“Spin Ice” is an exotic type of frustrated magnet realized in “pyrochlore” materials Ho2Ti2O7, Dy2Ti2O7, Ho2Sn2O7, and so forth, in which magnetic atoms (spins) reside on a sublattice made of the vertices of corner-sharing tetrahedra. Each spin is Ising-like with respect to a local axis which connects the centers of two tetrahedra sharing the vertex occupied by the spin. The macroscopically degenerate ground states of these magnets obey the “two-in two-out” “ice rule” within each tetrahedron. Magnetic monopoles and antimonopoles emerge as elementary excitations, “fractionalizing” the constituent magnetic dipoles. This system is also a novel type of statistical mechanical system. Here we introduce a conceptual generalization of “spin ice” to what we shall call “color-tripole ice,” in which three types of “color charges” can emerge as elementary excitations, which are Abelian approximations of the color charges introduced in high energy physics. Two two-dimensional (2D) models are introduced first, where the color charges are found to be 1D and constrained 2D, respectively. Generalizations of these two models to 3D are then briefly discussed. In the second one the color charges are likely 3D. Pauling-type estimates of the “residual (or zero-point) entropy” are also made for these models. 1. Introduction Frustration and fractionalization are two fundamental concepts in modern condensed matter physics. Frustration simply means the existence of competing interactions that cannot be minimized simultaneously. It could result from more than one kind of interactions present in the system, but when the interactions are all of one kind, frustration can still arise from the geometric arrangement of the constituent entities of the system (i.e., atoms, spins, etc.); it is then referred to as “geometric frustration” [1–3]. Spin systems have offered paradigmatic examples of these concepts: [4] spin glass is a simple example of a richly frustrated system, where ferromagnetic and antiferromagnetic bonds are randomly distributed in a spin system, leading to disordered, macroscopically degenerate ground states, and a finite residual or zero-point entropy. Antiferromagnetic Ising model on a two-dimensional triangular lattice is a simple example of geometric frustration, with also disordered, macroscopically degenerate ground states and finite residual entropy, even though there is only one kind of (antiferromagnetic) interaction in the system, acting between all nearest-neighbor pairs of “Ising spins,” which just mean quantized magnetic dipoles in strong uniaxial local
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