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@@ -70,60 +70,59 @@ \maketitle -The study of phase transitions is a central theme of condensed matter physics. -In many cases, a phase transition between different states of matter is marked -by a change in symmetry. In this paradigm, the breaking of symmetry in an -ordered phase corresponds to the condensation of an order parameter (\op) that -breaks the same symmetries. Near a second order phase transition, the physics -of the \op\ can often be described in the context of Landau--Ginzburg mean field -theory. However, to construct such a theory, one must know the symmetries of -the \op, i.e. the symmetry of the ordered state. - -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{hassinger_temperature-pressure_2008}, and at -sufficiently large hydrostatic pressures, both give way to local moment -antiferromagnetism (\afm). 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, ohkawa_quadrupole_1999, santini_crystal_1994, - kiss_group_2005, 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. Many 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{ghosh_single-component_nodate} 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 strain stiffness 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 +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. + +A paradigmatic example of a material with an ordered state whose broken +symmetry remains unknown is in \urusi. \urusi\ is a heavy fermion +superconductor in which superconductivity condenses out of a symmetry broken +state referred to as \emph{hidden order} (\ho) +\cite{hassinger_temperature-pressure_2008}, and at sufficiently large +hydrostatic pressures, both give way to local moment antiferromagnetism (\afm). +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, + ohkawa_quadrupole_1999, santini_crystal_1994, kiss_group_2005, + 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. 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. + +Recent work that studied the \ho\ transition using \emph{resonant ultrasound +spectroscopy} (\rus) was able to shed light on the symmetry of the ordered +state without the formulation of any microscopic model +\cite{ghosh_single-component_nodate}. \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 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. The resulting theory associates \ho\ with $\Bog$ order -\emph{modulated along the rotation axis}, \afm\ with uniform $\Bog$ order, and -a Lifshitz point with the triple point between them. - -We first present a phenomenological Landau--Ginzburg mean field theory of -strain coupled to an \op. We examine the phase diagrams predicted by this -theory for various \op\ symmetries and compare them 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 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{choi_pressure-induced_2018}. Finally, we discuss our conclusions and the -future experimental and theoretical work motivated by our results. +this information to place strict thermodynamic bounds on the dimension of the +\ho\ \op\ independent of any microscopic model. + +Motivated by these results, 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 features of the +experimental strain stiffness. That theory associates the \ho\ state with a +$\Bog$ \op\ \emph{modulated along the rotation axis}, the \afm\ state with +uniform $\Bog$ order, and the triple point between them with a Lifshitz point. +Besides the agreement with \rus\ data in the \ho\ state, the theory predicts +uniform $\Bog$ strain in the \afm\ state, which was recently seen in x-ray +scattering experiments \cite{choi_pressure-induced_2018}. The theory's +implications for the dependence of the strain stiffness on pressure and doping +strongly motivates future \rus\ experiments that could either further support +or falsify it. 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 |