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Polaron formation as the vertex function problem: From Dyck's paths to self-energy Feynman diagrams
by Tomislav Miškić, Juraj Krsnik, Andrey S. Mishchenko, Osor S. Barišić
Submission summary
| Authors (as registered SciPost users): | Juraj Krsnik · Tomislav Miškić |
| Submission information | |
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| Preprint Link: | scipost_202505_00059v1 (pdf) |
| Date submitted: | May 27, 2025, 2:03 p.m. |
| Submitted by: | Miškić, Tomislav |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We present a novel iterative method for generating all self-energy Feynman diagrams of any given order for the single polaron problem. This approach offers an effective tool for circumventing the sign problem that often arises in approximation-free numerical summations of Feynman diagrams. Each iterative step begins by rigorously listing all noncrossing diagrams using the graphical Dyck path representation of Stieltjes-Rogers polynomials, which exactly encode the Feynman diagram series. In the second phase, the Ward-Takahashi identity is used to uniquely identify the complete subset of vertex function contributions from the self-energy diagrams obtained in the previous iterative step. Finally, the noncrossing diagrams and vertex function contributions are combined to construct the full set of Feynman diagrams at a given order of the diagrammatic expansion, determining the number of diagrams of various types. This approach establishes a systematic procedure for generating the total sum of diagrams in a given order, enabling significant sign cancellation and making it broadly suitable for numerical summation techniques involving Feynman diagrams.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
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Strengths
1- The use of continued fractions and Dyck paths introduces a potentially elegant organizational principle for self-energy diagrams. *2-The authors provide a combinatorial framework that could in principle be useful for algorithmic generation of diagrammatic contributions.
Weaknesses
1- The scope is restricted to the single-polaron case, with no clear path to generalization. 2- The relevance of the method to the sign problem is speculative and unsupported by evidence. 3- Lack of benchmarking against established DMC methods in the same context. 4-Several technical and presentational issues that hinder clarity and accessibility.
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