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Register a new userFunctional renormalization group for fermion lattice models in three dimensions: application to the Hubbard model on the cubic lattice
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The channel-decomposed functional renormalization group (FRG) approach, most recently in the variant of truncated-unity-(TU-)FRG, has so far been used for various two-dimensional model systems. Yet, for many interesting material systems the third spatial dimension is of clear relevance. Therefore FRG schemes working in three spatial dimensions (3D) are definitely on the wishlist. Here we demonstrate that a 3D TUFRG scheme can be set up in straightforward extension of previous 2D codes and gives physically sensible results with affordable numerical effort, both regarding the qualitative as well as the quantitative description. The computed phase diagram of the three-dimensional Hubbard model at half filling or perfect nesting shows a phase transition to a ((pi,pi,pi))-ordered antiferromagnetic ground state for repulsive interactions at an energy scale that compares well with other numerical approaches in the literature. Furthermore, the method allowed us to detect a (d)-wave pairing and a concurring ((pi,pi,0)) antiferromagnetic ground state in the hole doped Hubbard model.
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We present a functional renormalization group analysis of superconductivity in the ground state of the attractive Hubbard model on a square lattice. Spontaneous symmetry breaking is treated in a purely fermionic setting via anomalous propagators and anomalous effective interactions. In addition to the anomalous interactions arising already in the reduced BCS model, effective interactions with three incoming legs and one outgoing leg (and vice versa) occur. We accomplish their integration into the usual diagrammatic formalism by introducing a Nambu matrix for the effective interactions. From a random-phase approximation generalized through use of this matrix we conclude that the impact of the 3+1 effective interactions is limited, especially considering the effective interactions important for the determination of the order parameter. The exact hierarchy of flow equations for one-particle irreducible vertex functions is truncated on the two-particle level, with higher-order self-energy corrections included in a scheme proposed by Katanin. Using a parametrization of effective interactions by patches in momentum space, the flow equations can be integrated numerically to the lowest scales without encountering divergences. Momentum-shell as well as interaction-flow cutoff functions are used, including a small external field or a large external field and a counterterm, respectively. Both approaches produce momentum-resolved order parameter values directly from the microscopic model. The size of the superconducting gap is in reasonable agreement with expectations from other studies.
Nematic order is an exotic property observed in several strongly correlated systems, such as the iron-based superconductors. Using large-scale density matrix renormalization group (DMRG) techniques, we study at zero-temperature the nematic spin liquid that competes with spin dipolar and quadrupolar orders. We use these nematic orders to characterize different quantum phases and quantum phase transitions. More specifically, we study a spin-$1$ bilinear-biquadratic Heisenberg model on the square lattice with couplings beyond nearest neighbors. We focus on parameter regions around the highly symmetric $SU(3)$ point where the bilinear and biquadratic interactions are equal. With growing further-neighbor biquadratic interactions, we identify different spin dipolar and quadrupolar orders. We find that the DMRG results on cylindrical geometries correctly detect nematicity in different quantum states and accurately characterize the quantum phase transitions among them. Therefore, spin-driven nematicity -- here defined as the spontaneous breaking of the lattice invariance under a 90$^o$ rotation -- is an order parameter which can be studied directly in DMRG calculations in two dimensions in different quantum states.
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We present a novel scheme for an unbiased and non-perturbative treatment of strongly correlated fermions. The proposed approach combines two of the most successful many-body methods, i.e., the dynamical mean field theory (DMFT) and the functional renormalization group (fRG). Physically, this allows for a systematic inclusion of non-local correlations via the flow equations of the fRG, after the local correlations are taken into account non-perturbatively by the DMFT. To demonstrate the feasibility of the approach, we present numerical results for the two-dimensional Hubbard model at half-filling.
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