Generalized hydrodynamics of integrable quantum circuits

The Authors extend the framework of generalized hydrodynamics to integrable quantum circuits. This framework is a »classical« description of large-scale phenomena in integrable models, which possess infinitely many local conservation laws (continuity equations). It allows one to treat exactly the bipartition protocols / Riemann problems in integrable models, for example. The topic is timely due to the advent of quantum computing devices which often implement circuits similar to the ones described by the Authors herein. Although the calculations are mostly generalization of the known results in the continuous-time models, one of the results is surprising: a localized perturbation may result in global change of the system’s state, which propagates (and persists) on ballistic scales.

The Authors have assessed that two conditions for SciPost physics are met: (1) Provide a novel and synergetic link between different research areas; (2) Detail a groundbreaking theoretical/experimental/computational discovery. I am not convinced that (2) is true, but I cannot argue that (1) is not (although, as pointed out by Referee 2, the »synergetic link« has essentially been done in ref. [33]). All in all, I would lean more towards the recommendation of Referee 2, that the paper might be more suitable for SciPost Physics Core. However, since my arguments against the claim that criterion (1) is met might not be deemed really strong, I am not strongly opposed to publication in SciPost Physics either. In any case, below I provide three remarks that can slightly improve the otherwise clear and well-written paper.

(1) At the end of section 2.1 the Authors remark on two families of conserved quantities which break the single-site translation symmetry. They claim that the latter maps one family into the other, referring to ref. [29]. Looking at eqs. (10), (11) in the latter, their statement does not seem obvious to me. Looking at those two eqs. in the reference, for example, the densities of the first two charges are indeed translated w.r.t. each other for one site, but they have a \pm sign difference in one of the terms. Could the Authors comment on this?


(2) After eq. (59), the Authors mention a global prefactor to the eigenvalue of the propagator. I assume that prefactor is not just a phase (-1)^L, otherwise it could have been written down. Where does is come from? Looking at Faddeev’s notes, eq. (412) in arXiv:hep-th/9605187, there is no L-dependent prefactor in the definition of the quasienergy, for example.

(3) The Authors mention refs. [99-101] as examples of local perturbation evolving into a macroscopic change of the system’s state. Some other examples of such non-dispersing localized perturbations: arXiv:1207.0862, arXiv:1909.02841, arXiv:2111.06325.

Accept in alternative Journal (see Report)

1. the paper is clear

2. the paper is correct

3. integrable quantum circuits are interesting in the context of quantum computing and quantum simulations

4. section 2 contains a useful review of TBA formulas for integrable quantum circuits

1. comparison with numerics limited to non-interacting cases

2. rather incremental results

In the past decade, many works have been dedicated to the real-time dynamics of integrable spin chains, such as the XXZ spin chain. In particular, the new theory of 'Generalized Hydrodynamics' was discovered.
Among all the works on Generalized Hydrodynamics that followed, many have focused on the so-called bipartition protocol, where two semi-infinite systems are suddenly put together, and where the predictions of Generalized Hydrodynamics are simple (because the results depend on a simple scaling variable $x/t$) and asymptotically exact.

Recently, 'integrable trotterizations' or 'integrable quantum circuits' have been introduced. These circuits are built from the 'diagonal-to-diagonal transfer matrices' known from the old algebraic Bethe Ansatz literature, with spectral parameters taken so that the transfer matrix is unitary and can be interpreted as an evolution operator. These integrable trotterizations have attracted a lot of interest recently in the context of digital quantum computation or simulation. In particular, the thermodynamic Bethe Ansatz for these systems was derived very recently ( by other people in Refs. [36-38] and by some of the authors in the Supplemental Material of Ref. [33]), as is nicely reviewed in Section 2 of this new paper.

No paper had yet looked at the bipartition protocol in integrable quantum circuits with Generalized Hydrodynamics. Now this is done, thanks to this new paper. The paper is correct, it can be published as it is.

Typos:
-page 4: 'Hamitlonian'
-page 5: 'masless'
-page 15: 'These points correspond to the either one of following choices'

Publish (easily meets expectations and criteria for this Journal; among top 50%)

The paper by Hübner et al. presents an interesting study of the nonequilibrium dynamics of a quantum spin chain, at the interface of three research areas that have received significant attention in recent years: (i) Trotter transitions, that is, qualitative changes in the dynamics that occur when the Trotter step size crosses a critical value; (ii) integrability and generalized hydrodynamics (GHD); (iii) quench dynamics, in particular, the bipartition protocol, which can lead to the emergence of nonequilibrium steady states. This study builds on and complements a previous work [33] that includes two of the authors of the present manuscript. The model system under investigation is an integrable quantum circuit that represents a Trotterization of the dynamics generated by the XXZ Hamiltonian. As initial states, the authors consider a single domain wall between the two halves of the spin chain in Néel, anti-Néel, or Majumdar-Ghosh states. To describe the dynamics in the space-time scaling limit, the authors build on recent developments in the theory of integrable quantum circuits, and derive the GHD equations for such systems. Apart from these technical developments, the key physical result of the paper is that for finite Trotter step sizes, the nonequilibrium macrostates appearing at late times can differ qualitatively from the continuous-time limit.

As stated above, the paper addresses a timely topic at the interface between different research areas. The results are interesting and clearly presented. I am not an expert on integrability and GHD, and, therefore, I find it hard to judge whether the paper is groundbreaking in terms of developing new techniques. However, my impression is that the foundations for the present work were laid in Ref. [33], which diminishes the claims to novelty of this work. Therefore, I believe that the paper is better suited for a less selective journal such as SciPost Physics Core.

1- As the authors explain, a domain wall between the Néel and anti-Néel states can be interpreted as a single localized defect, and such a defect does typically not lead to a change in the macroscopic nonequilibrium state for systems with single-site translation symmetry. In the model under consideration, this symmetry is broken due to the Trotterization. This observation, however, raises the question, whether the observed phenomenology is unique to Trotterized dynamics or could also be seen in autonomous systems with broken single-site translation symmetry.

2- I would welcome a discussion of micromotion. Do the calculated nonequilibrium states apply only at stroboscopic times? If that is the case, how do, e.g., the profiles of the staggered magnetization shown in Fig. 1 change during one driving period?

3- The paper is clearly written and overall accessible even to non experts. However, I have found the transition from the Bethe equations (9) to the TBA in Eq. (12) rather hard to follow. As far as I understand, the Bethe equations determine the spectrum of the Floquet operator, i.e., these equations are not specific to any state of the system. However, the distribution functions appearing in Eq. (12) describe a macrostate of the system. Which state is that and how did it appear in the Bethe equations?

4- The + sign in Eq. (54) should presumably be an =.

Accept in alternative Journal (see Report)