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1. Quantized Axial Charge of Staggered Fermions and the Chiral Anomaly

   https://doi.org/10.1103/PhysRevLett.134.021601  

   Chatterjee, Arkya; Pace, Salvatore D; Shao, Shu-Heng (January 2025, Physical Review Letters)

   In the $1+1\mathrm{D}$ultralocal lattice Hamiltonian for staggered fermions with a finite-dimensional Hilbert space, there are two conserved, integer-valued charges that flow in the continuum limit to the vector and axial charges of a massless Dirac fermion with a perturbative anomaly. Each of the two lattice charges generates an ordinary U(1) global symmetry that acts locally on operators and can be gauged individually. Interestingly, they do not commute on a finite lattice and generate the Onsager algebra, but their commutator goes to zero in the continuum limit. The chiral anomaly is matched by this non-Abelian algebra, which is consistent with the Nielsen-Ninomiya theorem. We further prove that the presence of these two conserved lattice charges forces the low-energy phase to be gapless, reminiscent of the consequence from perturbative anomalies of continuous global symmetries in continuum field theory. Upon bosonization, these two charges lead to two exact U(1) symmetries in the XX model that flow to the momentum and winding symmetries in the free boson conformal field theory. Published by the American Physical Society2025 

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2. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces

   https://doi.org/10.3934/cpaa.2021169  

   Jolly, Michael S.; Kumar, Anuj; Martinez, Vincent R. (January 2021, Communications on Pure & Applied Analysis)

   This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar \begin{document}$$ \theta $$\end{document} on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity \begin{document}$ u $$\end{document} is of lower singularity, i.e., \begin{document}$$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $$\end{document}, where \begin{document}$$ p $$\end{document} is a logarithmic smoothing operator and \begin{document}$$ \beta \in [0, 1] $$\end{document}. We complete this study by considering the more singular regime \begin{document}$$ \beta\in(1, 2) $$\end{document}$. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness. 

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3. Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators

   https://doi.org/10.1088/1361-6544/adc653  

   Liu, Wencai; Matos, Rodrigo; Treuer, John N (April 2025, Nonlinearity)

   Abstract This paper investigates uniqueness results for perturbed periodic Schrödinger operators on ${\mathbb{Z}}^d$. Specifically, we consider operators of the form $H=-\mathrm{\Delta }+V+v$, where Δ is the discrete Laplacian, $V:{\mathbb{Z}}^d\to \mathbb{R}$is a periodic potential, and $v:{\mathbb{Z}}^d\to \mathbb{C}$represents a decaying impurity. We establish quantitative conditions under which the equation $-\mathrm{\Delta }u+Vu+vu=\lambda u$, for $\lambda \in \mathbb{C}$, admits only the trivial solution $u\equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes ofλ. 

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4. Homeomorphism groups of telescoping $$2$$-manifolds are strongly distorted

   https://doi.org/10.4171/ggd/867  

   Vlamis, Nicholas G (February 2025, Groups, Geometry, and Dynamics)

   Building on the work of Mann and Rafi, we introduce an expanded definition of a telescoping2-manifold and proceed to study the homeomorphism group of a telescoping2-manifold. Our main result shows that it is strongly distorted. We then give a simple description of its commutator subgroup, which is index one, two, or four depending on the topology of the manifold. Moreover, we show its commutator subgroup is uniformly perfect with commutator width at most two, and we give a family of uniform normal generators for its commutator subgroup. As a consequence of the latter result, we show that for an (orientable) telescoping2-manifold, every (orientation-preserving) homeomorphism can be expressed as a product of at most 17 conjugate involutions. Finally, we provide analogous statements for mapping class groups. 

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5. A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality

   https://doi.org/10.1063/5.0091111  

   Carlen, Eric A.; Lieb, Elliott H. (June 2022, Journal of Mathematical Physics)

   The Golden–Thompson trace inequality, which states that Tr  e H+ K ≤ Tr  e H e K , has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or [Formula: see text] and K = potential, Tr  e H+(1− u) K e uK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr  X s Y t X 1− s Y 1− t ≤ Tr  XY for positive operators X, Y and for [Formula: see text], and [Formula: see text]. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the [Formula: see text] range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem. 

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