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Stronglyinteracting bosons at 2D1D Dimensional Crossover
by Hepeng Yao, Lorenzo Pizzino, Thierry Giamarchi
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Authors (as registered SciPost users):  Thierry Giamarchi · Lorenzo Pizzino · Hepeng Yao 
Submission information  

Preprint Link:  https://arxiv.org/abs/2204.02240v2 (pdf) 
Date submitted:  20220928 07:58 
Submitted by:  Yao, Hepeng 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approaches:  Theoretical, Computational 
Abstract
We study a two dimensional (2D) system of interacting quantum bosons, subjected to a continuous periodic potential in one direction. The correlation of such system exhibits a dimensional crossover between a canonical 2D behavior with BerezinskiKosterlitzThouless (BKT) properties and a onedimensional (1D) behavior when the potential is large and splits the system in essentially independent tubes. The later is in the universality class of TomonagaLuttinger liquids (TLL). Using a continuous quantum Monte Carlo method, we investigate this dimensional crossover by computing longitudinal and transverse superfluid fraction as well as the superfluid correlation as a function of temperature, interactions and potential. Especially, we find the correlation function evolves from BKT to TLL type, with special intermediate behaviors appearing at the dimensional crossover. We discuss how the consequences of the dimensional crossover can be investigated in cold atomic gases experiments.
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The manuscript meets the acceptance criterion, as the results are sound and have direct implications to experiments.
Author: Hepeng Yao on 20230207 [id 3323]
(in reply to Report 1 on 20221022)
We thank the Referee for the appreciation of our work. Below we reply to his/her questions.
Referee's comment/question 1 :"Why is there no complex conjugate term present in Eq. 5 for the tunneling part (Eq. 6)?"
Answer: We thank the Referee for pointing out this mistake. We have added the complex conjugate in Eq. 5.
Referee's comment/question 2 :"In the phase diagram (Fig. 1), the thermal regime is identified by the superfluid fractions being below 0.1 %. I am wondering if this can be corroborated by the decay of the correlation function (which in 2D changes from powerlaw to exponential in the thermal regime)? One might also expect a characteristic change in 1D?"
Answer: We thank the Referee for this interesting question. Indeed, the transition to thermal regime can be corroborated by the decay of correlation function, both in 2D and 1D regimes. In both cases, one expects an exponential decay in the thermal regime, which is totally different from the powerlaw decay observed at low temperature as in Fig. 4. To better explain this point, we run calculations at high temperature for V_y/E_r=0 and 30, namely the strictly2D and isolated1D regime. The calculations are performed at the temperature k_B T/E_r=0.2 where the system reaches thermal phase in both dimensionalities, see Fig. 5 of the revised manuscript. In both cases, we see a clear exponential decay which is different from the powerlaw decay observed in Fig. 4. It confirms the statement that the transition to thermal regime can be corroborated by the decay of the correlation function.
In the revised manuscript, we have added the discussion on this point in the last paragraph of Section 5, together with the numerical data in Fig. 5.
Answer to Referee's minor remarks: We thank the Referee for his/her careful reading and pointing out the minor remarks. We agree with all of them and we have addressed all of them carefully.
Anonymous on 20230118 [id 3246]
The manuscript studies with the quantum Moneta Carlo method stronglyinteracting bosons by subjecting them to a periodic potential in one direction. Depending on the transversal tunneling the system shows a dimensional crossover between a 2D behavior with BKT properties and a 1D behavior of the Tomonaga Luttinger liquid. The manuscript is concisely written and presents various interesting results, which characterize the emerging dimensional crossover. In view of possible experimental realizations of such a dimensional crossover in the realm of atomic quantum gases, these theoretical findings are important. Therefore, the manuscript should be published in SciPost Physics journal.
However, the authors should revise their manuscript by taking into account the following criticism. The BKT physics at finite V_y, when the superfluid becomes anisotropic, needs further analysis:
a) On page 7 the BKT critical temperature at V_y = 0 Er is compared with the quantum Monte Carlos results, which corresponds to the 2D regime. Here the question arises which BKT critical temperature emerges at a finite potential depth V_y, when the superfluidity becomes anisotropic. And how this BKT critical temperature compares with the results from the quantum Monte Carlo simulations.
b) Similar questions arise for Fig. 4(b2) and 4(c2). Does the powerlaw obtained from quantum Monte Carlos agree with the BKT predictions for the anisotropic superfluid regime?
Apart from that the authors should improve their manuscript along the following lines:
1) Abstract: The later is in the universality class → The latter is in the universality class
2) Page 2: The references [6, 33–36] could be extended by adding
P. Christodoulou, M. Gałka, N. Dogra, R. Lopes, J. Schmitt, and Z. Hadzibabic: Observation of first and second sound in a BKT superfluid, Nature 594, 191194 (2021)
3) Page 4: Instead of the unprecise comment
see detailed calculations in the appendix
it would be preferable to mention explicitly in which part of the Appendix this information is available. This also applies to other parts of the manuscript where only Ref. [50] is cited.
4) Caption of Fig. 2: and they follows a fit → and they follow a fit
5) Page 5: we can judge crossover potential > we can judge the crossover potential
6) Page 6: These parameters allows us –> These parameters allow us
7) Caption of Fig. 4: Subfigures (a1)(d1) shows > Subfigures (a1)(d1) show
8) Page 9: by the ground state of harmonic oscillator. > by the ground state of the harmonic oscillator.
Author: Hepeng Yao on 20230207 [id 3324]
(in reply to Anonymous Comment on 20230118 [id 3246])We thank the Referee for the appreciation of our work. Below we reply to his/her questions.
Referee's comment/question 1 :"On page 7 the BKT critical temperature at V_y = 0 Er is compared with the quantum Monte Carlos results, which corresponds to the 2D regime. Here the question arises which BKT critical temperature emerges at a finite potential depth V_y, when the superfluidity becomes anisotropic. And how this BKT critical temperature compares with the results from the quantum Monte Carlo simulations."
Answer: We thank the Referee for this interesting question. From the numerical aspect, now we perform the quantum Monte Carlo calculations for f_s^x as a function of T at various values of V_y and compute the crossover temperature to the thermal regime T_{cross}. Then, we plot T_{cross} as a function of V_y in Fig. 3(c) of the revised manuscript. At V_y=0 E_r, the value of T_{cross} fits with T_{2D} which is the estimated BKT critical temperature in strictly2D regime. Then, increasing V_y leads to decrease the value of T_{cross}. This suggests that the anisotropy induced by the unidirectional periodic lattice makes it harder for the system to reach quantum degeneracy. Further increase V_y up to 15 E_r, T_{cross} reaches the value T_{1D} which is estimated by the 1D TomonagaLuttinger theory.
From the analytical aspect, it will indeed be interesting to compare with the BKT temperature in anisotropic system. One possible solution is to perform selfconsistent harmonic approximation (SCHA). Such calculation has been performed in 2D anisotropic XY model, see “J.S. You et al Phys. Rev. A 86, 043612 (2012)” (Ref. [30] of the revised manuscript). The SCHA calculation shows that T_{BKT} decreases with the increase of anisotropy, which fits with the QMC results qualitatively. However, the value of T_c from the two calculations doesn’t match quantitatively. The results differ less in the limit of very anisotropic system and isotropic case. Notably, Ref. [30] considers the discrete case which is different from the continuous periodic potential we treat in the paper. More quantitative results adapted to our model, to the best of our knowledge, are still lacking. To find T_{BKT} at finite V_y in our model deserves a thorough calculation with methods like SCHA or renormalization group, which we feel is beyond the scope of the current manuscript and deserves to be an independent project. However, we agree that this is an interesting outlook based on the results in our paper and should definitely be mentioned in the manuscript.
We have added the numerical data in Fig. 3(c) and corresponding statement in the last paragraph of section 4 “longitudinal superfluidity”. We also added in the text: “It will be worth to investigating analytical calculations for the BKT temperature at anisotropic systems and compare it with the numerical data at finite size. Such calculations may be carried out by selfconsistent harmonic approximation, see for instance Ref. [30]”.
Referee's comment/question 2 :"Similar questions arise for Fig. 4(b2) and 4(c2). Does the powerlaw obtained from quantum Monte Carlos agree with the BKT predictions for the anisotropic superfluid regime?"
Answer: We thank the Referee for this interesting question. From the numerical point of view, the results in Fig. (b2) and (c2) suggest that the longrange behavior still maintains its BKT properties with a decay exponent similarly as the strictly2D case, while the shortrange behavior shows a clearly larger decay exponent due to the anisotropy induced by the periodic potential. Considering the analytical BKT predictions, we find such results to our specific model is also lacking in the known references. Although there are known calculations for 2D XY model such as Ref. [30], they cannot describe completely our current system due to the continuous and shallow periodic potential. As for the previous question, we agree that this is an interesting outlook to our paper and we have mentioned this in the revised manuscript.
In the revised version of our paper, we add in the second last paragraph of section 5 “correlation function” the sentence “One possible extension is to compute the BKT prediction of the correlation function for 2D systems in the presence of the unidirectional periodic potential, and compare them with the results here.”, in order to point out this interesting outlook based on our current results.
Answer to Referee's minor remarks: We thank the Referee for his/her careful reading and pointing out the minor remarks. We agree with all of them and we have hopefully addressed all of them carefully.
Matthew Davis on 20230119 [id 3249]
(in reply to Anonymous Comment on 20230118 [id 3246])As editor in charge of this submission, I note that this comment was submitted by one of the invited referees.