Loop updates for variational and projector quantum Monte Carlo simulations in the valence-bond basis

We show how efficient loop updates, originally developed for Monte Carlo simulations of quantum spin systems at finite temperature, can be combined with a ground-state projector scheme and variational calculations in the valence-bond basis. The methods are formulated in a combined space of spin $z$ components and valence bonds. Compared to schemes formulated purely in the valence-bond basis, the computational effort is reduced from up to $O\left(N^2\right)$ to $O\left(N\right)$ for variational calculations, where $N$ is the system size, and from $O\left(m^2\right)$ to $O\left(m\right)$ for projector simulations, where $m⪢N$ is the projection power. These improvements enable access to ground states of significantly larger lattices than previously. We demonstrate the efficiency of the approach by calculating the sublattice magnetization $M_s$ of the two-dimensional Heisenberg model to high precision, using systems with up to $256\times 256$ spins. Extrapolating the results to the thermodynamic limit gives $M_s=0.30743\left(1\right)$. We also discuss optimized variational amplitude-product states, which were used as trial states in the projector simulations, and compare results of projecting different types of trial states.

Figure 1

(Color online) Transposition graph of VB states on a $5\times 4$ lattice with sublattices $A$ and $B$ indicated with open and solid circles. Bond configurations of a bra $⟨V_l|$ and ket state $|V_r⟩$ are shown as dashed and solid lines, respectively. For clarity, all bonds here are of length one lattice spacing but in general, a bond can connect any pair of sites on different sublattices. The number of loops is $N_{\text{loop}}=3$ and the number of sites $N=20$, giving, according to Eq. (3), an overlap $⟨V_l|V_r⟩=2^{-7}$. A configuration of $\uparrow$ and $\downarrow$ spins compatible with both VB states is also shown.

Figure 2

(Color online) A two-bond update. For a pair of two sites on the same sublattice (here a, c, as indicated by the brackets), the two bonds connected to them can be reconfigured in a unique way while maintaining the restriction of bipartite bonds. Detailed balance is then satisfied for updates toggling between the two configurations, with the Metropolis acceptance probability [Eq. (12)].

Figure 3

(Color online) The first step in a bond-loop update. Open and solid circles indicate up and down spins compatible with the valence bonds. Given an initial site $j_0$, the site $i$ linked to it is identified. The bond ($j_0,i$) is then replaced by ($j,i)$, with $j$ on the opposite sublattice from $i$ and chosen probabilistically as discussed in the text. If $S_i^z=S_j^z$, the new bond configuration is not allowed, and a new $j$ should be generated. The bond move results in one site with no bond and one with two bonds (unless $j=j_0$ and the loop update is completed). The site linked by the old bond to the site with two bonds becomes site $i$ for the following step. These procedures are repeated until the loop closes $\left(j=j_0\right)$.

Figure 4

(Color online) A VB-spin-operator configuration contributing to $⟨\Psi |{\left(-H\right)}^{2m}|\Psi ⟩$ for a four-site system with $m=2$. The arcs to the left and right indicate VB states $⟨V_l|$, $|V_r⟩$ and the two columns of filled and open circles represent $\uparrow$ and $\downarrow$ spins of compatible spin states $⟨Z_j^l|$, $|Z_j^r⟩$. The spins at the four operators (vertices) are also indicated. There are three loops, part of which consist of VBs. Expectation values are evaluated at the midpoint indicated by the dashed line.

Figure 5

(Color online) The energy (lower panel) and the squared sublattice magnetization (upper panel) of the optimized variational and ground-state projected states.

Figure 6

(Color online) Convergence of the energy (lower panel) and the squared sublattice magnetization (upper panel) for $L=32$ states projected using different trial states; amplitude-product states with amplitudes $h\left(r\right)=1/r^p\left(p=2,3,4\right)$ as well as with $h\left(x,y\right)$ determined by minimizing the energy (in two independent optimizations, giving slightly different amplitudes).

Figure 7

(Color online) Convergence of the squared sublattice magnetization for $L=64$ ($L=20$ in the inset), using an optimized trial state. The dashed lines show the result $\pm$ error bar of SSE calculations (using loop updates) at very low temperatures ($\beta =8192$ in the case of $L=64$).

Figure 8

(Color online) Finite-size scaling of the sublattice magnetization. The curves are polynomials fitted to $16\le L\le 256$ data (cubic for $C$ and fourth order for $M_s^2$). The inset shows the deviation of the simulation results for $C\left(L/2,L/2\right)$ from the corresponding fit.

Figure 9

(Color online) Convergence of VB projector results for the energy (bottom panel) and the sublattice magnetization (top panel) for an $L=64$ system versus the power $m$ of the Hamiltonian used in the projection. The results at $m/N$ are compared with $T>0$ results graphed versus the normalized inverse temperature $0.6\beta$ (corresponding to a similar computational effort as the projection scheme at $m/N$).