Finite-size scaling and boundary effects in two-dimensional valence-bond solids

Various lattice geometries and boundary conditions are used to investigate valence-bond-solid (VBS) ordering in the ground state of an $S=1/2$ square-lattice quantum spin model—the $J$-$Q$ model, in which four- or six-spin interactions $Q$ are added to the standard Heisenberg exchange $J$. Ground-state results for finite systems (with up to thousands of spins) are obtained using an unbiased projector quantum Monte Carlo method. It is found that great care has to be taken when extrapolating the order parameter to infinite lattice size, in particular, in cylinder geometry. Even though strong VBS order exists in two dimensions, and is established clearly with increasing system size on $L\times L$ lattices (or $L_x\times L_y$ lattices with a fixed aspect ratio $L_x/L_y$ of order 1), only short-range VBS correlations are observed on long cylinders (when $L_x\to \infty$ at fixed $L_y$). The correlation length increases with the cylinder width, until long-range order sets in at a “critical” width. This width is very large even when the 2D order is relatively strong. For example, for a system in which the order parameter is $70\%$ of the largest possible value, $L_y=8$ is required for ordering. Extrapolations of the VBS order parameter based on correlation functions (the square of the order parameter) for small $L\times L$ lattices can also be misleading. For a $20\%$-ordered system, results for $L$ up to $\approx$20 appear to extrapolate clearly to a vanishing order parameter, while for larger lattices the scaling behavior crosses over and extrapolates to a nonzero value (with exponentially small finite-size corrections). The VBS order parameter also exhibits interesting edge effects related the known emergent U(1) symmetry close to a “deconfined” critical point, which, if not considered properly, can lead to wrong conclusions for the thermodynamic limit. The observed finite-size behavior for small $L\times L$ lattices and long cylinders is very similar to that predicted for a $Z_2$ spin liquid. The results therefore raise concerns about recent numerical work claiming $Z_2$ spin-liquid ground states in 2D frustrated quantum spin systems, in particular, the Heisenberg model with nearest and next-nearest-neighbor couplings. Based on the results presented so far, a VBS state in this system cannot be ruled out.

Figure 1

A cylindrical, semiperiodic 2D square lattice with open edges (left and right sides) and periodic boundary conditions in the vertical direction (i.e., the open links at the top and bottom are connected to each other).

Figure 2

$Q_2$ and $Q_3$ terms on the square lattice. The bars of length one lattice constant indicate the locations of singlet projectors $C(i,j)$ on site pairs $i,j$. The Hamiltonian contains all unique translations of these operators.

Figure 3

The VBS order parameter as a function of the projection power $m$ (normalized by the system size $N$) in simulations of the $Q_3$ model on a periodic $32\times 32$ lattice. The inset shows an exponential fit of the form (17).

Figure 4

Size dependence of the squared order parameter and its $x$ and $y$ components of the $Q_3$ model computed on periodic $L\times L$ and $2L\times L$ lattices. The curves passing through the $\langle D^2\rangle$ data are second-order polynomial fits (excluding the systems for which this form cannot be used). The error bars are much smaller than the plotting symbols (typically the standard deviation is $\approx$$2\times {10}^{-5}$).

Figure 5

Size dependence of the squared order parameters of the $Q_3$ model on cylindrical $2L\times L$ lattices (using the central $L\times L$ square for computing the expectation values). The smooth curves are second-order polynomials fitted to the $\langle D^2\rangle$ data for several of the largest system sizes.

Figure 6

Local columnar $x$ order parameter (18) of the $Q_3$ model computed at the center of a $2L\times L$ cylinder. The smooth curve is of the exponential form (17) and extrapolates to $0.264$. The inset shows the location dependent bond correlation function $\langle B_x\left(x\right)\rangle$ for a $32\times 16$ system.

Figure 7

Size dependence of the squared VBS order parameter of the pure $Q_2$ model on periodic $L\times L$ lattices. The solid black curve in the main graph shows a fit of the $L\le 12$ data to a second-order polynomial (which extrapolates to an unphysical negative value when $L\to \infty$). The solid red curve shows a fifth-order polynomial fit to all the data, while the dashed black curve shows a quadratic fit to only the $L\ge 20$ data. The inset shows the behavior for the largest systems on a more detailed scale.

Figure 8

Size dependence of the squared total VBS order parameter $\langle D^2\rangle$ and its individual $x$ and $y$ components of the pure $Q_2$ model on fully periodic $2L\times L$ lattices. The curve displayed for $1/L\le 0.125$ is a 4th order polynomial fit to the $\langle D^2\rangle$ data for $L\ge 16$.

Figure 9

Size dependence of the squared total order parameter $\langle D^2\rangle$ and its $x$ and $y$ components, computed for the $Q_2$ model on cylindrical $2L\times L$ lattices (including only the spins on the central $L\times L$ square in the definition of the order parameters). The curves are polynomial fits. In the case of $D_y^2$, no constant term was included. The inset shows the data for large systems on a more detailed scale.

Figure 10

Staggered component, Eq. (19), of the long-distance dimer correlations in the $Q_2$ model on periodic $L\times L$ and $2L\times L$ lattices. Here, $r$ is the longest distance on the lattices; $\mathbf{r}=(L_x/2,L_y/2)$. The curve through the $L\times L$ data is a high-order polynomial fit. The inset shows the $2L\times L$ data on a more detailed scale.

Figure 11

Open-edge induced order parameter of the $Q_2$ model at the center of cylindrical $L\times L$ and $2L\times L$ lattices. The horizontal line corresponds to the value of the infinite-size order parameter from the extrapolation in Fig. 10.

Figure 12

The staggered part, Eq. (19), of the long-distance correlation function (at $r_{\max }=\sqrt{2}L$) and the total dimer order parameter for the $J$-$Q_2$ model at $J/Q_2=0.03$ and $0.10$ on periodic $L\times L$ lattices.

Figure 13

Size dependence of the VBS (top) and Néel (bottom) order parameters of the $J$-$Q_2$ model at four different coupling ratios. The point $J/Q_2=0.0447$ should be very close to the quantum-critical value according to the scaling analysis of the spin stiffness carried out in Ref. 41. The straight lines fitted through the $J/Q_2=0.0447$ data (for system sizes $L\ge 32$) have slope $-1.27$ in both cases.

Figure 14

A cylindrical lattice with modified open edges favoring columnar VBS order with vertical bonds. The thick vertical bars represent $Q_2$ terms excluded from the summation in the Hamiltonian, Eq. (3).

Figure 15

Location-dependent expectation value of the VBS order parameter of the $Q_2$ model on $2L\times L$ cylinders with the left $(x=0)$ edge modified by the symmetry-breaking perturbation (inducing $y$-oriented order) illustrated in Fig. 14. The right edge is kept uniform. The top panel shows both the bare dimer expectation value $\langle B_{\widehat{x}}\left(x\right)\rangle$ and the dimer order parameter $\langle D_x\left(x\right)\rangle$ extracted from it according to Eq. (20) for $L=8$. The middle panel shows the $y$ order parameter defined according to Eq. (21) for $L=8,16$, and 32. The bottom graph shows the VBS angle extracted from the $x$ and $y$ order parameters according to Eq. (22).

Figure 16

Location dependent expectation values of the $x$ and $y$ VBS order-parameter components, Eqs. (20) and (21), of the $Q_2$ model on $2L\times L$ cylinders with both edges modified by a symmetry-breaking perturbation (favoring $y$-oriented bond order). The corresponding VBS angle extracted using Eq. (22) is shown in the bottom panel. The horizontal dashed line is at $\Theta =\pi /4$ (corresponding to equal $x$ and $y$ order parameter parameters).

Figure 17

Same as Fig. 16 for the $Q_3$ model.

Figure 18

The $x$ and $y$ components of the induced order parameter close to a modified edge of the $Q_3$ model on a $128\times 64$ lattice. These data are the same as those shown in the top ($x$) and middle ($y$) panels of Fig. 17 for smaller systems, but with the nonzero constant behavior at the center of the system subtracted off in the case of the $x$ component. The lines are exponential fits, giving decay lengths $1.9$ and $6.5$ for the $x$ and $y$ components, respectively.

Figure 19

VBS order parameter and correlations lengths on infinite ($L_x\to \infty$) $Q_2$ and $Q_3$ cylinders of width $L_y=4,6,8,10,12$. (a) The correlation length extracted using the dimer correlations with $y$-oriented bonds as a function of $L_y$ for those cylinders that have disordered ground states (the $x$ correlation lengths are about $5\text{--}10\%$ smaller). (b) The order parameter versus $L_y$, along with the corresponding 2D order parameters (shown with the horizontal lines).

Figure 20

VBS correlation functions, as defined in Eq. (24), for $y$-oriented dimers in the $Q_3$ model as a function of the separation in the $x$ direction on cylinders in the $L_x\to \infty$ limit. For $L_y=4,6$, fitted curves of the form $C\propto \exp (-x/\xi )/x^{\alpha }$ to the $x\ge 4$ data are also shown (with $\alpha \approx 0.5$ in both cases).

Figure 21

VBS correlation functions, as defined in Eqs. (23) and (24), for $x$- (top panel) and $y$-oriented (bottom panel) dimers in the $Q_2$ model as a function of the separation in the $x$ direction on cylinders in the $L_x\to \infty$ limit. Fits of the data for $x>4$ to the form $S\propto \exp (-x/\xi )/x^{\alpha }$ (with $\alpha \approx 0.5$ in all cases) are shown as solid curves.

Figure 22

Boundary induced $x$ and $y$ components of the dimer parameter of the $Q_3$ model on $64\times 4$ (top) and $128\times 8$ (bottom) lattices. Both edges are modified to induce $y$ order, as discussed in Sec. 4.

Figure 23

Finite-size scaling of the squared VBS order parameter (a) and staggered magnetization (b) calculated on the central $L\times L$ square of cylinders of size $2L\times L$. DMRG results for the $J_1$-$J_2$ model at $g=0.50$ and $0.56$, from Figs. 2a and 3a of Ref. 6, are compared with QMC results for the $J$-$Q_2$ model at its critical point, ${(J/Q_2)}_c=0.0447$, and at $J=0$. In (a), the VBS $y$ component of the $J_1$-$J_2$ model and the $x$ component (the larger component) of the $J$-$Q_2$ are shown. The line drawn close to the $J/Q_2=0.0447$ points has slope $-1.27$, corresponding to the critical exponent $\eta =0.27$ (as in Fig. 13), and that going through the $g=0.50$ points has slope $-1.8$. The dashed line has slope $-2$, corresponding to the expected asymptotic behavior in a non-VBS state. In (b), both lines have slope $-1.27$ ($\eta =0.27$).

Figure 24

VBS $y$ order parameter component induced by edges modified to break the $y$ translational symmetry (as explained in Fig. 14) in the $Q_2$ model. Here, $r$ is defined as the distance from the second column of spins away from the edge, since the edge modification extends to this location. The lines show exponential fits, with decay lengths $1.8$ ($L_y=4$), $3.2$ ($L_y=6$), $4.8$ ($L_y=8$), and $6.6$ ($L_y=10$).

Figure 25

VBS order parameter distribution $P(D_x,D_y)$ in the $Q_3$ model on periodic $L\times L$ lattices with $L=12$ (left) and $L=24$ (right). The size of both squares corresponds to the full space of possible values of the components $D_x,D_y\in [-D_{\max },D_{\max }]$, where $D_{\max }=3/8$ (for a perfect columnar VBS).

Figure 26

Size dependence of the columnar anisotropy weight, defined in Eq. (A1), of the VBS order parameter distribution in the $Q_3$ model.

Figure 27

VBS order parameter distribution $P(D_x,D_y)$ in the $Q_2$ model on periodic $L\times L$ lattices with $L=64$ (left) and $L=128$ (right). The size of both squares corresponds to $10\%$ of the maximum value $D_{\max }/10$ of the components, $D_x,D_y\in [-D_{\max },D_{\max }]$, where $D_{\max }=3/8$ (for a perfect columnar VBS).

Figure 28

Angular distribution of the VBS order parameter of the $Q_2$ model for system sizes $L=32$, 64, and 128. To improve the statistics, these results were obtained by symmetrizing the distributions using the expected ${90}^{\circ }$ rotational symmetry. The jaggedness of the curves (especially for $L=32$) is due to the discreteness of the allowed $(D_x,D_y)$ values (with $N$ possible values for each component).