INSTANTONS IN ABELIAN SYSTEMS
61
This fact guarantees the phase stability of (4.45). Notice that very
similar topological considerations could have been used to prove
stability of the kink solution of the preceeding section; it also can be
seen that in the case of a complex field the kink solution is unstable,
because the kink can be deformed to nothing by continuous phase
rotations.
The action for a single vortex is logarithmically divergent at large
distances, due to the last term in (4.48). However, if we consider a
neutral superposition of vortices and anti-vortices the total action will
be finite, and in the limit when all relative distances are larger than fiQ ^
it will be given by (4.39) with p replaced by
Our conclusion is that in the infrared limit we have three equivalent
descriptions of the 0(2) systems, given by the actions (4.21), (4.24) and
(4.30).
Remarks 1. The analogy between the ^ = 1 Ising model and
double-well quantum mechanics was first noted by Vaks and Larkin.
The theory of vortices in the planar 0(2) magnet and the effective
description by 0-functions was suggested by V. L. Berezinsky in his
PhD thesis (1970) and rediscovered by Koesterlitz and Thouless.
2. The 0(2) model is quite relevant to several physical problems.
First of all it describes by its definition planar magnetic systems with
0(2) symmetries. Secondly since the action (4.40) is the second quantized hamiltonian for the interacting Bose gas, the model decribes twodimensional "^He films at nonzero temperature (the static long range
properties). It is also operative for the theory of two dimensional
crystals. This occurs for the following reasons. It we denote by u^(x) a
displacement of the atom placed at the point x of the crystal then in the
infrared limit we have (according to the theory of elasticity):
(4.50)
(A, // are the so-called Lamé constants). In order to find a partition
function we have to compute:
Z = ^uj^x) e
(4.51)
This would have been an easy Gaussian integral if the fields were single
valued. However, just as phases {(p^,} were defined (mod 2n), displacements uj^x) are defined (mod bj where is a lattice vector, since on a
61
This fact guarantees the phase stability of (4.45). Notice that very
similar topological considerations could have been used to prove
stability of the kink solution of the preceeding section; it also can be
seen that in the case of a complex field the kink solution is unstable,
because the kink can be deformed to nothing by continuous phase
rotations.
The action for a single vortex is logarithmically divergent at large
distances, due to the last term in (4.48). However, if we consider a
neutral superposition of vortices and anti-vortices the total action will
be finite, and in the limit when all relative distances are larger than fiQ ^
it will be given by (4.39) with p replaced by
Our conclusion is that in the infrared limit we have three equivalent
descriptions of the 0(2) systems, given by the actions (4.21), (4.24) and
(4.30).
Remarks 1. The analogy between the ^ = 1 Ising model and
double-well quantum mechanics was first noted by Vaks and Larkin.
The theory of vortices in the planar 0(2) magnet and the effective
description by 0-functions was suggested by V. L. Berezinsky in his
PhD thesis (1970) and rediscovered by Koesterlitz and Thouless.
2. The 0(2) model is quite relevant to several physical problems.
First of all it describes by its definition planar magnetic systems with
0(2) symmetries. Secondly since the action (4.40) is the second quantized hamiltonian for the interacting Bose gas, the model decribes twodimensional "^He films at nonzero temperature (the static long range
properties). It is also operative for the theory of two dimensional
crystals. This occurs for the following reasons. It we denote by u^(x) a
displacement of the atom placed at the point x of the crystal then in the
infrared limit we have (according to the theory of elasticity):
(4.50)
(A, // are the so-called Lamé constants). In order to find a partition
function we have to compute:
Z = ^uj^x) e
(4.51)
This would have been an easy Gaussian integral if the fields were single
valued. However, just as phases {(p^,} were defined (mod 2n), displacements uj^x) are defined (mod bj where is a lattice vector, since on a