PHYSICAL REVIE%' 8 VOLUME 49, NUMBER 9 1 MARCH 1994-I Spin-rotationally-symmetric path-integral formulation of a generalized A/-orbital Hubbard model Prabal K. Maiti and Avinash Singh Department of Physics, Indian Institute of Technology, Kanpur 208016, India (Received 28 June 1993; revised manuscript received 22 October 1993) A spin-rotationally-symmetric path-integral formulation with a vector order parameter is pre- sented for a generalized Af-orbital Hubbard model which yields the correct Hartree-Fock result in the saddle-point approximation, and also the spin-wave mode representing transverse Gaussian Quc- tuations. A systematic expansion of the action in powers of inverse lV, wherein spin-rotational symmetry and hence the Goldstone mode are preserved order by order, is developed, and the first order O(1/A/) quantum corrections to the ground-state energy and sublattice magnetization ob- tained. Setting A' = 1 yields a quantitatively correct description of the antiferromagnetic state of the one-orbital Hubbard model at half-filling. I. INTRODUCTION Recently there has been interest in developing a spin- rotationally-symmetric path-integral formulation of the Hubbard model. Up until recently the situation has been that the process of taking the saddle-point ap- proximation has led to inconsistencies in the resulting theory. Thus if the Hubbard interaction term is decom- posed in terms of charge-density and spin-density terms, n, tn, t — — 4[(n, ~+ n, t) — (n, t — n, t)z], the resulting the- ory yields correct HF energies, however, the presence of a scalar magnetic order parameter (n, g — n, t) in the the- ory leads to the absence of Goldstone modes. On the other hand, the — 3 S, . S; representation introduces a vec- tor order parameter (S,), and hence Goldstone modes in the theory, however, because of the 3 factor, the single- particle spectrum obtained at the Hartree-Fock level is incorrect. In the present paper we show that, using a generalized A'-orbital Hubbard model, 4 s it is possible to develop a path-integral formulation wherein the e8'ective action is both spin-rotationally invariant and exhibits a correct Hartree-Fock result at the saddle point. For the Hub- bard model there is no formal expansion parameter in the problem to control the perturbative expansion, whereas in a perturbative treatment it is important to ensure that all conservation laws and symmetries are obeyed system- atically. In particular the Goldstone mode which fol- lows from spin-rotational symmetry must be preserved in perturbation theory. In the path-integral formulation of the generalized JV-orbital Hubbard model ~ acts as an expansion parameter and spin-rotational symmetry is guaranteed order by order in the expansion because the perturbative piece of the Hamiltonian is manifestly rotationally symmetric. In the saddle-point approxima- tion, which yields the classical action of O(1), we re- cover the correct HF result, and this is also exact in the limit JV ~ oo. The quantum fluctuation effects appear at higher orders and we consider Gaussian Huctuations around the saddle point yielding an O(l/A') correction to the action, from which we discuss evaluation of the quan- tum corrections to the ground-state energy and sublattice magnetization. We consider the following Hamiltonian for a general- ized JV orbital Hubbard model with A' orbitals on each site: ' '8 = t ) (at — as + H. c. ) (ij)oo. t +g (ag ag agpa gp+a~ a g'pa~pa g)'' imp Here n, P are orbital indices. The factor ~ is included in the interaction term to render energy density finite in the A' -+ oo limit. The two interaction terms are density- density and exchange type with respect to orbital indices, and together can be expressed as r t t t . t E ~rat a. t a~pa. ~p+at a. ~pa~pa. ~i) Si.si+n2 exp where 4', = (a, . & at& ), S, = P 4t — 4, is the total spin-density operator, and n, = g 4t — 4, the spin- averaged charge-density operator at site i. The interac- tion term, expressed as above with the expansion param- eter ~ in front, is manifestly spin-rotationally symmetric and hence the Goldstone mode is preserved order by or- der in a systematic ~ expansion. II. PATH-INTEGRAL FORMULATION To set up the pat¹integral formulation of the Hubbard model we go to the Grassmann-variable representation to 0163-1829/94/49(9}/6078{5}/$06. 00 49 1994 The American Physical Society