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arXiv:1306.1434 · Physical Review B

Origin of high- Tc superconductivity in doped Hubbard models and their extensions: Roles of uniform charge fluctuations

Takahiro Misawa, Masatoshi Imada

Published Sep 22, 2014• 99 citations
arXivHTMLPDF

Abstract

The doped Hubbard model is a simple model for high-${T}_{c}$ cuprate superconductors, while its ground state remains a challenge. Here, by performing state-of-the-art variational Monte Carlo calculations for the strong-coupling Hubbard model, we find evidence that the $d$-wave superconducting phase emerges always near the phase separation region and the superconducting order has one-to-one correspondence with the enhancement of charge compressibility. The order as well as the phase separation are vulnerable to realistic intersite Coulomb interaction, while the superexchange interaction enhances both. An appropriate combination of these two widens the stable superconducting phase.

DOI
10.1103/physrevb.90.115137

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The doped Hubbard model is a simple model for high-Tc{T}_{c}Tc​ cuprate superconductors, while its ground state remains a challenge. Here, by performing state-of-the-art variational Monte Carlo calculations for the strong-coupling Hubbard model, we find evidence that the ddd-wave superconducting phase emerges always near the phase separation region and the superconducting order has one-to-one correspondence with the enhancement of charge compressibility. The order as well as the phase separation are vulnerable to realistic intersite Coulomb interaction, while the superexchange interaction enhances both. An appropriate combination of these two widens the stable superconducting phase.
The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the ddd-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration ($\ensuremath{\delta}$) dependence of the Hubbard model in the ground states on a square lattice at strong coupling U/t=10U/t=10U/t=10, for the on-site interaction UUU and the transfer ttt, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of 0.01t0.01t0.01t. We have identified distinct phases including a uniform ddd-wave superconducting phase for $0.17\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.22$ and a stripe charge/spin ordered phase for $\ensuremath{\delta}\ensuremath{\lesssim}0.17$ with the stripe period depending on $\ensuremath{\delta}$, together with presumable spatially coexisting antiferromagnetic and stripe order for $\ensuremath{\delta}\ensuremath{\lesssim}0.07$ and coexisting stripe and ddd-wave superconductivity for $0.07\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.17$. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around $\ensuremath{\delta}=0.17$ in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.
Identification of the electronic state that appears upon doping a Mott insulator is important to understand the physics of cuprate high-temperature superconductors. Recent scanning tunneling microscopy of cuprates provides evidence that a charge-ordered state emerges before the superconducting state upon doping the parent compound. We study this phenomenon by computing the charge response function of the Hubbard model including frequency-dependent local vertex corrections that satisfy the compressibility sum rule. We find that upon approaching the Mott phase from the overdoped side, the charge fluctuations at wave vectors connecting hot spots are suppressed much faster than at the other wave vectors. It leads to a momentum dependence of the dressed charge susceptibility that is very different from either the bare susceptibility or from the susceptibility obtained from the random phase approximation. We also find that the paramagnetic lightly hole-doped Mott phase at finite temperature is unstable to charge ordering only at zero wave vector, confirming the results previously obtained from the compressibility. Charge order is driven by the frequency-dependent scattering processes that induce an attractive particle-hole interaction at large interaction strength and small doping.
By using variational wave functions and quantum Monte Carlo techniques, we investigate the interplay between electron-electron and electron-phonon interactions in the two-dimensional Hubbard-Holstein model. Here, the ground-state phase diagram is triggered by several energy scales, i.e., the electron hopping ttt, the on-site electron-electron interaction UUU, the phonon energy ${\ensuremath{\omega}}_{0}$, and the electron-phonon coupling ggg. At half filling, the ground state is an antiferromagnetic insulator for $U\ensuremath{\gtrsim}2{g}^{2}/{\ensuremath{\omega}}_{0}$, while it is a charge-density-wave (or bipolaronic) insulator for $U\ensuremath{\lesssim}2{g}^{2}/{\ensuremath{\omega}}_{0}$. In addition to these phases, we find a superconducting phase that intrudes between them. For ${\ensuremath{\omega}}_{0}/t=1$, superconductivity emerges when both U/tU/tU/t and $2{g}^{2}/t{\ensuremath{\omega}}_{0}$ are small; then, by increasing the value of the phonon energy ${\ensuremath{\omega}}_{0}$, it extends along the transition line between antiferromagnetic and charge-density-wave insulators. Away from half filling, phase separation occurs when doping the charge-density-wave insulator, while a uniform (superconducting) ground state is found when doping the superconducting phase. In the analysis of finite-size effects, it is extremely important to average over twisted boundary conditions, especially in the weak-coupling limit and in the doped case.
We investigate the ground-state properties of the two-dimensional Hubbard model, based on the off-diagonal wave function variational Monte Carlo method. We use an optimized wave function that is improved from an initial one-body wave function by multiplying by multiple correlation operators that are given by the form of exp(−S-S−S)-type where SSS is a suitable operator. We examine the inhomogeneous ground state at near 1/8 doping in the strongly correlated region where the on-site Coulomb interaction is larger than the bandwidth. The ddd-wave superconductivity with a spatially oscillating gap function can coexist with the charge ordering in the ground state without magnetic ordering, and superconducting condensation energy increases by this coexistence. We also show the ddd-wave pair correlation function as a function of lattice sites. The correlation function indicates that the long-range superconducting order indeed exists, and also that the strong electron correlation suppresses the pair correlation function.
Motivated by significant discrepancies between experimental observations of electron-doped cuprates and numerical results of the Hubbard and $t\text{\ensuremath{-}}J$ models, we investigate the role of intersite interactions VVV by studying the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model on square lattices. Based on large-scale density matrix renormalization group simulations, we identify the ground-state phase diagram across varying intersite interactions VVV and doping concentration $\ensuremath{\delta}$. We find that the phase diagram with finite intersite interactions $2\ensuremath{\lesssim}V/J\ensuremath{\lesssim}3$ offers a more accurate description of electron-doped cuprates than the conventional Hubbard and $t\text{\ensuremath{-}}J$ models. Moreover, we reveal the role of intersite interactions VVV at varying doping levels: at light doping, intersite interactions favor N\'eel antiferromagnetic order, and suppress both superconductivity and charge density wave; around optimal doping, these interactions support a pseudogap-like phase while suppressing superconductivity, and we further perform the slave-boson mean-field analysis to understand the numerical results microscopically; at higher doping, the effects of intersite interactions become insignificant, with our numerical predictions suggesting the emergence of an incommensurate spin density wave phase. Our specific focus around optimal doping with various intersite interactions identifies successive phases including phase separation, uniform ddd-wave SC, and a pseudogap-like phase, and reveals a relative insensitivity of the charge density wave to superconductivity. Our study suggests the $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$ model as the minimal model to capture the essential physics of electron-doped cuprates.