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Lecture notes on Diagrammatic Monte Carlo for the Fröhlich polaron

by Jonas Greitemann, Lode Pollet

This Submission thread is now published as

Submission summary

As Contributors: Jonas Greitemann
Arxiv Link: (pdf)
Date accepted: 2018-03-19
Date submitted: 2018-03-05 01:00
Submitted by: Greitemann, Jonas
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


These notes are intended as a detailed discussion on how to implement the diagrammatic Monte Carlo method for a physical system which is technically simple and where it works extremely well, namely the Fröhlich polaron problem. Sampling schemes for the Green function as well as the self-energy in the bare and skeleton (bold) expansion are disclosed in full detail. We discuss the Monte Carlo updates, possible implementations in terms of common data structures, as well as techniques on how to perform the Fourier transforms for functions with discontinuities. Control over the variety of parameters, especially in the bold scheme, is demonstrated. Sample codes are made available online along with extensive documentation. Towards the end, we discuss various extensions of the method and their applications. After working through these notes, the reader will be well equipped to explore the richness of the diagrammatic Monte Carlo method for quantum many-body systems.

Published as SciPost Phys. Lect. Notes 2 (2018)

Author comments upon resubmission

Dear Editor,

we would like to thank you for the communicating the referee reports and are grateful for having received such positive and detailed responses. We have carefully considered the referees' comments and would hereby like to submit a revised version of our manuscript. Below, we give our views on some of the points raised in the reports:

First referee:

Describing the slicing procedure (Fig.3) it would be of help for a reader to comment the method w.r.t the usual fixed size slicing (trotterization).

We don't really perform a "slicing procedure". We make it quite clear that this is a continuous time method. Discussing the absence of a systematic error from a non-existent discretization procedure seems rather pointless. Further, continuous time methods are quite prevalent these days, exactly for these advantages. We believe today Trotter-based methods are by no means the standard "gateway method" they used to be. Alluding to these methods this early in might give the reader the wrong idea. Thus, we respectfully disagree with the referee in this matter on pedagogical grounds.

Related to the discussion in sect. 3.4 the statistics of the diagram order should be improved according to my opinion. In both sections 2 and 3 a discussion on the sampling of the order of perturbation theory will be useful to the reader. In the datasets produced for the two level system (sect. 2) seems not to provide this information. Could the authors provide this information?

We believe we discuss the statistics of the diagram orders for the two-level system in Secs. 2.5 and 3.6. In the former, we give details on the percentages of the time spent in zeroth and higher orders; in the latter, Fig. 11 shows a histogram of the bare expansion orders.

In Eq. (25) but also in previous part of the paper the Weights, which are proportional to odd number of electron propagators are negative, perhaps a comment on this fact here or in the previous section should be needed.

We corrected this in the updated manuscript. The Green function and self-energy, and indeed all of the contributing diagrams, are strictly negative quantities which enables us to sample them without a sign problem by defining their MC weights with the opposite sign. This led us to "forget" about the physical minus sign in many places and indeed this minus sign does not appear in our code. However, for the benefit of readers unfamiliar with diagrammatic Monte Carlo, we agree that we should keep the notation in these notes consistent with the usual conventions found in many-body physics literature.

I also suggest the authors to supplement the paper with a list of exercises or propose exercises whenever possible. In the present of the paper on page 23,25 and 31 there are suggestions for possible exercises. I think it would be useful to regroup it on a devoted section or emphasize better in the text.

Perhaps a list of suggested exercises based on the codes (which are now proposed in the text) should be added at the end of the manuscript.

After careful consideration, we decided against amending the current manuscript with a dedicated section for exercises. The "exercises" mentioned in the text currently only concern technicalities and are mostly already "solved" in the code and would be of little pedagogical value to actually do. We considered developing a set of more meaningful exercises but see this outside the scope of this manuscript. In some sense, instructions to reproduce most of the figures are supplied as READMEs along with the code. We'd like to keep it that way, for ease of keeping them up-to-date with the corresponding source code. Smaller, pedagogical exercises for use in a workshop or school hands-on session would require additional effort to be of real use.

Second referee:

p.8 section 2.4: I fail to see why the second estimator is not really better than the first one when done "on the fly" after each update. Could the authors provide some insight?

Updating the integrated quantity "on-the-fly" is an O(1) operation, but one that has to be done after every single update. Since the number of elementary updates to achieve an independent configuration scales at least with the simulation "volume" beta, updating the integral on-the-fly is not cheaper than calculating it explicitly upon measurement. All in all, the correlations in imaginary time are typically strong enough to make the second estimator converge only marginally faster than the first, and certainly not fast enough to make it worth the computational effort. This is true also of other MC methods with an efficient update scheme like SSE where one only considers the full "time" information in the measurements if the observable is "dynamic". In the latter case, one has to be very careful to avoid the measurement dominating the computational effort.

p. 22. Why do you introduce formulas at finite temperature (Eqs. (31) and (32)). I think this might be confusing. It was mentioned earlier that calculations are done at zero temperature. Moreover, is β the same as τmax? I understand you want to introduce an equidistant grid so that FFT can be used subsequently, but this does not require to introduce a finite temperature all of a sudden.

Indeed, beta == tau_max. We are at zero temperature, the beta is merely introduced to allude to the commonly used well-known definition of the Matsubara frequencies. We have added a remark to that effect.

p. 25. I find the figure 16 quite confusing. There is E on the y-axis, on the x-axis and in the legend. I would like to see a more clear version of this figure.

We hope to have cleared up any confusion regarding Fig. 16 by labeling the abscissa E' and the lines with E'=E and E'=Phi(E), respectively.

p. 33 section 7: Please explain the sentence "In practice, bold DiagMC schemes rely on the Gˆ2W scheme."

This sentence summarizes the paragraph in the language of Ref. 34, namely that bold DiagMC as it has been introduced in these notes is based on a resummation of the self-energy and in principle also of the irreducible polarization which corresponds to an expansion in z=iG^2W, W being the screened interaction. As stated just prior, the latter does not play any role in the polaron problem because the infinite bath is not renormalized by a single particle.

The following paragraph then discusses the possibility of basing bold DiagMC on the G^2W Gamma^2 expansion of Ref. 34, which is impractical because of the numerical problems in treating a 3-point vertex as opposed to 2-point self-energy and Green function.

List of changes

* In the abstract, more aspects and extensions that are addressed in these notes are advertised as such.
* Sec. 1: more accurate citation of Ref. 11, as pointed out by referee #2
* Sec. 2.3: Stressed that W(X) and W(Y) refer to the weights of only the segment of the diagram that is being altered in the update as pointed out by referee #2.
* Sec. 3 (intro): Explained basis states for bra-ket notation and motivate path integral formalism, as requested by referee #2.
* Sec. 3.1: Derivation of asymptotic form of the imaginary time Green function as requested by referee #1. The equations for E_k and Z_k are also derived and referred to by Eq. number. A mistake in the equation for Z_k, pointed out by referee #2, as been corrected. The limits of the approximation, implicit in the derivation, are now discussed.
* Sec. 3.2: Clarified sentence concerning normalization of Green function and other quantities, as mentioned by referee #2.
* Included minus sign in the Monte Carlo weight of the diagram and its constituents. As correctly pointed out by referee #1, the Green function (and self-energy) are strictly negative quantities.
* Sec. 4.1: Added comment on the fictitious nature of beta in the Fourier transforms.
* Sec. 5.1: Made the notation more precise: the self-energy is viewed as a functional of the Green function. The split into bare Green function and corrections (and their corresponding self-energy contributions) is more explicit now (written as a proper equality) and the primed self-energy (contribution due to delta G) is defined as such.
* Sec. 5.3: Occurrences of "physical" / "unphysical" have been replaced with "two-particle irreducible" / "reducible" which is more precise and is what is meant in this context, as pointed out by referee #2.
* Sec. 8.3: more precise statement on tunable scattering lengths in cold atom experiments
* Various typos and other details
* Chose thicker line width and more distinguishable line colors in most plots.
* Relabeled the abscissa in Fig. 16 as E'.

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