Intertwined orders and exotic phase transitions in Dirac fermions

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Intertwined orders can compete or be mutually compatible. This statement acquires a deeper meaning when perceived from the point of view of Dirac fermions. Here orders correspond to mass terms and competition or mutual compatibility can be inferred from their commutation properties. In this grant proposal, we will design models where intertwined orders are dynamically generated in Dirac systems. The models we formulate are by construction free of the negative sign problem and large scale auxiliary field quantum Monte Carlo simulations can provide approximation free results in polynomial time. They will dynamically generate quantum spin Hall states, s-wave superconducting phases, antiferromagnetic insulators as well as Kekulé valence bond solids. Our aim is to understand transitions between any two of these phases in (2+1)-dimensions in light of i) the symmetry group of the respective phases and ii) the algebraic properties of the corresponding mass terms. Special attention will be placed on emergent symmetries as well as on the occurrence of topological and geometrical terms in the low energy effective field theory. Such terms are at the origin of phase transitions beyond the Ginzburg-Landau paradigm where topological defects of one phase carry the charge of the other phase and proliferate at the critical point. Another aspect of this grant proposal is to investigate alternative Monte Carlo sampling schemes aiming at reducing autocorrelation times for systems of fermions coupled to slowly varying fields in imaginary time.