PHYSICAL REVIEW B VOLUME 45, NUMBER 22 1 JUNE 1992-II Metallic and magnetic properties of the t-J model: One-site approximation Maciej Maska* Department of Theoretical Physics, Silesian University, Eato)vice, Poland (Received 8 April 1991;revised manuscript received 5 February 1992) The t-J model is analyzed by means of the Caron-Pratt approximation. With the use of a self- consistent cluster calculation, the magnetic phase diagram and the temperature and hole-concentration dependence of the antiferromagnetic and metallic order parameters are obtained. I. INTRODUCTION The theory of strongly correlated fermion systems on a two-dimensional lattice has been developed currently in connection with high-T, superconductivity. ' The mag- netic properties of the layered oxide superconductors with the Cu02 plane should contain important informa- tion about the mechanism of high-temperature supercon- ductivity. One of the simplified models which are con- sidered to include the essential physics of these materials is the t-J model. This model can be obtained as the large-U limit of the Hubbard model. The t-J Hamil- tonian is given by H= t g d; — d~ +J g (S;. S , 'n;nj— ) — —pgn;, (ij )o (ij) i where d; =(1 — n; )c;, S; = , 'c; o'~c, —tt, and n, = g c, c, . c; creates an electron of spin o at site i; n; is the particle-number operator. This effective Hamil- tonian is restricted to the Hilbert subspace with one or less electrons on each site, and the constraint of no dou- bly occupied site is ensured by the definition of the d; fermion operators. The parameters t and J describe the hopping matrix element and the Heisenberg coupling be- tween nearest-neighbor sites. The main results on the t J- model have been obtained either from mean-field- type " or numerical' ' calculations. In this work a self-consistent cluster approach, based on the Caron- Pratt approximation, ' is applied to investigate the magnetic and metallic properties of high-T, materials. The simple cluster calculations are performed to examine the phase diagram of the t-J model and the dependence of the antiferromagnetic and metallic order parameters on the doping concentration and temperature. II. CARON-PRATT APPROXIMATION In order to examine the antiferromagnetic properties of the Cu02 planes, we decompose the square lattice into two sublattices A and B in such a way that all neighbors of a site from sublattice A belong to sublattice B and vice versa. We propose to use a generalized cluster Caron- Pratt approximation' to obtain the metallic and mag- netic phase diagrams for the t-J model. In this approach we take into account a one-site cluster and self- consistently couple the cluster to its environment by con- sidering the surroundings as a particle reservoir which can exchange electrons with the cluster. The hopping be- tween the cluster and immediate neighborhood is re- placed by fermion source terms, which can create or an- nihilate electrons in the cluster. Namely, we replace the kinetic term in the Hamiltonian t g — d, d by zt(d; (d — )" ' '+H. c. ), where (i j) are the nearest neighbors and z is the coordination number. ( )" ' denotes the thermal average for the sublattice A (B). If site i belongs to sublattice A, all its nearest neighbors be- long to sublattice B and the thermal averages are calcu- lated for sublattice B and vice versa. The superex- change interaction and correct energy associated with the charge fiuctuations are treated in the mean-field approxi- mation with the same self-consistency condition as in the kinetic term [replacing g (S, . SJ , 'n;nj — ) —by z(S;(S')" ' ' — —, 'n, (n)" ' ')]. Then the resulting Ham- iltonian becomes a sum of one-site Hamiltonians, y HA (B) with HA (Bl t y [d~t (d )B (A)+((d )e)B (A)d ] +zJ( S( S)sB (A) ) n (n )B (A) pn ) Tr[exp( PH)0, ]— . 0; Tr exp( PH)— (4) p= 1 lk~ T, ktt is the Boltzmann constant, and the Hamil- tonian H is given by (2). The anticommutation properties of the operators d; and d. ~ imply that the averages (d; ) and (d. ~ ) cannot be simple c-numbers, but they must obey [d, , dt ] =I(d, ), dt, ] =[d;. , (d, '. )I=[(d;. &, (d, '. )]=0, where N is the number of sites. The sublattice index ( A or B) depends on which sublattice site i belongs to Un-. der the assumption that each atom in the environment behaves in the same way as the one we are looking at (within the framework of a given sublattice), the averages ( ) " ' ' can be calculated as expected values of the operators in the site i (for i belongs to sublattice A or B, respectively), (0) " ' '= (0; ) for 0; =d;~, d;~, Sf, n, , where 45 13 088 1992 The American Physical Society