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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


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:
Editor-in-charge assigned

Submission & Refereeing History

Submission scipost_201912_00047v1 on 30 December 2019

Reports on this Submission

Anonymous Report 1 on 2020-2-14 Invited Report


see report


see report: no obvious problems


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

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 Green’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

(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?)

  • validity: top
  • significance: top
  • originality: high
  • clarity: high
  • formatting: good
  • grammar: excellent

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