4.2 Instantons in the Global 0(2) Model
This model is described by the partition function:
54
GAUGE FIELDS AND STRINGS
Z =
n ^ exp()S X (cos( X
x,6
(4.21)
We shall consider its properties in the large-j? limit (weak coupling): the
natural thing to do would be to expand the cosine in (4.21) and to write:
^ = ^ Z < < P x - < P x+ «)^ * ^ d^x {V(pf
(4.22)
However, this expansion is not entirely correct. It restricts cp^ to be very
close to its neighbour (p^+a- This is physically reasonable since at large
jS neighbouring spins must be almost parallel. However, almost parallel
spins do not always mean cp ^, ^ cp^^^ since we can have as well the
situation:
(p^^n-e
< P x+ 6 ^ -n -he
e < 1
(4.23)
The configuration (4.23), for which ep^^^ — cp^ = 2e — In must be just
as important as configurations with cPx+s ~ (P x
expansion (4.22), where we have lost the periodicity of the cosine, (4.23) is
strongly suppressed. We must find the remedy for this unphysical
situation. There are several ways of doing it. The most elegant one is to
consider a continuum limit (4.22) but to allow (/? to be a multivalued
function, so that it has 27t jumps at certain branch cuts. We shall return
to this approach but first it is useful to work out the theory on a lattice,
and then to see how this multivalued field arises.
Our aim is to retain the harmonic approximation for the ep-fidd, but
to account for the configuration with (4.23). This aim can be achieved
by replacing (4.21) by:
n
Z =
j n ^ e x p ( ^ - ^ I( < P x - « ? > x + « + 27m.,,)^^
(4.24)