# Exact dynamical correlations of nonlocal operators in quadratic open Fermion systems: a characteristic function approach

### Submission summary

 As Contributors: Qing-Wei Wang Preprint link: scipost_202201_00041v2 Date accepted: 2022-03-30 Date submitted: 2022-03-23 04:05 Submitted by: Wang, Qing-Wei Submitted to: SciPost Physics Core Academic field: Physics Specialties: Atomic, Molecular and Optical Physics - Theory Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

The dynamical correlations of nonlocal operators in general quadratic open fermion systems is still a challenging problem. Here we tackle this problem by developing a new formulation of open fermion many-body systems, namely, the characteristic function approach. Illustrating the technique, we analyze a finite Kitaev chain with boundary dissipation and consider anyon-type nonlocal excitations. We give explicit formula for the Green's functions, demonstrating an asymmetric light cone induced by the anyon statistical parameter and an increasing relaxation rate with this parameter. We also analyze some other types of nonlocal operator correlations such as the full counting statistics of the charge number and the Loschmidt echo in a quench from the vacuum state. The former shows clear signature of a nonequilibrium quantum phase transition, while the later exhibits cusps at some critical times and hence demonstrates dynamical quantum phase transitions.

Published as SciPost Phys. Core 5, 027 (2022)

We have made revisions according to the comments of the three referees. We resubmit the revised version to SciPost Physics Core.

### List of changes

Note that the positions listed below refer to that in the old version [revised version] of the paper.

1. “statistical angle” is replaced by “statistical parameter” throughout the paper.

2. L8-9 [L8-9]: “an asymmetric light cone induced by the statistical angle ϕ and an increasing relaxation rate with ϕ” is changed to “an asymmetric light cone induced by the anyon satistical parameter and an increasing relaxation rate with this parameter”.

3.L53 [L54]: A typo is fixed, i.e., $\hat{c}_j^\dag \hat{c}_j$ should be $\hat{c}_l^\dag \hat{c}_l$.

4. L60-61 [L61-62]: “ a challenging and highly nontrivial theoretical problem” is changed to “a highly nontrivial theoretical problem”.

5.L107 [L108-110]: We add one sentence: Here we use the notations ...... ''

6.L112 [L114-115]: We add: where $D(\xi|\eta)\equiv e^{\bar{\xi}\eta-\bar{\eta}\xi}$ is the Grassmann analogy of the usual Fourier transformation kernel for complex variables''.

7. L114-117 [L118-138]. This paragraph is expanded into two paragraphs.

8. L119 [L140-141]: “quantum Lindblad master equation” is changed to “Gorini-Kossakorsky-Sudarshan-Lindblad (GKSL) equation”. And “Lindbladian” is changed to “Liouvillian”.

9. L129 [L150]: “a nontrivial and challenging problem” is changed to “a challenging problem”.

10. L133 [L154-159]: Three sentences are added: “See Appendix.B for the details of the derivation. We comment that for a general Liouvillian ... ... solved exactly by standard technique.”

11. L133 [L159]: “The solution with initial condition...” is changed to “The solution with an arbitrary initial condition...”.

12. L139 [L165-167]: One comment is added: “ We comment that the structure of the solution Eq.(9) is very similar to its bosonic counter part (see, for example, the work by T. Heinosaari et al. [81]).”.

13. L139-140 [L168-199]: Two more paragraphs are added just before Sec.2.2.

14. L143 [L203]: “by other methods” is changed to “by other methods[54-63]”.

15. L153-154 [L213-215]: We add: From the equation for $F(\bar{\xi},\xi)$ ... ... denotes anticommutation relation.''

16. L171 [L234-241]: We add: “We comment that the requirement ... ...otherwise the equation would be lengthy.”

17. L173-176 [L242-257]: These lines are rewritten by adding more arguments.

18. L216 [L297]: “the Kitaev chain” is changed to “the Kitaev chain[84]”.

19. Fig.1 [Fig.1] is re-plotted and the caption is changed correspondingly.

20. L222 [L303]: “The simplest nontrivial dissipations act only on the first and last site” is changed to “For simplicity of this illustrating example we take dissipations which act only on the first and last sites”.

21. L227 [L308-318]: Some more arguments are added: “We remark that the symmetric dissipative driving ... ... to illustrate the general technique developed above.”

22. L228-239 [L319-332]: This paragraph is rewritten in accordance with the change of Fig.1.

23.Eq.(28) [Eq.(28) ]: “Reλ” is changed to “Im(λ)”.

24. Eq.(30) [Eq.(30) ]: A typo is fixed.

25. L259-263 [L352-362]: These lines are rewritten.

26. L269 [L368]: We add: “where the average ⟨·⟩ is taken in the steady state.”

27. L275 [L375-378]: We add: When $t \neq 0$, these Green's functions tell us ... ... important quantities in both theoretical and experimental studies.''

28. L279 [L382-383]: We add “Spatial symmetry and temporal damping behaviors can be seen clearly.”

29. L282 [L385-387]: The sentence “This asymmetric propagation is induced by the statistical angle.” is changed to “This asymmetric propagation is induced by the statistical parameter, since the Hamiltonian and the dissipators are symmetric under the spatial reflection about the chain center.”

30. L284 [L389]: “that l is mapped” should be “ that l(j) is mapped”.

31. L285 [L390-393]: We add “We stress that this symmetry holds only for symmetric Hamiltonian and dissipators ... ... as observed elsewhere [57]. ”

32. L289-292 [L397-406]: These lines are rewritten.

33. L300-301 [L414-415 ]: A typo is fixed .

34. L303 [L417-419]: We add “In general the charge number ... ... that there are exactly n charge in A at time t.”

35. Appendix.A [Appendix.A] is expanded by including more formulas.

36. One new appendix is added, i.e., Appendix.B in the revised version.

37. L403-404 [L560-561]: One sentence is added: “This difference is due to .... ”.

38. L404 [L561]: “We should note that...” is changed to “However, we should note that...”.

39. L414-415 [L572-579]: One paragraph is added at the beginning of Appendix.C [Appendix.D]: “The dynamical correlation functions would reduce to static ones ...... which shows clear signatures of the NQPT.”

40. The following 14 publications are added into the bibliography:
• M.Michel, J. Gemmer, G. Mahler, Eur. Phys. J. B 42, 555 (2004).
• Z. Lenarcic and T. Prosen, Phys. Rev. E 91, 030103(R) (2015).
• Marko Znidaric, Phys. Rev. B 99, 035143 (2019).
• S. Morrison and A. S. Parkins, Phys. Rev. Lett. 100, 040403(2008).
• E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin, and J. I. Cirac, Phys. Rev. A 86, 012116 (2012).
• V. Balachandran, G. Benenti, E. Pereira, G. Casati, and Dario Poletti, Phys. Rev. Lett. 120, 200603 (2018).
• Chu Guo and Dario Poletti, Phys. Rev. A 94, 033610 (2016).
• K. Yamamoto, M. Nakagawa, N. Tsuji, M. Ueda, and N. Kawakami, Phys. Rev. Lett. 127, 055301 (2021).
• W. Berdanier, J. Marino, and E. Altman, Phys. Rev. Lett. 123, 230604 (2019).
• L. M. Vasiloiu, F. Carollo, and J. P. Garrahan, Phys. Rev. B 98, 094308 (2018).
• T. Heinosaari, A. S. Holevo, M. M. Wolf, Quantum Inf. Comp. 10: 0619-0635 (2010).
• A. Kitaev, Phys.-Usp. 44, 131 (2001).
• Qing-Wei Wang, arXiv:2202.06543.
• C. Itzykson, and J.-M. Drouffffe, Statistical Field Theory, Cambridge University Press, 1989.

### Submission & Refereeing History

#### Published as SciPost Phys. Core 5, 027 (2022)

Resubmission scipost_202201_00041v2 on 23 March 2022
Submission scipost_202201_00041v1 on 31 January 2022

## Reports on this Submission

### Report

I thank the authors for carefully considering all my comments and suggestions, and fixing the small issues. I can thus recommend it for publication as it is.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Report

I appreciate that the author thoroughly answered all remarks and clarified all points. In my opinion, the paper has been nicely improved is now ready to be published.

As a small detail, I would claim that point (ii) in L186 is wrong. Also in the third quantisation, all information is stored in the 2N * 2N matrix analogous to X_+ +i H mentioned in the manuscript, due to the block upper triangular form of the 4N * 4N matrix in the third quantisation.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -