SciPost Submission Page
Conformal field theory on top of a breathing one-dimensional gas of hard core bosons
by Paola Ruggiero, Yannis Brun, Jérome Dubail
This is not the current version.
|As Contributors:||Yannis Brun · Jerome Dubail · Paola Ruggiero|
|Submitted by:||Ruggiero, Paola|
|Submitted to:||SciPost Physics|
|Subject area:||Condensed Matter Physics - Theory|
The recent results of [J. Dubail, J.-M. St\'ephan, J. Viti, P. Calabrese, Scipost Phys. 2, 002 (2017)], which aim at providing access to large scale correlation functions of inhomogeneous critical one-dimensional quantum systems -- e.g. a gas of hard core bosons in a trapping potential -- are extended to a dynamical situation: a breathing gas in a time-dependent harmonic trap. Hard core bosons in a time-dependent harmonic potential are well known to be exactly solvable, and can thus be used as a benchmark for the approach. An extensive discussion of the approach and of its relation with classical and quantum hydrodynamics in one dimension is given, and new formulas for correlation functions, not easily obtainable by other methods, are derived. In particular, a remarkable formula for the large scale asymptotics of the bosonic $n$-particle function $\left< \Psi^\dagger (x_1,t_1) \dots \Psi^\dagger (x_n,t_n) \Psi(x_1',t_1') \dots \Psi(x_n',t_n') \right>$ is obtained. Numerical checks of the approach are carried out for the fermionic two-point function -- easier to access numerically in the microscopic model than the bosonic one -- with perfect agreement.
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Submission & Refereeing History
Reports on this Submission
Anonymous Report 2 on 2019-3-13 Invited Report
- Explicit and nice results
- Timely topic
- just few details left to clarify
This is a very nice paper with explicit and timely results. The authors take the classical hydrodynamics solution for the time evolution of density and velocity of a strongly interacting 1d quantum gas and on top of this develop a space and time dependent CFT to describe its fluctuations. I just have few question that I ask the authors to clarify in the paper:
-- how much their approach is different from local density approximation? Namely could they take simply the density and the momentum in eq 13,14 and from there write a CFT with the x,t -dependent geometry or there is more to this?
-- The authors focus on the correlation functions at the same time. Indeed dynamical correlation functions cannot be obtained by simple CFT and one would need something like non-linear Luttinger liquid. It would be interesting to know if their results in 4.2 for correlations at different times - that they obtain only for free fermion system - can be computed with a non linear Luttiger on top of the breathing cloud.
-- maybe in the introduction it would be good to stress that hydrodynamics well predict one-point functions but fails to predict two-point function (or more specifically, simply predicts them to be zero)
Anonymous Report 1 on 2019-3-12 Invited Report
1) This paper tackles a very timely subject, namely that of the out-of-equilibrium dynamics of a many-body quantum system. In particular, a rather complicated situation is considered namely, that of a a gas of hard-core bosons trapped in a potential that varies periodically in time. This set-up is close to experimentally realizable situations.
2) Remarkably the authors find closed analytic formulae for many-point functions of local operators at different times in the thermodynamic limit. This is a rather unique situation which is probably quite particular for this model and choice of potential, but which still gives a powerful benchmark for future work.
3) The paper is well written and well presented. Despite being rather technical and many different techniques being used, it manages to present the results in an accessible and clear way.
1) The main weakness is that the nice results obtained here seem only accessible in this form for this particular model, so the authors are treating a very simple and particular case and much of what they achieve here will not be accessible in other cases. At the same time, the authors are clear from the beginning about this and I believe their contribution is valuable as an example that stills captures features we may see in more general theories. So this is not a strong weakness.
2) There are also a few trivial typos that I report later.
This is a good paper which solves a clearly defined, interesting, and timely problem, namely computing correlation functions of operators at different times in a many-body quantum system which is driven out of equilibrium. The out-of-equilibrium dynamics is generated by the presence of a potential (trap) which varies periodically (harmonically) in time. The original system (before setting up the trap) is a hard-core boson gas which may be seen as a particular limit of the Lieb-Liniger model (an integrable model).
The paper is well written and well structured. Many different techniques are used (from hydrodynamic equations to CFT in curved space-time) and these are explained with the level of detail that is required to make the paper understandable.
The results are rather impresive as they are very explicit. In particular closed expressions for many point functions at different times are presented (the likes of which I have not seen for any other example) and successfully tested against numerical simulations.
I think that the paper could be published in its current form as I have no strong criticism. There are a couple of small suggestions I propose in the next section, but some of these are optional and (in part) a matter of taste.
I noticed some small inconsistencies in the use of capitals for words derived from people's names. For instance "galilean" is written both as "galilean" and as "Galilean" in different parts of the paper. Similarly "lagrangian" and "Laplacian" are both used. I think both uses are perfectly fine, but the authors should try to be consistent throughout the paper.
There are also a couple of equations that are too long (e.g. (67)) so should consider their presentation.
Finally, I have a tiny suggestion. Even though the paper is well written I have personally found it a bit difficult to follow at times, as different techniques are introduced at each stage, interrupting the flow of the presentation of original results. I personally find it quite useful in long papers to have a section after the introduction called something like "Summary of the main results" where you can report the main formulae (say for instance (52)) and give an indication of the techniques that are required to obtain them. I know some sort of summary is already given at the end of Section 1 but I think you could have an expanded version of that just before section 2.