INSTANTONS IN ABELIAN SYSTEMS
53
a mere superposition of kink and anti-kink will have the action:
Sik,k) ^ 5(fc) ^ 5(fc)
=
-f
(4.17)
where ii2 is the distance between k and k. This result implies that the
interaction between kinks is exponentially small. Since for small A the
average time distance between our objects is of the order of c~^
g2y 2/i3/3A ^ ^ - 1
above-mentioned interaction can be neglected.
Another simplification, possible for the same reason, is that the width of
the kink can be neglected. We can approximate the general configuration by:
( p i T i O = - 7 T n s g n ( t - T j)
yl^j=l
= N
3A
(4.18)
(4.19)
Ti < T2 < ... < Tn
The total effect on the correlation function is given by:
= T I (-C)^e-*l>’ idT,...dT^ = ^e-^l--'^l (4.20)
A N=0
J
^
min(ii, 12) < < Tj < '" < Tff < max(ti, 12) , A = C e
We have found that tunnelling trajectories (which are also called
instantons) remove the degeneracy of the ground state present on a
perturbative level. The symmetry
—cp gets restored and the system
acquires a finite, though large, correlation length
As we
discussed m Chapter 1, in the limit of large correlation length the
quantum theory with action (4.1) must be equivalent to the ^ = 1 Ising
model, t This equivalence is quite obvious from the present consideration: the moment we replaced the (p-field by the step function (4.16) we
actually started counting configurations of the Ising model described by
the picture:
t__ i__ L T~T~1 v~r
_ L J
and the counting of kinks in (4.20) is exactly the counting of spin
reversals in (1.18)
t This fact was realized long ago by Vaks and Larkin.
53
a mere superposition of kink and anti-kink will have the action:
Sik,k) ^ 5(fc) ^ 5(fc)
=
-f
(4.17)
where ii2 is the distance between k and k. This result implies that the
interaction between kinks is exponentially small. Since for small A the
average time distance between our objects is of the order of c~^
g2y 2/i3/3A ^ ^ - 1
above-mentioned interaction can be neglected.
Another simplification, possible for the same reason, is that the width of
the kink can be neglected. We can approximate the general configuration by:
( p i T i O = - 7 T n s g n ( t - T j)
yl^j=l
= N
3A
(4.18)
(4.19)
Ti < T2 < ... < Tn
The total effect on the correlation function is given by:
= T I (-C)^e-*l>’ idT,...dT^ = ^e-^l--'^l (4.20)
A N=0
J
^
min(ii, 12) < < Tj < '" < Tff < max(ti, 12) , A = C e
We have found that tunnelling trajectories (which are also called
instantons) remove the degeneracy of the ground state present on a
perturbative level. The symmetry
—cp gets restored and the system
acquires a finite, though large, correlation length
As we
discussed m Chapter 1, in the limit of large correlation length the
quantum theory with action (4.1) must be equivalent to the ^ = 1 Ising
model, t This equivalence is quite obvious from the present consideration: the moment we replaced the (p-field by the step function (4.16) we
actually started counting configurations of the Ising model described by
the picture:
t__ i__ L T~T~1 v~r
_ L J
and the counting of kinks in (4.20) is exactly the counting of spin
reversals in (1.18)
t This fact was realized long ago by Vaks and Larkin.