Publications at the Riemann Center
Geometry of matrix product states
Metric, parallel transport, and curvature
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.
Details
- Organisation(s)
-
Institute of Theoretical Physics
Riemann Center for Geometry and Physics
- External Organisation(s)
-
University of Vienna
Ghent University
- Type
- Article
- Journal
- Journal of Mathematical Physics
- Volume
- 55
- ISSN
- 0022-2488
- Publication date
- 06.02.2014
- Publication status
- Published
- Peer reviewed
- Yes
- ASJC Scopus subject areas
- Statistical and Nonlinear Physics, Mathematical Physics
- Electronic version(s)
-
https://doi.org/10.1063/1.4862851 (Access:
Unknown
)
