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

### Submission summary

 As Contributors: Daniel Chernowitz · Oleksandr Gamayun Arxiv Link: https://arxiv.org/abs/2110.08194v1 (pdf) Date submitted: 2021-10-26 11:58 Submitted by: Chernowitz, Daniel Submitted to: SciPost Physics Core Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Mathematical Physics Quantum Physics Statistical and Soft Matter Physics Approaches: Theoretical, Computational

### Abstract

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.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2110.08194v1 on 26 October 2021

## Reports on this Submission

### Report

The authors study the \tau-function for free fermions. Before doing so they discuss a simple physical setup, namely a many-particle version of the Aharonov-Bohm problem, where the \tau-function naturally appears. This section is very useful to clarify the definition, motivation and basic properties pf the \tau-function. In the following section they discuss why the \tau-function is hard to calculate by linking it to Andersons orthogonality catastrophe. After that they go on to analyse it using several complementary approaches. While to topic of the paper may seem a bit dry and the paper itself rather lengthy, I believe that such technical articles deserve publication in journals like SciPost Physics Core.

Regarding the presentation, my main concern is with the introduction, since I find the definition of the quantity of interest (1) not easy to understand. For example, the following points remain unclear: (i) Is there any restriction on the integers n_a? (ii) The k_a are defined as specific shifts of the g_a, in particular I deduce that there is precisely one state |k>. Why is there a summation over k then? (iii) It is stated that the summation is over ordered integers (the n_a I suppose) , but the n_a are given by |g>? Some of these points become clear in the following section, but at first the reader may be irritated. Thus I ask the authors to revise this part and make it more self-contained, or at least clearly refer to later parts of the manuscript.

Some further remarks:
-The authors mention that the number of terms in a form-factor expansion grows exponentially and thus cannot be performed at finite temperatures. There have been attempts to overcome this using some stochastic sampling of the Hilbert space [PRB 84, 094203 (2011); PRB 88, 085323 (2013)] which the authors may want to mention.
-It is unclear where (2) comes from. At least it should be pointed out that this formula will be derived later.
-Eqs. (1) and (3) use the same notation for \tau but the RHSs are different quantities.

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