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arXiv:cond-mat/0507242 · Physical Review Letters

Exact Transformation for Spin-Charge Separation of Spin- 1/2 Fermions without Constraints

Stellan S. Ostlund, Mats Granath

Published Feb 15, 2006• 31 citations
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Abstract

We demonstrate an exact local transformation which maps a purely Fermionic many-body system to a system of spinful bosons and spinless fermions, demonstrating a possible path to a non-Fermi-liquid state. We apply this to the half-filled Hubbard model and show how the transformation maps the ordinary spin half Fermionic degrees of freedom exactly and without introducing Hilbert space constraints to a chargelike quasicharge fermion and a spinlike quasispin Boson while preserving all the symmetries of the model. We present approximate solutions with localized charge which emerge naturally from the Hubbard model in this form. Our results strongly suggest that charge tends to remain localized for large values of the Hubbard U.

DOI
10.1103/physrevlett.96.066404

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Physical review. B, Condensed matter•Jan 1, 1995•7 citations
We demonstrate an exact local transformation which maps a purely Fermionic many-body system to a system of spinful bosons and spinless fermions, demonstrating a possible path to a non-Fermi-liquid state. We apply this to the half-filled Hubbard model and show how the transformation maps the ordinary spin half Fermionic degrees of freedom exactly and without introducing Hilbert space constraints to a chargelike quasicharge fermion and a spinlike quasispin Boson while preserving all the symmetries of the model. We present approximate solutions with localized charge which emerge naturally from the Hubbard model in this form. Our results strongly suggest that charge tends to remain localized for large values of the Hubbard U.
We propose a new fermion-spin transformation to implement the charge-spin separation in the large U Hubbard, or the equivalent t-J model. The charge degree of freedom is represented by a spinless fermion while the spin degree of freedom is represented by a hard-core boson. The local constraint for single occupancy is exactly satisfied. Very good agreement with exact solution is obtained for one-dimensional case in the mean field approximation, regarding the total energy, gapless spinon and holon spectra, and the momentum distribution of physical electrons. The same approximation yields good doping dependence of the staggered magnetization in the two-dimensional case.
We study the strongly correlated two-dimensional Hubbard model for general band filling by a transformation of the model to a spin-free representation. A pair transformation is constructed that maps a strong-coupling Hubbard problem in which the particles obey Fermi statistics onto a spin-free representation in which the charge degrees of freedom are spinless fermions, while constraining the physical system to an eigenstate of total spin. The Gutzwiller constraint is thus satisfied kinematically in the transformed problem for a class of Pauli-allowed many-fermion spatial wave functions. We find that the required pair transformation within the spin-singlet sector leads to correlated many-body states with broken time-reversal symmetry. For small band filling in the spin-singlet sector, the transformed problem for the charge degrees of freedom is formally identical to a problem for indistinguishable noninteracting 1/2 anyons (semions). For small deviations from half filling, the charge degrees of freedom (holes) in the singlet sector are again naturally represented as 1/2 anyons but in the presence of a strong uniform background U(1) gauge field. Exactly at half filling the charge degrees of freedom are quenched, and the formulation reduces to the spin-singlet wave function for the two-dimensional lattice first proposed by Kalmeyer and Laughlin. The one- and two-hole dynamics around this neutral chiral state on the two-dimensional lattice are considered in detail. Finally, numerical calculations are reported to test the accuracy of these states as approximations to the ground state of the Hubbard Hamiltonian. Over a range of band fillings we find that Gutzwiller projections of free-particle determinants provide improved hole kinetic energies, while a projection of the \ensuremath{\pi} flux phase provides improved spin-exchange energies for the pure Hubbard problem.
A fermion-spin transformation is used to implement the charge-spin separation, and is developed to study the low-dimensional t-J model. In this approach, the charge and spin degrees of freedom of the physical electron are separated, and the charge degree of freedom is represented by a spinless fermion while the spin degree of freedom is representd by a hard-core boson. The on-site local constraint for single occupancy is satisfied even in the mean-field approximation and the sum rule for the physical electron is obeyed. This approach can be applied to both one and two-dimensional systems. In the one-dimensional case, the spinon as well as the physical electron behave like Luttinger liquids. We have obtained a gapless charge and spin excitation spectrum, a good ground state energy, and a reasonable electron-momentum distribution within the mean-field approximation. The correct exponents of the correlation functions and momentum distribution are also obtained if the squeezing effect and rearrangement of the spin configurations are taken into account. In the two-dimensional case, within the mean-field approximation the magnetized flux state with a gap in the spinon spectrum has the lowest energy at half filling. The antiferromagnetic long-range order is destroyed by hole doping of the order \ensuremath{\sim}10-15 % for t/J=3-5 and a disordered flux state with gapless spinon spectrum becomes stable. The calculated specific heat is roughly consistent with observed results on copper oxide superconductors. The possible phase separation is also discussed at the mean-field level.
The general form of a mapping of the spin and charge degrees of freedom of electrons onto spinless fermions and local `spin'-1/2 operators is derived. The electron Hilbert space is mapped onto a tensor productspin-charge Hilbert space. The single occupancy condition of the t-J model is satisfied exactly without the constraints between the operators required with slave particle methods and the size of the Hilbert space (four states per site) is conserved. The connection and distinction between the physical electron spin and the ``pseudospin'' used in these maps is made explicit. Specifically the pseudospin generates rotations both in spin space and particle-hole space. A geometric description (up to sign) is provided using two component spinors. The form of the mapped t-J Hamiltonian involves the coupling of spin and spinless fermion currents, as one expects.
We review the slave-boson representations of the Hubbard model. When both spin and charge fermion degrees of freedom are represented by Bose fields, the approach is equivalent at the "self-consistent" saddle point level to the Gutzwiller approximation (GA). We show how the determination of the Lagrange multipliers, introduced to enforce the constraints, involves a Mott-Hubbard gap near half-filling for U > U c (localization edge). The quantum fluctuations are then considered within a renormalized basis of boson fields (restoring the spin-rotation invariance), which brings out two distinct channels, symmetric and antisymmetric. It is remarkable that both spin and charge fluctuations are obtained at the same level of the gaussian fluctuations, as oppossed to the standard 1/N expansion. This provides the microscopic basis for a Fermi liquid theory. Both dynamic effects and extension to finite temperature which were beyond the scope of the variational procedure can now be described by this approach. The physical implications of the model (dynamic correlation functions, T3Ln T term of the specific heat, superfluid instability) are finally considered, all of which reflect the Fermi liquid nature of the system.
An exact representation of the large-U Hubbard model is developed in terms of fermionic holon, fermionic doublon, and local spin operators which requires no constraint between the charge and spin operators. The Hamiltonian in the new representation explicitly expresses the Nagaoka theorem, and there is an exact one to one correspondence between the new representation and the large-U Lieb-Wu wave function for the case of one dimension. Using the new representation, we further show that the signature of spin-charge separation, in any dimension, is the characteristic wave vector 2kFSF{\mathbf{k}}_{\mathit{F}}^{\mathrm{SF}}kFSF​ in the static density-density correlation function (kFSF{\mathbf{k}}_{\mathit{F}}^{\mathrm{SF}}kFSF​ encloses a Fermi volume filled by 1-x spinless fermions per site, where x is the doping concentration). In addition, we derive the corresponding representation for the t-J-J' model, where the J' term describes the indirect hopping of the electrons, and the holon (doublon) propagator which explicitly shows the dependence of charge propagation on the spin state.