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@@ -94,28 +94,39 @@ transition, the physics of the OP can often be described in the context of Landa mean field theory. However, to construct such a theory, one must know the symmetries of the order parameter, i.e. the symmetry of the ordered state. -One quintessential case where the symmetry of an ordered phase remains unknown is in \urusi. +A paradigmatic example where the symmetry of an ordered phase remains unknown is in \urusi. \urusi is a heavy fermion superconductor in which superconductivity condenses out of a symmetry broken state referred to as hidden order (HO) [cite pd paper], and at sufficiently large [hydrostatic?] pressures, both give way to local moment antiferromagnetism. -Despite over thirty years of effort, the symmetry of the hidden order state remains unknown, and -modern theories [big citation chunk] propose a variety of possibilities. +Despite over thirty years of effort, the symmetry of the hidden order state remains unknown, and modern theories +\cite{kambe:pr2018a, haule:np2009a, kusunose:jpsj2011a, kung:s2015a,cricchio:prl2009a,ohkawa:jpcm1999a,santini:prl1994a,kiss:ap2004a,harima:jpsj2010a,thalmeier:pr2011a,tonegawa:prl2012a,rau:pr2012a,riggs:nc2015a,hoshino:jpsj2013a,ikeda:prl1998a,chandra:n2013a,harrison:apa2019a,ikeda:np2012a} +propose a variety of possibilities. Many [all?] of these theories rely on the formulation of a microscopic model for the HO state, but without direct experimental observation of the broken symmetry, none have been confirmed. -One case that does not rely on a microscopic model is recent work [cite RUS paper] +One case that does not rely on a microscopic model is recent work \cite{ghosh:apa2019a} that studies the HO transition using resonant ultrasound spectroscopy (RUS). RUS is an experimental technique that measures mechanical resonances of a sample. These resonances contain information about the full elastic tensor of the material. Moreover, the frequency locations of the resonances are sensitive to symmetry breaking at an electronic -phase transition due to electron-phonon coupling [cite]. Ref. [RUS paper] uses this information +phase transition due to electron-phonon coupling [cite]. Ref.~\cite{ghosh:apa2019a} uses this information to place strict thermodynamic bounds on the symmetry of the HO OP, again, independent of any microscopic model. Motivated by these results, in this paper we consider a mean field theory of an OP coupled to strain and the effect that the OP symmetry has on the elastic response in different symmetry channels. Our study finds that a single possible OP symmetry reproduces the experimental strain susceptibilities, and fits the experimental data well. +We first present a phenomenological Landau-Ginzburg mean field theory of strain coupled to an +order parameter. We examine the phase diagram predicted by this theory and compare it +to the experimentally obtained phase diagram of \urusi. +Then we compute the elastic response to strain, and examine the response function dependence on +the symmetry of the OP. +We proceed to compare the results from mean field theory with data from RUS experiments. +We further examine the consequences of our theory at non-zero applied pressure in comparison +with recent x-ray scattering experiments [cite]. +Finally, we discuss our conclusions and future experimental and theoretical work that our results motivate. + The point group of \urusi is \Dfh, and any coarse-grained theory must locally respect this symmetry. We will introduce a phenomenological free energy density in three parts: that of the strain, the order parameter, and their interaction. @@ -414,6 +425,7 @@ self-consistent. \end{acknowledgements} -\bibliography{hidden_order} +\bibliographystyle{apsrev4-1} +\bibliography{hidden_order,library} \end{document} |