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Generalized hydrodynamics of integrable quantum circuits
by Friedrich Hübner, Eric Vernier, Lorenzo Piroli
Submission summary
| Authors (as registered SciPost users): | Lorenzo Piroli |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2408.00474v3 (pdf) |
| Date submitted: | 2024-09-17 08:52 |
| Submitted by: | Piroli, Lorenzo |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Quantum circuits make it possible to simulate the continuous-time dynamics of a many-body Hamiltonian by implementing discrete Trotter steps of duration $\tau$. However, when $\tau$ is sufficiently large, the discrete dynamics exhibit qualitative differences compared to the original evolution, potentially displaying novel features and many-body effects. We study an interesting example of this phenomenon, by considering the integrable Trotterization of a prototypical integrable model, the XXZ Heisenberg spin chain. We focus on the well-known bipartition protocol, where two halves of a large system are prepared in different macrostates and suddenly joined together, yielding non-trivial nonequilibrium dynamics. Building upon recent results and adapting the generalized hydrodynamics (GHD) of integrable models, we develop an exact large-scale description of an explicit one-dimensional quantum-circuit setting, where the input left and right qubits are initialized in two distinct product states. We explore the phenomenology predicted by the GHD equations, which depend on the Trotter step and the gate parameters. In some phases of the parameter space, we show that the quantum-circuit large-scale dynamics is qualitatively different compared to the continuous-time evolution. In particular, we find that a single microscopic defect at the junction, such as the addition of a single qubit, may change the nonequilibrium macrostate appearing at late time.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
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.
Requested changes
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 =.
Recommendation
Accept in alternative Journal (see Report)