Spontaneous symmetry breaking in the O(4) scalar model on a lattice

The spontaneous symmetry breaking at zero temperature in the four-component four-dimensional scalar $\lambda \phi^4_4$ model (the O(4) model) is investigated on a lattice for different values of the coupling constant $\lambda$. A general method for dealing with such type dependence is developed. The Goldstone modes are integrated out in the spherical coordinates in the internal space of the scalar field by the saddle point method, and the initial functional integral of the model is reduced to an effective one-component theory convenient for lattice investigations. The partition function of this theory is calculated analytically in the static limit and demonstrates the $\lambda$-dependence which is characteristic for distinguishing symmetry breaking. Monte Carlo simulations are performed with the QCDGPU software package on a HGPU cluster. It is shown that the symmetry is spontaneously broken for $\lambda>\lambda_0 \simeq 10^{-5}$. For smaller coupling values, the scalar field vanishes on a lattice, which can be interpreted as instability of the homogeneous condensate or even instability of the model itself. The critical value $\lambda_0$ is independent of the lattice sizes $16^4$ and $32^4$ investigated.