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Physical review. B, Condensed matter

Representation of thet-Jmodel using decoupled spin and charge variables

Jan Loos

Published May 15, 1996• 6 citations
Publisher

Abstract

Using the mapping of the single-occupation space of the t-J model onto the tensor-product space of local spin-1/2 and holon states, the Hubbard operators and their products corresponding to physical quantities are expressed in terms of local spin and holon operators. All relevant operator relations of the t-J model are reproduced by this representation and no projection procedure is needed to remove the redundant degrees of freedom originating from the spin and charge decoupling. It is shown that the time-reversal symmetry requirement makes the essential restriction on the choice of the fictitious spin state associated with the electron-vacancy state in the mapping. The results are critically compared with the treatment of this topic in the paper of Wang and Rice [Phys. Rev. B 49, 4360 (1994)]. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.12556

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Physical review. B, Condensed matter•Jan 15, 1996•9 citations
Using the mapping of the single-occupation space of the t-J model onto the tensor-product space of local spin-1/2 and holon states, the Hubbard operators and their products corresponding to physical quantities are expressed in terms of local spin and holon operators. All relevant operator relations of the t-J model are reproduced by this representation and no projection procedure is needed to remove the redundant degrees of freedom originating from the spin and charge decoupling. It is shown that the time-reversal symmetry requirement makes the essential restriction on the choice of the fictitious spin state associated with the electron-vacancy state in the mapping. The results are critically compared with the treatment of this topic in the paper of Wang and Rice [Phys. Rev. B 49, 4360 (1994)]. \textcopyright{} 1996 The American Physical Society.
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.
An exact expression for the Hamiltonian H of the t-J model in terms of local spin (Si{\mathbf{S}}_{\mathit{i}}Si​) and fermionic holon (ei{\mathit{e}}_{\mathit{i}}ei​) operators is derived which requires no constraint between these operators. The result for the Hamiltonian H is H=-t tsumij{sum}_{\mathit{i}\mathit{j}}sumij​ei{\mathit{e}}_{\mathit{i}}ei​${\mathit{e}}_{\mathit{j}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(1/2+2Si{\mathbf{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Sj{\mathbf{S}}_{\mathit{j}}Sj​)+(J/2)t smij{\mathrm{sm}}_{\mathrm{ij}}smij​(1-${\mathit{e}}_{\mathit{i}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ei{\mathit{e}}_{\mathit{i}}ei​)(Si{\mathbf{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Sj{\mathbf{S}}_{\mathit{j}}Sj​-1/4)(1-${\mathit{e}}_{\mathit{j}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$e The number of electrons at site i is given by ni{\mathit{n}}_{\mathit{i}}ni​=1-${\mathit{e}}_{\mathit{i}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ei{\mathit{e}}_{\mathit{i}}ei​, and the true spin operator S${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{i}}$, is related to the local spin operator by S${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{i}}$=(1-${\mathit{e}}_{\mathit{i}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ei{\mathit{e}}_{\mathit{i}}ei​)Si{\mathbf{S}}_{\mathit{i}}Si​. The expression correctly produces the Nagaoka theorem in the limit J\ensuremath{\rightarrow}0, and preserves the time-reversal symmetry of the original model. For a bipartite lattice, H describes a competition between ferromagnetism, favored by the hopping term, and antiferromagnetism, favored by the Heisenberg term.
We have constructed an exact map of the U↦∞ one dimensional Hubbard Hamiltonian onto a system of free fermions related to the charge degrees of freedom (holons). The spin configuration only influences the energy in so far as the fermions move in a loop which is threaded by a fictitious magnetic flux proportional to the spin configuration's total momentum. A rather simple and physically appealing interpretation of some of the features of the Bethe Ansatz solution is possible in this representation. We illustrated its use with a calculation of the t/U corrections to the energy.
The equation for the charge vertex \ensuremath{\gamma} of the t-J model is derived and solved in leading order of a 1/N expansion, working directly in terms of Hubbard operators. Various quantities which depend crucially on \ensuremath{\gamma} are then calculated, such as the lifetime and the transport lifetime of electrons due to a charge coupling to other degrees of freedom and the charge-charge correlation function. Our results show that the static screening of charges and the dynamics of charge fluctuations depend only weakly on J and are mainly determined by the constraint of having no double occupancies of sites. \textcopyright{} 1996 The American Physical Society.
A generalized slave-particle scheme is introduced to study the t-J model with nonlocal effects included. A charge-spin decoupled state is found in this scheme, in which the nonlocal gauge fields play a role of attaching fictitious quantized fluxes to the particles as an effective way to describe antiferromagnetic correlation. As a result, the gauge fluctuation is suppressed in the long-wavelength limit, which shows the stability of flux binding, and the electromagnetic response function of such a state exhibits the Meissner effect. An important consequence of the present approach is that the electron, as a combination of the charge and spin degrees of freedom, does not show time-reversal and parity-symmetry-broken behavior, in contrast to the usual anyon theory. This is consistent with experimental results. The robustness of this conclusion relies on the exactness of the constraint between the charge and spin degrees of freedom as well as the flux binding condition. We also calculate the momentum distribution of electrons, which shows a characteristic scale equal to the Fermi momentum kf{\mathit{k}}_{\mathit{f}}kf​.
We show an exact representation of the t-J model by making use of a general Holstein-Primakoff transformation. The single occupancy constraint on the electrons is imposed strictly at every lattice site and the t-J Hamiltonian is transformed into an equivalent interacting boson and fermion system. The lowest-order approximation of a systematic perturbative expansion of the representation is shown to correspond to the mean-field theory of the Schwinger-boson approach. The second-order approximation gives the magnon-magnon, magnon-hole, and hole-hole interactions and the high-order antiferromagnetic spin polaron. In particular the second-order approximation takes further account of the renormalization of the magnon frequency and the antiferromagnetic spin polaron. The spectra of holes and magnons are calculated with the Green's function. The relevance of the obtained results to high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors is discussed. \textcopyright{} 1996 The American Physical Society.