Matrix multiplication

Quantum mechanics

Formalism (music)

Product (mathematics)

Engineering

Pure mathematics

Materials science

Tangent

Operating system

Computer science

Mathematical optimization

Visual arts

Hamiltonian (control theory)

Observable

Mechanical engineering

Matrix (chemical analysis)

Musical

Physics

Tangent space

Mathematics

Mathematical analysis

Geometry

Quantum

Composite material

Manifold (fluid mechanics)

Space (punctuation)

SciPost Physics Lecture Notes

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.

Tangent-space methods for uniform matrix product states