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| -rw-r--r-- | fig-stiffnesses.gplot | 66 | ||||
| -rw-r--r-- | fig-stiffnesses.pdf | bin | 90056 -> 70347 bytes | |||
| -rw-r--r-- | main.tex | 203 |
3 files changed, 117 insertions, 152 deletions
diff --git a/fig-stiffnesses.gplot b/fig-stiffnesses.gplot index 22dd499..e132869 100644 --- a/fig-stiffnesses.gplot +++ b/fig-stiffnesses.gplot @@ -7,62 +7,54 @@ cc4 = "#eb6235" set terminal epslatex size 8.68cm, 8.2cm standalone set output "fig-stiffnesses.tex" -set multiplot layout 3, 2 margins 0.13, 0.93, 0.1, 0.998 spacing 0.01, 0.01 +set multiplot layout 3, 2 margins 0.13, 0.9, 0.1, 0.998 spacing 0.01, 0.01 set nokey -set xrange [-25:315] +set xrange [0:315] set arrow 1 from 17.26,graph 0 to 17.26,graph 1 nohead lw 4 lc rgb cc2 +lam(T) = 71.1212 - 0.0104105 * T set format x "" set xtics 0, 100, 300 offset 0,0.5 set mxtics 2 -set ylabel '\footnotesize $\lambda / \mathrm{GPa}$' offset 4.5 +set ylabel '\tiny $C / \mathrm{GPa}$' offset 4.5 set format y '\tiny $%.1f$' set format y2 '\tiny $%.1f$' -set yrange [217.5:220.75] -set ytics 218,1,220 offset 0.5 -set title '\footnotesize (a) $\lambda^{(11)}_{\mathrm{A_{1\mathrm g}}}$' offset -1,-6 -plot "data/c11pc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3 - -set ylabel '' - -set format y2 '\tiny $%1.f$' -set yrange [306:317] -set y2tics 308,2,316 offset -0.5 mirror -set title '\footnotesize (b) $\lambda^{(22)}_{\mathrm{A_{1\mathrm g}}}$' offset -1,-6 -plot "data/c33.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3 -set format y2 '\tiny $%.1f$' - -set ylabel '\footnotesize $\lambda / \mathrm{GPa}$' - -set yrange [112.6:114] -set ytics 112.8,0.3,113.9 offset 0.5 -set title '\footnotesize (d) $\lambda^{(12)}_{\mathrm{A_{1\mathrm g}}}$' offset -1,-6 -plot "data/c13.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3 +set yrange [140:145] +set ytics 141,1,144 offset 0.5 +set title '\tiny (e) $C_{\mathrm{B_{2\mathrm g}}}$' offset -1,-6 +plot "data/c66.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \ + 144.345 - 0.019492 * x**2 / (120.462 + x) dt 3 lw 4 lc rgb cc4 set ylabel '' set yrange [65.05:65.7] set y2tics 62.1,0.1,65.6 offset -0.5 mirror -set title '\footnotesize (c) $\lambda^{(11)}_{\mathrm{B_{1\mathrm g}}}$' offset -1,-6 +set title '\tiny (c) $C_{\mathrm{B_{1\mathrm g}}}$' offset -1,-6 plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3 set format x '\tiny $%.0f$' -set ylabel '\footnotesize $\lambda / \mathrm{GPa}$' -set xlabel '\footnotesize $T / \mathrm K$' offset 0,1 +set ylabel '\tiny $C / \mathrm{GPa}$' +set xlabel '\tiny $T / \mathrm K$' offset 0,1 +unset y2tics -set yrange [140:145] -set ytics 141,1,144 offset 0.5 -set title '\footnotesize (e) $\lambda^{(11)}_{\mathrm{B_{2\mathrm g}}}$' offset -1,-6 -plot "data/c66.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3 +set yrange [65:71.5] +set ytics 63,1,72 +set title '\tiny (c) $C_{\mathrm{B_{1\mathrm g}}}$' offset -1,-6 +plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \ + lam(x) dt 3 lw 4 lc rgb cc4 set ylabel '' - -set format y2 '\tiny $%1.f$' -set yrange [101:106] -set y2tics 102,1,105 offset -0.5 mirror -set title '\footnotesize (f) $\lambda^{(11)}_{\mathrm{E_{\mathrm g}}}$' offset -1,-6 -plot "data/c44.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3 - +set y2label '\tiny $\log[(C^0 - C) / \mathrm{GPa}]$' offset -5 rotate by -90 +set xlabel '\tiny $\log(T / \mathrm K+D_\perp q_*^4/a)$' offset 0,1 + +set yrange [1:1.85] +set format x '\tiny $%0.1f$' +set xrange [5.5:6.35] +set y2tics 1,0.1,1.9 offset -0.7 mirror +set xtics 5.6,0.2,6.4 +set title '\tiny (c) $C_{\mathrm{B_{1\mathrm g}}}$' offset -1,-6 +plot "data/c11mc12.dat" using (log($1 + 1 / 0.00378087)):(log(lam($1) - 100 * $2)) with lines lw 3 lc rgb cc3, \ + 7.3 - x diff --git a/fig-stiffnesses.pdf b/fig-stiffnesses.pdf Binary files differindex 4c6cae0..5962c50 100644 --- a/fig-stiffnesses.pdf +++ b/fig-stiffnesses.pdf @@ -38,12 +38,11 @@ \def\A{\text{\r A}} % Other -\def\G{\text G} % Ginzburg \def\op{\textsc{op}} % order parameter \def\ho{\textsc{ho}} % hidden order -\def\rus{\textsc{rus}} % Resonant ultrasound spectroscopy -\def\afm{\textsc{afm}} % Antiferromagnetism +\def\rus{\textsc{rus}} % resonant ultrasound spectroscopy \def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy +\def\afm{\textsc{afm}} % antiferromagnetism \def\recip{{\{-1\}}} % functional reciprocal \begin{document} @@ -65,30 +64,7 @@ \maketitle -%The study of phase transitions is central to condensed matter physics. Phase -%transitions are often accompanied by a change in symmetry whose emergence can -%be described by the condensation of an order parameter (\op) that breaks the -%same symmetries. Near a continuous phase transition, the physics of the \op\ -%can often be qualitatively and sometimes quantitatively described by -%Landau--Ginzburg mean field theories. These depend on little more than the -%symmetries of the \op, and coincidence of their predictions with experimental -%signatures of the \op\ is evidence of the symmetry of the corresponding ordered -%state. - -% Many of these -%theories rely on the formulation of a microscopic model for the \ho\ state, but -%since there has not been direct experimental observation of the broken -%symmetry, none can been confirmed. - -%\Rus\ is an experimental technique that -%measures mechanical resonances of a sample. These resonances contain -%information about the sample's full strain stiffness tensor. Moreover, the -%frequency locations of the resonances are sensitive to symmetry breaking at an -%electronic phase transition due to electron-phonon coupling -%\cite{shekhter_bounding_2013}. Ref.~\cite{ghosh_single-component_nodate} uses -%this information to place strict thermodynamic bounds on the dimension of the -%\ho\ \op\ independent of any microscopic model. -\section{Introduction} +\emph{Introduction.} \urusi\ is a paradigmatic example of a material with an ordered state whose broken symmetry remains unknown. This state, known as \emph{hidden order} (\ho), sets the stage for unconventional superconductivity that emerges at even lower temperatures. At sufficiently large hydrostatic pressures, both superconductivity and \ho\ give way to local moment antiferromagnetism (\afm) \cite{hassinger_temperature-pressure_2008}. Despite over thirty years of effort, the symmetry of the \ho\ state remains @@ -102,53 +78,50 @@ ikeda_emergent_2012} propose a variety of possibilities. Our work here seeks to Recent \emph{resonant ultrasound spectroscopy} (\rus) measurements examined the thermodynamic discontinuities in the elastic moduli at T$_{\mathrm{HO}}$ \cite{ghosh_single-component_nodate}. The observation of discontinues only in compressional, or $\Aog$, elastic moduli requires that the point-group representation of \ho\ is one-dimensional. This rules out a large number of order parameter candidates \brad{cite those ruled out} in a model-free way, but still leaves the microscopic nature of \ho~ undecided. -Recent X-ray experiments discovered rotational symmetry breaking in \urusi\ under pressure \cite{choi_pressure-induced_2018}. Above 0.13--0.5 $\GPa$ (depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion. While it is still unclear as to whether this is a true thermodynamic phase transition, it may be related to the anomalous softening of the $\Bog$ elastic modulus---$(c_{11}-c_{12})/2$---that occurs over a broad temperature range at zero-pressure \brad{cite old ultrasound}. Motivated by these results, hinting at a $\Bog$ strain susceptibility associated with the \ho\ state, we construct a phenomenological mean field theory for an arbitrary \op\ coupled to strain, and the determine the effect of its phase transitions on the elastic response in different symmetry channels. +Recent X-ray experiments discovered rotational symmetry breaking in \urusi\ under pressure \cite{choi_pressure-induced_2018}. Above 0.13--0.5 $\GPa$ (depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion. While it is still unclear as to whether this is a true thermodynamic phase transition, it may be related to the anomalous softening of the $\Bog$ elastic modulus---$(c_{11}-c_{12})/2$ in Voigt notation---that occurs over a broad temperature range at zero-pressure \brad{cite old ultrasound}. Motivated by these results, hinting at a $\Bog$ strain susceptibility associated with the \ho\ state, we construct a phenomenological mean field theory for an arbitrary \op\ coupled to strain, and the determine the effect of its phase transitions on the elastic response in different symmetry channels. -We find that only one \op\ symmetry reproduces the anomalous $(c_{11}-c_{12})/2$ elastic modulus, which softens in a Curie-Weiss like manner from room temperature, but which cusps at T$_{\mathrm{HO}}$. That theory associates \ho\ with a $\Bog$ \op\ \emph{modulated along the $c$- axis}, the \afm\ state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. Besides the agreement with ultrasound data across a broad temperature range, the theory predicts uniform $\Bog$ strain at high pressure---the same distortion which was recently seen in x-ray scattering experiments \cite{choi_pressure-induced_2018}. This theory +We find that only one \op\ symmetry reproduces the anomalous $\Bog$ elastic modulus, which softens in a Curie-Weiss like manner from room temperature, but which cusps at T$_{\mathrm{HO}}$. That theory associates \ho\ with a $\Bog$ \op\ \emph{modulated along the $c$- axis}, the \afm\ state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. Besides the agreement with ultrasound data across a broad temperature range, the theory predicts uniform $\Bog$ strain at high pressure---the same distortion which was recently seen in x-ray scattering experiments \cite{choi_pressure-induced_2018}. This theory strongly motivates future ultrasound experiments under pressure approaching the Lifshitz point, which should find that the $(c_{11}-c_{12})/2$ diverges once the uniform $\Bog$ strain sets in. -%but the form of the bare strain -%stiffness tensor $C$ tensor is constrained by both the index symmetry of the -%strain tensor and by the point group symmetry \cite{landau_theory_1995} \brad{why %is there a Landau paper from 1995?!}. - - -\section{Model} +\emph{Model.} The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally respect this symmetry in the high-temperature phase. Our phenomenological free energy density contains three parts: the elastic free energy, the \op, and the interaction between strain and \op. The most general quadratic free energy of -the strain $\epsilon$ is $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$. Linear +the strain $\epsilon$ is $f_\e=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$. Linear combinations of the six independent components of strain form five irreducible components of strain as \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)}=2\epsilon_{12} \\ - \epsilon_\Eg^{(1)}=2\{\epsilon_{11},\epsilon_{22}\}. + & \epsilon_{\Aog,1}=\epsilon_{11}+\epsilon_{22} \hspace{0.15\columnwidth} && + \epsilon_\Bog=\epsilon_{11}-\epsilon_{22} \\ + & \epsilon_{\Aog,2}=\epsilon_{33} && + \epsilon_\Btg=2\epsilon_{12} \\ + & \epsilon_\Eg=2\{\epsilon_{11},\epsilon_{22}\}. \end{aligned} \label{eq:strain-components} \end{equation} All quadratic combinations of these irreducible strains that transform like $\Aog$ are included in the free energy, \begin{equation} - f_\e=\frac12\sum_\X C_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)}, + f_\e=\frac12\sum_\X C^0_{\X,ij}\epsilon_{\X,i}\epsilon_{\X,j}, \end{equation} where the sum is over irreducible representations of the point group and the -bare elastic moduli $C_\X^{(ij)}$ are \brad{I would write these in Voigt notation, so c1111 is just c11, c1122 is c12, etc.} +bare elastic moduli $C^0_\X$ are \begin{equation} \begin{aligned} - &C_{\Aog}^{(11)}=\tfrac12(C_{1111}+C_{1122}) && - C_{\Aog}^{(22)}=C_{3333} \\ - &C_{\Aog}^{(12)}=C_{1133} && - C_{\Bog}^{(11)}=\tfrac12(C_{1111}-C_{1122}) \\ - &C_{\Btg}^{(11)}=C_{1212} && - C_{\Eg}^{(11)}=C_{1313}. + & C^0_{\Aog,11}=\tfrac12(C^0_{1111}+C^0_{1122}) && + C^0_{\Bog}=\tfrac12(C^0_{1111}-C^0_{1122}) \\ + & C^0_{\Aog,22}=C^0_{3333} && + C^0_{\Btg}=C^0_{1212} \\ + & C^0_{\Aog,12}=C^0_{1133} && + C^0_{\Eg}=C^0_{1313}. \end{aligned} \end{equation} -The interaction between strain and an \op\ $\eta$ depends on the point group representation of $\eta$. If this representation is $\X$, the most general coupling to linear order is \brad{why the negative sign?} +The interaction between strain and an \op\ $\eta$ depends on the point group +representation of $\eta$. If this representation is $\X$, the most general +coupling to linear order is \begin{equation} f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta. \end{equation} @@ -163,7 +136,7 @@ attention on \op s that can produce linear couplings to strain. Looking at the components present in \eqref{eq:strain-components}, this rules out all of the \emph{u}-reps (which are odd under inversion) and the $\Atg$ irrep. -If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd terms are allowed in its free energy and any transition will be first-order and not continuous without fine-tuning. Since the \ho\ phase transition is second-order \brad{cite something}, we will henceforth rule out $\Aog$ \op s as well. +If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd terms are allowed in its free energy and any transition will be first order and not continuous without fine-tuning. Since the \ho\ phase transition is second-order \brad{cite something}, we will henceforth rule out $\Aog$ \op s as well. For the \op\ representation $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quadratic free energy density is @@ -190,14 +163,15 @@ The only strain relevant to the \op\ at linear coupling is $\epsilon_\X$, which 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(x)}\bigg|_{\epsilon=\epsilon_\star}=C_\X\epsilon^\star_\X(x) + 0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_\X(x)}\bigg|_{\epsilon=\epsilon_\star}=C^0_\X\epsilon^\star_\X(x) -b\eta(x) \end{equation} +\textbf{talk more about the functional-ness of these parameters!, also, why are we tracinig out strain?} gives the optimized strain conditional on the \op\ as -$\epsilon_\X^\star[\eta](x)=(b/C_\X)\eta(x)$ and $\epsilon_\Y^\star[\eta]=0$ +$\epsilon_\X^\star[\eta](x)=(b/C^0_\X)\eta(x)$ 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 $r\to\tilde r=r-b^2/2C_\X$. +$f_\op$ with $r\to\tilde r=r-b^2/2C^0_\X$. \begin{figure}[htpb] \includegraphics[width=\columnwidth]{phase_diagram_experiments} @@ -211,8 +185,8 @@ $f_\op$ with $r\to\tilde r=r-b^2/2C_\X$. superconducting phase) \cite{hassinger_temperature-pressure_2008} (b) mean field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean field theory of a two-component ($\Eg$) Lifshitz point. Solid lines denote - continuous transitions, while dashed lines denote abrupt transitions. - Later, when we fit the elastic stiffness predictions for a $\Bog$ \op\ to + continuous transitions, while dashed lines denote first order transitions. + Later, when we fit the elastic moduli predictions for a $\Bog$ \op\ to data along the zero (atmospheric) pressure line, we will take $\Delta\tilde r=\tilde r-\tilde r_c=a(T-T_c)$. } @@ -238,31 +212,30 @@ $q_*^2=-c_\perp/2D_\perp$ and =\frac{|\Delta\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 +transition between the uniform and modulated orderings is first order for a one-component \op\ and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde r/5}$. For a two-component \op\ ($\Eg$) we must also allow a relative phase between the two components of the \op. 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 $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. -The uniform--modulated transition is now continuous. This does not reproduce -the physics of \ho, which has a first-order \brad{I think we should keep the -language "First order" rather than "abrupt", which doesn't really mean anything -specific} transition between \ho\ and \afm, and so we will henceforth neglect +The uniform to modulated transition is now continuous. This does not reproduce +the physics of \ho, which has a first order transition between \ho\ and \afm, and so we will henceforth neglect the possibility of a multicomponent order parameter. The schematic phase diagrams for this model are shown in Figure~\ref{fig:phases}. -\section{Results} -We will now derive the \emph{effective elastic 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\ \brad{Why the mixed $\lambda$ and C notation? Why not C and C dagger or tilde or hat?}. Moreover, the effective strain stiffness \brad{I think "elastic moduli" is a lot more familiar to people than "Strain stiffness"} does not vanish at the unordered--modulated transition \brad{"unordered--modulated transition" is confusing language}---as it would if the transition were a $q=0$ structural phase transition---but instead 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 -strain stiffness using some tricks from functional calculus \brad{"tricks" will be a bit too colloquial for most referees}. - -The susceptibility of a single component ($\Bog$ or $\Btg$) \op\ $\eta$ to a -thermodynamically conjugate field (such as strain) \brad{it it clear why they are conjugate fields? Because they are linearlly coupled?} is given by +\emph{Results.} +We will now derive the \emph{effective elastic tensor} $C$ that results from +the coupling of strain to the \op. The ultimate result, found in +\eqref{eq:elastic.susceptibility}, is that $C_\X$ differs from its bare value +$C^0_\X$ only for the symmetry $\X$ of the \op. Moreover, the effective elastic +moduli does not vanish at the unordered to modulated transition---as it would +if the transition were a $q=0$ structural phase transition---but instead +exhibits a \emph{cusp}. To show this, we will first compute the susceptibility +of the \op, which will both be demonstrative of how the modulus is calculated +and prove useful in expressing the functional form of the modulus. Then we will +compute the elastic modulus using techniques from functional calculus. + +The generalized susceptibility of a single component ($\Bog$ or $\Btg$) \op\ is \begin{equation} \begin{aligned} &\chi^\recip(x,x') @@ -283,7 +256,7 @@ Taking the Fourier transform and integrating over $q'$ we have =\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4 +12u\sum_{q'}\langle\tilde\eta_{q'}\rangle\langle\tilde\eta_{-q'}\rangle\big)^{-1}. \end{equation} -Near the unordered--modulated transition \brad{again, this needs clearer language} this yields +Near the unordered to modulated transition this yields \begin{equation} \begin{aligned} \chi(q) @@ -300,18 +273,14 @@ 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 \brad{needs a descriptor like "in and perpendicular to the x-y plane" or something like that}. Notice that the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde -r|)^{-1}$ does not diverge at the unordered--modulated transition. Though it +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}. We must -emphasize that this is \emph{not} the magnetic susceptibility because a $\Bog$ -or $\Btg$ \op\ cannot couple linearly to a uniform magnetic field \brad{not sure that this reminder is important, I don't think anyone things we are dealing with magnetic fields.}. The object -defined in \eqref{eq:sus_def} is most readily interpreted as proportional to -the two-point connected correlation function -$\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$ \brad{is this important?}. - -The strain stiffness \brad{elastic modulus? Or is this elastic compliance?} 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 +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}. + +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 $\eta$ +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), @@ -319,7 +288,7 @@ effective free energy functional of $\epsilon$. Extremizing over $\eta$ yields \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 $\lambda_\X$ can be modified from its bare value $C_\X$. +alone, only the modulus $C_\X$ can be modified from its bare value $C^0_\X$. 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 @@ -345,44 +314,44 @@ 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')- + &=C^0_\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')- + &=C^0_\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')- + &=C^0_\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')}. + =C^0_\X\delta(x-x')-b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}. \end{aligned} \label{eq:big.boy} \end{equation} \end{widetext} -The strain stiffness is given by the second variation evaluated at the +The elastic modulus is given by the second variation evaluated at the 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 \begin{equation} \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip\bigg|_{\epsilon=\langle\epsilon\rangle} - =b^{-1}\chi^\recip(x,x')+\frac{b}{C_\X}\delta(x-x'), + =b^{-1}\chi^\recip(x,x')+\frac{b}{C^0_\X}\delta(x-x'), \label{eq:recip.deriv.op} \end{equation} 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} - \lambda_\X(q) - =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}. + C_\X(q) + =C^0_\X-b\bigg(\frac1{b\chi(q)}+\frac b{C^0_\X}\bigg)^{-1} + =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 stiffness measurements used here were taken, this -predicts a cusp in the static strain stiffness $\lambda_\X(0)$ of the form +$q=0$, 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} \begin{figure}[htpb] \centering @@ -391,33 +360,38 @@ $|\Delta\tilde r|^\gamma$ for $\gamma=1$. \brad{I think this last sentence, whi Resonant ultrasound spectroscopy measurements of the elastic moduli of \urusi\ as a function of temperature for the six independent components of strain. The vertical lines show the location of the \ho\ transition. + \textbf{ONE FIGURE: Just B2g and B1g, vashni fit for one, our fit for the other, something else} } \label{fig:data} \end{figure} -\section{Comparison to experiment} -\Rus\ experiments \cite{ghosh_single-component_nodate} yield the full elasticity tensor; the moduli broken into the irrep components defined in +\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 $\Btg$ -stiffness 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 $\Bog$ stiffness, 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 +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 +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 stiffness -$C_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in +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:fit}. 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 high temperature phase, suggesting that \ho\ can be described as a +$\Bog$-nematic phase that is modulated at finite $q$ along the $c-$axis. \begin{figure}[htpb] \includegraphics[width=\columnwidth]{fig-fit} \caption{ - Strain stiffness data for the $\Bog$ component of strain (solid) along with + Elastic modulus 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 - $C_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, + $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $b^2/D_\perp q_*^4\simeq6.2\,\GPa$, 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 @@ -428,12 +402,11 @@ the high temperature phase, suggesting that \ho\ can be described as a $\Bog$-ne \end{figure} 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. This theory has several other physical implications. First, +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 $\langle\epsilon_\Bog\rangle^2=b^2\tilde -r/4uC_\Bog^2$, which corresponds to an orthorhombic structural phase. Orthorhombic +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 @@ -447,7 +420,7 @@ 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 $\Bog$ stiffness should deepen +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\ @@ -484,10 +457,10 @@ 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. -\section{Conclusion and Outlook} +\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 stiffness. In this picture, the \ho\ phase is characterized by +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 |