The Mass Gap of the Nonlinear σ Model through the Finite Temperature Effective Action

The O(3) nonlinear σ model is studied in the disordered phase, using the techniques of the effective action and finite temperature field theory. The nonlinear constraint is implemented through a Lagrange multiplier. The finite temperature effective potential for this multiplier is calculated at one loop. The existence of a nontrivial minimum for this potential is the signal of a disordered phase in which the lowest excited state is a massive triplet. The mass gap is easily calculated as a function of temperature in dimensions 1, 2 and 3. In dimension 1, this gap is known as the Haldane gap, and its temperature dependence is compared with experimental results. 75.10.Jm, 11.10.Lm Typeset using REVTEX 1 The O(3) nonlinear σ model (NLσ) offers a good description of quantum antiferromagnets at long wavelengths, as suggested by renormalization group analyses. The mapping between the nonlinear σ model and the Heisenberg model was first established by Haldane, who also argued that a topological Hopf term should be present in the case of half-integer spins in one dimension (this distinction between half-integer and integer spins disappears in two dimensions). For integer spins, in which case the Hopf term is absent, Haldane argued that the lowest excitations should exhibit a mass gap. Accordingly, these excitations (the magnons) are no longer Goldstone modes and the rotation symmetry of the spins is no longer spontaneously broken: the symmetry has been dynamically restored by quantum fluctuations. This phenomenon has found an experimental realization in quasi one-dimensional antiferromagnets in their disordered phase. In this paper we calculate the temperature dependence of this mass gap in dimensions 1 to 3 within the framework of the finite temperature effective action of the nonlinear σ model. The use of this technique is facilitated by the Lorentz invariance of the NLσ model, which ensures that the magnon speed is not affected by quantum corrections. The O(3) nonlinear σ model describes the dynamics of a unit vector ~ φ representing the local direction of the staggered magnetization. The Euclidian action obtained from the Heisenberg model is