fixes and writing in the second half

1 files changed, 123 insertions, 78 deletions

diff --git a/main.tex b/main.tex
index bca1d8c..20f77d2 100644
--- a/main.tex
+++ b/main.tex
@@ -356,11 +356,12 @@ with $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}=\xi_{\perp0}|t|^{-1/4}$ and
 $\xi_\parallel=(|\Delta\tilde
 r|/c_\parallel)^{-1/2}=\xi_{\parallel0}|t|^{-1/2}$, where $t=(T-T_c)/T_c$ is
 the reduced temperature and $\xi_{\perp0}=(D_\perp/aT_c)^{1/4}$ and
-$\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths.
-Notice that the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde
-r|)^{-1}$ does not diverge at the unordered to modulated transition. Though it
-anticipates a transition with Curie--Weiss-like divergence at $\Delta\tilde
-r=-D_\perp q_*^4$, this is cut off with a cusp at $\Delta\tilde r=0$ \brad{this will all be clearer if you remind the reader that this is Tc, or the new renormalized Tc, or whatever it is}. 
+$\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths
+parallel and perpendicular to the axis of symmetry, respectively.  Notice that
+the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde r|)^{-1}$ does
+not diverge at the unordered to modulated transition. Though it anticipates a
+transition with Curie--Weiss-like divergence at $a(T-T_c)=\Delta\tilde
+r=-D_\perp q_*^4$, this is cut off with a cusp at $\Delta\tilde r=0$. 
 
 The elastic susceptibility, which corresponds with the reciprocal of the elastic modulus,
 is given in a similar way to the \op\ susceptibility: we must trace over $\eta$
@@ -434,116 +435,160 @@ the result, we finally arrive at
   =C^0_\X\bigg(1+\frac{b^2}{C^0_\X}\chi(q)\bigg)^{-1}.
   \label{eq:elastic.susceptibility}
 \end{equation}
-Though not relevant here, this result generalizes to multicomponent \op s.  At
-$q=0$, which is where the modulus measurements used here were taken, this
-predicts a cusp in the static elastic modulus $C_\X(0)$ of the form
-$|\Delta\tilde r|^\gamma$ for $\gamma=1$.  \brad{I think this last sentence, which is the point of the whole paper, needs to be expanded upon and emphasized. It needs to be clear that what we have done is consider a general OP of B1g or B2g type modulated along the c-axis. For a general Landau free energy, it will develop order at some finite q, but if you measure at q=0, which is what ultraound typically does, you still see "remnant" behaviour that cusps at the transition}
+Though not relevant here, this result generalizes to multicomponent \op s.
+
+What does \eqref{eq:elastic.susceptibility} predict in the vicinity of the
+\ho\ transition? Near the disordered to modulated transition, the
+static modulus is given by
+\begin{equation}
+  C_\X(0)=C_\X^0\bigg[1+\frac{b^2}{C_\X^0}\big(D_\perp q_*^4+|\Delta\tilde r|\big)^{-1}\bigg]^{-1}.
+  \label{eq:static_modulus}
+\end{equation}
+This corresponds to a softening in the $\X$-modulus at the transition that is
+cut off with a cusp of the form $|\Delta\tilde r|^\gamma\propto|T-T_c|^\gamma$
+with $\gamma=1$. This is our main result. The only \op\ irreps that couple
+linearly with strain and reproduce the topology of the \urusi\ phase diagram
+are $\Bog$ and $\Btg$. For either of these irreps, the transition into a
+modulated rather than uniform phase masks traditional signatures of a
+continuous transition by locating thermodynamic singularities at finite $q$.
+The remaining clue at $q=0$ is a particular kink in the corresponding modulus.
+
 \begin{figure}[htpb]
   \centering
   \includegraphics[width=\columnwidth]{fig-stiffnesses}
   \caption{
-   \Rus\ measurements of the elastic moduli of
-   \urusi\ as a function of temperature from \cite{ghosh_single-component_nodate} (green, solid) alongside fits to theory (red, dashed). The vertical yellow lines show the location of the \ho\ transition. (a) $\Btg$ modulus data and fit to standard form \cite{varshni_temperature_1970}. (b) $\Bog$ modulus data and fit
-    to \eqref{eq:elastic.susceptibility}. The fit gives
-    $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
-    $D_\perp q_*^4/b^2\simeq0.16\,\GPa^{-1}$, and $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of an additional parameter to fit the standard bare modulus \cite{varshni_temperature_1970} led to sloppy fits. (c) $\Bog$ modulus data and fit of \emph{bare}
-    $\Bog$ modulus. (d) $\Bog$ modulus data and fit transformed using
-    $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted from
-    \eqref{eq:susceptibility} and \eqref{eq:elastic.susceptibility} to be
-    $D_\perp q_*^4/b^2+a/b^2|T-T_c|$. 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
-    temperature is reached.
+   \Rus\ measurements of the elastic moduli of \urusi\ as a function of
+   temperature from \cite{ghosh_single-component_nodate} (blue, solid)
+   alongside fits to theory (magenta, dashed). The solid yellow region shows
+   the location of the \ho\ phase. (a) $\Btg$ modulus data and fit to standard
+   form \cite{varshni_temperature_1970}. (b) $\Bog$ modulus data and fit to
+   \eqref{eq:static_modulus}. The fit gives
+   $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $D_\perp
+   q_*^4/b^2\simeq0.16\,\GPa^{-1}$, and
+   $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of an additional
+   parameter to fit the standard bare modulus \cite{varshni_temperature_1970}
+   led to poorly constrained fits. (c) $\Bog$ modulus data and fit of
+   \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus data and fit transformed by
+   $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted from
+   \eqref{eq:static_modulus} to equal
+   $D_\perp q_*^4/b^2+a/b^2|T-T_c|$, e.g., an absolute value function. 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 temperature is reached.
   }
   \label{fig:data}
 \end{figure}
 
 \emph{Comparison to experiment.}
-\Rus\ experiments \cite{ghosh_single-component_nodate} yield the full
-elasticity tensor; the moduli broken into the irrep components defined in
-\eqref{eq:strain-components} is shown in Figure \ref{fig:data}.  The $\Btg$
-modulus doesn't appear to have any response to the presence of the
+\Rus\ experiments \cite{ghosh_single-component_nodate} yield the individual
+elastic moduli broken into irrep symmetry; the $\Bog$ and $\Btg$ components
+defined in \eqref{eq:strain-components} are shown in Figures \ref{fig:data}(a--b).
+The $\Btg$ modulus doesn't appear to have any response to the presence of the
 transition, exhibiting the expected linear stiffening upon cooling from room
 temperature, with a low-temperature cutoff at some fraction of the Debye
-temperature\cite{varshni_temperature_1970}. The $\Bog$ modulus, on the other
+temperature \cite{varshni_temperature_1970}. The $\Bog$ modulus, on the other
 hand, has a dramatic response, softening over the course of roughly $100\,\K$,
 and then cusping at the \ho\ 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 modulus
-$C^0_\Bog$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in
-Figure \ref{fig:data}. The data and theory appear quantitatively consistent in
-the high temperature phase, suggesting that \ho\ can be described as a
-$\Bog$-nematic phase that is modulated at finite $q$ along the $c-$axis.
+theory \eqref{eq:static_modulus}, with a linear background modulus $C^0_\Bog$
+and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in Figure
+\ref{fig:data}(b). The data and theory appear quantitatively consistent in the
+high temperature phase, suggesting that \ho\ can be described as a
+$\Bog$-nematic phase that is modulated at finite $q$ along the $c-$axis. The
+predicted softening appears over hundreds of Kelvin; Figures
+\ref{fig:data}(c--d) show the background modulus $C_\Bog^0$ and the
+\op--induced response isolated from each other.
 
 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 $\Bog$ elastic modulus 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
+of the \ho\ phase diagram and the temperature dependence of the $\Bog$ elastic
+modulus 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 $\langle\epsilon_\Bog\rangle^2=b^2\tilde
-r/4u(C^0_\Bog)^2$, which corresponds to an orthorhombic structural 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 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
-the lattice spacing $a_3\simeq9.68\,\A$, or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$ \cite{meng_imaging_2013,
+r/4u(C^0_\Bog)^2$, which corresponds to an orthorhombic structural 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 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 the lattice spacing $a_3\simeq9.68\,\A$, or
+$q_*=\pi/a_3\simeq0.328\,\A^{-1}$ \cite{meng_imaging_2013,
 broholm_magnetic_1991, wiebe_gapped_2007, bourdarot_precise_2010}. In between
 these two regimes, the ordering wavevector should shrink by jumping between
 ever-closer commensurate values in the style of the devil's staircase
 \cite{bak_commensurate_1982}. This motivates future \rus\ experiments done at
 pressure, where the depth of the cusp in the $\Bog$ modulus should deepen
 (perhaps with these commensurability jumps) at low pressure and approach zero
-like $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. \brad{Should also motivate x-ray and neutron-diffraction experiments to look for new q's - mentioning this is important if we want to get others interested, no one else does RUS...} Alternatively,
-\rus\ done at ambient pressure might examine the heavy fermi liquid to \afm\
-transition by doping. \brad{We have to be careful, someone did do some doping studies and it's not clear exctly what's going on}. The presence of spatial commensurability known to be irrelevant to the critical behavior at a one-component disordered to modulated transition, and therefore is not
-expected to modify the critical behavior otherwise
+like $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. \brad{Should also
+motivate x-ray and neutron-diffraction experiments to look for new q's -
+mentioning this is important if we want to get others interested, no one else
+does RUS...} Alternatively, \rus\ done at ambient pressure might examine the
+heavy fermi liquid to \afm\ transition by doping. \brad{We have to be careful,
+someone did do some doping studies and it's not clear exctly what's going on}.
+The presence of spatial commensurability known to be irrelevant to the critical
+behavior at a one-component disordered to modulated transition, and therefore
+is not expected to modify the critical behavior otherwise
 \cite{garel_commensurability_1976}. 
 
 There are two apparent discrepancies between the orthorhombic strain in the
 phase diagram presented by \cite{choi_pressure-induced_2018} and that predicted
 by our mean field theory.  The first is the apparent onset of the orthorhombic
 phase in the \ho\ state prior to the onset of \afm.  As
-\cite{choi_pressure-induced_2018} notes, this could be due to the lack of
-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
-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. \brad{I think this paragraph should probably be tigtened up a bit, we need to be more specific about "don't expect there to be one" and "fluctuations"}. 
-
-Three dimensions is below the upper critical dimension $4\frac12$ of a one-component disordered to modulated transition, and so mean
-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 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
-universality class of a uniaxial modulated one-component \op\ is $\mathrm O(2)$
+\cite{choi_pressure-induced_2018} notes, this could be due to the lack of 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 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. \brad{I think
+this paragraph should probably be tigtened up a bit, we need to be more
+specific about "don't expect there to be one" and "fluctuations"}. 
+
+Three dimensions is below the upper critical dimension $4\frac12$ of a
+one-component disordered to modulated transition, and so mean 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 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 universality class of
+a uniaxial modulated one-component \op\ is $\mathrm O(2)$
 \cite{garel_commensurability_1976}. 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.
 
-\emph{Conclusion and Outlook.}
-We have developed a general phenomenological treatment of  \ho\ \op s with the potential for linear coupling to strain. The two representations with mean
-field phase diagrams that are consistent with the phase diagram of \urusi\ are $\Bog$ and $\Btg$. Of these, only a staggered $\Bog$ \op is consistent with zero-pressure \rus\ data, with a cusp appearing in the
-associated elastic modulus. In this picture, the \ho\ phase is characterized by
-uniaxial modulated $\Bog$ order, while the \afm\ phase is characterized by
-uniform $\Bog$ order. \brad{We need to be a bit more explicit about what we think is going on with \afm - is it just a parasitic phase? Is our modulated phase somehow "moduluated \afm" (can you modualte AFM in such as way as to make it disappear? Some combination of orbitals?)} The corresponding prediction of uniform $\Bog$ symmetry
-breaking in the \afm\ phase is consistent with recent diffraction experiments
-\cite{choi_pressure-induced_2018} \brad{needs a caveat about temperature, so that we're being transparent}. This work motivates both further theoretical
-work regarding a microscopic theory with modulated $\Bog$ order, and preforming
-\rus\ experiments at pressure that could further support or falsify this idea.
+\emph{Conclusion and Outlook.} We have developed a general phenomenological
+treatment of  \ho\ \op s with the potential for linear coupling to strain. The
+two representations with mean field phase diagrams that are consistent with the
+phase diagram of \urusi\ are $\Bog$ and $\Btg$. Of these, only a staggered
+$\Bog$ \op is consistent with zero-pressure \rus\ data, with a cusp appearing
+in the associated elastic modulus. In this picture, the \ho\ phase is
+characterized by uniaxial modulated $\Bog$ order, while the \afm\ phase is
+characterized by uniform $\Bog$ order. \brad{We need to be a bit more explicit
+about what we think is going on with \afm - is it just a parasitic phase? Is
+our modulated phase somehow "moduluated \afm" (can you modualte AFM in such as
+way as to make it disappear? Some combination of orbitals?)} The corresponding
+prediction of uniform $\Bog$ symmetry breaking in the \afm\ phase is consistent
+with recent diffraction experiments \cite{choi_pressure-induced_2018}
+\brad{needs a caveat about temperature, so that we're being transparent}. This
+work motivates both further theoretical work regarding a microscopic theory
+with modulated $\Bog$ order, and preforming \rus\ experiments at pressure that
+could further support or falsify this idea.
 
 \begin{acknowledgements}
   This research was supported by NSF DMR-1719490 and DMR-1719875.