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Physical review. B, Condensed matter

Electron-electron interaction effects on Peierls dimerization in a half-filled band

S. N. Dixit, S. Mazumdar

Published Feb 15, 1984• 125 citations
Publisher

Abstract

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 $4{t}_{0}$ is the bandwidth of the uniform chain; (ii) the nearest-neighbor interaction ${V}_{1}$ further enhances the dimerization until ${V}_{1}\ensuremath{\lesssim}\frac{1}{2}U$, while ${V}_{1}>\frac{1}{2}U$ 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 ${V}_{2}$ 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 ${V}_{1}>\frac{1}{2}U$, ${V}_{2}$ 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.

DOI
10.1103/physrevb.29.1824

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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∼4t0U\ensuremath{\sim}4{t}_{0}U\ensuremath∼4t0​, 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 V1\ensuremath≲12U{V}_{1}\ensuremath{\lesssim}\frac{1}{2}UV1​\ensuremath≲21​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 V1\ensuremath≲12U{V}_{1}\ensuremath{\lesssim}\frac{1}{2}UV1​\ensuremath≲21​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 V1\ensuremath−V2{V}_{1}\ensuremath{-}{V}_{2}V1​\ensuremath−V2​; (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.
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.
In this last of three articles on the optical absorption of electrons in a half-filled Peierls-distorted chain we address the dimerized extended Hubbard model in the limit of a large on-site interaction U. When the Hubbard interaction is large both compared to the band width W and the nearest-neighbor interaction V the charge dynamics is properly described by the Harris-Lange model. This model can be exactly mapped onto a model of free spinless Fermions in parallel (Hubbard-)bands of width W which are eventually Peierls-split. To determine the coherent absorption features at low temperatures we design and employ the “no-recoil approximation ” in which we assume that the momentum transfer to the spin degrees of freedom can only be ∆qS = 0 or ∆qS = π/a during an optical excitation. We present explicit analytical results for the optical absorption in the presence of a lattice dimerization δ and a nearest-neighbor interaction V for the Néel and dimer state. We find
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=10U=10U=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=10U=10U=10 and $\ensuremath{\delta}=0.2$ where the spin-charge separation is still observable, increasing the nearest-neighbor interaction VVV 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 UUU 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 VVV can effectively open the single-particle gap and make the spinon and holon bands more prominent.
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=10U=10U=10, we observe that the band structure indicative of the spin-charge separation gradually transitions to four individual bands as the Peierls instability δδδ increases. At U=10U=10U=10 and δ=0.2δ=0.2δ=0.2 where the spin-charge separation is still observable, increasing the nearest-neighbor interaction VVV can drive the system to a charge-density-wave (CDW) state when V≳U/2V\gtrsim U/2V≳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 UUU results in only a very small gap in the single-particle spectrum because even for U=+∞U=+\inftyU=+∞, with the model transforming into a noninteracting half-filled dimerized tight-binding model, its gap determined by the Peierls instability δδδ remains small. Conversely, increasing VVV can effectively open the single-particle gap and make the spinon and holon bands more prominent.
In this, the last of three articles on the optical absorption of electrons in a half-filled Peierls-distorted chain, we address the dimerized extended Hubbard model in the limit of a large on-site interaction U. When the Hubbard interaction is large compared with both the bandwidth W and the nearest-neighbour interaction V, the charge dynamics are properly described by the Harris-Lange model. This model can be exactly mapped onto a model of free spinless fermions in parallel (Hubbard) bands of width W which are eventually Peierls split. To determine the coherent absorption features at low temperatures, we design and employ the ‘no-recoil approximation’ in which we assume that the momentum transfer to the spin degrees of freedom can only be Δqs = 0 or Δqs = π/a during an optical excitation. We present explicit analytical results for the optical absorption in the presence of a lattice dimerization δ and a nearest-neighbour interaction V for the Néel and dimer state. We find that the coherent part of the optical absorption for and the spin dimerization of the ground state.
We have carried out density-matrix renormalization group calculations on the ground state of long polyacene oligomers within a Pariser-Parr-Pople (PPP) Hamiltonian. The PPP model includes long-range electron correlations which are required for physically realistic modeling of conjugated polymers. We have obtained the ground-state energy as a function of the dimerization $\ensuremath{\delta}$ and various correlation functions and structure factors for $\ensuremath{\delta}=0$. From energetics, we find that the nature of the Peierls' instability in polyacene is conditional and strong electron correlations enhance the dimerization. The cis form of the distortion is favored over the trans form. However, from the analysis of correlation functions and associated structure factors, we find that polyacene is not susceptible to the formation of a bond order wave, spin-density wave, or a charge-density wave in the ground state.