Exact expression of thet-Jmodel in terms of local spin and fermionic holon operators

Abstract

An exact expression for the Hamiltonian H of the t-J model in terms of local spin (${\mathbf{S}}_{\mathit{i}}$) and fermionic holon (${\mathit{e}}_{\mathit{i}}$) operators is derived which requires no constraint between these operators. The result for the Hamiltonian H is H=-t t${sum}_{\mathit{i}\mathit{j}}$${\mathit{e}}_{\mathit{i}}$${\mathit{e}}_{\mathit{j}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(1/2+2${\mathbf{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathbf{S}}_{\mathit{j}}$)+(J/2)t ${\mathrm{sm}}_{\mathrm{ij}}$(1-${\mathit{e}}_{\mathit{i}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{e}}_{\mathit{i}}$)(${\mathbf{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathbf{S}}_{\mathit{j}}$-1/4)(1-${\mathit{e}}_{\mathit{j}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$e The number of electrons at site i is given by ${\mathit{n}}_{\mathit{i}}$=1-${\mathit{e}}_{\mathit{i}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{e}}_{\mathit{i}}$, and the true spin operator S${\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{}}}$${\mathit{e}}_{\mathit{i}}$)${\mathbf{S}}_{\mathit{i}}$. 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.