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arXiv:2307.02360 · Physical review. B./Physical review. B

Emergent nearest-neighbor attraction in the fully renormalized interactions of the single-band repulsive Hubbard model at weak coupling

Daria Gazizova, J. P. F. LeBlanc

Published Oct 27, 2023• 5 citations
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Abstract

We compute the perturbative expansion for the effective interaction $W$ of the half-filled 2-dimensional Hubbard model. We derive extensions of standard random-phase-approximation resummations that include arbitrarily high order contributions in the ${W}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}$ and ${W}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ basis. Using algorithmic tools we explore the static $Q$-dependent interaction as well as the same-time quantity both in momentum and real space. We emphasize the absence of screening in the Hubbard interaction where we find an enhanced repulsive local ${W}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ with a nonzero attractive ${W}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}$. Finally, starting from only a locally repulsive bare interaction we find an emergent nonlocal nearest-neighbor attraction for low temperatures at sufficiently large values of $U/t$ which may be key to understanding pairing processes in the model.

DOI
10.1103/physrevb.108.165149

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We compute the perturbative expansion for the effective interaction WWW of the half-filled 2-dimensional Hubbard model. We derive extensions of standard random-phase-approximation resummations that include arbitrarily high order contributions in the W\ensuremath↑\ensuremath↑{W}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}W\ensuremath↑\ensuremath↑​ and W\ensuremath↑\ensuremath↓{W}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}W\ensuremath↑\ensuremath↓​ basis. Using algorithmic tools we explore the static QQQ-dependent interaction as well as the same-time quantity both in momentum and real space. We emphasize the absence of screening in the Hubbard interaction where we find an enhanced repulsive local W\ensuremath↑\ensuremath↓{W}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}W\ensuremath↑\ensuremath↓​ with a nonzero attractive W\ensuremath↑\ensuremath↑{W}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}W\ensuremath↑\ensuremath↑​. Finally, starting from only a locally repulsive bare interaction we find an emergent nonlocal nearest-neighbor attraction for low temperatures at sufficiently large values of U/tU/tU/t which may be key to understanding pairing processes in the model.
We describe a recent implementation of the combined GWGWGW and dynamical mean field method (GWGWGW+DMFT) for the two-dimensional Hubbard model with onsite and nearest-neighbor repulsion. We clarify the relation of the GWGWGW+DMFT scheme to alternative approaches in the literature, and discuss the corresponding approximations to the free-energy functional of the model. Furthermore, we describe a numerically exact technique for the solution of the GWGWGW+DMFT equations, namely, the hybridization expansion continuous-time algorithm for impurity models with retarded interactions. We compute the low-temperature phase diagram of the half-filled extended Hubbard model, addressing the metal-insulator transition at small intersite interactions and the transition to a charge-ordered state for stronger intersite repulsions. GWGWGW+DMFT introduces a nontrivial momentum dependence into the many-body self-energy and polarization. We find that the charge fluctuations included in the present approach have a larger impact on the latter than on the former. Finally, within the GWGWGW+DMFT framework, as in extended DMFT, the intersite repulsion translates into a frequency dependence of the local effective interaction. We analyze this dependence and show how it affects the local spectral function.
We develop a systematic self-consistent perturbative expansion for the self-energy of Hubbard-like models. The interaction lines in the Feynman diagrams are dynamically screened by the charge fluctuations in the system. Although the formal expansion is exact---assuming that the model under the study is perturbative---only if diagrams to all orders are included, it is shown that for large-on-site-Coulomb-repulsion-UUU systems weak-coupling expansions to a few orders may already converge. In order to test the approximation at intermediate-to-high temperatures, we use the exact charge-fluctuation susceptibility from quantum Monte Carlo (QMC) simulation studies as input, which determines the exact screened interaction, and compare our results for the self-energy to the QMC results. We also make comparisons with fluctuation-exchange approximation. We show that the screened interaction for the large-UUU system can be vanishingly small at a certain intermediate electron filling, and it is found that our approximation for the imaginary part of the one-particle self-energy agrees well with the QMC results in the low-energy scales at this particular filling. But the usefulness of the approximation is hindered by the fact that it has the incorrect filling dependence when the filling deviates from this value. We also calculate the exact QMC Fermi surfaces for the two-dimensional (2D) Hubbard model for several fillings. Our results near half filling show extreme violation of the concepts of the band theory; in fact, instead of growing, the Fermi surface vanishes when doped toward the half-filled Mott-Hubbard insulator. Sufficiently away from half filling, noninteractinglike Fermi surfaces are recovered. These results combined with the Luttinger theorem might show that diagrammatic expansions for the nearly-half-filled Hubbard model are unlikely to be possible; however, the nonperturbative part of the solution seems to be less important as the filling gradually moves away from one half. Results for the 2D one-band Hubbard model for several hole dopings are presented. Implications of this study for the high-temperature superconductors are also discussed.
The half-filled extended Hubbard model (including nearest-neighbor interaction) is considered in perturbation theory at weak coupling on a d-dimensional hypercubic lattice (d\ensuremath{\gtrsim}3). It is shown that the exact critical temperature and the exact order parameter in the weak-coupling limit differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion. The results depend sensitively on dimension and on the relative strength of the nearest-neighbor interaction. The renormalization factor in d=3, estimated from the 1/d expansion, is q\ensuremath{\simeq}0.282 for the pure Hubbard model, q\ensuremath{\simeq}0.179 near the spin-density-wave to charge-density-wave transition and q\ensuremath{\simeq}0.830 in the limit of pure nearest-neighbor interaction.
We report on a nonperturbative approach to the one-dimensional (1D) and two-dimensional (2D) Hubbard models that is capable of recovering both strong- $(U\ensuremath{\gg}t)$ and weak-coupling $(U\ensuremath{\ll}t)$ limits, with U the on-site Coulomb repulsion and t the kinetic energy. Dynamical corrections to the electron self-energy in the single particle Green function are explicitly included by expanding in terms of the 16 eigenstates that characterize two nearest neighbor sites. We first show that even when U is much smaller than the bandwidth, the Mott-Hubbard gap never closes at half filling in both 1D and 2D. Consequently, the Hubbard model at half filling is always in the strong-coupling nonperturbative regime. For both large and small U, we find that the population of nearest-neighbor singlet states approaches a value of order unity as $\stackrel{\ensuremath{\rightarrow}}{T}0$ as would be expected for antiferromagnetic order. We also find that the double occupancy is a smooth monotonic function of U and approaches the anticipated noninteracting limit of 1/41/41/4 as $\stackrel{\ensuremath{\rightarrow}}{U}0$ and vanishes as $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}.$ Finally, we compute the heat capacity [C(T,U)][C(T,U)][C(T,U)] for both 1D and 2D. Our results for 1D at moderate to high temperatures are in quantitative agreement with those of the exact Bethe ansatz solution, differing by no more than 11%.1 In addition, we find that in 2D, the C(T,U)C(T,U)C(T,U) curves vs T for different values of U exhibit a universal crossing point at two characteristic temperatures $T\ensuremath{\approx}1.7t\ifmmode\pm\else\textpm\fi{}0.1t$ and $T\ensuremath{\approx}0.4\ifmmode\pm\else\textpm\fi{}0.1t$ as is seen universally in Hubbard models and experimentally in a wide range of strongly correlated systems such as 3He,{}^{3}\mathrm{He},3He, UBe3,{\mathrm{UBe}}_{3},UBe3​, and ${\mathrm{CeCu}}_{6\ensuremath{-}x}{\mathrm{Al}}_{x}.$ The success of this method in recovering well-established results that stem fundamentally from the Coulomb interaction suggests that local dynamics are at the heart of the physics of strongly correlated systems.
Salmhofer [Commun. Math. Phys. 194, 249 (1998)] recently developed a new renormalization-group method for interacting Fermi systems, where the complete flow from the bare action of a microscopic model to the effective low-energy action, as a function of a continuously decreasing infrared cutoff, is given by a differential flow equation which is local in the flow parameter. We apply this approach to the repulsive two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping amplitudes. The flow equation for the effective interaction is evaluated numerically on a one-loop level. The effective interactions diverge at a finite-energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions we extend Salmhofer's renormalization group for the calculation of susceptibilities. We compute the singlet superconducting susceptibilities for various pairing symmetries, and also charge- and spin-density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength, and band filling) we find commensurate and incommensurate antiferromagnetic instabilities or d-wave superconductivity as leading instability. We present the resulting phase diagram in the vicinity of half-filling, and also results for the density dependence of the critical energy scale.
A half-filled one-dimensional (1D) extended Hubbard model (U-V_⊥-V_∥) with on-site and nearest-neighbor spin-dependent interactions is studied analytically. The ground state phase diagram is obtained in the weak-coupling regime by performing the bosonization and renormalization group methods. The result shows that the nearest-neighbor spin-dependent attraction influences the low-energy physics of the 1D correlated electron system, and changes the topological structure of the ground state phase diagram of the U-V model.