Topological Route to New and Unusual Coulomb Spin Liquids

Coulomb spin liquids are topological magnetic states obeying an emergent Gauss law. Little distinction has been made between different kinds of Coulomb liquids. Here we show how a series of distinct Coulomb liquids can be generated straightforwardly by varying the constraints on a classical spin system. This leads to pair creation, and coalescence, of topological defects of an underlying vector field. The latter makes higher-rank spin liquids, of recent interest in the context of fracton theories, with attendant multifold pinch points in the structure factor, appear naturally. New Coulomb liquids with an abundance of pinch points also arise. We thus establish a new and general route to uncovering exotic Coulomb liquids, via the manipulation of topological defects in momentum space.

Figure 1

Honeycomb lattice model exhibiting a series of distinct classical spin liquids. (a) The local constraint defining the ground states of the model. The sum of spins on each hexagon (black, filled circles) plus a coefficient $\gamma$ multiplied by the sum of spins linked to the exterior of the hexagon (red, open circles) must vanish on every hexagon [Eqs. (8) and (9)]. (b) Phase diagram as a function of $\gamma$, showing a series of algebraically correlated CSLs. They are distinguished by the number and arrangement of topological defects of the constraint vector $\mathbf{L}(\mathbf{q})$ in the Brillouin zone, with these defects giving rise to pinch points in the structure factor $S(\mathbf{q})$ (Fig. 2). Transitions between spin liquids occur either by creation or annihilation of pairs of defects ($\gamma =1/3$) or by coalescence of defects with a net charge ($\gamma =1/2$) leading to a higher-rank Coulomb liquid with multifold pinch points.

Figure 2

Evolution of $S(\mathbf{q})$ across distinct honeycomb spin liquids [Fig. 1]. For small values of $\gamma$ there are vortices in $\mathbf{L}(\mathbf{q})$ and corresponding pinch points in $S(\mathbf{q})$ at the Brillouin zone corners. At $\gamma =1/3$ there is a nucleation of pairs of oppositely charged topological defects at the zone boundary which then migrate toward the zone corners. At $\gamma =1/2$ the coalescence of these defects into objects with charge $Q=\pm 2$ manifests in the appearance of fourfold pinch points in the structure factor, associated to the emergence of a rank-2 Coulomb liquid. On increasing $\gamma$ further these defects separate and the system enters a new spin liquid with six pinch points in the interior of the Brillouin zone, as well as at the zone corners. The left half of each panel is the result of a Monte Carlo simulation of $N=1920$ spins at $T=0.002J$ and the right half is a calculation using the projection approach [Eq. (6)].

Figure 3

Construction and phase diagram of octochlore spin liquids. (a) The octochlore lattice, a network of corner sharing octahedra, is the medial lattice of the simple cubic lattice. (b) The local constraint applying to each octahedron in the lattice. The sum of the spins on the central octahedron (red), plus $\alpha$ times the sum of the spins neighboring that octahedron (blue), plus $\beta$ times the sum of remaining spins on the surrounding octahedra (yellow) must vanish [Eq. (15)]. (c) The phase diagram of distinct Coulomb spin liquids obtained by varying $\alpha$ and $\beta$, distinguished by the number and arrangement of topological defects in $\mathbf{L}(\mathbf{q})$. For the definition of each spin liquid see Table 1.

Figure 4

Abundant and multifold pinch points in octochlore spin liquids. (a) $S(\mathbf{q})$ in region IV of Fig. 3. $\mathbf{L}(\mathbf{q})$ exhibits topological defects at $(\pi ,\pi ,\pi )$ and at wave vectors removed along the (1,0,0), (1,1,0), and (1,1,1) directions from that point. The structure factor exhibits pinch point singularities at the corresponding positions. (b) $S(\mathbf{q})$ at the boundary between regions II and VI of Fig. 3. There are $Q=7$ topological defects at the zone corners which manifest as multifold pinch points, associated to a higher-rank Coulomb liquid. The left half of each panel is the result of a Monte Carlo simulation of $N=31944$ spins at $T=0.01J$ and the right half is a calculation using the projection approach.