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@@ -609,23 +609,31 @@ is characterized by uniform $\Bog$ order. The staggered nematic of \ho\ is similar to the striped superconducting phase found in LBCO and other cuperates.\cite{Berg_2009b} +% {\color{blue} +% We can also connect our abstract order parameter to a physical picture of multipolar +% ordering. +% The U-5f electrons in URu$_2$Si$_2$ exhibit a moderate degree of localization [cite], which is +% reflected in partial occupancy of many electronic states. Motivated by the results of refs [cite], +% we assume that the dominant U state consists of $j = 5/2$ electrons in the U-5f2 configuration, which has +% total angular momentum $J = 4$. Within the $J=4$ multiplet, the precise energetic ordering +% of the $D_{4h}$ crystal field states still remains a matter of debate [cite]. In a simple +% framework of localized $j = 5/2$ electrons in the 5f2 configuration, our phenomenological theory +% is consistent with the ground state being the B$_{1g}$ crystal field state with +% order parameter +% \[ +% H = \eta (J_x^2 - J_y^2) +% \] +% corresponding to hexadecapolar orbital order, +% where here $\eta$ is taken to be modulated at $\vec{Q} = (0, 0, 1)$. +% The result of non-zero $\eta$ is a nematic distortion of the B1g orbitals, alternating along the c-axis. +% } {\color{blue} -We can also connect our abstract order parameter to a physical picture of multipolar -ordering. -The U-5f electrons in URu$_2$Si$_2$ exhibit a moderate degree of localization [cite], which is -reflected in partial occupancy of many electronic states. Motivated by the results of refs [cite], -we assume that the dominant U state consists of $j = 5/2$ electrons in the U-5f2 configuration, which has -total angular momentum $J = 4$. Within the $J=4$ multiplet, the precise energetic ordering -of the $D_{4h}$ crystal field states still remains a matter of debate [cite]. In a simple -framework of localized $j = 5/2$ electrons in the 5f2 configuration, our phenomenological theory -is consistent with the ground state being the B$_{1g}$ crystal field state with -order parameter -\[ - H = \eta (J_x^2 - J_y^2) -\] -corresponding to hexadecapolar orbital order, -where here $\eta$ is taken to be modulated at $\vec{Q} = (0, 0, 1)$. -The result of non-zero $\eta$ is a nematic distortion of the B1g orbitals, alternating along the c-axis. +We can also connect our results to the large body of work concerning various multipolar +orders as candidate states for HO (e.g. refs.~\cite{Haule_2009,Ohkawa_1999,Santini_1994,Kiss_2005,Kung_2015,Kusunose_2011_On}). +Physically, our phenomenological order parameter could correspond to $\Bog$\ multipolar +ordering originating from the localized component of the U-5f electrons. For the crystal +field states of \urusi, this could correspond either to electric quadropolar or +hexadecapolar order based on the available multipolar operators \cite{Kusunose_2011_On}. } The coincidence of our theory's orthorhombic high-pressure phase and \urusi's |