Geometry of Matrix Product States: metric, parallel transport and curvature

By J. Haegeman, Michael Marien, Tobias J. Osborne, Frank Verstraete. 2012

89 citations recorded by the source corpus.

We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e. the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a K\"ahler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mathematics, Nuclear and High Energy Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

In Computational Mathematics

In Nuclear and High Energy Physics

Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “The Kähler manifold structure for complex MPS with open and periodic boundary conditions has been studied in [119].”
Tangent-space methods for uniform matrix product states (2018, methodology): “First, we interpret the set of uniform MPS with a given bond dimension as a manifold [11] within the full Hilbert space of the system we are investigating.”
Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, background): “This can be done by shifting the orthogonality as explained in (87).”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, background): “More precisely, the set of uniform MPS forms a Kähler manifold (Haegeman et al., 2014).”
Geometric Structures in Tensor Representations (Final Release) (2013, unknown): “and Tyrtyshnikov (see [28], [29], and [30]) and called ”TT decomposition”. For matrix product states (MPS), the diﬀerential geometry in a ﬁnite-dimensional complex Hilbert space setting is covered in [17]. As we will show below, the Tucker and the HT formats are completely characterised by a rooted ”
The geometry of algorithms using hierarchical tensors (2013, background): “Also, a generalization to Hilbert spaces is possible; see [73,49,74].”