constant between string and defect, $K_2$, as a function of their separation $\delta$. The black circles and red squares are the fitting data obtained for the strings I and II, respectively.

Figure 8

Magnetic field intensity of the system's ground state. At the center of the plaquettes and at the centers of the vertices, indicated by the red crosses, the field goes to zero (white regions). These points were used to obtain the magnetic field produced by the excitations alone. At the right side of the figure the color palette for the magnetic field intensity is shown in normalized units.

Figure 9

This figure exhibits the magnetic field lines of a pair of Nambu monopoles and its string (the spins located between the dark gray vertices or red spins) considering the field produced by the excitation alone. Only a small portion of the system is shown for clarity.

Figure 10

(a) Electric-field lines produced by four electric charges: two of unity magnitude representing the Nambu monopoles and two of one-half magnitude representing the vacancy's charges. (b) Magnetic field lines of the spin configuration obtained by considering the field produced by the excitation alone. The red spins (the spins located between the dark gray vertices) were flipped to produce the Nambu monopoles and its string. Only a small portion of the system is shown. The vacancy is placed approximately at $(-1,2)$.