Topological paramagnetism in frustrated spin-1 Mott insulators

Time-reversal-protected three-dimensional (3D) topological paramagnets are magnetic analogs of the celebrated 3D topological insulators. Such paramagnets have a bulk gap and no exotic bulk excitations, but have non-trivial surface states protected by symmetry. We propose that frustrated spin-1 quantum magnets are a natural setting for realizing such states in three dimensions. We describe a physical picture of the ground-state wave function for such a spin-1 topological paramagnet in terms of loops of fluctuating Haldane chains with nontrivial linking phases. We illustrate some aspects of such loop gases with simple exactly solvable models. We also show how 3D topological paramagnets can be very naturally accessed within a slave particle description of a spin-1 magnet. Specifically, we construct slave-particle mean-field states which are naturally driven into the topological paramagnet upon including fluctuations. We propose bulk projected wave functions for the topological paramagnet based on this slave-particle description. An alternate slave-particle construction leads to a stable $\mathrm{U}\left(1\right)$ quantum spin liquid from which a topological paramagnet may be accessed by condensing the emergent magnetic monopole excitation of the spin liquid.

Figure 1

Two species of AKLT loops, one on each sublattice of the diamond lattice (blue and red). Note that loops live on the links of the fcc sublattices, i.e., on next-nearest-neighbor bonds of diamond.

Figure 2

For appropriate boundary conditions, end points of $A$ and $B$ chains (red and blue, respectively) give surface excitations with mutual semionic statistics. Braiding the anyons on the surface (first arrow) changes the sign of the wave function for consistency with the rule that configurations related by passing an $A$ strand through a $B$ strand in the bulk (second arrow) appear in the wave function with opposite sign.

Figure 3

(left) Loops on interpenetrating cubic lattices $A$ and $B$. State $\left|\mathrm{\Psi }\right.〉$ is a superposition of such configurations with signs determined by linking $A$ and $B$ loops. (right) The product of Pauli matrices defining the flip term $\mathcal{F}$ on a plaquette [see Eq. (6)].

Figure 4

After a basis change, $\left|\mathrm{\Psi }\right.〉$ is a superposition of membrane configurations $({\tau }^x=-1$ on shaded plaquettes) with red loops (where ${\sigma }^z=-1)$ glued to membrane boundaries. (The red loops are $\sigma$-electric lines, and the membrane boundaries are $\tau$-magnetic lines.)

Figure 5

String operators creating surface excitations. (left) Acting with a chain of ${\sigma }^x$ operators on the links of the upper layer $(A$ lattice surface) gives a pair of $e$ excitations (i.e., end points of bulk $A$ strings). (right) A pair of $m$ excitations (i.e., end points of $B$ strings) are created by a chain of ${\tau }^x$ operators (thick green strand) on the lower layer $(B$ surface), together with ${\sigma }^z$ operators on the corresponding links in the upper layer (thick purple links).

Figure 6

Under the mapping (11), a ${\sigma }^z$ $\left({\tau }^z\right)$ operator on a link is exchanged with a product of ${\tau }^x$ $\left({\sigma }^x\right)$ operators on the surrounding links of the other lattice. (Links of one lattice can equally be thought of as plaquettes of the other.)