Renormaliztion-Group Approach to Spin-Wave Theory of Quantum Heisenberg Ferromagnet

The renormalization-group method is used to analyze the low-temperature behaviour of a two-dimentional, spin-s quantum Heisenberg ferromagnet. A set of recursion equations is derived in an one-loop approximation. The lowtemperature asymptotics of the correlation length and the uniform susceptibility are obtained. For small spins (s = 1/2, 1) the results are essentialy different from those in the spin-wave theory. 75.10.Jm,75.30.Ds.75.40.Cx Typeset using REVTEX 1 The revival of the interest in the theory of quantum magnetism has led to creation of new approaches of investigation. An effective continuum field theory, which is quantummechanical generalization of the classical nonlinear σ-model was derived from lattice, large spin Heisenberg model of antiferromagnets [1]. The two dimensional antiferromagnet on a square lattice was treated also by means of the Schwinger boson representation of the spin algebra. This representation allows an appropriate mean-field theory of the low-temperature behaviour of the sytem [2]. At the same time, a modified spin-wave theory of Heisenberg (anti)ferromagnets was formulated [3,4]. The usual spin-wave theory was suplemented with the constraint that the magnetization at each site is zero. This ensures that the sublattice rotational symmetry is not broken. The results obtained in the papers are in quantitative agreement, but, as was recognized by the authors, it is difficult to continue them to smaller values of the spin. The aim of the present investigation is to obtain a better understanding of a small spin quantum ferromagnet. The renormalization-group method is used to analyze the lowtemperature behaviour of a two dimensional, spin-s quantum Heisenberg ferromagnet. A set of recursion equations is derived in an one-loop approximation. The spin-recursion relation shows that the effective spin increases (the spin rescaling factor is two) and hence in the limit of infinitely many recursions, the spin-wave approximation can be used. The low-temperature asymptotics of the correlation length and the uniform susceptibility are obtained solving the recursion relations. For small spins (s = 1/2, 1) the results are essentialy different from those in the spin-wave theory. The spin-s quantum Heisenberg ferromagnet is defined by the hamiltonian ĥ = −J ∑ ~̂ Si · ~̂ Sj (1) where ~̂ Si are spin operators on site i(j) of a two dimentional square lattice with number of sites N and lattice spacing a. By < i, j > I denote the sum over the nearest neighbours. To preserve the sublattice rotational symmetry in spin-wave theory of the disordered phase, Takahashy [3] imposed a condition of zero sublattice magnetization 〈Ŝ3 i 〉 = 0. Using 2 Holstein-Primakoff representation for the spin operators, one can rewrite it as a condition that the total number, on average, of spin-waves per site is s. To enfors the constraint, one introduces a new term in the hamiltonian ĥ → ĥ − μ∑ i Ŝ i , with the chemical potential μ to be determined from the condition. The system is disordered at any temperature T 6= 0 and the chemical potential is positive μ(T ) > 0. When temperature goes to zero μ(T ) reaches 0 and the spin-wave bosons Bose-condense at zero wave-vector. Due to Bose condensation at T = 0, the long-range ferromagnetic correlation is present. The low-temperature features are determined by the behaviour of the system near the origin of the Brillouin zone. Following the renormalization group method one devides the Brillouin zone into two parts. Then the half of the zone which contains small wave-vectors is of interest to us, and the theory should be reformulated using the reduced Brilloun zone. In the site representation, this is equivalent to introduce two sublattices, A and B. I identify the sublattices and refer to the new lattice, with number of sites N1 = 1 2 N and lattice spacing a1 = √ 2a , as L1. The hamiltonian (1) can be rewritten in the form ĥ = −J ∑