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arXiv:1301.0746 · Linear and Multilinear Algebra

Exploiting matrix symmetries and physical symmetries in matrix product states and tensor trains

Thomas Huckle, Konrad Waldherr, Thomas Schulte‐Herbrüggen

Published Mar 29, 2012• 13 citations

Abstract

We focus on symmetries related to matrices and vectors appearing in the simulation of quantum many-body systems. Spin Hamiltonians have special matrix-symmetry properties such as persymmetry. Furthermore, the systems may exhibit physical symmetries translating into symmetry properties of the eigenvectors of interest. Both types of symmetry can be exploited in sparse representation formats such as Matrix Product States (MPS) for the desired eigenvectors. This article summarizes symmetries of Hamiltonians for typical physical systems such as the Ising model and lists resulting properties of the related eigenvectors. Based on an overview of MPS (Tensor Trains or Tensor Chains) and their canonical normal forms, we show how symmetry properties of the vector translate into relations between the MPS matrices and, in turn, which symmetry properties result from relations within the MPS matrices. In this context, we analyse different kinds of symmetries and derive appropriate normal forms for MPS representing these symmetries. Exploiting such symmetries by using these normal forms will lead to a reduction in the number of degrees of freedom in the MPS matrices. This article provides a uniform platform for both well-known and new results which are presented from the (multi-)linear algebra point of view.

DOI: 10.1080/03081087.2012.663371

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This chapter discusses the properties of matrix product state (MPS). It starts with a simple proof that the wave function generated by DMRG is an MPS. Then three different but equivalent canonical forms or representations of MPS are introduced. An MPS generally has redundant gauge degrees of freedom on each bond linking two neighboring local tensors. One can convert it into a canonical form by taking a canonical transformation to remove the gauge redundancy in the local tensors. Finally, the implementation of symmetries, including both the U(1) and SU(2) symmetries, is discussed.