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Inhomogeneous quantum quenches in the sine-Gordon theory
by D. X. Horváth, M. Kormos, S. Sotiriadis, G. Takács
This Submission thread is now published as
|Authors (as Contributors):||Márton Kormos · Spyros Sotiriadis|
|Arxiv Link:||https://arxiv.org/abs/2109.06869v3 (pdf)|
|Date submitted:||2022-03-24 14:09|
|Submitted by:||Kormos, Márton|
|Submitted to:||SciPost Physics|
We study inhomogeneous quantum quenches in the attractive regime of the sine-Gordon model. In our protocol, the system is prepared in an inhomogeneous initial state in finite volume by coupling the topological charge density operator to a Gaussian external field. After switching off the external field, the subsequent time evolution is governed by the homogeneous sine-Gordon Hamiltonian. Varying either the interaction strength of the sine-Gordon model or the amplitude of the external source field, an interesting transition is observed in the expectation value of the soliton density. This affects both the initial profile of the density and its time evolution and can be summarised as a steep transition between behaviours reminiscent of the Klein-Gordon, and the free massive Dirac fermion theory with initial external fields of high enough magnitude. The transition in the initial state is also displayed by the classical sine-Gordon theory and hence can be understood by semi-classical considerations in terms of the presence of small amplitude field configurations and the appearance of soliton excitations, which are naturally associated with bosonic and fermionic excitations on the quantum level, respectively. Features of the quantum dynamics are also consistent with this correspondence and comparing them to the classical evolution of the density profile reveals that quantum effects become markedly pronounced during the time evolution. These results suggest a crossover between the dominance of bosonic and fermionic degrees of freedom whose precise identification in terms of the fundamental particle excitations can be rather non-trivial. Nevertheless, their interplay is expected to influence the sine-Gordon dynamics in arbitrary inhomogeneous settings.
Published as SciPost Phys. 12, 144 (2022)
Author comments upon resubmission
We thank the referees for their constructive suggestions and the questions they raised. Below we answer each point separately and explain the changes we have made to the paper.
- I think it is essential to add a fully quantitative comparison between TSCA and the exact free fermion result. The color plots in Fig. IV.9 show that TSCA appears to be generally in good agreement with the numerically exact result. However, I think readers will want to have a much more quantitative comparison. A simple way of providing this would be to show representative "cuts" for fixed x/L as a function of time.
- The authors stress the failure of the local density approximation. As they discuss, it is well known that LDA has a regime of validity and is not expected to work outside it. The criteria for LDA to hold in the free fermion case are given in Appendix C.5 and I think it would be useful to phrase the discussion in IV.B. in terms of these criteria. Perhaps the authors will want to comment about using the LDA in combination with the Bethe Ansatz solution away from the free fermion point.
- A natural question is whether there are any easily identifiable features associated with the first breather in the weakly attractive regime. It seems that the answer is no (as no such feature is mentioned), but I think it would be helpful for the readers if this was spelled out clearly.
We thank Referee 1 for the valuable remarks. Our replies are the following:
Following the advice of the Referee, we have added an additional figure (IV.12) to demonstrate the comparison between the exact FF dynamics and that of TCSA. In addition, we now also show the extrapolated time evolved quantities to have a better insight into the accuracy and limitations of TCSA. However, we would like to stress again that the use of our extrapolating procedure is not strictly justified for time-evolved quantities (unlike for equilibrium ones), nevertheless we think such quantities may be useful to understand better TCSA in out-of-equilibrium settings.
We have rewritten some parts of Appendix C.5 and we now explicitly quote the two conditions for the applicability of LDA in Sec. IV.B, see Eqs. (IV.4) and (IV.5) and the discussion around them. We refrain from extending our LDA calculations away from the free fermion point as this requires much more sophisticated methods that are out of the scope of the present work; however, we may return to this problem in the near future.
We are grateful for drawing our attention to this point and have added some additional text in Section IV.C (first two paragraphs on p. 17) to better explain the effect of the first breather close to the free-fermion point or in other words in a weakly interacting regime (which is not to be confused with the free-boson point of the theory). We have demonstrated that exactly at the free-fermion point a fermion and an antifermion (soliton-antisoliton pair) form a bosonic collective excitation of mass 2M, where M is the soliton mass. When moving away from this point, the 1st breather, an inherent bosonic excitation indeed enters the spectrum. Nevertheless its mass is almost equal to 2M, and as indicated by the wave function renormalisation constant, its matrix elements with respect to the canonical field (or its derivatives) are small. These factors mean the impact of the 1st breather on observables that are functions of the fundamental field is very small. The disentangling of such a small effect, in principle, can be possible via a proper Fourier analysis of the time evolved quantities. However, due to the minor mass difference between the mass of the 1st breather and soliton-antisoliton excitation, one needs much longer time windows to carry out such an analysis than the ones in which we are able to perform our simulations.
Report: 1- I suggest to clarify a bit more what is meant by oscillatory behaviour. In particular on page 16, last paragraph the authors say: "Indeed, it can be easily seen from Figs. IV.6 and IV.9 that the dominant oscillation frequency approximately equals 2π which is associated with a single boson at rest". I must admit this statement is not very obvious... My understanding of this oscillation is a colour change on these plots as we move forward in time (is it correct?). Please clarify this. Related question is why the lighter breathers do not show up in this oscillatory behaviour (of course in the regime of the β parameter where they are present)? 2- Can the setup and the results in the paper be modelled by the integrable hydrodynamics approach? If so, what would be the relevant time scale for the integrable hydrodynamics to work? Please comment on this. 3- I think the prefactor in eq. C.60 is wrong (it should be L/(2 π) 2 π/L instead, I think), otherwise we do not get eq. C.61 as it is written now (which I think is correct). 4- In the introduction I would recommend to mention commensurate-incommensurate transition (and so the old papers by Haldane, Japaridze and Nersessyan) in the limiting case where profile is constant in space (homogeneous), so that the perturbation is just proportional to the topological charge density. 5-I suggest to go through the text once again to avoid some trivial repetitions of words and to improve some formulations.
We thank Referee 2 for the valuable suggestions. Our replies are the following:
- We have added an additional figure to better highlight that the period of the oscillations after the initial bump splits equals 2 π. To this end, we plotted the time evolution of the expectation value of the topological charge density at a fixed position x=0 (Fig. IV. 11).
The effect of the other breathers is generically less pronounced in various quantities during the time evolution. One way to identify their effect is via Fourier analysis (a.k.a. “quench spectroscopy”), however it requires longer time windows than those we were able to simulate. In addition, these much longer simulation times are also needed to achieve precision sufficient to distinguish numerically the frequencies of higher breathers from those of multi-particle states containing lower breathers.
Integrable hydrodynamic approaches applicable for the sine-Gordon have only been developed for the repulsive regime of the model and for very particular initial states. The quench studied in our work takes place in the attractive regime and it is not known how the particular characterisation of the inhomogeneous initial state that is necessary for integrable hydrodynamical approaches can be carried out. We would like to stress that to the best of our knowledge, the precise conditions of hydrodynamical approaches, such as from what time scales they provide an accurate description of the physical system, have not yet been explored.
We have corrected the prefactor.
We added the references to the end of paragraph 4 of the Introduction. In addition, we added a reference to a related work by Pokrovsky and Talapov (also mentioned in the Conclusions).
We have gone through the text and fixed a number of such repetitions, and improved formulations.
List of changes
- For the changes indicated in our reply to the referee reports see above. These included adding further text, two new figures (Figs. IV.11 and IV.12) and additional references (Ref. [64-68]).
- We have also added a new reference (), which presents in detail the numerical method used in this work.
- Lastly, we went through the text and corrected misprints, typos and also made some improvements to the style.
Submission & Refereeing History
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