Ground State Projection of Quantum Spin Systems in the Valence-Bond Basis

A Monte Carlo method for quantum spin systems is formulated in the valence-bond basis. The nonorthogonality allows for an efficient importance-sampled projection of the ground state out of an arbitrary state. The method provides access to resonating valence-bond physics, enables a direct estimator for the singlet-triplet gap, and extends the class of models that can be studied without negative-sign problems. As a demonstration, the valence-bond distribution in the ground state of the 2D Heisenberg antiferromagnet is calculated. Generalizations of the method to fermion systems are also discussed.

Figure 1

Action of a bond operator on two VB states.

Figure 2

Singlet convention for a system with sublattices $A$ and $B$. The arrows indicate the order of the spins in the singlets, e.g., $c\to b$ means $(c,b)=({\uparrow }_c{\downarrow }_b-{\downarrow }_c{\uparrow }_b)/\sqrt{2}$.

Figure 3

Two VB states in two dimensions and their overlap in terms of loops formed by superimposing the two bond configurations. In this case, there are $N_v=8$ valence bonds and $N_l=3$ loops, and $⟨S_{\alpha }|S_{\beta }⟩=2^{N_l-N_v}=1/32$ [22].

Figure 4

The bond-length probability for bonds along the line $(x,0)$ in the 2D Heisenberg ground state wave function.