# A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model

### Submission summary

 As Contributors: Fabian Essler · Etienne Granet Preprint link: scipost_202008_00011v2 Date accepted: 2020-11-23 Date submitted: 2020-11-12 17:08 Submitted by: Granet, Etienne Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Mathematical Physics

### Abstract

We introduce a framework for calculating dynamical correlations in the Lieb-Liniger model in arbitrary energy eigenstates and for all space and time, that combines a Lehmann representation with a $1/c$ expansion. The $n^{\rm th}$ term of the expansion is of order $1/c^n$ and takes into account all $\lfloor \tfrac{n}{2}\rfloor+1$ particle-hole excitations over the averaging eigenstate. Importantly, in contrast to a bare" $1/c$ expansion it is uniform in space and time. The framework is based on a method for taking the thermodynamic limit of sums of form factors that exhibit non integrable singularities. We expect our framework to be applicable to any local operator.\\ We determine the first three terms of this expansion and obtain an explicit expression for the density-density dynamical correlations and the dynamical structure factor at order $1/c^2$. We apply these to finite-temperature equilibrium states and non-equilibrium steady states after quantum quenches. We recover predictions of (nonlinear) Luttinger liquid theory and generalized hydrodynamics in the appropriate limits, and are able to compute sub-leading corrections to these.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

We are grateful to the three referees for their careful reading of the manuscript and helpful comments. We appreciate their very positive reports. In the following we answer the minor points raised by the first and second referees.

— Reply to the first referee —

1- We added the argument mentioned by the referee in paragraph 5.3.1.

2- The referee asks whether the dressing of one-particle-hole excitations by two-particle- hole excitations could be done directly at the level of the form factor. This is indeed a very interesting question. However, this is a rather subtle point that goes beyond the scope of the paper. We are working on higher-order corrections in 1/c and should be able to address this question in future work.

3- The referee asks why some terms contribute negatively to the dynamical structure factor. This is indeed a relevant remark. We added a paragraph at the end of 6.2 to explain this feature. As noted by the referee, this comes from the fact that what we call one and two-particle-hole excitations are terms that result from cross-cancellations of divergent parts in their bare spectral sum, and these resulting pieces need not be positive. The association of the resulting terms with one or two-particle-hole excitations is based on whether they are expressed as double or quadruple integrals over the root densities. As we further note in the added paragraph, negative spectral weights are expected to arise in individual terms of any perturbative expansions of the Lehmann representation.

4- We added the reference mentioned by the referee in Section 7.2.2.

— Reply to the second referee —

1- This n(λ) was a typo and has been changed into θ(λ).

2- We added the reference mentioned by the referee in Section 7.1.1.

### Submission & Refereeing History

Resubmission scipost_202008_00011v2 on 12 November 2020
Submission scipost_202008_00011v1 on 19 August 2020

## Reports on this Submission

### Report

As I said in my report, I was already satisfied with the first version of this manuscript. It has been further improved by taking into account the tiny suggestions of the referees and is ready for publication in its present form.

• validity: -
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### Report

I'm satisfied with the new version of the manuscript and consider the paper ready to be published in its present form.

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