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arXiv:cond-mat/0111162 · Physical review. B, Condensed matter

Incommensurate charge-density waves in the adiabatic Hubbard-Holstein model

Vieri Mastropietro

Published Feb 1, 2002• 1 citation
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Abstract

The adiabatic Holstein-Hubbard model describes electrons on a chain with step a interacting with themselves (with coupling U) and with a classical phonon field ${\ensuremath{\varphi}}_{x}$ (with coupling \ensuremath{\lambda}). There is Peierls instability if the electronic ground-state energy $F(\ensuremath{\varphi})$ as a functional of ${\ensuremath{\varphi}}_{x}$ has a minimum which corresponds to a periodic function with period $\ensuremath{\pi}{/p}_{F},$ where ${p}_{F}$ is the Fermi momentum. We consider ${p}_{F}/\ensuremath{\pi}a$ irrational so that the charge-density wave is incommensurate with the chain. We prove in a rigorous way in the spinless case, when \ensuremath{\lambda}, U are small and $U/\ensuremath{\lambda}$ large, that (a) when the electronic interaction is attractive $U<0$ there is no Peierls instability and (b) when the interaction is repulsive $U>0$ there is Peierls instability in the sense that our convergent expansion for $F(\ensuremath{\varphi}),$ truncated at second order has a minimum which corresponds to an analytical and $\ensuremath{\pi}{/p}_{F}$ periodic ${\ensuremath{\varphi}}_{x}.$ Such a minimum is found solving an infinite set of coupled self-consistent equations, one for each of the infinite Fourier modes of ${\ensuremath{\varphi}}_{x}.$

DOI
10.1103/physrevb.65.075113

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The adiabatic Holstein-Hubbard model describes electrons on a chain with step a interacting with themselves (with coupling U) and with a classical phonon field \ensuremathφx{\ensuremath{\varphi}}_{x}\ensuremathφx​ (with coupling \ensuremath{\lambda}). There is Peierls instability if the electronic ground-state energy F(\ensuremathφ)F(\ensuremath{\varphi})F(\ensuremathφ) as a functional of \ensuremathφx{\ensuremath{\varphi}}_{x}\ensuremathφx​ has a minimum which corresponds to a periodic function with period \ensuremathπ/pF,\ensuremath{\pi}{/p}_{F},\ensuremathπ/pF​, where pF{p}_{F}pF​ is the Fermi momentum. We consider pF/\ensuremathπa{p}_{F}/\ensuremath{\pi}apF​/\ensuremathπa irrational so that the charge-density wave is incommensurate with the chain. We prove in a rigorous way in the spinless case, when \ensuremath{\lambda}, U are small and U/\ensuremathλU/\ensuremath{\lambda}U/\ensuremathλ large, that (a) when the electronic interaction is attractive U<0U<0U<0 there is no Peierls instability and (b) when the interaction is repulsive U>0U>0U>0 there is Peierls instability in the sense that our convergent expansion for F(\ensuremathφ),F(\ensuremath{\varphi}),F(\ensuremathφ), truncated at second order has a minimum which corresponds to an analytical and \ensuremathπ/pF\ensuremath{\pi}{/p}_{F}\ensuremathπ/pF​ periodic \ensuremathφx.{\ensuremath{\varphi}}_{x}.\ensuremathφx​. Such a minimum is found solving an infinite set of coupled self-consistent equations, one for each of the infinite Fourier modes of \ensuremathφx.{\ensuremath{\varphi}}_{x}.\ensuremathφx​.
We consider the Hubbard-Holstein model in the adiabatic limit to investigate the effects of electron-electron interactions on the electron-phonon coupling. To this aim we compute at any momentum and filling the static charge susceptibility of the Hubbard model within the Gutzwiller approximation, and we find that electron-electron correlations effectively screen the electron coupling to the lattice. This screening is more effective at large momenta, and as a consequence, the charge-density-wave phase due to the usual Peierls instability of the Fermi-surface momenta is replaced by a phase-separation instability when the correlations are sizable.
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 derive the Luttinger liquid to charge density wave (CDW) transition condition, exact to second order in a blocked perturbative approach, for the spinless one-dimensional Holstein model in the adiabatic regime. The small parameter is the ratio of the electron-phonon coupling ggg and the adiabaticity parameter $t∕{\ensuremath{\omega}}_{0}$. Here we correct the mean-field criterion for Peierls instability, by replacing the static noninteracting susceptibility at twice the Fermi momentum ${\ensuremath{\chi}}_{0}(2{k}_{F},0)$ with the dynamic one, and get $1+2{g}^{2}{\ensuremath{\omega}}_{0}{\ensuremath{\chi}}_{0}(2{k}_{F},{\ensuremath{\omega}}_{0})=0$. At non-half-filling, we present the phase diagram showing the surprising result that the CDW occurs in a more restricted region of the two-parameter (${g}^{2}{\ensuremath{\omega}}_{0}∕t$ and $t∕{\ensuremath{\omega}}_{0}$) space than at half-filling.
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.
Monte Carlo simulations are performed to examine superconductivity and charge-density-wave fluctuations in the infinite-dimensional electron-phonon Holstein model. The electron-phonon system interpolates between an attractive, static, Falicov-Kimball model that always exhibits charge-density-wave order and an attractive, instantaneous, Hubbard model that always superconducts as a function of phonon frequency. The maximum charge-density-wave transition temperature at half-filling is an order of magnitude smaller than the effective electronic bandwidth and is virtually independent of the phonon frequency. The maximum superconducting transition temperature depends strongly on phonon frequency and is bounded from above by the maximum charge-density-wave transition temperature. The exact solution is compared to both weak-coupling expansions and strong-coupling expansions. The effective phonon potential becomes anharmonic and develops a double-well structure that deepens as the electron-phonon interaction increases.