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Physical review. B, Condensed matter

Modeling electron-electron interactions in reduced-dimensional materials: Bond-charge Coulomb repulsion and dimerization in Peierls-Hubbard models

D. K. Campbell, J. Tinka Gammel, Eugene Loh

Published Jul 1, 1990• 132 citations
Publisher

Abstract

To the conventional Peierls-Hubbard model, involving both on-site (U) and nearest-neighbor (V) Coulomb repulsions, we add ``off-diagonal'' terms, not expressible purely in terms of site densities, representing bond-bond (W) and mixed bond-site (X) electron-electron repulsive interactions involving nearest neighbors. We review earlier analyses of these interactions and discuss relative magnitudes of the parameters in applications to real materials. As a specific illustration, we investigate the effects of the off-diagonal W and X terms on dimerization in the one-dimensional, half-filled-band Peierls-Hubbard models, which have been widely applied to conjugated polymers (such as trans-polyacetylene) and to related quasi-one-dimensional charge-density wave (CDW) systems. Using both weak- and strong-coupling perturbation theory for large systems and exact diagonalizations of small systems, we investigate thoroughly the nature of the ground state of the model. For a broad range of the site-diagonal Hubbard parameters (U,V), including the values believed to be relevant to trans-polyacetylene, we find that the off-diagonal terms (W,X) initially enhance dimerization, thereby stabilizing the dimerized [or bond-order-wave (BOW)] ground state. For (unphysically) large values of W relative to U and V, dimerization is destroyed, and the BOW ground state goes over to a ferromagnetic ground state or a CDW ground state, depending on the relative sizes of U, V, and W. We conclude with a general discussion of the applicability of the Peierls-Hubbard models to quasi-one-dimensional materials, including the potential importance of the breaking of charge conjugation (``particle-hole'') symmetry by the X term.

DOI
10.1103/physrevb.42.475

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To the conventional Peierls-Hubbard model, involving both on-site (U) and nearest-neighbor (V) Coulomb repulsions, we add off-diagonal'' terms, not expressible purely in terms of site densities, representing bond-bond (W) and mixed bond-site (X) electron-electron repulsive interactions involving nearest neighbors. We review earlier analyses of these interactions and discuss relative magnitudes of the parameters in applications to real materials. As a specific illustration, we investigate the effects of the off-diagonal W and X terms on dimerization in the one-dimensional, half-filled-band Peierls-Hubbard models, which have been widely applied to conjugated polymers (such as trans-polyacetylene) and to related quasi-one-dimensional charge-density wave (CDW) systems. Using both weak- and strong-coupling perturbation theory for large systems and exact diagonalizations of small systems, we investigate thoroughly the nature of the ground state of the model. For a broad range of the site-diagonal Hubbard parameters (U,V), including the values believed to be relevant to trans-polyacetylene, we find that the off-diagonal terms (W,X) initially enhance dimerization, thereby stabilizing the dimerized [or bond-order-wave (BOW)] ground state. For (unphysically) large values of W relative to U and V, dimerization is destroyed, and the BOW ground state goes over to a ferromagnetic ground state or a CDW ground state, depending on the relative sizes of U, V, and W. We conclude with a general discussion of the applicability of the Peierls-Hubbard models to quasi-one-dimensional materials, including the potential importance of the breaking of charge conjugation (particle-hole'') symmetry by the X term.
We investigate the general applicability of the standard Peierls-Hubbard model, involving both on-site (UUU) and nearest-neighbor (VVV) Coulomb repulsions, by adding nearest-neighbor off-diagonal terms representing the bond (WWW) and mixed bond-site (XXX) electron-electron repulsive interactions. For a broad range of the site-diagonal Hubbard parameters (U,VU,VU,V), we find that the off-diagonal terms (W,XW,XW,X) initially enhance dimerization, thereby stabilizing the dimerized ground state.
A novel real-space approach to dimerization in a half-filled band is developed to investigate effects of electron-electron interactions on the Peierls instability. Dimerization is shown to be a result of imperfect resonance between pairs of electron configurations related to each other by a mirror-plane symmetry passing through the longest diagonal of the infinite ring, and the kinetic- and potential-energy contributions to the barrier to resonance are identified separately. The effects of including the on-site, nearest-neighbor, and next-nearest-neighbor interactions are investigated, and in each case it is shown that the enhancement or reduction in dimerization can be predicted from elementary physical arguments. These predictions are then substantiated by exact numerical calculations on a ten-site ring, and finite-size effects are shown to be small. The principal results that are obtained are the following: (i) The on-site correlation UUU strongly enhances the dimerization, the enhancement being strongest for $U\ensuremath{\sim}4{t}_{0}$, where 4t04{t}_{0}4t0​ is the bandwidth of the uniform chain; (ii) the nearest-neighbor interaction V1{V}_{1}V1​ further enhances the dimerization until ${V}_{1}\ensuremath{\lesssim}\frac{1}{2}U$, while V1>12U{V}_{1}>\frac{1}{2}UV1​>21​U favors a uniform chain with a different broken-symmetry ground state, an on-site charge-density wave; (iii) for ${V}_{1}\ensuremath{\lesssim}\frac{1}{2}U$, the second-neighbor interaction V2{V}_{2}V2​ reduces the dimerization slightly, although the dimerization is still stronger than that with an effective nearest-neighbor interaction ${V}_{1}\ensuremath{-}{V}_{2}$; (iv) for V1>12U{V}_{1}>\frac{1}{2}UV1​>21​U, V2{V}_{2}V2​ destroys the on-site charge-density wave and the ground state is strongly dimerized again. The complete Parisier-Parr-Pople (PPP) Hamiltonian is discussed, and it is pointed out that the above results, together with the excited-state orderings in the PPP Hamiltonian, strongly indicate that the ground state of the PPP Hamiltonian is the dimerized state. The excited-state orderings in finite polyenes, spin-density distributions in polyacetylene, and our theoretical results all indicate then that explicit inclusion of Coulomb interactions may be necessary for an accurate description of the ground and excited states in polyacetylene.
Using the Lanczos method for exact diagonalization on systems up to 14 sites, combined with a novel phase randomization'' technique for extracting more information from these small systems, we investigate several aspects of the one-dimensional Peierls-Hubbard Hamiltonian, in the context of trans-polyacetylene: the dependence of the ground state dimerization on the strength of the electron-electron interactions, including the effects of off-diagonal'' Coulomb terms generally ignored in the Hubbard model; the phonon vibrational frequencies and dispersion relations, and the optical absorption properties, including the spectrum of absorptions as a function of photon energy. These three different observables provide considerable insight into the effects of electron-electron interactions on the properties of real materials and thus into the nature of strongly correlated electron systems. 29 refs., 11 figs.
An exact-diagonalization technique on small clusters is used to study the ground states and single-particle excitations of the Hubbard model with on-site (U) and nearest-neighbor (V) Coulombic repulsive interactions. It is shown that the long-range charge-density-wave state realized in a half-filled two-dimensional square lattice for 4V\ensuremath{\gtrsim}U persists up to quarter filling, and that the doped carriers form a small Fermi surface that encloses an area corresponding to the number of doped carriers rather than the total number of electrons involved in the system. The present result offers a well-defined counterexample in the discussion of the Fermi-surface dimensions of strongly correlated electron systems, and may provide a complementary understanding of the large Fermi surface believed to exist in the ordinary Hubbard model.
We revisit the ground-state phase diagram of the one-dimensional half-filled extended Hubbard model with on-site (U) and nearest-neighbor (V) repulsive interactions. In the first half of the paper, using the weak-coupling renormalization-group approach (g(g(g-ology) including second-order corrections to the coupling constants, we show that bond-charge-density-wave (BCDW) phase exists for $U\ensuremath{\approx}2V$ in between charge-density-wave (CDW) and spin-density-wave (SDW) phases. We find that the umklapp scattering of parallel-spin electrons disfavors the BCDW state and leads to a bicritical point where the CDW-BCDW and SDW-BCDW continuous-transition lines merge into the CDW-SDW first-order transition line. In the second half of the paper, we investigate the phase diagram of the extended Hubbard model with either additional staggered site potential $\ensuremath{\Delta}$ or bond alternation $\ensuremath{\delta}.$ Although the alternating site potential $\ensuremath{\Delta}$ strongly favors the CDW state (that is, a band insulator), the BCDW state is not destroyed completely and occupies a finite region in the phase diagram. Our result is a natural generalization of the work by Fabrizio, Gogolin, and Nersesyan [Phys. Rev. Lett. 83,83,83, 2014 (1999)], who predicted the existence of a spontaneously dimerized insulating state between a band insulator and a Mott insulator in the phase diagram of the ionic Hubbard model. The bond alternation $\ensuremath{\delta}$ destroys the SDW state and changes it into the BCDW state (or Peierls insulating state). As a result the phase diagram of the model with $\ensuremath{\delta}$ contains only a single critical line separating the Peierls insulator phase and the CDW phase. The addition of $\ensuremath{\Delta}$ or $\ensuremath{\delta}$ changes the universality class of the CDW-BCDW transition from the Gaussian transition into the Ising transition.
We study extended Hubbard and Peierls-Hubbard models with up to N=10 sites and half-filled bands with modified boundary conditions. For the extended Hubbard model, the finite-size dependence of the transition from the charge-density-wave regime (CDW) to the spin-density-wave (SDW) regime and of the condensation transition is critically examined, clarifying some previous results. For the Peierls-Hubbard model and N\ensuremath{\le}10 with modified periodic boundary conditions, we always see a decrease of dimerization with U. Extrapolating these results to infinite N, we find that dimerization is enhanced by U for small electron-phonon coupling \ensuremath{\alpha}\ifmmode \tilde{}\else \~{}\fi{}0.75, whereas it decreases with U for 0.75\ensuremath{\le}1. By extending this model by a nearest-neighbor repulsion V, the amplitude of dimerization reaches a maximum at the same value U\ensuremath{\le}2V, where the CDW-to-SDW transition occurs in the extended Hubbard model.