From ddc18664974154798ab4c865589f90cd60d20264 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 21 Aug 2019 15:06:30 -0400
Subject: added comments on irreps

---
 hidden_order.bib | 14 ++++++++++++++
 main.tex         |  3 ++-
 2 files changed, 16 insertions(+), 1 deletion(-)

diff --git a/hidden_order.bib b/hidden_order.bib
index d17c22c..3b779ed 100644
--- a/hidden_order.bib
+++ b/hidden_order.bib
@@ -500,4 +500,18 @@ thermodynamics from a hidden order parameter.},
   file = {/home/pants/Zotero/storage/CDTQB6PI/Inoue et al_2001_High-field magnetization of URu2Si2 under high pressure.pdf;/home/pants/Zotero/storage/323PS9NS/S0921452600006578.html}
 }
 
+@article{hornreich_critical_1975,
+  title = {Critical {{Behavior}} at the {{Onset}} of \$\textbackslash{}stackrel\{\textbackslash{}ensuremath\{\textbackslash{}rightarrow\}\}\{\textbackslash{}mathrm\{k\}\}\$-{{Space Instability}} on the \$\textbackslash{}ensuremath\{\textbackslash{}lambda\}\$ {{Line}}},
+  volume = {35},
+  abstract = {We calculate the critical behavior of systems having a multicritical point of a new type, hereafter called a Lifshitz point, which separates ordered phases with \textrightarrowk=0 and \textrightarrowk{$\not =$}0 along the {$\lambda$} line. For anisotropic systems, the correlation function is described in terms of four critical exponents, whereas for isotropic systems two exponents suffice. Critical exponents are calculated using an {$\epsilon$}-type expansion.},
+  number = {25},
+  journal = {Physical Review Letters},
+  doi = {10.1103/PhysRevLett.35.1678},
+  author = {Hornreich, R. M. and Luban, Marshall and Shtrikman, S.},
+  month = dec,
+  year = {1975},
+  pages = {1678-1681},
+  file = {/home/pants/Zotero/storage/GBYIESIW/Hornreich et al_1975_Critical Behavior at the Onset of.pdf;/home/pants/Zotero/storage/KBYQHWSH/PhysRevLett.35.html}
+}
+
 
diff --git a/main.tex b/main.tex
index 0a31ff4..a245429 100644
--- a/main.tex
+++ b/main.tex
@@ -155,6 +155,7 @@ action of the point group, or
     \epsilon_\Btg^{(1)}=2\epsilon_{12}               \\
     \epsilon_\Eg^{(1)}=2\{\epsilon_{11},\epsilon_{22}\}.
   \end{aligned}
+  \label{eq:strain-components}
 \end{equation}
 Next, all quadratic combinations of these irreducible strains that transform
 like $\Aog$ are included in the free energy as
@@ -183,7 +184,7 @@ If $\X$ is a representation not present in the strain there can be no linear
 coupling, and the effect of $\eta$ going through a continuous phase transition
 is to produce a jump in the $\Aog$ strain stiffness. We will therefore focus
 our attention on order parameter symmetries that produce linear couplings to
-strain.
+strain. Looking at the components present in \eqref{eq:strain-components}, this rules out all of the u-reps (odd under inversion) and the $\Atg$ irrep as having any anticipatory response in the strain stiffness.
 
 If the order parameter transforms like $\Aog$, odd terms are allowed in its
 free energy and any transition will be abrupt and not continuous without
-- 
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