280
GAUGE FIELDS AND STRINGS
Kramers-Wannier duality ensures that the partition functions (10.74)
and (10.54) are the same. The average of (x(x) variables (which are
disorder variables with respect to (10.74)) is defined as follows. Attach
an infinite path to the point jc and change the sign of P on all plaquettes
intersected by the path. Then:
P j
=
(10.76)
After short meditation one concludes that (10.75), just as (10.56), does
not depend on the choice of the path P^. Indeed, let us consider a short
closed path intersecting a bunch of four plaquettes having one common
link. Changing the sign of the spin fi on this link we get a new
configuration which is equivalent to the one in the model without the
path. Therefore for this small closed path the modified partition
function coincides with the old one. Since any large path can be
composed from small ones, we conclude that a closed loop of dislocations does not change the system.
We are in a position now to obtain the desired equation for
«2(0 * Let us use the identities
Q-imep) ^ cosh(2^) - sinh(2p)nidP)
(10.77)
n \ d P ) = 1
and imagine a tail S attached to a(x^ -f
which gives a contribution:
+ e j - n e
2fin(dP)
(where the product goes along all the plaquettes intersected by the tail
S). Using (10.77) (which is analogous to 10.59) we obtain:
..... a .( 0 = cosh(2^)T,^..... ..................... ,J C )
-sinh(2^)T,^..... + U J (10.78)
Here the loop C -f 11^^ is obtained from the original loop by removing
the link 5 and attaching instead the letter H oriented in the direction a^.
The loop C -h
has length equal to L -h 2, and on the two extra links
we place indices a'^ and
-h 3, so that the corresponding a are both
placed in middle of the plaquette H. Since
= 1 this does not affect
our relations.
We observe now a remarkable analogy between equation (10.78) and
equation (10.60). Indeed (10.78) implies that if we concentrate on some
link s then it propagates in the plane orthogonal to itself exactly as a
GAUGE FIELDS AND STRINGS
Kramers-Wannier duality ensures that the partition functions (10.74)
and (10.54) are the same. The average of (x(x) variables (which are
disorder variables with respect to (10.74)) is defined as follows. Attach
an infinite path to the point jc and change the sign of P on all plaquettes
intersected by the path. Then:
P j
=
(10.76)
After short meditation one concludes that (10.75), just as (10.56), does
not depend on the choice of the path P^. Indeed, let us consider a short
closed path intersecting a bunch of four plaquettes having one common
link. Changing the sign of the spin fi on this link we get a new
configuration which is equivalent to the one in the model without the
path. Therefore for this small closed path the modified partition
function coincides with the old one. Since any large path can be
composed from small ones, we conclude that a closed loop of dislocations does not change the system.
We are in a position now to obtain the desired equation for
«2(0 * Let us use the identities
Q-imep) ^ cosh(2^) - sinh(2p)nidP)
(10.77)
n \ d P ) = 1
and imagine a tail S attached to a(x^ -f
which gives a contribution:
+ e j - n e
2fin(dP)
(where the product goes along all the plaquettes intersected by the tail
S). Using (10.77) (which is analogous to 10.59) we obtain:
..... a .( 0 = cosh(2^)T,^..... ..................... ,J C )
-sinh(2^)T,^..... + U J (10.78)
Here the loop C -f 11^^ is obtained from the original loop by removing
the link 5 and attaching instead the letter H oriented in the direction a^.
The loop C -h
has length equal to L -h 2, and on the two extra links
we place indices a'^ and
-h 3, so that the corresponding a are both
placed in middle of the plaquette H. Since
= 1 this does not affect
our relations.
We observe now a remarkable analogy between equation (10.78) and
equation (10.60). Indeed (10.78) implies that if we concentrate on some
link s then it propagates in the plane orthogonal to itself exactly as a