Constructing Quantum Spin Liquids Using Combinatorial Gauge Symmetry

We introduce the notion of combinatorial gauge symmetry—a local transformation that includes single spin rotations plus permutations of spins (or swaps of their quantum states)—that preserve the commutation and anticommutation relations among the spins. We show that Hamiltonians with simple two-body interactions contain this symmetry if the coupling matrix is a Hadamard matrix, with the combinatorial gauge symmetry being associated with the automorphism of these matrices with respect to monomial transformations. Armed with this symmetry, we address the physical problem of how to build quantum spin liquids with physically accessible interactions. In addition to its intrinsic physical significance, the problem is also tied to that of how to build topological qubits.

Figure 1

(a) A single site (star) of the ${\mathbb{Z}}_2$ gauge theory, with 4 matter spins ${\mu }_a$ on the site, and 4 gauge spins ${\sigma }_i$ on the links. (b) The full lattice.

Figure 2

(a) Operator generating the local ${\mathbb{Z}}_2$ gauge transformation on an elementary plaquette, $G_p$ in Eq. (6). (b) A closed loop operator along a path ${\gamma }_C$. (c) An open string operator along a path in a system with boundaries.

Figure 3

(a) A single plaquette of the ${\mathbb{Z}}_2$ gauge theory, with 4 gauge spins ${\sigma }_i$ on the links, 4 matter spins ${\mu }_a$ on the site, and 4 additional matter spins ${\tau }_b$ at the center of the plaquette. (b) A single star operator $F_s$.