Temperature behavior of the magnon modes of the square lattice antiferromagnet

A spin-wave theory of short-range order in the square lattice Heisenberg antiferromagnet is formulated. With growing temperature from T = 0 a gapless mode is shown to arise simultaneously with opening a gap in the conventional spin-wave mode. The spectral intensity is redistributed from the latter mode to the former. For low temperatures the theory reproduces results of the modified spin-wave theory by M. Takahashi, J. E. Hirsch et al. and without fitting parameters gives values of observables in good agreement with Monte Carlo results in the temperature range 0 ≤ T ∼ 0.8J where J is the exchange constant. Typeset using REVTEX 1 Properties of the spin2 quantum Heisenberg antiferromagnet on a square lattice attract much attention in connection with the investigation of cuprate-perovskite high-temperature superconductors. In accord with the existing theories1–5 the spectrum of this antiferromagnet contains the doubly degenerate (in the magnetic Brillouin zone) magnon mode which is gapless at zero temperature. For T > 0 in this mode a gap opens near the center of the zone. The appearance of this gap is connected with the short-range antiferromagnetic ordering which is established in two-dimensional antiferromagnets at nonzero temperature. Of special note are the spin-wave theory of Refs. 3,4 where without fitting parameters values of many observables were calculated in good agreement with the exact diagonalization and Monte Carlo calculations in the temperature range 0 ≤ T < ∼ 0.6J . Results of these works can be obtained with the mean-field decoupling of terms of the Hamiltonian which are quartic in the magnon operators or, equivalently, by decoupling of many-particle Green’s functions in the equations of motion for the one-particle Green’s functions. In the theory of Refs. 3,4 the gap in the finite-temperature magnon spectrum appears with imposing the constraint of zero staggered magnetization to retain the sublattice symmetry in short-range order and to ensure zero site magnetization in the absence of magnetic field. Notice that this spin-wave approximation is not rotationally invariant. In the present paper we try to obtain a better approximation for the magnon Green’s functions by transferring the decoupling to higher order equations of motion. This allows us to expand the temperature range where the theory conforms with Monte Carlo data up to T ≈ 0.8J . Besides, in our theory, as soon as the temperature exceeds zero an additional gapless mode arises simultaneously with opening a gap in the conventional magnon mode. For large crystals and low temperatures the spectral intensity of the gapless mode is weak and our theory reproduces results of Refs. 3,4. With growing temperature the spectral intensity is redistributed from the conventional to the gapless mode. We consider the antiferromagnetic Heisenberg model on a plane square lattice with the Hamiltonian H = J ∑ la SlSl+a. (1) Here Sl is the spin1 2 operator, l runs over sites of one of two sublattices, and a are vectors of the four nearest neighbors of site zero. In the following discussion the exchange constant J is taken as the unit of energy. The Dyson-Maleev transformation is used to represent the spin operators by boson operators al and bm on the two sublattices A and B S l = a † l , S + l = ( 1− a†lal ) al, S z l = 1 2 − a†lal, l ∈ A, (2) S m = −bm, S m = −bm (