Depleted pyrochlore antiferromagnets

I consider the class of “depleted pyrochlore” lattices of corner-sharing triangles, made by removing spins from a pyrochlore lattice such that every tetrahedron loses exactly one. Previously known examples are the “hyperkagome” and “kagome staircase”. I give criteria in terms of loops for whether a given depleted lattice can order analogous to the kagome “ √ 3× √ 3” state, and also show how the pseudo-dipolar correlations (due to local constraints) generalize to even the random depleted case. Consider “bisimplex” antiferromagnets [1], meaning that every spin is shared between exactly two triangles or tetrahedraLet there be only the nearest-neighbour coupling, so the system is highly frustrated. After the kagomé [2], pyrochlore [3, 4], and garnet [5] lattices, further such systems were discovered such as the “kagomé staircase” (realised in e.g. Ni3V2O8 [6]) and the “hyperkagomé” lattice [8, 7] (realised in Na4Ir3O8 [9]). Both magnetic lattices are obtained from the pyrochlore lattice by removing 1/4 of the sites so as to leave a network of cornersharing triangles. This paper considers the entire family of such “depleted” structures, and the equilibrium states of classical or semiclassical spins on them. 1 The pyrochlore lattice is most simply visualised as the “medial graph” (bond midpoints) of a diamond lattice. A depleted lattice is made by placing a dimer covering on the diamond lattice bonds, then removing spins from the covered sites. This constraint is plausible: in real spinels with magnetic B sites (= a “pyrochlore” magnetic lattice), dilution is achieved by substituting a nonmagnetic species X on B sites. Due to size (or maybe charge) imbalance, X ions repel; hence the lattice gas of X ions maps to an Ising antiferromagnet [3] in a field. That is itself a highly frustrated problem, with a degenerate ground state (all those dimer coverings) – provided the lattice gas has only nearest neighbour interactions. Further neighbour ion terms presumably select a specific depletion pattern. After some examples of depleted lattices, I address two questions (for the periodic and random cases): (i) What is the pattern of magnetic order (if any); (iii) How do we generalize the disordered classical liquid with pseudo-dipolar correlations due to the constraints? 1. Regularly depleted pyrochlore lattices In this section, I catalog some highly symmetrical dimer coverings of the diamond lattice, each of which specifies a different depleted pyrochlore lattice. 1 I will not consider other depleted lattices, except to observe the hidden symmetry (considering nearestneighbour bonds) whereby the hyperkagomé lattice [9] is equivalent to a garnet lattice [5], is reminiscent of the hidden equivalence of the 1/5-depleted square lattice of CaV4O9 [10] to the 4-8 lattice [11]. These can be conveniently be visualized in two possible ways: (i) projecting the conventional cubic cell (containing four diamond sites, i.e. four pyrochlore tetrahedra) in the (100) direction; or, (ii) expressing the diamond lattice as a stacking of puckered honeycomb layers, in which the (odd) sites have an additional bond extending upwards (downwards) to the next layer. 1.1. Cubic conventional cell Consider the family of patterns preserving the periodicity of the conventional cubic cell, with four dimers per cell. There are only three symmetry-inequivalent ways to place them. Pattern 1 has all dimers oriented the same; the diamond lattice separates into disconnected (puckered) honeycomb layers, and the spins form stacks of uncoupled kagomé lattices, as induced by a field in “kagomé ice” [12]. Pattern 2 has two dimers in one orientation and two in another orientation. This again separates the lattice into disjoint slabs, now transverse to a (110) axis. The depleted lattice is a “kagomé staircase”, which has the topology of a kagomé lattice, but folded so the hexagons alternate between two (111) type orientations. Imagine a canonical spinel AB2O4 with cubic lattice constant ac ≈ 8.2 Å; depletion by B-site vacancies gives the formula A1B1.5O4 with x = 1/4. Pattern 2 makes the structure orthorhombic, with a ≈ ac/sqrt2, b ≈ √ 2ac, and c ≈ ac. This nearly describes the “kagomé staircase” compounds e.g. VCo1.5O4, except each successive slab is slid a half cell relative to the one before. Pattern 3 uses each of the four possible orientations once, yielding the “hyperkagomé” arrangement with cubic symmetry. as realized [9] in the spinel Na4Ir3O8, i.e. “(Na1.5)1(Ir3/4Na1/4)2O4” in our framework, 1.2. Kagomé layer stacking approach I will focus on the layered patterns made by viewing the diamond lattice as a stack of puckered honeycomb layers (a kagomé layer of spin sites) connected by vertical linking bonds (forming a triangular lattice). Say a fraction xlink of linking bonds is depleted; then the honeycomb layer has depletion xkag ≡ (1 − xlink)/3; its depletion pattern is a dimer covering having monomers at the endpoints of the depleted linking bonds. If either kind of layer lacks threefold point symmetry, we restore it by stacking the layers with a rotation (producing a screw axis). Table 1. Some depleted lattices, with properties of the shortest loops (Loop density is per site; in the tags, interlayer bonds are underlined for comparison to Fig. 1.) Fig. 1 xlink xkag Lloop loop Colouring Loop is part density tags colourable? a) 1/3 2/9 8 1/2 bbbsbbbb no b) 1/4 1/4 10 3/4 (bbsbs) yes c) 1/4 1/4 8 1/16 bbbbbssb no d) 1/7 2/7 6 1/14 bbbbbb yes Fig. 1 shows some examples; there are many more, e.g. in Fig. 1(c) a different honeycomb dimer pattern could be used. Fig. 1(b) is just the cubic “hyperkagomé” structure. With xlink = 0 and xkag = 1/3 we get a dimer covering of the honeycomb lattice. Figure 1. Depleted lattices built by stacking puckered layers along a 3-fold axis. Diamond lattice are shown bonds as lines (in the layer) or as circles (for linking bonds: up=shaded, down=black). Removed bonds are indicated by dashed lines or empty circles. In (a) and (c), successive layers are stacked with a 2π/3 rotation around the dotted triangle. (a). a √ 3 × √ 3 pattern; (b). hyperkagomé lattice; (c). another 2× 2 pattern; (d). a √ 7× √ 7 pattern. 2. Magnetic ground state The magnetic Hamiltonian is assumed to be H = J ∑〈ij〉 si · sj where only nearest-neighbour isotropic interactions are included, and {si} are either classical unit vectors or quantum spins with S ≫ 1. Let us review the known story for the kagomé or garnet/hyperkagomé) antiferromagnets. The classical ground states are the (many) configurations in which the spins differ by angles 2π/3 on every triangle. It then transpires that (i) a (still highly degenerate) subset of “coplanar” states gets selected at harmonic order, which amounts to a 3-colouring; (ii) a specific coplanar ordered state is selected by anharmonic fluctuations, which was the “ √ 3× √ 3” state in the kagomé case [14, 13, 15] and Lawler’s state in the hyperkagomé/garnet case [7]. Do these results generalise? 2.1. Coplanar states At harmonic order, thermal fluctuations (in the classical case [2, 4]) or quantum fluctuations [13, 16] select “coplanar” ground states, such that spins in different triangles lie in the same plane of spin space. 2 This selection should carry over to arbitrary corner-sharing triangle networks. A coplanar ground state has every spin in one of three directions so it is effectively a 3colouring of the sites (equivalently, diamond-lattice bonds) by colours A,B,C, such that every triangle (i.e. bonds meeting at one diamond vertex) has one of each colour. By König’s theorem [17] of graph theory, a 3-colouring exists on any bipartite graph with coordination z = 3, and hence on any depleted lattice. Any 3-colouring of a depleted lattice can usefully be reimagined as a 4-colouring of the original pyrochlore lattice, with the fourth color corresponding to the depleted sites. Thus, any 3-colouring in fact generates four different ways to build a depleted lattice (along with a sample 3-colouring of each), depending on which color we select for removal. (In Lawler’s state [7], all four colours are equivalent; each colour forms a “trillium” lattice [18].) 2.2. Ising mapping and effective Hamiltonian As is well known, at harmonic order all the coplanar states have equivalent Hamiltonians [2, 13]. Hence, fluctuations distinguish among them only at anharmonic order. Observe too that coplanar configurations cannot distinguished on the loop-free “cactus lattice” – the medial graph of a z = 3 Bethe lattice – since they are all symmetry-equivalent by permutations of the sites. Loops are essential to state selection [19, 20]. On the kagomé lattice, coplanar states are more transparently represented by “chiralities” ηα = ±1, Ising variables defined on triangle centers α, and equal to +1 (−1); if the colours 2 The constraint counting argument of [4] carries over independent of how the triangles are arranged. ABC run counterclockwise (clockwise) around the triangle. The colouring can be uniquely reconstructured (modulo trivial symmetries) from {ηα}, but not every configuration of {ηα} corresponds to a colouring. With approximations, one can obtain an effective Hamiltonian of form Heff = − ∑ αβ Jαβηαηβ in the quantum case [15], and also in the classical case [21] at small T , based on [22]. The nearest neighbor Ising coupling J1 < 0, so the optimum state is an antiferromagnetic pattern of ηα on the honeycomb vertices. What about d = 3? Let the index α label triangles, or equivalently diamond-lattice vertices. A gauge choice is necessary on every triangle to define which sense of its normal vector is “up”, before the spin chirality can be defined. I conjecture that, in general, the sign of J1 is the opposite of the projection of the normal vectors of the respective triangles. Then the ground state has an alternating chirality pattern (which is always possible, since the diamond vertices are bipartite). For the hyperkagomé lattice – the most regular depleted example – this gives the correct answer: Lawler’s state, favored in a large-n calculation [7] and found in simulations [8].) Whereas the “ √ 3× √ 3” state of the kagomé had alternating colours (e.g. ABABAB) around loops and triple colors (ABCABC...) along lines, and has a nonzero ordering vector, Lawler’s state has alternating colours along lines and ordering vector Q = 0. 3 The basis for believing J1 < 0 is general is that in the kagomé case, it was expressed [15, 21] in terms of expectations of spin-wave fluctuations of “soft” modes, which have only anharmonicorder restoring forces. Such soft modes are generic to the coplanar state on any of our lattices. They sum to zero on every triangle, a “zero-divergence” constraint that implies generic powerlaw correlations of the fluctuations, whether classical [23, 24, 25] or quantum [15, 26]. We obtain J1 < 0 if the sign depends on orientation the same way it does asympotically. A caveat is that at root, the crucial spin-wave correlations must depend on the loops (as noted above, only loops distinguish among coplanar states); the pseudo-dipolar correlations are just a coarse-grained way to incorporate the net effects of many long loops. An alternate local derivation of Heff , based on a loop expansion [19, 20, 28] might better capture the differences among depleted lattices.