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

1- the paper is technically precise
2- the results are correct
3- the paper provides an overview of the subject
4- the previous results are put into a nice algebraic structure

- The paper is hard to read.

This is about the recent high-profile works on building exact nonequilibrium steady states in driven quantum integrable spin chains. The manuscript reviews some of the work, and puts it in a common algebraic framework. The structure identified is based on Yangians.

Again, this is an interesting work, it is important to clarify the underlying algebraic structure in order to eventually connects with advanced concepts of integrability. There is a lot of material, and some interesting ideas.

However, there is no way out of the fact that the paper is extremely hard to read. Many aspects are developed in different parts, and it is easy to get lost (in the notation, in the concepts, in what it is that the author is trying to do). Perhaps, if there was an overview section where all aspects were put together, all explained, so that we could have a big picture, that would be better. But I don't know if this is possible at all. I understand that the author has made an effort to improve the text, it is more readable then previous version (I think everything is defined, at least somewhere, even if sometimes we have to search for the definition).

Here are some little things that I found unclear / inaccurate:

p 7, bottom: can the author be more precise about what a "representation label" is.

p10: the dissipator acting on Omega cancels right the hand side of (20) (that is, it gives a term that equates the negative of rhs of (20); it does not annihilate rhs of (20) in the sense that acting on rhs of (20) with it does not give zero)

p 9 before eq 18: the hamiltonian density does not coincide with the permutation, as in 18 there is an extra factor of derivative of R.

p9: "the differential equation (19)" : should be (18) instead? "A general solution to eq 18" should be eq 17 instead?

p9 in paragraph after eq 18: which one is physical space and which is auxiliary space? please specify

p9 just after eq 19: in expression of ${\cal H}_{\rm aux}$ why is index now $n$ instead of $n+m$?

p10: what does it mean that two copies of a Hilbert space are "mutually conjugate" (just above eq 21)? A Hilbert space is either isomorphic or not to another Hilbert space, but I don't see how it can be conjugate.

p 11 top: "pertains to the conjugate dual" -> does it really mean "is the conjugate dual"? I don't understand the use of "pertain".

p 11 bottom: "a general solution of condition (19)"... but (19) is not a condition, it is a definition.

p12 fig 5: it is a bit strange to put 2A where it is: why not a factor 2 for this term instead?

p14: why are level-0 grassmann numbers? they seem to commute in eq 32.

p14: how is eq 33 linear in z? there is z^{-1}, so it is rather linear in z^{-1}

p 20, footnote 13: where is the Jordan-Wigner transformation used? It is not clear in the main text where it would be involved. Why is it involved in the dissipator?

At this moment I do not know what changes to ask, even though the reading is difficult. I'd say, then, only clarify the small things listed in the report and repeated here:

p 7, bottom: can the author be more precise about what a "representation label" is.

p10: the dissipator acting on Omega cancels right the hand side of (20) (that is, it gives a term that equates the negative of rhs of (20); it does not annihilate rhs of (20) in the sense that acting on rhs of (20) with it does not give zero)

p 9 before eq 18: the hamiltonian density does not coincide with the permutation, as in 18 there is an extra factor of derivative of R.

p9: "the differential equation (19)" : should be (18) instead? "A general solution to eq 18" should be eq 17 instead?

p9 in paragraph after eq 18: which one is physical space and which is auxiliary space? please specify

p9 just after eq 19: in expression of ${\cal H}_{\rm aux}$ why is index now $n$ instead of $n+m$?

p10: what does it mean that two copies of a Hilbert space are "mutually conjugate" (just above eq 21)? A Hilbert space is either isomorphic or not to another Hilbert space, but I don't see how it can be conjugate.

p 11 top: "pertains to the conjugate dual" -> does it really mean "is the conjugate dual"? I don't understand the use of "pertain".

p 11 bottom: "a general solution of condition (19)"... but (19) is not a condition, it is a definition.

p12 fig 5: it is a bit strange to put 2A where it is: why not a factor 2 for this term instead?

p14: why are level-0 grassmann numbers? they seem to commute in eq 32.

p14: how is eq 33 linear in z? there is z^{-1}, so it is rather linear in z^{-1}

p 20, footnote 13: where is the Jordan-Wigner transformation used? It is not clear in the main text where it would be involved. Why is it involved in the dissipator?

The subject and the aim of the paper are interesting.

The presentation of several main objects is poor and somewhere uncertain.

The author address the interesting issue to provide an algebraic framework to the derivation of the steady-state density matrix for some classes of open quantum systems with dissipation driven localized on the boundaries.

The approach presented rely on the integrable structure of the Hamiltonian associated to the open quantum system. More in detail, the author considers this problem for a class of models associated to graded Sl(n|m) Hamiltonians. The steady-state density matrix is then given in a factorized form as a product of two conjugate "amplitude operators" which in turns are written as the expectation values of a product of "Partonic" Lax operators, of references [85-88], over a so-called "vacuum state".

The manuscript is quite well written in its introduction, conclusion and section 3 and also in the part where are reproduced the graded representations and the "Partonic" Lax operators. On the other hand, I find that the presentation of several main objects is poor and somewhere uncertain leaving the reader missing of several operative definitions and so unable to reproduce by himself the material presented in the manuscript.

Let me start from my main problem, that is the definition of the vacuum states (one of the two fundamental ingredients to characterize the steady-state density matrix in the author's proposal).

The most clear definition that I have found of it is the following general statement given at page 5 of section 2:

"In all the instances addressed in this work, the vacuum state pertains to a product of highest- or lowest-weight vectors from irreducible spaces associated to the auxiliary algebra of the L-operator."

The problem for me is that in section 4 where concrete examples of open quantum models are taken the vacuum state is not explicitly written; the author should present just one equation where it is defined for any considered model. For example one should have not only the definition of the Lax operator (45) but also of the vacuum state to show that the condition (47) allows to satisfy the equation (19); and the same is required for all the other cases presented in the manuscript.

There are moreover unclear notations in section 2, just to make some examples, the definition of the Hamiltonian above equation (6) is not explicit, in equation (7) it is written a graded tensor product of L matrix but the number of them is not explicitly written; are they N like the number of quantum site of the chain? The same problem is in equation (17). In equation (24) it is unclear the index k which values takes.

Once these clarifications are done there are still some conceptual points that have to be clarified. The Hamiltonian of the models considered in this manuscripts should be open ones, no closing integrable conditions are given but a priori it should be very important to clarify if some integrable boundary conditions are considered. Indeed, the conserved charges of the model depends from these boundary conditions and they should play an important role in the determination of the steady-state density matrix according to the author statement at page 5 of section 2:
" This suggests that the algebra of (possibly non-local) commuting operators belonging to H_{n|m} might be a good place to look for an appropriate Ω-amplitude of Eq. (5). "

and to the discussion in section 2.4.

Furthermore, in section 5 the author seems to justify his introduction of the "vacuum Q-operator" (expectation value on the "vacuum state" of a product of Lax matrices) as for open quantum system "taking (super) traces over auxiliary spaces cease to make sense " for the absence of translation invariance. Well here it is important to remark that the integrable structure of an open quantum system codified by a reflection equation and its dual allow to overcome such a problem. This is already clear in the seminal work of Sklyanin "Boundary conditions for integrable quantum systems " of the 1988.

I want also to mention that at least a couple of statements in the paper are out of place, at page 3 the author write:

"However, to best of our knowledge, these objects have mostly been used as a formal tool in the Bethe Ansatz diagonalization techniques, while their role in concrete physics applications remains rather unclear"

there are several physical applications of the knowledge of the Baxter Q-operator, for example it is an intermediate step in the computation of correlation functions and dynamical structure factors.

at page 19 of the section 5, referring to the Baxter Q-operator the author write:

"The concept was resurrected more than two decades later in the context of CFTs"

well between the seminal work of Baxter of the 1972 and the series of works of Bazhanov, Lukyanov and Zamolodchikov there is a vast literature in integrable quantum models about the Baxter Q-operator, just to make an example let me mention the connection that it allows to establish between two a priori very different models like the 6-vertex model and the chiral Potts model, as pioneered in the paper of Bazhanov and Stroganov of the 1989.

In order the paper can be accepted for publication on SciPost I suggest that the author implement the clarifications above mentioned and make more explicit some of the content of section 2.

the paper is technically precise
the results are correct
the paper provides an overview of the subject
the previous results are put into a nice algebraic structure

integrability is not reviewed enough, making the paper hard to read; section 2 should be improved
it is not clear what is new

This is about the recent high-profile works on building exact nonequilibrium steady states in driven quantum integrable spin chains. The manuscript reviews some of the work, and puts it in a common algebraic framework. This is a very interesting endeavour, it is important to clarify the underlying algebraic structure in order to eventually connects with advanced concepts of integrability. The structure identified is based on Yangians.

Since this is mostly reviewing previous works, the technical accuracy of the results is not in doubt. The most important thing is to put the results within a framework. I think this is relatively well done in sections 3 and 4. But section 2 needs improvement, I would say especially section 2.4. I think a lot of knowledge of the structures of integrability is assumed, and it makes the paper rather hard to read.

As specific examples, what is $\mathcal{O}^\nu$ on page 9? A linear space of an operator? The term "$\Omega$-amplitude" is used without really being defined. Also, the signature $n|m$ is explained in section 3 but used already in 2.4. Also, footnote (4) is unclear (I don't know what it means). These are small points, but in addition to these, overall I would be happy to see the discussion improved and clarified in section 2.

Another point is that it is not entirely clear what is new, and what is not new. Emphasizing the new structures identified would be good.

See report.

1. carefully written
2. technically strong

1. no clear separation between innovations brought by this paper, and results previously obtained by others
2. the bibliography is not selective at all, and does not orient the reader towards references that are relevant to appreciate the results of this paper
3. no new result about any concrete physical observable
4. no pictures (while algebraic manipulations of Matrix Product Operators/monodromy matrices are sometimes best illustrated with diagrams)

The author investigates algebraic structures that underly the study of steady states in a class of spin chains evolving with Lindblad dynamics. The dynamics is generated by an integrable hamiltonian in the bulk, and dissipators that act only on the boundaries of the chain. This class of exactly solvable models has been studied extensively in the past years (Refs. [72-80]) with techniques that exploit both integrability of the Hamiltonian and the possibility of finding exact steady states in the form of matrix product states (dating back to Refs. [47,48]). This paper provides a generalization of those works in the case of $\mathfrak{gl}(n|m)$ super-spin chains, and discusses connections between various algebraic structures that appear in this model.

In my opinion, even though it is carefully written, the manuscript is difficult to read. It is addressed only to an expert audience, with a sharp knowledge of the existing literature. The main claim of the author is that his framework is more general than the existing one, and that it offers a unified perspective on all previous results, such as the ones of Refs. [72-80]. But the paper is written in such a way that it is very difficult to distinguish the new ideas from those that are simply taken from Refs. [72-80]. As a consequence, the relevance of the whole construction has remained quite unclear to me. In its present form, I cannot recommend this paper for publication in Scipost.

I have several objections to the way the paper is presented.


1. The main problem is, in my opinion, that the formalism of integrable spin chains driven by boundary dissipators is not properly reviewed in the introduction, or in a separate section at the beginning of the paper. It should be explained clearly that this problem was already solved, for instance, for the XXZ chain, and that such key objects as the matrix product operator $\Omega_N$ were introduced in that context. This pre-existing formalism should be recalled, and the reader should be oriented towards relevant (selected) references.

Instead, the author mixes the introduction of the model he’s studying (a spin chain with $\mathbb{C}^{n|m}$ degrees of freedom) with important pre-existing ideas and formalism, without properly referencing those. This generates confusion, and for a non-expert reader, it is hard to see what the new material is here.

I think it would be much better to have two clearly separated sections, one with a review of previous results in standard spin chains (i.e. not $\mathbb{Z}_2$ graded), and the other with a clear presentation of the new ones in the context of $\mathbb{Z}_2$ grading.



2. A related problem is the fact that, apart from the last paragraph and the outline, the introduction is not very informative, and does not introduce the true topic of the paper to the reader. Instead, it contains a large number of citations to works that are only very remotely connected to the content of the manuscript (the very first paragraph, for instance, contains no less 36 references to papers that have essentially nothing to do with the present one).

I would find it helpful if the introduction and the list of references were more focused on the true topic of the paper, as in the last paragraph ‘’In this work we shall focus exclusively on formal algebraic aspects related to a class of boundary-driven integrable spin chains. […]’’. The introduction could focus on the presentation of this class of models, and on some of the key ideas that have appeared in that context, like the factorization property $\rho_\infty = \Omega_N \Omega^\dagger_N$, that are really needed later in the paper.


3. There is some confusion in the way the discussion is organized between sections 2 and 3. Some of the notations for $\mathbb{Z}_2$ grading are being used in section 2 (for instance in Eq. (6)) without a definition; they are defined only later in section 3 (or also in appendix A).


4. When reading all the algebraic relations in the paper, the reader always needs to be careful about whether operators act in physical space or in auxiliary space. This is of course inherent to the topic itself, but usually, drawing diagrams helps a lot. I think it would really help the reader follow the logic of the paper if the author could represent the key equations diagrammatically.