Physics

Pure mathematics

Mathematics

Manifold (fluid mechanics)

Musical

Operating system

Matrix (chemical analysis)

Mathematical optimization

Engineering

Visual arts

Hamiltonian (control theory)

Mathematical analysis

Tangent

Matrix multiplication

Tangent space

Composite material

Materials science

Geometry

Space (punctuation)

Computer science

Quantum mechanics

Formalism (music)

Product (mathematics)

Observable

Mechanical engineering

Quantum

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