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# Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems

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### Abstract

This article reviews recent developments in the theoretical understanding and the numerical implementation of variational renormalization group methods using matrix product states and projected entangled pair states.

### Most cited references66

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### Boson localization and the superfluid-insulator transition

(1989)
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### Fault-tolerant quantum computation by anyons

(1997)
A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature.
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### Cold bosonic atoms in optical lattices

(1998)
The dynamics of an ultracold dilute gas of bosonic atoms in an optical lattice can be described by a Bose-Hubbard model where the system parameters are controlled by laser light. We study the continuous (zero temperature) quantum phase transition from the superfluid to the Mott insulator phase induced by varying the depth of the optical potential, where the Mott insulator phase corresponds to a commensurate filling of the lattice (optical crystal''). Examples for formation of Mott structures in optical lattices with a superimposed harmonic trap, and in optical superlattices are presented.
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### Author and article information

###### Journal
16 July 2009
###### Article
10.1080/14789940801912366
0907.2796