Two-dimensional Hubbard-Holstein Model Research Articles

Stripe correlations in the two-dimensional Hubbard-Holstein model

Several state-of-the-art numerical methods have observed static or fluctuating spin and charge stripes in doped two-dimensional Hubbard models, suggesting that these orders play a significant role in shaping the cuprate phase diagram. Many experiments, however, also indicate that the cuprates have strong electron-phonon (e-ph) coupling, and it is unclear how this interaction influences stripe correlations. We study static and fluctuating stripe orders in the doped single-band Hubbard-Holstein model using zero temperature variational Monte Carlo and finite temperature determinant quantum Monte Carlo. We find that the lattice couples more strongly with the charge component of the stripes, leading to an enhancement or suppression of stripe correlations, depending on model parameters like the next-nearest-neighbor hopping {t}^{{prime} } or phonon energy Ω. Our results help elucidate how the e-ph interaction can tip the delicate balance between stripe and superconducting correlations in the Hubbard-Holstein model with implications for our understanding of the high-Tc cuprates.

Zero-temperature phases of the two-dimensional Hubbard-Holstein model: A non-Gaussian exact diagonalization study

We propose a numerical method which embeds the variational non-Gaussian wavefunction approach within exact diagonalization, allowing for efficient treatment of correlated systems with both electron-electron and electron-phonon interactions. Using a generalized polaron transformation, we construct a variational wavefunction that absorbs entanglement between electrons and phonons into a variational non-Gaussian transformation; exact diagonalization is then used to treat the electronic part of the wavefunction exactly, thus taking into account high-order correlation effects beyond the Gaussian level. Keeping the full electronic Hilbert space, the complexity is increased only by a polynomial scaling factor relative to the exact diagonalization calculation for pure electrons. As an example, we use this method to study ground-state properties of the two-dimensional Hubbard-Holstein model, providing evidence for the existence of intervening phases between the spin and charge-ordered states. In particular, we find one of the intervening phases has strong charge susceptibility and binding energy, but is distinct from a charge-density-wave ordered state, while the other intervening phase displays superconductivity at weak couplings. This method, as a general framework, can be extended to treat excited states and dynamics, as well as a wide range of systems with both electron-electron and electron-boson interactions.

Phase diagram of the two-dimensional Hubbard-Holstein model

The electron–electron and electron–phonon interactions play an important role in correlated materials, being key features for spin, charge and pair correlations. Thus, here we investigate their effects 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, favouring (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.

Study of supersolidity in the two-dimensional Hubbard–Holstein model

We derive an effective Hamiltonian for the two-dimensional Hubbard-Holstein model in the regimes of strong electron-electron and strong electron-phonon interactions by using a nonperturbative approach. In the parameter region where the system manifests the existence of a correlated singlet phase, the effective Hamiltonian transforms to a $t_1-V_1-V_2-V_3$ Hamiltonian for hard-core-bosons on a checkerboard lattice. We employ quantum Monte Carlo simulations, involving stochastic-series-expansion technique, to obtain the ground state phase diagram. At filling $1/8$, as the strength of off-site repulsion increases, the system undergoes a first-order transition from a superfluid to a diagonal striped solid with ordering wavevector $\vec{Q}=(\pi/4,3\pi/4)$ or $(\pi/4,5\pi/4)$. Unlike the one-dimensional situation, our results in the two-dimensional case reveal a supersolid phase (corresponding to the diagonal striped solid) around filling $1/8$ and at large off-site repulsions. Furthermore, for small off-site repulsions, we witness a valence bond solid at one-fourth filling and tiny phase-separated regions at slightly higher fillings.

Superconductivity, charge-density waves, antiferromagnetism, and phase separation in the Hubbard-Holstein model

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 $\omega_0$, and the electron-phonon coupling $g$. At half filling, the ground state is an antiferromagnetic insulator for $U \gtrsim 2g^2/\omega_0$, while it is a charge-density-wave (or bi-polaronic) insulator for $U \lesssim 2g^2/\omega_0$. In addition to these phases, we find a superconducting phase that intrudes between them. For $\omega_0/t=1$, superconductivity emerges when both $U/t$ and $2g^2/t\omega_0$ are small; then, by increasing the value of the phonon energy $\omega_0$, 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.

Renormalization of spectra by phase competition in the half-filled Hubbard-Holstein model

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 ($e$-$e$) and electron-phonon ($e$-$ph$) interaction strengths, we show the electron spectral function evolving between antiferromagnetic insulating, metallic, and charge density wave insulating phases. The phonon spectra concurrently gain a strong momentum dependence and soften in energy upon approaching the charge density wave phase. In particular, we study how the $e$-$e$ and $e$-$ph$ interactions renormalize the spectra, and analyze how the interplay of these interactions influence the spectral renormalizations. We find that the presence of both interactions suppresses the amount of renormalization at low energy, thus allowing the emergence of a metallic phase. These findings demonstrate the importance of considering the influence of multiple interactions in spectroscopically determining any one interaction strength in strongly correlated materials.

Charge ordering and magnetism in quarter-filled Hubbard-Holstein model

We study a two-dimensional Hubbard-Holstein model with phonons treated in the adiabatic limit. A Hartree-Fock decomposition is employed for the Hubbard term. A range of electronic densities are discussed with special emphasis on the quarter filling $(n=0.5)$. We argue that the quarter-filled system is relevant for the electronic properties observed at the interface between ${\text{LaAlO}}_{3}$ and ${\text{SrTiO}}_{3}$, where half-electron per unit cell is transferred to the ${\text{TiO}}_{2}$ layer as a consequence of the polar discontinuity at the interface. In addition to presenting the overall phase diagrams, we identify an interesting charge-ordered antiferromagnetic phase for $n=0.5$, which was also reported recently in the ab initio study of the ${\text{LaAlO}}_{3}{\text{-SrTiO}}_{3}$ interface.

COEXISTENCE OF CHARGES TRAPPED IN LOCAL LATTICE DISTORTIONS AND FREE CARRIERS IN CUPRATES

The interplay of strong local electron-lattice and electron-electron interaction is analyzed in the two-dimensional Hubbard-Holstein model using an effective single band Hamiltonian. The effect of the strain field acting on the CuO2 plane and the local cooperative pseudo Jahn Teller vibronic electron-lattice coupling appropriate to cuprate superconductors at optimum doping (i.e.; at doping larger than 1/8) are included via the effective next near neighbor electronic hopping integral t′ (determined by the strain field) close to the local vibronic frequency ω0 (involved in the pseudo Jahn Teller electron lattice coupling). The system shows a characteristic feature of the physics of complex systems, the competition and fluctuation between two states: 1) a first phase of charges self trapped in domains with local lattice distortions (a generalization to a many polaron system of the idea of single intermediate polaron) and 2) a second phase of free carriers.

Two-dimensional Hubbard-Holstein model.

The interplay of electron-phonon coupling and strong electronic correlations is studied in the frame of the two-dimensional Hubbard-Holstein model. Static and dynamic properties are determined by quantum Monte Carlo simulations and by Migdal-Eliashberg theory. The comparison allows us to assess the diagrammatic approach. The competition between the phonon-mediated electron-electron attraction and the local Coulomb repulsion leads to a rich phase diagram, which we study in detail for a wide range of parameters. We address the question, to which extent the systems can be described by an effective negative-U Hubbard model.