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Timedependent variational Monte Carlo study of the dynamic structure factor for bosons in an optical lattice
by Mathias Gartner, Ferran Mazzanti, Robert E. Zillich
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Submission summary
Authors (as registered SciPost users):  Mathias Gartner · Ferran Mazzanti · Robert Zillich 
Submission information  

Preprint Link:  https://arxiv.org/abs/2202.07535v1 (pdf) 
Date submitted:  20220301 14:19 
Submitted by:  Gartner, Mathias 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We study the dynamics of a onedimensional Bose gas at unit filling in both shallow and deep optical lattices and obtain the dynamic structure factor $S(k,\omega)$ by monitoring the linear response to a weak probe pulse. In order to do that we introduce a new procedure, based on the timedependent variational Monte Carlo method (tVMC), which allows to evolve the system in real time, using as a variational model a JastrowFeenberg wave function that includes pair correlations. Comparison with exact diagonalization results of $S(k,\omega)$ obtained on a lattice in the BoseHubbard limit shows good agreement of the dispersion relation for sufficiently deep optical lattices, while for shallow lattices we observe the influence of higher Bloch bands. We also investigate nonlinear response and obtain the excitation spectrum, albeit broadened, by higher harmonic generation after a strong pulse with a single low wave number. As a remarkable feature of our simulations we demonstrate that the full excitation spectrum can be retrieved from the stochastic noise inherent in any Monte Carlo method, without applying an actual perturbation.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 202247 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2202.07535v1, delivered 20220407, doi: 10.21468/SciPost.Report.4885
Strengths
1. First calculation (to my knowledge) of the dynamical structure factor using tvmc
2. Ability to access the regime of nonlinear perturbation
Weaknesses
1. The strategy to compute S(q,omega) with "noise only" is not too well justified in the paper
Report
This is a very interesting work studying the time evolution of interacting bosons . The work is novel, especially on the technical level, and I strongly recommend publishing it once the points below have been addressed.
1. In order to understand the accuracy of the method, it might be useful to first report the accuracy on the ground state results in the absence of the external field. What is the accuracy compared to the Bethe ansatz solution of Lieb Liniger model?
2. While the form of the external potential is explicitly given, the authors do not specify how the delta interaction is implemented/ regularized in practice. Do the authors use an effective twobody potential or something similar?
3. The discussion on obtaining the S(q,omega) directly from the monte carlo noise is very interesting and suggestive. However, I find that the lack of mathematical analysis and justification somehow problematic, indeed the main issue I see is that the noise among different parameters is going to be correlated among different parameters, and the propagation of the noise in the ODE is also highly nonlinear and nontrivial. I am not sure this going to yield an unbiased estimator of S(q,omega). I would suggest to either rephrase the discussion to reflect this formal uncertainty or remove entirely this part.
4. (Optional) Would it be possible to compare the dynamics more quantitatively to what obtained with the Bose Hubbard model ? The authors report about qualitative similarities, but I believe it would be nice to see also a direct comparison with exact diagonalization results for the dynamics of the Bose Hubbard, also to understand how accurate is the tVMC in this specific case.
Other points
a. How is the matrix S regularized / is it regularized at all?
b. Since there is no specific length limitation here (I believe) it would be useful to add a few more details on the sampling procedure used / ODE solver etc , also in the interest of reproducibility
Report #2 by Anonymous (Referee 2) on 202246 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2202.07535v1, delivered 20220406, doi: 10.21468/SciPost.Report.4872
Strengths
1. Nice if somewhat limited extension of Carleo's timedependent variational Monte Carlo work on bosons in one dimension to allow calculations of the dynamic structure factor.
2. Interesting comparison of the effects of applying very small and very large timedependent perturbations, exploring the linear and highly nonlinear regimes and showing how the linear response can be seen even in the nonlinear regime.
3. Interesting use of the stochastic noise inherent in Monte Carlo simulations to calculate the dynamic structure factor without applying a timedependent perturbation.
Weaknesses
1. Little new physics. Advances reported are primarily technical.
2. Unnecessarily confusing definition of the wave vectors of the periodic applied potentials.
Report
This paper uses the timedependent variational quantum Monte Carlo method, introduced (I believe?) by Giuseppe Carleo, to calculate the dynamic structure factor of a system of bosons moving in a onedimensional sinusoidal potential and interacting via deltafunction contact interactions. For the structure factor calculations, an additional timedependent sinusoidal potential is applied. Although the model studied is idealized, it is not a bad description of bosonic atoms in a cigarshaped trap aligned with a onedimensional optical lattice. All results presented are for unit filling, in which there is one boson per potential well.
The model studied  the LiebLiniger model with a sinusoidal applied potential  can be solved exactly using the Bethe Ansatz when the applied potential is zero and reduces to the BoseHubbard model when the applied potential is very strong. It has been so well investigated that I doubt this paper tells us anything new about the physics it represents.
The methods used are similar to those used in Ref. [23]: G. Carleo, L. Cevolani, L. SanchezPalencia, and M. Holzmann, "Unitary Dynamics of Strongly Interacting Bose Gases with the TimeDependent Variational Monte Carlo Method in Continuous Space", Phys. Rev. X 7, 031026 (2017). Carleo and coworkers also studied the LiebLiniger model, although they did not calculate the dynamic structure factor or apply a sinusoidal potential. The current work improves on Carleo's approach by using splines to represent the one and twoparticle correlation functions, $f_1$ and $f_2$, and by the introduction of new methods to calculate the dynamic structure factor. The simulation cell contains only 20 bosons, which is probably too few to obtain results representative of the largesystem limit; Carleo looked at up to 100 bosons.
On the whole, although the number of new physics results reported in this manuscript is limited, the structure factor calculations are new and technically interesting, especially the comparison of results obtained in the linear response (small perturbation) and highly nonlinear regimes. The paper is quite well written and I enjoyed reading it. I think it deserves publication without major changes.
Requested changes
1. For reasons I don't understand, you choose to write the lattice potential as\[V_0 \sin^2(k_L x) = (V_0/2) (1  \cos(2 k_L x)),\]so the wave vector of the applied potential is $2k_L$, not $k_L$ as readers will expect. This explains, for example, why the spectra in Fig. 2 are plotted over the domain $0 \leq k/k_L < 2$ instead of $0 \leq k/k_L < 1$. The same confusing use of $\sin^2$ functions appears in the definition of the timedependent perturbation. This small issue of notation confused me repeatedly as I read the manuscript and made it significantly harder to follow than it should have been. Since the constant shift in the $1  \cos(2 k_L x)$ factor is of no importance, why not make your readers' lives easier by writing the applied potential as $V_0 \cos(k_L x)$ or $V_0 \sin(k_L x)$ and redefining $V_0$ and $k_L$ accordingly? If you would rather make fewer changes, you could write the potential as $V_0 \sin^2 (\frac{k_L x}{2})$.
2. On p4., you discuss the interesting observation that you can use the stochastic noise inherent in the Monte Carlo simulation to calculate the dynamic structure factor without applying a periodic perturbation. (Is this related to the fluctuationdissipation theorem?) Have you thought about what would happen in the limit in which the trial wavefunction is exact, so that the local energy $\mathcal{E}(\mathbf{x})$ is a constant, equal to the groundstate energy regardless of the boson position vector $\mathbf{x}$. According to Eq. (2), the coefficients of the wavefunction then cease to evolve in time, which makes sense as you already have the exact ground state and are not applying a timedependent perturbation. It also implies, however, that the fluctuations you are using to see the structure factor have disappeared. I suspect that your "no perturbation" method only works because your wave function is **not** very accurate, and that it will become harder and harder to use as your trial wavefunction improves. If that is correct, the noperturbation approach is an interesting curiosity but may not be as useful as you hope. This issue may merit some discussion.
3. On p2, the lead up to Eq.(3) says: "The correlation functions $f_1$ and $f_2$ depend on timedependent complex variational parameters .... We can thus write (2) as ..." Strictly, I think Eq.(3) requires more than this: it relies on the fact that the **logarithms** of $f_1$ and $f_2$ depend **linearly** on the timedependent variational parameters, so the current wording is misleading. Please amend.
4. The expression for the local energy $\mathcal{E}$ given on the penultimate line of p2 is wrong: it implies that $\mathcal{E}$ is a number, but it is actually a function of the many boson position vector $\mathbf{x}$. Please fix this.
Author: Mathias Gartner on 20220603 [id 2554]
(in reply to Report 2 on 20220406)
We thank the referee for reading the manuscript and his/her report. In the following we address the questions and comments raised:

We understand that these could indeed be a bit confusing. The reason for writing the potential in this form is the usual convention in cold atomic systems where the optical lattice potential is generated by counter propagating laser beams with wave vector $k_L$, which results in the lattice of the given form. Basically, we follow the literature (eg. Bloch et al. 10.1103/RevModPhys.80.885, Greiner et al. 10.1038/415039a) and books (eg. Lewenstein et al. “Ultracold Atoms in Optical Lattices”). We therefore want to keep this notation. However, we understand the possible confusion this may lead to, and have added a sentence explaining its origin in the revised version.

The referee raises valid questions about obtaining information on the excitations by relying only on Monte Carlo noise. If we would sample from the exact trial wavefunction, the local energy would be the exact ground state energy and have no variance. Then the right hand side of the differential equation (6) for the variational parameters would vanish (while $S_{KK’}$ would still fluctuate). Since $S_{KK’}$ is invertible, all $\dot\alpha_K$ vanish, hence indeed all $\alpha_K$ stay constant exactly (in contrast to approximate trial functions where $\alpha_K$ keeps on fluctuating slightly during the time evolution). When we sample the density from this exact ground state, the stochastic noise of the density will be purely due to the Metropolis sampling, and particularly will be uncorrelated in time. We have tried this for noninteracting particles where the exact wave function is known, and indeed we do not see the single particle dispersion in the power spectrum of the density fluctuation. So the referee is completely right in that when working with the exact ground state one cannot use the MC noise to get the excitation spectrum. We note however, that for most manybody problems the exact ground state is beyond reach. We have improved the trial wave function by using more parameters $\alpha_K$ (and still using the Jastrow ansatz) and find that the reduced variance of the local energy indeed lowers the signaltonoise ratio for the excitation peaks in the density fluctuation power spectrum, however, the peak positions do not change. If on the other hand we used more samples, the whole power spectrum is scaled down but the signal to noise ratio is not affected by sample size. We have added a comprehensive discussion in the manuscript in the second paragraph of the new section 3.3.

To give a more clear statement about the choice of the trial wavefunction we write the Jastrow ansatz as exponentials of correlation functions  which is still a completely general ansatz. Furthermore we extend the paragraph where we link the parametrized wavefunction to this ansatz.

Thank you for pointing this out. We fixed it.
Report #1 by Anonymous (Referee 1) on 2022330 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2202.07535v1, delivered 20220330, doi: 10.21468/SciPost.Report.4823
Report
In the submitted manuscript, Gartner et al. discuss the applicability of the timedependent variational Monte Carlo (tVMC) method to calculate the dynamic structure factor S(k,w) of strongly interacting bosons in continuous space in both linear and nonlinear regime, and in absence of perturbation.
The paper is well written but the discussion of the results is a bit brief. However, I may give a positive recommendation for a properly revised manuscript.
Requested changes
1) In the abstract, introduction and conclusions, authors claim to compute the dynamic structure factor S(k,w) with the tVMC method. However, when they discuss the nonlinear and in absence of perturbation cases, they compute and show the space and time Fourier transform of the density $\delta\tilde{\rho}(k,w)$ only. I feel this is misleading. They should be clearer in both the introduction and the abstract.
2) In Fig.2 a color legend should be added in order to allow the reader to compare the results with Ref.[35].
Authors should explain while, as claimed in the manuscript, "the shape of S(k,w), however, differs somewhat from the results in [35], where a more complex structure was obtained for N=16".
They also claim that "the broadening of the tVMC dispersion, as well as the ringing oscillations, are artifacts resulting from the Fourier transform over a finite simulation time window of length T=10t_0". Could they increase the simulation time in order to remove such effects or there are compulational limitation?
3) As the authors are presenting a new way to calculate the dynamic structure factor, I feel like they do not make clear the potential of their approach, in particular compared to exact diagonalization method (Ref.[35]), in terms of accuracy, limit on the size of the system and computation time.
4) Authors claim that "methods that allow for time dependent simulations of strongly correlated manybody systems which can describe the linear, but also nonlinear response to perturbation, are in demand". I agree with them, for this reason I feel like the more interesting part, the non linear regime, is too rushed and should be extended in order to make the manuscript more appealing.
Author: Mathias Gartner on 20220603 [id 2553]
(in reply to Report 1 on 20220330)
We thank the referee for reading the manuscript and his/her report. In the following we address the questions and comments raised:
1) We agree with the referee that this was indeed a bit confusing. The dynamic structure factor $S(k,\omega)$ is the ratio of the density fluctuation $\delta \tilde \rho(k, \omega)$ to the perturbation $\delta \tilde V_p(k, \omega)$ causing it, in the linear regime. In the case of the noisegenerated signal, the Monte Carlo noise itself, which is not given as external perturbation, is the source of density fluctuations. Therefore, for comparison with the results from weak and strong pulses, we choose to present $\delta \tilde \rho(k, \omega)$, i.e. the square root of the power spectrum of the density fluctuations (which we call that way in the revised manuscript). For the strong pulses, we can still calculate $\operatorname{Im} \left[ \delta\tilde\rho(k, \omega) / \delta \tilde V_p(k, \omega) \right]$, but it would be misleading as the system would no longer be in the linear regime where different modes contribute independently. For example a strong pulse generates higher harmonics of $k$, without actually perturbing with higher $k$. In order to clarify this point, in the introduction, we now introduce the 3 cases (linear, nonlinear, no perturbation) and explain the choice of $\delta \tilde \rho(k, \omega)$ or $S(k,\omega)$, and also changed the abstract.
2) We added a color legend in Fig.2, however, a direct comparison with Ref.[35] (now [39]) will still not be possible due to the lack of such a color legend therein. The differences in $S(k,\omega)$ can be attributed to the fact that we use a variational method, a slightly different number of particles and a continuous Hamiltonian (in contrast to the BHM model of [35], now [39]). We add a more extensive discussion of this in the revised version. Regarding the second question,we have chosen a total simulation time of $T=10 t_0$ because for this time window we get enough energy resolution to see the relevant features present in the dynamic structure factor. There is no a priori technical restriction on increasing the simulation time. In order to ensure energy conservation for very long simulation times, a small propagation time step $\delta t$ is required, which additionally increases the computational workload. Increasing the total time $T$ reduces the strength of the artificial oscillations caused by the cutoff at $T=10 t_0$, but the cost associated with preserving the accuracy of the calculation would increase considerably. We added a sentence mentioning this explicitly.
3) The tVMC method is both variational and a Monte Carlo technique. Being variational, the accuracy of the prediction is ultimately limited by the accuracy of the wave function employed, which is determined by its functional form and by the actual number of parameters in the model. The accuracy of exact diagonalization methods are not limited by a model, but they are limited by the number of basis vectors employed, which increases exponentially with the number of degrees of freedom. That drastically limits the dimensionality, type of Hamiltonian, and number of particles that can be treated reliably, e.g. a lattice Hamiltonian and $N=16$ particles in case of Ref.[35], now [39]. Monte Carlo methods do not have these severe limitations, even less if they are variational as in tVMC. Monte Carlo methods are known to scale well with both the number of particles and the dimensionality of space, and are efficient for both discrete and continuous Hamiltonians. Our implementation of the tVMC method is readily portable to 2D and 3D geometries. We are currently working on that.
4) We agree with the referee that nonlinear dynamics is a very interesting point that deserves a deeper treatment. The number of possibilities for nonlinear perturbation as well as the aspects to study is large. We have already performed and are currently performing further simulations of the nonlinear response of bosons in optical lattices, among other systems. The technical challenges are similar to the linear case in that we need to make sure that all numerical parameters are under control (sample size and sample equilibration, time step, number of parameters for our Jastrow ansatz, simulation length), however the interpretation and assessment of the results is harder because nonequilibrium manybody dynamics is much less explored than linear response. The preliminary results are promising and look very interesting, but have to stand up against the mentioned numerical checks, which requires plenty of computer time, and also have to be critically assessed regarding the limitations of the variational ansatz. This is why it is prudent to put more work into the nonlinear studies and eventually devote a separate publication to nonlinear manybody dynamics in optical lattices. In the present paper, we therefore present only one nonlinear result, where we have afforded all the numerical checks.
Author: Mathias Gartner on 20220603 [id 2555]
(in reply to Report 3 on 20220407)We thank the referee for reading the manuscript and his/her report. In the following we address the questions and comments raised:
The variational ground state energy of the Lieb Liniger model compares very well with Bethe ansatz calculations. We benchmarked our results to the energy fit given in the publication by Lang et al. (10.21468/SciPostPhys.3.1.003), equation (10), and obtained a relative energy error for our variational wavefunctions of less than 0.4 percent for the interaction strengths that are relevant in this work. We include this check in the revised manuscript.
We briefly mentioned in section 2.1 that we incorporate the contact interaction as a boundary condition on the wavefunction. To clarify this, we expand this paragraph by giving the required boundary condition and mention that the variational parameters are restricted in such a way that this condition is always fulfilled. In particular we use $\frac{1}{\Phi} \frac{\partial}{\partial x_i}\Phi = \frac{1}{4} k_L g$.
Our results show that the timedependent density correlations due to noise provide the same location for peaks associated with excitation as does $S(k,\omega)$. This may not be surprising, because the random fluctuations on top of the variational ground state are just a superposition of manybody exciations. But we do not claim that we get the whole $S(k,\omega)$ only from noisy ground state propagation in tVMC: to calculate $\operatorname{Im} \left[ \delta\tilde\rho(k, \omega) / \delta \tilde V_p(k, \omega) \right]$, we would need $\delta \tilde V_p(k,\omega)$ which we cannot quantify if we use Monte Carlo noise. For this reason we show the power spectrum $\delta\tilde\rho(k, \omega)$ of the density fluctuations, which has excitation peaks where $S(k,\omega)$ has peaks. Also in reply to a question of referee 2 about the noise method, we discuss this much more comprehensively in the new section 3.3. In particular we state explicitly that for calculating $S(k,\omega)$ with all the correct spectral weights rather than just peak locations, we use the linear response method.
The dynamic structure function $S(k,\omega)$ for the Bose Hubbard model can be found in Roux et al (previously [35], now [39]). We use the same $U/J$ ratio to allow the comparison between our variational result on a continuous optical lattice and the presumably exact BHM result in [39] obtained with exact diagonalization, albeit with smaller $N=16$ than our $N=20$. As discussed in the manuscript, the overall shape of $S(k,\omega)$ and resulting excitation energies in [39] are essentially the same as our result, but the former is more structured, with multiple close peaks for $k$ around the edge of the Brioullin zone, compared to our single peak in $S(k,\omega)$. This may be due to the limitation of our Jastrow ansatz, but also due to different particle numbers. Also note that [39] accounts for only the lowest band.
Other points: a. We precondition and regularize the matrix $S$ by scaling the entries according to ${S_{KK'}' = S_{KK'}/\sqrt{S_{KK}S_{K'K'}}}$ and adding a small value $\varepsilon = 10^{4}$ to the diagonal. b. We use QR decomposition for solving the equation system (6) and a fourth order Runge Kutta scheme to propagate the ODE in time. All the expectation values are estimated by means of Monte Carlo integration where we use the MetropolisHastings algorithm for generating uncorrelated samples. To provide this information in the revised submission, we have added a new section 2.2 where we give further details on the numerical methods.