Chapter 19
Examples
To illustrate the above general ideas, we discuss simple concrete models exhibiting
spontaneous symmetry breaking.
19.1 Heisenberg Ferromagnet
The Heisenberg model for spin 1/2 Ferromagnets is described by the following finite
lattice Hamiltonian
H V = −
i, j∈V
J i j σ i · σ j − h ·
j∈V
σ j ,
(19.1)
where V denotes the finite three-dimensional lattice, h is an external uniform magnetic field, i, j label the lattice points and J i j is the positive coupling constant or
“potential”, invariant under lattice translations and of short range, e.g. a nearest
neighbour coupling (see Sect. 17.1).
As discussed in Sect. 17.1, the algebraic dynamics α t is defined as the norm limit
of the finite volume dynamics α
V
t generated by H V . The spin rotations define a threeparameter group of
∗ -automorphisms or algebraic symmetries of the quasi-local spin
algebra A, which commute with the time translations α t in the limit h = |h| → 0.
For finite V , the ground state Ψ 0
V,h (defined on A V and by Hahn–Banach extension
on A) is characterized by all the spins pointing in the direction of n ≡ h/|h|, i.e.
σ j · n Ψ 0
V,h
= Ψ 0
V,h
.
The correlation functions of Ψ 0
V,h converge as V → ∞ and define a state
h
0 on
A, which is invariant under space translations and under α t . In fact, thanks to the
uniform convergence of α
V
t , one has
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_19
129
Examples
To illustrate the above general ideas, we discuss simple concrete models exhibiting
spontaneous symmetry breaking.
19.1 Heisenberg Ferromagnet
The Heisenberg model for spin 1/2 Ferromagnets is described by the following finite
lattice Hamiltonian
H V = −
i, j∈V
J i j σ i · σ j − h ·
j∈V
σ j ,
(19.1)
where V denotes the finite three-dimensional lattice, h is an external uniform magnetic field, i, j label the lattice points and J i j is the positive coupling constant or
“potential”, invariant under lattice translations and of short range, e.g. a nearest
neighbour coupling (see Sect. 17.1).
As discussed in Sect. 17.1, the algebraic dynamics α t is defined as the norm limit
of the finite volume dynamics α
V
t generated by H V . The spin rotations define a threeparameter group of
∗ -automorphisms or algebraic symmetries of the quasi-local spin
algebra A, which commute with the time translations α t in the limit h = |h| → 0.
For finite V , the ground state Ψ 0
V,h (defined on A V and by Hahn–Banach extension
on A) is characterized by all the spins pointing in the direction of n ≡ h/|h|, i.e.
σ j · n Ψ 0
V,h
= Ψ 0
V,h
.
The correlation functions of Ψ 0
V,h converge as V → ∞ and define a state
h
0 on
A, which is invariant under space translations and under α t . In fact, thanks to the
uniform convergence of α
V
t , one has
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_19
129