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On the Dynamics of Free-Fermionic Tau-Functions at Finite Temperature

by Daniel Chernowitz, Oleksandr Gamayun

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Submission summary

Authors (as registered SciPost users): Daniel Chernowitz · Oleksandr Gamayun
Submission information
Preprint Link:  (pdf)
Code repository:
Date accepted: 2022-01-28
Date submitted: 2022-01-20 04:10
Submitted by: Chernowitz, Daniel
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
  • Mathematical Physics
  • Quantum Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


In this work we explore an instance of the $\tau$-function of vertex type operators, specified in terms of a constant phase shift in a free-fermionic basis. From the physical point of view this $\tau$-function has multiple interpretations: as a correlator of Jordan-Wigner strings, a Loschmidt Echo in the Aharonov-Bohm effect, or the generating function of the local densities in the Tonks-Girardeau gas. We present the $\tau$-function as a form-factors series and tackle it from four vantage points: (i) we perform an exact summation and express it in terms of a Fredholm determinant in the thermodynamic limit, (ii) we use bosonization techniques to perform partial summations of soft modes around the Fermi surface to acquire the scaling at zero temperature, (iii) we derive large space and time asymptotic behavior for the thermal Fredholm determinant by relating it to effective form-factors with an asymptotically similar kernel, and (iv) we identify and sum the important basis elements directly through a tailor-made numerical algorithm for finite-entropy states in a free-fermionic Hilbert space. All methods confirm each other. We find that, in addition to the exponential decay in the finite-temperature case the dynamic correlation functions exhibit an extra power law in time, universal over any distribution and time scale.

Author comments upon resubmission

Individual reply to anonymous report nr 1:

We thank the reviewer for their time and attention. They were correct to point out the double use of the n_a in the notation. This has been remedied (the summed over Hilbert space will now be written with m_a as its integers). We have added a few sentences to help the reader understand where the introductory formulae originate. We agree with and have incorporated all of the 'further remarks'.

Individual reply to anonymous report nr 2:

We thank the reviewer for their time and attention. In regards to the weaknesses, indeed not everywhere have we kept quantitative track of the errors. However, where it is crucial (in the main text) we have estimated the error where it depends on, e.g. the \epsilon parameter which is taken to 0 in the thermodynamic limit. The conflicting circumlocution and coarseness probably splits down the line separating the two authors. I imagine the 'lack of details' is referring to CH5: written mainly by OG who is the more senior academic. I, DC, might do better to get to the point more quickly. This is fair feedback.

Individual reply to report by Pieter Claeys, nr 3:

The authors thank Pieter for his meticulous and fair review. We will address most of his feedback. About weaknesses: Indeed ch5 is somewhat disconnected, however it has been a long-term project of OG and we feel this paper is the best home for the content. It does show a completely separate machinery to arrive at a signal that is the limiting (T=0) case of the main topic of the paper. In that sense it serves as another confirmation of the validity of the techniques in the paper. It is true that a printed paper might not be the best way to distribute Python code, however we feel there is merit in understanding the algorithm on the ground level. Not everyone will be interested in ch 7.2 and it can safely be skipped by those who aren't. No other parts depend on it. In regards to the enumerated list of requested changes, we agree with and have incorporated them straightforwardly, or have cleared up the confusion. Exceptions are addressed below. 2- We have not succeeded in proving (by means of some integer/discrete mathematics) that the diagonal overlaps are largest. It is most certainly true, though, as evinced by a large amount of numerics. We think a proof should be possible. 3- That line was plotted, as the black dots. There is no need for numerical sampling of the ground state, there is only one ground state. The caption was, however, confusing, and we amended it. 4- We believe a contour integral is still the preferred way to address the sum. If anyone has a way to short-circuit the calculation with a smart expansion or identity, please cite us. 7- Indeed, the peaks in both panels (on the left they are not clearly visible) represent singularities in the asymptotic formulas that occur at the "light" cone $x=\pm k_F t$. In reality, these singularities are absent meaning that the asymptotic expansion should be trusted only far from the "light" cones. Our plots demonstrate that, in fact, asymptotic expansion starts to be relevant already in the close vicinity of the light cone. The asymptotic exactly on the light cone is discussed below in chapter 5. As for the numeric position: the time is fixed to $t =100 E_F = 50 k_F^2 $, which means that for $k_F=1$ the singularities are located at $x=\pm 50$. (We have expanded caption of the picture to clarify this fact) 8- This would be too much work to remedy, apologies. 12- We have left ch 7.2 as is, and beg for leniency, as restructuring would probably introduce more errors than is warranted. DC has, however, created a github and linked to it. That is most likely practical for prospective users.

List of changes

- Signposted the importance of various quantities, such as the shift \nu, the origin of certain expressions such as the definition of \tau, or the roles of certain sections such as chapters 5 and 7, more clearly.
- Cleared up definition issue in the summation of Hilbert space (n_a -> m_a).
- Improved the captions of certain figures, sich as figures 3 and 7.
- Improved references at various points.
- Clarified the scale on which poles of the effective form factors can deviate from the real line.
- Improved clarity of some ambiguous formulations.
- Created and linked to a github for the codebase.
- Removed assorted typos.

Published as SciPost Phys. Core 5, 006 (2022)

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