Tangent-space methods for uniform matrix product states

TL;DR

This paper provides a comprehensive overview of tangent-space techniques for uniform matrix product states, covering optimization, evolution, excitations, and extensions, with applications to both lattice models and field theories.

Contribution

It introduces the tangent-space formalism for uniform matrix product states and demonstrates its application to various computational tasks in quantum many-body physics.

Findings

Efficient methods for ground-state optimization and real-time evolution.

Framework for elementary excitations in one-dimensional models.

Extension of techniques to continuous matrix product states.

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.