## SciPost Submission Page

# Tangent-space methods for uniform matrix product states

### by Laurens Vanderstraeten, Jutho Haegeman, Frank Verstraete

#### - Published as SciPost Phys. Lect. Notes 7 (2019)

### Submission summary

As Contributors: | Laurens Vanderstraeten |

Arxiv Link: | https://arxiv.org/abs/1810.07006v3 |

Date accepted: | 2019-01-14 |

Date submitted: | 2019-01-08 |

Submitted by: | Vanderstraeten, Laurens |

Submitted to: | SciPost Physics Lecture Notes |

Domain(s): | Computational |

Subject area: | Condensed Matter Physics - Theory |

### 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.

###### Current status:

### Ontology / Topics

See full Ontology or Topics database.### Author comments upon resubmission

The authors would like to thank the referee for the careful reading of the manuscript. We have changed the manuscript according to the referee's comments, and hope that the paper can be published.

### List of changes

In response to the referee's comments we have changed the manuscript as follows:

1- equation (46) : there is an extra n in the second term of the right-hand site of the first line -> we have omitted the extra n

2- In the paragraph below eq (76) there is a typo ("recudes") -> corrected typo

3- Just below eq (185) I think q should be replaced by p -> corrected typo

4- It seems that the link to arXiv of ref [57] does not send me to the right webpage -> this paper is published by now, we have changed the reference

5- Is it possible to add a comment about the feasibility (or not) of applying these methods to study periodic systems in finite-size ? -> we have added a sentence in the outlook concerning periodic boundary conditions, and added a reference that is relevant for this matter