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To resolve disputes or determine the order of things, people commonly use binary choices such as tossing a coin, even though it is obscure whether the empirical probability equals to the theoretical probability. The geomagnetic field (GMF) is broadly applied as a sensory cue for various movements in many organisms including humans, although our understanding is limited. Here we reveal a GMF-modulated probabilistic abstract decision-making in humans and the underlying mechanism, exploiting the zero-sum binary stone choice of Go game as a proof-of-principle. The large-scale data analyses of professional Go matches and in situ stone choice games showed that the empirical probabilities of the stone selections were remarkably different from the theoretical probability. In laboratory experiments, experimental probability in the decision-making was significantly influenced by GMF conditions and specific magnetic resonance frequency. Time series and stepwise systematic analyses pinpointed the intentionally uncontrollable decision-making as a primary modulating target. Notably, the continuum of GMF lines and anisotropic magnetic interplay between players were crucial to influence the magnetic field resonance-mediated abstract decision-making. Our findings provide unique insights into the impact of sensing GMF in decision-makings at tipping points and the quantum mechanical mechanism for manifesting the gap between theoretical and empirical probability in 3-dimensional living space., Comment: 32 pages, 5 figures, 4 supplementary figures, 2 supplementary tables, and separate 15 raw data files

Motivated by recent discovery of correlated insulating and superconducting behavior in twisted bilayer graphene, we revisit graphene's honeycomb lattice doped close to the van Hove singularity, using the truncated unity functional renormalization group approach. We consider an extended Hubbard model on the honeycomb lattice including on-site and nearest-neighbor Coulomb repulsions, and nearest-neighbor ferromagnetic exchange and pair hopping interactions. By varying the strength of the nearest-neighbor exchange coupling and Coulomb repulsion as free parameters, we present rich ground-state phase diagrams which contain the spin-triplet $f$-wave and spin-singlet chiral $d$-wave superconducting phases, the commensurate and incommensurate spin- and charge-density-wave phases, and the ferromagnetic phase. In the absence of the exchange coupling and for small value of the nearest-neighbor repulsion, the four-sublattice spin-density-wave phase is generated right around the van Hove filling, while the chiral $d$-wave superconductivity emerges slightly away from it. Surprisingly, the chiral $d$-wave superconductivity is strongly suppressed by weak nearest-neighbor exchange coupling in our calculations. We argue that this suppression might be one of the reasons why the chiral superconductivity proposed for doped graphene has not yet been observed experimentally., Comment: 20 pages, 11 figures, 2 tables

The impact of local and nonlocal density-density interactions on the electronic instabilities in the honeycomb lattice is widely investigated. Some early studies proposed the emergence of interaction-induced topologically nontrivial phases, but recently, it was denied in several works including renormalization group calculations with refined momentum resolution. We use the truncated unity functional renormalization group to study the many-body instabilities of electrons on the half-filled honeycomb lattice, focusing on the effect of the exchange interaction. We show that varying the next-nearest-neighbor repulsion and nearest-neighbor exchange integral can lead to diverse ordered phases, namely, the quantum spin Hall, the spin-Kekul\'e, and some spin- and charge-density-wave phases. The quantum spin Hall phase can be induced by a combination of the ferromagnetic exchange and pair hopping interactions. Another exotic phase, the spin-Kekul\'e phase, develops in a very small region of the parameter space considered. We encounter the three-sublattice charge-density-wave phase in a large part of the parameter space. It is replaced by the incommensurate charge density wave when increasing the exchange integral. In order to reduce the computational effort, we derive the explicit symmetry relations for the bosonic propagators of the effective interaction and propose a linear-response-based approach for identifying the form factor of order parameter. Their efficiencies are confirmed by numerical calculations in our work., Comment: 13 pages, 6 figures, 1 Table