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1. The Eisenstein ideal at prime-square level has constant rank

   https://doi.org/10.1073/pnas.2500729122  

   Lang, Jaclyn; Wake, Preston (July 2025, Proceedings of the National Academy of Sciences)

   LetNandpbe prime numbers with $p\ge 5$such that $p\mid \mid (N+1)$. In a previous paper, we showed that there is a cuspformfof weight 2 and level ${\mathrm{\Gamma }}_0\left(N^2\right)$whoseℓ-th Fourier coefficient is congruent to $\ell +1$modulo a prime abovepfor all primesℓ. In this paper, we prove that this formfis unique up to Galois conjugacy, and the extension of ${\mathbb{Z}}_p$generated by the coefficients offis exactly ${\mathbb{Z}}_p[{\zeta }_p+{\zeta }_p^{-1}]$. We also prove similar results when a higher power ofpdivides $N+1$. 

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2. Detection of anyon braiding through pump–probe spectroscopy

   https://doi.org/10.1073/pnas.2511917122  

   Yang, Xu; Buechele, Ryan; Trivedi, Nandini (November 2025, Proceedings of the National Academy of Sciences)

   We show that the braiding of anyons in a quantum spin liquid leaves a distinct dynamical signature in the nonlinear pump–probe response. Using a combination of exact diagonalization and matrix product state techniques, we study the nonlinear pump–probe response of the toric code in a magnetic field, a model with mobile electriceand magneticmanyonic excitations. While the linear response signal oscillates and decays with time like $\sim t^{-1.3}$, the amplitude of the nonlinear signal for ${\chi }_{\mathit{XZZ}}^{\left(3\right)}$features a linear-in-time enhancement at early times and a stronger enhancement $t^{\alpha }$with $\alpha >1$at later times. The comparison between ${\chi }_{\mathit{XZZ}}^{\left(3\right)}$, which involves nontrivial braiding ofeandmanyons, and ${\chi }_{\mathit{XXX}}^{\left(3\right)}$that involves trivial braiding of the same types of anyons, distinguishes the braiding statistics of anyons. We support our analysis with a hard-core anyon model with statistical gauge fields to develop further insights into the time dependence of the pump–probe response. Pump–probe spectroscopy provides a distinctive new probe of quantum spin liquid states, beyond the inconclusive broad features observed in single spin-flip inelastic neutron scattering. 

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3. Development of broad modulus profile upon polymer–polymer interface formation between immiscible glassy–rubbery domains

   https://doi.org/10.1073/pnas.2312533120  

   Gagnon, Yannic J; Burton, Justin C; Roth, Connie B (January 2024, Proceedings of the National Academy of Sciences)

   Interfaces of glassy materials such as thin films, blends, and composites create strong unidirectional gradients to the local heterogeneous dynamics that can be used to elucidate the length scales and mechanisms associated with the dynamic heterogeneity of glasses. We focus on bilayer films of two different polymers with very different glass transition temperatures ( $T_{\text{g}}$) where previous work has demonstrated a long-range (∼200 nm) profile in local $T_{\text{g}}\left(z\right)$is established between immiscible glassy and rubbery polymer domains when the polymer–polymer interface is formed to equilibrium. Here, we demonstrate that an equally long-ranged gradient in local modulus $\tilde{G}\left(z\right)$is established when the polymer–polymer interface ( $\approx$5 nm) is formed between domains of glassy polystyrene (PS) and rubbery poly(butadiene) (PB), consistent with previous reports of a broad $T_{\text{g}}\left(z\right)$profile in this system. A continuum physics model for the shear wave propagation caused by a quartz crystal microbalance across a PB/PS bilayer film is used to measure the viscoelastic properties of the bilayer during the evolution of the PB/PS interface showing the development of a broad gradient in local modulus $\tilde{G}\left(z\right)$spanning $\approx$180 nm between the glassy and rubbery domains of PS and PB. We suggest these broad profiles in $T_{\text{g}}\left(z\right)$and $\tilde{G}\left(z\right)$arise from a coupling of the spectrum of vibrational modes across the polymer–polymer interface as a result of acoustic impedance matching of sound waves with $\lambda \sim 5$nm during interface broadening that can then trigger density fluctuations in the neighboring domain. 

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4. Generalizations of the quadratic bound optimization principle

   https://doi.org/10.1073/pnas.2525320123  

   Li, Xun-Jian; Tian, Guo-Liang; Zhou, Hua; Lange, Kenneth (January 2026, Proceedings of the National Academy of Sciences)

   The quadratic bound (QB) principle proposed by Böhning and Lindsay in 1988 is an important special case of the majorization–minimization or minorization-maximization optimization principle. The quadratic upper-bound (QUB) principle is pertinent to minimization; the analogous quadratic lower-bound principle is pertinent to maximization. Unfortunately, in minimizing a loss $f\left(\mathit{\theta }\right)$, the QUB principle is limited by the difficulty of finding a constant positive definite matrix $\mathit{B}$such that $\mathit{B}-d^{2}f\left(\mathit{\theta }\right)$is positive semidefinite for all $\mathit{\theta }$. This paper proposes a generalization of the QB principle that avoids this limitation. In particular, we construct QUB algorithms by replacing the matrix $\mathit{B}$by a continuous matrix-valued function $B_n\left(\mathit{\theta }\right)$that dominates the Hessian $d^{2}f\left(\mathit{\theta }\right)$and depends on the both the current iterate ${\mathit{\theta }}_n$and the next potential iterate $\mathit{\theta }$. In practice, we require $B_n\left(\mathit{\theta }\right)$to be diagonal with its diagonal entries separated in $\mathit{\theta }$. In other words, the $i$th diagonal entry of $B_n\left(\mathit{\theta }\right)$depends on $\mathit{\theta }$only through its $i$th entry ${\theta }_i$. Theoretical analysis confirms that this class of generalized QB algorithms enjoys global convergence in favorable circumstances. For the scalar case, the tangency condition that the second derivative equals the bound function at the current point promotes superlinear convergence. Several numerical experiments implemented in Julia illustrate the power of the generalized QB principle. 

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5. Perturbative second-order optical susceptibility of bulk materials: a symmetry-enforced return to non-orthogonal localized basis sets

   https://doi.org/10.1088/1361-648X/ae35fa  

   Huamán, Angiolo; Enrique_Rosas-Hernandez, Luis; Barraza-Lopez, Salvador (January 2026, Journal of Physics: Condensed Matter)

   Abstract The second-order optical susceptibility of semiconductors ${\chi }_{ijk}^{\left(2\right)}(-2\omega ;\omega ,\omega )$finds application in metrology, spectroscopy, telecommunications, material characterization, and quantum information. Pioneering calculations of ${\chi }_{ijk}^{\left(2\right)}(-2\omega ;\omega ,\omega )$utilized non-orthogonal Gaussian orbitals centered at atoms. That formulation transitioned into plane-wave-based algorithms as time went by. As of late, nevertheless, multiple tools for calculating optical susceptibilities have recast the problem using Wannier (i.e.localized) orbitals, making a comeback onto frameworks based on localized basis sets. Here, we present an approach for calculating ${\chi }_{ijk}^{\left(2\right)}(-2\omega ;\omega ,\omega )$reliant on numerical pseudo-atomic orbitals (PAOs) within perturbation theory in the velocity gauge. Its salient feature is a calculation of ‘Slater–Koster-like’ two-center integrals of the momentum operator in between PAOs identified by symmetry. The approach was successfully tested on paradigmatic cubic silicon carbide (3C-SiC) and gallium arsenide, for which linear responses are contributed as well. 

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