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A field theory representation of sum of powers of principal minors and physical applications
by Morteza Nattagh Najafi, Abolfazl Ramezanpour, Mohammad Ali Rajabpour
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Morteza Nattagh Najafi · Mohammad Ali Rajabpour |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202410_00035v1 (pdf) |
| Date submitted: | Oct. 15, 2024, 11:02 a.m. |
| Submitted by: | Nattagh Najafi, Morteza |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We introduce a novel field theory representation for the Sum of Powers of Principal Minors (SPPM), a mathematical construct with profound implications in quantum mechanics and statistical physics. We begin by establishing a Berezin integral formulation of the SPPM problem, showcasing its versatility through various symmetries including $SU(n)$, its subgroups, and particle-hole symmetry. This representation not only facilitates new analytical approaches but also offers deeper insights into the symmetries of complex quantum systems. For instance, it enables the representation of the Hubbard model's partition function in terms of the SPPM problem. We further develop three mean field techniques to approximate SPPM, each providing unique perspectives and utilities: the first method focuses on the evolution of symmetries post-mean field approximation, the second, based on the bosonic representation, enhances our understanding of the stability of mean field results, and the third employs a variational approach to establish a lower bound for SPPM. These methods converge to identical consistency relations and values for SPPM, illustrating their robustness. The practical applications of our theoretical advancements are demonstrated through two compelling case studies. First, we exactly solve the SPPM problem for the Laplacian matrix of a chain, a symmetric tridiagonal matrix, allowing for precise benchmarking of mean-field theory results. Second, we present the first analytical calculation of the Shannon-R\'enyi entropy for the transverse field Ising chain, revealing critical insights into phase transitions and symmetry breaking in the ferromagnetic phase. This work not only bridges theoretical gaps in understanding principal minors within quantum systems but also sets the stage for future explorations in more complex quantum and statistical physics models.
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
Current status:
Reports on this Submission
Strengths
1- Thorough theoretical examination of sums of powers of principal minors, with motivations in quantum lattice models 2- Mean field approximation gives good results
Weaknesses
1- No exact results except for the discrete Laplacian
Report
The main achievement is a Grassman integral representation for the SPPM, to which a mean field treatment can be applied, leading to reasonable approximations for large matrix sizes (long chains in the quantum context). This approximation correctly predicts the well known quantum phase transition occurring in the transverse field Ising model.
Overall the paper is quite technical and long, with also many appendices. Most computations are clearly presented. The results obtained by the authors also represent a small but worthy progress on an otherwise difficult problem. For this reason I recommend publication in Scipost Physics, provided the following questions are addressed.
Requested changes
1- It might be worth stating right after (1) that the SPPM for n=1 is simply related to the characteristic polynomial.
2- Page 3, second paragraph: 'We have managed to calculate this quantity analytically'. Technically this holds only at mean field level, so it is not an analytical formula for the entropy.
3- Page 4, top left. The sentence ending with 'those sites that there is a fermion' reads awkwardly.
4- Page 5, top left: 'DPP can be defined easily using the concept of formation probabilites'. For quadratic fermionic Hamiltonians.
5- Page 6, left 'that we also classify them' remove them.
6- The notation in equation (22) is not consistent for n=1 with that in (21), which is confusing.
7- I find the discussion in section VII. A to be a bit confusing. Where exactly is the analytical solution for large $L$ SPPM?
8- In particle conserving quadratic fermionic Hamiltonians, the ground state has a fixed number of particles. In the formalism of the present paper, this would mean summing over all minors of a given size. Could the author comment on this case.
Recommendation
Ask for minor revision