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International Journal of Modern Physics B

PERTURBATIVE EXPANSION FOR THE t-J MODEL CONSTRUCTED FROM THE GENERATORS OF THE SUPERSYMMETRIC HUBBARD ALGEBRA

C. Repetto, Oscar P. Zandron

Published Jun 2, 2009• 0 citations
Publisher

Abstract

By using the Hubbard [Formula: see text]-operators as field variables along with the supersymmetric version of the Faddeev–Jackiw symplectic formalism, a family of first-order constrained Lagrangians for the t-J model is found. In order to satisfy the Hubbard [Formula: see text]-operator commutation rules satisfying the graded algebra spl(2,1), the number and kind of constraints that must be included in a classical first-order Lagrangian formalism for this model are presented. The model is also analyzed via path-integral formalism, where the correlation-generating functional and the effective Lagrangian are constructed. In this context, the introduction of a proper ghost field is needed to render the model renormalizable. The perturbative Lagrangian formalism is developed and it is shown how propagators and vertices can be renormalized to each order. In particular, the renormalized ferromagnetic magnon propagator arising in the present formalism is discussed. As an example, the thermal softening of the magnon frequency is computed.

DOI
10.1142/s0217979209052856

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Physical review. B, Condensed matter•Jan 15, 1996•9 citations
By using the Hubbard [Formula: see text]-operators as field variables along with the supersymmetric version of the Faddeev–Jackiw symplectic formalism, a family of first-order constrained Lagrangians for the t-J model is found. In order to satisfy the Hubbard [Formula: see text]-operator commutation rules satisfying the graded algebra spl(2,1), the number and kind of constraints that must be included in a classical first-order Lagrangian formalism for this model are presented. The model is also analyzed via path-integral formalism, where the correlation-generating functional and the effective Lagrangian are constructed. In this context, the introduction of a proper ghost field is needed to render the model renormalizable. The perturbative Lagrangian formalism is developed and it is shown how propagators and vertices can be renormalized to each order. In particular, the renormalized ferromagnetic magnon propagator arising in the present formalism is discussed. As an example, the thermal softening of the magnon frequency is computed.
By using the supersymmetric version of the Faddeev-Jackiw symplectic formalism, a family of first-order constrained Lagrangians for the t-J model is found. In this approach the Hubbard ˆ X-operators are used as field variables. In this framework, we first study the spinless fermion model which satisfies the graded algebra spl(1,1). Later on, in order to satisfy the Hubbard ˆ X-operators commutation rules satisfiying the graded algebra spl(2,1), the number and kind of constraints that must be included in a classical first-order Lagrangian formalism for the t-J model are found. This model is also analyzed in the context of the path- integral formalism, and so the correlation generating functional and the effective Lagrangian are constructed.
In this paper it is shown that the family of first-order Lagrangians for the t - J model and the corresponding correlation-generating functional previously found can be exactly mapped into the slave-fermion decoupled representation. Next, by means of the Faddeev-Jackiw symplectic method, a different family of Lagrangians is constructed and it is shown how the corresponding correlation-generating functional can be mapped into the slave-boson representation. Finally, in order to define the propagation of fermion modes we discuss two alternative ways to treat the fermionic sector in the path-integral formalism for the t - J model.
In the present work we have developed a large-NNN expansion for the $t\ensuremath{-}J$ model based on the path integral formulation for Hubbard operators. Our large-NNN expansion formulation contains diagrammatic rules, in which the propagators and vertex are written in term of Hubbard operators. Using our large-NNN formulation we have calculated, for J=0,J=0,J=0, the renormalized O(1/N)O(1/N)O(1/N) boson propagator. We also have calculated the spin-spin and charge-charge correlation functions to leading order 1/N.1/N.1/N. We have compared our diagram technique and results with existing ones in the literature.
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.
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.