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@@ -1,3 +1,4 @@ + \documentclass[aps,prl,reprint]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,graphicx,upgreek,amssymb} @@ -39,16 +40,27 @@ \begin{document} -\title{\urusi mft} +\title{Elastic properties of \urusi are reproduced by modulated $\Bog$ order} \author{Jaron Kent-Dobias} \author{Michael Matty} \author{Brad Ramshaw} -\affiliation{Laboratory of Atomic \& Solid State Physics, Cornell University, Ithaca, NY, USA} +\affiliation{ + Laboratory of Atomic \& Solid State Physics, Cornell University, + Ithaca, NY, USA +} \date\today \begin{abstract} - blah blah blah its-a abstract + We develop a phenomenological theory for the elastic response of materials + with a \Dfh\ point group through phase transitions. The physics is + generically that of Lifshitz points, with disordered, uniform ordered, and + modulated ordered phases. Several experimental features of \urusi are + reproduced when the order parameter has $\Bog$ symmetry: the topology of the + temperature--pressure phase diagram, the response of the strain stiffness + tensor above the hidden-order transition, and the strain response in the + antiferromagnetic phase. In this scenario, the hidden order is a version of + the high-pressure antiferromagnetic order modulated along the symmetry axis. \end{abstract} \maketitle @@ -104,38 +116,58 @@ of an OP coupled to strain and the effect that the OP symmetry has on the elasti 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 +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} \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth} \epsilon_\Aog^{(2)}=\epsilon_{33} \\ \epsilon_\Bog^{(1)}=\epsilon_{11}-\epsilon_{22} && - \epsilon_\Btg^{(1)}=\epsilon_{12} \\ - \epsilon_\Eg^{(1)} =\{\epsilon_{11},\epsilon_{22}\}. + \epsilon_\Btg^{(1)}=2\epsilon_{12} \\ + \epsilon_\Eg^{(1)}=2\{\epsilon_{11},\epsilon_{22}\}. \end{aligned} \end{equation} -Next, all quadratic combinations of these irreducible strains that transform like $\Aog$ are included in the free energy as +Next, all quadratic combinations of these irreducible strains that transform +like $\Aog$ are included in the free energy as \begin{equation} f_\e=\frac12\sum_\X\lambda_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)}, \end{equation} -where the sum is over irreducible representations of the point group and the $\lambda_\X^{(ij)}$ are +where the sum is over irreducible representations of the point group and the +stiffnesses $\lambda_\X^{(ij)}$ are \begin{equation} \begin{aligned} &\lambda_{\Aog}^{(11)}=\tfrac12(\lambda_{1111}+\lambda_{1122}) && \lambda_{\Aog}^{(22)}=\lambda_{3333} \\ &\lambda_{\Aog}^{(12)}=\lambda_{1133} && \lambda_{\Bog}^{(11)}=\tfrac12(\lambda_{1111}-\lambda_{1122}) \\ - &\lambda_{\Btg}^{(11)}=4\lambda_{1212} && - \lambda_{\Eg}^{(11)}=4\lambda_{1313}. + &\lambda_{\Btg}^{(11)}=\lambda_{1212} && + \lambda_{\Eg}^{(11)}=\lambda_{1313}. \end{aligned} \end{equation} -The interaction between strain and the order parameter $\eta$ depends on the representation of the point group that $\eta$ transforms as. If this representation is $\X$, then the most general coupling to linear order is +The interaction between strain and the order parameter $\eta$ depends on the +representation of the point group that $\eta$ transforms as. If this +representation is $\X$, then the most general coupling to linear order is \begin{equation} f_\i=b^{(i)}\epsilon_\X^{(i)}\eta \end{equation} -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. +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. -If the order parameter transforms like $\Aog$, odd terms are allow in its free energy and any transition will be abrupt and not continuous without tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic free energy density is +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 +tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic +free energy density is \begin{equation} \begin{aligned} f_\o=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2 @@ -145,29 +177,56 @@ If the order parameter transforms like $\Aog$, odd terms are allow in its free e \end{aligned} \label{eq:fo} \end{equation} -where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and $\nabla_\perp=\partial_3$ transforms like $\Atu$. We'll take $D_\parallel=0$ since this does not affect the physics at hand. Neglecting interaction terms higher than quadratic order, the only strain relevant to the problem is $\epsilon_\X$, and this can be traced out of the problem exactly, since +where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and +$\nabla_\perp=\partial_3$ transforms like $\Atu$. We'll take $D_\parallel=0$ +since this does not affect the physics at hand. Neglecting interaction terms +higher than quadratic order, the only strain relevant to the problem is +$\epsilon_\X$, and this can be traced out of the problem exactly, since \begin{equation} - 0=\frac{\delta F}{\delta\epsilon_{\X i}(x)}=\lambda_\X\epsilon_{\X i}(x)+\frac12b\eta_i(x) + 0=\frac{\delta F}{\delta\epsilon_{\X i}(x)}=\lambda_\X\epsilon_{\X i}(x) + +\frac12b\eta_i(x) \end{equation} -gives $\epsilon_\X(x)=-(b/2\lambda_\X)\eta(x)$. Upon substitution into the free energy, tracing out $\epsilon_\X$ has the effect of shifting $r$ in $f_\o$, with $r\to\tilde r=r-b^2/4\lambda_\X$. +gives $\epsilon_\X(x)=-(b/2\lambda_\X)\eta(x)$. Upon substitution into the free +energy, tracing out $\epsilon_\X$ has the effect of shifting $r$ in $f_\o$, +with $r\to\tilde r=r-b^2/4\lambda_\X$. -With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942, lifshitz_theory_1942-1}. For a scalar order parameter ($\Bog$ or $\Btg$) it is traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. For $\tilde r>0$ and $c_\perp>0$, or $\tilde r<c_\perp^2/4D_\perp$ and $c_\perp<0$, the only stable solution is $\eta_*=q_*=0$ and the system is unordered. For $\tilde r<0$ there are free energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has uniform order. For $c_\perp<0$ and $\tilde r<c_\perp^2/4D_\perp$ there are free energy minima for $q_*^2=-c_\perp/2D_\perp$ and +With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz +point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942, +lifshitz_theory_1942-1}. For a scalar order parameter ($\Bog$ or $\Btg$) it is +traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. For $\tilde +r>0$ and $c_\perp>0$, or $\tilde r<c_\perp^2/4D_\perp$ and $c_\perp<0$, the +only stable solution is $\eta_*=q_*=0$ and the system is unordered. For $\tilde +r<0$ there are free energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and +this system has uniform order. For $c_\perp<0$ and $\tilde +r<c_\perp^2/4D_\perp$ there are free energy minima for +$q_*^2=-c_\perp/2D_\perp$ and \begin{equation} - \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u}=\frac{\tilde r_c-\tilde r}{3u} + \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u} + =\frac{\tilde r_c-\tilde r}{3u} \end{equation} -with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The transition between the uniform and modulated orderings is abrupt for a scalar field and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde r/5}$. For a vector order parameter ($\Eg$) we must also allow a relative phase between the two components of the field. In this case the uniform ordered phase is only stable for $c_\perp>0$, and the modulated phase is now characterized by helical order with $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$ and +with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The +transition between the uniform and modulated orderings is abrupt for a scalar +field and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde r/5}$. For a +vector order parameter ($\Eg$) we must also allow a relative phase between the +two components of the field. In this case the uniform ordered phase is only +stable for $c_\perp>0$, and the modulated phase is now characterized by helical +order with $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$ and \begin{equation} - \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{16D_\perp u}=\frac{\tilde r_c-\tilde r}{4u} + \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{16D_\perp u} + =\frac{\tilde r_c-\tilde r}{4u} \end{equation} -The uniform--modulated transition is now continuous. The schematic phase diagrams for this model are shown in Figure \ref{fig:phases}. +The uniform--modulated transition is now continuous. The schematic phase +diagrams for this model are shown in Figure \ref{fig:phases}. \begin{figure}[htpb] \includegraphics[width=0.51\columnwidth]{phases_scalar}\hspace{-1.5em} \includegraphics[width=0.51\columnwidth]{phases_vector} - \caption{Schematic phase diagrams for this model. Solid lines denote - continuous transitions, while dashed lines indicated abrupt transitions. (a) - The phases for a scalar ($\Bog$ or $\Btg$). (b) The phases for a vector - ($\Eg$).} + \caption{ + Schematic phase diagrams for this model. Solid lines denote continuous + transitions, while dashed lines indicated abrupt transitions. (a) The + phases for a scalar ($\Bog$ or $\Btg$). (b) The phases for a vector + ($\Eg$). + } \label{fig:phases} \end{figure} @@ -176,7 +235,8 @@ The susceptibility is given by \begin{aligned} &\chi_{ij}^{-1}(x,x') =\frac{\delta^2F}{\delta\eta_i(x)\delta\eta_j(x')} \\ - &\quad=\Big[\big(\tilde r-c_\parallel\nabla_\parallel^2-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+4u\eta^2(x)\big)\delta_{ij} \\ + &\quad=\Big[\big(\tilde r-c_\parallel\nabla_\parallel^2 + -c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+4u\eta^2(x)\big)\delta_{ij} \\ &\qquad\qquad +8u\eta_i(x)\eta_j(x)\Big]\delta(x-x'), \end{aligned} \end{equation} @@ -184,27 +244,37 @@ or in Fourier space, \begin{equation} \begin{aligned} \chi_{ij}^{-1}(q) - &=8u\sum_{q'}\tilde\eta_i(q')\eta_j(-q')+\bigg(\tilde r+c_\parallel q_\parallel^2-c_\perp q_\perp^2 \\ - &\qquad+D_\perp q_\perp^4+4u\sum_{q'}\tilde\eta_k(q')\tilde\eta_k(-q')\bigg)\delta_{ij}. + &=8u\sum_{q'}\tilde\eta_i(q')\eta_j(-q')+\bigg(\tilde r + +c_\parallel q_\parallel^2-c_\perp q_\perp^2 \\ + &\qquad+D_\perp q_\perp^4+4u\sum_{q'}\tilde\eta_k(q')\tilde\eta_k(-q')\bigg) + \delta_{ij}. \end{aligned} \end{equation} Near the unordered--modulated transition this yields \begin{equation} \begin{aligned} - \chi(q) - &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2+|\tilde r-\tilde r_c|} \\ - &=\frac1{D_\perp}\frac{\xi_\perp^4}{1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2}, + \chi_{ij}(q) + &=\frac{\delta_{ij}}{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2 + +|\tilde r-\tilde r_c|} \\ + &=\frac{\delta_{ij}}{D_\perp}\frac{\xi_\perp^4} + {1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2}, \end{aligned} \label{eq:susceptibility} \end{equation} -with $\xi_\perp=(|\tilde r-\tilde r_c|/D_\perp)^{-1/4}$ and $\xi_\parallel=(|\tilde r-\tilde r_c|/c_\parallel)^{-1/2}$. +with $\xi_\perp=(|\tilde r-\tilde r_c|/D_\perp)^{-1/4}$ and +$\xi_\parallel=(|\tilde r-\tilde r_c|/c_\parallel)^{-1/2}$. -The elastic susceptibility (inverse stiffness) is given in the same way: we must trace over $\eta$ and take the second variation of the resulting free energy. Extremizing over $\eta$ yields +The elastic susceptibility (inverse stiffness) is given in the same way: we +must trace over $\eta$ and take the second variation of the resulting free +energy. Extremizing over $\eta$ yields \begin{equation} - 0=\frac{\delta F}{\delta\eta_i(x)}=\frac{\delta F_\o}{\delta\eta_i(x)}+\frac12b\epsilon_{\X i}(x), + 0=\frac{\delta F}{\delta\eta_i(x)}=\frac{\delta F_\o}{\delta\eta_i(x)} + +\frac12b\epsilon_{\X i}(x), \label{eq:implicit.eta} \end{equation} -which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this cannot be solved explicitly, we can make use of the inverse function theorem to write +which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this +cannot be solved explicitly, we can make use of the inverse function theorem to +write \begin{equation} \begin{aligned} \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1} @@ -214,7 +284,8 @@ which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this cann \end{aligned} \label{eq:inv.func} \end{equation} -It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the susceptibility of the material to $\epsilon_\X$ strain is given by +It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the +susceptibility of the material to $\epsilon_\X$ strain is given by \begin{widetext} \begin{equation} \begin{aligned} @@ -241,16 +312,13 @@ whose Fourier transform follows from \eqref{eq:inv.func} as \chi_{\X ij}(q)=\frac{\delta_{ij}}{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q). \label{eq:elastic.susceptibility} \end{equation} -At $q=0$, which is where the stiffness measurements used here were taken, this predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde r_c|^\gamma$ for $\gamma=1$. +At $q=0$, which is where the stiffness measurements used here were taken, this +predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde +r_c|^\gamma$ for $\gamma=1$. \begin{figure}[htpb] \centering - \includegraphics[width=0.49\columnwidth]{stiff_a11.pdf} - \includegraphics[width=0.49\columnwidth]{stiff_a22.pdf} - \includegraphics[width=0.49\columnwidth]{stiff_a12.pdf} - \includegraphics[width=0.49\columnwidth]{stiff_b1.pdf} - \includegraphics[width=0.49\columnwidth]{stiff_b2.pdf} - \includegraphics[width=0.49\columnwidth]{stiff_e.pdf} + \includegraphics[width=\columnwidth]{fig-stiffnesses} \caption{ Measurements of the effective strain stiffness as a function of temperature for the six independent components of strain from ultrasound. The vertical @@ -259,40 +327,88 @@ At $q=0$, which is where the stiffness measurements used here were taken, this p \label{fig:data} \end{figure} -We have seen that mean field theory predicts that whatever component of strain transforms like the order parameter will see a $t^{-1}$ softening in the stiffness that ends in a cusp. Ultrasound experiments \textbf{[Elaborate???]} yield the strain stiffness for various components of the strain; this data is shown in Figure \ref{fig:data}. The $\Btg$ and $\Eg$ stiffnesses don't appear to have any response to the presence of the transition, exhibiting the expected linear stiffening with a low-temperature cutoff \textbf{[What's this called? Citation?]}. The $\Bog$ stiffness has a dramatic response, softening over the course of roughly $100\,\K$. There is a kink in the curve right at the 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 high-temperature phase can be fit to the theory \eqref{eq:elastic.susceptibility}, with a linear background stiffness $\lambda_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in Figure \ref{fig:fit}. The data and theory appear consistent. +We have seen that mean field theory predicts that whatever component of strain +transforms like the order parameter will see a $t^{-1}$ softening in the +stiffness that ends in a cusp. Ultrasound experiments \textbf{[Elaborate???]} +yield the strain stiffness for various components of the strain; this data is +shown in Figure \ref{fig:data}. The $\Btg$ and $\Eg$ stiffnesses don't appear +to have any response to the presence of the transition, exhibiting the expected +linear stiffening with a low-temperature cutoff \textbf{[What's this called? +Citation?]}. The $\Bog$ stiffness has a dramatic response, softening over the +course of roughly $100\,\K$. There is a kink in the curve right at the +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 high-temperature phase can be fit to the theory +\eqref{eq:elastic.susceptibility}, with a linear background stiffness +$\lambda_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is +shown in Figure \ref{fig:fit}. The data and theory appear consistent. \begin{figure}[htpb] - \includegraphics[width=\columnwidth]{cusp} + \includegraphics[width=\columnwidth]{fig-fit} \caption{ Strain stiffness data for the $\Bog$ component of strain (solid) along with - a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$ (dashed). The fit gives $\lambda_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp q_*^4\simeq0.0038\,\K^{-1}$. + a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$ + (dashed). The fit gives + $\lambda_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, + $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp + q_*^4\simeq0.0038\,\K^{-1}$. } \label{fig:fit} \end{figure} -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. 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 $k_BT\chi(x,x')$, +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. 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 +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 \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} + 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}, + \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 $\mathcal I$ is defined by +where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I$ +is defined by \begin{equation} - \mathcal I(x)=\frac1\pi\int_{-\infty}^\infty dy\,\frac{\sin\tfrac y2}y\bigg(\frac1{1+(y^2-x^2)^2}-\frac{K_1(\sqrt{1+(y^2-x^2)^2})}{\sqrt{1+(y^2-x^2)^2}}\bigg) + \mathcal I(x)=\frac1\pi\int_{-\infty}^\infty dy\,\frac{\sin\tfrac y2}y + \bigg(\frac1{1+(y^2-x^2)^2} + -\frac{K_1(\sqrt{1+(y^2-x^2)^2})}{\sqrt{1+(y^2-x^2)^2}}\bigg) \end{equation} For large argument, $\mathcal I(x)\sim x^{-4}$, yielding \begin{equation} t_\G^{9/4}\sim\frac{2k_B}{\pi\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}^5q_*^4} \end{equation} -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 \textbf{[Why???]}. 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. +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 +\textbf{[Why???]}. 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. \begin{acknowledgements} |