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Brief synopsis of the RG in two dimensions

For completeness, I will summarize the structure of the RG treatment in two dimensions. The interested reader is encouraged to check the literature for discussions of alternative approaches and the various controversial points. The marginality of temperature in this case leads to the formation of a finite temperature fixed line instead of the usual isolated T=0 fixed point. The RG has been studied quite some time ago by Cardy and Ostlund. There are both complications and simplifications over the d>2 case.

First, we may no longer ignore the tex2html_wrap_inline3730 term, which is itself marginal in two dimensions. This of course complicates the analysis. However, because one is working at finite temperature, it turns out to be possible to keep only the first harmonic in a Fourier expansion of tex2html_wrap_inline3658 , i.e.

equation1343

Likewise, we may decompose

equation1345

where tex2html_wrap_inline3734 is a traceless matrix. Only tex2html_wrap_inline3310 plays an important role in the RG. The Cardy-Ostlund analysis gives

eqnarray1350

where A and C are cut-off dependent, but tex2html_wrap_inline3742 , and

equation1358

Here we have kept T fixed, as is appropriate for a finite temperature fixed line.

For tex2html_wrap_inline3746 , i.e. tex2html_wrap_inline3748 , the system is in a thermal phase, and the disorder g flows to zero. A residual finite value of tex2html_wrap_inline3310 leads to some small distortions of the phase.

For tex2html_wrap_inline3754 , the system is in a glassy phase. For small tex2html_wrap_inline3756 , the RG is controlled and g flows to a fixed point tex2html_wrap_inline3760 . The vector field tex2html_wrap_inline3730 then grows without bound. This, however, can be shown not to invalidate the RG, and leads only to a large, scale-dependent ``background'' distortion of phase. It is this background distortion that shows up as the tex2html_wrap_inline3718 growth of phase fluctuations.

Some nice extensions of the RG to calculate other quantities are described in T. Hwa and D. S. Fisher, PRL 72, 2466 (1994).


next up previous contents
Next: Beyond the elastic approximation Up: Behavior in d=2 Previous: Behavior in d=2

Leon Balents
Thu May 30 08:21:44 PDT 1996