Motivated by the study of Polyakov lines in gauge theories, Hanada and Watanabe Hanada and Watanabe [1] recently presented a conjectured formula for the distribution of eigenphases of Haar-distributed random SU(N) matrices (β = 2), supported by explicit examples at small N and by numerical samplings at larger N. In this letter, I spell out a concise proof of their formula, and present its orthogonal and symplectic counterparts, i.e. the eigenphase distributions of Haar-random unimodular symmetric (β = 1) and selfdual (β = 4) unitary matrices parametrizing SU(N)/SO(N) and SU(2N)/Sp(2N), respectively.
{"title":"Eigenphase distributions of unimodular circular ensembles","authors":"Shinsuke Nishigaki","doi":"10.1093/ptep/ptae018","DOIUrl":"https://doi.org/10.1093/ptep/ptae018","url":null,"abstract":"Motivated by the study of Polyakov lines in gauge theories, Hanada and Watanabe Hanada and Watanabe [1] recently presented a conjectured formula for the distribution of eigenphases of Haar-distributed random SU(N) matrices (β = 2), supported by explicit examples at small N and by numerical samplings at larger N. In this letter, I spell out a concise proof of their formula, and present its orthogonal and symplectic counterparts, i.e. the eigenphase distributions of Haar-random unimodular symmetric (β = 1) and selfdual (β = 4) unitary matrices parametrizing SU(N)/SO(N) and SU(2N)/Sp(2N), respectively.","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"39 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139768985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The relation of the pole and running heavy quark masses of order O $(alpha _s^4)$ in perturbative quantum chromodynamics (pQCD) can be obtained using the Principle of Maximum Conformality (PMC), a formalism that provides a rigorous method for eliminating renormalization scale and scheme ambiguities for observables in pQCD. Using PMC, an optimal renormalization scale for the heavy quark mass ratio is determined, independent of the renormalization scale and scheme up to order $alpha _s^4$. Precise values are then obtained for the PMC pole masses of the heavy quarks $M_b^{text{PMC}}=4.86^{+0.03}_{-0.02}$ GeV, $M_t^{text{PMC}}=172.3pm 0.6$ GeV, and the running mass $overline{m}_t^{text{PMC}}=162.6pm 0.7$ GeV at the PMC scale.
{"title":"Relation between pole and running masses of heavy quarks using the principle of maximum conformality","authors":"Daniel Salinas-Arizmendi, Iván Schmidt","doi":"10.1093/ptep/ptae020","DOIUrl":"https://doi.org/10.1093/ptep/ptae020","url":null,"abstract":"The relation of the pole and running heavy quark masses of order O $(alpha _s^4)$ in perturbative quantum chromodynamics (pQCD) can be obtained using the Principle of Maximum Conformality (PMC), a formalism that provides a rigorous method for eliminating renormalization scale and scheme ambiguities for observables in pQCD. Using PMC, an optimal renormalization scale for the heavy quark mass ratio is determined, independent of the renormalization scale and scheme up to order $alpha _s^4$. Precise values are then obtained for the PMC pole masses of the heavy quarks $M_b^{text{PMC}}=4.86^{+0.03}_{-0.02}$ GeV, $M_t^{text{PMC}}=172.3pm 0.6$ GeV, and the running mass $overline{m}_t^{text{PMC}}=162.6pm 0.7$ GeV at the PMC scale.","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"4 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139768983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G Amelino-Camelia, V D’Esposito, G Fabiano, D Frattulillo, P A Höhn, F Mercati
Quantum gravity is expected to introduce quantum aspects into the description of reference frames. Here we begin exploring how quantum gravity induced deformations of classical symmetries could modify the transformation laws among reference frames in an effective regime. We invoke the quantum group SUq(2) as a description of deformed spatial rotations and interpret states of a representation of its algebra as describing the relative orientation between two reference frames. This leads to a quantization of one of the Euler angles and to an aspect of agency dependence: space is reconstructed as a collection of fuzzy points, exclusive to each agent, which depends on their choice of reference frame. Each agent can choose only one direction in which points can be sharp, while points in all other directions become fuzzy in a way that depends on this choice. Two agents making different choices will thus observe the same points with different degrees of fuzziness.
{"title":"Quantum euler angles and agency-dependent spacetime","authors":"G Amelino-Camelia, V D’Esposito, G Fabiano, D Frattulillo, P A Höhn, F Mercati","doi":"10.1093/ptep/ptae015","DOIUrl":"https://doi.org/10.1093/ptep/ptae015","url":null,"abstract":"Quantum gravity is expected to introduce quantum aspects into the description of reference frames. Here we begin exploring how quantum gravity induced deformations of classical symmetries could modify the transformation laws among reference frames in an effective regime. We invoke the quantum group SUq(2) as a description of deformed spatial rotations and interpret states of a representation of its algebra as describing the relative orientation between two reference frames. This leads to a quantization of one of the Euler angles and to an aspect of agency dependence: space is reconstructed as a collection of fuzzy points, exclusive to each agent, which depends on their choice of reference frame. Each agent can choose only one direction in which points can be sharp, while points in all other directions become fuzzy in a way that depends on this choice. Two agents making different choices will thus observe the same points with different degrees of fuzziness.","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"144 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139579607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the effect of the Lorentz violation on the Casimir energy and pressure of charged Dirac in a background uniform magnetic field. In the model, the Lorentz violation is parameterized not only by the intensity but also by its direction. We investigate two cases of the direction of violation, namely, time-like and space-like vector cases. We use the boundary condition of the MIT bag model to represent the property of the plates. We show how the Lorentz violation and the magnetic field affect the structure of the Casimir energy and its pressure. We also investigate the weak and strong magnetic field cases with two different limits, heavy and light masses. In addition, we compute the ratio of the influence of the strong magnetic field to that of the weak ones for the Casimir energy and its pressure. We find that the strong magnetic field enhances the magnitude of the Casimir energy and its pressure, where the parameter of the intensity of Lorentz violation could scale the plate’s distance.
我们研究了洛伦兹违反对背景均匀磁场中带电狄拉克的卡西米尔能和压力的影响。在模型中，洛伦兹违反不仅以强度为参数，还以其方向为参数。我们研究了违反方向的两种情况，即类时间和类空间矢量情况。我们使用 MIT 袋模型的边界条件来表示板块的属性。我们展示了洛伦兹违反和磁场如何影响卡西米尔能的结构及其压力。我们还研究了弱磁场和强磁场情况下的两种不同限制：重质量和轻质量。此外，我们还计算了强磁场与弱磁场对卡西米尔能及其压力的影响比例。我们发现，强磁场增强了卡西米尔能及其压力的大小，其中洛伦兹违反强度参数可以缩放板的距离。
{"title":"Casimir effect of Lorentz-violating charged Dirac in background magnetic field","authors":"Ar Rohim, Arista Romadani, Apriadi Salim Adam","doi":"10.1093/ptep/ptae016","DOIUrl":"https://doi.org/10.1093/ptep/ptae016","url":null,"abstract":"We study the effect of the Lorentz violation on the Casimir energy and pressure of charged Dirac in a background uniform magnetic field. In the model, the Lorentz violation is parameterized not only by the intensity but also by its direction. We investigate two cases of the direction of violation, namely, time-like and space-like vector cases. We use the boundary condition of the MIT bag model to represent the property of the plates. We show how the Lorentz violation and the magnetic field affect the structure of the Casimir energy and its pressure. We also investigate the weak and strong magnetic field cases with two different limits, heavy and light masses. In addition, we compute the ratio of the influence of the strong magnetic field to that of the weak ones for the Casimir energy and its pressure. We find that the strong magnetic field enhances the magnitude of the Casimir energy and its pressure, where the parameter of the intensity of Lorentz violation could scale the plate’s distance.","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"2 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139587355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hauke Fischer, Christian Käding, Hartmut Lemmel, Stephan Sponar, Mario Pitschmann
We use previously obtained experimental results by neutron interferometry to effectively constrain the parameter space of several prominent dark energy models. This investigation encompasses the environment-dependent dilaton field, a compelling contender for dark energy that emerges naturally within the strong coupling limit of string theory, alongside symmetron and chameleon fields. Our study presents substantial improvements over previous constraints of the dilaton and symmetron fields, improving parameter constraints by several orders of magnitude. However, the analysis does not yield any new constraints on the chameleon field. Furthermore, we establish constraints for the projected neutron split interferometer, which has recently concluded a decisive proof-of-principle demonstration. Our symmetron simulations reveal that depending on the parameter values there are multiple static solutions with increasing number of nodes and increasing energy inside a cylindrical vacuum chamber. This agrees with results obtained earlier in the literature for infinitely parallel plates. Interestingly, while these multiple solutions can correspond to domain walls forming inside the vacuum chamber, we also find solutions that do not reach their vacuum expectation value inside the vacuum chamber, but display multiple nodes nonetheless.
{"title":"Search for dark energy with neutron interferometry","authors":"Hauke Fischer, Christian Käding, Hartmut Lemmel, Stephan Sponar, Mario Pitschmann","doi":"10.1093/ptep/ptae014","DOIUrl":"https://doi.org/10.1093/ptep/ptae014","url":null,"abstract":"We use previously obtained experimental results by neutron interferometry to effectively constrain the parameter space of several prominent dark energy models. This investigation encompasses the environment-dependent dilaton field, a compelling contender for dark energy that emerges naturally within the strong coupling limit of string theory, alongside symmetron and chameleon fields. Our study presents substantial improvements over previous constraints of the dilaton and symmetron fields, improving parameter constraints by several orders of magnitude. However, the analysis does not yield any new constraints on the chameleon field. Furthermore, we establish constraints for the projected neutron split interferometer, which has recently concluded a decisive proof-of-principle demonstration. Our symmetron simulations reveal that depending on the parameter values there are multiple static solutions with increasing number of nodes and increasing energy inside a cylindrical vacuum chamber. This agrees with results obtained earlier in the literature for infinitely parallel plates. Interestingly, while these multiple solutions can correspond to domain walls forming inside the vacuum chamber, we also find solutions that do not reach their vacuum expectation value inside the vacuum chamber, but display multiple nodes nonetheless.","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"9 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139587377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Much research in recent years has focused on circular accelerators that accelerate and store secondary particles with a large momentum spread, such as muons, unstable nuclei and heavy ions with different charge states. A fixed field alternating gradient (FFAG) accelerator with large transverse and momentum acceptance has obvious advantages for such requirements. A versatile beam injection method is required to accelerate secondary particles with a large momentum spread and different charge states with an FFAG accelerator. In the present study, a method for charge exchange injection of positive heavy ions using the large momentum acceptance of an FFAG accelerator is proposed. A charge injection system, which converts a He1 + beam to a He2 + beam, is developed for a 150 MeV FFAG accelerator at the Center for Accelerator and Beam Applied Science (CABAS) of Kyushu University. As the first step to verify the injection method, orbits shift from one charge state to the other is demonstrated. This is the first demonstration of heavy ion injection using an FFAG accelerator.
{"title":"Heavy ion injection of fixed-field alternating gradient accelerator","authors":"Yujiro Yonemura, Hidehiko Arima, Hiroki Nishibata, Takashi Teranishi, Tomotsugu Wakasa, Nobuo Ikeda, Kenichi Watanabe, Nobuhiro Shigyo, Tatsunori Iwamura, Kyosuke Adachi, Koki Takamatsu, Motoki Kotani, Hisato Tanaka, Rintaro Matsunaga, Taichi Matsumoto, Kyohei Takenaka, Takafumi Kajihara, Sotaro Matsunaga, Yusuke Shinohara, Yoshiharu Mori","doi":"10.1093/ptep/ptae017","DOIUrl":"https://doi.org/10.1093/ptep/ptae017","url":null,"abstract":"Much research in recent years has focused on circular accelerators that accelerate and store secondary particles with a large momentum spread, such as muons, unstable nuclei and heavy ions with different charge states. A fixed field alternating gradient (FFAG) accelerator with large transverse and momentum acceptance has obvious advantages for such requirements. A versatile beam injection method is required to accelerate secondary particles with a large momentum spread and different charge states with an FFAG accelerator. In the present study, a method for charge exchange injection of positive heavy ions using the large momentum acceptance of an FFAG accelerator is proposed. A charge injection system, which converts a He1 + beam to a He2 + beam, is developed for a 150 MeV FFAG accelerator at the Center for Accelerator and Beam Applied Science (CABAS) of Kyushu University. As the first step to verify the injection method, orbits shift from one charge state to the other is demonstrated. This is the first demonstration of heavy ion injection using an FFAG accelerator.","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"9 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139587187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by a new model of nonlinear electrodynamics known as Modified Maxwell (ModMax) theory, an exact analytical solution for black holes is obtained by coupling ModMax nonlinear electrodynamics and F(R) gravity. Then, the effects of the system’s parameters (F(R)-ModMax gravity parameters) on the event horizons are analyzed. The obtained black holes thermodynamic properties in the F(R)-ModMax theory are investigated by extracting their thermodynamic quantities such as Hawking temperature, electric charge, electric potential, entropy, and also total mass. The first law of thermodynamics for the system under study is evaluated. Next, by considering these black holes, the impact of various parameters on both the local stability and global stability are investigated by examining the heat capacity and the Helmholtz free energy, respectively. Finally, the thermodynamic geometry of the black hole in F(R)-ModMax gravity is investigated by applying the thermodynamic metric (the HPEM metric).
{"title":"Analytic electrical charged black holes in F(R)-ModMax theory","authors":"Behzad Eslam Panah","doi":"10.1093/ptep/ptae012","DOIUrl":"https://doi.org/10.1093/ptep/ptae012","url":null,"abstract":"Motivated by a new model of nonlinear electrodynamics known as Modified Maxwell (ModMax) theory, an exact analytical solution for black holes is obtained by coupling ModMax nonlinear electrodynamics and F(R) gravity. Then, the effects of the system’s parameters (F(R)-ModMax gravity parameters) on the event horizons are analyzed. The obtained black holes thermodynamic properties in the F(R)-ModMax theory are investigated by extracting their thermodynamic quantities such as Hawking temperature, electric charge, electric potential, entropy, and also total mass. The first law of thermodynamics for the system under study is evaluated. Next, by considering these black holes, the impact of various parameters on both the local stability and global stability are investigated by examining the heat capacity and the Helmholtz free energy, respectively. Finally, the thermodynamic geometry of the black hole in F(R)-ModMax gravity is investigated by applying the thermodynamic metric (the HPEM metric).","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"66 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using a classical approach, we theoretically investigate the frustrated double ionization (FDI) of Ar atoms under elliptically polarized (EP) laser fields. Its FDI yield is calculated as a function of laser intensity with different carrier-envelope phases (CEPs), and, similar to the carrier-envelope phase (CEP) dependence of nonsequential double ionization (NSDI), several knee-shaped structures appear. Moreover, both recollision and direct ionization trajectories are observed in FDI events for the “knee” structure regime. The footprints of which channel leads to FDI are encoded in the photoelectron momentum distributions. FDI events related to recollision excitation with subsequent ionization channels and recollision impact ionization channels can be controlled by CEP in EP laser fields.
{"title":"Control of frustrated nonsequential double ionization channels with the carrier-envelope phase of elliptically polarized laser pulses","authors":"Tong-Tong Xu, Lian-Lian Zhang, Wei-Jiang Gong","doi":"10.1093/ptep/ptae002","DOIUrl":"https://doi.org/10.1093/ptep/ptae002","url":null,"abstract":"Using a classical approach, we theoretically investigate the frustrated double ionization (FDI) of Ar atoms under elliptically polarized (EP) laser fields. Its FDI yield is calculated as a function of laser intensity with different carrier-envelope phases (CEPs), and, similar to the carrier-envelope phase (CEP) dependence of nonsequential double ionization (NSDI), several knee-shaped structures appear. Moreover, both recollision and direct ionization trajectories are observed in FDI events for the “knee” structure regime. The footprints of which channel leads to FDI are encoded in the photoelectron momentum distributions. FDI events related to recollision excitation with subsequent ionization channels and recollision impact ionization channels can be controlled by CEP in EP laser fields.","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"10 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The kinematics of (p, pX) knockout reactions is overviewed. Consistent descriptions of the kinematical quantities for normal and inverse kinematics are given. Starting from the kinematics of p-X elastic scatterings, which are elementary processes of the (p, pX) knockout reactions, effects of the separation energy of the knocked-out particle SX, its in-nucleus momentum $vec{k}_{F}$, and a momentum transfer q are investigated. The three quantities are called “knockout feature quantities” in this article. The main conclusions are: SX dependences of the (p, pX) kinematics are mainly through changes in the p-X center-of-momentum energy $sqrt{s_{pX}}$ for which no difference exists between normal and inverse kinematics. Effects of longitudinal-to-the-beam component of $vec{k}_{F}$ cause a significant difference between normal and inverse kinematics through changes in βpX and $sqrt{s_{pX}}$. On the other hand, the transverse-to-the-beam component of $vec{k}_{F}$ changes energy sharing between the scattered proton and the knocked-out particle in both normal and inverse kinematics.
{"title":"On the kinematics of (p, pX) knockout reactions in normal and inverse kinematics","authors":"Tomohiro Uesaka","doi":"10.1093/ptep/ptae013","DOIUrl":"https://doi.org/10.1093/ptep/ptae013","url":null,"abstract":"The kinematics of (p, pX) knockout reactions is overviewed. Consistent descriptions of the kinematical quantities for normal and inverse kinematics are given. Starting from the kinematics of p-X elastic scatterings, which are elementary processes of the (p, pX) knockout reactions, effects of the separation energy of the knocked-out particle SX, its in-nucleus momentum $vec{k}_{F}$, and a momentum transfer q are investigated. The three quantities are called “knockout feature quantities” in this article. The main conclusions are: SX dependences of the (p, pX) kinematics are mainly through changes in the p-X center-of-momentum energy $sqrt{s_{pX}}$ for which no difference exists between normal and inverse kinematics. Effects of longitudinal-to-the-beam component of $vec{k}_{F}$ cause a significant difference between normal and inverse kinematics through changes in βpX and $sqrt{s_{pX}}$. On the other hand, the transverse-to-the-beam component of $vec{k}_{F}$ changes energy sharing between the scattered proton and the knocked-out particle in both normal and inverse kinematics.","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"195 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix named as “anti-symmetrized adjacency matrix”, specifically tailored for cycle digraphs (T1 lattice) and simple directed paths (B1 lattice). The nontrivial relation between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the anti-symmetrized adjacency matrix, along with its extensions to higher dimensions, are equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a cartesian product of cycle digraphs (T1 lattice) and simple directed paths (B1 lattice).
我们探讨了晶格场理论与图论之间的关系，特别强调了频谱图论领域中的狄拉克和标量晶格算子与矩阵之间的相互作用。除了深入研究谱图理论的基本概念，如邻接矩阵和拉普拉斯矩阵，我们还介绍了一种名为 "反对称邻接矩阵 "的新型矩阵，它专门为循环数图（T1 格）和简单有向路径（B1 格）量身定制。图论矩阵和晶格算子之间的非对称关系表明，图拉普拉斯矩阵反映了晶格标量算子和晶格费米子中的威尔逊项，而反对称邻接矩阵及其向更高维度的扩展等价于天真晶格狄拉克算子。基于这些联系，我们为两个关键论断提供了严格的证明：（i）自由晶格标量算子的零模数与底层图（晶格）的贝蒂数重合。(ii) 当 D 维图由循环数图（T1 格）和简单有向路径（B1 格）的笛卡尔积而成时，自由格费米子算子中的狄拉克零模的最大计数等同于所有贝蒂数的累积和。
{"title":"Equivalence of lattice operators and graph matrices","authors":"Jun Yumoto, Tatsuhiro Misumi","doi":"10.1093/ptep/ptae009","DOIUrl":"https://doi.org/10.1093/ptep/ptae009","url":null,"abstract":"We explore the relationship between lattice field theory and graph theory, placing special emphasis on the interplay between Dirac and scalar lattice operators and matrices within the realm of spectral graph theory. Beyond delving into fundamental concepts of spectral graph theory, such as adjacency and Laplacian matrices, we introduce a novel matrix named as “anti-symmetrized adjacency matrix”, specifically tailored for cycle digraphs (T1 lattice) and simple directed paths (B1 lattice). The nontrivial relation between graph theory matrices and lattice operators shows that the graph Laplacian matrix mirrors the lattice scalar operator and the Wilson term in lattice fermions, while the anti-symmetrized adjacency matrix, along with its extensions to higher dimensions, are equivalent to naive lattice Dirac operators. Building upon these connections, we provide rigorous proofs for two key assertions: (i) The count of zero-modes in a free lattice scalar operator coincides with the zeroth Betti number of the underlying graph (lattice). (ii) The maximum count of Dirac zero-modes in a free lattice fermion operator is equivalent to the cumulative sum of all Betti numbers when the D-dimensional graph results from a cartesian product of cycle digraphs (T1 lattice) and simple directed paths (B1 lattice).","PeriodicalId":20710,"journal":{"name":"Progress of Theoretical and Experimental Physics","volume":"3 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139507137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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