## SciPost Submission Page

# Yang-Baxter integrable Lindblad equations

### by Aleksandra A. Ziolkowska and Fabian H.L. Essler

### Submission summary

As Contributors: | Fabian Essler |

Preprint link: | scipost_201912_00047v1 |

Date submitted: | 2019-12-30 |

Submitted by: | Essler, Fabian |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Quantum Physics |

Approach: | Theoretical |

### Abstract

We consider Lindblad equations for one dimensional fermionic models and quantum spin chains. By employing a (graded) super-operator formalism we identify a number of Lindblad equations than can be mapped onto non-Hermitian interacting Yang-Baxter integrable models. Employing Bethe Ansatz techniques we show that the late-time dynamics of some of these models is diffusive.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 1 on 2020-2-14 Invited Report

### Strengths

see report

### Weaknesses

see report: no obvious problems

### Report

The authors present a very interesting study of Lindblad equations resulting in

integrable Liouvillians. The manuscript presents concrete results, path breaking

ideas, and open problems. I surely recommend the manuscript for publication in

SciPost.

The authors describe their approach as "direct". This means that they set up

Lindblad equations with or "without" a Hamiltonian, they add suitable jump

operators that result in a Liouvillian which in a natural manner can be

understood as a Hamiltonian on a doubled system (respectively a 2-leg ladder)

and "appear" to be known integrable Hamiltonians. Such an identification, of

course, may involve some similarity transformations.

The authors start by revisiting the appearance of the Hubbard model with

imaginary U parameter as Liouvillian. This is "obtained" by considering a

tight-binding model coupled to an environment by jump operators. A new result

is the use of a modified jump operator which leads to the Umeno, Shiroishi and

Wadati model.

In a similar manner, generalized Hubbard models like GL(N, M) Maassarani

models are used. A notable example is the 4-state Maassarani model. Also

GL(N^2) magnets are found to have associated Lindblad equations. An important

example of this is the GL(4) spin ladder.

Other examples of integrable quantum chains with associated Lindblad equations

are graded magnets. However, generalizations therof on the basis of

q-deformations seem to not qualify as integrable Liouvillians. And likewise

the Alcaraz-Bariev model "A" does not qualify. However, suitable similarity

transformations may change this understanding.

Some other open problem is posed by the continuum limit. The authors argue

that the only consistent scaling limit exists for vanishing interaction term

(a general feature independent of integrability).

The authors point out that n-particle Greenâ€™s functions fulfil simple, closed

evolution equations. Some of the constructed models, using Bethe Ansatz

techniques, show diffusive late-time dynamics.

I am convinced that the manuscript will serve as a starting point for many

future investigations of the presented models and for the search of new ones.

### Requested changes

none

(some typos exist, but I think the authors will spot them)

A possibly incomplete list:

-- The customary approach in (-> is)

-- As we have mention (ed)

-- but must keep it finite in order to have describe (drop have?)