ATTEMPT AT A SYNTHESIS
279
surfaces. Let us introduce the dual lattice which is formed by the centres
of the cubes of the original one. Consider a closed loop C on the dual
lattice and form an arbitrary surface 5^ (on this lattice) bounded by this
loop. Define the distorted partition function Z(C, Sc) by changing the
signs of all couplings on the original links intersected by 5^. Then fi(C)
is defined by :
Z(C, Sc)
< K Q > =
(10.71)
The definition (10.71) is a straightforward generalization of (10.56).
Here again it appears that does not depend on the choice of the
surface
because a closed surface of dislocation does not change Z (it
is possible to reverse spins in the drop, bounded by S). Therefore we can
form loop Green functions like:
(10.72)
If also c t (jc) are present then these functions are defined up to a sign
which is physically irrelevant.
As in the 2d case, we shall obtain simple equations not for or
<(j(jc)> but for certain mixed objects which will be called “fermionic
string” or “barbed wire”.
Let us supply the middle of each link(s) of C with one of the vectors
= 1, 2, 3, 4) which lie in the plane orthogonal to the link under
consideration. Let us consider the following object:
s c l ) = m c l ) n
+ «<..)
(10.73)
Here L is the length of the loop C and jc^ is the middle of the link s. In
order to obtain an equation most closely resembling equations (10.67)
it is convenient to notice that the average <'I^ai...a^(Q)> can be
computed in two ways: first with the definition of the average by (10.54)
and fi given above, but also the dual way of computation is possible. By
that we mean that we can define a dual Ising model by introducing
variables
± 1 attached to the links of the dual lattice and
considering the partition function:
(10.74)
Here we have denoted by P plaquettes of the dual lattice and by ii{dP)
the product of
around the plaquette. The dual temperature P is
given by:
e ~ 2 ^ = ta n h )5
(10.75)
279
surfaces. Let us introduce the dual lattice which is formed by the centres
of the cubes of the original one. Consider a closed loop C on the dual
lattice and form an arbitrary surface 5^ (on this lattice) bounded by this
loop. Define the distorted partition function Z(C, Sc) by changing the
signs of all couplings on the original links intersected by 5^. Then fi(C)
is defined by :
Z(C, Sc)
< K Q > =
(10.71)
The definition (10.71) is a straightforward generalization of (10.56).
Here again it appears that does not depend on the choice of the
surface
because a closed surface of dislocation does not change Z (it
is possible to reverse spins in the drop, bounded by S). Therefore we can
form loop Green functions like:
(10.72)
If also c t (jc) are present then these functions are defined up to a sign
which is physically irrelevant.
As in the 2d case, we shall obtain simple equations not for or
<(j(jc)> but for certain mixed objects which will be called “fermionic
string” or “barbed wire”.
Let us supply the middle of each link(s) of C with one of the vectors
= 1, 2, 3, 4) which lie in the plane orthogonal to the link under
consideration. Let us consider the following object:
s c l ) = m c l ) n
+ «<..)
(10.73)
Here L is the length of the loop C and jc^ is the middle of the link s. In
order to obtain an equation most closely resembling equations (10.67)
it is convenient to notice that the average <'I^ai...a^(Q)> can be
computed in two ways: first with the definition of the average by (10.54)
and fi given above, but also the dual way of computation is possible. By
that we mean that we can define a dual Ising model by introducing
variables
± 1 attached to the links of the dual lattice and
considering the partition function:
(10.74)
Here we have denoted by P plaquettes of the dual lattice and by ii{dP)
the product of
around the plaquette. The dual temperature P is
given by:
e ~ 2 ^ = ta n h )5
(10.75)