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1. Superconducting Phases in Lithium Decorated Graphene LiC6

   https://doi.org/10.1038/s41598-018-32050-9  

   Gholami, Rouhollah ; Moradian, Rostam ; Moradian, Sina ; Pickett, Warren E. ( September 2018 , Scientific Reports)

   A study of possible superconducting phases of graphene has been constructed in detail. A realistic tight binding model, fit to ab initio calculations, accounts for the Li-decoration of graphene with broken lattice symmetry, and includessanddsymmetry Bloch character that influences the gap symmetries that can arise. The resulting seven hybridized Li-C orbitals that support nine possible bond pairing amplitudes. The gap equation is solved for all possible gap symmetries. One band is weakly dispersive near the Fermi energy along Γ → Mwhere its Bloch wave function has linear combination of$${d}_{{x}^{2}-{y}^{2}}$$$d_{x^2-y^2}$andd_xycharacter, and is responsible for$${d}_{{x}^{2}-{y}^{2}}$$$d_{x^2-y^2}$andd_xypairing with lowest pairing energy in our model. These symmetries almost preserve properties from a two band model of pristine graphene. Another part of this band, alongK → Γ, is nearly degenerate with uppersband that favors extendedswave pairing which is not found in two band model. Upon electron doping to a critical chemical potentialμ₁ = 0.22 eVthe pairing potential decreases, then increases until a second critical valueμ₂ = 1.3 eV at which a phase transition to a distorteds-wave occurs. The distortion ofd- or s-wave phases are a consequence of decoration which is not appear in two band pristine model. In the pristine graphene these phases convert to usuald-wave or extendeds-wave pairing.

    

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2. Spin-resolved topology and partial axion angles in three-dimensional insulators

   https://doi.org/10.1038/s41467-024-44762-w  

   Lin, Kuan-Sen ; Palumbo, Giandomenico ; Guo, Zhaopeng ; Hwang, Yoonseok ; Blackburn, Jeremy ; Shoemaker, Daniel P. ; Mahmood, Fahad ; Wang, Zhijun ; Fiete, Gregory A. ; Wieder, Benjamin J. ; et al ( January 2024 , Nature Communications)

   Symmetry-protected topological crystalline insulators (TCIs) have primarily been characterized by their gapless boundary states. However, in time-reversal- ($${{{{{{{\mathcal{T}}}}}}}}$$$T$-) invariant (helical) 3D TCIs—termed higher-order TCIs (HOTIs)—the boundary signatures can manifest as a sample-dependent network of 1D hinge states. We here introduce nested spin-resolved Wilson loops and layer constructions as tools to characterize the intrinsic bulk topological properties of spinful 3D insulators. We discover that helical HOTIs realize one of three spin-resolved phases with distinct responses that are quantitatively robust to large deformations of the bulk spin-orbital texture: 3D quantum spin Hall insulators (QSHIs), “spin-Weyl” semimetals, and$${{{{{{{\mathcal{T}}}}}}}}$$$T$-doubled axion insulator (T-DAXI) states with nontrivial partial axion angles indicative of a 3D spin-magnetoelectric bulk response and half-quantized 2D TI surface states originating from a partial parity anomaly. Using ab-initio calculations, we demonstrate thatβ-MoTe₂realizes a spin-Weyl state and thatα-BiBr hosts both 3D QSHI and T-DAXI regimes.

    

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3. New lower bounds for matrix multiplication and

   https://doi.org/10.1017/fmp.2023.14  

   Conner, Austin ; Harper, Alicia ; Landsberg, J.M. ( January 2023 , Forum of Mathematics, Pi)

   Let$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$denote the matrix multiplication tensor (and write$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$), and let$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$denote the determinant polynomial considered as a tensor. For a tensorT, let$\underline {\mathbf {R}}(T)$denote its border rank. We (i) give the first hand-checkable algebraic proof that$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$, (ii) prove$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$and$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$M_{\langle 2\rangle }$, (iii) prove$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$, (iv) prove$\underline {\mathbf {R}}(\operatorname {det}_3)=17$, improving the previous lower bound of$12$, (v) prove$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$for all$\mathbf {n}\geq 25$, where previously only$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$was known, as well as lower bounds for$4\leq \mathbf {n}\leq 25$, and (vi) prove$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$for all$\mathbf {n} \ge 18$, where previously only$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors.

   The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, calledborder apolaritydeveloped by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorTand an integerr, in a finite number of steps, either outputs that there is no border rankrdecomposition forTor produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenThas a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.

    

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4. Twisted-bilayer FeSe and the Fe-based superlattices

   https://doi.org/10.21468/SciPostPhys.15.3.081  

   Eugenio, Paul Myles ; Vafek, Oskar ( January 2023 , SciPost Physics)

   We derive BM-like continuum models for the bands of superlattice heterostructures formed out of Fe-chalcogenide monolayers: (I) a single monolayer experiencing an external periodic potential, and (II) twisted bilayers with long-range moire tunneling. A symmetry derivation for the inter-layer moire tunnelling is provided for both the\Gamma$\Gamma$andM$M$high-symmetry points. In this paper, we focus on moire bands formed from hole-band maxima centered on\Gamma$\Gamma$, and show the possibility of moire bands withC=0$C=0$or±1$\pm 1$topological quantum numbers without breaking time-reversal symmetry. In theC=0$C=0$region for\theta→0$\theta \to 0$(and similarly in the limit of large superlattice period for I), the system becomes a square lattice of 2D harmonic oscillators. We fit our model to FeSe and argue that it is a viable platform for the simulation of the square Hubbard model with tunable interaction strength.

    

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5. Rates of convergence for regression with the graph poly-Laplacian

   https://doi.org/10.1007/s43670-023-00075-5  

   Trillos, Nicolás García ; Murray, Ryan ; Thorpe, Matthew ( November 2023 , Sampling Theory, Signal Processing, and Data Analysis)

   In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset$$\{x_i\}_{i=1}^n$$${\left\{x_i\right\}}_{i=1}^n$and a set of noisy labels$$\{y_i\}_{i=1}^n\subset \mathbb {R}$$${\left\{y_i\right\}}_{i=1}^n\subset R$we let$$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$$u_n:{\left\{x_i\right\}}_{i=1}^n\to R$be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When$$y_i = g(x_i)+\xi _i$$$y_i=g\left(x_i\right)+{\xi }_i$, for iid noise$$\xi _i$$${\xi }_i$, and using the geometric random graph, we identify (with high probability) the rate of convergence of$$u_n$$$u_n$togin the large data limit$$n\rightarrow \infty $$$n\to \infty$. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.

    

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