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arXiv:cond-mat/0105012 · Physical review. B, Condensed matter

Spin-charge decoupling and orthofermi quantum statistics

A. K. Mishra

Published Mar 12, 2001• 12 citations
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Abstract

Currently Gutzwiller projection technique and nested Bethe ansatz are two main methods used to handle electronic systems in the U infinity limit. We demonstrate that these two approaches describe two distinct physical systems. In the nested Bethe ansatz solutions, there is a decoupling between the spin and charge degrees of freedom. Such a decoupling is absent in the Gutzwiller projection technique. Whereas in the Gutzwiller approach, the usual antisymmetry of space and spin coordinates is maintained, we show that the Bethe ansatz wave function is compatible with a form of quantum statistics, viz., orthofermi statistics. In these statistics, the wave function is antisymmetric in spatial coordinates alone. This feature ultimately leads to spin-charge decoupling.

DOI
10.1103/physrevb.63.132405

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Currently Gutzwiller projection technique and nested Bethe ansatz are two main methods used to handle electronic systems in the U infinity limit. We demonstrate that these two approaches describe two distinct physical systems. In the nested Bethe ansatz solutions, there is a decoupling between the spin and charge degrees of freedom. Such a decoupling is absent in the Gutzwiller projection technique. Whereas in the Gutzwiller approach, the usual antisymmetry of space and spin coordinates is maintained, we show that the Bethe ansatz wave function is compatible with a form of quantum statistics, viz., orthofermi statistics. In these statistics, the wave function is antisymmetric in spatial coordinates alone. This feature ultimately leads to spin-charge decoupling.
Abstract The Bethe ansatz can be generalized to problems where particles have internal degrees of freedom. The generalized method can be viewed as two Bethe ansätze executed one after the other: nested Bethe ansatz. Electronic systems are the most relevant examples for condensed matter physics. Prominent electronic many-particle systems in one dimension solvable by a nested Bethe ansatz are the one-dimensional δ-Fermi gas, the one-dimensional Hubbard model, and the Kondo model. The major difference to the Bethe ansatz for one component systems is a second, spin, eigenvalue problem, which has the same form in all cases and is solvable by a second Bethe ansatz, e.g. an algebraic Bethe ansatz. A quantum dot tuned to Kondo resonance and coupled to an isolated metallic ring presents an application of the coupled sets of Bethe ansatz equations of the nested Bethe ansatz.
The momentum distribution function n(k) and spin-correlation function S(k) are determined for the one-dimensional large-U Hubbard model with various electron densities. The present work is featured with an application of the Bethe-ansatz wave function, which has a simple form in the large-U limit for any electron density. Namely, its charge degrees of freedom are expressed as a Slater determinant of spinless fermions, while its spin degrees of freedom are equivalent to the one-dimensional S=(1/2 Heisenberg model. The singularity of n(k) at k=kF{k}_{F}kF​ is analyzed from the system size dependence. In addition to the kF{k}_{F}kF​ singularity, n(k) has a weak singularity at k=3kF{k}_{F}kF​; however, no detectable singularity is present at 2kF{k}_{F}kF​, which one might expect from the spinless fermion wave function. The singularity of S(k) at 2kF{k}_{F}kF​ is also examined in detail. It is concluded from the size dependence that, when the system is away from half-filling, the nature of the singularity at 2kF{k}_{F}kF​ is different from that in the Heisenberg model. The results are compared with the behavior in the weak-correlation regime examined with the perturbation calculation, the prediction of g-ology, and recent Monte Carlo calculations.
An exact expression for the Green's function G(k,\ensuremath{\omega}) of one hole in the U\ensuremath{\rightarrow}\ensuremath{\infty} one-dimensional Hubbard model is derived from the known Bethe-ansatz solution of the model. Due to the spin-charge decoupling of the one-hole excitation spectrum and the Fermi-like statistics of the spin excitations, G shows branch-cut singularities at \ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}2 sink and a nontrivial dependence on the momentum of the hole. As a result, the hole can propagate although the quasiparticle weight vanishes.
Orthofermi statistics is characterized by an exclusion principle which is more “exclusive” than Pauli’s exclusion principle: an orbital state shall not contain more than one particle, no matter what the spin direction is. The wavefunction is antisymmetric in spatial indices alone with arbitrary symmetry in the spin indices. Orthobose statistics is corresponding Bose analog: the wavefunction is symmetric in spatial indices, with arbitrary symmetry in spin indices. We construct the quantum field theory of particles obeying these new kinds of quantum statistics. Non-relativistic as well as relativistic quantum field theories with interactions are considered.
This chapter revisits the construction of the Bethe wave function for the Lieb–Liniger model with spin degrees of freedom, which has traditionally followed a two-step approach: independent determination of coordinate and spin wave functions, later combined to enforce correct particle statistics. While this procedure is effective under the assumption that the hamiltonian is spin-independent, it becomes inadequate when spin-dependent interactions are present. Two key challenges arise: the need for a complete wave function without splitting it into separate coordinate and spin components, and the treatment of integrable systems with spin-dependent hamiltonians. To address these, the chapter introduces an efficient algebraic method based on the Faddeev–Zamolodchikov algebra, which provides a unified framework for handling coordinate and spin variables in integrable models with factorized scattering. This algebraic approach enables direct construction of the full wave function, offering a natural extension to systems involving both bosonic and fermionic degrees of freedom. The resulting spin chain representation of the transfer matrix not only simplifies the analysis but also yields a deeper understanding of the underlying algebraic structure governing these quantum systems.
Orthofermi statistics is characterized by an exclusion principle which is more ``exclusive'' than Pauli's exclusion principle: an orbital state shall not contain more than one particle, no matter what the spin direction is. The wavefunction is antisymmetric in spatial indices alone with arbitrary symmetry in the spin indices. Orthobose statistics is corresponding Bose analog: the wavefunction is symmetric in spatial indices, with arbitrary symmetry in spin indices. We construct the quantum field theory of particles obeying these new kinds of quantum statistics. Non-relativistic as well as relativistic quantum field theories with interactions are considered.