From af87e6ab65294b4a89f29fa57ed6928bba7167e6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 17 Sep 2019 12:12:41 -0400
Subject: more improvements to Ginzburg section

---
 main.tex | 32 +++++++++-----------------------
 1 file changed, 9 insertions(+), 23 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index d7cd7cf..98266fc 100644
--- a/main.tex
+++ b/main.tex
@@ -122,10 +122,8 @@ strain, and examine the response function dependence on the symmetry of the
 \op.  We compare the results from mean field theory with data from \rus\
 experiments.  We further examine the consequences of our theory at non-zero
 applied pressure in comparison with recent x-ray scattering experiments
-\cite{choi_pressure-induced_2018}. We preform a consistency check for the
-applicability of our mean field theory for the \rus\ data.  Finally, we discuss
-our conclusions and the future experimental and theoretical work motivated by
-our results.
+\cite{choi_pressure-induced_2018}. Finally, we discuss our conclusions and the
+future experimental and theoretical work motivated by our results.
 
 The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally
 respect this symmetry. We will introduce a phenomenological free energy density
@@ -471,30 +469,18 @@ phase.
 
 Three dimensions is below the upper critical dimension $4\frac12$, and so mean
 field theory should break down sufficiently close to the critical point due to
-fluctuations \cite{hornreich_lifshitz_1980}. Mean field theory neglects the
-effect of fluctuations on critical behavior, yet also predicts the magnitude of
-those fluctuations. This allows a mean field theory to undergo an internal
-consistency check to ensure the predicted fluctuations are indeed negligible in the region under consideration.
-This is typically done by computing the Ginzburg temperature
-\cite{ginzburg_remarks_1961}, which gives the proximity to the critical point
-$t=(T-T_c)/T_c$ at which mean field theory is expected to break down by
-comparing the magnitude of fluctuations in a correlation-length sized box to
-the magnitude of the field.
-
+fluctuations, at the Ginzburg temperature \cite{hornreich_lifshitz_1980, ginzburg_remarks_1961}. Magnetic phase transitions tend to have Ginzburg temperature of order one.
 Our fit above gives $\xi_{\perp0}q_*=(D_\perp
-q_*^4/aT_c)^{1/4}\simeq2$. The estimate here predicts
-that an experiment may begin to see deviations from mean field behavior within
+q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of $q_*\simeq\pi/a_3$ puts the bare correlation length $\xi_{\perp0}$ at about what one would expect for a generic magnetic transition.
+The argeement of this data in the $t\sim0.1$--10 range with the mean field exponent suggests that this region is outside the Ginzburg region, but an experiment may begin to see deviations from mean field behavior within
 around several degrees Kelvin of the critical point. A \rus\ experiment with more precise
 temperature resolution near the critical point may be able to resolve a
 modified cusp exponent $\gamma\simeq1.31$ \cite{guida_critical_1998}, since the
 universality class of a uniaxial modulated scalar \op\ is $\mathrm O(2)$
-\cite{garel_commensurability_1976}. Our work here appears self--consistent,
-given that our fit is mostly concerned with temperatures ten to hundreds of
-Kelvin from the critical point. This analysis also indicates that we should not
-expect any quantitative agreement between mean field theory and experiment in
-the low temperature phase since, by the point the Ginzburg criterion is
-satisfied, $\eta$ is order one and the Landau--Ginzburg free energy expansion
-is no longer valid.
+\cite{garel_commensurability_1976}. We should not expect any quantitative
+agreement between mean field theory and experiment in the low temperature phase
+since, by the point the Ginzburg criterion is satisfied, $\eta$ is order one
+and the Landau--Ginzburg free energy expansion is no longer valid.
 
 We have preformed a general treatment of phenomenological \ho\ \op s with the
 potential for linear coupling to strain. The possibilities with consistent mean
-- 
cgit v1.2.3-70-g09d2

