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 Lidecoration of graphene with broken lattice symmetry, and includes
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Dominant shapes naturally emerge in atomic nuclei from first principles, thereby establishing the shapepreserving symplectic Sp(3,
 NSFPAR ID:
 10475431
 Publisher / Repository:
 SciPost
 Date Published:
 Journal Name:
 SciPost Physics Proceedings
 Issue:
 14
 ISSN:
 26664003
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract s andd symmetry Bloch character that influences the gap symmetries that can arise. The resulting seven hybridized LiC 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 →M where its Bloch wave function has linear combination of and$${d}_{{x}^{2}{y}^{2}}$$ ${d}_{{x}^{2}{y}^{2}}$d _{xy}character, and is responsible for and$${d}_{{x}^{2}{y}^{2}}$$ ${d}_{{x}^{2}{y}^{2}}$d _{xy}pairing 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 uppers band that favors extendeds wave pairing which is not found in two band model. Upon electron doping to a critical chemical potentialμ _{1} = 0.22eV the pairing potential decreases, then increases until a second critical valueμ _{2} = 1.3 eV at which a phase transition to a distorteds wave occurs. The distortion ofd  or swave 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. 
Abstract Symmetryprotected topological crystalline insulators (TCIs) have primarily been characterized by their gapless boundary states. However, in timereversal (
) invariant (helical) 3D TCIs—termed higherorder TCIs (HOTIs)—the boundary signatures can manifest as a sampledependent network of 1D hinge states. We here introduce nested spinresolved 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 spinresolved phases with distinct responses that are quantitatively robust to large deformations of the bulk spinorbital texture: 3D quantum spin Hall insulators (QSHIs), “spinWeyl” semimetals, and$${{{{{{{\mathcal{T}}}}}}}}$$ $T$ doubled axion insulator (TDAXI) states with nontrivial partial axion angles indicative of a 3D spinmagnetoelectric bulk response and halfquantized 2D TI surface states originating from a partial parity anomaly. Using abinitio calculations, we demonstrate that$${{{{{{{\mathcal{T}}}}}}}}$$ $T$β MoTe_{2}realizes a spinWeyl state and thatα BiBr hosts both 3D QSHI and TDAXI regimes. 
Abstract Let
denote the matrix multiplication tensor (and write$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$ ), and let$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$ denote the determinant polynomial considered as a tensor. For a tensor$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$ T , let denote its border rank. We (i) give the first handcheckable algebraic proof that$\underline {\mathbf {R}}(T)$ , (ii) prove$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$ and$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$ , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$ , (iii) prove$M_{\langle 2\rangle }$ , (iv) prove$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$ , improving the previous lower bound of$\underline {\mathbf {R}}(\operatorname {det}_3)=17$ , (v) prove$12$ for all$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$ , where previously only$\mathbf {n}\geq 25$ was known, as well as lower bounds for$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$ , and (vi) prove$4\leq \mathbf {n}\leq 25$ for all$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$ , where previously only$\mathbf {n} \ge 18$ 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.$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$ 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, called
border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorT and an integerr , in a finite number of steps, either outputs that there is no border rankr decomposition forT or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenT has 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. 
Abstract 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 polyLaplacian regulariser. The methodology is readily adapted to graphs and here we consider graph polyLaplacian regularization in a fully supervised, nonparametric, noise corrupted, regression problem. In particular, given a dataset
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