Abstract
This chapter outlines the fundamental construction of the Stochastic Series Expansion, a highly efficient and easily implementable quantum Monte Carlo method for quantum lattice models. Originally devised as a finite-temperature simulation based on a Taylor expansion of the partition function, the method has recently been recast in the formalism of a zero-temperature projector method, where a large power of the Hamiltonian is applied to a trial wavefunction to project out the groundstate. Although these two methods appear formally quite different, their implementation via non-local loop or cluster algorithms reveals their underlying fundamental similarity. Here, we briefly review the finite- and zero-temperature formalisms, and discuss concrete manifestations of the algorithm for the spin 1/2 Heisenberg and transverse field Ising models.
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Notes
- 1.
or alternatively, Sandvik’s Simple and Efficient QMC.
- 2.
and, of course, the infamous “sign problem”.
- 3.
Alternatively, with the imaginary-time evolution operator \(e^{-\beta H}\), where \(\beta \) is large. The two methods are essentially equivalent however, since the exponential can be Taylor expanded.
- 4.
A generalization of this basis rotation on non-bipartite lattices would amount to a solution to the aforementioned “sign-problem”—and likely a Nobel prize for its architect.
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Acknowledgments
This work would not have been possible without the continuing collaboration of A. Sandvik, who is proprietor of almost every algorithmic advance outlined in the Chapter, and without who’s willingness to communicate these ideas privately would have made this work impossible. I am also indebted to A. Kallin and S. Inglis for contributions to all sections of this chapter, especially the figures, and many critical readings.
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Melko, R.G. (2013). Stochastic Series Expansion Quantum Monte Carlo. In: Avella, A., Mancini, F. (eds) Strongly Correlated Systems. Springer Series in Solid-State Sciences, vol 176. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35106-8_7
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