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| -rw-r--r-- | main.tex | 51 |
1 files changed, 25 insertions, 26 deletions
@@ -166,7 +166,7 @@ The interaction between strain and an \op\ $\eta$ depends on the representation of the point group that $\eta$ transforms as. If this representation is $\X$, the most general coupling to linear order is \begin{equation} - f_\i=b^{(i)}\epsilon_\X^{(i)}\eta. + f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta. \end{equation} If the representation $\X$ is not present in the strain there can be no linear coupling, and the effect of the \op\ condensing at a continuous phase @@ -207,14 +207,14 @@ The only strain relevant to the \op\ is $\epsilon_\X$, which can be traced out of the problem exactly in mean field theory. Extremizing with respect to $\epsilon_\X$, \begin{equation} - 0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_{\X i}(x)}\bigg|_{\epsilon=\epsilon_\star}=C_\X\epsilon^\star_{\X i}(x) - +\frac12b\eta_i(x) + 0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_\X(x)}\bigg|_{\epsilon=\epsilon_\star}=C_\X\epsilon^\star_\X(x) + -b\eta(x) \end{equation} gives the optimized strain conditional on the \op\ as -$\epsilon_\X^\star[\eta](x)=-(b/2C_\X)\eta(x)$ and $\epsilon_\Y^\star[\eta]=0$ +$\epsilon_\X^\star[\eta](x)=(b/C_\X)\eta(x)$ and $\epsilon_\Y^\star[\eta]=0$ for all other $\Y$. Upon substitution into the free energy, the resulting effective free energy $F[\eta,\epsilon_\star[\eta]]$ has a density identical to -$f_\op$ with $r\to\tilde r=r-b^2/4C_\X$. +$f_\op$ with $r\to\tilde r=r-b^2/2C_\X$. \begin{figure}[htpb] \includegraphics[width=\columnwidth]{phase_diagram_experiments} @@ -317,7 +317,7 @@ must trace over $\eta$ and take the second variation of the resulting free energy functional of $\epsilon$. Extremizing over $\eta$ yields \begin{equation} 0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}= - \frac12b\epsilon_\X(x)+\frac{\delta F_\op[\eta]}{\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 $\epsilon_\X$ @@ -328,7 +328,7 @@ $\eta_\star^{-1}[\eta]$ the inverse functional of $\eta_\star$ implied by \eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$ corresponding to each solution of \eqref{eq:implicit.eta} it receives. This we can immediately identify from \eqref{eq:implicit.eta} as -$\eta^{-1}_\star[\eta](x)=-2/b(\delta F_\op[\eta]/\delta\eta(x))$. Now, we use +$\eta^{-1}_\star[\eta](x)=b^{-1}(\delta F_\op[\eta]/\delta\eta(x))$. Now, we use the inverse function theorem to relate the functional reciprocal of the derivative of $\eta_\star[\epsilon]$ with respect to $\epsilon_\X$ to the derivative of $\eta^{-1}_\star[\eta]$ with respect to $\eta$, yielding @@ -336,7 +336,7 @@ derivative of $\eta^{-1}_\star[\eta]$ with respect to $\eta$, yielding \begin{aligned} \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip &=\frac{\delta\eta_\star^{-1}[\eta](x)}{\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]} - =-\frac2b\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]}. + =b^{-1}\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]}. \end{aligned} \label{eq:inv.func} \end{equation} @@ -347,19 +347,18 @@ the second variation \begin{equation} \begin{aligned} \frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} - &=C_\X\delta(x-x')+ - b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')} - +\frac12b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') \\ - &\quad+\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')}\frac{\delta^2F_\op[\eta]}{\delta\eta(x'')\delta\eta(x''')}\bigg|_{\eta=\eta_\star[\epsilon]} - +\int dx''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}\frac{\delta F_\op[\eta]}{\delta\eta(x'')}\bigg|_{\eta=\eta_\star[\epsilon]} \\ - &=C_\X\delta(x-x')+ - b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')} - -\frac12b\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')}\bigg(\frac{\partial\eta_\star[\epsilon](x'')}{\partial\epsilon_\X(x''')}\bigg)^\recip \\ - &=C_\X\delta(x-x')+ - b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')} - -\frac12b\int dx''\,\delta(x-x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x')} - =C_\X\delta(x-x')+ - \frac12b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}. + &=C_\X\delta(x-x')- + 2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')} + -b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') +\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}\frac{\delta F_\op[\eta]}{\delta\eta(x'')}\bigg|_{\eta=\eta_\star[\epsilon]}\\ + &\qquad\qquad+\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')}\frac{\delta^2F_\op[\eta]}{\delta\eta(x'')\delta\eta(x''')}\bigg|_{\eta=\eta_\star[\epsilon]} \\ + &=C_\X\delta(x-x')- + 2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')} + -b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') +\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}(b\epsilon_\X(x''))\\ + &\qquad\qquad+b\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')} \bigg(\frac{\partial\eta_\star[\epsilon](x'')}{\partial\epsilon_\X(x''')}\bigg)^\recip\\ + &=C_\X\delta(x-x')- + 2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')} + +b\int dx''\,\delta(x-x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x')} + =C_\X\delta(x-x')-b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}. \end{aligned} \label{eq:big.boy} \end{equation} @@ -371,7 +370,7 @@ $\langle\epsilon\rangle$ (or $\eta_\star(\langle\epsilon\rangle)=\langle\eta\rangle$) yields \begin{equation} \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip\bigg|_{\epsilon=\langle\epsilon\rangle} - =-\frac2b\chi^\recip(x,x')-\frac{b}{2C_\X}\delta(x-x'), + =b^{-1}\chi^\recip(x,x')+\frac{b}{C_\X}\delta(x-x'), \label{eq:recip.deriv.op} \end{equation} where $\chi^\recip$ is the \op\ susceptibilty given by \eqref{eq:sus_def}. @@ -379,8 +378,8 @@ Upon substitution into \eqref{eq:big.boy} and taking the Fourier transform of the result, we finally arrive at \begin{equation} \lambda_\X(q) - =C_\X-\frac b2\bigg(\frac2{b\chi(q)}+\frac b{2C_\X}\bigg)^{-1} - =C_\X\bigg(1+\frac{b^2}{4C_\X}\chi(q)\bigg)^{-1}. + =C_\X-b\bigg(\frac1{b\chi(q)}+\frac b{C_\X}\bigg)^{-1} + =C_\X\bigg(1+\frac{b^2}{C_\X}\chi(q)\bigg)^{-1}. \label{eq:elastic.susceptibility} \end{equation} Though not relevant here, this result generalizes to multicomponent \op s. At @@ -420,7 +419,7 @@ shown in Figure \ref{fig:fit}. The data and theory appear consistent. a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$ (dashed). The fit gives $C_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, - $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp + $b^2/D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp q_*^4\simeq0.0038\,\K^{-1}$. 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 @@ -435,7 +434,7 @@ stiffness at zero pressure. There are several implications of this theory. 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 $\langle\epsilon_\Bog\rangle^2=b^2\tilde -r/16uC_\Bog^2$, which corresponds to an orthorhombic phase. Orthorhombic +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 |