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The DMRG method is very effective at finding ground states of 1D quantum systems in practice, but it is a heuristic method, and there is no known proof for when it works.
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Recall that μ β \mu_{\beta} corresponds to a Schmidt coefficient in a Schmidt decomposition that coincides with the standard basis
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This can also be seen by analyzing the tree-width of that tensor-network and using the analysis of Ref. markov2008simulating
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