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@@ -60,7 +60,7 @@ \date\today \begin{abstract} - We develop a phenomenological mean field theory of the hidden order phase in \urusi as a ``staggered nematic" order. Several experimental features are reproduced when the order parameter is a nematic of the $\Bog$ representation, staggered along the c-axis: the topology of the temperature--pressure phase diagram, the response of the elastic modulus $(c_{11}-c_{12})/2$ above the hidden-order transition at zero pressure, and orthorhombic symmetry breaking in the high-pressure antiferromagnetic phase. In this scenario, hidden order is characterized by broken rotational symmetry that is modulated along the $c$-axis, the primary order of the high-pressure phase is an unmodulated nematic state, and the triple point joining those two phases with the high-temperature paramagnetic phase is a Lifshitz point. + We develop a phenomenological mean field theory of the hidden order phase in \urusi\ as a ``staggered nematic" order. Several experimental features are reproduced when the order parameter is a nematic of the $\Bog$ representation, staggered along the c-axis: the topology of the temperature--pressure phase diagram, the response of the elastic modulus $(c_{11}-c_{12})/2$ above the hidden-order transition at zero pressure, and orthorhombic symmetry breaking in the high-pressure antiferromagnetic phase. In this scenario, hidden order is characterized by broken rotational symmetry that is modulated along the $c$-axis, the primary order of the high-pressure phase is an unmodulated nematic state, and the triple point joining those two phases with the high-temperature paramagnetic phase is a Lifshitz point. \end{abstract} \maketitle @@ -89,8 +89,8 @@ %this information to place strict thermodynamic bounds on the dimension of the %\ho\ \op\ independent of any microscopic model. \section{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}. +\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 unknown, and modern theories \cite{kambe_odd-parity_2018, haule_arrested_2009, kusunose_hidden_2011, kung_chirality_2015, cricchio_itinerant_2009, @@ -98,11 +98,11 @@ unknown, and modern theories \cite{kambe_odd-parity_2018, haule_arrested_2009, harima_why_2010, thalmeier_signatures_2011, tonegawa_cyclotron_2012, rau_hidden_2012, riggs_evidence_2015, hoshino_resolution_2013, ikeda_theory_1998, chandra_hastatic_2013, harrison_hidden_nodate, -ikeda_emergent_2012} propose a variety of possibilities. Our work here seeks to unify two experimental observations: one, the $\Bog$ ``nematic" elastic susceptibility $(c_{11}-c_{12})/2$ softens anomalously from room temperature down to T$_{\mathrm{HO}}=17.5~$ K \brad{find old citations for this data}; and two, a $\Bog$ nematic distortion is observed by x-ray scattering under sufficient pressure to destroy the \ho state \brad{cite johan}. +ikeda_emergent_2012} propose a variety of possibilities. Our work here seeks to unify two experimental observations: one, the $\Bog$ ``nematic" elastic susceptibility $(c_{11}-c_{12})/2$ softens anomalously from room temperature down to T$_{\mathrm{HO}}=17.5~$ K \brad{find old citations for this data}; and two, a $\Bog$ nematic distortion is observed by x-ray scattering under sufficient pressure to destroy the \ho\ state \brad{cite johan}. -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 \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 \brad{cite Johan}. Above \brad{whatever pressure they find it at...}, \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 \brad{cite Johan}. Above \brad{whatever pressure they find it at...}, \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. 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 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. @@ -230,7 +230,7 @@ energy minima for $q_*^2=-c_\perp/2D_\perp$ and \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 -one-component \op and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde +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 field \brad{Unless there is a specific reason, we should probably stick to \op\ instead of "field"}. In this case the uniform ordered phase is only stable for $c_\perp>0$, and the modulated phase is now characterized by @@ -240,7 +240,7 @@ helical order with $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. 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 +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 @@ -373,7 +373,7 @@ $|\Delta\tilde r|^\gamma$ for $\gamma=1$. \brad{I think this last sentence, whi \centering \includegraphics[width=\columnwidth]{fig-stiffnesses} \caption{ - Resonant ultrasound spectroscopy measurements of the elastic moduli of \urusi as a function of temperature + 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. \brad{Can you move the labels on the right-hand panels over to the right-hand axis? Also, can you make the labels smaller and the actual panels bigger} } @@ -471,7 +471,7 @@ and the Landau--Ginzburg free energy expansion is no longer valid. \section{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 +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 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 |