Abstract
In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.
Highlights
We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space
We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories
That we have introduced the manifold of matrix product states and the concept of the tangent space, we should explain how to find the point in the manifold that provides the best approximation for the ground state of a given hamiltonian H
Summary
The quantum many-body problem is of central importance in diverse fields of physics such as quantum chemistry, condensed-matter physics and quantum field theory. The natural way of describing those novel tensor network methods is through the low-dimensional manifold that those states span in the full Hilbert space This manifold picture provides a unifying framework by which both DMRG and time-dependent and spectral MPS methods can be understood. The main objective of these lecture notes is to highlight the novel aspects of quantum tensor networks that are made apparent by looking at them through the lens of this manifold picture and, by studying the tangent spaces of this manifold Those tangent spaces play a central role as they parameterize the directions in Hilbert space which are accessible from within the manifold.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
