Goldstone modes in statistical mechanics
- From: beheiger <beheiger@xxxxxxxxxxx>
- Date: Mon, 29 Oct 2007 15:17:39 +0000 (UTC)
Hi there,
I would like to ask a question to the statistical mechanics experts.
For a O(n) symmetric spin system like the O(n) Landau Ginzburg model
defined on, say, a d-dimensional lattice, one can define a longitidinal as
well as a transverse order parameter susceptibility.
Below T_c, it is easy to see that while the longitudinal susceptibility
is finite in the mean field approximation (MFA), the transverse
susceptibility diverges
in the MFA due to the vanishing "mass" of the Goldstone modes. Fine.
So I always naively assumed that the finiteness of the longitudinal
susceptibility is not ruined by fluctuation corrections.
However, reading the (excellent!) book "Statistical Theory of Fields"
by Kardar (Cambridge University Press), I now find a calculation which
shows that statement in first order perturbation theory not only the
transverse but also the longitudinal susceptibility diverges well below (!)
T_c, the reason being the coupling between longitudinal and transverse
modes!
That puzzles me. Is this a physical fact, or is it possible to e.g.perform
some
"renormalization" i.e. readjustment of parameters to get rid of this
divergence?
For if the above statement would be true to all orders of perturbation
theory,
how could there be any ordered state below T_c?
On the other hand, I am well aware that it is not possible to define a
finite
susceptibility below T_c for the spherical model, which, after all is just
the
large n limit of the O(n) Landau Ginzburg model...
Please help to resolve my confusion. Thanks in advance for any help,
Andy
.
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