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arXiv:1306.4751 · Physical Review B

Interacting-Holstein and extended-Holstein bipolarons

Monodeep Chakraborty, Masaki Tezuka, Byung Il Min

Published Jan 30, 2014• 20 citations
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Abstract

Employing the recently developed self-consistent variational basis generation scheme, we have investigated the bipolaron-bipolaron interaction within the purview of the Holstein-Hubbard and the extended-Holstein-Hubbard (F2H) models on a discrete one-dimensional lattice. The density-matrix renormalization group method has also been used for the Holstein-Hubbard model. We have shown that there exists no bipolaron-bipolaron attraction in the Holstein-Hubbard model. In contrast, we have obtained clear-cut bipolaron-bipolaron attraction in the F2H model. Composite bipolarons are formed above a critical electron-phonon coupling strength, which can survive the finite Hubbard $U$ effect. We have constructed the phase diagram of F2H polarons and bipolarons, and discussed the phase separation in terms of the formation of composite bipolarons.

DOI
10.1103/physrevb.89.035146

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The bound states of two electrons in the adiabatic Holstein-Hubbard model are studied numerically in one and two dimensions from the anticontinuous limit. This model involves a competition between a local electron-phonon coupling (with a classical lattice) which tends to form pairs of electrons and the repulsive Hubbard interaction $U \geq 0$ which tends to break them. In 1D, the ground-state always consists in a pair of localized polarons in a singlet state. They are located at the same site for U=0. Increasing U, there is a first order transition at which the bipolaron becomes a spin singlet pair of two polarons bounded by a magnetic interaction. The pinning mode of the bipolaron soften in the vicinity of this transition leading to a higher mobility of the bipolaron which is tested numerically. In 2D, and for any $U$, the electron-phonon coupling needs to be large enough in order to form small polarons or bipolarons instead of extended electrons. We calculate the phase diagram of the bipolaron involving first order transitions lines with a triple point. A pair of polarons can form three types of bipolarons: a) on a single site at small $U$, b) a spin singlet state on two nearest neighbor sites for larger $U$ as in 1D and c) a new intermediate state obtained as the resonant combination of four 2-sites singlet states sharing a central site, called quadrisinglet. The breathing and pinning internal modes of bipolarons in 2D generally only weakly soften and thus, they are practically not mobile. On the opposite, in the vicinity of the triple point involving the quadrisinglet, both modes exhibit a significant softening. However, it was not sufficient for allowing the existence of a classical mobile bipolaron (at least in that model).

•Jan 1, 1998•0 citations
Employing the recently developed self-consistent variational basis generation scheme, we have investigated the bipolaron-bipolaron interaction within the purview of the Holstein-Hubbard and the extended-Holstein-Hubbard (F2H) models on a discrete one-dimensional lattice. The density-matrix renormalization group method has also been used for the Holstein-Hubbard model. We have shown that there exists no bipolaron-bipolaron attraction in the Holstein-Hubbard model. In contrast, we have obtained clear-cut bipolaron-bipolaron attraction in the F2H model. Composite bipolarons are formed above a critical electron-phonon coupling strength, which can survive the finite Hubbard UUU effect. We have constructed the phase diagram of F2H polarons and bipolarons, and discussed the phase separation in terms of the formation of composite bipolarons.
Employing the recently developed self-consistent variational basis generation scheme, we have investigated the bipolaron-bipolaron interaction within the purview of Holstein-Hubbard and the Froehlich-Hubbard model on a discrete one-dimensional lattice. The density-matrix renormalization group (DMRG) method has also been used for the Holstein-Hubbard model. We have shown that there exists no bipolaron-bipolaron attraction in the Holstein-Hubbard model. In contrast, we have obtained clear-cut bipolaron-bipolaron attraction in the Froehlich-Hubbard model. Composite bipolarons are formed above a critical electron-phonon coupling strength, which can survive the finite Hubbard U effect. We have constructed the phase diagram of Froehlich-Hubbard polarons and bipolarons, and discussed the phase separation in terms of the formation of composite bipolarons.
In this paper we have studied the properties of a two-fermion system interacting with the phonon field in the framework of the Holstein Hubbard model in one, two and three dimensions. We have chosen the modified Lang-Firsov variational wave function with on-site and nearest-neighbour lattice distortion for the phonon subsystem to obtain an effective electronic Hamiltonian. This effective electronic Hamiltonian is then solved exactly to obtain the binding energy and effective mass of the bipolaron. We investigated the stability of the bipolaronic phase by comparing the ground-state energy of a bipolaron and two free polarons. We observed a critical repulsive on-site Hubbard interaction U c below which the bipolaronic phases are stable for a fixed electron-phonon coupling g . In the absence of on-site repulsion, bipolaronic phases are stable over the entire range of electron-phonon coupling for one dimension, whereas there is a critical electron-phonon coupling g c for formation of a stable bipolaron in two and three dimensions.
We have investigated bipolaron formation in the Holstein-Hubbard and the Fr\"ohlich-Hubbard model on a discrete one-dimensional lattice. Phonon enriched basis has been generated by exploiting the idea of the Lang-Firsov transformation, which results in better convergence in the strong coupling regime. The incorporation of the finite-range electron-phonon (e-ph) interaction reduces the bipolaron effective mass. We systematically increase the spatial extent of the e-ph interaction to study the effect of the long-range tail of the Fr\"ohlich model. The electron-electron (e-e) interaction also plays a crucial role in lowering the effective mass for both the Holstein and the Fr\"ohlich bipolarons. In the presence of the extended e-e interaction, the intersite S1 Holstein bipolaron becomes unstable, while the effective mass of the Fr\"ohlich bipolaron becomes significantly reduced for both the spin singlet and triplet cases.
We numerically and analytically calculate the properties of the bipolaron in an extended Hubbard-Holstein model, which has a longer-range electron-phonon coupling like the Fr\"ohlich model. In the strong-coupling regime, the effective mass of the bipolaron in the extended model is much smaller than the Holstein bipolaron mass. In contrast to the Holstein bipolaron, the bipolaron in the extended model has a lower binding energy and remains bound with substantial binding energy even in the large-UUU limit. In comparison with the Holstein model where only a singlet bipolaron is bound, in the extended Holstein model a triplet bipolaron can also form a bound state. We discuss the possibility of phase separation in the case of finite electron doping.
We present a diagrammatic Monte Carlo study of the properties of the Hubbard-Holstein bipolaron on a two-dimensional square lattice. With a small Coulomb repulsion UUU and with increasing electron-phonon interaction, and when reaching a value about two times smaller than the one corresponding to the transition of light polaron to heavy polaron, the system suffers a sharp transition from a state formed by two weakly bound light polarons to a heavy, strongly bound on-site bipolaron. Aside from this rather conventional bipolaron a new bipolaron state is found for large UUU at intermediate and large electron-phonon coupling, corresponding to two polarons bound on nearest-neighbor sites. We discuss both the properties of the different bipolaron states and the transition from one state to another. We present a phase diagram in parameter space defined by the electron-phonon coupling and UUU. Our numerical method does not use any artificial approximation and can be easily modified to other bipolaron models with longer range electron-phonon and∕or electron-electron interaction.
The bound states of two electrons in the adiabatic Holstein-Hubbard model are studied numerically in one and two dimensions from the anticontinuous limit. This model involves a competition between a local electron-phonon coupling (with a classical lattice) which tends to form pairs of electrons and the repulsive Hubbard interaction U≥0U \geq 0U≥0 which tends to break them. In 1D, the ground-state always consists in a pair of localized polarons in a singlet state. They are located at the same site for U=0. Increasing U, there is a first order transition at which the bipolaron becomes a spin singlet pair of two polarons bounded by a magnetic interaction. The pinning mode of the bipolaron soften in the vicinity of this transition leading to a higher mobility of the bipolaron which is tested numerically. In 2D, and for any UUU, the electron-phonon coupling needs to be large enough in order to form small polarons or bipolarons instead of extended electrons. We calculate the phase diagram of the bipolaron involving first order transitions lines with a triple point. A pair of polarons can form three types of bipolarons: a) on a single site at small UUU, b) a spin singlet state on two nearest neighbor sites for larger UUU as in 1D and c) a new intermediate state obtained as the resonant combination of four 2-sites singlet states sharing a central site, called quadrisinglet. The breathing and pinning internal modes of bipolarons in 2D generally only weakly soften and thus, they are practically not mobile. On the opposite, in the vicinity of the triple point involving the quadrisinglet, both modes exhibit a significant softening. However, it was not sufficient for allowing the existence of a classical mobile bipolaron (at least in that model).