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Cambridge University Press eBooks

Matrix Product States

Tao Xiang

Published Aug 31, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1017/9781009398671.011

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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.
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The canonical form of Matrix Product States (MPS) and the associated fundamental theorem, which relates different MPS representations of a state, are the theoretical framework underlying many of the analytical results derived through MPS, such as the classification of symmetry-protected phases in one dimension. Yet, the canonical form is only defined for MPS without non-trivial periods and thus cannot fully capture paradigmatic states such as the antiferromagnet. Here, we introduce a new standard form for MPS, the irreducible form, which is defined for arbitrary MPS, including periodic states, and show that any tensor can be transformed into a tensor in irreducible form describing the same MPS. We then prove a fundamental theorem for MPS in irreducible form: If two tensors in irreducible form give rise to the same MPS, then they must be related by a similarity transform, together with a matrix of phases. We provide two applications of this result: an equivalence between the refinement properties of a state and the divisibility properties of its transfer matrix, and a more general characterisation of tensors that give rise to matrix product states with symmetries.
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.