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| -rw-r--r-- | main.tex | 8 |
1 files changed, 4 insertions, 4 deletions
@@ -237,7 +237,7 @@ $f_\op$ with $r\to\tilde r=r-b^2/2C_\X$. \label{fig:phases} \end{figure} -With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz +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 one-component \op\ ($\Bog$ or $\Btg$) it is traditional to make the field ansatz @@ -266,7 +266,7 @@ reproduce the physics of \ho, which has an abrupt transition between \ho\ and \a We will now proceed to derive the \emph{effective strain stiffness tensor} $\lambda$ that results from the coupling of strain to the \op. The ultimate result, found in \eqref{eq:elastic.susceptibility}, is that $\lambda_\X$ -differs from its bare value $C_\X$ only for the symmetry $\X$ of the \op. To +differs from its bare value $C_\X$ only for the symmetry $\X$ of the \op. Moreover, the effective strain stiffness does not vanish at the unordered--modulated transition, but exhibits a \emph{cusp}. To show this, we will first compute the susceptibility of the \op, which will both be demonstrative of how the stiffness is calculated and prove useful in expressing the functional form of the stiffness. Then we will compute the @@ -381,7 +381,7 @@ $\eta_\star(\langle\epsilon\rangle)=\langle\eta\rangle$) yields =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}. +where $\chi^\recip$ is the \op\ susceptibility given by \eqref{eq:sus_def}. Upon substitution into \eqref{eq:big.boy} and taking the Fourier transform of the result, we finally arrive at \begin{equation} @@ -481,7 +481,7 @@ field theory should break down sufficiently close to the critical point due to fluctuations, at the Ginzburg temperature \cite{hornreich_lifshitz_1980, ginzburg_remarks_1961}. Magnetic phase transitions tend to have Ginzburg temperature of order one. Our fit above gives $\xi_{\perp0}q_*=(D_\perp q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of $q_*\simeq\pi/a_3$ puts the bare correlation length $\xi_{\perp0}$ at about what one would expect for a generic magnetic transition. -The argeement of this data in the $t\sim0.1$--10 range with the mean field exponent suggests that this region is outside the Ginzburg region, but an experiment may begin to see deviations from mean field behavior within +The agreement of this data in the $t\sim0.1$--10 range with the mean field exponent suggests that this region is outside the Ginzburg region, but an experiment may begin to see deviations from mean field behavior within around several degrees Kelvin of the critical point. A \rus\ 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 |