redid Ginzburg criterion section

6 files changed, 162 insertions, 45 deletions

diff --git a/fig-fit.gplot b/fig-fit.gplot
index f35fde7..7f33134 100644
--- a/fig-fit.gplot
+++ b/fig-fit.gplot
@@ -1,4 +1,9 @@
 
+cc1 = "#5e81b5"
+cc2 = "#e19c24"
+cc3 = "#8fb032"
+cc4 = "#eb6235"
+
 set terminal epslatex size 8.68cm, 5.5cm standalone
 set output "fig-fit.tex"
 
@@ -12,16 +17,15 @@ set xrange [-25:300]
 set format x '\scriptsize $%g$'
 set format y '\scriptsize $%g$'
 
-set arrow 1 from 17.26,graph 0 to 17.26,graph 1 nohead lw 3 lc 'red' dt 3
+set arrow 1 from 17.26,graph 0 to 17.26,graph 1 nohead lw 4 lc rgb cc2
 
 set xlabel '\footnotesize $T / \mathrm K$'
 set ylabel '\footnotesize $\tilde\lambda_{\mathrm B_{1\mathrm g}} / \mathrm{GPa}$' offset 2
 set yrange [65.05:65.7]
 
-plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc 'medium-blue'
-
-lam(T) = 71.1083 - 0.0103972 * T
+lam(T) = 71.1212 - 0.0104105 * T
 
-plot lam(x) / (1 + 6.24582 / (1 + 0.0037839 * abs(x - 17.26)) / lam(x)) dt 4 lc 'goldenrod' lw 4
+plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \
+      lam(x) / (1 + 6.2662 / (1 + 0.00378087 * abs(x - 17.26)) / lam(x)) dt 3 lw 4 lc rgb cc4
 
 
diff --git a/fig-fit.pdf b/fig-fit.pdf
index 592942e..6064f6f 100644
--- a/fig-fit.pdf
+++ b/fig-fit.pdf
diff --git a/fig-stiffnesses.gplot b/fig-stiffnesses.gplot
index 15ae95c..84aab79 100644
--- a/fig-stiffnesses.gplot
+++ b/fig-stiffnesses.gplot
@@ -1,4 +1,9 @@
 
+cc1 = "#5e81b5"
+cc2 = "#e19c24"
+cc3 = "#8fb032"
+cc4 = "#eb6235"
+
 set terminal epslatex size 8.68cm, 7.5cm standalone
 set output "fig-stiffnesses.tex"
 
@@ -6,7 +11,7 @@ set multiplot  layout 3, 2  margins 0.185, 0.97, 0.15, 0.998  spacing 0.13, 0.01
 
 set nokey
 set xrange [-25:300]
-set arrow 1 from 17.26,graph 0 to 17.26,graph 1 nohead lw 3 lc 'red' dt 3
+set arrow 1 from 17.26,graph 0 to 17.26,graph 1 nohead lw 4 lc rgb cc2
 
 
 set format x ""
@@ -17,28 +22,28 @@ set format y '\scriptsize $%.1f$'
 set yrange [217.5:220.75]
 set ytics 218,1,220
 set title '\footnotesize (a) $\tilde\lambda^{(11)}_{\mathrm{A_{1\mathrm g}}}$' offset -1,-5.2
-plot "data/c11pc12.dat" using 1:(100 * $2) with lines lw 3
+plot "data/c11pc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3
 
 set ylabel ''
 
 set yrange [306:317]
 set ytics 308,2,316
 set title '\footnotesize (b) $\tilde\lambda^{(22)}_{\mathrm{A_{1\mathrm g}}}$' offset -1,-5.2
-plot "data/c33.dat"     using 1:(100 * $2) with lines lw 3
+plot "data/c33.dat"     using 1:(100 * $2) with lines lw 3 lc rgb cc3
 
 set ylabel '\footnotesize $\tilde\lambda / \mathrm{GPa}$'
 
 set yrange [112.6:114]
 set ytics 112.8,0.3,113.9
 set title '\footnotesize (d) $\tilde\lambda^{(12)}_{\mathrm{A_{1\mathrm g}}}$' offset -1,-5.2
-plot "data/c13.dat"     using 1:(100 * $2) with lines lw 3
+plot "data/c13.dat"     using 1:(100 * $2) with lines lw 3 lc rgb cc3
 
 set ylabel ''
 
 set yrange [65.05:65.7]
 set ytics 62.1,0.1,65.6
 set title '\footnotesize (c) $\tilde\lambda^{(11)}_{\mathrm{B_{1\mathrm g}}}$' offset -1,-5.2
-plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3
+plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3
 
 set format x '\scriptsize $%.0f$'
 set ylabel '\footnotesize $\tilde \lambda / \mathrm{GPa}$'
@@ -47,12 +52,12 @@ set xlabel '\footnotesize $T / \mathrm K$'
 set yrange [140:145]
 set ytics 141,1,144
 set title '\footnotesize (e) $\tilde\lambda^{(11)}_{\mathrm{B_{2\mathrm g}}}$' offset -1,-5.2
-plot "data/c66.dat"     using 1:(100 * $2) with lines lw 3
+plot "data/c66.dat"     using 1:(100 * $2) with lines lw 3 lc rgb cc3
 
 set ylabel ''
 
 set yrange [101:106]
 set ytics 102,1,105
 set title '\footnotesize (f) $\tilde\lambda^{(11)}_{\mathrm{E_{\mathrm g}}}$' offset -1,-5.2
-plot "data/c44.dat"     using 1:(100 * $2) with lines lw 3
+plot "data/c44.dat"     using 1:(100 * $2) with lines lw 3 lc rgb cc3
 
diff --git a/fig-stiffnesses.pdf b/fig-stiffnesses.pdf
index 9baa3d1..de3b765 100644
--- a/fig-stiffnesses.pdf
+++ b/fig-stiffnesses.pdf
diff --git a/hidden_order.bib b/hidden_order.bib
index 3b779ed..2194a59 100644
--- a/hidden_order.bib
+++ b/hidden_order.bib
@@ -92,6 +92,22 @@
   file = {/home/pants/Zotero/storage/FQWHY9TF/Hornreich - 1980 - The Lifshitz point Phase diagrams and critical be.pdf}
 }
 
+@article{guida_critical_1998,
+  title = {Critical Exponents of the {{N}}-Vector Model},
+  volume = {31},
+  issn = {0305-4470},
+  abstract = {Recently the series for two renormalization group functions (corresponding to the anomalous dimensions of the fields \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img1.gif] and \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img2.gif] ) of the three-dimensional \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img3.gif] field theory have been extended to next order (seven loops) by Murray and Nickel. We examine the influence of these additional terms on the estimates of critical exponents of the N -vector model, using some new ideas in the context of the Borel summation techniques. The estimates have slightly changed, but remain within the errors of the previous evaluation. Exponents such as \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img4.gif] (related to the field anomalous dimension), which were poorly determined in the previous evaluation of Le Guillou-Zinn-Justin, have seen their apparent errors significantly decrease. More importantly, perhaps, summation errors are better determined. The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values. Finally, because an error has been discovered in the last order of the published \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img5.gif] expansions (order \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img6.gif] ), we have also re-analysed the determination of exponents from the \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img7.gif] -expansion. The conclusion is that the general agreement between \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img7.gif] -expansion and three-dimensional series has improved with respect to Le Guillou-Zinn-Justin.},
+  language = {en},
+  number = {40},
+  journal = {Journal of Physics A: Mathematical and General},
+  doi = {10.1088/0305-4470/31/40/006},
+  author = {Guida, R. and {Zinn-Justin}, J.},
+  year = {1998},
+  keywords = {_tablet},
+  pages = {8103},
+  file = {/home/pants/Zotero/storage/K468APXL/Guida and Zinn-Justin - 1998 - Critical exponents of the N-vector model.pdf}
+}
+
 @article{varshni_temperature_1970,
   title = {Temperature {{Dependence}} of the {{Elastic Constants}}},
   volume = {2},
@@ -514,4 +530,96 @@ thermodynamics from a hidden order parameter.},
   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}
 }
 
+@article{selke_monte_1978,
+  title = {Monte Carlo Calculations near a Uniaxial {{Lifshitz}} Point},
+  volume = {29},
+  issn = {1431-584X},
+  abstract = {The Monte Carlo method is applied to a threedimensional Ising model with nearest neighbour ferromagnetic interactions and next nearest neighbour antiferromagnetic interactions along one axis only. Special emphasis is given to the critical behaviour near the Lifshitz point.},
+  language = {en},
+  number = {2},
+  journal = {Zeitschrift f{\"u}r Physik B Condensed Matter},
+  doi = {10.1007/BF01313198},
+  author = {Selke, Walter},
+  month = jun,
+  year = {1978},
+  keywords = {Complex System,Neural Network,Spectroscopy,State Physics,Monte Carlo Method},
+  pages = {133-137},
+  file = {/home/pants/Zotero/storage/5NRZEWP8/Selke_1978_Monte carlo calculations near a uniaxial Lifshitz point.pdf}
+}
+
+@article{hornreich_critical_1975-1,
+  title = {Critical Exponents at a {{Lifshitz}} Point to {{O}}(1/n)},
+  volume = {55},
+  issn = {0375-9601},
+  abstract = {The critical exponents at a general Lifshitz point are calculated in the spherical model limit, as are those of an isotropic Lifshitz point to O(1/n). These results are in exact agreement in the overlap region with those obtained using an {$\epsilon$}-expansion.},
+  number = {5},
+  journal = {Physics Letters A},
+  doi = {10.1016/0375-9601(75)90465-X},
+  author = {Hornreich, R. M. and Luban, M. and Shtrikman, S.},
+  month = dec,
+  year = {1975},
+  pages = {269-270},
+  file = {/home/pants/Zotero/storage/RED39SK4/Hornreich et al_1975_Critical exponents at a Lifshitz point to O(1-n).pdf;/home/pants/Zotero/storage/X8UJ5CHZ/037596017590465X.html}
+}
+
+@article{hornreich_exactly_1977,
+  title = {Exactly Solvable Model Exhibiting a Multicritical Point},
+  volume = {86},
+  issn = {0378-4371},
+  abstract = {A hypercubic d-dimensional lattice of spins with nearest neighbor ferromagnetic coupling and next nearest neighbor antiferromagnetic coupling along a single axis is studied in the spherical model limit (n\textrightarrow{$\infty$}) and is found to exhibit a multicritical point of the uniaxial Lifshitz type. The shape of the {$\lambda$} line is calculated explicitly in the vicinity of the multicritical point, and analytic expressions are given for the shift exponent {$\psi$}(d) and its amplitudes A{$\pm$}(d). The amplitude A\_(d) changes sign for d = 3.},
+  number = {2},
+  journal = {Physica A: Statistical Mechanics and its Applications},
+  doi = {10.1016/0378-4371(77)90042-5},
+  author = {Hornreich, R. M. and Luban, Marshall and Shtrikman, S.},
+  month = feb,
+  year = {1977},
+  pages = {465-470},
+  file = {/home/pants/Zotero/storage/5MFN7M9Z/Hornreich et al_1977_Exactly solvable model exhibiting a multicritical point.pdf;/home/pants/Zotero/storage/CZNV72TI/0378437177900425.html}
+}
+
+@article{nicoll_renormalization_1976,
+  title = {Renormalization Group Calculation for Critical Points of Higher Order with General Propagator},
+  volume = {58},
+  issn = {0375-9601},
+  abstract = {We give first order perturbation results for the critical point exponents at order O critical points with anisotropic propagators. The exponent {$\eta$} is calculated to second order for isotropic propagators, and all O; 1/n expansion results are given for O = 2.},
+  number = {1},
+  journal = {Physics Letters A},
+  doi = {10.1016/0375-9601(76)90527-2},
+  author = {Nicoll, J. F. and Tuthill, G. F. and Chang, T. S. and Stanley, H. E.},
+  month = jul,
+  year = {1976},
+  pages = {1-2},
+  file = {/home/pants/Zotero/storage/55AS69UD/Nicoll et al_1976_Renormalization group calculation for critical points of higher order with.pdf;/home/pants/Zotero/storage/L6WH4D36/0375960176905272.html}
+}
+
+@article{nicoll_onset_1977,
+  title = {Onset of Helical Order},
+  volume = {86-88},
+  issn = {0378-4363},
+  abstract = {Renormalization group methods are used to describe systems which model critical phenomena at the onset of helical order. This onset is marked by a change in the ``bare propagator'' used in perturbation theory from a k2-dependence to a more general form. We consider systems which in the non-helical region exhibit O simultaneously critical phases. Results are given to first order in an {$\epsilon$}-expansion. For the isotropic case of k2L dependence and O = 2, we give {$\eta$} to first order in 1/n for d- {$\leqslant$} d {$\leqslant$} d+ where d+- are upper and lower borderline dimensions.},
+  journal = {Physica B+C},
+  doi = {10.1016/0378-4363(77)90620-9},
+  author = {Nicoll, J. F. and Tuthill, G. F. and Chang, T. S. and Stanley, H. E.},
+  month = jan,
+  year = {1977},
+  pages = {618-620},
+  file = {/home/pants/Zotero/storage/ZLV5YFH6/Nicoll et al_1977_Onset of helical order.pdf;/home/pants/Zotero/storage/84ZZT6CN/0378436377906209.html}
+}
+
+@article{garel_commensurability_1976,
+  title = {Commensurability Effects on the Critical Behaviour of Systems with Helical Ordering},
+  volume = {9},
+  issn = {0022-3719},
+  abstract = {The critical behaviour of an m-component spin system with helical ordering is studied using the renormalization group method to order epsilon 2 (where epsilon =4-d). For m=1 and 2 the system is equivalent to a 2m-vector model. For m=3 a first-order transition is expected. The effect of the commensurability of the helical structure with the lattice has been considered and is shown in certain situations to change the order of the transition.},
+  language = {en},
+  number = {10},
+  journal = {Journal of Physics C: Solid State Physics},
+  doi = {10.1088/0022-3719/9/10/001},
+  author = {Garel, T. and Pfeuty, P.},
+  month = may,
+  year = {1976},
+  pages = {L245--L249},
+  file = {/home/pants/Zotero/storage/34KTXA6I/Garel_Pfeuty_1976_Commensurability effects on the critical behaviour of systems with helical.pdf}
+}
+
 
diff --git a/main.tex b/main.tex
index a245429..3f9c01f 100644
--- a/main.tex
+++ b/main.tex
@@ -33,7 +33,7 @@
 \def\m{\text m}
 \def\K{\text K}
 \def\GPa{\text{GPa}}
-\def\A{\text{\c A}}
+\def\A{\text{\r A}}
 
 % Other
 \def\G{\text G} % Ginzburg
@@ -349,7 +349,7 @@ r_c|^\gamma$ for $\gamma=1$.
   \caption{
     Measurements of the effective strain stiffness as a function of temperature
     for the six independent components of strain from ultrasound. The vertical
-    dashed lines show the location of the hidden order transition.
+    lines show the location of the hidden order transition.
   }
   \label{fig:data}
 \end{figure}
@@ -391,43 +391,43 @@ fluctuations are indeed negligible. This is typically done by computing the
 Ginzburg criterion \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, or since the correlation function is
-$k_BT\chi(x,x')$,
-\begin{equation}
-  V_\xi^{-1}k_BT\int_{V_\xi}d^3x\,\chi(x,0)
-  =\langle\delta\eta^2\rangle_{V_\xi}
-  \lesssim\frac12\eta_*^2=\frac{|\Delta\tilde r|}{6u}
-\end{equation}
-with $V_\xi$ the correlation volume, which we will take to be a cylinder of
-radius $\xi_\parallel/2$ and height $\xi_\perp$. Upon substitution of
-\eqref{eq:susceptibility} and using the jump in the specific heat at the
-transition from
+sized box to the magnitude of the field. In the modulated phase the spatially
+averaged magnitude is zero, and so we will instead compare fluctuations in the
+\emph{amplitude} at $q_*$ to the magnitude of that amplitude. Defining the
+field $\alpha$ by $\eta(x)=\alpha(x)e^{-iq_*x_3}$, it follows that in the
+modulated phase $\alpha(x)=\alpha_0$ for $\alpha_0^2=|\delta \tilde r|/4u$. In
+the modulated phase, the $q$-dependant fluctuations in $\alpha$ are given by
+\[
+  G_\alpha(q)=k_BT\chi_\alpha(q)=\frac1{c_\parallel q_\parallel^2+D_\perp(4q_*^2q_\perp^2+q_\perp^4)+2|\delta r|},
+\]
+An estimate of the Ginzburg criterion is then given by the temperature at which
+$V_\xi^{-1}\int_{V_\xi}G(0,x)\,dx=\langle\delta\alpha^2\rangle\simeq\langle\alpha\rangle^2=\alpha_0^2$,
+where $V_\xi=\xi_\perp\xi_\parallel^2$ is a correlation volume. The parameter $u$
+can be replaced in favor of the jump in the specific heat at the transition
+using
 \begin{equation}
   c_V=-T\frac{\partial^2f}{\partial T^2}
-    =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c,\end{cases}
-\end{equation}
-this expression can be brought to the form
-\begin{equation}
-  \frac{2k_B}{\pi\Delta c_V\xi_{\perp0}\xi_{\parallel0}^2}
-    \mathcal I(\xi_{\perp0} q_*|t|^{-1/4})
-  \lesssim |t|^{13/4},
-\end{equation}
-where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I(x)\sim x^{-4}$ for large $x$, yielding
-\begin{equation}
-  t_\G^{9/4}\sim\frac{2k_B}{\pi\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}^5q_*^4}
+    =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c.\end{cases}
 \end{equation}
+The integral over the correlation function $G_\alpha$ can be preformed up to one integral analytically using a Gaussian-bounded correlation volume, yielding $\langle\delta\tilde r\rangle^2\simeq k_BTV_\xi^{-1}\delta r^{-1}\mathcal I(\xi_\perp q_*)$ for
+\[
+  \mathcal I(x)=-2^{3/2}\pi^{5/2}\int dy\,e^{[2+(4x^2-1)y^2+y^4]/2}
+\mathop{\mathrm{Ei}}\big(-(1+2x^2y^2+\tfrac12y^4)\big)
+\]
+This gives a transcendental equation
+\[
+  \mathcal I(\xi_{\perp0}q_*\delta t_\G)\simeq3k_B^{-1}\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}\delta t_\G^{3/4}.
+\]
 Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$
 \cite{fisher_specific_1990}, and our fit above gives $\xi_{\perp0}q_*=(D_\perp
-q_*^4/aT_c)^{1/4}\sim2$. We have reason to believe that at zero pressure, very
-far from the Lifshitz point, $q_*$ is roughly the inverse lattice spacing
+q_*^4/aT_c)^{1/4}\simeq2$. We have reason to believe that at zero pressure, very
+far from the Lifshitz point, the half-wavelength of the modulation should be commensurate with the lattice, giving $q_*\simeq0.328\,\A^{-1}$ 
 \cite{meng_imaging_2013}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$,
-we find $t_\G\sim0.04$, so that an experiment would need to be within
-$\sim1\,\K$ to detect a deviation from mean field behavior. An ultrasound
-experiment able to capture data over several decades within this vicinity of
-$T_c$ may be able to measure a cusp with $|t|^\gamma$ for
-$\gamma=\text{\textbf{???}}$, the empirical exponent \textbf{[Citation???]}.
-Our analysis has looked at behavior for $T-T_c>1\,\K$, and so it remains
-self-consistent.
+we find $\delta t_\G\sim0.4$, though this estimate is sensitive to uncertainty in $\xi_{\parallel0}$: varying our estimate for $\xi_{\parallel0}$ over one order of magnitude yields changes in $\delta t_\G$ over nearly four orders of magnitude.
+The estimate here predicts that an experiment may begin to see deviations from
+mean field behavior within around $5\,\K$ of the critical point. An ultrasound
+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 order parameter is $\mathrm O(2)$ \cite{garel_commensurability_1976}. Our work here appears self--consistent, given that our fit is mostly concerned with temperatures farther than this 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.
 
 There are two apparent discrepancies between the phase diagram presented in 
 \cite{choi_pressure-induced_2018} and that predicted by our mean field theory. The first is the apparent