It is widely believed that in the adiabatic limit the Fermi liquid state of an electron-phonon system, described by Migdal-Eliashberg theory, remains stable until the dressed phonon softens. Our variational and analytic analysis of the prototypical Holstein model shows that, in a wide range of fillings both in 3D and 2D, a polaronic/bipolaronic state emerges before phonon softening; at small filling in 3D this happens already at weak coupling. We show that a polaronic/bipolaronic state emerges, upon increasing coupling, via an intermediate pseudogap-type mixed state, in which some fermions retain Fermi liquid behavior, yet Luttinger's theorem is violated. At even larger couplings the density of states gradually approaches its form in the atomic limit.It is widely believed that in the adiabatic limit the Fermi liquid state of an electron-phonon system, described by Migdal-Eliashberg theory, remains stable until the dressed phonon softens. Our variational and analytic analysis of the prototypical Holstein model shows that, in a wide range of fillings both in 3D and 2D, a polaronic/bipolaronic state emerges before phonon softening; at small filling in 3D this happens already at weak coupling. We show that a polaronic/bipolaronic state emerges, upon increasing coupling, via an intermediate pseudogap-type mixed state, in which some fermions retain Fermi liquid behavior, yet Luttinger's theorem is violated. At even larger couplings the density of states gradually approaches its form in the atomic limit.

arXiv:2604.14293Apr 15, 20261 citation

Small Bipolarons in the 2-dimensional Holstein-Hubbard Model I The Adiabatic Limit

Laurent Proville, S. Aubry

Abstract. The spatially localized bound states of two electrons in the adiabatic two-dimensional Holstein-Hubbard model on a square lattice are investigated both numerically and analytically. The interplay between the electron-phonon coupling g, which tends to form bipolarons and the repulsive Hubbard interaction υ ≥ 0, which tends to break them, generates many different ground-states. There are four domains in the g,υ phase diagram delimited by first order transition lines. Except for the domain at weak electronphonon coupling (small g) where the electrons remain free, the electrons form bipolarons which can 1) be mostly located on a single site (small υ, large g); 2) be an anisotropic pair of polarons lying on two neighboring sites in the magnetic singlet state (large υ, large g); or 3) be a ”quadrisinglet state ” which is the superposition of 4 electronic singlets with a common central site. This quadrisinglet bipolaron is the most stable in a small central domain in between the three other phases. The pinning modes and the Peierls-Nabarro barrier of each of these bipolarons are calculated and the barrier is found to be strongly depressed in the region of stability of the quadrisinglet bipolaron. PACS. 71.10.Fd Lattice fermion models (Hubbard model, etc.) – 71.38.+i Polarons and electron phonon interactions – 74.20.Mn Nonconventional mechanisms (spin fluctuations, polarons and bipolarons, resonating valence bond model, anyon mechanism, marginal Fermi liquid, Luttinger liquid, etc.) – 74.25.Jb Electronic structureAbstract. The spatially localized bound states of two electrons in the adiabatic two-dimensional Holstein-Hubbard model on a square lattice are investigated both numerically and analytically. The interplay between the electron-phonon coupling g, which tends to form bipolarons and the repulsive Hubbard interaction υ ≥ 0, which tends to break them, generates many different ground-states. There are four domains in the g,υ phase diagram delimited by first order transition lines. Except for the domain at weak electronphonon coupling (small g) where the electrons remain free, the electrons form bipolarons which can 1) be mostly located on a single site (small υ, large g); 2) be an anisotropic pair of polarons lying on two neighboring sites in the magnetic singlet state (large υ, large g); or 3) be a ”quadrisinglet state ” which is the superposition of 4 electronic singlets with a common central site. This quadrisinglet bipolaron is the most stable in a small central domain in between the three other phases. The pinning modes and the Peierls-Nabarro barrier of each of these bipolarons are calculated and the barrier is found to be strongly depressed in the region of stability of the quadrisinglet bipolaron. PACS. 71.10.Fd Lattice fermion models (Hubbard model, etc.) – 71.38.+i Polarons and electron phonon interactions – 74.20.Mn Nonconventional mechanisms (spin fluctuations, polarons and bipolarons, resonating valence bond model, anyon mechanism, marginal Fermi liquid, Luttinger liquid, etc.) – 74.25.Jb Electronic structure

Jan 1, 199827 citations

Dynamical mean field theory of polarons and bipolarons in the half-filled Holstein model

Massimo Capone, P. Carta, Sergio Ciuchi

The evolution of the properties of a finite density electronic system as a function of the electron-phonon coupling is investigated in the Holstein model using the dynamical mean-field theory (DMFT) that becomes exact in infinite dimensions. We compare the spinless fermion case, in which only isolated polarons can be formed, with the spinful model in which the polarons can bind and form bipolarons. In the latter case, the bipolaronic binding occurs within DMFT as a metal-insulator transition. In the adiabatic regime in which the phonon energy is small with respect to the electron hopping we compare numerically exact DMFT results with an analytical scheme inspired by the Born-Oppenheimer procedure. Within the latter approach, a truncation of the phononic Hilbert space leads to a mapping of the original model onto an Anderson spin-fermion model. In the anti-adiabatic regime (where the phonon energy exceeds the electronic scales) the standard treatment based on Lang-Firsov canonical transformation allows one to map the original model on to an attractive Hubbard model in the spinful case. The separate analysis of the two regimes supports the numerical evidence that the presence of well-defined polaronic lattice displacements is not necessarily associated to a metal-insulator transition, which is instead due to pairing between the carriers. The finite-dimensionality effects neglected in DMFT may lead to a finite conductivity in the bipolaronic state which is, however, not always associated with polaronic distortions. At the polaron crossover the Born-Oppenheimer approximation is shown to break down due to the entanglement of the electron-phonon state.The evolution of the properties of a finite density electronic system as a function of the electron-phonon coupling is investigated in the Holstein model using the dynamical mean-field theory (DMFT) that becomes exact in infinite dimensions. We compare the spinless fermion case, in which only isolated polarons can be formed, with the spinful model in which the polarons can bind and form bipolarons. In the latter case, the bipolaronic binding occurs within DMFT as a metal-insulator transition. In the adiabatic regime in which the phonon energy is small with respect to the electron hopping we compare numerically exact DMFT results with an analytical scheme inspired by the Born-Oppenheimer procedure. Within the latter approach, a truncation of the phononic Hilbert space leads to a mapping of the original model onto an Anderson spin-fermion model. In the anti-adiabatic regime (where the phonon energy exceeds the electronic scales) the standard treatment based on Lang-Firsov canonical transformation allows one to map the original model on to an attractive Hubbard model in the spinful case. The separate analysis of the two regimes supports the numerical evidence that the presence of well-defined polaronic lattice displacements is not necessarily associated to a metal-insulator transition, which is instead due to pairing between the carriers. The finite-dimensionality effects neglected in DMFT may lead to a finite conductivity in the bipolaronic state which is, however, not always associated with polaronic distortions. At the polaron crossover the Born-Oppenheimer approximation is shown to break down due to the entanglement of the electron-phonon state.

arXiv:cond-mat/0509542Physical Review BJul 10, 200638 citations

Mobile Bipolarons in the Adiabatic Holstein-Hubbard Model in 1 and 2 dimensions

Laurent Proville, S. Aubry

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).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, 19980 citations

Polarons and bipolarons in strongly interacting electron-phonon systems

Gerhard Wellein, Heinrich Röder, Holger Fehske

The Holstein Hubbard and Holstein t-J models are studied for a wide range of phonon frequencies, electron-electron, and electron-phonon interaction strengths on finite lattices with up to ten sites by means of direct Lanczos diagonalization. Previously the necessary truncation of the phononic Hilbert space caused serious limitations to either very small systems (four or even two sites) or to weak electron-phonon coupling, in particular in the adiabatic regime. Using parallel computers we were able to investigate the transition from ``large'' to ``small'' polarons in detail. By resolving the low-lying eigenstates of the Hamiltonian and by calculating the spectral function, we can identify a polaron band in the strong-coupling case, whose dispersion deviates from the free-particle dispersion at low and intermediate phonon frequencies. For two electrons (holes) we establish the existence of bipolaronic states and discuss the formation of a bipolaron band. For the two-dimensional Holstein t-J model, we demonstrate that the formation of hole polarons is favored by strong Coulomb correlations. Analyzing hole-hole correlation functions, we find that hole binding is enhanced as a dynamical effect of the electron-phonon interaction. \textcopyright{} 1996 The American Physical Society.The Holstein Hubbard and Holstein t-J models are studied for a wide range of phonon frequencies, electron-electron, and electron-phonon interaction strengths on finite lattices with up to ten sites by means of direct Lanczos diagonalization. Previously the necessary truncation of the phononic Hilbert space caused serious limitations to either very small systems (four or even two sites) or to weak electron-phonon coupling, in particular in the adiabatic regime. Using parallel computers we were able to investigate the transition from ``large'' to ``small'' polarons in detail. By resolving the low-lying eigenstates of the Hamiltonian and by calculating the spectral function, we can identify a polaron band in the strong-coupling case, whose dispersion deviates from the free-particle dispersion at low and intermediate phonon frequencies. For two electrons (holes) we establish the existence of bipolaronic states and discuss the formation of a bipolaron band. For the two-dimensional Holstein t-J model, we demonstrate that the formation of hole polarons is favored by strong Coulomb correlations. Analyzing hole-hole correlation functions, we find that hole binding is enhanced as a dynamical effect of the electron-phonon interaction. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9512143Physical review. B, Condensed matterApr 15, 1996163 citations

Bipolaronic charge density waves and polaronic spin density waves in the Holstein-Hubbard model

S. Aubry

At large electron-phonon coupling, the phonon of the Holstein-Hubbard model can be (generally) treated classically. Exact results can be obtained for this model at any band filling and any finite dimension where the Hubbard term with coefficient U is treated rigorously. The existence of chaotic bipolaronic states is proven when U is smaller than a certain value Uc and for an electron-phonon coupling k sufficiently large. The existence of magnetic polaronic states is also proven for positive U and k sufficiently large. For 0<U<Uc, there exists an overlap region with chaotic mixed polaronic-bipolaronic states. There is a region for U=Uc/2 and large k of special interest, where the bipolarons in their ground-state become singlet pairs of polarons bounded by a magnetic resonance. Then, the Peierls-Nabarro barrier which usually pins polarons and bipolarons to the lattice depresses to almost zero and the validity of the classical lattice approximation breaks down so that the initial exact results do not apply in that specific region. The formation of a superconducting state with possibly a high critical temperature, is then conjectured at a large coupling k in a narrow domain in-between the two insulating domains corresponding to Bipolaronic Charge Density Waves and to magnetic Polaronic Spin Density Waves respectively.At large electron-phonon coupling, the phonon of the Holstein-Hubbard model can be (generally) treated classically. Exact results can be obtained for this model at any band filling and any finite dimension where the Hubbard term with coefficient U is treated rigorously. The existence of chaotic bipolaronic states is proven when U is smaller than a certain value Uc and for an electron-phonon coupling k sufficiently large. The existence of magnetic polaronic states is also proven for positive U and k sufficiently large. For 0<U<Uc, there exists an overlap region with chaotic mixed polaronic-bipolaronic states. There is a region for U=Uc/2 and large k of special interest, where the bipolarons in their ground-state become singlet pairs of polarons bounded by a magnetic resonance. Then, the Peierls-Nabarro barrier which usually pins polarons and bipolarons to the lattice depresses to almost zero and the validity of the classical lattice approximation breaks down so that the initial exact results do not apply in that specific region. The formation of a superconducting state with possibly a high critical temperature, is then conjectured at a large coupling k in a narrow domain in-between the two insulating domains corresponding to Bipolaronic Charge Density Waves and to magnetic Polaronic Spin Density Waves respectively.

Journal de Physique IV (Proceedings)Jul 1, 19937 citations

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).