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arXiv:1111.2982 · Physical Review B

Effects of longer-range interactions on unconventional superconductivity

S. Raghu, Erez Berg, Andrey V. Chubukov, Steven Allan Kivelson

Published Jan 9, 2012• 71 citations
arXivHTMLPDF

Abstract

We analyze the effect of the nonvanishing range of electron-electron repulsion on the mechanism of unconventional superconductivity. We present asymptotically exact weak-coupling results for dilute electrons in the continuum and for the 2D extended Hubbard model, as well as density-matrix renormalization group results for the two-leg extended Hubbard model at intermediate couplings, and approximate results for the case of realistically screened Coulomb interactions. We show that ${T}_{c}$ is generally suppressed in some pairing channels as longer range interactions increase in strength, but superconductivity is not destroyed. Our results confirm that electron-electron interaction can lead to unconventional superconductivity under physically realistic circumstances.

DOI
10.1103/physrevb.85.024516

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In superconducting phases, repulsive interactions between the electrons usually lead to higher-angular momentum Cooper pairing. Here, the authors employ the weak-coupling renormalization group approach to study unconventional superconducting phases emerging in the extended, repulsive Hubbard model on paradigmatic two-dimensional lattices. Among their findings are extended regions of chiral topological superconductivity on the square, triangular, and honeycomb lattices. They provide detailed superconducting phase diagrams as well as the coupling strengths that quantify the corresponding critical temperatures depending on the band-structure parameters, band filling, and interaction parameters. The importance of the resolution and accuracy of the numerical integration for the method is discussed and a way to higher numerical efficiency provided.
We study the phase diagram of the Hubbard model in the limit where UUU, the onsite repulsive interaction, is much smaller than the bandwidth. We present an asymptotically exact expression for Tc{T}_{c}Tc​, the superconducting transition temperature, in terms of the correlation functions of the noninteracting system which is valid for arbitrary densities so long as the interactions are sufficiently small. Our strategy for computing Tc{T}_{c}Tc​ involves first integrating out all degrees of freedom having energy higher than an unphysical initial cutoff ${\ensuremath{\Omega}}_{0}$. Then, the renormalization group (RG) flows of the resulting effective action are computed and Tc{T}_{c}Tc​ is obtained by determining the scale below which the RG flows in the Cooper channel diverge. We prove that Tc{T}_{c}Tc​ is independent of ${\ensuremath{\Omega}}_{0}$. Using this method, we find a variety of unconventional superconducting ground states in two- and three-dimensional lattice systems, and present explicit results for Tc{T}_{c}Tc​ and pairing symmetries as a function of the electron concentration.
Charge ordering is often found in the phase diagram of unconventional superconductors in close proximity to the superconducting state. This has led to the suggestion that fluctuations of charge order can mediate superconducting pairing. While several mechanisms can lead to charge order, one common mechanism is the long-range Coulomb interaction, resulting in a Wigner crystal charge ordered state. For an electron density of 0.5 per site, we investigate the extended Hubbard model on the two-dimensional square lattice using exact diagonalization and density matrix renormalization group methods. Our results show that the strength of pairing decreases with the nearest-neighbor Coulomb interaction strength VVV and remains weaker than the tendency of pairing for noninteracting electrons.
We investigate the influence of nonlocal interactions on superconducting correlations within the extended Hubbard model. In addition to the on-site Coulomb interaction and nearest-neighbor hopping, we include next-nearest-neighbor hopping together with several physically relevant nonlocal terms, namely nearest-neighbor Coulomb interaction, correlated hopping, exchange interaction, and pair hopping. Using Lanczos exact diagonalization on a 4×44\times44×4 cluster, supported by projector quantum Monte Carlo simulations for selected parameter regimes, we analyze pairing correlations in both the s- and d-wave channels. We demonstrate that nonlocal interactions exert a highly nontrivial and symmetry-dependent influence on superconducting correlations. While the on-site repulsion in cooperation with next-nearest-neighbor hopping enhances d-wave pairing tendencies, correlated hopping and nearest-neighbor Coulomb interaction strongly promote s-wave correlations, whereas exchange and pair-hopping interactions can efficiently suppress superconductivity beyond relatively small critical strengths. When all nonlocal interactions are considered simultaneously, the resulting phase diagrams reveal a complex interplay and competition between different pairing symmetries. Our results highlight the crucial role of extended interactions in shaping the pairing landscape of strongly correlated systems and demonstrate that a comprehensive treatment beyond the minimal Hubbard model is essential for a realistic description of unconventional superconductivity.
Abstract We investigate the influence of nonlocal interactions on superconducting correlations within the extended Hubbard model. In addition to the on-site Coulomb interaction and nearest-neighbor hopping, we include next-nearest-neighbor hopping together with several physically relevant nonlocal terms, namely nearest-neighbor Coulomb interaction, correlated hopping, exchange interaction, and pair hopping. Using Lanczos exact diagonalization on a 4×44\times 44×4 4 × 4 cluster, supported by projector quantum Monte Carlo simulations for selected parameter regimes, we analyze pairing correlations in both the s - and d -wave channels. We demonstrate that nonlocal interactions exert a highly nontrivial and symmetry-dependent influence on superconducting correlations. While the on-site repulsion in cooperation with next-nearest-neighbor hopping enhances d -wave pairing tendencies, correlated hopping and nearest-neighbor Coulomb interaction strongly promote s -wave correlations, whereas exchange and pair-hopping interactions can efficiently suppress superconductivity beyond relatively small critical strengths. When all nonlocal interactions are considered simultaneously, the resulting phase diagrams reveal a complex interplay and competition between different pairing symmetries. Our results highlight the crucial role of extended interactions in shaping the pairing landscape of strongly correlated systems and demonstrate that a comprehensive treatment beyond the minimal Hubbard model is essential for a realistic description of unconventional superconductivity.
The Hubbard and related models serve as a fundamental starting point in understanding the novel experimental phenomena in correlated electron materials, such as superconductivity, Mott insulator, magnetism. Recent numerical simulations demonstrate that the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ plays a key role for the superconductivity. However, the impacts of long-range intersite interactions in such a ${t}^{\ensuremath{'}}$-Hubbard model are less explored. Using the state-of-art density-matrix renormalization group method, we investigate the ${t}^{\ensuremath{'}}$-Hubbard model on a two-leg ladder with the intersite interactions extended to the fourth neighbors. The numerical results show that the quasi-long-range superconducting correlation remains stable under the repulsive intersite interactions and these long-range repulsive interactions only change the ground state quantitatively. In addition, inspired by recent experiments on one-dimensional cuprate chain ${\mathrm{Ba}}_{2\ensuremath{-}\mathrm{x}}{\mathrm{Sr}}_{\mathrm{x}}{\mathrm{CuO}}_{3+\ensuremath{\delta}}$, we find that the nearest-neighbor attractive interaction significantly enhances the superconducting correlation when it is comparable to the nearest-neighbor hopping ttt. Stronger attraction drives the system into an electron-hole phase separation. Finally, we discuss the effects of the on-site interaction on superconductivity.