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Authors: Gang Li and Werner Hanke
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**Geometric frustration and its interplay with strong electronic correlations has long been a subject of very active research, since Anderson suggested that a spin-1/2 Heisenberg antiferromagnet (AF) on the triangular lattice would have a gapless quantum spin-liquid (QSL) ground state, i.e. a resonant valence bond (RVB) state. Quite generally, this interplay is believed to be the source of a variety of exotic phases in triangular systems: examples span from the possible QSL state in -(ET) _{2}Cu_{2}(CN) _{3} over to a magnetically ordered Mott insulator, which provides the parent phase for the superconductivity (SC) found in -(BEDT-TTF) _{2}X organics and Na_{x}CoO_{2.y}H_{2}O, with the latter having possibly a topological SC phase, i.e. a gap structure breaking time-reversal symmetry.**

To verify the validity of a microscopic, i.e. triangular Hubbard model in this context, we explore its phase diagram using a combination of advanced many-body approaches, such as the dynamical cluster approach (DCA), the dual-fermion (DF) approach, as well as the variational cluster approach and the functional renormalization-group method. It is shown that this model contains the essential phases resolved experimentally in the organic salts, such as a paramagnetic metal for smaller interactions, a paramagnetic insulator for larger interactions and higher temperatures, as well as a Mott insulator with spiral AF correlations for large interactions and lower temperatures. In addition, we found evidence for the existence of a non-magnetic insulator (which can be a candidate of a SL) at intermediate interaction and low temperature. We also examined two thermodynamic relations with respect to entropy and found the constant-entropy curve to monotonically decrease in the metallic state with increasing of interactions. Thus, "adiabatic cooling" should be possible in a triangular lattice and be used to reach a magnetic ordered phase in a corresponding cold-atom system, which is a topic much under study presently. Doping the triangular system, we found the entropy to be maximized at the electron doped side with a concentration which coincides with the maximum Tc for SC found in Na_{x}CoO_{2.y}H_{2}O. The large entropy near optimal doping emerges from the interplay (crossover) of localized spins (remnants of the Mott insulator) and charge degress of freedom via doping. The existence of a topological SC phase (with a d+id symmetry) in this doped regime is also verified by our calculations.

One topical issue, which has recently been intensively studied, is the possibility of a magnetically ordered state appearing despite the frustration and, in particular, how to pin down its appearance in a convincing theoretical and experimental manner. In our work, we show that the one-electron spectral function is the key quantity to extract detailed information on the complex spin pattern in a frustrated magnetic system. This is demonstrated here by a detailed comparison of theory, which combines a priori density-functional (LDA) with cluster many-body (LDA + DCA) calculations, with high-precision angle-resolved photoelectron spectroscopy (ARPES). The role model in this work is the isotropic triangular antiferromagnetic adatom system Sn/Si(111). Its geometric frustration and strong electronic correlations are shown at low temperatures to combine to an unexpected magnetic, i.e. collinear order, and not the possible spiral (120^{o}antiferromagnetic order or a disordered spin-liquid phase.

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