A field theory representation of sum of powers of principal minors and physical applications
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...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SciPost
2025-08-01
|
| Series: | SciPost Physics Core |
| Online Access: | https://scipost.org/SciPostPhysCore.8.3.051 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849389926787842048 |
|---|---|
| author | Morteza Nattagh Najafi, Abolfazl Ramezanpour, Mohammad Ali Rajabpour |
| author_facet | Morteza Nattagh Najafi, Abolfazl Ramezanpour, Mohammad Ali Rajabpour |
| author_sort | Morteza Nattagh Najafi, Abolfazl Ramezanpour, Mohammad Ali Rajabpour |
| collection | DOAJ |
| description | 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-Rényi 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. |
| format | Article |
| id | doaj-art-8eae4dd4e2d14deca57b976a7cb78e6e |
| institution | Kabale University |
| issn | 2666-9366 |
| language | English |
| publishDate | 2025-08-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics Core |
| spelling | doaj-art-8eae4dd4e2d14deca57b976a7cb78e6e2025-08-20T03:41:49ZengSciPostSciPost Physics Core2666-93662025-08-018305110.21468/SciPostPhysCore.8.3.051A field theory representation of sum of powers of principal minors and physical applicationsMorteza Nattagh Najafi, Abolfazl Ramezanpour, Mohammad Ali RajabpourWe 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-Rényi 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.https://scipost.org/SciPostPhysCore.8.3.051 |
| spellingShingle | Morteza Nattagh Najafi, Abolfazl Ramezanpour, Mohammad Ali Rajabpour A field theory representation of sum of powers of principal minors and physical applications SciPost Physics Core |
| title | A field theory representation of sum of powers of principal minors and physical applications |
| title_full | A field theory representation of sum of powers of principal minors and physical applications |
| title_fullStr | A field theory representation of sum of powers of principal minors and physical applications |
| title_full_unstemmed | A field theory representation of sum of powers of principal minors and physical applications |
| title_short | A field theory representation of sum of powers of principal minors and physical applications |
| title_sort | field theory representation of sum of powers of principal minors and physical applications |
| url | https://scipost.org/SciPostPhysCore.8.3.051 |
| work_keys_str_mv | AT mortezanattaghnajafiabolfazlramezanpourmohammadalirajabpour afieldtheoryrepresentationofsumofpowersofprincipalminorsandphysicalapplications AT mortezanattaghnajafiabolfazlramezanpourmohammadalirajabpour fieldtheoryrepresentationofsumofpowersofprincipalminorsandphysicalapplications |