## SciPost Submission Page

# Twisted and untwisted negativity spectrum of free fermions

### by Hassan Shapourian, Paola Ruggiero, Shinsei Ryu, Pasquale Calabrese

#### - Published as SciPost Phys. 7, 037 (2019)

### Submission summary

As Contributors: | Paola Ruggiero · Hassan Shapourian |

Arxiv Link: | https://arxiv.org/abs/1906.04211v1 |

Date accepted: | 2019-09-04 |

Date submitted: | 2019-06-26 |

Submitted by: | Ruggiero, Paola |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Theory |

Approach: | Theoretical |

### Abstract

A basic diagnostic of entanglement in mixed quantum states is known as the positive partial transpose (PT) criterion. Such criterion is based on the observation that the spectrum of the partially transposed density matrix of an entangled state contains negative eigenvalues, in turn, used to define an entanglement measure called the logarithmic negativity. Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the operation of PT to fermionic systems remained a technical challenge until recently when a more natural definition of PT for fermions that accounts for the Fermi statistics has been put forward. In this paper, we study the many-body spectrum of the reduced density matrix of two adjacent intervals for one-dimensional free fermions after applying the fermionic PT. We show that in general there is a freedom in the definition of such operation which leads to two different definitions of PT: the resulting density matrix is Hermitian in one case, while it becomes pseudo-Hermitian in the other case. Using the path-integral formalism, we analytically compute the leading order term of the moments in both cases and derive the distribution of the corresponding eigenvalues over the complex plane. We further verify our analytical findings by checking them against numerical lattice calculations.

### Ontology / Topics

See full Ontology or Topics database.Published as SciPost Phys. 7, 037 (2019)

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 2 on 2019-8-30 Invited Report

- Cite as: Anonymous, Report on arXiv:1906.04211v1, delivered 2019-08-30, doi: 10.21468/SciPost.Report.1140

### Report

The authors continue an exploration of modified measures of negativity for free, one dimensional fermions. In this manuscript, their emphasis is on the eigenvalue spectrum of the ``partial transpose'' of the density matrix. Partial transpose is placed in scare quotes because the naive notion of partial transpose has some undesirable features for free fermions -- for example it does not produce a Gaussian matrix. The authors here study two variants of the partial transpose, both of which involve also time reversal, and one of which is further partially twisted by $(-1)^{F}$ where $F$ is the fermion number operator. From a replica point of view, the usual Renyi negativity can be expressed as a sum over spin structures on a $n$-fold cover of the original manifold. The $n^{\rm th}$ moments of the two ``partial transposed'' density matrices considered here give two special terms in this sum, as emphasized in ref. [110].

Moving forward, it will be interesting to see if this modified notion of negativity remains useful in exploring disordered and interacting systems, in higher dimensions, and also in time dependent scenarios that involve quenches and thermalization.

I recommend publication.

### Anonymous Report 1 on 2019-8-26 Invited Report

- Cite as: Anonymous, Report on arXiv:1906.04211v1, delivered 2019-08-26, doi: 10.21468/SciPost.Report.1131

### Strengths

- New and interesting results

- Readable and self-contained presentation

### Report

The authors study the negativity spectrum of free fermions, defined

via the fermionic partial transpose (PT) of the density matrix.

This is constructed on a replicated spacetime manifold, by inserting

appropriate twist operators along the cuts (corresponding to the subsystems),

which encode the glueing conditions. It is pointed out, that there is a

freedom in the construction which can be used to define two different PT

operations, one of them being twisted. The authors then analyze the

so-called tail eigenvalue distribution for both of the PT density matrices.

This can be obtained by first calculating the corresponding moments,

and applying a Stieltjes transform, similarly as done for the simple

reduced density matrix.

The main results are as follows: while the twisted PT is a Hermitian

operator, having real (positive and negative) eigenvalues, the

untwisted PT is only pseudo-Hermitian and thus yields complex

eigenvalues. The latter ones are found to be distributed along

quantized phases of $\pi/3$ on the complex plane. The tail distribution

is then calculated along each of the branches for both twisted and

untwisted operators and the results are compared to lattice calculations

with a good agreement.

The results are interesting and the manuscript is nicely written

therefore I recommend its publication. I have only one question

which the authors may consider commenting on.

### Requested changes

- The results for the tail distribution of the PT are given only for

adjacent intervals, although the moments for the disjoint case are

given explicitly in appendix B. Is the transformation to $P(\lambda)$

much more difficult to handle in this case?

- Stieltjes is misspelled before Eq. (21).

We appreciate the referee’s positive feedback. We also thank the referee for pointing out the typo.

Regarding the referee’s question about the negativity spectrum of disjoint intervals: The method used in this paper can be applied to the case of two disjoint intervals similarly. However, one of our motivations to study adjacent intervals is that, in this case, the negativity calculations can be written in terms of three-point correlators of twist fields which are universal in CFTs. In contrast, the negativity of disjoint intervals involves four-point correlators of twist fields which in general depend on the full operator content of CFTs. Therefore, we think that our findings for the negativity spectrum of two adjacent intervals may apply to other fermionic CFTs although our calculations were done for free fermions.

We added a comment concerning this point in the conclusions of the paper.

(in reply to Report 2 on 2019-08-30)

We thank the referee for his feedback.

We just want to remark that the result of Ref.[110] mentioned by the referee

for the untwisted negativity spectrum of two adjacent intervals was actually overlooked there (and indeed the result given was misinterpreted). We comment about this point in a paragraph before eq.(70) on page 16, where we also provide its solution. More details were discussed in Appendix B.