Geometry of matrix product states

Metric, parallel transport, and curvature

authored by

Jutho Haegeman, Michaël Marien, Tobias J. Osborne, Frank Verstraete

Abstract

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 Kähler 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.

Organisation(s)

Institut für Theoretische Physik
Riemann Center for Geometry and Physics

External Organisation(s)

Universität Wien
Universiteit Gent

Type

Artikel

Journal

Journal of Mathematical Physics

Volume

55

ISSN

0022-2488

Publication date

06.02.2014

Publication status

Veröffentlicht

Peer reviewed

Yes

ASJC Scopus Sachgebiete

Statistische und nichtlineare Physik, Mathematische Physik

Electronic version(s)

https://doi.org/10.1063/1.4862851 (Access: Unbekannt)