VOLUME 60, NUMBER 26 PHYSICAL REVIEW LETTERS Spectral Function of Holes in a Quantum Antiferromagnet 27 JUNE 1988 S. Schmitt-Rink and C. M. Varma AT& T Bell Laboratories, Murray Hill, We~ Jersey 07974 and A. E. Ruckenstein Department of Pltysies, University of California at San Diego, La Jolla, California 92093 (Received 21 March 1988) By use of an effective Hamiltonian which takes into account the constraints on the motion of a hole in a quantum antiferromagnet, the spectral function of the hole is calculated. For small exchange and away from the antiferromagnetic zone boundary, it is found to be dominated by incoherent multiple- spin-wave processes. The dispersion of the quasiparticle part and the possible implications for disorder- ing of the quantum antiferromagnet are discussed. PACS numbers: 75. 10. Jm, 71. 28.+d It is becoming increasingly clear that the new oxide superconductors are insulating antiferromagnets for compositions corresponding to one electron per Cu site. ' The magnetic correlations are strongly two-dimensional (2D) and the three-dimensional ordering at quite high temperatures T is a parasitic effect. The observed tem- perature dependence of the correlation length is now well understood in terms of the fluctuations of a spin- —, ' 2D Heisenberg model with a Neel-type order as T 0. 2 These discoveries have revived the classic problem of the transition from the Mott insulating antiferromagnet- ic state to the conducting nonmagnetic state as the densi- ty of electrons is varied away from one electron per (Cu) site. An essential first step in the solution of this prob- lem is to understand the motion of a hole in the quantum antiferromagnetic (QAFM) Heisenberg model. 3' In the case of highly anisotropic, Ising spin interac- tions, the motion of the hole always leaves behind a "string" of overturned spins which can be healed only by retracing of the original path, thus leading to self- trapped states centered at the original hole position. These effects were discussed in detail by Nagaev and co- workers, ' and rediscovered by Hirsch, 6 Shraiman and Siggia, and Trugman, in their work on hole pairing in AFM insulators. As recognized by most of these au- thors, the physics is qualitatively different when the quantum fluctuations associated with the transverse ex- change interactions are included from the start. In that case, in a QAFM with a hole, the ground-state wave function is a linear combination of the Neel state and other basis states with different multiple spin deviations. As the hole hops to a neighboring site, it also creates spin deviations, so that the new state is a different linear com- bination of the Neel state and spin-deviated states — with finite overlap with the earlier state, thus allowing for a Bloch-wave-type solution. Using wave functions which bear close resemblance to that of liquid He, we show that the corresponding hole spectral function can easily be calculated. The Hamiltonian H„which describes the hopping of holes from site to site, takes on a relatively simple form if we consider the Neel state IN) as the vacuum state. We define hole operators h; (that obey Fermi statistics) so that h; =c;tt on the t sublattice and c;tj on the ) sub- lattice. Furthermore, we define hard-core boson opera- tors b;, such that b; I N& 0, b;t S; on the t sublattice and S;+ on the ) sublattice. With these, it may be seen that the hopping part of the Hamiltonian H, =t h tht. [btt(l bit. bt)+ (1 b~tb~ )bt, ]. i, j& po-exp X k; b;tbj IN). (2) In linear spin-wave theory (i. e. , treating the b's as ideal bosons) po gives results identical to those obtained from the leading term of the Holstein-Primakoff transforma- tion. In the same framework, the Hamiltonian (1) can be rewritten as satisfies all the constraints on the motion of a hole. In particular, it commutes with the number of doubly occu- pied sites, g; h;th;b;tb;, i.e. , it conserves the constraint of no double occupancy, h;b; =0. It also properly describes the alteration of spin configurations as the hole moves. The Heisenberg Hamiltonian HJ may similarly be writ- ten in terms of b*s and projection operators 1 — h;th;. A good ground-state wave function for QAFM's, which includes the zero-point spin deviations, is the Bo- golyubov wave function Ht =, l2 g hghg — q[ttq(ttqyp — q+ vqytr) +a — q(vqyQ — q+ ttqyg)], Ez N' )cq (3) 2793