This talk will review the conceptual foundations of the Density Matrix Renormalization Group (DMRG) algorithm using the language of Matrix Product States (MPS), focusing on the practical implications of the MPS approach for numerical algorithms. Two recent developments will be presented in detail: The extension of DMRG to an efficient algorithm for the thermodynamic limit of infinite-size translationally invariant or periodic Hamiltonians will be described, uncovering an interesting relationship to the infinite-size variant of Time Evolving Block Decimation. Secondly, I will show how to utilize a Matrix Product Operator representation for observables, which has many practical benefits including the ability to perform exact arithmetic on operators, for example to obtain higher moments of an observable.
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