Spontaneous symmetry breaking by local or partial fields

The basic problem how spontaneous symmetry breaks is studied rigorously for local fields in a finite domain. Another basic problem is to investigate whether or not there exist lower dimensions (of the domain to which a field H is applied) below which no spontaneous symmetry breaking (SSB) occurs. This is studied using scaling, renormalization group and mean-field theory for partial fields in an infinite domain. Our main results are the following: (i) The total magnetization is proportional to the square of the spontaneous magnetization when a field is applied only to a finite region (an extension of Yang's result [Phys. Rev. 85 (1952) 808], M = Nms2 for a fixed single spin). A general formulation is given to calculate the proportionality coefficient which is shown to vanish as the field goes to zero. This indicates no SSB in this situation. (ii) On the other hand, if we apply partial fields to a d'-dimensional region and make them vanish, we find that SSB occurs, at least, for d' > d'c = β/ν. In the region 0 < d' ≤ d'c, the situation is subtle, as is exemplified by an exactly soluble model with such a field on one-side boundary as depends on the system-size. (author)