20 Constructive Symmetry Breaking
137
The criterion of symmetry breaking associated with the Bogoliubov strategy is
then the following: if in the thermodynamical limit the so obtained correlation functions do not depend on the direction along which h → 0, one has only one phase
and no symmetry breaking. On the other hand, if different directions of h give rise
to different limits, one obtains non-invariant correlation functions and spontaneous
symmetry breaking. Indeed, the Bogoliubov procedure closely corresponds to the
way by which one operationally produces a non-trivial magnetization in a given
direction.
The non-uniqueness of the thermodynamical limit, in the strategies discussed
above, is an indication of a sort of dynamical instability, since an infinitesimally small
interaction (a surface or boundary term or a vanishingly small volume interaction with
an external field) is capable of drastically changing the state in the thermodynamical
limit and the physical behaviour of the system.
137
The criterion of symmetry breaking associated with the Bogoliubov strategy is
then the following: if in the thermodynamical limit the so obtained correlation functions do not depend on the direction along which h → 0, one has only one phase
and no symmetry breaking. On the other hand, if different directions of h give rise
to different limits, one obtains non-invariant correlation functions and spontaneous
symmetry breaking. Indeed, the Bogoliubov procedure closely corresponds to the
way by which one operationally produces a non-trivial magnetization in a given
direction.
The non-uniqueness of the thermodynamical limit, in the strategies discussed
above, is an indication of a sort of dynamical instability, since an infinitesimally small
interaction (a surface or boundary term or a vanishingly small volume interaction with
an external field) is capable of drastically changing the state in the thermodynamical
limit and the physical behaviour of the system.