Appendix C: Anderson Model of Superconductivity
247
with no appearance of variables at infinity; α
t
π = α
t and the symmetry β
λ is not
broken. The time evolution α
t
π (σ
i
) is a periodic motion corresponding to rotations
around n 3 with frequency 2ε. The case ω 0 (σ) = (0, 0, 1) (selected by the positive energy spectrum) describes a ground state with unoccupied Cooper pairs and
corresponds to a normal metal.
In the second case Eq. (C.2) reduces to
i
d
dt
σ
i
= −2T c σ
∞
∧ σ
i
,
(C.3)
of exactly the same form of Eq. (A.10), with h = 0, for the time evolution of the
Curie–Weiss model.
In each factorial representation π n with π(σ ∞ ) = n = (n 1 , n 2 , ε/T c ), β
λ is not a
symmetry of the effective dynamics α
t
π n
and it is broken in π n . The time evolution
α
t
π n
(σ) is a periodic motion corresponding to rotations around n with frequency
w = 2T c .
The covariance group generated by β
λ and by α
t
π n
, i.e. by rotations around the
third axis and by rotations around n, is not a symmetry of the finite volume dynamics
(and therefore not of α
t ).
An interesting phenomenon is however displayed: the symmetry breaking in the
presence of long range interactions generates exact symmetries of the effective
dynamics α
t
π n
, (here rotations around n = π n (σ ∞ )), which did not exist as symmetries of the finite volume Hamiltonian (dynamical generation of exact symmetries
by boundary effects).
It is instructive to check that the Goldstone spectrum corresponding to the breaking
of β
λ describes quasi-particle excitations with an energy gap ω = 2T c as k → 0, i.e.
the standard Goldstone theorem is evaded.
247
with no appearance of variables at infinity; α
t
π = α
t and the symmetry β
λ is not
broken. The time evolution α
t
π (σ
i
) is a periodic motion corresponding to rotations
around n 3 with frequency 2ε. The case ω 0 (σ) = (0, 0, 1) (selected by the positive energy spectrum) describes a ground state with unoccupied Cooper pairs and
corresponds to a normal metal.
In the second case Eq. (C.2) reduces to
i
d
dt
σ
i
= −2T c σ
∞
∧ σ
i
,
(C.3)
of exactly the same form of Eq. (A.10), with h = 0, for the time evolution of the
Curie–Weiss model.
In each factorial representation π n with π(σ ∞ ) = n = (n 1 , n 2 , ε/T c ), β
λ is not a
symmetry of the effective dynamics α
t
π n
and it is broken in π n . The time evolution
α
t
π n
(σ) is a periodic motion corresponding to rotations around n with frequency
w = 2T c .
The covariance group generated by β
λ and by α
t
π n
, i.e. by rotations around the
third axis and by rotations around n, is not a symmetry of the finite volume dynamics
(and therefore not of α
t ).
An interesting phenomenon is however displayed: the symmetry breaking in the
presence of long range interactions generates exact symmetries of the effective
dynamics α
t
π n
, (here rotations around n = π n (σ ∞ )), which did not exist as symmetries of the finite volume Hamiltonian (dynamical generation of exact symmetries
by boundary effects).
It is instructive to check that the Goldstone spectrum corresponding to the breaking
of β
λ describes quasi-particle excitations with an energy gap ω = 2T c as k → 0, i.e.
the standard Goldstone theorem is evaded.