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“Spin Ice” is an exotic type of frustrated magnet realized in “pyrochlore” materials Ho_{2}Ti_{2}O_{7}, Dy_{2}Ti_{2}O_{7}, Ho_{2}Sn_{2}O_{7}, 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.

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” [

“Spin ice” is an exotic type of magnet in which both geometric frustration and fractionalization are realized. To the knowledge of this author, it is the first three-dimensional system known to exhibit fractionalization. It occurs in pyrochlore materials Ho_{2}Ti_{2}O_{7}, Dy_{2}Ti_{2}O_{7}, Ho_{2}Sn_{2}O_{7}, and so forth [

Spin ice is already a well studied subarea of condensed matter physics, with more than eleven hundred research works already published. In order to demonstrate that this is a very rich and fundamental subarea, with many research directions to pursue, I mention here some more published general reviews [

In this work, I will present a conceptual generalization of spin ice to what we shall call “color-tripole ice”, which also has macroscopically degenerate ground states, and nonvanishing zero-point entropy. But it will have three kinds of (so far Abelian) “color charges” (to be defined in Section

In Section

We begin by introducing a (in general hypothetical) two-component Coulomb charge:

Next, we will define a color tripole of strength

The twelve configurations of color tripole needed to construct the two 2D models of color-tripole ice. The first six configurations, (a)–(f), are needed in both models. The next six configurations, (a′)–(f′), are needed in the second model only.

In Section

Explicit numerical study of these models will be given in a future publication. Here we concentrate on the qualitative properties of these models.

To construct this model we begin with a triangular lattice viewed as many downward-pointing triangles sharing vertices. Each vertex is seen to be shared by three downward-pointing triangles. Thus we place at each vertex an Ising color tripole which takes only one of the six configurations (a)–(f) in Figure

The first two-dimensional model of color-tripole ice. (a) A color-tripole-crystal state constructed using configuration (a) in Figure

However, we shall argue below that, like spin ice, the ground states of this model of color-tripole ice are macroscopically degenerate, if we assume that the interaction between any pair of color tripoles in this model is the color-tripole-tripole interaction discussed in the previous section, which is a straightforward generalization of the magnetic dipole-dipole interaction in spin ice. We do not consider any possible generalized exchange interaction between nearest neighbor color tripoles, as in the case of spin ice, since spins have a quantum origin and can give rise to an exchange interaction, whereas the color tripoles introduced here so far are purely classical objects. Thus we can use an argument similar to the one introduced by Castelnovo et al. [

Starting with the color-tripole crystal state introduced already, which obeys the generalized ice rule (or any of the other five color-tripole crystal state which can be obtained from this crystal state by applying one of the five non-trivial symmetry operations of an equilateral triangle to all color tripoles in the lattice at the same time) we can see how other ground states can be generated, which also obey the generalized ice rule. First, we can swap green and blue charges within each of a

The first two-dimensional model of color-tripole ice—continuation. (a) Two type-II Y-intersections of color-swap lines introduced in the color-tripole crystal state of Figure

If any of the color-swap lines has a finite length and terminates at two downward-pointing triangles of the lattice, we would have created two color-charged elementary excitations in the lattice, one having the anticolor of the other. That is, they can combine to color neutrality. For example, if a green/blue swap line is terminated on both ends at two downward-pointing triangles, then its right-end downward triangle will have a net color charge of green-plus-anti-blue, or in short,

In this model we begin with a hexagonal lattice. We note that each vertex in this lattice is shared by three hexagons. However, we also note that all vertices of this lattice can be classified into two types: a type-A vertex and its three nearest neighbor vertices form an upright Y. At each type-A vertex one must put a color tripole which is among the configurations (a) through (f) in Figure

The second two-dimensional model of color-tripole ice. (a) One of the 36 color-tripole crystal states that obey the generalized ice rule for color-tripole ice and preserve the translational symmetries of the lattice. It is made of configurations (a) and (a′) of Figure

Note that without breaking the translational symmetries of this lattice, one can form

Now suppose we start with a different color-tripole crystal state, which is constructed with configurations (a) and (e′) of Figure

The second two-dimensional model of color-tripole ice—continuation. (a) A different one of the 36 color-tripole crystal states that obey the generalized ice rule for color-tripole ice and preserve the translational symmetries of the lattice. It is made of configurations (a) and (e′) of Figure

For this purpose we begin with the so-called “trillium lattice” [

The trillium lattice cubic unit cell (from Figure

For this purpose we begin with simply the “pyrochlore lattice” already mentioned in association with spin ice. We note that one view of this lattice is that each vertex is shared by two tetrahedra, but another view of this lattice is that each vertex is shared by six equilateral triangles, three of which are the faces of one tetrahedron, and the other three of which are the faces of the other tetrahedron. Thus to construct the 3D generalization of our second model of color-tripole ice, we must put a color tripole at the center of each equilateral triangle of this lattice, in the orientation of that triangle. In the trumbbell approximation, this color tripole will again be replaced by three color charges, one red, one green, one blue, each of which is deposited at a separate vertex of the triangle. Each vertex will then have six color charges deposited at this site, originating from the six trumbbells replacing the six color tripoles located at the centers of the six triangles sharing this vertex, These six color charges, in the ground state, will again combine to color neutrality, thus obeying the generalized ice rule defined in this work for color-tripole ice. These six color charges must then be two red, two green, and two blue. Thus once a color-swap line enters a vertex, it now has two choices on how it can exit the vertex—in much analogy with the pyrochlore spin ice, where once a spin-flip line enters a tetrahedron, to preserve the ice rule it can have two choices on how it can exit the tetrahedron, because each tetrahedron has four vertices where spins reside, with two of them pointing into, and the remaining two of them pointing out of the tetrahedron.

We shall defer to a future work to introduce even a color-tripole crystal state in this model and to give explicit examples of the color-swap lines. These can be combined with a numerical study of the statistical mechanical properties of this model, which is likely very rich in properties. We only note here that the color-charged elementary excitations, which can emerge in this model, will most likely be 3D, much like the monopole excitations in pyrochlore spin ice, because of the properties of any color swap line described in the previous paragraph.

There may be even better lattices to realize this 3D model of color-tripole ice, where each vertex is shared by six (or nine? twelve? fifteen?…) equilateral triangles that are more evenly distributed around this vertex, instead of being grouped into those of two (or more?) tetrahedra. But so far we have not found them. The readers with more experiences with exotic 3D lattices might be able to find them. Or perhaps mathematicians might be able to prove the nonexistence of any of such cases.

We first review the Pauling estimate of the zero-point entropy of pyrochlore spin ice. For this purpose the interactions between the spins are totally neglected, and are simply replaced by the ice rule. If the system has

We now apply similar reasonings in order to estimate the zero-point entropy of the four models of color-tripole ice introduced here. For this purpose we begin by assuming that there are

Next, we consider the 3D generalization of this model. Referring to Figure

Next, we consider the second 2D model. In this model, we have each vertex shared by three “hexagons” and each hexagon having six vertices. Thus to

Finally, we consider the 3D generalization of the second 2D model. Here we must consider the “dual lattice” of the pyrochlore lattice, with each “dual-lattice” point being the center of an equilateral-triangle face of a tetrahedron in the pyrochlore lattice. Since each equilateral triangle has three original lattice points as its vertices, and each original lattice point is shared by six equilateral triangles, we see that

Note that the infinite-temperature molar entropies of these four models of color-tripole ice are all equal to

Our estimate of the zero-point entropies presented above are all based on a generalization of the original Pauling estimate. Whether this approximation is reasonably accurate or much cruder for the color-tripole ice than for the spin ice must await an essentially exact numerical study of of the color-tripole-ice models.

In this work, we have introduced a conceptual generalization of pyrochlore spin ice to what we have called “color-tripole ice.” It has macroscopically degenerate ground states and nonvanishing molar zero-point entropy just like spin ice. However, unlike spin ice, which has magnetic monopole/antimonopole elementary excitations, fractionalizing magnetic dipoles which are the elementary building blocks of that system, in the case of color-tripole ice, what are fractionalized are what we called “color tripoles,” resulting in three kinds of color-charged elementary excitations, and their corresponding anticolor-charged elementary excitations. These color charges are analogous to the color charges introduced in high energy physics, except that the former are Abelian so far, whereas the latter are well-known to be non-Abelian. Further generalization to non-Abelian color charges may be possible, but the chance to realize them in condensed matter systems (natural or artificially fabricated) would be even more difficult, so we have not yet given it much thought so far. Instead, we have concentrated on finding ways to realize the Abelian models introduced here. Thus before we conclude this section, we would like to give some general suggestions to this goal. We have proposed in this work two two-dimensional (2D) models and then discussed their possible three-dimensional (3D) generalizations. We deemed that realizing either of the two 3D models will be much more difficult than realizing the corresponding 2D models. So below we will only discuss the possible ways to realize the two 2D models by artificial fabrication. To arrange entities in a triangular or hexagonal lattice can presumably be done in the laboratories by a number of methods, including optical and e-beam lithography, masks and etching, and even by the use of a scanning tunneling tip to put atoms in atomic-scale patterns. Thus we need only concentrate on the realization of the individual color tripoles which should be arranged in the prescribed locations in the said lattices as described in the two 2D models. For this purpose we must first realize the two-component, fictitious, “vector Coulomb charges” introduced in Section

There may exist various generalizations (extensions) of the idea presented here. At first sight, it may appear that instead of having three distinct colors that can combine to color neutrality, we can also have five, or seven, or even larger prime numbers of, distinct colors that can combine to color neutrality that are still based on two-component vector Coulomb charges. One can simply replace the equilateral triangle used in the definition of the three fundamental color charges presented in Section

Finite temperature properties of the color-tripole-ice models including possible phase transitions in the absence or presence of external fields coupled to the two components of the vector Coulomb charges will be investigated in a future publication. Such systems will likely have very rich and novel behavior.

_{2}Ti

_{2}O

_{7}: a dipolar Spin Ice system

_{2}Ti

_{2}O

_{7}under a magnetic field

_{2}Ti

_{2}O

_{7}in a magnetic field

_{2}Ti

_{2}O

_{7}

_{22}O

_{7}: kagome Ice behavior

_{2}Ti

_{2}O

_{7}

_{2}Ti

_{2}O

_{7}in a [111] magnetic field

_{3}-BaTiO

_{3}thin films at room temperature

_{2}Ta

_{2}O

_{11}film