From 3151527d24e96846c10cc2818b6351bbecfe33c2 Mon Sep 17 00:00:00 2001
From: mfm94 <mfm94@cornell.edu>
Date: Tue, 9 Jul 2019 20:05:26 +0000
Subject: Update on Overleaf.

---
 main.tex | 32 +++++++++++++++++++++++++++++++-
 1 file changed, 31 insertions(+), 1 deletion(-)

diff --git a/main.tex b/main.tex
index fb73629..8d9955e 100644
--- a/main.tex
+++ b/main.tex
@@ -41,7 +41,7 @@
 
 \title{\urusi mft}
 \author{Jaron Kent-Dobias}
-\author{Mike Matty}
+\author{Michael Matty}
 \author{Brad Ramshaw}
 \affiliation{Laboratory of Atomic \& Solid State Physics, Cornell University, Ithaca, NY, USA}
 
@@ -75,6 +75,36 @@
     \item Talk about more cool stuff like AFM C4 breaking etc
 \end{enumerate}
 
+The study of phase transitions is a central theme of condensed matter physics. In many
+cases, a phase transition between different states of matter is marked by a change in symmetry.
+In this paradigm, the breaking of symmetry in an ordered phase corresponds to the condensation
+of an order parameter (OP) that breaks the same symmetries. Near a second order phase
+transition, the physics of the OP can often be described in the context of Landau-Ginzburg 
+mean field theory. However, to construct such a theory, one must know the symmetries
+of the order parameter, i.e. the symmetry of the ordered state.
+
+One quintessential case where the symmetry of an ordered phase remains unknown is in \urusi.
+\urusi is a heavy fermion superconductor in which superconductivity condenses out of a 
+symmetry broken state referred to as hidden order (HO) [cite pd paper], and at sufficiently
+large [hydrostatic?] pressures, both give way to local moment antiferromagnetism. 
+Despite over thirty years of effort, the symmetry of the hidden order state remains unknown, and
+modern theories [big citation chunk] propose a variety of possibilities. 
+Many [all?] of these theories rely on the formulation of a microscopic model for the 
+HO state, but without direct experimental observation of the broken symmetry, none 
+have been confirmed. 
+
+One case that does not rely on a microscopic model is recent work [cite RUS paper] 
+that studies the HO transition using resonant ultrasound spectroscopy (RUS). 
+RUS is an experimental technique that measures mechanical resonances of a sample. These
+resonances contain information about the full elastic tensor of the material. Moreover, 
+the frequency locations of the resonances are sensitive to symmetry breaking at an electronic
+phase transition due to electron-phonon coupling [cite]. Ref. [RUS paper] uses this information
+to place strict thermodynamic bounds on the symmetry of the HO OP, again, independent of 
+any microscopic model. Motivated by these results, in this paper we consider a mean field theory
+of an OP coupled to strain and the effect that the OP symmetry has on the elastic response
+in different symmetry channels. Our study finds that a single possible OP symmetry
+reproduces the experimental strain susceptibilities, and fits the experimental data well.
+
 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 in three parts: that of the strain, the order parameter, and their interaction. The most general quadratic free energy of the strain $\epsilon$ is $f_\e=\lambda_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the $\lambda$ tensor is constrained by both that $\epsilon$ is a symmetric tensor and by the point group symmetry \cite{landau_theory_1995}. The latter can be seen in a systematic way. First, the six independent components of strain are written as linear combinations that behave like irreducible representations under the action of the point group, or
 \begin{equation}
   \begin{aligned}
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