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arXiv:2304.09001 · SciPost Physics

Quasi-particle functional renormalisation group calculations in the two-dimensional t-t'-Hubbard model

Daniel Rohe

Published Nov 13, 2023• 2 citations
arXivHTMLPDF

Abstract

We extend and apply a recently introduced quasi-particle functional renormalisation group scheme to the two-dimensional Hubbard model with next-nearest-neighbour hopping and away from half filling. We confirm the generation of superconducting correlations in some regions of the phase diagram, but also find that the inclusion of self-energy feedback by means of a decreasing quasi-particle weight can suppress superconducting tendencies more than anti-ferromagnetic correlations by which they are generated. As a supplement, we provide sample results for the self-energy in second-order perturbation theory and address some conceptual matters.

DOI
10.21468/scipostphys.15.5.192

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We extend and apply a recently introduced quasi-particle functional renormalisation group scheme to the two-dimensional Hubbard model with next-nearest-neighbour hopping and away from half filling. We confirm the generation of superconducting correlations in some regions of the phase diagram, but also find that the inclusion of self-energy feedback by means of a decreasing quasi-particle weight can suppress superconducting tendencies more than anti-ferromagnetic correlations by which they are generated. As a supplement, we provide sample results for the self-energy in second-order perturbation theory and address some conceptual matters.
In this thesis we apply the functional renormalisation group (fRG) to the two-dimensional Hubbard model (2D-HM). We do this by solving approximate renormalisation group equations for the self energy and the two-particle interaction numerically. The purpose of these calculations is two-fold in the sense that we want to learn about the reliability of the method on the one hand, and about the physics of the 2D-HM on the other. We find that different fRG schemes agree in describing the mutual interplay and the resulting competition between different correlations, but also that the results sometimes differ quantitatively. This sharpens our perception when it comes to interpreting fRG results, and tells us that we have to separate the indicative character from the quantitative aspects of the results. The main line of arguments presented in previous fRG studies is confirmed, that in the weak-coupling region particle-hole correlations feed back into the particle-particle channel, triggering strong superconducting correlations in a non-zero parameter range. Concerning the physics of the 2D-HM show new results for the single-particle spectral functions for the case of a Fermi surface which is not perfectly nested and intersects the Umklapp surface at so-called hot spots. These results are obtained from the Wick-ordered fRG scheme, and the calculations are done directly on the real frequency axis. We find that close to a critical temperature TcT_cTc​ where correlations become strong, a dip appears in the single-particle spectral functions for momenta close to and at the hot spots. Away from the van Hove scale, this is traced back to strong correlations in the particle-hole channel and attributed to antiferromagnetic fluctuations, in agreement with numerous previous studies. For a Fermi surface at or very close to the van Hove scale, there exists a degeneracy in the sense that the strongest correlations are found around a point in coupling space which is relevant for both superconductivity and antiferromagnetism, and we can thus not distinguish between the importance of the respective fluctuations.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2{t}_{2}t2​\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.
A frequency- and momentum-renormalization-group acceleration together with an analytical approach is used to obtain the retarded Green's function in the self-consistent and conserving fluctuation-exchange (FLEX) approximation for the two-dimensional Hubbard model in the normal state and in the superconducting state. The analytical expressions for this approach are given. For band-fillings near half filling the self-energy in the normal state exhibits Fermi-liquid behaviour for, low temperatures and frequencies near the chemical potential, if the momentum is chosen near the Fermi-surface. However due to the presence of large many body effects the observed Fermi-liquid region near the chemical potential and near the Fermi-surface is very small. Results for the single particle density of states, the occupation number and the spectral function are presented. The superconducting state with symmetry is obtained for U = 2 to U = 6 and a (U, n)-phase diagram for the transition temperature Tc is presented. A maximum Tc/t of 0.0275 is obtained for U = 6 near half filling.
Recent refinements of analytical and numerical methods have improved our understanding of the ground-state phase diagram of the two-dimensional (2D) Hubbard model. Here, we focus on variational approaches, but comparisons with both quantum cluster and Gaussian Monte Carlo methods are also made. Our own ansatz leads to an antiferromagnetic ground state at half filling with a slightly reduced staggered order parameter (as compared to simple mean-field theory). Away from half filling, we find d-wave superconductivity, but confined to densities where the Fermi surface passes through the antiferromagnetic zone boundary (if hopping between both nearest-neighbour and next-nearest-neighbour sites is considered). Our results agree surprisingly well with recent numerical studies using the quantum cluster method. An interesting trend is found by comparing gap parameters Δ (antiferromagnetic or superconducting) obtained with different variational wave functions. Δ varies by an order of magnitude and thus cannot be taken as a characteristic energy scale. In contrast, the order parameter is much less sensitive to the degree of sophistication of the variational schemes, at least at and near half filling.