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1. Long‐diagonal pentagram maps

   https://doi.org/10.1112/blms.12792  

   Izosimov, Anton; Khesin, Boris (January 2023, Bulletin of the London Mathematical Society)

   Abstract The pentagram map on polygons in the projective plane was introduced by R. Schwartz in 1992 and is by now one of the most popular and classical discrete integrable systems. In the present paper we introduce and prove integrability of long‐diagonal pentagram maps on polygons in , by now the most universal pentagram‐type map encompassing all known integrable cases. We also establish an equivalence of long‐diagonal and bi‐diagonal maps and present a simple self‐contained construction of the Lax form for both. Finally, we prove that the continuous limit of all these maps is equivalent to the ‐KdV equation, generalizing the Boussinesq equation for . 

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2. Matsubara dynamics approximation for generalized multi-time correlation functions

   https://doi.org/10.1063/5.0146654  

   Videla, Pablo E.; Batista, Victor S. (May 2023, The Journal of Chemical Physics)

   We introduce a semi-classical approximation for calculating generalized multi-time correlation functions based on Matsubara dynamics, a classical dynamics approach that conserves the quantum Boltzmann distribution. This method is exact for the zero time and harmonic limits and reduces to classical dynamics when only one Matsubara mode is considered (i.e., the centroid). Generalized multi-time correlation functions can be expressed as canonical phase-space integrals, involving classically evolved observables coupled through Poisson brackets in a smooth Matsubara space. Numerical tests on a simple potential show that the Matsubara approximation exhibits better agreement with exact results than classical dynamics, providing a bridge between the purely quantum and classical descriptions of multi-time correlation functions. Despite the phase problem that prevents practical applications of Matsubara dynamics, the reported work provides a benchmark theory for the future development of quantum-Boltzmann-preserving semi-classical approximations for studies of chemical dynamics in condensed phase systems. 

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3. Auxiliary field deformations of (semi-)symmetric space sigma models

   https://doi.org/10.1007/JHEP01(2025)096  

   Bielli, Daniele; Ferko, Christian; Smith, Liam; Tartaglino-Mazzucchelli, Gabriele (January 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We generalize the auxiliary field deformations of the principal chiral model (PCM) introduced in [1] and [2] to sigma models whose target manifolds are symmetric or semi-symmetric spaces, including a Wess-Zumino term in the latter case. This gives rise to a new infinite family of classically integrable ℤ₂and ℤ₄coset models of the form which are of interest in applications of integrability to worldsheet string theory and holography. We demonstrate that every theory in this infinite class admits a zero-curvature representation for its equations of motion by exhibiting a Lax connection. 

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4. Lax dynamics for Cartan decomposition with applications to Hamiltonian simulation

   https://doi.org/10.1093/imanum/drad018  

   Chu, Moody T (April 2023, IMA Journal of Numerical Analysis)

   Simulating the time evolution of a Hamiltonian system on a classical computer is hard—The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge—Assuming that a quantum system composing of spin-½ particles can be manipulated at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution, is therefore essential for algorithm design and circuit implementation. Most current techniques are fallible because of truncation errors or the stagnation at local solutions. This work offers an innovative avenue by tackling the Cartan decomposition with the notion of Lax dynamics. Within the integration errors that is controllable, this approach gives rise to a genuine unitary synthesis that not only is numerically feasible, but also can be utilized to gauge the quality of results produced by other means, and extend the knowledge to a wide range of applications. This paper aims at establishing the theoretic and algorithmic foundations by exploiting the geometric properties of Hamiltonian subalgebras and describing a common mechanism for deriving the Lax dynamics. 

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5. Kinetic theory for spin-polarized relativistic plasmas

   https://doi.org/10.1063/5.0165836  

   Seipt, Daniel; Thomas, Alec_G_R (September 2023, Physics of Plasmas)

   The investigation of spin and polarization effects in ultra-high intensity laser–plasma and laser–beam interactions has become an emergent topic in high-field science recently. In this paper, we derive a relativistic kinetic description of spin-polarized plasmas, where quantum-electrodynamics effects are taken into account via Boltzmann-type collision operators under the local constant field approximation. The emergence of anomalous precession is derived from one-loop self-energy contributions in a strong background field. We are interested, in particular, in the interplay between radiation reaction effects and the spin polarization of the radiating particles. For this, we derive equations for spin-polarized quantum radiation reaction from moments of the spin-polarized kinetic equations. By comparing with the classical theory, we identify and discuss the spin-dependent radiation reaction terms and radiative contributions to spin dynamics. 

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