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arXiv:2010.06267 · Physical review. B./Physical review. B

Dynamical functional renormalization group computation of order parameters and critical temperatures in the two-dimensional Hubbard model

Demetrio Vilardi, Pietro M. Bonetti, Walter Metzner

Published Dec 18, 2020• 33 citations
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Abstract

We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping $p$ and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with N\'eel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the $(p,T)$ phase diagram centered around 15% hole doping.

DOI
10.1103/physrevb.102.245128

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Abstract The interplay and competition of magnetic and superconducting correlations in the weakly interacting two‐dimensional Hubbard Model is investigated by means of the functional renormalization group. At zero temperature the flow of interactions in one‐loop approximation evolves into a strong coupling regime at low energy scales, signalling the possible onset of spontaneous symmetry breaking. This is further analyzed by a mean‐field treatment of the strong renormalized interactions which takes into account magnetic and superconducting order simultaneously. The effect of strong correlations on single‐particle properties in the normal phase is studied by calculating the flow of the self‐energy. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping ppp and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with N'eel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the (p,T)(p,T)(p,T) phase diagram centered around 15% hole doping.
Using the dynamical mean-field theory (DMFT) as a ``booster rocket'', the functional renormalization group (fRG) can be upgraded from a weak-coupling method to a powerful computation tool for strongly interacting fermion systems. The strong local correlations are treated nonperturbatively by the DMFT, while the fRG flow can be formulated such that it is driven exclusively by nonlocal correlations, which are more amenable to approximations. We show that the full frequency dependence of the two-particle vertex needs to be taken into account in this approach, and demonstrate that this is actually possible---in spite of the singular frequency dependence of the vertex at strong coupling. We are thus able to present results obtained from the DMFT-boosted fRG for the two-dimensional Hubbard model in the strongly interacting regime. We find strong antiferromagnetic correlations from half filling to 18% hole doping and at the lowest temperature we can access, a sizable ddd-wave pairing interaction driven by magnetic correlations at the edge of the antiferromagnetic regime.
We analyze the competition between antiferromagnetism and superconductivity in the two-dimensional Hubbard model by combining a functional renormalization group flow with a mean-field theory for spontaneous symmetry breaking. Effective interactions are computed by integrating out states above a scale ${\ensuremath{\Lambda}}_{\mathrm{MF}}$ in one-loop approximation, which captures in particular the generation of an attraction in the ddd-wave Cooper channel from fluctuations in the particle-hole channel. These effective interactions are then used as an input for a mean-field treatment of the remaining low-energy states, with antiferromagnetism, singlet superconductivity, and triplet $\ensuremath{\pi}$ pairing as the possible order parameters. Antiferromagnetism and superconductivity suppress each other, leaving only a small region in parameter space where both orders can coexist with a sizable order parameter for each. Triplet $\ensuremath{\pi}$ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.
The interplay and competition of magnetic and superconducting correlations in the weakly interacting two-dimensional Hubbard Model is investigated by means of the functional renormalization group. At zero temperature the flow of interactions in one-loop approximation evolves into a strong coupling regime at low energy scales, signalling the possible onset of spontaneous symmetry breaking. This is further analyzed by a mean-field treatment of the strong renormalized interactions which takes into account magnetic and superconducting order simultaneously. The effect of strong correlations on single-particle properties in the normal phase is studied by calculating the flow of the self-energy. 1
The phases with spontaneously broken symmetries corresponding to antiferromagnetic and ddd-wave superconducting order in the two-dimensional $t\ensuremath{-}{t}^{'}$-Hubbard model are investigated by means of the functional renormalization group. The introduction of composite boson fields in the magnetic, charge density, and superconducting channels allows an efficient parametrization of the four-fermion vertex and the study of regimes where either the antiferromagnetic or superconducting order parameter, or both, are nonzero. We compute the phase diagram and the temperature dependence of the order parameter below the critical temperature, where antiferromagnetic and superconducting order show a tendency of mutual exclusion.
The treatment of strongly interacting two-dimensional Fermi systems constitutes one of the most challenging problems in the field of condensed matter physics. Many theoretical works focused on the strongly coupled Hubbard model, since it is expected to capture the most important physics of cuprate superconductors. Due to our methodological improvements and understanding of the frequency dependence of the two-particle vertex function, the functional renormalization group combined with the dynamical mean field theory can now be used to study competing correlations in the strongly interacting regime. While limited to moderate interaction strengths, the functional renormalization group describes efficiently systems with a hierarchy of different energy scales and competing correlations. For instance, it provides definite evidence for d-wave superconductivity in the two-dimensional Hubbard model at moderate coupling. In a first project, we study the frequency dependence of the one-particle irreducible vertex function generated by the functional renormalization group flow. The frequency dependence, which becomes singular for strong interactions, appears to be important already for moderate couplings, and it cannot be represented by separate channels each depending on a single linear combination of frequencies. For strongly interacting systems, the dynamical mean field theory captures strong local correlation effects nonperturbatively. With this approximation, we study the impact of local correlations on the magnetic susceptibility. The local dynamics strongly affects the spin response function and its momentum dependence. In contrast to the widely used random-phase approximation with self-energy corrections that predicts Néel antiferromagnetic order, the local vertex corrections favor an incommensurate order similar to the ordering instability predicted by the Fermi surface geometry, as for weakly interacting systems. The dynamical mean field theory is also used as starting point for the functional renormalization group flow. We demonstrate that, due to our improvements for the parametrization of the vertex function, this approach is actually able to access the strong coupling physics. Moreover, we derive a flow scheme that conserves the local contributions and reduces the truncation error of the flow equations. In the strongly interacting regime, we capture strong d-wave pairing correlations driven by magnetic fluctuations with a mechanism similar to the one observed in the weakly interacting system.
Abstract The interplay and competition of magnetic and superconducting correlations in the weakly interacting two‐dimensional Hubbard Model is investigated by means of the functional renormalization group. At zero temperature the flow of interactions in one‐loop approximation evolves into a strong coupling regime at low energy scales, signalling the possible onset of spontaneous symmetry breaking. This is further analyzed by a mean‐field treatment of the strong renormalized interactions which takes into account magnetic and superconducting order simultaneously. The effect of strong correlations on single‐particle properties in the normal phase is studied by calculating the flow of the self‐energy. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)