Dissipation-driven integrable fermionic systems: from graded Yangians to exact nonequilibrium steady states

We again thank all the referees for carefully reading the manuscript and for providing valuable feedback. We have taken their suggestions into account and accordingly prepared a revised version of the manuscript. The main changes are summarized along with our response to the criticism.

1- The primary purpose of Section 3 has precisely been to introduce the basic ideas and concepts, and to familiarize the reader with the general structure of the algebraic construction before eventually proceeding with more technical tools (Section 4) and finally presenting explicit results (Section 5). Perhaps the presentation flow was not optimal, but all the key ideas were already there in the previous version. In an attempt to improve readability, in the present version we completely reorganized Section 3 and added many additional remarks to make the global picture clearer. We also included the introductory paragraph to Section 3 by briefly explaining the core idea of splitting the problem into the bulk (kinematic) part and the boundary matching conditions. Since there is no well-defined causal order in which the concepts enter the story, we hope that the reader will find this version easier to digest.

On the graded vs. non-graded issue: In our opinion, the readers should appreciate the fact that the so-called fundamental integrable quantum chains with rational R-matrices nicely fit in a common group-theoretic framework. This not only calls for a unified treatment but also conforms with the general philosophy of this work. On the other hand, we cannot recognize any practical value for the splitting suggested by one of our referees: the non-graded cases represent a subclass of solutions which can readily be obtained by simply disregarding the Z_2 phase factor. The auxiliary spaces then become, as explained in the manuscript, those of non-compact sl(2) spins and canonical bosons. The two simplest instances for the non-graded algebras of rank one and two have been explicitly treated in section 5.1.

If our understanding holds, the sole logic behind the splitting is based on the objection by one of the referees claiming that the results for the non-graded su(n)-symmetric quantum chains have been derived in the previous literature and should be separately reviewed. This must be a misconception and we hope that a few remarks below can clarify this. First off, previous works [73,74] (and our subsequent paper [81]) only deal with the solution to the simplest su(2) case (the anisotropic version and slightly more general dissipative boundaries to be more precise). The former two papers, however, aside from having a different scope, do not recognize integrability or make direct use of quantum symmetries. This has been accomplished later in [81]. While the latter is a precursor to our work, extending the construction to higher-rank models was not completely obvious. In fact, in [82] the authors obtained the solution to the su(3) case, but once again have not been able to explain its curious structure on the basis of the underlying quantum symmetry. We nevertheless sincerely believe these previous studies have been acknowledged at appropriate places in the manuscript.

Our manuscript now offers a universal symmetric formulation for a general solution for arbitrary higher-rank algebra, fermionizes the construction and discusses representation-theoretic aspects and the notion of vacuum Q-operators. All of these all original contributions of our work. We believe that this has been effectively communicated to the reader in the abstract and in the final part of the introduction.

2- After carefully re-reading the (previous version of the) manuscript, we have not been able to find any missing 'operative definitions'. We have nevertheless decided to make several extra clarifications regarding the definition of the auxiliary vacua and their internal structure, while also explaining the subtle role of infinite dimensional sl(2) module of highest and lowest weight type. In particular, we now stress multiple times (twice in section 5.1., and in section 5.2.) that the class of solutions under scrutiny always comes with the auxiliary vacuum states which are product states of highest or lowest weight vectors in each irreducible auxiliary component which constitutes the auxiliary algebra of the Lax operator. This in particular means that the solution at hand is uniquely determined after supplying (in addition to Dynkin labels) a set of binary labels which assign the correct types of representations to each irreducible component. As said in our previous reply, this unambiguously defines the vacuum state. Of course, one has to be careful as the very notion of highest- and lowest-weight states is a matter of convention as it suffers from the ambiguities of redefining the generators using algebra automorphisms. For this reason, we give explicit definitions for Verma modules and canonical Fock spaces.

3- We hope that the revised version of the manuscript is typo-free.

1- Slightly revised Introduction.
2- Restructured and refined Section 3.
3- Extra clarifications on the auxiliary vacuum states (Section 5).
4- Removed typos.