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arXiv:1004.3677 · Physical Review B

Competition between antiferromagnetic and charge order in the Hubbard-Holstein model

Johannes Bauer, Alex C. Hewson

Published Jun 11, 2010• 51 citations
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Abstract

We study the competition between an instantaneous local Coulomb repulsion and a boson mediated retarded attraction, as described by the Hubbard-Holstein model. Restricting to the case of half-filling, the ground-state phase diagram and the transitions from antiferromagnetically ordered states to charge ordered states are analyzed. The calculations are based on the model in large dimensions, so that dynamical mean-field theory can be applied, and the associated impurity problem is solved using the numerical renormalization group method. The transition is found to occur when electron-electron coupling strength $U$ and the induced interaction $\ensuremath{\lambda}$ due to electron-phonon coupling approximately coincide, $U\ensuremath{\simeq}\ensuremath{\lambda}$. We find a continuous transition for small coupling and large ${\ensuremath{\omega}}_{0}$, and a discontinuous one for large coupling and/or small ${\ensuremath{\omega}}_{0}$. We present results for the order parameters, the static expectation values for the electrons and phonons, and the corresponding spectral functions. They illustrate the different types of behavior to be seen near the transitions. Additionally, the quasiparticle properties are calculated in the normal state, which leads to a consistent interpretation of the low-energy excitations.

DOI
10.1103/physrevb.81.235113

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We study the competition between an instantaneous local Coulomb repulsion and a boson mediated retarded attraction, as described by the Hubbard-Holstein model. Restricting to the case of half-filling, the ground-state phase diagram and the transitions from antiferromagnetically ordered states to charge ordered states are analyzed. The calculations are based on the model in large dimensions, so that dynamical mean-field theory can be applied, and the associated impurity problem is solved using the numerical renormalization group method. The transition is found to occur when electron-electron coupling strength UUU and the induced interaction \ensuremathλ\ensuremath{\lambda}\ensuremathλ due to electron-phonon coupling approximately coincide, U\ensuremath≃\ensuremathλU\ensuremath{\simeq}\ensuremath{\lambda}U\ensuremath≃\ensuremathλ. We find a continuous transition for small coupling and large \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​, and a discontinuous one for large coupling and/or small \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​. We present results for the order parameters, the static expectation values for the electrons and phonons, and the corresponding spectral functions. They illustrate the different types of behavior to be seen near the transitions. Additionally, the quasiparticle properties are calculated in the normal state, which leads to a consistent interpretation of the low-energy excitations.
Competing interactions are often responsible for intriguing phase diagrams in correlated electron systems. Here we analyze the competition of instantaneous short-range Coulomb interaction U with the retarded electron-electron interaction induced by an electron-phonon coupling g as described by the Hubbard-Holstein model. The ground-state phase diagram of this model in the limit of large dimensions at half-filling is established. The study is based on dynamical mean-field theory combined with the numerical renormalization group. Depending on U , g , and the phonon frequency ω 0 , the ground state is antiferromagnetically (AFM) or charge ordered (CO). We find quantum phase transitions from the AFM to CO state to occur when U-λ≃0, where λ characterizes the phonon-induced effective attraction. The transition is continuous for small couplings and large phonon frequencies ω 0 and becomes discontinuous for large couplings and small values of ω 0 . We comment on the possible relevance of this work for Ba 1- x K x BiO 3 .
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{T}_{c}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{T}_{c}Tc​ dome against the effective electron-electron interaction Ueff{U}_{\text{eff}}Ueff​ given by the Hubbard UUU 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{U}_{\mathrm{eff}}=0Ueff​=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.
We investigate ordered phases (s-wave superconducting (SC), antiferromagnetic (AF) and charge ordered (CO) phases) of the half-filled Holstein-Hubbard model using the dynamical mean-field theory in combination with a continuous time quantum Monte Carlo impurity solver. We particularly focus on the parameter regime where the electron-electron Coulomb interaction and the phonon mediated retarded attractive interaction are both comparable to the bare electron bandwidth to understand the interplay of these interactions and the retardation effects due to finite phonon frequency ω_0. Results for the transition temperatures for SC and AF as a function of the effective interaction U_eff(=U-λwith U the Coulomb repulsion and λthe phonon-mediated attraction) show deviations from the the corresponding results in the anti-adiabatic limit. These are analyzed by comparing with an effective static model in the polaron picture. We also determine the phase diagram around U_eff=0 at low, but nonzero temperatures to show that paramagnetic phases appear between AF and CO when U and λare small or large, while AF and CO directly compete with each other at moderate U and λ.
Abstract We explore the quantum phase transitions between two ordered states in the infinite dimensional Hubbard–Holstein model at half filling. Our study is based on the dynamical mean field theory (DMFT) combined with the numerical renormalization group (NRG), which allows us to handle both strong electron–electron and strong electron–phonon interactions. The transition line is characterized by an effective electron–electron interaction. Depending on this effective interaction and the phonon frequency ω0 one finds either a continuous transition or discontinuous transition. Here, the analysis focuses on the behavior of the system when the electron–electron repulsion U and the phonon‐mediated attraction λ are equal. We first discuss the adiabatic and antiadiabatic limiting cases. For finite ω0 we study the differences between the antiferromagnetic (AFM) and charge order, and find that when present the AFM state has a lower energy on the line.
We present a determinant quantum Monte Carlo study of the competition between instantaneous on-site Coulomb repulsion and retarded phonon-mediated attraction between electrons, as described by the two-dimensional Hubbard-Holstein model. At half filling, we find a strong competition between antiferromagnetism (AFM) and charge-density-wave (CDW) order. We demonstrate that a simple picture of AFM-CDW competition that incorporates the phonon-mediated attraction into an effective-U Hubbard model requires significant refinement. Specifically, retardation effects slow the onset of charge order so that CDW order remains absent even when the effective U is negative. This delay opens a window where neither AFM nor CDW order is well established and where there are signatures of a possible metallic phase.
The Hubbard-Holstein model is one of the simplest to incorporate both electron-electron and electron-phonon interactions. In one dimension at half filling, the Holstein electron-phonon coupling promotes on-site pairs of electrons and a Peierls charge-density wave, while the Hubbard on-site Coulomb repulsion UUU promotes antiferromagnetic correlations and a Mott insulating state. Recent numerical studies have found a possible third intermediate phase between Peierls and Mott states. From direct calculations of charge and spin susceptibilities, we show that (i) as the electron-phonon coupling is increased, first a spin gap opens, followed by the Peierls transition. Between these two transitions, the metallic intermediate phase has a spin gap, no charge gap, and properties similar to the negative-UUU Hubbard model. (ii) The transitions between Mott/intermediate and intermediate/Peierls states are of the Kosterlitz-Thouless form. (iii) For larger UUU, the two transitions merge at a tricritical point into a single first-order Mott/Peierls transition. In addition, we show that an intermediate phase also occurs in the quarter-filled model.