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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
Submission summary
| Authors (as registered SciPost users): | Morteza Nattagh Najafi · Mohammad Ali Rajabpour |
| Submission information | |
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| Preprint Link: | scipost_202410_00035v2 (pdf) |
| Date accepted: | July 10, 2025 |
| Date submitted: | July 5, 2025, 5:43 p.m. |
| Submitted by: | Morteza Nattagh Najafi |
| Submitted to: | SciPost Physics Core |
| 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 comments upon resubmission
Comments of the reviewer:
[1] It might be worth stating right after (1) that the SPPM for n=1 is simply related to the characteristic polynomial.
We thank the reviewer for comments. Although it has been mentioned in Eq. 22c, we state it also right after Eq. 1.
[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.
We changed it to “… we have managed to calculate this quantity analytically at the mean field level for the first time”.
[3] Page 4, top left. The sentence ending with 'those sites that there is a fermion' reads awkwardly.
We corrected this sentence.
[4] Page 5, top left: 'DPP can be defined easily using the concept of formation probabilites'. For quadratic fermionic Hamiltonians.
Corrected.
[5] Page 6, left 'that we also classify them' remove them.
Corrected.
[6] The notation in equation (22) is not consistent for n=1 with that in (21), which is confusing.
The notation is correct; please note that in Eq. (22) we write M^(n=1)(A) for a given matrix A. In Eq. (21) we have M^(n=2)(B) for a matrix B which comes from the Hubbard problem.
[7] I find the discussion in section VII. A to be a bit confusing. Where exactly is the analytical solution for large L SPPM?
The exact analytic expression for the partition function G(n,μ) (in the thermodynamic limit) is provided in Eq. (154). Its generalization G(n,μ,λl) with Lagrange multipliers λl is given in Eq. (161). The partition function is enough to obtain for instance the average energy and entropy in Eqs. (156) and (157).
A sentence is added to the beginning of this sub-section.
[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.
In the conclusion, we incorporated an additional sentence and reference, explicitly noting both the significance of this problem and our ongoing efforts to resolve the computational methodology.
List of changes
1- We processed all the comments given by the referee, highlighted in blue. 2- In the SEC. VIIA we added a description for the analytical results, as requested by the referee.
Published as SciPost Phys. Core 8, 051 (2025)