RENORMALIZATION-GROUP TREATMENT OF THE QUANTUM NONLINEAR SIGMA-MODEL

The quantum nonlinear sigma-model in (d + 1)-dimensional space-time is investigated by a renormalization group approach. The beta-functions for the coupling g and the temperature t are given. The renormalisation group equations of the N-point functions are derived for finite coupling and finite temperature. It is known that the model shows a phase transition at zero temperature at some critical coupling g(c). The behaviour near this critical point is investigated. The crossover exponent-phi, describing the crossover between different regimes near the critical point is calculated, verifying a conjecture by Chakravarty, Halperin and Nelson, who have argued that phi in d dimensions should have the same value as the critical exponent v of the correlation length in a (d + 1)-dimensional classical system. A subtraction scheme appropriate to calculate the renormalisation factors and from these the beta-functions at finite temperature and finite coupling constant will be introduced. Using this method the beta-functions will be calculated to order two loops. The exponents obtained this way are in good agreement with the values found on other ways.