Critical correlations for short-range valence-bond wave functions on the square lattice

We investigate the arguably simplest $SU\left(2\right)$-invariant wave functions capable of accounting for spin-liquid behavior, expressed in terms of nearest-neighbor valence-bond states on the square lattice and characterized by different topological invariants. While such wave functions are known to exhibit short-range spin correlations, we perform Monte Carlo simulations and show that four-point correlations decay algebraically with an exponent 1.16(4). This is reminiscent of the classical dimer problem, albeit with a slower decay. Furthermore, these correlators are found to be spatially modulated according to a wave vector related to the topological invariants. We conclude that a recently proposed spin Hamiltonian that stabilizes the here considered wave function(s) as its (degenerate) ground state(s) should exhibit gapped spin and gapless nonmagnetic excitations.

Figure 1

(Color online) (a) Transition graph between two NN-VB configurations on the square lattice (sublattices $\mathcal{A}/\mathcal{B}$ are indicated by filled/open circles). Reference lines for the winding numbers $\mathbf{w}=\left(w_x,w_y\right)$ are indicated by dashed lines: configurations with $\mathbf{w}=\left(1,0\right)$ (red lines) and $\mathbf{w}=\left(0,1\right)$ (dark blue lines) are shown. Trivial loops with coinciding VBs are depicted as thick-black lines and a loop winding in both directions is evident. (b) NN spin correlations versus $L^{-1}$ (from MC simulations) for winding sectors $\mathbf{w}=\left(0,0\right)$, (0,1), (1,0), and (1,1). For $\mathbf{w}\ne \left(0,0\right)$, correlations along $\pm {\mathbf{e}}_x$ and $\pm {\mathbf{e}}_y$ are discriminated. Lines are linear-quadratic fits.

Figure 2

(Color online) (a) Spin correlation ${\left(-1\right)}^{\mathbf{r}}⟨{\mathbf{S}}_{\mathbf{0}}\cdot {\mathbf{S}}_{\mathbf{r}}⟩$ between the spin at the origin and spins located at $\mathbf{r}=x{\mathbf{e}}_x+y{\mathbf{e}}_y$. (b) ${\left(-1\right)}^{\mathbf{r}}⟨{\mathbf{S}}_{\mathbf{0}}\cdot {\mathbf{S}}_{\mathbf{r}}⟩$ versus distance $r$. An exponential fit, ${\left(-1\right)}^{\mathbf{r}}⟨{\mathbf{S}}_{\mathbf{0}}\cdot {\mathbf{S}}_{\mathbf{r}}⟩\sim \text{exp}\left(-r/\xi \right)$, yields the correlation length $\xi =1.35\left(1\right)$. Data for $L=128$ and $\mathbf{w}=\left(0,0\right)$.

Figure 3

(Color online) MC results for $r{C}^{ijkl}$, for $L=16$ (PBC) and indicated $\mathbf{w}$. In all panels, the reference bond is indicated by a thick-black line and the thickness of the remaining ones is proportional to $r{C}^{ijkl}$: blue (pale-red) lines indicate positive (negative) values. Lines along which $C_{\parallel }^{ijkl}$ is strongest are indicated by black circles.

Figure 4

(Color online) MC results for $C^{ijkl}$ as a function of distance. (a) Longitudinal correlations $C_{\parallel }^{ijkl}$ for $\mathbf{w}=\left(0,0\right)$ and various system sizes. (b) $C_{\parallel }^{ijkl}$ along zero-phase antinodal directions (filled circles in Fig. 3) for $L=128$ and $\mathbf{w}=\left(0,0\right)$, (1,0), (0,1), and (1,1). (c) $C_{\parallel }^{ijkl}\left(r\right)/\text{cos}\left(\mathbf{Q}\cdot \mathbf{r}\right)$ and $C_{\perp }^{ijkl}\left(r\right)/\text{sin}\left(\mathbf{Q}\cdot \mathbf{r}\right)$ for $\mathbf{w}=\left(0,1\right)$ and $L=128$ (data separated from a nodal line by a parallel displacement $dx<8$ are excluded). (d) Absolute value of transverse correlations $C_{\perp }^{ijkl}$ for $\mathbf{w}=\left(0,0\right)$ and $L=128$, along the line highlighted in Fig. 3a. In [(a)–(c)] the line indicates our best fit yielding the exponent $\alpha =1.16\left(4\right)$ and in (d) the fit yielding ${\alpha }^{\prime }=2.53\left(5\right)$.

Figure 5

(Color online) Histogram $P\left(D_x,D_y\right)$ for $L=96$ and $\mathbf{w}=\left(0,0\right)$.