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Free fermions with dephasing and boundary driving: Bethe Ansatz results

by Vincenzo Alba

Submission summary

Authors (as registered SciPost users): Vincenzo Alba
Submission information
Preprint Link: scipost_202310_00015v1  (pdf)
Date submitted: 2023-10-15 10:25
Submitted by: Alba, Vincenzo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


By employing the Lindblad equation, we derive the evolution of the two-point correlator for a free-fermion chain of length $L$ subject to bulk dephasing and boundary losses. We use the Bethe ansatz to diagonalize the Liouvillian ${\mathcal L}^{\scriptscriptstyle(2)}$ governing the dynamics of the correlator. The majority of its energy levels are complex. Precisely, $L(L-1)/2$ complex energies do not depend on dephasing, apart from a trivial shift. The remaining complex levels are perturbatively related to the dephasing-independent ones for large $L$. The long-time dynamics is governed by a band of real energies, which contains an extensive number of levels. They give rise to diffusive scaling at intermediate times, when boundaries can be neglected. Moreover, they encode the breaking of diffusion at asymptotically long times. Interestingly, for large loss rate two boundary modes appear in the spectrum. The real energies correspond to string solutions of the Bethe equations, and can be treated effectively for large chains. This allows us to derive compact formulas for the dynamics of the fermionic density. We check our results against exact diagonalization, finding perfect agreement.

Current status:
Awaiting resubmission

Reports on this Submission

Anonymous Report 2 on 2023-12-2 (Invited Report)


1 - very comprehensive generalization of recent results on Bethe ansatz results for open quantum systems
2- new method for explicitly computing 2 point correlators


The author studies a free-fermion chain subjected to bulk Markovian local dephasing and boundary loss. The full Liouvillian is seemingly not integrable. Despite this, the author manages to use Bethe ansatz to solve the 2-point correlation dynamics, which I find very interesting. I find the work both conceptually and technically innovative and recommend publication, but I would ask the author to address the below question.

Requested changes

1 - Is the solvability of the pure loss boundaries related to the triangular form of the Liouvillian as exploited in [13,17]? It would be nice for the author to comment on this.
2- If so, would this approach work for bulk loss?

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

Anonymous Report 1 on 2023-11-17 (Invited Report)


1- The paper presents an interesting new application of the Bethe ansatz to obtain the spectrum in the single-particle sector of a Liouvillian that is not integrable. Based on the results obtained from the Bethe ansatz, the time evolution of the two-point function is studied as an important application.

2- The calculations presented in the paper are technically very challenging and the results are impressive.

3- The presentation is very clear and even though the subject is rather technical, the paper is relatively easy to read.

4- Results obtained from the Bethe ansatz are backed up by excellent agreement with exact numerics.


1- The progress that is being made in this paper is rather technical than conceptual, and it is unclear to which extent the new results can lead to new physical insights.


The paper presents an interesting new application of the Bethe ansatz to study a fermionic tight-binding chain with bulk dephasing and losses at the boundary, described by a quantum master equation in Lindblad form. While the full Liouvillian is not integrable, there is a closed equation of motion for the equal-time two-point function, and the author employs the Bethe ansatz to determine the spectrum of the superoperator that generates the dynamics of the two-point function. To illustrate the usefulness of the results, diffusive dynamics of a particle that is initially localized at the center of the chain are demonstrated.

The technical developments presented in the paper are impressive. I appreciate very much that even though the subject is very technical, the text is rather accessible. However, I am not convinced that the new results represent a groundbreaking discovery. In particular, it is not clear to me which important novel physical rather than technical insights could arise from the results of the paper. Therefore, I believe that the manuscript would be more suitable for a less selective journal such as SciPost Physics Core.

I do not have substantial criticism regarding the contents of the paper. The points listed below are rather minor.

Requested changes

1- The term "energies" for the eigenvalues of the Liouvillian is somewhat ambiguous. In particular, when "energy levels" are first mentioned in the abstract, it is not clear whether the real or imaginary parts of the energies correspond to decay rates. This should be clarified.

2- In the introduction and below Eq. (26) it is stated that for $\gamma^- = 0$, a specific set of energies is purely imaginary. However, as far as I understand, they are not purely imaginary but have real part $- \gamma$.

3- How should the following statement be interpreted: "For $x_1 = x_2$, $G_{x_1, x_2}$ is given by the first row in (9)." Is it really only the first row or the full term that multiplies the first Heaviside function? Does this statement imply a particular choice of the value $\Theta(0)$?

4- What is the physical meaning of the symmetry $\mathcal{R}$?

5- Maybe I am overlooking something here, but for the ansatz in Eq. (9) it seems to me that $G_{x_2, x_1} = \sigma G_{x_1, x_2}$ without the factor $( - 1)^{x_1 + x_2}$ on the left-hand side.

6- Is inversion symmetry of the setup reflected in the ansatz Eq. (9)?

7- It is not quite clear what is meant with "Let us now impose the “contact” condition obtained by fixing x1 = x2 in (9)." Is this the equality of the values of $G_{x_1, x_2}$ for $x_1$ approaching $x_2$ from above and from below?

8- In the discussion preceding Eq. (26), it is not quite clear where the restrictions on the values of $k_1$ and $k_2$ for finite $\gamma$ come from.

9- In Eq. (26), why is there $L + 1$ in the denominator? To the given order in $L$, the $+ 1$ is negligible.

10- Below Eq. (26), I suppose the reference should be to Fig. 2.

11- The title of Sec. 3.4, "Solutions with vanishing imaginary parts," is somewhat confusing. It would help to clarify that the imaginary parts of the momenta $k_1$ and $k_2$ and not of the eigenvalues $\varepsilon$ vanish.

12- Also in Eqs. (29), (30), and (31), I believe that $L + 1$ can be replaced by $L$.

13- Above Eq. (42), it is stated that "This is obtained by considering the energy $\varepsilon$ with the smallest nonzero real part." Actually, it should be the largest nonzero real part.

14- I find the formulation above Eq. (48), that "it is possible to determine a more convenient choice" for the left eigenvectors, somewhat misleading. For a given eigenvalue, the left eigenvectors are determined by the eigenvalue equation, and there is no choice in how to define them.

15- I believe that in Eq. (63) there is a factor $f(x)$ missing in the sum.

16- Below Eq. (64), there is a typo: "see section (24)."

17- In Sec. 3.5, it does not become quite clear what the string hypothesis actually is or where exactly it is being used. A brief discussion of these points would be helpful.

18- The Liouvillian gap is defined twice, in Eqs. (42) and (77), and the definitions do not agree.

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

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