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arXiv:cond-mat/9512090 · Physical review. B, Condensed matter

Composite quasiparticle formation and the low-energy effective Hamiltonians for the one- and two-dimensional Hubbard model

Junwu Gan, D. H. Lee, Per Hedegård

Published Sep 15, 1996• 0 citations
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Abstract

We investigate the effect of hole doping on the strong-coupling Hubbard model at half filling in spatial dimensions D\ensuremath{\ge}1. We start with an antiferromagnetic mean-field description of the insulating state, and show that doping creates solitons in the antiferromagnetic background. In one dimension, the soliton is topological, spinless, and decoupled from the background antiferromagnetic fluctuations at low energies. In two dimensions and above, the soliton is nontopological, has spin quantum number 1/2, and is strongly coupled to the antiferromagnetic fluctuations. We derive the effective action governing the quasiparticle motion, study the properties of a single carrier, and comment on a possible description at finite concentration. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.7737

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We investigate the effect of hole doping on the strong-coupling Hubbard model at half filling in spatial dimensions D\ensuremath{\ge}1. We start with an antiferromagnetic mean-field description of the insulating state, and show that doping creates solitons in the antiferromagnetic background. In one dimension, the soliton is topological, spinless, and decoupled from the background antiferromagnetic fluctuations at low energies. In two dimensions and above, the soliton is nontopological, has spin quantum number 1/2, and is strongly coupled to the antiferromagnetic fluctuations. We derive the effective action governing the quasiparticle motion, study the properties of a single carrier, and comment on a possible description at finite concentration. \textcopyright{} 1996 The American Physical Society.
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The ground state (GS) properties of the quasi-one-dimensional AB2AB_2AB2​ Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. In the strong-coupling regime, we derive a low-energy Lagrangian suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice magnetizations, and Lieb total spin per unit cell of the effective quantum Heisenberg model, in very good agreement with previous results. In the challenging hole doping regime away from half filling, we derive the corresponding t-Jt\textrm{-}Jt-J Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially modulated spin structures and the emergence of phase separation in the presence of resonating-valence-bond states. We also calculate the doping-dependent GS energy and total spin per unit cell, in which case it is shown that the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in the doped regime are in very good agreement with density matrix renormalization group studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the magnetic structure.
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In the first part of the present Thesis, the ground state (GS) properties of the quasi-one-dimensional AB₂ Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. Using a functional integral approach, combined with a perturbative expansion in the strong-coupling regime, we obtain the Lagrangian density associated with the charge (Grassmann fields) and spin [SU(2) gauge fields] degrees of freedom. In the strong-coupling regime, we derive a perturbative low-energy theory suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice magnetizations, and total spin per unit cell in the Lieb ferrimagnetic GS of the effective quantum Heisenberg model, in very good agreement with previous results. In the challenging hole doping regime away from half filling, we derive the corresponding t-J Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially modulated spin structures and the emergence of phase separation in the presence of resonatingvalence-bond (RVB) states. We also calculate the doping-dependent GS energy and total spin per unit cell, including both Zeeman and orbital contributions, in which case it is shown that the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in the doped regime are in very good agreement with density matrix renormalization group (DMRG) studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the magnetic structure. In the second part, motivated by analogy with photonic lattices, we examine the edge states of a one-dimensional trimer lattice in the phases with and without inversion symmetry protection. In contrast to the Su-Schrieffer-Heeger model, we show that the edge states in the inversion-symmetry broken phase of the trimer model turn out to be chiral, i.e., instead of appearing in pairs localized at opposite edges they can appear at a single edge. Interestingly, these chiral edge states remain robust to large amounts of disorder. In addition, we use the Zak phase to characterize the emergence of degenerate edge states in the inversion-symmetric phase of the trimer model. Furthermore, we capture the essentials of the whole family of trimers through a mapping onto the commensurate off-diagonal Aubry-André-Harper model, which allows us to establish a direct connection between chiral edge modes in the two models, including the calculation of Chern numbers.We thus suggest that the chiral edge modes of the trimer lattice have a topological origin inherited from this effective mapping. Also, we find a nontrivial connection between the topological phase transition point in the trimer lattice and the one in its associated two-dimensional parent system, in agreement with results in the context of Thouless pumping in photonic lattices.
Field-theoretic methods are used to investigate the large-UUU Hubbard model on the honeycomb lattice at half-filling and in the hole-doped regime. Within the framework of a functional-integral approach, we obtain the Lagrangian density associated with the charge and spin degrees of freedom. The Hamiltonian related to the charge degrees of freedom is exactly diagonalized. In the strong-coupling regime, we derive a perturbative low-energy theory suitable to describe the quantum antiferromagnetic phase (AF) as a function of hole doping. At half-filling, we deal with the underlying spin degrees of freedom of the quantum AF Heisenberg model by employing a second-order spin-wave analysis, in which case we have calculated the ground-state energy and the staggered magnetization; the results are in very good agreement with previous studies. Further, in the continuum, we derive a nonlinear $\ensuremath{\sigma}$ model with a topological Hopf term that describes the AF-VBS (valence bond solid) competition. Lastly, in the challenging doped regime, our approach allows the derivation of a $t\text{\ensuremath{-}}J$ Hamiltonian, and the analysis of the role played by charge and spin quantum fluctuations on the ground-state energy and, particularly, on the breakdown of the AF order at a critical hole doping; the results are benchmarked against recent Grassmann tensor product state simulations.
We study the temperature and doping dependence of the two-dimensional (2D) Hubbard model in the intermediate coupling regime (U=4).(U=4).(U=4). As temperature is decreased for a half-filled band, 2D critical spin fluctuations give rise to a strong local maximum in $|\mathrm{Im}\ensuremath{\Sigma}({k}_{F},\ensuremath{\omega})|$ at $\ensuremath{\omega}=0,$ leading to a split peak in the spectral function. These strong fluctuations dominating near half filling are completely destroyed by 12% doping concentration. Below this concentration the imaginary part of the self-energy vanishes nearly quadratically in frequency at the Fermi energy, a Fermi-liquid-like feature. With varying the temperature at this concentration, we present numerical evidence for the band Kondo effect in the doped 2D Hubbard model by showing at low temperatures a growing peak in the density of states, nearly quadratically vanishing (in frequency) scattering rates at the Fermi energy and a linearly vanishing (in temperature) effective magnetic moment.
A path-integral approach to the Hubbard model is developed for the whole range of the coupling strength U. At half filling, the strong-coupling results are readily reproduced within the simple Gaussian fluctuations. The low-lying spin wave is shown to be described by the nonlinear \ensuremath{\sigma} model. The effective coupling of the doped hole with the background fluctuations also agrees with that obtained from the t-J model in the small-doping limit. At finite doping, such a formalism may provide a starting point for investigating the short-range spin-liquid state.