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1. 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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2. Designing multicomponent hydrides with potential high T _c superconductivity

   https://doi.org/10.1073/pnas.2413096121  

   Denchfield, Adam; Park, Hyowon; Hemley, Russell J (November 2024, Proceedings of the National Academy of Sciences)

   While hydrogen-rich materials have been demonstrated to exhibit high T_csuperconductivity at high pressures, there is an ongoing search for ternary, quaternary, and more chemically complex hydrides that achieve such high critical temperatures at much lower pressures. First-principles searches are impeded by the computational complexity of solving the Eliashberg equations for large, complex crystal structures. Here, we adopt a simplified approach using electronic indicators previously established to be correlated with superconductivity in hydrides. This is used to study complex hydride structures, which are predicted to exhibit promisingly high critical temperatures for superconductivity. In particular, we propose three classes of hydrides inspired by the Fm $\overline{3}$m RH ${}_3$structures that exhibit strong hydrogen network connectivity, as defined through the electron localization function. The first class [RH ${}_{11}$X ${}_3$Y] is based on a Pm $\overline{3}$m structure showing moderately high T_c, where the T_cestimate from electronic properties is compared with direct Eliashberg calculations and found to be surprisingly accurate. The second class of structures [(RH ${}_{11}$) ${}_2$X ${}_6$YZ] improves on this with promisingly high density of states with dominant hydrogen character at the Fermi energy, typically enhancing T_c. The third class [(R ${}^1$H ${}_{11}$)(R ${}^2$H ${}_{11}$)X ${}_6$YZ] improves the strong hydrogen network connectivity by introducing anisotropy in the hydrogen network through a specific doping pattern. These design principles and associated model structures provide flexibility to optimize both T_cand the structural stability of complex hydrides. 

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3. 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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4. Stabilizing Ti ₃ C ₂ T _x MXene flakes in air by removing confined water

   https://doi.org/10.1073/pnas.2400084121  

   Fang, Hui; Thakur, Anupma; Zahmatkeshsaredorahi, Amirhossein; Fang, Zhenyao; Rad, Vahid; Shamsabadi, Ahmad A; Pereyra, Claudia; Soroush, Masoud; Rappe, Andrew M; Xu, Xiaoji G; et al (July 2024, Proceedings of the National Academy of Sciences)

   MXenes have demonstrated potential for various applications owing to their tunable surface chemistry and metallic conductivity. However, high temperatures can accelerate MXene film oxidation in air. Understanding the mechanisms of MXene oxidation at elevated temperatures, which is still limited, is critical in improving their thermal stability for high-temperature applications. Here, we demonstrate that Ti ${}_3$C ${}_2$T ${}_x$MXene monoflakes have exceptional thermal stability at temperatures up to 600 ${}^{°}$C in air, while multiflakes readily oxidize in air at 300 ${}^{°}$C. Density functional theory calculations indicate that confined water between Ti ${}_3$C ${}_2$T ${}_x$flakes has higher removal energy than surface water and can thus persist to higher temperatures, leading to oxidation. We demonstrate that the amount of confined water correlates with the degree of oxidation in stacked flakes. Confined water can be fully removed by vacuum annealing Ti ${}_3$C ${}_2$T ${}_x$films at 600 ${}^{°}$C, resulting in substantial stability improvement in multiflake films (can withstand 600 ${}^{°}$C in air). These findings provide fundamental insights into the kinetics of confined water and its role in Ti ${}_3$C ${}_2$T ${}_x$oxidation. This work enables the use of stable monoflake MXenes in high-temperature applications and provides guidelines for proper vacuum annealing of multiflake films to enhance their stability. 

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5. 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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