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

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.
Authors:
;  [1] ;  [2] ;  [3] ;  [4] ;  [5]
  1. Vienna Center for Quantum Science and Technology, Faculty of Physics, University of Vienna (Austria)
  2. (Belgium)
  3. Faculty of Physics and Astronomy, University of Ghent, Krijgslaan 281 S9, 9000 Gent (Belgium)
  4. Leibniz Universität Hannover, Institute of Theoretical Physics, Appelstrasse 2, D-30167 Hannover (Germany)
  5. (Germany)
Publication Date:
OSTI Identifier:
22251028
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 2; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; GEOMETRY; HILBERT SPACE; MATRICES; METRICS; RIEMANN SPACE; TIME DEPENDENCE; VECTORS