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1. 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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2. 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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3. Convolution identities for divisor sums and modular forms

   https://doi.org/10.1073/pnas.2322320121  

   Fedosova, Ksenia; Klinger-Logan, Kim; Radchenko, Danylo (October 2024, Proceedings of the National Academy of Sciences)

   Ribet, Kenneth (Ed.)

   We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums $\sum _{n_1\in \mathbb{Z}\setminus \{0,n\}}\phi (n_1,n-n_1){\sigma }_{2m_1}\left(n_1\right){\sigma }_{2m_2}(n-n_1),$ where $\phi (n_1,n_2)$is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to $n_1\in \{0,n\}$, do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms. 

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4. Imaging nuclear shape through anisotropic and radial flow in high-energy heavy-ion collisions

   https://doi.org/10.1088/1361-6633/ae0fc3  

   Collaboration, The STAR (October 2025, Reports on Progress in Physics)

   Abstract Most atomic nuclei exhibit ellipsoidal shapes characterized by quadrupole deformationβ₂and triaxialityγ, and sometimes even a pear-like octupole deformationβ₃. The STAR experiment introduced a new ‘imaging-by-smashing’ technique ((STAR Collaboration) 2024Nature63567; Jia 2025Rep. Prog. Phys.88092301) to image the nuclear global shape by colliding nuclei at ultra-relativistic speeds and analyzing outgoing debris. Features of nuclear shape manifest in collective observables like anisotropic flowv_nand radial flow via mean transverse momentum $\left[p_{\mathrm{T}}\right]$. We present new measurements of the variances ofv_n(n = 2, 3, and 4) and $\left[p_{\mathrm{T}}\right]$, and the covariance of $v_n^2$with $\left[p_{\mathrm{T}}\right]$, in collisions of highly deformed²³⁸U and nearly spherical¹⁹⁷Au. Ratios of these observables between the two systems effectively suppress common final-state effects, isolating the strong impact of uranium’s deformation. By comparing results with state-of-the-art hydrodynamic model calculations, we extract ${\beta }_{2\mathrm{U}}$and ${\gamma }_{\mathrm{U}}$values consistent with those deduced from low-energy nuclear structure measurements. Measurements ofv₃and its correlation with $\left[p_{\mathrm{T}}\right]$also provide the first experimental suggestion of a possible octupole deformation for²³⁸U. These findings provide significant support for using high-energy collisions to explore nuclear shapes on femtosecond timescales, with implications for both nuclear structure and quark-gluon plasma studies. 

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5. Structural phase purification of bulk HfO ₂ :Y through pressure cycling

   https://doi.org/10.1073/pnas.2312571121  

   Musfeldt, J. L.; Singh, Sobhit; Fan, Shiyu; Gu, Yanhong; Xu, Xianghan; Cheong, S.-W.; Liu, Z.; Vanderbilt, David; Rabe, Karin M. (January 2024, Proceedings of the National Academy of Sciences)

   We combine synchrotron-based infrared absorption and Raman scattering spectroscopies with diamond anvil cell techniques and first-principles calculations to explore the properties of hafnia under compression. We find that pressure drives HfO ${}_2$:7%Y from the mixed monoclinic ( $P{2}_1/c$) $+$antipolar orthorhombic ( $\mathit{Pbca}$) phase to pure antipolar orthorhombic ( $\mathit{Pbca}$) phase at approximately 6.3 GPa. This transformation is irreversible, meaning that upon release, the material is kinetically trapped in the $\mathit{Pbca}$metastable state at 300 K. Compression also drives polar orthorhombic ( $Pca{2}_1$) hafnia into the tetragonal ( $P{4}_2/nmc$) phase, although the latter is not metastable upon release. These results are unified by an analysis of the energy landscape. The fact that pressure allows us to stabilize targeted metastable structures with less Y stabilizer is important to preserving the flat phonon band physics of pure HfO ${}_2$. 

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