GAUGE FIELDS AND STRINGS
mean that if one considers a large but finite system and fixes the value of
at the boundary B by the condition:
,= 1
(1.14)
we have the average value of inside the system vanishing as the
size of the system goes to infinity. To prove this, let us compute the
correlation function in the small jS limit. We have:
= Z-^
(1.15)
(In (1.15) we have expanded the exponent in jS and left the lowest
nonvanishing order obtained by the string of
along the
shortest path connecting the points 0 and R). We conclude that, since
the correlation length is small, being of the order (log(l/jS))“ \ the
influence of the boundary condition inside the system must also be
small. So, one expects that for small j?:
<^X> >-Llog(l/^) 0
(1.16)
(L being the size of the system).
Now let us look at the case of large P (low temperature phase). The
maximal contribution to (1.13) in this case will be given by the
configuration with all tx, = 1- The probability for a spin to flip is of the
order of
so one expects:
(1.17)
Here ^ is the dimensionality of space and IQ) is equal to the number
of nearest neighbours. However, (1.17) is not completely true. For
^ = 1 the entropy effects spoil the order completely for all p. In order
to see how this happens, let us examine a one dimensional Ising chain.
In the ground state all the spins point up. The general configuration can
be described by marking the links that connect opposite spins. If there
are n such links, then the energy factor of the system is just
but the
number of such configurations is 2(N\/n\(N — n)\). (N is the total
number of links). As a result:
^ =
(1.18)
V n\{N-n)\
mean that if one considers a large but finite system and fixes the value of
at the boundary B by the condition:
,= 1
(1.14)
we have the average value of
size of the system goes to infinity. To prove this, let us compute the
correlation function in the small jS limit. We have:
(1.15)
(In (1.15) we have expanded the exponent in jS and left the lowest
nonvanishing order obtained by the string of
along the
shortest path connecting the points 0 and R). We conclude that, since
the correlation length is small, being of the order (log(l/jS))“ \ the
influence of the boundary condition inside the system must also be
small. So, one expects that for small j?:
<^X> >-Llog(l/^) 0
(1.16)
(L being the size of the system).
Now let us look at the case of large P (low temperature phase). The
maximal contribution to (1.13) in this case will be given by the
configuration with all tx, = 1- The probability for a spin to flip is of the
order of
so one expects:
(1.17)
Here ^ is the dimensionality of space and IQ) is equal to the number
of nearest neighbours. However, (1.17) is not completely true. For
^ = 1 the entropy effects spoil the order completely for all p. In order
to see how this happens, let us examine a one dimensional Ising chain.
In the ground state all the spins point up. The general configuration can
be described by marking the links that connect opposite spins. If there
are n such links, then the energy factor of the system is just
but the
number of such configurations is 2(N\/n\(N — n)\). (N is the total
number of links). As a result:
^ =
(1.18)
V n\{N-n)\