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arXiv:cond-mat/0612273 · Physical Review B

Phase diagram of the one-dimensional Hubbard-Holstein model at half and quarter filling

Rahul P. Hardikar, Rudolf Torsten Clay

Published Jun 6, 2007• 73 citations
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Abstract

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 $U$ 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-$U$ Hubbard model. (ii) The transitions between Mott/intermediate and intermediate/Peierls states are of the Kosterlitz-Thouless form. (iii) For larger $U$, 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.

DOI
10.1103/physrevb.75.245103

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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.
We present a numerical study of the Hubbard-Holstein model in one dimension at half filling, including finite-frequency quantum phonons. At half filling, the effects of the electron-phonon and electron-electron interactions compete with the Holstein phonon coupling acting as an effective negative Hubbard on-site interaction U that promotes on-site electron pairs and a Peierls charge-density wave state. Most previous work on this model has assumed that only Peierls or Mott phases are possible at half filling. However, there has been speculation that a third metallic phase exists between the Peierls and Mott phases. We confirm the intermediate phase, and show that the Luttinger liquid correlation exponent K(rho) >1 in this region, indicating dominant superconducting pair correlations. We explore the full phase diagram as a function of Hubbard U, phonon coupling constant, and phonon frequency.
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.
The renormalization group technique is applied to one-dimensional electron-phonon Hubbard models at half filling and zero temperature. For the Holstein-Hubbard model, the results of one-loop calculations are congruent with the phase diagram obtained by quantum Monte Carlo simulations in the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane for the phonon-mediated interaction gph{g}_{\mathrm{ph}}gph​ and the Coulomb interaction UUU. The incursion of an intermediate phase between a fully gapped charge-density-wave state and a Mott antiferromagnet is supported along with the growth of its size with the molecular phonon frequency ${\ensuremath{\omega}}_{0}$. We find additional phases enfolding the base boundary of the intermediate phase. A Luttinger liquid line is found below some critical ${U}^{*}\ensuremath{\approx}{g}_{\mathrm{ph}}^{*}$, followed at larger $U\ensuremath{\sim}{g}_{\mathrm{ph}}$ by a narrow region of bond-order-wave ordering which is either charge or spin gapped depending on UUU. For the Peierls-Hubbard model, the region of the (U,gph)(U,{g}_{\mathrm{ph}})(U,gph​) plane with a fully gapped Peierls-bond-order-wave state shows a growing domination over the Mott gapped antiferromagnet as the Debye frequency ${\ensuremath{\omega}}_{D}$ decreases. A power-law dependence ${g}_{\mathrm{ph}}\ensuremath{\sim}{U}^{2\ensuremath{\eta}}$ is found to map out the boundary between the two phases, whose exponent is in good agreement with the existing quantum Monte Carlo simulations performed when a finite nearest-neighbor repulsion term VVV is added to the Hubbard interaction.
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.
We derive an effective Hamiltonian for the one-dimensional Hubbard-Holstein model, valid in a regime of both strong electron-electron (e-e) and electron-phonon (e-ph) interactions and in the nonadiabatic limit ($t/{\ensuremath{\omega}}_{0}\ensuremath{\le}1$), by using a nonperturbative approach. We obtain the phase diagram at quarter filling by employing a modified Lanczos method and studying various density-density correlations. The spin-spin AF (antiferromagnetic) interactions and nearest-neighbor repulsion, resulting from the e-e and the e-ph interactions, respectively, are the dominant terms (compared to hopping) and compete to determine the various correlated phases. As e-e interaction (U/t)(U/t)(U/t) is increased, the system transits from an AF cluster to a correlated singlet phase through a discontinuous transition at all strong e-ph couplings $2\ensuremath{\le}g\ensuremath{\le}3$ considered. At higher values of U/tU/tU/t and moderately strong e-ph interactions ($2\ensuremath{\le}g\ensuremath{\le}2.6$), the singlets break up to form an AF order and then to a paramagnetic order all in a single sublattice; whereas at larger values of ggg (>>>2.62.62.6), the system jumps directly to the spin disordered charge-density-wave (CDW) phase.
The Hubbard-Holstein model is one of the central models that describe the competition between electron-electron and electron-phonon interactions. In one dimension and at half-filling, the interplay between an electronic spin-density wave and a phonon-driven charge-density wave is considered to stabilize an intermediate Luther-Emery liquid. Here we show that, once the extended metallic regime disappears, a narrow bond-order-wave phase emerges. We use an exact directed-loop quantum Monte Carlo method for retarded interactions to simulate system sizes of several hundred sites, necessary to observe the weak signatures of this novel phase. Our results suggest a second-order quantum phase transition between the two charge orders that only turns first-order at strong coupling. Our findings are reminiscent of the extended Hubbard model, but they are driven by competing frequency dependencies of the same dynamically-screened Hubbard interaction.