Geometry of matrix product states

Autor(en)

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

Abstrakt

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(en)

Quantenoptik, Quantennanophysik und Quanteninformation

Externe Organisation(en)

Gottfried Wilhelm Leibniz Universität Hannover, Ghent University

Journal

Journal of Mathematical Physics

Band

55

Anzahl der Seiten

50

ISSN

0022-2488

DOI

https://doi.org/10.1063/1.4862851

Publikationsdatum

02-2014

Peer-reviewed

Ja

ÖFOS 2012

103025 Quantenmechanik

Schlagwörter

ASJC Scopus Sachgebiete

Statistical and Nonlinear Physics, Mathematical Physics

Link zum Portal

https://ucrisportal.univie.ac.at/de/publications/23a40699-6914-42c9-a923-a4cfd11818f2