Coexistence of Long-Range and Algebraic Correlations for Short-Range Valence-Bond Wave Functions in Three Dimensions

We investigate nearest-neighbor valence-bond wave functions on bipartite three-dimensional lattices. By performing large-scale Monte Carlo simulations, we find that long-range magnetic order coexists with dipolar four-spin correlations on the cubic lattice, this latter feature being reminiscent of the Coulomb phase for classical dimers on the same geometry. Similar properties are found for the lower-coordination diamond lattice. While this suggests that the coexistence of magnetic order and dipolar four-spin correlations is generic for such states on bipartite three-dimensional lattices, we show that simple generalizations of these wave functions can encode different ordering behaviors.

Figure 1

(Color online) (a) Cubic and (b) diamond lattices considered in the present work. Thick light-shaded links in (a) represent singlets in a NN-VB state [Eq. (1)]. We also indicate special directions (with respect to a reference dimer depicted as a thick dark line), along which four-point correlations are computed.

Figure 2

(Color online) (a) Two-spin staggered correlations as a function of distance $r$ for the NN-VB state on the cubic and diamond lattices (linear size is $L=16$). (b) Squared staggered magnetization density versus inverse system size $L^{-1}$. Extrapolated values, obtained from linear fits to the three left-most data points, are $⟨m_{\mathrm{S}}^2{⟩}_{\infty }=0.02306(8)$ (cubic) and $⟨m_{\mathrm{S}}^2{⟩}_{\infty }=0.0076(2)$ (diamond).

Figure 3

(Color online) Probability $P(l)$ of observing a loop of length $l$ in the transition graph between two NN-VB states when sampling the function [Eq. (1)] on the cubic lattice, for various system sizes $L$. Two regimes can be distinguished for short and long loops with power-law decays with, respectively, best-fit exponents $\alpha =2.51(2)$ and $\beta =1.07(4)$ (lines), in agreement [44] with the scaling found in Ref. 21 for a spin ice system. Error bars are smaller than symbols.

Figure 4

(Color online) Four-point connected correlations Eq. (4) for the NN-VB state on the cubic lattice, multiplied by the sublattice sign factor $\sigma$ and the cube of the distance between dimers, $r^3$, for (a) $L=64$ and specified directions and (b) various system sizes and fixed direction ${\mathbf{e}}_0$ [special directions are highlighted in Fig. 1a]. In both panels, symbols indicate data from MC simulations, and lines correspond to the finite-size expression Eq. (6).

Figure 5

(Color online) MC data for $r^3\sigma C_{ijkl}$ Eq. (4) for the NN-VB state on the diamond lattice as a function of $r$ along special directions indicated in Fig. 1b. (a) Data for $L=24$ and specified directions for classical dimers (filled symbols) and the NN-VB wave function (open symbols; data multiplied by 2). (b) Correlations for the NN-VB wave function along the direction ${\mathbf{e}}_0$ [see Fig. 1b] for various system sizes. Lines are guides to the eye.