Added Appendix with calculation of modulus with higher-order interaction, made figure bigger.

4 files changed, 336 insertions, 113 deletions

diff --git a/fig-stiffnesses.gplot b/fig-stiffnesses.gplot
index 24d4e17..c91d0b8 100644
--- a/fig-stiffnesses.gplot
+++ b/fig-stiffnesses.gplot
@@ -8,13 +8,21 @@ cc4 = "#d41159"
 
 Tc = 17.26
 
-a1 = 71.1212
-b1 = 0.0104105
-c1 = 0.00378087
-d1 = 6.2662
+a1 = 71.13597475161484
+b1 = 0.010425748328804992
+c1 = 1665.064389018724
+d1 = 6.283065722894796
+e1 = 14.579664883988315
 
 C10(T) = a1 - b1 * T
-C1(T) = C10(T) / (1 + d1 / (1 + c1 * abs(T - Tc)) / C10(T))
+C1(T) = C10(T) / (1 + 1 / (1 / d1 + abs(T - Tc) / c1) / C10(T))
+C11(T) = C10(T) \
+  + e1**2 / (2 * C10(T)) \
+  + e1**2 * ((T - Tc) / c1) / 12 \
+  - e1**2 * (9 * C10(T) + d1) / (6 * C10(T) * (9 * C10(T) + d1 + C10(T) * d1 * (T - Tc) / c1)) \
+  + d1 * (C10(T) + d1) * (12 * C10(T)**2 + 5 * e1**2) / (12 * C10(T) * ((C10(T) + d1) + C10(T) * d1 * (T - Tc) / c1)**2) \
+  - (24 * C10(T)**2 * d1 + (4 * C10(T) + 9 * d1) * e1**2) / (12 * C10(T) * (C10(T) + d1 + C10(T) * d1 * (T - Tc) / c1))
+C12(T) = T < Tc ? C11(T) : C1(T)
 
 a2 = 144.345
 b2 = 0.019492
@@ -22,11 +30,11 @@ c2 = 120.462
 
 C20(T) = a2 - b2 * T**2 / (c2 + T)
 
-set terminal epslatex size 8.68cm, 6.7cm standalone header \
+set terminal epslatex size 18.00cm, 11.12cm standalone header \
   '\usepackage{xcolor}\definecolor{mathc3}{HTML}{1a85ff}\definecolor{mathc4}{HTML}{d41159}'
 set output "fig-stiffnesses.tex"
 
-set multiplot  layout 2, 2  margins 0.1, 0.88, 0.125, 0.99  spacing 0.01, 0.01
+set multiplot  layout 2, 2  margins 0.075, 0.915, 0.1, 0.99  spacing 0.01, 0.01
 
 set nokey
 
@@ -35,10 +43,10 @@ set mxtics 2
 set mx2tics 2
 set mytics 2
 set my2tics 2
-set format y '\tiny $%1.f$'
-set format y2 '\tiny $%.1f$'
-set format x '\tiny $%.0f$'
-#set format x2 '\tiny $%.0f$'
+set format y '\scriptsize $%1.f$'
+set format y2 '\scriptsize $%.1f$'
+set format x '\scriptsize $%.0f$'
+#set format x2 '\scripsize $%.0f$'
 set format x2 ''
 
 set style rect back fc rgb cc2l
@@ -52,44 +60,47 @@ set x2tics 50, 50, 250 offset 0,-0.5 mirror
 #set x2label '\tiny $T / \mathrm K$' offset 0,-1.0
 set xrange [0:300]
 
-set title '(a)' offset 5,-2.7
-set ylabel '\scriptsize $C_{\mathrm{B_{2\mathrm g}}} / \mathrm{GPa}$' offset 4.3
+set title '(a)' offset 12,-2.7
+set ylabel '$C_{\mathrm{B_{2\mathrm g}}} / \mathrm{GPa}$' offset 3
 set yrange [140:145]
 set ytics 141,1,144 offset 0.5 mirror
-plot "data/c66.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \
+plot "data/c66.dat" using 1:(100 * $2) with lines lw 6 lc rgb cc3, \
      C20(x) dt 3 lw 4 lc rgb cc4
 
-set title '(b)' offset 5,-2.7
+set title '(b)' offset 12,-2.7
 unset ylabel
-set y2label '\scriptsize $C_{\mathrm{B_{1\mathrm g}}} / \mathrm{GPa}$' offset -5 rotate by -90
+set y2label '$C_{\mathrm{B_{1\mathrm g}}} / \mathrm{GPa}$' offset -4 rotate by -90
 set yrange [65.05:65.7]
 set y2tics 62.1,0.1,65.6 offset -0.5 mirror
-plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \
+plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 6 lc rgb cc3, \
+     C12(x) dt 1 lw 2 lc black, \
      C1(x) dt 3 lw 4 lc rgb cc4
 
 
+
 unset x2label
 unset x2tics
 
-set title '(c)' offset 5,-2.7
-set xlabel '\scriptsize $T / \mathrm K$\\~' offset 0,0.78
+set title '(c)' offset 12,-2.7
+set xlabel '$T / \mathrm K$\\~' offset 0,0.5
 set xtics 50, 50, 250 offset 0,0.5 mirror
 unset y2tics
 set ytics 63,1,72
 unset y2label
-set ylabel '\scriptsize $\textcolor{mathc3}{C_{\mathrm{B_{1\mathrm g}}}},\textcolor{mathc4}{C_{\mathrm{B_{1\mathrm g}}}^0} / \mathrm{GPa}$' offset 3.7
+set ylabel '$\textcolor{mathc3}{C_{\mathrm{B_{1\mathrm g}}}},\textcolor{mathc4}{C_{\mathrm{B_{1\mathrm g}}}^0} / \mathrm{GPa}$' offset 3
 
 set yrange [64.5:71.5]
-plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \
+plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 6 lc rgb cc3, \
      C10(x) dt 3 lw 4 lc rgb cc4
 
 set ylabel ''
-set y2label '\scriptsize $[C^0(C^0/C - 1)]^{-1} / \mathrm{GPa}^{-1}$' offset -5.5 rotate by -90
+set y2label '$\Big[C^0_{\mathrm{B_{1\mathrm g}}}(C^0_{\mathrm{B_{1\mathrm g}}}/C_{\mathrm{B_{1\mathrm g}}} - 1)\Big]^{-1} / \mathrm{GPa}^{-1}$' offset -3 rotate by -90
 
-set title '(d)' offset 5,-2.7
-set format y2 '\tiny $%0.2f$'
-set format y '\tiny $%0.2f$'
+set title '(d)' offset 12,-2.7
+set format y2 '\scriptsize$%0.2f$'
+set format y '\scriptsize$%0.2f$'
 set yrange [0.12:0.38]
 set y2tics 0.15,0.05,0.39 offset -0.7 mirror
-plot "data/c11mc12.dat" using 1:(1 / (C10($1)*(C10($1) / (100 * $2) - 1))) with lines lw 3 lc rgb cc3, \
-     1/(C10(x) * (C10(x) / C1(x) - 1)) dt 3 lw 4 lc rgb cc4
+plot "data/c11mc12.dat" using 1:(1 / (C10($1)*(C10($1) / (100 * $2) - 1))) with lines lw 6 lc rgb cc3, \
+     1/(C10(x) * (C10(x) / C12(x) - 1)) dt 1 lw 2 lc black, \
+     1/(C10(x) * (C10(x) / C1(x) - 1)) dt 2 lw 4 lc rgb cc4
diff --git a/fig-stiffnesses.pdf b/fig-stiffnesses.pdf
index a297327..6013abe 100644
--- a/fig-stiffnesses.pdf
+++ b/fig-stiffnesses.pdf
diff --git a/hidden_order.bib b/hidden_order.bib
index b3b6e38..e2ed334 100644
--- a/hidden_order.bib
+++ b/hidden_order.bib
@@ -86,7 +86,7 @@ Ultrasound Spectroscopy and Machine Learning},
  isbn = {9780511813467}
 }
 
-@article{Chandra_2013a,
+@article{Chandra_2013_Hastatic,
  author = {Chandra, Premala and Coleman, Piers and Flint, Rebecca},
  title = {Hastatic order in the heavy-fermion compound {URu$_2$Si$_2$}},
  journal = {Nature},
@@ -100,13 +100,13 @@ Ultrasound Spectroscopy and Machine Learning},
  doi = {10.1038/nature11820}
 }
 
-@article{Chandra_2013b,
+@article{Chandra_2013_Origin,
  author = {Chandra, Premala and Coleman, Piers and Flint, Rebecca},
- title = {Origin of the Large Anisotropy in the $\chi_3$ Anomaly in {URu$_2$Si$_2$}},
+ title = {Origin of the Large Anisotropy in the {$\chi_3$} Anomaly in {URu$_2$Si$_2$}},
  journal = {Journal of Physics: Conference Series},
  publisher = {IOP Publishing},
  year = {2013},
- month = {July},
+ month = {7},
  volume = {449},
  pages = {012026},
  url = {https://doi.org/10.1088%2F1742-6596%2F449%2F1%2F012026},
@@ -130,7 +130,7 @@ Ultrasound Spectroscopy and Machine Learning},
 @article{Cricchio_2009,
  author = {Cricchio, Francesco and Bultmark, Fredrik and Grånäs, Oscar and Nordström, Lars},
  title = {Itinerant Magnetic Multipole Moments of Rank Five as the Hidden Order in 
-URu$_2$Si$_2$},
+{URu$_2$Si$_2$}},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2009},
@@ -144,7 +144,7 @@ URu$_2$Si$_2$},
 
 @article{deVisser_1986,
  author = {de Visser, A. and Kayzel, F. E. and Menovsky, A. A. and Franse, J. J. M. and van den Berg, J. and Nieuwenhuys, G. J.},
- title = {Thermal expansion and specific heat of monocrystalline URu$_2$Si$_2$},
+ title = {Thermal expansion and specific heat of monocrystalline {URu$_2$Si$_2$}},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {1986},
@@ -198,7 +198,7 @@ URu$_2$Si$_2$},
 
 @article{Guida_1998,
  author = {Guida, R. and Zinn-Justin, Jean},
- title = {Critical exponents of the $N$-vector model},
+ title = {Critical exponents of the {$N$}-vector model},
  journal = {Journal of Physics A: Mathematical and General},
  publisher = {IOP Publishing},
  year = {1998},
@@ -212,7 +212,7 @@ URu$_2$Si$_2$},
 
 @article{Harima_2010,
  author = {Harima, Hisatomo and Miyake, Kazumasa and Flouquet, Jacques},
- title = {Why the Hidden Order in URu$_2$Si$_2$ Is Still Hidden---One Simple Answer},
+ title = {Why the Hidden Order in {URu$_2$Si$_2$} Is Still Hidden---One Simple Answer},
  journal = {Journal of the Physical Society of Japan},
  publisher = {Physical Society of Japan},
  year = {2010},
@@ -226,7 +226,7 @@ URu$_2$Si$_2$},
 
 @article{Hassinger_2008,
  author = {Hassinger, E. and Knebel, G. and Izawa, K. and Lejay, P. and Salce, B. and Flouquet, J.},
- title = {Temperature-pressure phase diagram of URu$_\text2$Si$_\text2$ from resistivity measurements and ac calorimetry: Hidden order and Fermi-surface nesting},
+ title = {Temperature-pressure phase diagram of {URu$_2$Si$_2$} from resistivity measurements and ac calorimetry: Hidden order and {Fermi}-surface nesting},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {2008},
@@ -254,7 +254,7 @@ URu$_2$Si$_2$},
 
 @article{Haule_2009,
  author = {Haule, Kristjan and Kotliar, Gabriel},
- title = {Arrested Kondo effect and hidden order in URu$_\text2$Si$_\text2$},
+ title = {Arrested {Kondo} effect and hidden order in {URu$_2$Si$_2$}},
  journal = {Nature Physics},
  publisher = {Springer Science and Business Media LLC},
  year = {2009},
@@ -268,7 +268,7 @@ URu$_2$Si$_2$},
 
 @article{Hornreich_1980,
  author = {Hornreich, R. M.},
- title = {The Lifshitz point: Phase diagrams and critical behavior},
+ title = {The {Lifshitz} point: Phase diagrams and critical behavior},
  journal = {Journal of Magnetism and Magnetic Materials},
  publisher = {Elsevier BV},
  year = {1980},
@@ -279,13 +279,13 @@ URu$_2$Si$_2$},
  doi = {10.1016/0304-8853(80)91100-2}
 }
 
-@article{Hoshino_2013,
+@article{Hoshino_2013_Resolution,
  author = {Hoshino, Shintaro and Otsuki, Junya and Kuramoto, Yoshio},
- title = {Resolution of Entropy $łn\sqrt2$ by Ordering in Two-Channel Kondo Lattice},
+ title = {Resolution of Entropy {$\ln\sqrt2$} by Ordering in Two-Channel {Kondo} Lattice},
  journal = {Journal of the Physical Society of Japan},
  publisher = {Physical Society of Japan},
  year = {2013},
- month = {April},
+ month = {4},
  number = {4},
  volume = {82},
  pages = {044707},
@@ -293,13 +293,13 @@ URu$_2$Si$_2$},
  doi = {10.7566/jpsj.82.044707}
 }
 
-@article{Ikeda_1998,
+@article{Ikeda_1998_Theory,
  author = {Ikeda, Hiroaki and Ohashi, Yoji},
- title = {Theory of Unconventional Spin Density Wave: A Possible Mechanism of the Micromagnetism in U-based Heavy Fermion Compounds},
+ title = {Theory of Unconventional Spin Density Wave: A Possible Mechanism of the Micromagnetism in {U}-based Heavy Fermion Compounds},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {1998},
- month = {October},
+ month = {10},
  number = {17},
  volume = {81},
  pages = {3723--3726},
@@ -309,7 +309,7 @@ URu$_2$Si$_2$},
 
 @article{Ikeda_2012,
  author = {Ikeda, Hiroaki and Suzuki, Michi-To and Arita, Ryotaro and Takimoto, Tetsuya and Shibauchi, Takasada and Matsuda, Yuji},
- title = {Emergent rank-5 nematic order in URu$_\text2$Si$_\text2$},
+ title = {Emergent rank-5 nematic order in {URu$_2$Si$_2$}},
  journal = {Nature Physics},
  publisher = {Springer Science and Business Media LLC},
  year = {2012},
@@ -323,7 +323,7 @@ URu$_2$Si$_2$},
 
 @article{Inoue_2001,
  author = {Inoue, T. and Kindo, K. and Okuni, H. and Sugiyama, K. and Haga, Y. and Yamamoto, E. and Kobayashi, T. C. and Uwatoko, Y. and Onuki, Y.},
- title = {High-field magnetization of URu$_\text2$Si$_\text2$ under high pressure},
+ title = {High-field magnetization of {URu$_2$Si$_2$} under high pressure},
  journal = {Physica B: Condensed Matter},
  publisher = {Elsevier BV},
  year = {2001},
@@ -336,7 +336,7 @@ URu$_2$Si$_2$},
 
 @article{Kambe_2018,
  author = {Kambe, S. and Tokunaga, Y. and Sakai, H. and Hattori, T. and Higa, N. and Matsuda, T. D. and Haga, Y. and Walstedt, R. E. and Harima, H.},
- title = {Odd-parity electronic multipolar ordering in URu$_\text2$Si$_\text2$: Conclusions from Si and Ru NMR measurements},
+ title = {Odd-parity electronic multipolar ordering in {URu$_2$Si$_2$}: Conclusions from {Si} and {Ru} {NMR} measurements},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {2018},
@@ -350,7 +350,7 @@ URu$_2$Si$_2$},
 
 @article{Kiss_2005,
  author = {Kiss, Annamária and Fazekas, Patrik},
- title = {Group theory and octupolar order in URu$_\text2$Si$_\text2$},
+ title = {Group theory and octupolar order in {URu$_2$Si$_2$}},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {2005},
@@ -364,7 +364,7 @@ URu$_2$Si$_2$},
 
 @article{Kung_2015,
  author = {Kung, H.-H. and Baumbach, R. E. and Bauer, E. D. and Thorsmølle, V. K. and Zhang, W.-L. and Haule, K. and Mydosh, J. A. and Blumberg, G.},
- title = {Chirality density wave of the ``hidden order" phase in URu$_\text2$Si$_\text2$},
+ title = {Chirality density wave of the ``hidden order" phase in {URu$_2$Si$_2$}},
  journal = {Science},
  publisher = {American Association for the Advancement of Science (AAAS)},
  year = {2015},
@@ -376,9 +376,9 @@ URu$_2$Si$_2$},
  doi = {10.1126/science.1259729}
 }
 
-@article{Kusunose_2011,
+@article{Kusunose_2011_On,
  author = {Kusunose, Hiroaki and Harima, Hisatomo},
- title = {On the Hidden Order in URu$_\text2$Si$_\text2$---Antiferro Hexadecapole Order and Its Consequences},
+ title = {On the Hidden Order in {URu$_2$Si$_2$}---Antiferro Hexadecapole Order and Its Consequences},
  journal = {Journal of the Physical Society of Japan},
  publisher = {Physical Society of Japan},
  year = {2011},
@@ -392,7 +392,7 @@ URu$_2$Si$_2$},
 
 @article{Kuwahara_1997,
  author = {Kuwahara, Keitaro and Amitsuka, Hiroshi and Sakakibara, Toshiro and Suzuki, Osamu and Nakamura, Shintaro and Goto, Terutaka and Mihalik, Marián and Menovsky, Alois~A. and de~Visser, Anne and Franse, Jaap~J. ~M.},
- title = {Lattice Instability and Elastic Response in the Heavy Electron System URu$_\text2$Si$_\text2$},
+ title = {Lattice Instability and Elastic Response in the Heavy Electron System {URu$_2$Si$_2$}},
  journal = {Journal of the Physical Society of Japan},
  publisher = {Physical Society of Japan},
  year = {1997},
@@ -406,20 +406,22 @@ URu$_2$Si$_2$},
 
 @article{Lifshitz_1942a,
  author = {Lifshitz, Evgeniĭ Mikhaĭlovich},
- title = {On the theory of phase transitions of the second order {I.}~{Changes} of the elementary cell of a crystal in phase transitions of the second order},
+ title = {On the theory of phase transitions of the second order {I}},
  journal = {Proceedings of the USSR Academy of Sciences Journal of Physics},
  year = {1942},
  volume = {6},
- pages = {61}
+ pages = {61},
+ subtitle = {Changes of the elementary cell of a crystal in phase transitions of the second order}
 }
 
 @article{Lifshitz_1942b,
  author = {Lifshitz, Evgeniĭ Mikhaĭlovich},
- title = {On the theory of phase transitions of the second order {II.}~{Phase} transitions of the second order in alloys},
+ title = {On the theory of phase transitions of the second order {II}},
  journal = {Proceedings of the USSR Academy of Sciences Journal of Physics},
  year = {1942},
  volume = {6},
- pages = {251}
+ pages = {251},
+ subtitle = {Phase transitions of the second order in alloys}
 }
 
 @article{Luthi_1970,
@@ -438,7 +440,7 @@ URu$_2$Si$_2$},
 
 @article{Meng_2013,
  author = {Meng, Jian-Qiao and Oppeneer, Peter M. and Mydosh, John A. and Riseborough, Peter S. and Gofryk, Krzysztof and Joyce, John J. and Bauer, Eric D. and Li, Yinwan and Durakiewicz, Tomasz},
- title = {Imaging the Three-Dimensional Fermi-Surface Pairing near the Hidden-Order Transition in URu$_\text2$Si$_\text2$ Using Angle-Resolved Photoemission Spectroscopy},
+ title = {Imaging the Three-Dimensional {Fermi}-Surface Pairing near the Hidden-Order Transition in {URu$_2$Si$_2$} Using Angle-Resolved Photoemission Spectroscopy},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2013},
@@ -452,8 +454,8 @@ URu$_2$Si$_2$},
 
 @article{Ohkawa_1999,
  author = {Ohkawa, Fusayoshi J and Shimizu, Hirofumi},
- title = {Quadrupole and dipole orders in URu$_\text2$Si$_\text2$},
- journal = {Journal of Physics: Condensed Matter},
+ title = {Quadrupole and dipole orders in {URu$_2$Si$_2$}},
+ journal = {Journal of Physics},
  publisher = {IOP Publishing},
  year = {1999},
  month = {November},
@@ -461,7 +463,8 @@ URu$_2$Si$_2$},
  volume = {11},
  pages = {L519--L524},
  url = {https://doi.org/10.1088%2F0953-8984%2F11%2F46%2F101},
- doi = {10.1088/0953-8984/11/46/101}
+ doi = {10.1088/0953-8984/11/46/101},
+ journalsubtitle = {Condensed Matter}
 }
 
 @article{Ramshaw_2015,
@@ -478,13 +481,13 @@ URu$_2$Si$_2$},
  doi = {10.1073/pnas.1421174112}
 }
 
-@article{Rau_2012,
+@article{Rau_2012_Hidden,
  author = {Rau, Jeffrey G. and Kee, Hae-Young},
  title = {Hidden and antiferromagnetic order as a rank-5 superspin in {URu$_2$Si$_2$}},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {2012},
- month = {June},
+ month = {6},
  number = {24},
  volume = {85},
  pages = {245112},
@@ -492,13 +495,13 @@ URu$_2$Si$_2$},
  doi = {10.1103/physrevb.85.245112}
 }
 
-@article{Riggs_2015,
+@article{Riggs_2015_Evidence,
  author = {Riggs, Scott C. and Shapiro, M. C. and Maharaj, Akash V and Raghu, S. and Bauer, E. D. and Baumbach, R. E. and Giraldo-Gallo, P. and Wartenbe, Mark and Fisher, I. R.},
- title = {Evidence for a nematic component to the hidden-order parameter in URu$_2$Si$_2$ from differential elastoresistance measurements},
+ title = {Evidence for a nematic component to the hidden-order parameter in {URu$_2$Si$_2$} from differential elastoresistance measurements},
  journal = {Nature Communications},
  publisher = {Springer Science and Business Media LLC},
  year = {2015},
- month = {March},
+ month = {3},
  number = {1},
  volume = {6},
  pages = {6425},
@@ -508,7 +511,7 @@ URu$_2$Si$_2$},
 
 @article{Santini_1994,
  author = {Santini, P. and Amoretti, G.},
- title = {Crystal Field Model of the Magnetic Properties of URu$_\text2$Si$_\text2$},
+ title = {Crystal Field Model of the Magnetic Properties of {URu$_2$Si$_2$}},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {1994},
@@ -522,7 +525,7 @@ URu$_2$Si$_2$},
 
 @article{Shekhter_2013,
  author = {Shekhter, Arkady and Ramshaw, B. J. and Liang, Ruixing and Hardy, W. N. and Bonn, D. A. and Balakirev, Fedor F. and McDonald, Ross D. and Betts, Jon B. and Riggs, Scott C. and Migliori, Albert},
- title = {Bounding the pseudogap with a line of phase transitions in YBa$_\text2$Cu$_\text3$O$_\text6+δ$},
+ title = {Bounding the pseudogap with a line of phase transitions in {YBa$_2$Cu$_3$O$_{6+\delta}$}},
  journal = {Nature},
  publisher = {Springer Science and Business Media LLC},
  year = {2013},
@@ -536,7 +539,7 @@ URu$_2$Si$_2$},
 
 @article{Thalmeier_2011,
  author = {Thalmeier, Peter and Takimoto, Tetsuya},
- title = {Signatures of hidden-order symmetry in torque oscillations, elastic constant anomalies, and field-induced moments in URu$_\text2$Si$_\text2$},
+ title = {Signatures of hidden-order symmetry in torque oscillations, elastic constant anomalies, and field-induced moments in {URu$_2$Si$_2$}},
  journal = {Physical Review B},
  publisher = {American Physical Society (APS)},
  year = {2011},
@@ -548,13 +551,13 @@ URu$_2$Si$_2$},
  doi = {10.1103/physrevb.83.165110}
 }
 
-@article{Tonegawa_2012,
+@article{Tonegawa_2012_Cyclotron,
  author = {Tonegawa, S. and Hashimoto, K. and Ikada, K. and Lin, Y. -H. and Shishido, H. and Haga, Y. and Matsuda, T. D. and Yamamoto, E. and Onuki, Y. and Ikeda, H. and Matsuda, Y. and Shibauchi, T.},
- title = {Cyclotron Resonance in the Hidden-Order Phase of URu$_\text2$Si$_\text2$},
+ title = {Cyclotron Resonance in the Hidden-Order Phase of {URu$_2$Si$_2$}},
  journal = {Physical Review Letters},
  publisher = {American Physical Society (APS)},
  year = {2012},
- month = {July},
+ month = {7},
  number = {3},
  volume = {109},
  pages = {036401},
@@ -577,8 +580,8 @@ URu$_2$Si$_2$},
 }
 
 @article{Wiebe_2007,
- author = {Wiebe, C. R. and Janik, J. A. and MacDougall, G. J. and Luke, G. M. and Garrett, J. D. and Zhou, H. D. and Jo, Y. -J. and Balicas, L. and Qiu, Y. and Copley, J. R. D. and Yamani, Z. and Buyers, W. J. L.},
- title = {Gapped itinerant spin excitations account for missing entropy in the hidden-order state of URu$_\text2$Si$_\text2$},
+ author = {Wiebe, C. R. and Janik, J. A. and MacDougall, G. J. and Luke, G. M. and Garrett, J. D. and Zhou, H. D. and Jo, Y.-J. and Balicas, L. and Qiu, Y. and Copley, J. R. D. and Yamani, Z. and Buyers, W. J. L.},
+ title = {Gapped itinerant spin excitations account for missing entropy in the hidden-order state of {URu$_2$Si$_2$}},
  journal = {Nature Physics},
  publisher = {Springer Science and Business Media LLC},
  year = {2007},
@@ -592,7 +595,7 @@ URu$_2$Si$_2$},
 
 @article{Wolf_1994,
  author = {Wolf, B. and Sixl, W. and Graf, R. and Finsterbusch, D. and Bruls, G. and Lüthi, B. and Knetsch, E. A. and Menovsky, A. A. and Mydosh, J. A.},
- title = {Elastic properties of the heavy fermion superconductor URu$_\text2$Si$_\text2$},
+ title = {Elastic properties of the heavy fermion superconductor {URu$_2$Si$_2$}},
  journal = {Journal of Low Temperature Physics},
  publisher = {Springer Science and Business Media LLC},
  year = {1994},
diff --git a/main.tex b/main.tex
index 7e934bb..b815c79 100644
--- a/main.tex
+++ b/main.tex
@@ -1,5 +1,5 @@
 
-\documentclass[aps,prb,reprint,longbibliography,floatfix]{revtex4-1}
+\documentclass[aps,prb,reprint,longbibliography,floatfix,fleqn]{revtex4-1}
 \usepackage[utf8]{inputenc}
 \usepackage{amsmath,graphicx,upgreek,amssymb,xcolor}
 \usepackage[colorlinks=true,urlcolor=purple,citecolor=purple,filecolor=purple,linkcolor=purple]{hyperref}
@@ -111,9 +111,10 @@ broken symmetry remains unknown. This state, known as \emph{hidden order}
 lower temperatures.  At sufficiently large hydrostatic pressures, both
 superconductivity and \ho\ give way to local moment antiferromagnetism
 (\afm).\cite{Hassinger_2008}  Modern theories~\cite{Kambe_2018, Haule_2009,
-  Kusunose_2011, Kung_2015, Cricchio_2009, Ohkawa_1999, Santini_1994,
-Kiss_2005, Harima_2010, Thalmeier_2011, Tonegawa_2012, Rau_2012, Riggs_2015,
-Hoshino_2013, Ikeda_1998, Chandra_2013a, 1902.06588v2, Ikeda_2012} propose
+Kusunose_2011_On, Kung_2015, Cricchio_2009, Ohkawa_1999, Santini_1994,
+Kiss_2005, Harima_2010, Thalmeier_2011, Tonegawa_2012_Cyclotron,
+Rau_2012_Hidden, Riggs_2015_Evidence, Hoshino_2013_Resolution,
+Ikeda_1998_Theory, Chandra_2013_Hastatic, 1902.06588v2, Ikeda_2012} propose
 associating any of a variety of broken symmetries with \ho. This work analyzes
 a family of phenomenological models with order parameters of general symmetry
 that couple linearly to strain. Of these, only one is compatible with two
@@ -128,9 +129,10 @@ examine the thermodynamic discontinuities in the elastic moduli at
 $T_{\text{\ho}}$.\cite{1903.00552v1} The observation of discontinues only in
 compressional, or $\Aog$, elastic moduli requires that the point-group
 representation of \ho\ be one-dimensional. This rules out many order parameter
-candidates~\cite{Thalmeier_2011, Tonegawa_2012, Rau_2012, Riggs_2015,
-Hoshino_2013, Ikeda_2012, Chandra_2013b} in a model-independent way, but
-doesn't differentiate between those that remain. 
+candidates~\cite{Thalmeier_2011, Tonegawa_2012_Cyclotron, Rau_2012_Hidden,
+Riggs_2015_Evidence, Hoshino_2013_Resolution, Ikeda_2012, Chandra_2013_Origin}
+in a model-independent way, but doesn't differentiate between those that
+remain. 
 
 Recent x-ray experiments discovered rotational symmetry breaking in \urusi\
 under pressure.\cite{Choi_2018} Above 0.13--0.5 $\GPa$ (depending on
@@ -462,33 +464,34 @@ signatures of a continuous transition by locating thermodynamic singularities
 at nonzero $q=q_*$.  The remaining clue at $q=0$ is a particular kink in the
 corresponding modulus.
 
-\begin{figure}[htpb]
+\section{Comparison to experiment}
+
+\begin{figure*}[htpb]
   \centering
-  \includegraphics[width=\columnwidth]{fig-stiffnesses}
+  \includegraphics{fig-stiffnesses}
   \caption{
-   \Rus\ measurements of the elastic moduli of \urusi\ at ambient pressure as a
-   function of temperature from recent experiments\cite{1903.00552v1} (blue,
-   solid) alongside fits to theory (magenta, dashed). The solid yellow region
-   shows the location of the \ho\ phase. (a) $\Btg$ modulus data and a fit to
-   the standard form.\cite{Varshni_1970} (b) $\Bog$ modulus data and a fit to
-   \eqref{eq:static_modulus}. The fit gives
-   $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $D_\perp
-   q_*^4/b^2\simeq0.16\,\GPa^{-1}$, and
-   $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of a quadratic
-   term in $C^0_\Bog$ was here not needed for the fit.\cite{Varshni_1970} (c)
-   $\Bog$ modulus data and the fit of the \emph{bare} $\Bog$ modulus. (d)
-   $\Bog$ modulus data and the fit transformed by
-   $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted from
-   \eqref{eq:static_modulus} to equal $D_\perp q_*^4/b^2+a/b^2|T-T_c|$, e.g.,
-   an absolute value function. The failure of the Ginzburg--Landau prediction
-   below the transition is expected on the grounds that the \op\ is too large
-   for the free energy expansion to be valid by the time the Ginzburg
-   temperature is reached.
+    \Rus\ measurements of the elastic moduli of \urusi\ at ambient pressure as a
+    function of temperature from recent experiments\cite{1903.00552v1} (blue,
+    solid) alongside fits to theory (magenta, dashed). The solid yellow region
+    shows the location of the \ho\ phase. (a) $\Btg$ modulus data and a fit to
+    the standard form.\cite{Varshni_1970} (b) $\Bog$ modulus data and a fit to
+    \eqref{eq:static_modulus}. The fit gives
+    $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $D_\perp
+    q_*^4/b^2\simeq0.16\,\GPa^{-1}$, and
+    $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of a quadratic
+    term in $C^0_\Bog$ was here not needed for the fit.\cite{Varshni_1970} (c)
+    $\Bog$ modulus data and the fit of the \emph{bare} $\Bog$ modulus. (d)
+    $\Bog$ modulus data and the fit transformed by
+    $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted from
+    \eqref{eq:static_modulus} to equal $D_\perp q_*^4/b^2+a/b^2|T-T_c|$, e.g.,
+    an absolute value function. The failure of the Ginzburg--Landau prediction
+    below the transition is expected on the grounds that the \op\ is too large
+    for the free energy expansion to be valid by the time the Ginzburg
+    temperature is reached.
   }
   \label{fig:data}
-\end{figure}
+\end{figure*}
 
-\section{Comparison to experiment}
 \Rus\ experiments~\cite{1903.00552v1} yield the individual elastic moduli
 broken into irreps; data for the $\Bog$ and $\Btg$ components defined in
 \eqref{eq:strain-components} are shown in Figures \ref{fig:data}(a--b).  The
@@ -497,18 +500,27 @@ the presence of the transition, exhibiting the expected linear stiffening upon
 cooling from room temperature, with a low-temperature cutoff at some fraction
 of the Debye temperature.\cite{Varshni_1970} The $\Bog$ modulus
 Fig.~\ref{fig:data}(b) has a dramatic response, softening over the course of
-roughly $100\,\K$ and then cusping at the \ho\ transition. While the
-low-temperature response is not as dramatic as the theory predicts, mean field
-theory---which is based on a small-$\eta$ expansion---will not work
-quantitatively far below the transition where $\eta$ has a large nonzero value
-and higher powers in the free energy become important. The data in the
+roughly $100\,\K$ and then cusping at the \ho\ transition. The data in the
 high-temperature phase can be fit to the theory \eqref{eq:static_modulus}, with
 a linear background modulus $C^0_\Bog$ and $\tilde r-\tilde r_c=a(T-T_c)$, and
-the result is shown in Figure \ref{fig:data}(b). The data and theory appear
-quantitatively consistent in the high temperature phase, suggesting that \ho\
-can be described as a $\Bog$-nematic phase that is modulated at finite $q$
-along the $c-$axis. The predicted softening appears over hundreds of Kelvin;
-Figures \ref{fig:data}(c--d) show the background modulus $C_\Bog^0$ and the
+the result is shown in Figure \ref{fig:data}(b).
+
+The behavior of the modulus below the transition does not match
+\eqref{eq:static_modulus} well, but this is because of the truncation of the
+free energy expansion used above.  Higher order terms like $\eta^2\epsilon^2$
+contribute to the modulus starting at order $\eta_*^2$, and therefore while
+they do not affect the behavior above the transition, they change the behavior
+below it. To demonstrate this, in Appendix~\ref{sec:higher-order} we compute
+the modulus in a theory where the interaction free energy is truncated after
+fourth order with new term $\frac12g\eta^2\epsilon^2$. The thin solid black
+line in Fig.~\ref{fig:data} shows the fit of the \rus\ data to \eqref{eq:C0}
+and shows that high-order corrections can account for the low-temperature
+behavior. 
+
+The data and theory appear quantitatively consistent, suggesting that \ho\ can
+be described as a $\Bog$-nematic phase that is modulated at finite $q$ along
+the $c-$axis. The predicted softening appears over hundreds of Kelvin; Figures
+\ref{fig:data}(c--d) show the background modulus $C_\Bog^0$ and the
 \op--induced response isolated from each other. 
 
 We have seen that the mean-field theory of a $\Bog$ \op\ recreates the topology
@@ -624,6 +636,203 @@ such as ultrasound, that could further support or falsify this idea.
   Elena Hassinger. We thank Sayak Ghosh for \rus\ data.
 \end{acknowledgements}
 
+\appendix
+
+\section{Adding a higher-order interaction}
+\label{sec:higher-order}
+
+In this appendix, we compute the $\Bog$ modulus for a theory with a high-order
+interaction truncation to better match the low-temperature behavior.  Consider
+the free energy density $f=f_\e+f_\i+f_\op$ with
+\begin{equation}
+  \begin{aligned}
+    f_\e&=\frac12C_0\epsilon^2 \\
+    f_\i&=-b\epsilon\eta+\frac12g\epsilon^2\eta^2 \\
+    f_\op&=\frac12\big[r\eta^2+c_\parallel(\nabla_\parallel\eta)^2+c_\perp(\nabla_\perp\eta)^2+D(\nabla_\perp^2\eta)^2\big]+u\eta^4.
+  \end{aligned}
+  \label{eq:new_free_energy}
+\end{equation}
+The mean-field stain conditioned on the order parameter is found from
+\begin{equation}
+  \begin{aligned}
+    0
+    &=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon(x)}\bigg|_{\epsilon=\epsilon_\star[\eta]} \\
+    &=C_0\epsilon_\star[\eta](x)-b\eta(x)+g\epsilon_\star[\eta](x)\eta(x)^2,
+  \end{aligned}
+\end{equation}
+which yields
+\begin{equation}
+  \epsilon_\star[\eta](x)=\frac{b\eta(x)}{C_0+g\eta(x)^2}.
+  \label{eq:epsilon_star}
+\end{equation}
+Upon substitution into \eqref{eq:new_free_energy} and expanded to fourth order
+in $\eta$, $F[\eta,\epsilon_\star[\eta]]$ can be written in the form
+$F_\op[\eta]$ alone with $r\to\tilde r=r-b^2/C_0$ and $u\to\tilde
+u=u+b^2g/2C_0^2$. The phase diagram in $\eta$ follows as before with the
+shifted coefficients, and namely $\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$ for
+$\tilde r<c_\perp^2/4D=\tilde r_c$ with $q_*^2=-c_\perp/2D$ and
+\begin{equation}
+  \eta_*^2=\frac{c_\perp^2-4D\tilde r}{12D\tilde u}
+  =\frac{|\Delta\tilde r|}{3\tilde u}.
+\end{equation}
+We would like to calculate the $q$-dependant modulus
+\begin{equation}
+  C(q)
+  =\frac1V\int dx\,dx'\,C(x,x')e^{-iq(x-x')},
+\end{equation}
+where
+\begin{widetext}
+\begin{equation}
+  C(x,x')
+  =\frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}\bigg|_{\epsilon=\langle\epsilon\rangle}
+    =\frac{\delta^2F_\e[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}+
+     \frac{\delta^2F_\i[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}+
+     \frac{\delta^2F_\op[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}
+     \bigg|_{\epsilon=\langle\epsilon\rangle}
+\end{equation}
+and $\eta_\star$ is the mean-field order parameter conditioned on the strain defined implicitly by
+\begin{equation}
+  0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star[\epsilon]}
+  =-b\epsilon(x)+g\epsilon(x)^2\eta_\star[\epsilon](x)+\frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star[\epsilon]}.
+  \label{eq:eta_star}
+\end{equation}
+We will work this out term by term. The elastic term is the most straightforward, giving
+\begin{equation}
+  \frac{\delta^2F_\e[\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}
+  =\frac12C_0\frac{\delta^2}{\delta\epsilon(x)\delta\epsilon(x')}\int dx''\,\epsilon(x'')^2
+  =C_0\delta(x-x').
+\end{equation}
+The interaction term gives
+\begin{equation}
+  \begin{aligned}
+    \frac{\delta^2F_\i[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}
+    &=-b\frac{\delta^2}{\delta\epsilon(x)\delta\epsilon(x')}\int dx''\,\epsilon(x'')\eta_\star[\epsilon](x'')
+      +\frac12g\frac{\delta^2}{\delta\epsilon(x)\delta\epsilon(x')}\int dx''\,\epsilon(x'')^2\eta_\star[\epsilon](x'')^2 \\
+    &=-b\frac{\delta\eta_\star[\epsilon](x')}{\delta\epsilon(x)}
+    -b\frac{\delta}{\delta\epsilon(x)}\int dx''\,\epsilon(x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon(x')}
+    +g\frac{\delta}{\delta\epsilon(x)}\big[\epsilon(x')\eta_\star[\epsilon](x')^2\big] \\
+    &\qquad+g\frac{\delta}{\delta\epsilon(x)}\int dx''\,\epsilon(x'')^2\eta_\star[\epsilon](x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon(x')} \\
+    &=-2(b-2g\epsilon(x)\eta_\star[\epsilon](x))\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon(x')}-b\int dx''\,\epsilon(x'')\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon(x)\delta\epsilon(x')}
+    +g\eta_\star[\epsilon](x)^2\delta(x-x') \\
+    &\qquad+g\int dx''\,\epsilon(x'')^2\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon(x)}\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon(x')}
+    +g\int dx''\,\epsilon(x'')^2\eta_\star[\epsilon](x'')\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon(x)\delta\epsilon(x')}.
+  \end{aligned}
+\end{equation}
+The order parameter term relies on some other identities. First, \eqref{eq:eta_star} implies
+\begin{equation}
+  \frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star[\epsilon]}
+    =b\epsilon(x)-g\epsilon(x)^2\eta_\star[\epsilon](x),
+  \label{eq:dFodeta}
+\end{equation}
+and therefore that the functional inverse $\eta_\star^{-1}[\eta]$ is
+\begin{equation}
+  \eta_\star^{-1}[\eta](x)=\frac b{2g\eta(x)}\Bigg(1-\sqrt{1-\frac{4g\eta(x)}{b^2}\frac{\delta F_\op[\eta]}{\delta\eta(x)}}\Bigg).
+\end{equation}
+The inverse function theorem further implies (with substitution of \eqref{eq:dFodeta} after the derivative is evaluated) that
+\begin{equation}
+  \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon(x')}\bigg)^{\{-1\}}
+  =\frac{\delta\eta_\star^{-1}[\eta](x)}{\delta\eta(x')}\bigg|_{\eta=\eta_\star[\epsilon]}
+  =\frac{g\epsilon(x)^2\delta(x-x')+\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\big|_{\eta=\eta_\star[\epsilon]}}{b-2g\epsilon(x)\eta_\star[\epsilon](x)}
+\end{equation}
+and therefore that
+\begin{equation}
+  \frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta_\star[\epsilon]}
+  =(b-2g\epsilon(x)\eta_\star[\epsilon](x))\bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon(x')}\bigg)^{\{-1\}}
+  -g\epsilon(x)^2\delta(x-x').
+  \label{eq:d2Fodetadeta}
+\end{equation}
+Finally, we evaluate the order parameter term, using \eqref{eq:dFodeta} and \eqref{eq:d2Fodetadeta} which give
+\begin{equation}
+  \begin{aligned}
+    \frac{\delta^2F_\op[\eta_\star[\epsilon]]}{\delta\epsilon(x)\delta\epsilon(x')}
+    &=\frac{\delta}{\delta\epsilon(x)}\int dx''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon(x')}\frac{\delta F_\op[\eta]}{\delta\eta(x'')}\bigg|_{\eta=\eta_\star[\epsilon]} \\
+    &=\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon(x)\delta\epsilon(x')}\frac{\delta F_\op[\eta]}{\delta\eta(x'')}\bigg|_{\eta=\eta_\star[\epsilon]}
+    +\int dx''dx'''\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon(x')}\frac{\delta^2F_\op[\eta]}{\delta\eta(x'')\delta\eta(x''')}\bigg|_{\eta=\eta_\star[\epsilon]} \\
+    &=\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon(x)\delta\epsilon(x')}\big(b\epsilon(x)-g\epsilon(x)^2\eta_\star[\epsilon](x)\big)
+    +(b-2g\epsilon(x)\eta_\star[\epsilon](x))\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon(x')} \\
+    &\qquad-g\int dx''\,\epsilon(x'')^2\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon(x)}\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon(x')}.
+  \end{aligned}
+\end{equation}
+Summing all three terms, we see a great deal of cancellation, with
+\[
+  \frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}=C_0\delta(x-x')+g\eta_\star[\epsilon](x)^2\delta(x-x')-(b-2g\epsilon(x)\eta_\star[\epsilon](x))\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon(x')}.
+\]
+We new need to evaluate this at $\langle\epsilon\rangle$. First, $\eta_\star[\langle\epsilon\rangle]=\langle\eta\rangle$, and
+\[
+  \frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}\bigg|_{\epsilon=\langle\epsilon\rangle}=C_0\delta(x-x')+g\langle\eta(x)\rangle^2\delta(x-x')-(b-2g\langle\epsilon(x)\rangle\langle\eta(x)\rangle)\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon(x')}\bigg|_{\epsilon=\langle\epsilon\rangle}.
+\]
+Computing the final functional derivative is the most challenging part. We will
+first compute its functional inverse, take the Fourier transform of that, and
+then use the basic relationship between Fourier functional inverses to find the
+form of the non-inverse. First, we note
+\begin{equation}
+  \frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\langle\eta\rangle}
+  =\big[r-c_\perp\nabla_\perp^2-c_\parallel\nabla_\parallel^2+D\nabla_\perp^4+12u\langle\eta(x)\rangle^2\big]\delta(x-x'),
+\end{equation}
+which gives
+\begin{equation}
+  \begin{aligned}
+    \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon(x')}\bigg)^{\{-1\}}\bigg|_{\epsilon=\langle\epsilon\rangle}
+    &=\frac1{b-2g\langle\epsilon(x)\rangle\langle\eta(x)\rangle}\bigg[g\langle\epsilon(x)\rangle^2\delta(x-x')+\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg]_{\eta=\langle\eta\rangle} \\
+    &=\frac1{b-2g\langle\epsilon(x)\rangle\langle\eta(x)\rangle}\Big[
+      g\langle\epsilon(x)\rangle^2+r-c_\perp\nabla_\perp^2-c_\parallel\nabla_\parallel^2+D\nabla_\perp^4+12u\langle\eta(x)\rangle^2
+    \Big]\delta(x-x').
+  \end{aligned}
+\end{equation}
+Upon substitution of \eqref{eq:epsilon_star} and expansion to quadratic order it $\langle\eta(x)\rangle$, we find
+\begin{equation}
+  \begin{aligned}
+    \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon(x')}\bigg)^{\{-1\}}\bigg|_{\epsilon=\langle\epsilon\rangle}
+    &=\frac1b\Bigg\{r-c_\perp\nabla_\perp^2-c_\parallel\nabla_\parallel^2+D\nabla_\perp^4\\
+    &\qquad\qquad+\langle\eta(x)\rangle^2\bigg[12u+\frac{b^2g}{C_0^2}+\frac{2g}{C_0}(r-c_\perp\nabla_\perp^2-c_\parallel\nabla_\parallel^2+D\nabla_\perp^4)\bigg]+O(\langle\eta\rangle^4)\Bigg\}\delta(x-x').
+  \end{aligned}
+\end{equation}
+Defining $\widehat{\langle\eta\rangle^2}=\int dq'\,\langle\hat\eta(q')\rangle\langle\hat\eta(-q')\rangle$, its Fourier transform is then
+\begin{equation}
+  \begin{aligned}
+    G(q)
+    &=\frac1V\int dx\,dx'\,e^{-iq(x-x')}\bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon(x')}\bigg)^{\{-1\}}\bigg|_{\epsilon=\langle\epsilon\rangle} \\
+    &=\frac1b\Bigg\{r+c_\perp q_\perp^2+c_\parallel q_\parallel^2+Dq_\perp^4+\widehat{\langle\eta\rangle^2}\bigg[12u+\frac{b^2g}{C_0^2}+\frac{2g}{C_0}(r+c_\perp q_\perp^2+c_\parallel q_\parallel^2+Dq_\perp^4)\bigg]+O(\langle\hat\eta\rangle^4)\Bigg\}.
+  \end{aligned}
+\end{equation}
+We can now compute $C(q)$ by taking its Fourier transform, using the convolution theorem for the second term:
+\begin{equation}
+  \begin{aligned}
+    C(q)
+    &=C_0+g\widehat{\langle\eta\rangle^2}-\int dq''\bigg(b\delta(q'')-\frac{gb}{C_0}\int dq'\langle\hat\eta_{q'}\rangle\langle\hat\eta_{q''-q'}\rangle\bigg)/G(q-q'') \\
+    &=C_0+g\widehat{\langle\eta\rangle^2}-b^2\bigg(\frac1{r+c_\perp q_\perp^2+c_\parallel q_\parallel^2+Dq_\perp^4}-\widehat{\langle\eta\rangle^2}\frac{12u+b^2g/C_0^2+\frac{2g}{C_0}(r+c_\perp q^2+c_\parallel q_\parallel^2+Dq_\perp^4)}{(r+c_\perp q_\perp^2+c_\parallel q_\parallel^2+Dq_\perp^4)^2}\bigg)\\
+    &\qquad+\frac{gb^2}{C_0}\int dq'\,dq''\frac{\langle\hat\eta_{q'}\rangle\langle\hat\eta_{q''-q'}\rangle}{r+c_\perp(q_\perp-q_\perp'')^2+c_\parallel(q_\parallel-q_\parallel'')^2+D(q_\perp-q_\perp'')^4}+O(\langle\hat\eta\rangle^4).
+  \end{aligned}
+\end{equation}
+Upon substitution of $\langle\hat\eta_q\rangle=\frac12\eta_*\big[\delta(q_\perp-q_*)+\delta(q_\perp+q_*)\big]\delta(q_\parallel)$, we have
+\begin{equation}
+  \begin{aligned}
+    C(q)
+    &=C_0+\frac14g\eta_*^2-b^2\bigg(\frac1{r+c_\perp q_\perp^2+c_\parallel q_\parallel^2+Dq_\perp^4}-\frac{\eta_*^2}4\frac{12u+b^2g/C_0^2+\frac{2g}{C_0}(r+c_\perp q^2+c_\parallel q_\parallel^2+Dq_\perp^4)}{(r+c_\perp q_\perp^2+c_\parallel q_\parallel^2+Dq_\perp^4)^2}\bigg)\\
+    &\qquad+\frac{gb^2\eta_*^2}{4C_0}\bigg(\frac2{r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+Dq_\perp^4}+\frac1{r+c_\parallel q_\parallel^2+c_\perp(q_\perp-2q_*)^2+D(q_\perp-2q_*)^4} \\
+    &\qquad\qquad+\frac1{r+c_\parallel q_\parallel^2+c_\perp(q_\perp+2q_*)^2+D(q_\perp+2q_*)^4}\bigg)+O(\eta_*^4).
+  \end{aligned}
+\end{equation}
+Evaluating at $q=0$, we have
+\begin{equation}
+  \begin{aligned}
+    C(0)
+    &=C_0-\frac{b^2}r+\frac{\eta_*^2}4\bigg(g+\frac{b^2}{r^2}(12u+b^2g/C_0^2)+\frac{2gb^2}{C_0r}\frac{16Dq_*^4+3r}{8Dq_*^4+r}\bigg)
+  \end{aligned}
+  \label{eq:C0}
+\end{equation}
+Above the transition this has exactly the form of \eqref{eq:static_modulus} for
+any $g$; below the transition it has the same form at $g=0$ to order
+$\eta_*^2$. With $r=a\Delta T+c^2/4D+b^2/C_0$, $u=\tilde u-b^2g/2C_0^2$, and
+\begin{equation}
+  \eta_*^2=\begin{cases}
+    0 & \Delta T > 0 \\
+    -a\Delta T/3\tilde u & \Delta T \leq 0,
+  \end{cases}
+\end{equation}
+we can fit the ratios $b^2/a=1665\,\mathrm{GPa}\,\mathrm K$, $b^2/Dq_*^4=6.28\,\mathrm{GPa}$, and $b\sqrt{-g/\tilde u}=14.58\,\mathrm{GPa}$ with $C_0=(71.14-(0.010426\times T)/\mathrm K)\,\mathrm{GPa}$. The resulting fit the thin solid black line in Fig.~\ref{fig:data}.
+\end{widetext}
+
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