Chiral Spin Liquid Wave Function and the Lieb-Schultz-Mattis Theorem

We study a chiral spin liquid wave function defined as a Gutzwiller projected BCS state with a complex pairing function. After projection, spontaneous dimerization is found for any odd but finite number of chains, thus satisfying the Lieb-Schultz-Mattis theorem, whereas for an even number of chains there is no dimerization. The two-dimensional thermodynamic limit is consistently reached for a large number of chains since the dimer order parameter vanishes in this limit. This property clearly supports the possibility of a spin liquid ground state in two dimensions with a gap to all physical excitations and with no broken translation symmetry.

Figure 1

Dimer-dimer correlation functions as a function of the distance for three and four chains with PBC on both directions.

Figure 2

(left panel) Finite-size scaling of the dimer order parameter in the rectangular geometries with an odd number of chains. ( right panel) Dimer order parameter as a function of the number of chains. The full circles in the right panel refer to the short-range resonating valence bond state, defined as the superposition of all nearest-neighbor dimer coverings of the lattice with the same weights, having $\xi \to \infty$ in 2D.

Figure 3

(a) Chiral order parameter as a function of the number of chains. (b) Difference of the nearest-neighbor total energy $\Delta E_{nn}=N[⟨{\widehat{\mathbf{S}}}_i·{\widehat{\mathbf{S}}}_j{⟩}_e-⟨{\widehat{\mathbf{S}}}_i·{\widehat{\mathbf{S}}}_j{⟩}_o]$ between two orthogonal states $|e⟩$ and $|o⟩$ predicted by the LSM theorem. (c) Finite-size scaling of the dimer order parameter in rectangular lattices with finite aspect ratio. For (b) and (c), $L_y=L_x+1$.