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One-dimensional Peierls phase separation in the dilute carrier density limit

Alberto Nocera, John Sous, Adrian E. Feiguin, Mona Berciu

Published Aug 7, 2020• 3 citations
OpenAlex

Abstract

We use Density Matrix Renormalization Group to study a one-dimensional chain with Peierls electron-phonon coupling describing the modulation of the electron hopping due to lattice distortion. We demonstrate the appearance of an exotic phase-separated state, which we call Peierls phase separation, in the limit of very dilute electron densities, for sufficiently large couplings and small phonon frequencies. This is unexpected, given that Peierls coupling mediates effective pair-hopping interactions that disfavor phase clustering. The Peierls phase separation consists of a homogenous, dimerized, electron-rich region surrounded by electron-poor regions, which we show to be energetically more favorable than a dilute liquid of bipolarons. This mechanism qualitatively differs from that of typical phase separation in conventional electron-phonon models that describe the modulation of the electron's potential energy due to lattice distortions. Surprisingly, the electron-rich region always stabilizes a dimerized pattern at fractional densities, hinting at a non-perturbative correlation-driven mechanism behind phase separation.

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We use the density matrix renormalization group method to study a one-dimensional chain with Peierls electron-phonon coupling, which describes the modulation of the electron hopping by lattice distortions. We demonstrate that this system is stable against phase separation in the dilute density limit. We only find phase separation numerically for large couplings for which the linear approximation for the electron-phonon coupling becomes invalid; this behavior can be stabilized in a narrow sliver of the physical parameter space if the dispersion of the phonons is carefully tuned. These results indicate that in the dilute electron density limit, Peierls bipolaron liquids are generically stable, unlike in other models of electron-phonon coupling. We show that this behavior extends to finite carrier concentrations of up to quarter filling. This stability of low-density, light-mass bipolaron liquids in the Peierls model opens a path to high-${T}_{c}$ superconductivity based on a bipolaronic mechanism, in higher dimensions.

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Non-2kFcharge density wave induced by phonon dispersion in one-dimensional Peierls conductors

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Standard Peierls theory predicts that the wave-vector of the charge density wave (CDW) induced by the electron-phonon coupling in the ground-state of a one-dimensional conductor is twice the Fermi wave-vector 2k F . At moderately large electron-phonon coupling, beyond the TBA (transition by breaking of analyticity), it is known that this CDW becomes an array of interacting bipolarons. In the simplest models, such as the 1D adiabatic Holstein model, the bipolaron interactions are repulsive and the ground state remains a 2k F CDW of equidistant bipolarons. We show that (apparently) minor changes in the phonon spectrum of the model induce extra elastic forces between the bipolarons, which could break the bipolaronic 2k F CDW ordering. This effect is studied in detail in the modified 1D Holstein model, where the optical phonon has a non-zero dispersion which is chosen positive in order that an extra force that appears between the bipolarons be attractive. The Coulomb forces between the bipolarons are initially neglected. An accurate numerical study modelled on the standard analysis of multiphase points confirms that, in a large part of the phase diagram, the CDW ground state separates in two phases, which correspond to two CDW with different 2k F wave-vectors (which are generally commensurate) and thus different electronic densities. These separations, into phases called either 'parent' or 'non-parent', are studied on the basis of the Farey construction of rational numbers. The long-range Coulomb interaction between the charged bipolarons, forbids any macroscopic phase separation in two phases with different electronic densities. The bipolaron structure then has to be a periodic sequence of alternate domains of the two CDW phases on the microscopic scale. The resulting structure is a new CDW with a modulation wave-vector that is not 2k F . We suggest that the aberrant wave-vector observed in the real CDW system (TaSe 4 ) 2 I could be interpreted along the ideas developed in this paper. Although this compound shares many of the usual properties of quasi-one-dimensional conductors with CDW, the component of the modulation wave-vector 0.085c* in the chain direction is, unusually, quite different from the value 2k F =c* expected from band filling. The CDW in (TaSe 4 ) 2 I could correspond to a nonstandard bipolaron ordering due to a strong attractive elastic interaction between the bipolarons along the chain competing with the long-range Coulomb forces.

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Metallicity in the half-filled Holstein-Hubbard model

Holger Fehske, Georg Hager, Eric Jeckelmann

We re-examine the Peierls insulator to Mott insulator transition scenario in the one-dimensional Holstein-Hubbard model where, at half-filling, electron-phonon and electron-electron interactions compete for establishing charge- and spin-density-wave states, respectively. By means of large-scale density-matrix renormalization group calculations we determine the spin, single-particle and two-particle excitation gaps and prove —in the course of a careful finite-size scaling analysis— recent claims for an intervening metallic phase in the weak-coupling regime. We show that for large phonon frequencies the metallic region is even more extended than previously expected, and subdivided into ordinary Luttinger-liquid and bipolaronic-liquid phases.

arXiv:0808.1675•Europhysics Letters (EPL)•Nov 24, 2008•80 citations
We use Density Matrix Renormalization Group to study a one-dimensional chain with Peierls electron-phonon coupling describing the modulation of the electron hopping due to lattice distortion. We demonstrate the appearance of an exotic phase-separated state, which we call Peierls phase separation, in the limit of very dilute electron densities, for sufficiently large couplings and small phonon frequencies. This is unexpected, given that Peierls coupling mediates effective pair-hopping interactions that disfavor phase clustering. The Peierls phase separation consists of a homogenous, dimerized, electron-rich region surrounded by electron-poor regions, which we show to be energetically more favorable than a dilute liquid of bipolarons. This mechanism qualitatively differs from that of typical phase separation in conventional electron-phonon models that describe the modulation of the electron's potential energy due to lattice distortions. Surprisingly, the electron-rich region always stabilizes a dimerized pattern at fractional densities, hinting at a non-perturbative correlation-driven mechanism behind phase separation.
We use the density matrix renormalization group method to study a one-dimensional chain with Peierls electron-phonon coupling, which describes the modulation of the electron hopping by lattice distortions. We demonstrate that this system is stable against phase separation in the dilute density limit. We only find phase separation numerically for large couplings for which the linear approximation for the electron-phonon coupling becomes invalid; this behavior can be stabilized in a narrow sliver of the physical parameter space if the dispersion of the phonons is carefully tuned. These results indicate that in the dilute electron density limit, Peierls bipolaron liquids are generically stable, unlike in other models of electron-phonon coupling. We show that this behavior extends to finite carrier concentrations of up to quarter filling. This stability of low-density, light-mass bipolaron liquids in the Peierls model opens a path to high-Tc{T}_{c}Tc​ superconductivity based on a bipolaronic mechanism, in higher dimensions.
We study the one-dimensional Holstein model with spin-12\frac{1}{2}21​ electrons at half filling. Ground-state properties are calculated for long chains with great accuracy using the density-matrix renormalization-group method and extrapolated to the thermodynamic limit. We show that for small electron-phonon coupling or large phonon frequency, the insulating Peierls ground state predicted by mean-field theory is destroyed by quantum lattice fluctuations and that the system remains in a metallic phase with a nondegenerate ground state and power-law electronic and phononic correlations. When the electron-phonon coupling becomes large or the phonon frequency small, the system undergoes a transition to an insulating Peierls phase with a twofold degenerate ground state, long-range charge-density-wave order, a dimerized lattice structure, and a gap in the electronic excitation spectrum.
Standard Peierls theory predicts that the wave-vector of the charge density wave (CDW) induced by the electron-phonon coupling in the ground-state of a one-dimensional conductor is twice the Fermi wave-vector 2k F . At moderately large electron-phonon coupling, beyond the TBA (transition by breaking of analyticity), it is known that this CDW becomes an array of interacting bipolarons. In the simplest models, such as the 1D adiabatic Holstein model, the bipolaron interactions are repulsive and the ground state remains a 2k F CDW of equidistant bipolarons. We show that (apparently) minor changes in the phonon spectrum of the model induce extra elastic forces between the bipolarons, which could break the bipolaronic 2k F CDW ordering. This effect is studied in detail in the modified 1D Holstein model, where the optical phonon has a non-zero dispersion which is chosen positive in order that an extra force that appears between the bipolarons be attractive. The Coulomb forces between the bipolarons are initially neglected. An accurate numerical study modelled on the standard analysis of multiphase points confirms that, in a large part of the phase diagram, the CDW ground state separates in two phases, which correspond to two CDW with different 2k F wave-vectors (which are generally commensurate) and thus different electronic densities. These separations, into phases called either 'parent' or 'non-parent', are studied on the basis of the Farey construction of rational numbers. The long-range Coulomb interaction between the charged bipolarons, forbids any macroscopic phase separation in two phases with different electronic densities. The bipolaron structure then has to be a periodic sequence of alternate domains of the two CDW phases on the microscopic scale. The resulting structure is a new CDW with a modulation wave-vector that is not 2k F . We suggest that the aberrant wave-vector observed in the real CDW system (TaSe 4 ) 2 I could be interpreted along the ideas developed in this paper. Although this compound shares many of the usual properties of quasi-one-dimensional conductors with CDW, the component of the modulation wave-vector 0.085c* in the chain direction is, unusually, quite different from the value 2k F =c* expected from band filling. The CDW in (TaSe 4 ) 2 I could correspond to a nonstandard bipolaron ordering due to a strong attractive elastic interaction between the bipolarons along the chain competing with the long-range Coulomb forces.
The ground state of the one-dimensional $t\text{\ensuremath{-}}J$ model coupled with phonons in the adiabatic limit is numerically investigated by use of the Lanczos technique at quarter filling. Due to the interplay between the electron-electron Coulomb repulsion and electron-phonon interaction, this model shows a sequence of lattice distortions leading to the formation of charge-density waves and bond-order waves. Moderate electron-electron and electron-lattice coupling may lead to the coexistence of dimerization and tetramerization in the distortion pattern. Dimerization leads to the formation of an ``antiferromagnetic'' Mott insulator, while tetramerization gives rise to a spin-Peierls phase. By increasing the super exchange coupling, antiferromagnetism is inhibited due to the change of the distortion periodicity.
The Hubbard-Holstein model describes fermions on a discrete lattice, with on-site repulsion between fermions and a coupling to phonons that are localized on sites. Generally, at half-filling, increasing the coupling ggg to the phonons drives the system towards a Peierls charge density wave state, whereas increasing the electron-electron interaction UUU drives the fermions into a Mott antiferromagnet. At low ggg and UUU, or when doped, the system is metallic. In one dimension, using quantum Monte Carlo simulations, we study the case where fermions have a long-range coupling to phonons, with characteristic range $\ensuremath{\xi}$, interpolating between the Holstein and Fr\"ohlich limits. Without electron-electron interaction, the fermions adopt a Peierls state when the coupling to the phonons is strong enough. This state is destabilized by a small coupling range $\ensuremath{\xi}$ and leads to a collapse of the fermions, and, consequently, phase separation. Increasing interaction UUU will drive any of these three phases (metallic, Peierls, phase separation) into a Mott insulator phase. The phase separation region is once again present in the $U\ensuremath{\ne}0$ case, even for small values of the coupling range.
We re-examine the Peierls insulator to Mott insulator transition scenario in the one-dimensional Holstein-Hubbard model where, at half-filling, electron-phonon and electron-electron interactions compete for establishing charge- and spin-density-wave states, respectively. By means of large-scale density-matrix renormalization group calculations we determine the spin, single-particle and two-particle excitation gaps and prove —in the course of a careful finite-size scaling analysis— recent claims for an intervening metallic phase in the weak-coupling regime. We show that for large phonon frequencies the metallic region is even more extended than previously expected, and subdivided into ordinary Luttinger-liquid and bipolaronic-liquid phases.