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
t, the on-site electron-electron interaction
U, the phonon energy ${\ensuremath{\omega}}_{0}$, and the electron-phonon coupling
g. 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/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 study the Holstein-Hubbard model at half filling to explore ordered phases including superconductivity (SC), antiferromagnetism (AF), and charge order (CO) in situations where the electron-electron and electron-phonon interactions are strong (comparable to the electronic bandwidth). The model is solved in the dynamical mean-field approximation with a continuous-time quantum Monte Carlo impurity solver. We determine the superconducting transition temperature
Tc and the SC order parameter and show that the phonon-induced retardation or the strong Coulomb interaction leads to a significant reduction and shift of the
Tc dome against the effective electron-electron interaction
Ueff given by the Hubbard
U reduced by the phonon-mediated attraction in the static limit. This behavior is analyzed by comparison to an effective static model in the polaron representation with a renormalized bandwidth. In addition, we discuss the superconducting gap $\ensuremath{\Delta}$ and $2\ensuremath{\Delta}/{T}_{c}$ to reveal the effect of the retardation and the Coulomb interaction. We also determine the finite-temperature phase diagram including AF and CO. In the moderate-coupling regime, there is a hysteretic region of AF and CO around
Ueff=0, while the two phases are separated by a paramagnetic metal in the weak-coupling regime and a paramagnetic insulator in the strong-coupling regime.
Recently, a flurry of works have found strong competition between charge density wave (CDW) and superconductivity (SC) in the doped Hubbard and $t\text{\ensuremath{-}}J$ models on the square lattice. Interestingly, some recent results suggest that the electron-phonon coupling may suppress CDW order and enhance SC. In this work, we consider the square-lattice Hubbard model with the Holstein or Su-Schrieffer-Heeger electron-phonon coupling at the large-
U and antiadiabatic (infinite phonon frequency) limits, which gives an effective $t\text{\ensuremath{-}}J$ model with either a density attractive interaction
V or a
JP term that contributes a larger spin exchange and a density repulsive interaction. To explore how these effective couplings may tune the competition between CDW and SC, we implement the density matrix renormalization group simulations on the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model with
V or
JP coupling. We focus on the six-leg cylinder system with the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, which hosts partially filled stripe and
d-wave SC in phase diagram. By tuning ${t}^{\ensuremath{'}}/t>0$ and
V or
JP, we establish two quantum phase diagrams at
121 doping ratio. In the SC phases, the increased
V or
JP coupling can enhance the quasi-long-range SC order, consistent with some previous findings. Nonetheless, no SC emerges when the partially filled stripe phase disappears with increased
V or
JP. Instead, the system has a transition to either a phase-separation-like regime or a filled stripe phase. On the other hand, with increased ${t}^{\ensuremath{'}}/t$, not only the partially filled stripe but the phase separation and filled stripe can also be tuned to SC phase. Our results suggest that although
V and
JP couplings may strengthen hole binding, the hole dynamics controlled by ${t}^{\ensuremath{'}}/t$ appears to play a more crucial role for obtaining a SC in the $t\text{\ensuremath{-}}J$ model.
We investigate the role of electron-electron and electron-phonon interactions in strongly correlated systems by performing unbiased quantum Monte Carlo simulations in the square lattice Hubbard-Holstein model at half-filling. We study the competition and interplay between antiferromagnetism (AFM) and charge-density wave (CDW), establishing its very rich phase diagram. In the region between AFM and CDW phases, we have found an enhancement of superconducting pairing correlations, favoring (nonlocal)
s-wave pairs. Our study sheds light over past inconsistencies in the literature, in particular the emergence of CDW in the pure Holstein model case.
We present electron and phonon spectral functions calculated from determinant quantum Monte Carlo simulations of the half-filled two-dimensional Hubbard-Holstein model on a square lattice. By tuning the relative electron-electron $(e\ensuremath{-}e)$ and electron-phonon $(e\ensuremath{-}ph)$ interaction strengths, we show the electron spectral function evolving between antiferromagnetic insulating, metallic, and charge-density-wave (CDW) insulating phases. The phonon spectra concurrently gain a strong momentum dependence and soften in energy upon approaching the CDW phase. In particular, we study how the $e\ensuremath{-}e$ and $e\ensuremath{-}ph$ interactions renormalize the spectra and find that the presence of both interactions suppresses the amount of renormalization at low energy, thus allowing the emergence of a metallic phase at intermediate coupling strengths. In addition, we find a modest enhancement of the
d-wave pairing susceptibility in the metallic regime, although spin and charge correlations are still dominant at the temperatures considered in our study. These findings demonstrate the importance of considering the influence of multiple interactions in spectroscopically determining any one interaction strength in strongly correlated materials.
We investigate the magnetotransport properties of the one-band Hubbard-Holstein model at half-filling in three dimensions (3D) using exact diagonalization based semi-classical Monte Carlo simulations with phonons treated in the adiabatic limit. The low-temperature electronic correlation
U vs electron-phonon coupling
V phase diagram reveals two insulating phases--antiferromagnetic (AF-I) and charge-ordered (CO-I)--separated by a first-order transition, with no metallic phase observed at their intersection, indicating robustness of these phases in 3D. For
U∼bandwidth, the
V vs temperature
T phase diagram exhibits multiple phases including AF-I, CO-I, Mott-Hubbard insulator, bipolaronic insulator, and two bipolaronic metallic states. Several first-order transitions occur near
V∼3.75. Above the ordering temperature, the density of states shows universal behavior dominated by electronic contribution, while susceptibility and DOS analyses reveal pseudogap features. Magnetic and transport properties along the phase boundary highlight strong proximity effects between competing phases, suggesting routes for tuning correlated materials and emergent electronic states.