Singlet product states provide an alternative description of quantum magnets in which each state corresponds to a particular grouping of the spins into entangled singlet pairs called valence bonds. The set of all bond patterns forms an overcomplete and maximally nonorthogonal basis for the low-energy sector of the Hilbert space. The unusual properties of the basis invalidate most conventional algorithms but have also opened the door to several powerful new Monte Carlo simulation techniques developed over the past few years.
Valence bond states are useful for constructing expressive trial wave functions. Moreover, the "space-time" loop structure of their matrix elements allow for very efficient simulation of SU(N) Heisenberg-type models. Even frustrated models, which are not amenable to Monte Carlo simulation because of the infamous sign problem, are somewhat better behaved in this basis. As a result of overcompleteness, updates have a many-to-one property that allows for their consolidation around plaquets in such a way that the negative sampling weights are mostly eliminated.
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