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We introduce a class of topological pairing orders characterized by a half-integer pair monopole charge, leading to Berry phase enforced half-integer partial wave symmetry. This exotic spinor pairing order can emerge from pairing between Fermi surfaces with Chern numbers differing by an odd integer. Using tight-binding models, we demonstrate spinor superconducting orders with monopole charges ±1/2, featuring a single gap node and nontrivial surface states. Additionally, the superfluid velocity follows a fractionalized Mermin-Ho relation in spatially inhomogeneous pairing orders. The concept extends to spinor density waves and excitons. 

We demonstrate that coupling itinerant electrons to a noncollinear classical exchange field can induce anisotropic Josephson coupling between superconducting d vectors, analogous to the Dzyaloshinskii-Moriya and Γ-type interactions in magnetism. Using perturbative methods, we analyze an s-d model on a geometrically frustrated lattice. Noncollinear local spin textures generate spin triplet pairing correlations and can favor spatially varying superconducting order due to anisotropic Josephson couplings between d vectors, endowing a “pliability” to the pairing order that competes with the superfluid stiffness. For nonunitary pairing, this spatial texture of d vectors can give rise to anomalous vortices in the absence of an external magnetic field. We further predict a Josephson diode effect with efficiency proportional to the spin chirality of the underlying magnetic texture. These results establish a link between frustrated magnetism and spatial textures of triplet superconducting pairing, with implications for a range of materials such as Mn3Ge and 4Hb-TaS2, where superconductivity can be proximity induced or intrinsic. 

ABSTRACT In many genomic studies, gene co-expression graphs are influenced by subject-level covariates like single nucleotide polymorphisms. Traditional Gaussian graphical models ignore these covariates and estimate only population-level networks, potentially masking important heterogeneity. Covariate-dependent Gaussian graphical regressions address this limitation by regressing the precision matrix on covariates, thereby modeling how graph structures vary with high-dimensional subject-specific covariates. To fit the model, we adopt a multi-task learning approach that achieves lower error rates than node-wise regressions. Yet, the important problem of statistical inference in this setting remains largely unexplored. We propose a class of debiased estimators based on multi-task learners, which can be computed quickly and separately. In a key step, we introduce a novel projection technique for estimating the inverse covariance matrix, reducing optimization costs to scale with the sample size n. Our debiased estimators achieve fast convergence and asymptotic normality, enabling valid inference. Simulations demonstrate the utility of the method, and an application to a brain cancer gene-expression dataset reveals meaningful biological relationships.