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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