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
U strongly enhances the dimerization, the enhancement being strongest for $U\ensuremath{\sim}4{t}_{0}$, where
4t0 is the bandwidth of the uniform chain; (ii) the nearest-neighbor interaction
V1 further enhances the dimerization until ${V}_{1}\ensuremath{\lesssim}\frac{1}{2}U$, while
V1>21U 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 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>21U,
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.
By utilizing the twisted boundary conditions in the exact diagonalization method, we investigate the single-particle spectral function of the extended Peierls-Hubbard model at both half-filling and quarter-filling. In one-dimensional (1D) interacting systems, the spin-charge separation can typically be identified in the single-particle spectral function by observing the distinct spinon and holon bands. At half-filling, starting from the pure 1D Hubbard model with the on-site interaction
U=10, we observe that the band structure indicative of the spin-charge separation gradually transitions to four individual bands as the Peierls instability $\ensuremath{\delta}$ increases. At
U=10 and $\ensuremath{\delta}=0.2$ where the spin-charge separation is still observable, increasing the nearest-neighbor interaction
V can drive the system to a charge-density-wave state when $V\ensuremath{\gtrsim}U/2$. At quarter-filling, on the other hand, the ground state of Peierls-Hubbard model manifests an antiferromagnetic Mott insulator in units of dimers. Increasing
U results in only a very small gap in the single-particle spectrum because even for $U=+\ensuremath{\infty}$, with the model transforming into a noninteracting half-filled dimerized tight-binding model, its gap determined by the Peierls instability $\ensuremath{\delta}$ remains small. Conversely, increasing
V can effectively open the single-particle gap and make the spinon and holon bands more prominent.
We study the nature of the ground state of a one-dimensional electron-phonon model for molecular crystals: The phonons are assumed dispersionless and couple to the local electronic density. We consider the half-filled-band sector and discuss the stability of the Peierls-dimerized ground state as a function of the phonon frequency ($\ensuremath{\omega}$), electron-phonon coupling constant ($\ensuremath{\lambda}$), and number of components of the electron spin (
n). First, we discuss the properties of the model in the limiting cases of zero frequency and infinite frequency. We then perform a strong coupling expansion which maps the system onto a spinless fermion model with nearest-neighbor repulsion for both spinless and spin-\textonehalf{} electrons, but with different parameters in both cases. Finally, we perform a numerical study of the system using a Monte Carlo technique. We study the behavior of the order parameter and of correlation functions for various points in parameter space. We also perform a finite-size---scaling analysis of the numerical data. The conclusions of our study are the following: For the case of spinless electrons, there exists always a disordered phase for small coupling constant, and the system undergoes an infinite-order transition to a Peierls-dimerized state as the coupling constant increases beyond a critical value. The phase diagram is divided between a disordered and an ordered region by a line that connects the points ($\ensuremath{\omega}=0$, $\ensuremath{\lambda}=0$) and ($\ensuremath{\omega}=\ensuremath{\infty}$, $\ensuremath{\lambda}=\ensuremath{\infty}$). For the case of spin-\textonehalf{} electrons, the system is dimerized for arbitrary $\ensuremath{\lambda}$ and $\ensuremath{\omega}$ except in the limit $\ensuremath{\omega}=\ensuremath{\infty}$.
A Green-function formalism is presented to study a Peierls-Hubbard Hamiltonian in two dimensions. The lattice consists of parallel dimerized chains with alternating nearest-neighbor hoppings t and ${\mathit{t}}_{\mathrm{\ensuremath{\parallel}}}$ and another hopping ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ between different chains. The method treats the interdimer hopping as a perturbation and yields exact results in the uncorrelated case and for isolated dimers. The calculated spectral functions exhibit a number of narrow subbands with typical low-dimensional singularities. The dependence of the gap at the Fermi level on the electron-electron interaction U agrees qualitatively with the exact result in the known one-dimensional nondimerized limit. The paramagnetic susceptibility shows a maximum structure at low temperatures that is enhanced by U and by dimerization and a Curie-Weiss behavior at high temperatures.
By utilizing the twisted boundary conditions in the exact diagonalization method, we investigate the single-particle spectral function of the extended Peierls-Hubbard model at both half-filling and quarter filling. In one-dimensional (1D) interacting systems, the spin-charge separation can typically be identified in the single-particle spectral function by observing the distinct spinon and holon bands. At half filling, starting from the pure 1D Hubbard model with the on-site interaction
U=10, we observe that the band structure indicative of the spin-charge separation gradually transitions to four individual bands as the Peierls instability
δ increases. At
U=10 and
δ=0.2 where the spin-charge separation is still observable, increasing the nearest-neighbor interaction
V can drive the system to a charge-density-wave (CDW) state when
V≳U/2, without the obeservation of spinon and holon bands. At quarter-filling, on the other hand, the ground state of Peierls-Hubbard model manifests an antiferromagnetic Mott insulator in units of dimers. Increasing
U results in only a very small gap in the single-particle spectrum because even for
U=+∞, with the model transforming into a noninteracting half-filled dimerized tight-binding model, its gap determined by the Peierls instability
δ remains small. Conversely, increasing
V can effectively open the single-particle gap and make the spinon and holon bands more prominent.