Strongly-interacting bosons at 2D-1D Dimensional Crossover

Dear editor,

We thank you for transmitting the Referees’ reports. We thank the Referees for their work on our manuscript and constructive comments/suggestions. Here, we submit a revised version of the manuscript by addressing all the comments/suggestions. Referee report 3 already concludes our manuscript “fit for the journal and recommend its publication” and doesn’t propose any further changes. Below, we provide our answers to the Referee report 4 and indicate the corresponding changes. For Referee report 2, we have provided a detailed point-to-point response in our reply [id 3432], where we propose several changes to be carried out in the resubmission. Since the points [1]-[4] in the report 4 are exactly the same points than for report 2, we provide here the responses and modifications to the manuscript in response to the questions [1]-[4] as answer to both Referee report 2 and 4.

We hope that with the changes, the paper is now suitable for publication in SciPost Physics.

Sincerely yours, Hepeng Yao, Lorenzo Pizzino and Thierry Giamarchi

Answer to Referee report 4:

Referee's comment/question 1 :" In their response from 07.02.2023 the authors have not properly answered my comment from 18.01.2023 concerning the anisotropy of the superfluidity density and its impact upon the BKT transition. In the literature it is established that in an anisotropic situation in 2D the proper BKT transition criterion is given by the geometric mean of the two superfluid densities, see for instance PRL 113, 165304 (2014) and the references therein. Thus, to properly determine the BKT transition in the anisotropic system at hand, boils down to determine numerically both superfluid densities. A separate RG calculation, which would indeed be an independent project, is not necessary. I suggest that the authors look into this during the resubmission process and improve the manuscript accordingly. "

Answer: We thank the Referee for raising this interesting question. We also thank the Referee for pointing out the method for estimating of BKT temperature in anisotropic systems presented in PRL 113, 165304 (2014) and the reference therein (Refs. [59-62] in the resubmitted paper). Indeed, we can compute the superfluid fraction along the two directions and decide the BKT temperature based on the geometric mean of the two values. In the updated manuscript, we have computed the superfluid fraction in the two directions with the QMC calculations and estimated the corresponded BKT temperature T_{cross}^{BKT}, see green curve in Fig. 3(c). The inset plot shows one example of how this BKT temperature is determined. We found its value is always lower than T_{cross} and the difference gets larger at larger V_y. For the range 0<V_y/E_r<10, this difference should originate from the finite size effect for the T_{cross}. For V/Er>10, the quantum degeneracy parameter \tilde{D}s will hit zero already when the system enters the I-1D regime, instead of the crossover temperature to the thermal regime. This is due to the fact that we have both f_s^x=0 and f_s^y=0 in the thermal regime while f_s^x>=0 and f_s^y=0 in the I-1D regime (for the finite size system). Thus, we always find a much lower T^{BKT} in this regime which is even lower than the crossover line between C-1D and I-1D. It’s not clear whether these two curves should collapse in the thermodynamic limit, and we leave it as an open question in our current manuscript.

Moreover, as suggested in the first report of the Referee, we also look at the correlation function in Fig 4 (b) and (c). According to the BKT property of anisotropic superfluid, one may expect an algebraic decay g1~x^{-1/D} with D=\tilde{n_s}\lamT^2 and \tilde{n_s} the superfluid density under geometric average. However, we find from QMC a behavior characterized by two decay exponents. Therefore, we suggest further calculations such as a field theory approach to tackle this issue and get a better understanding of what happens in the large Vy regime.

In the revised manuscript, we have updated Fig. 4(c) and added the corresponding discussions in the last paragraph of “section 4 longitudinal superfluidity”. Since now we have computed the crossover temperature between 2D and TH regime with a better accuracy (green curve in Fig. 3(c)), we also update Fig. 1 with the new data. Moreover, we also add the discussions about its comparison with correlation functions in the second last paragraph of “section 5 correlation functions”.

Referee's comment/question 2 :" This would give the authors also the possibility to take into account the various suggestions of the other referees. In particular, it seems to be appropriate to clarify more precisely. 1) why the system studied here can be considered as "strongly interacting" 

Answer: We thank the Referee for this comment. Indeed, we think it’s also a good chance to make modifications concerning suggestions of the other Referees. Below, we list the modifications we made to the manuscript in answer to the listed points. This answers both the questions of Referee report 2 and 4.

For the judgement of strongly-interacting regime, we follow strictly the criteria identified in the review papers Refs. [5–7,55], namely \tilde{g}{2D}>=1 and \gamma>>1. This is the criteria widely used in the cold atom community, and strongly interacting properties beyond mean field have been found in these regimes, see for instance Refs. [34, 36, 54, 56]. For illustrating this point, we have provided the general statement on this in the main paper together with technical calculations in the appendix.

In the revised manuscript, we add more statement for the strongly interacting regime in the first paragraph of “section 3 phase diagram”. We feel the detailed calculations of \tilde{g} and \gamma is too technical, and we keep them in the appendix.

Referee's comment/question 3:" to which extend one reaches a 1D regime or, a wording which I would prefer more, a quasi-1D regime" 

Answer: We thank the Referee for pointing out this misleading point. The 1D regime is defined as the regime where the system can be incoherently coupled due to the interplay of temperature, tunneling and interactions. In this case, most of the properties of the system can be described by a purely 1D Hamiltonian. This is in fact how 1D systems are created in practice in various cold atom experiments, see for instance Refs [9, 15, 34, 36, 57]. Also, at the limit T=0, indeed the system will never reach the 1D regime since the critical V_y goes to infinity.

To make a better clarification on this point, we have made several updates in the paper. In the second paragraph of “section 3 phase diagram”, we now add “In I-1D regime, the tunneling is small enough that the system can be described by a purely 1D Hamiltonian \hat{H}=\hat{H}_{1D}. “ and “Notably, the tubes can be incoherently coupled due to the interplay of temperature, tunneling and interactions. This is in fact how 1D systems are created in practice in a various of cold atom experiments, see for instance Refs [9, 15, 34, 36, 57].” We also add in the second last paragraph of the same section “Notably, the scaling we found here indicates the critical Vy tends to infinity at T = 0. This suggests the system will always remain two-dimensional at the zero-temperature limit. “

Referee's comment/question 4 :"why the considered system sizes are considered to be sufficient from an experimental point of view, although from a theoretical point of view a more extensive finite-size scaling would have been preferable" 

Answer: We thank the Referee for pointing out this issue. Our calculation is limited by finite size due to the consideration of the continuous potential and strong interactions. On one hand, we benefit from the presence of the continuous potential which allows us to obtain physical properties beyond mean-field description and provides a description adapted to the cold atom experiments. On the other hand, a continuous potential can cause a very small resolution in the position space as well as very small imaginary time step in the PIMC algorithm, which makes the calculation limited to the current system sizes we considered here. Even if so, the system sizes we considered is large enough since they are at the typical size scale in actual cold atom experiments (10a - 100a), see for instance Refs. [34, 36, 57].

In the revised version of the paper, we now add in the first paragraph of section 3 “Here, we study the finite size effect for our continuous systems up to anisotropic sizes Lx, Ly =50a,10a and isotropic sizes Lx = Ly = 20a. We argue that the system sizes we considered is large enough since they are at the typical size scale in actual cold atom experiments (10a − 100a) [34, 36, 57]. “

Referee's comment/question 5:" the literature of dimensional crossover by comparing, for instance, the 1D-2D crossover also with the 1D-3D crossover: Phys. Rev. Lett. 113, 215301 (2014), Phys. Rev. Lett. 117, 235301 (2016), Phys. Rev. Lett. 130, 123401 (2023)" 

Answer: We thank the Referee for drawing our attention on these references concerning experimental outcome for 3D-1D dimensional crossover. We add statement related to these references as well as comparison between 2D-1D and 3D-1D dimensional crossover in 3 positions:

1. They are added as Refs. [9,15,18] in the second paragraph of introduction, where we listed examples of dimensional crossover systems.

2. The main difference between 3D-1D and 2D-1D crossover originates from the different properties between 3D and 2D system. Especially, 3D systems can have long-range correlation order at low temperature while 2D systems exhibit quasi-long-range order and BKT properties. Naturally, such differences lead to a strong impact during the crossover process. To further point out it, we add in the 4th paragraph of introduction by saying “In previous works, various quantum coherence properties have been studied for the 3D- 1D crossover [15, 18, 27, 29]. Here, we argue that this question is also particularly relevant for 2D bosonic systems which have been recently realized in a homogeneous box potential, where the BKT physics was clearly identified [6, 37–41]. ” And also, in the 3rd paragraph of section 5, we write “Different from the 3D case where the correlation holds as constant for low enough temperature, it exhibits a BKT type of decay”.

3. For the behavior of crossover temperature as a function of transverse tunneling T_c~t^{-\nu(K)}, the 3D-1D and 2D-1D crossover share the similar decay exponent \nu(K) according to the analysis in Refs. [27,29] and also linked with the experimental study in Ref. [15]. This exponent is further obtained for the 2D-1D case by our numerical data in Fig. 2(b). To better clarify this point, we now write in the second last paragraph of section 3 “For both 3D-1D and 2D-1D dimensional crossover, the mean field prediction indicates \nu_{MF} = 2K/(4K −1) ≃ 0.67 for the discrete model in the thermodynamic limit [15,27,29]. Remarkably, the exponent we found here is less than 10% different from this value.”

Answer to Referee report 2:

We thank the Referee for his/her work on our manuscript. We have provided a point by point answer to his/her comments in our reply [id 3432]. In the reply, we have mentioned several modifications to be carried out in the resubmission. Since the points [1]-[4] mentioned by Referee report 4 are based on the points raised by referee report 2, we provide in the response to the report 4 the modifications that are done in answer to both referee report 2 and referee report 4.

Additional changes: We realize the rescaling unit is missing in the vertical axis of Fig. 5. We have added it in the revised manuscript.

List of changes:
- In section 4 and 5, we add discussions about comparison with BKT properties of anisotropic superfluid
- In section 3, we add details about strongly interacting regime, finite system size and crossover between C-1D and I-1D regime
- In section 1, 3 and 5, we add discussions about comparison with 3D-1D dimensional crossover
- Figure 1, 3 and 5 are updated