The theory of fermion phase transitions is reviewed from a unified field theoretic standpoint, based on the diagrammatic perturbation expansion of a generalized matrix propagator. Transitions from a normal to a condensed phase are characterized by the spontaneous appearance of long-range order and (in the presence of a suitable infinitesimal external field) broken symmetry. This is illustrated by the ferromagnetic, solid, superconducting and spindensity wave ground states. The phenomenon is explained qualitatively as caused by the creation of a long-range internal field, F, due to the interactions between particles. This field establishes long-range order in the system, and is in turn itself established by the long-range order, in a self-consistent fashion. The mechanism here is expressed quantitatively in terms of a self-consistent Dyson equation relating a generalized matrix propagator, G, to a proper self-energy matrix, Σ. The off-diagonal elements of G describe ‘anomalous’ propagation processes which are characteristic for the condensed phase, and they yield directly the long-range order parameters. The Σ-matrix is just the potential of the internal field. The method is illustrated by applying it to the ferromagnetic phase of a system with δ-function interaction between particles. Finally, the technique is used to derive the vertex part equation for the transition point.
We argue that ground states of 1D spin chains can spontaneously break U(1) ``easy-plane'' spin rotation symmetry, via true long-range order of
(Sx,Sy), at the phase transition between two quasi-long-range-ordered phases. The critical point can be reached by tuning a single parameter in a Hamiltonian with the same symmetry as the XXZ model, without further fine-tuning. Equivalently, it can arise in systems of bosons with particle-hole symmetry, as a long-range-ordered transition point between two quasi-long-range-ordered superfluids. Our approach is to start with the continuum field theory of the isotropic Heisenberg ferromagnet and consider generic perturbations that respect easy-plane symmetry. We argue for a renormalization-group flow to a critical point where long-range order in
(Sx,Sy) is enabled by coexisting critical fluctuations of
Sz. (We also discuss multicritical points where further parameters are tuned to zero.) These results show that it is much easier to break continuous symmetries in 1D than standard lore would suggest. The failure of standard intuition for 1D chains (based on the quantum--classical correspondence) can be attributed to Berry phases, which prevent the 1+1D system from mapping to a classical 2D spin model. The present theory also gives an example of an ordered state whose Goldstone mode is interacting even in the infra-red, rather than becoming a free field.