Geometry of matrix product states: Metric, parallel transport, and curvature

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

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

Externe Organisationen

  • Universität Wien
  • Universiteit Gent
Forschungs-netzwerk anzeigen

Details

OriginalspracheEnglisch
Aufsatznummer021902
FachzeitschriftJournal of Mathematical Physics
Jahrgang55
Ausgabenummer2
PublikationsstatusVeröffentlicht - 6 Feb. 2014

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.

ASJC Scopus Sachgebiete

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Geometry of matrix product states: Metric, parallel transport, and curvature. / Haegeman, Jutho; Marien, Michaël; Osborne, Tobias J. et al.
in: Journal of Mathematical Physics, Jahrgang 55, Nr. 2, 021902, 06.02.2014.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Haegeman, J., Marien, M., Osborne, T. J., & Verstraete, F. (2014). Geometry of matrix product states: Metric, parallel transport, and curvature. Journal of Mathematical Physics, 55(2), Artikel 021902. https://doi.org/10.1063/1.4862851
Haegeman J, Marien M, Osborne TJ, Verstraete F. Geometry of matrix product states: Metric, parallel transport, and curvature. Journal of Mathematical Physics. 2014 Feb 6;55(2):021902. doi: 10.1063/1.4862851
Haegeman, Jutho ; Marien, Michaël ; Osborne, Tobias J. et al. / Geometry of matrix product states : Metric, parallel transport, and curvature. in: Journal of Mathematical Physics. 2014 ; Jahrgang 55, Nr. 2.
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