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| -rw-r--r-- | main.tex | 53 |
1 files changed, 28 insertions, 25 deletions
@@ -320,16 +320,17 @@ defined in \eqref{eq:sus_def} is most readily interpreted as proportional to the two-point connected correlation function $\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$. -The strain stiffness is given in a similar way to the inverse susceptibility: we -must trace over $\eta$ and take the second variation of the resulting free -energy functional of $\epsilon$. Extremizing over $\eta$ yields +The strain stiffness is given in a similar way to the inverse susceptibility: +we must trace over $\eta$ and take the second variation of the resulting +effective free energy functional of $\epsilon$. Extremizing over $\eta$ yields \begin{equation} 0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}= \frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}-b\epsilon_\X(x), \label{eq:implicit.eta} \end{equation} -which implicitly gives $\eta_\star[\epsilon]$, the optimized \op\ conditioned on the strain. Since $\eta_\star$ is a functional of $\epsilon_\X$ -alone, only the stiffness $\lambda_\X$ is modified from its bare value $C_\X$. +which implicitly gives $\eta_\star[\epsilon]$, the optimized \op\ conditioned +on the strain. Since $\eta_\star$ is a functional of $\epsilon_\X$ +alone, only the stiffness $\lambda_\X$ can be modified from its bare value $C_\X$. Though this differential equation for $\eta_*$ cannot be solved explicitly, we can make use of the inverse function theorem. First, denote by $\eta_\star^{-1}[\eta]$ the inverse functional of $\eta_\star$ implied by @@ -372,7 +373,7 @@ the second variation \end{equation} \end{widetext} The strain stiffness is given by the second variation evaluated at the -extremized solution $\langle\epsilon\rangle$. To calculate it, note that +extremized strain $\langle\epsilon\rangle$. To calculate it, note that evaluating the second variation of $F_\op$ in \eqref{eq:inv.func} at $\langle\epsilon\rangle$ (or $\eta_\star(\langle\epsilon\rangle)=\langle\eta\rangle$) yields @@ -392,8 +393,8 @@ the result, we finally arrive at \end{equation} Though not relevant here, this result generalizes to multicomponent \op s. At $q=0$, which is where the stiffness measurements used here were taken, this -predicts a cusp in the strain stiffness of the form $|\Delta\tilde -r|^\gamma$ for $\gamma=1$. +predicts a cusp in the static strain stiffness $\lambda_\X(0)$ of the form +$|\Delta\tilde r|^\gamma$ for $\gamma=1$. \begin{figure}[htpb] \centering \includegraphics[width=\columnwidth]{fig-stiffnesses} @@ -406,19 +407,21 @@ r|^\gamma$ for $\gamma=1$. \end{figure} \Rus\ experiments \cite{ghosh_single-component_nodate} yield the strain -stiffness for various components of the strain; this data is shown in Figure -\ref{fig:data}. The $\Btg$ stiffness doesn't appear to have any response to -the presence of the transition, exhibiting the expected linear stiffening with -a low-temperature cutoff \cite{varshni_temperature_1970}. 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 $C_\Bog^{(11)}$ 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. +stiffness tensor; the data broken into the irrep components defined in +\eqref{eq:strain-components} is shown in Figure \ref{fig:data}. The $\Btg$ +stiffness doesn't appear to have any response to the presence of the +transition, exhibiting the expected linear stiffening with a low-temperature +cutoff \cite{varshni_temperature_1970}. 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 +$C_\Bog^{(11)}$ 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 quantitatively consistent in +the high temperature phase. \begin{figure}[htpb] \includegraphics[width=\columnwidth]{fig-fit} @@ -438,7 +441,7 @@ shown in Figure \ref{fig:fit}. The data and theory appear consistent. We have seen that the mean-field theory of a $\Bog$ \op\ recreates the topology of the \ho\ phase diagram and the temperature dependence of the $\Bog$ strain -stiffness at zero pressure. There are several implications of this theory. First, +stiffness at zero pressure. This theory has several other physical implications. First, the association of a modulated $\Bog$ order with the \ho\ phase implies a \emph{uniform} $\Bog$ order associated with the \afm\ phase, and moreover a uniform $\Bog$ strain of magnitude $\langle\epsilon_\Bog\rangle^2=b^2\tilde @@ -446,8 +449,8 @@ r/4uC_\Bog^2$, which corresponds to an orthorhombic phase. Orthorhombic symmetry breaking was recently detected in the \afm\ phase of \urusi\ using x-ray diffraction, a further consistency of this theory with the phenomenology of \urusi\ \cite{choi_pressure-induced_2018}. Second, as the Lifshitz point is -approached from low pressure this theory predicts the modulation wavevector -$q_*$ should continuously vanish. Far from the Lifshitz point we expect the +approached from low pressure, this theory predicts that the modulation wavevector +$q_*$ should vanish continuously. Far from the Lifshitz point we expect the wavevector to lock into values commensurate with the space group of the lattice, and moreover that at zero pressure, where the \rus\ data here was collected, the half-wavelength of the modulation should be commensurate with @@ -471,7 +474,7 @@ an ambient pressure calibration for the lattice constant. The second discrepancy is the onset of orthorhombicity at higher temperatures than the onset of \afm. Susceptibility data sees no trace of another phase transition at these higher temperatures \cite{inoue_high-field_2001}, and therefore we don't -in fact expect there to be one. We do expect that this could be due to the high +expect there to be one. We do expect that this could be due to the high energy nature of x-rays as an experimental probe: orthorhombic fluctuations could appear at higher temperatures than the true onset of an orthorhombic phase. |