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International Journal of Modern Physics B

One-dimensional Large-U Hubbard Model in Strong Coupling: Charge and Spin Separation

Zheng-Yu Weng, D. N. Sheng, C. S. Ting

Published Jun 1, 1991• 0 citations
Publisher

Abstract

A path-integral formalism of the Hubbard model is used to study the one-dimensional large-U case. It is shown that the bare electron (hole) becomes a composite particle of two decoupled excitations, holon and spinon, together with the non-local string fields. Various correlation functions are analytically derived. The results strongly suggest a U*=∞ fixed point of Hubbard model which is distinct from the weak coupling case.

DOI
10.1142/s021797929100170x

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A path-integral formalism of the Hubbard model is used to study the one-dimensional large-U case. It is shown that the bare electron (hole) becomes a composite particle of two decoupled excitations, holon and spinon, together with the non-local string fields. Various correlation functions are analytically derived. The results strongly suggest a U*=∞ fixed point of Hubbard model which is distinct from the weak coupling case.
A path-integral formalism for the one-dimensional Hubbard model in the strong-coupling regime, which is equivalent to the t-J model in t/U expansion but without any explicit constraint, is developed. Based on this formalism, the zero-temperature properties of the Hubbard chain are systematically studied. In the infinite-U limit, the charge and spin degrees of freedom are shown to be completely separated. Such a separation at U=\ensuremath{\infty} provides a controllable perturbative scheme to study the strong-coupling case. In the large-U regime, the spin degree of freedom is represented by a ``squeezed'' Heisenberg chain. We solve the (squeezed) Heisenberg chain by introducing a soliton representation. Both the charge and spin excitations are found to agree well with the exact solution. The bare electron (hole) is shown to be a composite particle of these basic excitations, i.e., holon and spinon, together with a nonlocal string field. This string field makes the electron behave like a ``semion'' and plays an important role in determining various correlation functions. We analytically derive the asymptotic forms of the spin-spin and density-density correlation functions as well as the single-electron and the pair-electron Green's functions. The implications of the present work to the two-dimensional model are discussed.
The ground state of the one-dimensional Hubbard Hamiltonian is discussed on the basis of the Bethe ansatz solution of Lieb and Wu. A simple analytical representation of the distribution functions for charge and spin degrees of freedom is derived. The elementary excitations are associated with electron-hole pairs of two pseudo-fermion systems, one for the charge, the other for the magnetic excitations. Spinons and holons correspond to ground state configurations of odd-numbered rings. It is argued that holons will form bound pairs in weakly coupled chains.
We study the charge and spin currents transported by the elementary excitations of the one-dimensional (1D) Hubbard model and derive the corresponding current spectra. We present results both for finite-size systems and in the thermodynamic limit. This includes finding the couplings of both the low-energy and finite-energy (string) excitations to external charge and spin probes. At zero magnetic field the general structure of the charge-spin separation survives at all energy scales and the effective charges of both the low-energy and finite-energy charge excitations are studied as functions of the on-site Coulomb interaction U , electronic density n , and applied magnetic field H . In some limits the effective charge of the low-energy excitations equals that of the electrons, whereas that of the finite-energy charge-string excitations of rapidity length γ is found to be 2γ times the electronic charge. At U = ∞ the spin excitations do not contribute to spin transport, whereas the low-energy charge excitations feel an effective flux given by (ϕ ↑ N ↑ -ϕ ↓ N ↓ )/( N ↑ + N ↓ ), where N σ is the number of electrons of spin σ and ϕ σ is a spin-dependent flux. This reveals that at zero magnetic field and U = ∞ there is no spin transport, while at finite magnetic field the low-energy charge excitations also carry spin. In the U >> t limit the spin is carried both by holons and spinons. Finally, we find that the charge- and spin-current spectra can be derived from a semi-classical approach.
We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.
A Hamiltonian for the motion of oxygen holes strongly coupled to the Cu spins in a one-dimensional (1D) Cu-O lattice is solved exactly for all dopings. This Hamiltonian retains the dominant O-hole hopping term, that is, spin-exchange hopping of the hole spin with the intervening Cu spin and neglects the weaker non-spin-exchange hopping term. The excitations naturally separate into two independent quasiparticles: spinons and holons. It is this Hamiltonian that should be taken as the correct zeroth-order Hamiltonian from which perturbation theory should be applied. We show that the spinon spectrum is the spectrum of the 1D antiferromagnet incommensurate with the Cu lattice and the holons are spinless noninteracting fermions in a cosine band. The Cu-Cu spin correlation in the ground state increases linearly with doping from antiferromagnetic (-0.443) at x=0 to ferromagnetic (\ensuremath{\approxeq}0.19) at x=1.