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Abstract Kramers degeneracy is one fundamental embodiment of the quantum mechanical nature of particles with half-integer spin under time reversal symmetry. Under the chiral and noncentrosymmetric achiral crystalline symmetries, Kramers degeneracy emerges respectively as topological quasiparticles of Weyl fermions and Kramers nodal lines (KNLs), anchoring the Berry phase-related physics of electrons. However, an experimental demonstration for ideal KNLs well isolated at the Fermi level is lacking. Here, we establish a class of noncentrosymmetric achiral intercalated transition metal dichalcogenide superconductors with large Ising-type spin-orbit coupling, represented by In_xTaS₂, to host an ideal KNL phase. We provide evidence from angle-resolved photoemission spectroscopy with spin resolution, angle-dependent quantum oscillation measurements, and ab-initio calculations. Our work not only provides a realistic platform for realizing and tuning KNLs in layered materials, but also paves the way for exploring the interplay between KNLs and superconductivity, as well as applications pertaining to spintronics, valleytronics, and nonlinear transport. 

A<sc>bstract</sc> The eigenstate thermalization hypothesis (ETH) is the leading conjecture for the emergence of statistical mechanics in generic isolated quantum systems and is formulated in terms of the matrix elements of operators. An analog known as the ergodic bipartition (EB) describes entanglement and locality and is formulated in terms of the components of eigenstates. In this paper, we significantly generalize the EB and unify it with the ETH, extending the EB to study higher correlations and systems out of equilibrium. Our main result is a diagrammatic formalism that computes arbitrary correlations between eigenstates and operators based on a recently uncovered connection between the ETH and free probability theory. We refer to the connected components of our diagrams as generalized free cumulants. We apply our formalism in several ways. First, we focus on chaotic eigenstates and establish the so-called subsystem ETH and the Page curve as consequences of our construction. We also improve known calculations for thermal reduced density matrices and comment on an inherently free probabilistic aspect of the replica approach to entanglement entropy previously noticed in a calculation for the Page curve of an evaporating black hole. Next, we turn to chaotic quantum dynamics and demonstrate the ETH as a sufficient mechanism for thermalization, in general. In particular, we show that reduced density matrices relax to their equilibrium form and that systems obey the Page curve at late times. We also demonstrate that the different phases of entanglement growth are encoded in higher correlations of the EB. Lastly, we examine the chaotic structure of eigenstates and operators together and reveal previously overlooked correlations between them. Crucially, these correlations encode butterfly velocities, a well-known dynamical property of interacting quantum systems. 

The nonlinear Hall effect (NLHE), an emergent response in systems with broken inversion symmetry, provides a powerful tool for probing topological transport properties. In this context, we investigate copper-substituted lead apatite (LK-99), a material that initially garnered attention for its controversial claim of room-temperature superconductivity. Despite the unresolved nature of its superconducting properties, LK-99’s unique electronic structure characterized by flat bands near the Fermi level and broken inversion symmetry makes it a promising candidate for exploring Berry curvature-driven phenomena, such as the NLHE. Using first-principles density functional theory and an augmented tight-binding Hamiltonian model, we investigate LK-99’s band topology and transport properties. Our calculations indicate that spin–orbit coupling in LK-99 generates multiple Weyl points near the Fermi level, thereby enhancing the Berry curvature distribution by further splitting the bands. Crucially, the absence of inversion symmetry in LK-99 leads to a net Berry curvature dipole (BCD), producing a nonlinear Hall current that scales quadratically with the applied electric field. The nonlinear Hall effect is solely due to the BCD, as the contributions from the Drude weight and quantum metric are zero due to time reversal symmetry. Moreover, we demonstrate that the NLHE in LK-99 can be tuned by varying the direction of the applied electric field, underscoring its potential as a versatile platform for exploring topological transport phenomena and designing next-generation nonlinear electronic devices. 

Dataset for publication "Kramers nodal lines in intercalated TaS2 superconductors". Detailed descriptions are in the README text documents in subdirectories.