Article
Physics, Multidisciplinary
Vincent Caudrelier, Matteo Stoppato
Summary: Recent studies have discovered new properties of space-time duality in certain integrable classical field theories, leading to their reformulation using ideas from covariant Hamiltonian field theory. By extending these results to the whole hierarchy, specifically focusing on the AKNS hierarchy, and introducing a Lagrangian multiform, important objects such as a symplectic multiform and Hamiltonian multiform were explicitly constructed. These constructions help prove crucial results, including the rational classical r-matrix structure, multiform Hamilton equations, and a method to characterize an infinite set of conservation laws akin to the familiar criterion for a first integral.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2021)
Review
Mathematics, Applied
A. Zabrodin
Summary: This is a short review of the construction of quasi-periodic solutions, which are expressed through Riemann's theta-functions associated with algebraic curves, and how they can be treated within the framework of the integrable hierarchies developed by the Kyoto school.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Mathematics, Applied
Xing Li, Da-jun Zhang
Summary: We propose a bilinear framework for elliptic soliton solutions composed of Lame-type plane wave factors. We derive tau functions in Hirota's form and present vertex operators that generate these tau functions. Bilinear identities are constructed and a calculation algorithm for residues and bilinear equations is formulated. We investigate these concepts in detail for the KdV equation and provide a brief overview for the KP hierarchy. Degenerations by the periods of elliptic functions are explored, leading to a bilinear framework associated with trigonometric/hyperbolic and rational functions. Reductions by dispersion relation are considered through the use of elliptic N-th roots of unity, resulting in tau functions, vertex operators, and bilinear equations of the KdV hierarchy and Boussinesq equation obtained from those of the KP hierarchy. We also propose two methods to calculate bilinear derivatives involving Lame-type plane wave factors, which demonstrate the quasi-gauge property of bilinear equations.
JOURNAL OF NONLINEAR SCIENCE
(2022)
Article
Physics, Multidisciplinary
Sh Khachatryan, A. Sedrakyan
Summary: We construct the exact spectral parameter dependent vertex R-matrix for the classical 3D N-state chiral Potts models, convenient for considering the model in the context of the Bethe ansatz. The R-matrix is defined on the N-4 dimensional space V-N circle times V-N circle times V-N circle times V-N, appropriate for consideration by means of the cubeequations defined in Khachatryan et al. (2015). We present the 2D quantum spin Hamiltonians for the general case and, at N = 2, a fermionic lattice action representation corresponding to 3D Ising's statistical model.
Article
Physics, Multidisciplinary
Jean-Francois De Kemmeterl, Bryan Debin, Philippe Ruelle
Summary: This paper investigates the six-vertex model with partial domain wall boundary conditions and domino tilings of double Aztec rectangles. The analytic expression of the arctic curve is derived and confirmed through extensive numerical simulations.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Astronomy & Astrophysics
A. Melikyan, G. Weber
Summary: Starting from the fermionic formulation of Bazhanov-Stroganov's three-parameter elliptic parametrization for the R-operator, we derive the Lax connection of the free fermion model on a lattice. This leads to the Yang-Baxter and decorated Yang-Baxter equations of difference type in one of the spectral parameters, providing the most suitable form for obtaining any relativistic model of free fermions in the continuous limit.
Article
Physics, Multidisciplinary
Bryant Cox, Blake Sisson
Summary: This article introduces the additional symmetries of the extended bigraded Toda hierarchy and the properties of the Lax operator fixed under these symmetries. It determines the unique Lax operator in the special case of the extended Toda hierarchy and presents the differential equations and general solutions for the wave functions, suggesting a possible connection to the bispectral problem. Moreover, the article finds the form of the tau function for the Lax operator and computes a second solution to the extended Toda hierarchy using a Darboux transformation.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Materials Science, Multidisciplinary
Gabriel Matos, Andrew Hallam, Aydin Deger, Zlatko Papic, Jiannis K. Pachos
Summary: Research has shown that systems of interacting fermions can exhibit ground state correlations similar to those of free fermions in the thermodynamic limit. By establishing the relation between system size and correlation length, the emergence of fermionic gaussianity can be quantitatively analyzed. Through the applicability of Wick's theorem, this behavior can be observed experimentally and remains insensitive to variations in interaction range, coupling inhomogeneities, and local random potentials.
Article
Physics, Multidisciplinary
Philippe Di Francesco
Summary: The tangent method of Colomo and Sportiello was applied to predict the arctic curves of the six vertex model with reflecting boundary and the related twenty vertex model with suitable domain wall boundary conditions on a quadrangle, both in their disordered phase.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2021)
Article
Multidisciplinary Sciences
Binlu Feng, Yufeng Zhang, Hongyi Zhang
Summary: Based on the R-matrix theory, this paper investigates a series of generalized integrable systems and their corresponding solution types by abstracting Lax pairs, exhibiting zero-curvature equations, and using nonisospectral techniques.
Article
Materials Science, Multidisciplinary
Brenden Roberts, Shenghan Jiang, Olexei Motrunich
Summary: Researchers continue to explore examples of deconfined quantum criticality in one-dimensional models, specifically studying the transition between a Z(3) ferromagnet and a valence bond solid (VBS) phase. Evidence suggests a possible second-order or weakly first-order transition, with an integrable lattice model in the parameter space indicating an extremely weak first-order transition with a long correlation length. This transition is proposed to be part of a family of deconfined quantum critical points described by renormalization group flows.
Article
Mathematics, Applied
Ahmed Bakhet, Mohra Zayed
Summary: In this paper, the incomplete exponential type of R-matrix functions is established and some properties of the incomplete exponential matrix functions are identified, including integral representation, derivative formula, and generating functions. Special cases of the results are also pointed out.
Article
Mechanics
R. S. Vieira, A. Lima-Santos
Summary: The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters yield two systems of differential equations, which can be simplified into two systems of polynomial equations by eliminating the derivatives of the R matrix elements. The polynomial systems have a non-zero Hilbert dimension and allow for solving some unknowns through simple differential equations, ensuring the uniqueness and generality of the solutions.
JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
(2021)
Article
Physics, Multidisciplinary
Mattia Cafasso, Di Yang
Summary: In this study, we extended the matrix-resolvent method to compute logarithmic derivatives of tau-functions in the Ablowitz-Ladik hierarchy. We derived a formula for the generating series of logarithmic derivatives using matrix resolvents. As an application, we introduced a method to compute certain integrals over the unitary group.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Physics, Multidisciplinary
A. V. Belitsky, L. V. Bork, A. F. Pikelner, V. A. Smirnov
Summary: We study the Sudakov form factor in the off-shell kinematic regime of planar N=4 supersymmetric Yang-Mills theory, which is achieved by considering the theory on its Coulomb branch. We show that both the infrared-divergent and finite terms exponentiate up to three loops, with the coefficient determined by the octagon anomalous dimension. This behavior contradicts previous conjectural accounts. We also find that the logarithm of the Sudakov form factor is equivalent to twice the logarithm of the null octagon O0, which has a closed form expression for all values of the 't Hooft coupling constant and kinematical parameters.
PHYSICAL REVIEW LETTERS
(2023)