Figure 4: Indistinguishability of local observables.

(a) To demonstrate local indistinguishability of the states on the torus in the thermodynamic limit, we consider the set of all spin operators that act on a plaquette of four spins. Note that all plaquettes are equivalent due to the translational invariance. Using symmetries and the properties of spin operators, the correlators of all such local operators can be expressed in terms of the eight correlators depicted (the uppermost drawing, for example, represents the correlator , where the spins on the plaquette are labelled n₁, n₂, n₃, n₄ in the counter clockwise direction starting from the upper right corner). For the special case L_x=L_y, the correlators displayed in black are not needed. (b) and (c), Dependence of the relative difference d=|2(c₀−c₁)/(c₀+c₁)| between the correlators c₀≡c() and c₁≡c() on the size of the lattice for even-by-even and even-by-odd lattices (we use the markers indicated in (a)). The extra set of smaller fainter symbols for 4 × 4 in (b) and 4 × 3, 4 × 5 and 6 × 5 in (c) (see the upper axes) show the same for the exact ground states of the Hamiltonian with φ₁=0.07 × 2π and φ₂=0.03 × 2π. The results are obtained by exact computations for L_xL_y≤30 and from Monte Carlo simulations for L_xL_y≥36. The error bars of d are given as d±δd, where , and the variance is taken over the outcome of independent Monte Carlo trajectories.